TC0<\/sub> = \u03b2 [\u03c3\u03b7\/A<\/i>)]1\/2<\/sup> \u03b6\u22121<\/sup> <\/big>,<\/p>\n where \u03c3 is the fractional charge per formula unit in a layer of the type I reservoir (an outer layer in cases of \u03bd = 2), corresponding to surface area A<\/em>. The interaction occurs between two charge reservoirs (type I and type II) of opposite sign, separated by the nearest-neighbor interaction distance \u03b6 , where \u03b7 is the number of charge-carrying layers in the type II reservoir, \u2113\u22122<\/sup> \u2261 [\u03c3\u03b7\/A<\/em>] is the areal charge density per type I layer per formula unit for participating carriers, and \u03b2 [= k<\/em>B<\/sub>-1<\/sup> (0.1075 \u00b1 0.0003 eV-\u00c52<\/sup>) \u2261 e2<\/sup>\u039b \/ k<\/em>B<\/sub>-1<\/sup>] is a universal constant. Note that the length \u039b is approximately twice the reduced electron Compton wavelength. There are two methods of determining \u03c3 as described below.<\/p>\n Charge Allocation Methodology Whereas A<\/em> and \u03b6 are readily determined from crystal structure and \u03b7 is determined by inspection, the fractional charge \u03c3 is much more elusive. In general, doping may be either cation or anion, and can occur in the type I reservoir, as in the case of La2\u2212x<\/sub>Srx<\/sub>CuO4<\/sub>, the type II reservoir, e.g. Ba2<\/sub>Y(Ru1\u2212x<\/sub>Cux<\/sub>)O6<\/sub> or in both; the binary and ternary iron chalcogenide systems (e.g. Kx<\/sub>Fe2\u2212y<\/sub>Se2<\/sub> [2]) have as many as two dopant ions, one from each reservoir. Thus, for the compounds discussed herein, \u03c3 can be determined by considering the cation and anion doping according to,<\/p>\n \u03c3 = \u03b3[|vI<\/sub>(x \u2013 x0<\/sub>)I<\/sub>| + |vII<\/sub>(x \u2013 x0<\/sub>)II<\/sub>|]<\/big> ,<\/p>\n where vi<\/sub><\/em> is the net charge due to dopants (typically the valence difference between the dopant and the native ion) in reservoir i, (x \u2212 x0<\/sub>)i<\/sub><\/em> is the generic doping factor in which x denotes the content of the dopant species (e.g. x in A<\/em>x<\/sub>B<\/em>) and x0<\/sub> is the minimum value of x required for superconductivity. The above equation is a generalization of equation (2.4a) from Ref. [1], which incorporates the possibility of non-unit valence doping and contains two terms corresponding to the two charge reservoirs; the absolute values confine the summation to the magnitudes of the individual contributions to \u03c3. The factor \u03b3 derives from the allocation of the dopant charge by considering a given compound\u2019s structure. To calculate \u03c3 for a given compound, the modifying factor \u03b3 is determined by application of the set of rules developed in [1]; the following (first set of) charge allocation rules are apply:<\/p>\n (1a) Sharing between N (typically 2) ions or structural layers introduces a factor of 1\/N in \u03b3<\/em>.<\/em><\/strong><\/p>\n (1b) The doping is shared equally between hole and electron reservoirs resulting in a factor of 1\/2.<\/em><\/strong><\/p>\n Multiple charge sources in opposing reservoirs (e.g. determined by K and Fe stoichiometries in Kz<\/sub>Fe2\u2212y<\/sub>Se2<\/sub>) are treated as a single contribution to \u03c3 (hence the absolute values), with \u03b3 determined by rules (1a) and (1b). Numerous high-TC<\/sub> materials share the same value of \u03c3 as optimal YBa2<\/sub>Cu3<\/sub>O6.92<\/sub>, which we have denoted \u03c30<\/sub> and determined to have the value 0.228. For those compounds where the (x \u2212 x0<\/sub>)i<\/em><\/sub> cannot be discerned independently through doping, \u03c3 can be calculated by scaling to \u03c30<\/sub> according to \u03c3 = \u03b3 \u03c30<\/sub> , where \u03b3 is defined in conjunction with a second set of (stoichiometric) scaling rules that are discussed below.