**YBa _{2}Cu_{3}O_{6.92}** (

*Pmmm*, T

_{C0}

^{meas}= 93.78 K)

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, one then has [1]:

- Type I reservoir: BaO-CuO-BaO
- Type II reservoir: CuO
_{2}-Y-CuO_{2} - σ = σ
_{0}= 1.14/5 = 0.228 - η = 2; ν = 2
*A*= 14.8596 Å^{2}; ζ = 2.2677 Å- T
_{C0}^{calc}= 93.36 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011).

**YBa _{2}Cu_{3}O_{6.60}** (

*Pmmm*, T

_{C0}

^{meas}= 63 K)

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]:

- Type I reservoir: BaO-CuO-BaO
- Type II reservoir: CuO
_{2}-Y-CuO_{2} - σ = [(6.60 – 6.35)/(6.92 – 6.35)] σ
_{0}= 0.439 σ_{0}= 0.1 - η = 2; ν = 2
*A*= 14.8990 Å^{2}; ζ = 2.2324 Å- T
_{C0}^{calc}= 64.77 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011).

**LaBa _{2}Cu_{3}O_{7–δ}** (

*Pmmm*, T

_{C0}

^{meas}= 97 K)

For this material, the valences and stoichiomentries are unchanged with respect to YBa_{2}Cu_{3}O_{6.92}, such that γ = 1. Therefore, the optimal transition temperature for this and all other REBa_{2}Cu_{3}O_{7–δ} compounds is completely determined by the structural parameters, *A* and ζ [1]:

- Type I reservoir: BaO-CuO-BaO
- Type II reservoir: CuO
_{2}-Y-CuO_{2} - σ = σ
_{0} - η = 2; ν = 2
*A*= 15.3306 Å^{2}; ζ = 2.1952 Å- T
_{C0}^{calc}= 98.00 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011).;**23**349501 (2011).

**(Ca _{0.45}La_{0.55})(Ba_{1.3}La_{0.7})Cu_{3}O_{y}** (

*Pmmm*, T

_{C0}

^{meas}= 80.5 K)

The charge compensated compound, (Ca_{x}La_{1–x})(Ba_{1.75–x}La_{0.25+x})Cu_{3}O_{y} (or CLBLCO), optimizes for x = 0.45. To determine γ, one considers only the type I reservoir and its stoichiometric relationship to that of YBa_{2}Cu_{3}O_{6.92}. From rule (2b) γ is defined by the ratio of the Ba content of CLBLCO (i.e., 1.75 – x) divided by the 2 Ba ions contained in YBa_{2}Cu_{3}O_{6.92}. Thus, one obtains [1]:

- Type I reservoir: (Ba
_{1.3}La_{0.7})O-CuO-(Ba_{1.3}La_{0.7})O - Type II reservoir: CuO
_{2}-(Ca_{0.45}La_{0.55})-CuO_{2} - σ = [(1.75 – 0.45)/2] σ
_{0}= 0.148 σ_{0} - η = 2; ν=2
*A*= 15.0118 Å^{2}; ζ = 2.1297 Å- T
_{C0}^{calc}= 82.29 K

*Verifying Eq. (2.3) of Ref. [2] *– The equilibrium assertion of equation, νσ_{I} = ησ_{II}, defines the requirement for achieving an optimal high-*T*_{C} superconducting state. It is, therefore, important that its validity be tested. An ideal material for this exercise is the charge-compensated compound (Ca_{x}La_{01–x})(Ba_{1.75–x}La_{0.25+x})Cu_{3}O_{y}, for which the doping parameters for La and O are well established [1]. Accepting that the La^{+3} substituting for Ba^{+2} and the excess oxygen are associated with the type I reservoir and Ca^{+2} substituting for La^{+3} is associated with the type II reservoir, one can write,

νσ_{I }=ησ_{II} → ν(γ_{Ba/La}) (γ_{O}) σ_{0} = η(γ_{La/Ca}) σ_{0} ,

where, in this case, ν = η = 2. The three γ-factors derive from the valence scaling rules as follows:

- γ
_{Ba/La}= (1.75 –*x*)/2; derived utilizing rule (2b), which scales the outer-layer Ba^{+2}content [= (1.75 –*x*)] to that of YBa_{2}Cu_{3}O_{92}(= 2) [1]. - γ
_{O}= [(*y*–*y*_{0})/(6.92 – 6.35)]; deduced by taking*y*= 7.15 ± 0.02 and*y*_{0}= 6.88 (at*x*= 0.40, near optimal stoichiometry) [1], and scaling to the corresponding oxygen content of YBa_{2}Cu_{3}O_{6.92}participating in superconductivity [2]. - γ
_{La/Ca}= [1 + (2–3)/3]*x*= 2*x*/3; follows from rule (2a) for Ca^{+2}substituting for La^{+3}in the type II reservoir. Since both Ca and La have very low and nearly identical electronegativities (1.0 and 1.1, respectively), no additional factors come into play.

