_{1}

^{*}

The empirical relation of
between the transition temperature of optimum doped superconductors T
_{co} and the mean cationic charge
, a physical paradox, can be recast to strongly support fractal theories of high-T
_{c} superconductors, thereby applying the finding that the optimum hole concentration of σ
_{o} = 0.229 can be linked with the universal fractal constant δ
_{1} = 8.72109… of the renormalized quadratic Hénon map. The transition temperature obviously increases steeply with a domain structure of ever narrower size, characterized by Fibonacci numbers. However, also conventional BCS superconductors can be scaled with δ
_{1}, exemplified through the energy gap relation k
_{B}T
_{c} ≈ 5Δ
_{0}/δ
_{1}, suggesting a revision of the entire theory of superconductivity. A low mean cationic charge allows the development of a frustrated nano-sized fractal structure of possibly ferroelastic nature delivering nano-channels for very fast charge transport, in common for both high-T
_{c} superconductor and organic-inorganic halide perovskite solar materials. With this backing superconductivity above room temperature can be conceived for synthetic sandwich structures of
less than 2+. For instance, composites of tenorite and cuprite respectively tenorite and CuI (CuBr, CuCl) onto AuCu alloys are proposed. This specification is suggested by previously described filamentary superconductivity of “bulk” CuO1
﹣x samples. In addition, cesium substitution in the Tl-1223 compound is an option.

The recent discovery of conventional superconductivity at the highest until today known transition temperature of 190 K on hydrogen sulfide at a high pressure >150 GPa by Drozdov, Eremets and Troyan (2014) [_{2}S was solved, indicating an anti-perovskite structure in the sense that SH^{−} represents the A site and _{3} substrate (Schönberger et al., 1991) [_{2} and CuO, respectively, besides the superconducting main phase YBa_{2}Cu_{3}O_{7}_{−}_{δ}. A comparable result with a large resistivity drop at 220 K has been published little earlier by Azzoni et al. (1990) [_{2}O besides CuO. Later Osipov et al. (2001) [^{5} even at 300 K. Today it is assumed that the observed rapid drop of the electric resistivity at 220 K and 300 K is caused by superconductivity of oxygen deficient CuO_{1}_{−}_{δ} filaments [

Interestingly, artificial interfaces between insolating perovskites that indicate superconducting response are described in 2007 by Reyen et al. [

A hypothetical BaCuO_{2} phase with puckered T-CuO nets, in contrast to planar CuO_{2} nets of the BaCuO_{2 }infinite layer phase, was recently proposed by the present author as an alternative high-T_{c} candidate together with a large cupric oxide cluster compound [_{co} of optimum doped superconductors with the mean cationic charge _{2} nets should envisage physicists to realize a dream of mankind, superconductivity above room temperature.

This work was pre-published in arXiv just as the message about an apparently successful synthesis of a room temperature superconductor (with far withheld information about chemical composition!) astonishing the scientific community [

The most abundant mineral in Earth is MgSiO_{3}, a perovskite phase confined to the high pressure region of the Earth mantle, but also discovered in a shocked meteorite and now named bridgmanite [

Beginning with the last mentioned challenge, the development of efficient solar cells, you just witnessed the great progress in organic inorganic halide perovskite solar cells. Their very high solar efficiency is caused by the low cationic charge of

A less restricted freedom of electronic movement may also be suggested for the family of superconductors related to oxide perovskites. The rise of the transition temperature T_{co} goes inversely proportional with the fourth power of the mean cationic charge_{co}, somewhat simplified compared to previous results [_{1−x} as hole conductor (p-type conductivity), and with the result for H_{2}S under high pressure as electron conductor (n-type conductivity). The curve for electron transport will be discussed in a forthcoming contribution, especially considering the fractal structure of BCS superconductivity. The magenta colored curve for hole superconductors (