<\/p>\n An example of the application of charge allocation is provided by La1.837<\/sub>Sr0.163<\/sub>CuO4\u2013\u03b4<\/sub> (TC0<\/sub>meas<\/sup> = 38 K). Since x0<\/sub> = 0, the total charge doping is 0.163. This charge is accordingly distribution between the type I and type II reservoirs, thereby introducing a factor of 1\/2 in \u03b3 [rule (1b)]. As there are two SrO layers in the type I reservoir (i.e., \u03bd = 2) rule (1a) provides an additional factor of 1\/2 [1]:<\/p>\n Stoichiometric Scaling Methodology Also termed “Valency Scaling,” this approach assumes that the optimal fractional charge \u03c3 for all cuprate superconductors are proportional to that of YBa2<\/sub>Cu3<\/sub>O6.92<\/sub> (denoted as \u03c30<\/sub>), where the scaling factor \u03b3 is dependent upon electronic and structural parameters of a given cuprate in relation to YBa2<\/sub>Cu3<\/sub>O6.92<\/sub>. Thus one can write [1]:<\/p>\n \u03c3 = \u03b3\u03c30<\/sub><\/big> ,<\/p>\n where \u03c30<\/sub> is considered a fundamental quantity. To determine \u03c30<\/sub>, one considers oxygen content x<\/em> above the minimum value (x<\/em>0<\/sub> = 6.35 [1]) required for superconductivity to occur. The total oxygen content associated with the optimal superconductive state is then, (6.92 \u2013 6.35) = 0.57. Given a valence of \u20132 per oxygen ion, the total number of carriers available to dope the superconducting structure is 2\u00d70.57 = 1.14. As there are five oxygen-containing layers, two CuO2<\/sub> layers, two BaO layers and one CuO layer, and assuming that all five layers are populated equally, \u03c30<\/sub> = 1.14\/5 = 0.228.<\/p>\n Since \u03b7 = 2, A<\/em> = 14.8596 \u00c5 and \u03b6 = 2.2677 \u00c5 [1], TC0<\/sub>calc<\/sup> is found to be 96.36 K, which agrees well with the measured value of TC0<\/sub>meas<\/sup> = 93.78 K. There are three rules governing the value of \u03b3 [1,3]:<\/p>\n (2a) Heterovalent substitution in the type I inner layer(s) of a +3 (or +1) ion mapped to a +2 ion, corresponding to the YBa2<\/sub>Cu3<\/sub>O6.92<\/sub> structural type, introduces a factor of 1\/2 (or 2) in \u03b3<\/em>.<\/em><\/strong><\/p>\n (2b) The factor \u03b3 scales with the +2 (\u20132) cation (anion) structural and charge stoichiometry of the participating charge<\/em>.<\/em><\/strong><\/p>\n (2c) The factor \u03b3 scales with the net valence of the undoped mediating layer.<\/strong> <\/em><\/p>\n Rules (2a) and (2b) are specific to optimal cuprates superconductors. Moreover, the electronegativity \u03c7 of the substituted cation, relative to that of Cu (\u03c7 = 1.90), can affect charge-transfer efficacy: High electronegativity cations, such as Pb (\u03c7 = 2.33) and Bi (\u03c7 = 2.02), suppress charge transfer, resulting in an additional factor of 1\/2 in \u03b3. Cations with comparatively low electronegativities, such as Tl (\u03c7 = 1.62), do not impede charge transfer, and thus do not induce an additional \u03b3-factor [3]. This understanding provides an explanation for the dichotomy in optimal transition temperatures between Tl-and Bi-based homologues. Rule (2c) allows scaling between YBa2<\/sub>Cu3<\/sub>O6.92<\/sub> and other superconducting families, given a one-to-one structural correspondence [1].<\/p>\n <\/a><\/p>\n An example of the application of stoichiometric scaling is provided by YBa2<\/sub>Cu3<\/sub>O6.60<\/sub>. Using rule (2b), \u03c3 for the \u201c60 K\u201d phase material can be similarly obtained as with the \u201c90 K\u201d material, but with a participating charge of 2 (6.60 \u2013 6.35)\/5 = 0.1. However, one obtains the same answer by simply scaling the oxygen (anion) content above x0<\/sub> = 6.35 relative to that of YBa2<\/sub>Cu3<\/sub>O6.92<\/sub>. Therefore, one has [1]:<\/p>\n For high-TC superconductors, the optimal transition temperature TC0 is given by the algebraic expression:<\/p>\n TC0 = \u03b2 [\u03c3\u03b7\/A<\/i>)]1\/2 \u03b6\u22121 ,<\/p>\n where \u03c3 is the fractional charge per formula unit in a layer of the type I reservoir (an outer layer in cases of \u03bd = 2), corresponding to surface area \u2026 Continue reading ![\"(LaBa)2CuO4-d\"](\"http:\/\/physikon.net\/wp-content\/uploads\/2015\/07\/LaBa2CuO4-d.gif\")
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