By inserting the value for y – y_{0} (= 0.27) into the above equation, and solving for *x*, one obtains a value of 0.46 which is in excellent agreement with average value of *x* = 0.45, thus verifying Eq. (2.3) of Ref. [2].

- D. R. Harshman and A. T. Fiory, Phys. Rev. B
**86**, 144533 (2012). - D. R. Harshman, A. T. Fiory and J. D. Dow, Phys. Rev. B
**86**, 144533 (2012);**23**349501 (2011).

**YBa _{2}Cu_{4}O_{8} (12 Gpa)** (

*Ammm*, T

_{C0}

^{meas}= 104 K)

For this material, the valences and stoichiomentries are unchanged with respect to YBa_{2}Cu_{3}O_{6.92}, such that γ = 1. Therefore, the optimal transition temperature for this and all other REBa_{2}Cu_{3}O_{7–δ} compounds is completely determined by the structural parameters, *A* and ζ [1]:

- Type I reservoir: BaO-CuO-CuO-BaO
- Type II reservoir: CuO
_{2}-Y-CuO_{2} - σ = σ
_{0} - η = 2; ν=2
*A*= 14.2060 Å^{2}; ζ = 2.1658 Å- T
_{C0}^{calc}= 103.19 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011).

**Tl _{2}Ba_{2}CuO_{6}** (

*I4/mmm*, T

_{C0}

^{meas}= 80 K)

Given the +3 charge state of Tl in Tl_{2}Ba_{2}CuO_{6}, compared to Cu^{+2} in the CuO chain layer of YBa_{2}Cu_{3}O_{6.92}, rule (2a) introduces a factor of 1/2 in γ. The presence of the double TlO layer structure, in place of the single CuO chain layer in YBa_{2}Cu_{3}O_{6.92}, invokes an additional factor of 2 from rule (2b). Combining the two factors then yields [1,2]:

- Type I reservoir: BaO-TlO-TlO-BaO
- Type II reservoir: CuO
_{2} - σ = (1/2)(2) σ
_{0}= σ_{0}^{[2]} - η = 1; ν=2
*A*= 14.9460 Å^{2}; ζ = 1.9291 Å- T
_{C0}^{calc}= 79.86 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011). - D. R. Harshman and A. T. Fiory, J. Phys. Chem. Solids
**85**, 106 (2015).

**Tl _{2}Ba_{2}CaCu_{2}O_{8}** (

*I4/mmm*, T

_{C0}

^{meas}= 110 K)

Given the +3 charge state of Tl in Tl_{2}Ba_{2}CaCu_{2}O_{8}, compared to Cu^{+2} in the CuO chain layer of YBa_{2}Cu_{3}O_{6.92}, rule (2a) introduces a factor of 1/2 in γ. The presence of the double TlO layer structure, in place of the single CuO chain layer in YBa_{2}Cu_{3}O_{6.92}, invokes an additional factor of 2 from rule (2b). Combining the two factors then yields [1,2]:

- Type I reservoir: BaO-TlO-TlO-BaO
- Type II reservoir: CuO
_{2}-Ca-CuO_{2} - σ = (1/2)(2) σ
_{0}= σ_{0}^{[2]} - η = 2; ν=2
*A*= 14.8610 Å^{2}; ζ = 2.0139 Å- T
_{C0}^{calc}= 108.50 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011). - D. R. Harshman and A. T. Fiory, J. Phys. Chem. Solids
**85**, 106 (2015).

**Tl _{2}Ba_{2}Ca_{2}Cu_{3}O_{10}** (

*I4/mmm*, T

_{C0}

^{meas}= 130 K)

Given the +3 charge state of Tl in Tl_{2}Ba_{2}Ca_{2}Cu_{3}O_{10}, compared to Cu^{+2} in the CuO chain layer of YBa_{2}Cu_{3}O_{6.92}, rule (2a) introduces a factor of 1/2 in γ. The presence of the double TlO layer structure, in place of the single CuO chain layer in YBa_{2}Cu_{3}O_{6.92}, invokes an additional factor of 2 from rule (2b). Combining the two factors then yields [1,2]:

- Type I reservoir: BaO-TlO-TlO-BaO
- Type II reservoir: CuO
_{2}-Ca-CuO_{2}-Ca-CuO_{2} - σ = (1/2)(2) σ
_{0}= σ_{0}^{[2]} - η = 3; ν=2
*A*= 14.8248 Å^{2}; ζ = 2.0559 Å- T
_{C0}^{calc}= 130.33 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011). - D. R. Harshman and A. T. Fiory, J. Phys. Chem. Solids
**85**, 106 (2015).