Surprisingly, _{co} asymptotic limit. With a successive reduction of the mean cationic charge in direction of

tetragonal BaCuO_{2} with puckered T-CuO nets [

The scale factor of Equation (1) of s_{c} = 2740 K can intuitively traced back to give

where

charge, h the Planck constant, v_{F} the Fermi speed, ε the permittivity of the medium, ε_{o} the permittivity of free space, and m_{e} the electron mass, respectively. With ε = 5.04 and an assumed Fermi speed of v_{F} = 2.5 × 10^{5} m/s, respectively, the scaling factor agrees well with the fitted parameter of 2740 K. If a higher effective mass of the quasiparticles is chosen, for instance m_{h} ≈ 1.5m_{e} (or m_{h} = φ∙m_{e} with φ = 1.61803), then the Fermi velocity has to be reduced correspondingly, as was observed experimentally in the region of optimal doping. With

This represents rather the total energy of a system of oscillating charged quasiparticles, dispersed in a medium of permittivity ε, than solely coulomb energy. The dimensionless variable_{co} are composed of highly polarizable heavy atom ions. How the matter might be, the relation between T_{co} and _{co}.

Some ideas should be passed on that are related to optimum values of the hole concentration found, and to obvious accumulations in certain T_{co} values. Beginning with the value of optimum h^{+} of σ_{o} = 0.229 (or multiples) [_{c} superconductors based on Tl or Hg, respectively. Surprisingly, the multiplier, which would give two holes needed to create a pair, emerges as the number

known as a universal scaling constant for two-dimensional maps in the theory of fractal systems or chaotic ones, with the precise value of δ_{1} = 8.7210972∙∙∙ [_{1} (and δ_{2} = 2) have been determined as eigenvalues of the matrix containing the existence intervals of two subsequent cycles of the periodic-doubling cascade in the parameterized version of a quadratic Hénon map with renormalized (x, y)-parameters.

The optimum concentration of holes is given per cuprate layer per unit-cell. Maintaining two dimensions along cuprate layers, then two electrons reside in a slab of the extension δ_{1}a^{2} = πl^{2}a^{2}. The “domain” extension multiplier is then given by the quotient of Fibonacci numbers

Pnictide superconductors yield an optimum hole concentration of about σ_{o}/4 [_{1} too. Further, although it is already difficult to identify optimum doping, picking out accumulations in the T_{co} ranking is the more difficult. Nevertheless, one can try it as Mitin [_{co} groups with domain widths. This author related the transition temperature T_{co} to assumed zig-zag bosonic stripe domains of width w = η∙a that are connected with oxygen interstitials. He obtained

where h represents the Planck constant, k_{B} the Boltzmann constant, m_{e} the rest mass of the electron, a an elementary length related to neighboring cations, r_{o} = 2.72 Å a string distance along O-O bonds, and

Going a step further, one can associate such T_{co} clumps with Fibonacci numbers f_{i} determining the extension of domains. Possible mixed domain states can be described by the mean of consecutive f_{i}’s. _{c} superconductivity. Using Fibonacci numbers f_{i} one can roughly formulate (see

Upon substitution of Equation (1) into Equation (6) one gets

Surprisingly, the emerging factor corresponds numerically to the charge of σ_{o} = 0.228 [

With this result one can finally express the energy in different and scale-free forms:

η | d_{i } | Mean of consecutive f_{i}’s | T_{co } | T_{co}∙d_{i}^{ } | _{ } | |
---|---|---|---|---|---|---|

2η^{2} + η | f_{i} | |||||

8 | (1500) | 12,000 | - | |||

2 | 10 | 10.5 | (1200) | 12,000 | - | |

13 | (923) | 12,000 | - | |||

3 | 21 | 21 | 571 | 11,991 | 1.48 | |

4 | 36 | 34 | 353 334 | 12,000 12,024 | 1.67 1.70 | |

5 | 55 | 55 | 219 | 12,045 | 1.87 | |

6 | 78 | 72 | 167 155^{*} | 12,024 12,090 | 2.01 2.05 | |

89 | 135 | 12,015 | 2.12 | |||

7 | 105 | 115 | 12,075 | 2.21 | ||

116.5 | 103 | 12,000 | 2.26 | |||

8 | 136 | 144 | 89 84 | 12,104 12,096 | 2.35 2.39 | |

9 | 171 | 70 | 11,970 | 2.50 | ||

188.5 | 64 | 12,064 | 2.55 | |||

10 | 210 | 233 | 57 52 | 12,000 12,116 | 2.63 2.70 | |

11 | 253 | 47 | 12,000 | 2.76 | ||

12 | 300 | 305 | 40 | 12,000 | 2.88 |

^{*}High pressure applied.

