For high-T_{C} superconductors, the optimal transition temperature T_{C0} is given by the algebraic expression:

T_{C0} = *k*_{B}^{-1} *β* [ση/*A*)]^{1/2} ζ^{−1} ,

where σ is the fractional charge per formula unit in a layer of the type I reservoir (an outer layer in cases of ν = 2), corresponding to basal-plane area *A*. The interaction occurs between two charge reservoirs (type I and type II) of opposite sign, separated by the nearest-neighbor interaction distance ζ , where η is the number of charge-carrying layers in the type II reservoir, ℓ^{−2} ≡ [ση/*A*] is the areal charge density per type I layer per formula unit for participating carriers, and β (= 0.1075 ± 0.0003 eV Å) ≡ e^{2}Λ is a universal constant. Note that the length Λ is approximately twice the reduced electron Compton wavelength. There are two methods of determining σ as described below.

Whereas *A* and ζ are readily determined from crystal structure and η is determined by inspection, the fractional charge σ 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 La_{2−x}Sr_{x}CuO_{4}, the type II reservoir, e.g. Ba_{2}Y(Ru_{1−x}Cu_{x})O_{6} or in both; the binary and ternary iron chalcogenide systems (e.g. K_{x}Fe_{2−y}Se_{2} [2]) have as many as two dopant ions, one from each reservoir. Thus, for the compounds discussed herein, σ can be determined by considering the cation and anion doping according to,

σ = γ[|v_{I}(x – x_{0})_{I}| + |v_{II}(x – x_{0})_{II}|] ,

where v* _{i}* is the net charge due to dopants (typically the valence difference between the dopant and the native ion) in reservoir i, (x − x

_{0})

*is the generic doping factor in which x denotes the content of the dopant species (e.g. x in*

_{i}*A*

_{x}

*B*) and x

_{0}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 σ. The factor γ derives from the allocation of the dopant charge by considering a given compound’s structure. To calculate σ for a given compound, the modifying factor γ is determined by application of the set of rules developed in [1]; the following (first set of) charge allocation rules are apply:

*(1a) Sharing between N (typically 2) ions or structural layers introduces a factor of 1/N in γ**.*

*(1b) The doping is shared equally between hole and electron reservoirs resulting in a factor of 1/2.*

Multiple charge sources in opposing reservoirs (e.g. determined by K and Fe stoichiometries in K_{z}Fe_{2−y}Se_{2}) are treated as a single contribution to σ (hence the absolute values), with γ determined by rules (1a) and (1b). Numerous high-T_{C} materials share the same value of σ as optimal YBa_{2}Cu_{3}O_{6.92}, which we have denoted σ_{0} and determined to have the value 0.228. For those compounds where the (x − x_{0})_{i} cannot be discerned independently through doping, σ can be calculated by scaling to σ_{0} according to σ = γ σ_{0} , where γ is defined in conjunction with a second set of (stoichiometric) scaling rules that are discussed below.

An example of the application of charge allocation is provided by La_{1.837}Sr_{0.163}CuO_{4–δ} (T_{C0}^{meas} = 38 K). Since x_{0} = 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 γ [rule (1b)]. As there are two SrO layers in the type I reservoir (i.e., ν = 2) rule (1a) provides an additional factor of 1/2 [1]:

- σ = (0.163)(1/2)(1/2) = 0.048
- η = 1; ν = 2
*A*= 14.2268 Å^{2}; ζ = 1.7828 Å- T
_{C0}^{calc}= 37.47 K

Stoichiometric Scaling Methodology

Also termed “Valency Scaling,” this approach assumes that the optimal fractional charge σ for all cuprate superconductors are proportional to that of YBa_{2}Cu_{3}O_{6.92} (denoted as σ_{0}), where the scaling factor γ is dependent upon electronic and structural parameters of a given cuprate in relation to YBa_{2}Cu_{3}O_{6.92}. Thus one can write [1]:

σ = γσ_{0} ,

where σ_{0} is considered a fundamental quantity. To determine σ_{0}, one considers oxygen content *x* above the minimum value (*x*_{0} = 6.35 [1]) required for superconductivity to occur. The total oxygen content associated with the optimal superconductive state is then, (6.92 – 6.35) = 0.57. Given a valence of –2 per oxygen ion, the total number of carriers available to dope the superconducting structure is 2×0.57 = 1.14. As there are five oxygen-containing layers, two CuO_{2} layers, two BaO layers and one CuO layer, and assuming that all five layers are populated equally, σ_{0} = 1.14/5 = 0.228. Since η = 2, *A* = 14.8596 Å and ζ = 2.2677 Å [1], T_{C0}^{calc} is found to be 96.36 K, which agrees well with the measured value of T_{C0}^{meas} = 93.78 K. There are three rules governing the value of γ [1,3]:

**( 2a) Heterovalent substitution in the type I inner layer(s) of a +3 (or +1) ion mapped to a +2 ion, corresponding to the YBa_{2}Cu_{3}O_{6.92} structural type, introduces a factor of 1/2 (or 2) in γ.**

*(2b) The factor γ scales with the +2 (–2) cation (anion) structural and charge stoichiometry of the participating charge**.*

**(2c) The factor γ scales with the net valence of the undoped mediating layer.**

Rules (2a) and (2b) are specific to optimal cuprates superconductors. Moreover, the electronegativity χ of the substituted cation, relative to that of Cu (χ = 1.90), can affect charge-transfer efficacy: High electronegativity cations, such as Pb (χ = 2.33) and Bi (χ = 2.02), suppress charge transfer, resulting in an additional factor of 1/2 in γ. Cations with comparatively low electronegativities, such as Tl (χ = 1.62), do not impede charge transfer, and thus do not induce an additional γ-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 YBa_{2}Cu_{3}O_{6.92} and other superconducting families, given a one-to-one structural correspondence [1].

An example of the application of stoichiometric scaling is provided by YBa_{2}Cu_{3}O_{6.60}. Using rule (2b), σ for the “60 K” phase material can be similarly obtained as with the “90 K” material, but with a participating charge of 2 (6.60 – 6.35)/5 = 0.1. However, one obtains the same answer by simply scaling the oxygen (anion) content above x_{0} = 6.35 relative to that of YBa_{2}Cu_{3}O_{6.92}. Therefore, one has [1]:

- σ = [(6.60 – 6.35)/(6.92 – 6.35)] σ
_{0}= 0.439 σ_{0}= 0.1 - η = 2
*A*= 14.8990 Å^{2}; ζ = 2.2324 Å- T
_{C0}^{meas}= 63 K; T_{C0}^{calc}= 64.77 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, Journal of Physics: Condensed Matter
**23**, 295701 (2011). - D. R. Harshman and A. T. Fiory, J. Phys.: Condens. Matter
**24**, 135701 (2012). - D. R. Harshman and A. T. Fiory, Journal of Physics and Chemistry of Solids
**85**, 106 (2015).