**TlBa _{1.2}La_{0.8}CuO_{5}** (P4/mmm, T

_{C0}

^{meas}= 45.4 K)

Given the +3 charge state of Tl in TlBa_{1.2}La_{0.8}CuO_{5}, compared to Cu^{+2} in the CuO chain layer of YBa_{2}Cu_{3}O_{6.92}, rule (2a) introduces a factor of 1/2 in γ. The presence of only 1.2 Ba ions in the outer layers, compare to two in YBa_{2}Cu_{3}O_{6.92}, invokes an additional γ-factor of 1.2/2 = 0.6 from rule (2b). Combining the two factors then yields [1,2]:

- Type I reservoir: (Ba
_{1.2}La_{0.8})O-TlO-(Ba_{1.2}La_{0.8})O - Type II reservoir: CuO
_{2} - σ = (1/2)(0.6) σ
_{0}= 0.300 σ_{0}^{[2]} - η = 1; ν=2
*A*= 14.7475 Å^{2}; ζ = 1.9038 Å- T
_{C0}^{calc}= 44.62 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011). - D. R. Harshman and A. T. Fiory, J. Phys. Chem. Solids
**85**, 106 (2015).

**Tl _{0.7}LaSrCuO_{5}** (P4/mmm, T

_{C0}

^{meas}= 37 K)

Given the +3 charge state of Tl in Tl_{0.7}LaSrCuO_{5}, compared to Cu^{+2} in the CuO chain layer of YBa_{2}Cu_{3}O_{6.92}, and treating vacancies as non-contributors (i.e., equivalent to a factor of 1/4 – see Bi-2212), rule (2a) introduces a γ-factor of [0.7(1/2) + 0.3(1/4)]. The presence of only 1 Sr^{+2} ion in the outer layers, compare to two Ba^{+2} ions in YBa_{2}Cu_{3}O_{6.92}, invokes an additional γ-factor of 1/2 from rule (2b). Combining the two factors then yields [1,2]:

- Type I reservoir: LaSrO-Tl
_{0.7}O-LaSrO - Type II reservoir: CuO
_{2} - σ = (1/2)[0.7(1/2) + 0.3(1/4)] σ
_{0}= 0.2125 σ_{0 }^{[2]} - η = 1; ν=2
*A*= 14.2453 Å^{2}; ζ = 1.8368 Å- T
_{C0}^{calc}= 39.63 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011). - D. R. Harshman and A. T. Fiory, J. Phys. Chem. Solids
**85**, 106 (2015).

**TlBa _{2}CaCu_{2}O_{7–δ}** (

*P4/mmm*, T

_{C0}

^{meas}= 103 K)

In the case of the two Tl-based compounds containing a single TlO layer, TlBa_{2}CaCu_{2}O_{7–δ} (Tl-1212) and TlBa_{2}Ca_{2}Cu_{3}O_{9–δ} (Tl-1223), the mixed-valence nature of Tl places the average Tl oxidation state between *f*_{+1} and *f*_{+3}. As a consequence, σ becomes a function of the fractional distribution of the monovalent and trivalent Tl oxidation states, which is measured experimentally. Denoting *f*_{+1} to be the fraction of monovalent cations in the inner TlO layer (i.e., Tl^{+1}), and apportioning a mixture of +1 and +3 valencies in rule (2a), one has γ = (2)*f*_{+1} + (1/2)(1 – *f*_{+1}) for these optimal compounds; therefore, σ = γ σ_{0} = (1.5 *f*_{+1} + 0.5) σ_{0}. Knowing *f*_{+1} thus determines σ or, alternatively, *f*_{+1} can be deduced for a given value of σ. Given the absence of +3 cations in the outer type I layers, one may assume for argument’s sake that, as with their double-TlO layer counterparts, charge depletion below σ_{0} does not occur for these compounds; taking optimal σ = σ_{0} one has [1,2]:

- Type I reservoir: BaO-TlO-BaO
- Type II reservoir: CuO
_{2}-Ca-CuO_{2} - σ = (1.5
*f*_{+1}+ 0.5) σ_{0}= σ_{0}(assumed the same as Tl-1223)^{[2]} *f*_{+1}= 1/3;*f*_{+3}= 2/3^{[2]}- η = 2; ν=2
*A*= 14.8734 Å^{2}; ζ = 2.0815 Å- T
_{C0}^{calc}= 104.93 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011). - D. R. Harshman and A. T. Fiory, J. Phys. Chem. Solids
**85**, 106 (2015).