When using the conductance quantum G_{o} = 2e^{2}/h = 7.748091 × 10^{−5 }(S) one can write down the energy as

The multiple energy representation is given by the special nesting property of fractals. Resolving the paradox finally leads to the result that high-T_{c} superconductors obey fractal conductance behavior, which is intrinsically accompanied with self-similarity and scale-free characteristics. This finding of possible fractality and self-simi- larity of high-T_{c} superconductors supports the seminal investigations of Fratini et al. [_{c} superconductors of Phillips [

If the domain width lessens, then the ratio of its surface to volume will be enlarged. This ratio is thought to be scaled by the golden mean,

It remains to be emphasized that the energy k_{B}T_{c} of conventional BCS superconductors can also be formulated using an inverse δ_{1} scaling applied to the energy gap Δ_{0} at T = 0 K [

2Δ_{0} is the energy required to break a Cooper pair. The BCS superconducting gap has s-wave symmetry and shows minimal momentum dependence.

Recently, Mushkolaj [_{c} functions related to an elastic atom collision model in contrast to an elastic spring model, respectively, given by

where_{i} = atom or electron masses, Δx = distances. However, the picture of colliding point particles should be replaced by nested highly coherent vibrations of most irrational frequency, a superposition of inwardly and outwardly extending spherical quantum waves in the sense of Wollf’s conjecture for the electron [

ers. Analogously, the most irrational number of the golden mean,

its inverse by φ − φ^{−1} = 1, or generalized, by φ^{i} ± φ^{−i} = f_{i} + f_{i}_{+2 }with Fibonacci sequence numbers f_{i} = 0, 1, 1, 2, 3, 5, 8∙∙∙ (±: +if i = even else −). This divine number is connected with chaos respectively fractal structures and self-similarity. The transfinite golden mean is most suitable to describe comprehensively our self-similar universe because of the proper connection of that number with its inverse. Philosophically formulated, since the big bang the diverging matter (and energy) is retracted again at the same time by an inverse process.

Both universal constants δ_{1} and the golden mean φ, respectively, can be connected by the approximate relation

Using this result, one can approximately express δ_{1} as a quite catchy product of three fundamental constants together with a denominator as a product of Fibonacci sequence numbers:

where

represents the inverse of Sommerfeld’s dimensionless fine structure constant as given in El Naschie’s E-infinity theory [_{1}. The number 137 itself is the 33^{rd} prime number.

The golden mean respectively

Further, the fourth power of the golden mean φ^{4} applied to the υ_{τ} lepton mass m(υ_{τ}) = 15.4214 MeV/c^{2} gives the mass of the μ lepton:

The mass ratio of the υ_{τ} particle to the electron yields

In this way, the universal Feigenbaum constant δ_{1} of the quadratic Hénon map, explained before, is apparently involved in mass relations between elementary lepton particles.

A majority of high-T_{c} superconducting compounds shows a pronounced deviation from the tetragonal symmetry. This supports the formation of ferroelastic domains with walls as charge carrier sink. If the itinerant holes formed by doping travel to the walls, it results at the first moment a stack of positively charged walls separating insulating regions. The repulsive forces have to keep in balance and may strengthen orthorhombicity. However, when the concentration of holes exceeds a certain limit, then repulsive forces will be reduces by the formation of bosons, confining preformed pairs of holes in one-dimensional bosonic stripes [

However, in which way interstitial oxygen ions could be involved in the pairing scenario? A possible modified exciton pairing mechanism may be as follows. In the vicinity of the large and highly polarizable Ba^{2+} ions the oxygen interstitials may show a “chameleon” feature, being once −1 charged and then again −2, respectively. It means that this oxygen atom is able to easily expel or souk a charge carrier. Apical oxygen atoms on the interlayer between cuprate layers and spacer layer can be involved in this process. Just when the apical oxygen atom is displaced towards the Cu center, enough space is provided to create the larger O^{1}^{−} ion. An exciton may bind a just expelled (hot) hole to form a metastable three-particle-entity that would rapidly decay, leaving a boson and the mediating electron, but could also survive as quantum wave composite and would then be observable. It needs only few electrons to form a cascade of bosons, and the probability it happens is higher than for an exciton-exciton process. The pairing process can take place locally near interstitial oxygen atoms or within the strain field of ferroelastic domain wall sinks, involving strain effect as important too.

When the mass of holes is assumed to be 1.5m_{e} in accordance with experimental results [_{e}. Intuitively, this special mass multiple of an electron can be depicted alternatively by a third power golden mean representation, because

The given ideas about fractality of high-T_{c} superconductors and a modified pairing mechanics, respectively, should be considered in a modified and more founded theory. However, despite the undoubtedly achieved great success of research on high-T_{c} superconductivity, the impression is still given that the forest is not seen for the trees.

So other concepts may be chosen to go ahead. For instance, a heretical concept may assume an influence of high-T_{c} superconductivity from the cold. As noticed before, high-T_{c} superconductors may act as extremely effective “particle” detectors. It remains open to be conjectured that a (less divine) ghostly subatomic “particle” could help to overcome teething problems of pairing, so to speak, an interaction from the dark universe to upset physicists.

If such influence would ever be conceivable, then it would clearly lead to a chaotic response, for instance filamentary-chaotic conductance with self-similar characteristic. Furthermore, a scaling of the pairing vigor down the flight of stairs determined by Fibonacci levels (domains) is easy to imagine.

Until then we can work with the auxiliary variable

First, the T_{co} versus mean cationic relation will be applied to make progress with known Tl-based superconductors with CuO_{2}-based nets. Following this, tenorite-based composites are prospected in detail with twice as many copper atoms per layer.

In short, a resume of the crystal-chemistry of Tl-based superconductors will be given. Crystal-chemical data from own structure determinations on Tl-based cuprates were summarized in ^{1+}, and the enhanced value at the Ca position was attributed to a partial Tl^{3+} substitution. The Cu^{1+} content in the calculated amount was confirmed by electron energy loss spectroscopy (EELS) on single crystals [^{2+} or Tl^{3+} attain the lower oxidation state with lone electron pairs and associated dipole momentum? The properties of Bi^{3+} are dominated by both the space that the lone electron pair needed and the resulting dipole momentum. Therefore, Bi^{3+} will be shared in Aurivillius double-layers, triple domains or channels to minimize the dipole momentum. Indeed, only double layer Bi-based superconductors are found. In the case of Hg^{1+} two ions join to form ^{3+} by Tl^{1+} in high-T_{c} superconductors, again one is faced with the effect of the dipole momentum of Tl^{1+}, especially if mono-layer compounds such as Tl-1223 are considered. The replacement should be significantly less than halve the Tl content as observed, but may be restricted to the Tl site and not to the Ca site substitution. Because Tl-1223, given as example in ^{1+} compared to Tl^{3+}. The changes are recently summarized [

Phase symbol | Tl-2201 | Tl-2212 | Tl-2223 | Tl-1223 |
---|---|---|---|---|

a (Å) | 3.8656 (3) | 3.8565 (4) | 3.8498 (4) | 3.848 (4) |

c (Å) | 23.2247 (18) | 29.326 (3) | 35.638 (4) | 15.890 (10) |

n | 1 | 2 | 3 | 3 |

f | 0 | 0 | 0 | 0.092 |

x | 0.11 | 0.31 | 0.42 | 0 |

y | - | 0.10 | 0.07 | 0.069 |

δ | 0 | 0 | 0 | 0.18 |

h^{+ } | 0.220 | 0.52 | 0.700 | 0.686 |

h^{+}/n | 0.220 | 0.26 | 0.233 | 0.229 |

2.353 | 2.211 | 2.144 | 2.12 | |

ζ (Å) Ba-O || c | 1.945 | 2.014 | 2.010 | 2.073 |

^{ } | 88.9 | 114.6 | 130 | 136 |

77.8 | 115.8 | 135 | 130 | |

T_{c} (K) measured | 80 | 110 | 130 | 133 |

Refs. | [ | [ | [ | [ |

Notation | Cations | Anions | |||||||
---|---|---|---|---|---|---|---|---|---|

Formula subscripts | 1 | 2 | 2 | 3 | 9 | ||||

Elements | Tl^{3+ } | Cs^{1+ } | Ba^{2+ } | Cs^{1+ } | Ca^{2+}_{ } | Tl^{3+ } | Cu^{2+ } | O^{2− } | F^{1- } |

Substitution | 0.908 | 0.092 | 2 | − | 1.862 | 0.138 | 3 | 8.82 | − |

Charge | +2.724 | 0.092 | +4 | − | +3.724 | +0.414 | +6 | −17.64 | − |

h^{+} = 0.686 | T_{co} = 136 K | ||||||||

Substitution | 0.75 | 0.25 | 2 | − | 1.862 | 0.138 | 3 | 8.33 | 0.67 |

Charge | +2.25 | +0.25 | +4 | − | +3.724 | +0.414 | +6 | −16.6 | −0.7 |

h^{+} = 0.692 | T_{co} = 146 K | ||||||||

Substitution | 0.5 | 0.5 | 2 | − | 1.862 | 0.138 | 3 | 7.82 | 1.18 |

Charge | +1.5 | +0.5 | +4 | − | +3.724 | +0.414 | +6 | −15.64 | −1.18 |

h^{+} = 0.682 | T_{co} = 166 K | ||||||||

Substitution | − | 1 | 0.9 | 1.1 | 2 | − | 3 | 5.59 | 3.41 |

Charge | − | +1 | +1.8 | +1.1 | +4 | − | +6 | −11.18 | −3.41 |

h^{+} = 0.690 | T_{co} = 300 K |

The complex chemical formula for the homologous series of Tl^{3+}-based superconductors may be written as

m = 1:TlO mono-layers (space group P4/mmm), m = 2:TlO bi-layers (space group I4/mmm). The space group notation is that of the “averaged” structures.

The hole concentration h^{+} then yields

Subtracting h^{+} from the total charge of oxygen, taken as O^{2−}, the total cationic charge results. The mean cationic charge is calculated by division of the number of cations as

The amount of lethally toxic Tl^{1+} besides Tl^{3+}may easily be replaced by the environmentally benign Rb^{1+} or Cs^{1+} cations of comparable size (^{137}Cs isotope, without alter the charge balance. However, further replacement of Tl^{3+} and Ba^{2+}, respectively, by these alkali earth cations would reduce the mean cationic charge _{c}. Such replacement may be further supported by substitution of some oxygen by group VII anions like F^{1−} or more polarizable ones such as Br^{1−} or even I^{1−}. Fluorine substitution was already successfully applied in Bi-based superconductors [_{co} known [_{c} to its optimum due to internal “chemical” pressure. In ^{1+} and F^{1−} substitutions are proposed assuming the possible adaption of a stable perovskite-related crystal structure.

A Cs-1223 compound of the formula Cs(Cs_{1.1}Ba_{0.9})Ca_{2}Cu_{3}O_{5.6}F_{3.4} with an optimum hole concentration of h^{+} ≈ 0.7 = 3 × 0.233 and a mean cationic charge of _{co} = 300 K. The effect of crystal lattice swelling due to large cation substitution may be compensated by a larger permittivity of the compound (see Equation (3)).

Recently I proposed an Cs-elpasolite as candidate for solar cell application, namely Cs_{2}(Na,Cu,Ag)_{1}Bi_{1}(I,Br)_{6} [^{1−} or Br^{1−} by some

O^{2−}? A chemical content near Cs_{2}(Na,Cu,Ag)_{1}Bi_{1}(I,Br)_{5.4}(O,S)_{0.6} looks very interesting, though it shows an optimum h^{+} concentration and would reach a T_{c} ≈ 370 K, counting the formally enhanced oxidation state of Cu or Ag, respectively. Alternative sulfide replacement is considered owing to the well adapted ionic radius near that of I^{1−} or Br^{1−}. A vision is that both properties, high solar efficiency and superconductivity, occur together in slightly altered parts of a single compound and can be shared in future.

The discovery of multiferroic properties of slightly acentric CuO (tenorite) [_{3} substrate [_{N1} = 213 K and T_{N2} = 230 K [_{c} = 230 K together with the development of a spiral-magnetic order and spontaneous electric polarization P_{s} along the b axis. Removing of the glide mirror plane would lead to the acentric point group 2. It is natural to identify the temperature of T_{N2} = 230 K with the 220 K temperature found for filamentary superconductivity on tenorite samples. However, the intrinsically physical properties and the high permittivity near the phase transition may also support a real superconducting transition, if applied strain would enhance the small concentration of holes that already exists. Notably, the smallest observed Cu-Cu distance of 2.900 Å (

In tenorite and T-CuO, respectively, are twice as many copper atoms in the atomic layers compared to the CuO_{2} nets, and the copper to copper distances are similar to the oxygen-oxygen distances of about 2.73 Å. Both nets are compared in _{2}O) and metallic AuCu alloy supports will be investigated. In addition, also CuBr or CuI are considered as material using [

The monoclinic crystal structure of tenorite is indeed a strongly folded and distorted rocksalt structure formed by unit cell twinning along (010) planes (

Twinning of the rocksalt structure type was first described in the system PbS-Bi_{2}S_{3} by the present author and may serve as an example for CuO twinning too, although it does not be “chemical” twinning, because contrary to the PbS-Bi_{2}S_{3} system, the chemical content remains unchanged [

Starting with the T-CuO structure, twin-folding to tenorite may be initiated by local Cu^{1+}-Cu^{3+} charge disproportionation, which would remove the Jahn-Teller distortion of Cu^{2+} caused by its d^{9} electron configuration [^{2 }configuration rather than a d^{9} configuration [^{1+}-Cu^{3+} pairs. Not surprisingly, a very small quantity of Cu^{3+} was found as relict in tenorite [

Turning now to copper oxide composites, room temperature superconductivity is proposed for a composition near 6CuO∙Cu_{2}O with_{3} substrate may be first coated with an AuCu I thin film, onto which a tenorite layer is deposited along [

An open question is, whether self-doping is possible. If cuprite is considered as reservoir layer, one find short

layer distances between copper layers and oxygen layers of c/4 = 4.2698/4 Å = 1.0674 Å, whereas the oxygen distances in cuprite are extreme large (

1) Cu-Cu metallic bonding Cu^{2+} + Cu^{2+} ® (Cu-Cu)^{2+} + 2h^{+} (formally)

2) Charge disproportionation 6CuO∙Cu_{2}O ® Cu_{2}O_{3}∙2CuO・2Cu_{2}O ® 4CuO∙2Cu_{2}O + 2h^{+}

The composition is chosen to satisfy the _{(002)} = 2.5289 Å. Using these geometrical specifications, hole stripes along [

be formed. In the empty space, corresponding to the acute Penrose tiles, dangling bonds and charge alteration of oxygen, respectively, can easily deliver holes.

Harshman, Fiory and Dow [_{co} results (

The HFD rule gives

where h^{+} is the hole carrier concentration, d (Å) the lattice parameter (Cu-Cu distance in the ab plane) and ζ (Å) represents a critical length down the charge reservoir layer, typically the Ba-O distance projected down [_{c}. Interestingly, with the change from CuO_{2} nets to CuO ones the value of d is reduced by a factor of_{co} would be enhanced by this factor, if the other parameters remain unchanged.

Identifying the lattice plane spacing d_{(004)} = 1.0674 Å of cuprite as the critical length ξ of a “charge reservoir layer”, then the HFD rule can be applied.

Surprisingly, the mysterious critical temperatures of previous experiments, namely 220 K [_{co} ratio of about

Alternatively, other orientations of thin film deposition may be tested. For instance, (001)-layers of Au or AuCu can be deposited on a cleaved mica sheet, succeeded by a (111)-oriented cuprite layer (see

In this way one avoids that not wanted T-CuO forms at all.

Thus, two possible orientations for epitaxial layer growth of copper oxides are indicated as consequence of this modeling: along (010) of tenorite and along (111) of cuprite, respectively. For an elaborated experiment, also tenorite single crystals could be cut in the wanted orientation. Under less controlled conditions, the deposition of (010)-oriented layers likely happens with unfortunate domain extension normal to the film surface. The extremely short filaments would lead to unstable current transport.

Tetragonal CuO (siemonsite), which may initially be formed, is not wanted because of its large layer separation of c = 5.30 Å. Tetragonal CuO should be given the opportunity to collapse into the “dying swan” structure of tenorite by varying the coating thickness. Also paramelaconite, Cu_{4}O_{3−x}, is not considered as superconductor material [

Finally, attention needs to be given to nano-structures that may be built up from tenorite and CuI (or CuBr, CuCl), respectively. Adequate doping implied, a composition of 2CuO∙CuI with a Fibonnaci mean cationic quotient of _{c} ≈ 355 K. As pursued recently for solar cell application [

The hypothetical compound BaCuO_{2} with puckered T-CuO nets [

Domain extension d(Å) Tenorite | Critical length ξ (Å) Cuprite | Hole concentration h^{+ } | T_{co} (K) |
---|---|---|---|

d_{(002)} = 2.5289 | d_{(004)} = 1.0674 | 0.228 | 221 |

2 × 0.228 | 312 | ||

d_{(202)} = 1.5803 | d_{(222)} = 1.2325 | 0.228 | 306 |

2 × 0.228 | 433 |

substrate, itself deposited onto (100)-SrTiO_{3}. Cs^{1+} substitution for Ba^{2+} could deliver holes. The mean cationic charge for a composition of Ba_{0.772}Cs_{0.228}CuO_{2} would yield_{co} ≈ 278 K for the assumed critical temperature. However, the critical length ξ down the “charge reservoir” would be too large to uphold this result by an alternative calculation, using the HFD rule [

In conclusion, there are strong arguments for a reappraisal of tenorite-cuprite as well as tenorite-CuI (CuBr, CuCl) sandwich structures as candidates for room temperature superconductivity.

Based on the recent investigation of the crystal structure of the H_{2}S based high-pressure superconductor by Gordon et al. [_{3}GeO_{5} and presented it as anti-perovskite structure type, where the large (GeO_{4})^{4}^{−} units occupy the A-sites and (OPb_{3})^{4+} the B-site of a perovskite type [_{2}S modification may as well be understood as anti-perovskite with (SH)^{−} occupying the A-site and (SH_{3})^{+} the B-site, respectively.

The Pb_{3}GeO_{5} lead germanate is ferroelastic with a pronounced domain structure. Domains can easily be restored in the other orientation states. When hole doping would be possible, nesting of the charge carriers in the ferroelasic domain wall sinks could be proposed. Crystal-chemically interesting would be a composition of (SiO_{2}F_{2})^{2}^{−}(BrRb_{3})^{2+} with a mean cationic charge of_{co} ≈ 292 K, if optimal doped. The realization of such compound is doubtful. However, high pressure may be a mean to outsmart limiting factors.

From an empirical relation between the critical temperature T_{co} of optimum doped superconductors and the mean cationic charge _{c }superconductivity. The optimum hole concentration of σ_{o} = 0.229 can be linked with the universal fractal constant of δ_{1} = 8.72109 characteristic for the renormalized quadratic Hénon map, giving σ_{o} = 2/δ_{1}. In addition, the width of superconducting domains is governed by Fibonacci numbers, including mixed domain states (applying the mean of consecutive numbers). Inverse scaling with δ_{1} also occurs in the gap relation for conventional superconductors, k_{B}T_{c} (BCS) ≈ 5Δ_{0}/δ_{1}. Experiment and theory in this field of science may be steered into new paths, in accordance with fractal conductance behavior and the conjecture that superconductivity may come from the cold. There is evidence to favor low ^{1+} substitution in the known Tl-1223 cuprate, especially composites consisting of multiferroic tenorite and cuprite layers, respectively, were recommended because of previously reported filamentary superconductivity of such composites. Tenorite with twice as many copper atoms in the structural layers compared to CuO_{2}-based nets is highly interesting. Cesium is an element of the green future and has already found its way into smart perovskite solar cells. Modeled on the anti-perovskite high-pressure modification of H_{2}S, a (SiO_{2}F_{2})^{2}^{−}(BrRb_{3})^{2+} anti-perovskite may be a further option for high-T_{c}. Once again it should be stressed that a low cationic charge is optimal for both solar cells and superconductors.

Hans Hermann Otto, (2016) A Different Approach to High-Tc Superconductivity: Indication of Filamentary-Chaotic Conductance and Possible Routes to Room Temperature Superconductivity. World Journal of Condensed Matter Physics,06,244-260. doi: 10.4236/wjcmp.2016.63023

A frequently used substrate for epitaxial growth of thin films of superconductors is (100)-SrTiO_{3}. It can be used for the deposition of both types of copper oxide based nets. If one thinks about a metallic substrate, AuCu alloys would be best adapted to an epitaxial growth of any copper oxide composite. Au has the tendency to favor (111) layers, if deposited on smooth surfaces. However, deposition onto a (100)-SrTiO_{3} would favor (100) growth. The cubic AuCu L1_{0} phase with an atomic ratio of 1:1 has a lattice parameter of a = 3.859 Å. The atomic distance yields _{2}O (cuprite) is (100)-MgO due to its cubic lattice parameter of a = 4.212 Å compared to a = 4.2696 Å for cuprite.

If simply extruded foils of pure copper are used as a substrate for preliminary experiments, their annealing under protective hydrogen atmosphere at 600˚C provides a highly ductile product that does not break, even if most frequently bent.

Phase | copper | tenorite | “siemonsite” | cuprite | paramelaconite |
---|---|---|---|---|---|

Formula | Cu | CuO | T-CuO | Cu_{2}O | Cu_{4}O_{3-x } |

Space group | Fm3m | C2/c (293 K) | Pn3m | I4_{1}/amd | |

a (Å) | 3.6147 | 4.6837 | 3.905 | 4.2696 | 5.837 |

b (Å) | 3.4226 | ||||

c (Å) | 5.1288 | 5.30 | 9.932 | ||

β (°) | 99.54 | ||||

V (Å^{3}) | 81.080 | 80.82 | 77.832 | 4 × 84.598 | |

ρ (kg/m^{3}) | 8.937 | 6.516 | 6.54 | 6.106 | 5.93 |

Cu-Cu (Å) | 2.5560 | 2.9005 3.0830 3.1733 | 2.761 | 3.0191 | 2.9185 |

ε | 5.9 || [ | 7.11 ε(0) 6.46 ε(∞) | 4.24 ε(∞)^{*} | ||

Θ_{D} (K) | 347 (0 K) 310 (298 K) | 391 | 188 | ||

v_{s} (m/s) | 4.76 × 10^{3 } longitudinal 2.33 × 10^{3} transverse | 6.4 × 10^{3} [^{3} [^{3} [^{3} [^{3} 6.8 × 10^{3} [ | |||

T_{N} (K) | 213 230 | 45 - 55 120 | |||

E_{g} (eV) | 1.2 | 2.137 | 1.34 (indirect) 2.47 (direct) |

^{*}Calculated from the refractivity index. Θ_{D} Debye temperature, v_{s} sound velocity, ε permittivity, ρ density, T_{N} Neel temperature, E_{g} energy gap.