**TlBa _{2}Ca_{2}Cu_{3}O_{9–δ}** (

*P4/mmm*, T

_{C0}

^{meas}= 133.5 K)

In the case of the two Tl-based compounds containing a single TlO layer, TlBa_{2}CaCu_{2}O_{7–δ} (Tl-1212) and TlBa_{2}Ca_{2}Cu_{3}O_{9–δ} (Tl-1223), the mixed-valence nature of Tl places the average Tl oxidation state between *f*_{+1} and *f*_{+3}. As a consequence, σ becomes a function of the fractional distribution of the monovalent and trivalent Tl oxidation states, which is measured experimentally. Denoting *f*_{+1} to be the fraction of monovalent cations in the inner TlO layer (i.e., Tl^{+1}), and apportioning a mixture of +1 and +3 valencies in rule (2a), one has γ = (2)*f*_{+1} + (1/2)(1 – *f*_{+1}) for these optimal compounds; therefore, σ = γ σ_{0} = (1.5 *f*_{+1} + 0.5) σ_{0}. Knowing *f*_{+1} thus determines σ or, alternatively, *f*_{+1} can be deduced for a given value of σ. Given the absence of +3 cations in the outer type I layers, one may assume for argument’s sake that, as with their double-TlO layer counterparts, charge depletion below σ_{0} does not occur for these compounds; taking optimal σ = σ_{0} one has [1,2]:

- Type I reservoir: BaO-TlO-BaO
- Type II reservoir: CuO
_{2}-Ca-CuO_{2}-Ca-CuO_{2} - σ = (1.5
*f*_{+1}+ 0.5) σ_{0}= σ_{0}(assumed)^{[2]} *f*_{+1}= 1/3;*f*_{+3}= 2/3^{[2]}(shown experimentally)^{[2]}- η = 3; ν=2
*A*= 14.7686 Å^{2}; ζ = 2.0315 Å- T
_{C0}^{calc}= 132.14 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011). - D. R. Harshman and A. T. Fiory, J. Phys. Chem. Solids
**85**, 106 (2015).

**HgBa _{2}Ca_{2}Cu_{3}O_{8+δ}** (

*P4/mmm*, δ = 0.27±0.04, T

_{C0}

^{meas}= 134.43 K)

For this material, it is assumed that σ = σ_{0}. Thus one has [1]:

- Type I reservoir: BaO-HgO
_{x}-BaO - Type II reservoir: CuO
_{2}-Ca-CuO_{2}-Ca-CuO_{2} - σ = σ
_{0}(assumed) - η = 3; ν=2
*A*= 14.8060 Å^{2}; ζ = 1.9959 Å- T
_{C0}^{calc}= 134.33 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011).

**HgBa _{2}Ca_{2}Cu_{3}O_{8+δ}**

**(25 GPa)**(

*P4/mmm*, δ = 0.27±0.04, T

_{C0}

^{meas}= 145 K)

For this material, it is assumed that σ = σ_{0}, where the increase in T_{C0}^{calc} over the 0 GPa value is directly related to the decrease in *A* and ζ . Thus one has [1]:

- Type I reservoir: BaO-HgO
_{x}-BaO - Type II reservoir: CuO
_{2}-Ca-CuO_{2}-Ca-CuO_{2} - σ = σ
_{0}(assumed) - η = 3; ν=2
*A*= 13.6449 Å^{2}; ζ = 1.9326 Å- T
_{C0}^{calc}= 144.51 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011).

**HgBa _{2}CuO_{4.15}** (

*P4/mmm*, T

_{C0}

^{meas}= 95 K)

We know of at least two optimal compounds for which σ > σ_{0}, HgBa_{2}CuO_{4.15} and HgBa_{2}CaCu_{2}O_{6.22}, T_{C0} = 95 K and 127 K, respectively, where the two electrons per excess oxygen (0.15 and 0.22, respectively) are equally distributed among the (3+η) layers (this is equivalent to dividing the carriers between the two reservoirs, and then by the average number of layers per reservoir). For HgBa_{2}CuO_{4.15}, one then finds that σ = σ_{0} + 2(0.15)/(3+1) = 0.3030. This enhancement in T_{C0}^{meas} is attributed to the unique structure of the Hg compounds which provides vacancy locations for excess oxygen and the ≤ 2 valence of Hg. Thus one has [1]:

- Type I reservoir: BaO-HgO-BaO
- Type II reservoir: CuO
_{2} - σ = σ = σ
_{0}+ 2(0.15)/(3+1) = σ_{0}+ 0.075 = 0.3030 - η = 1; ν=2
*A*= 15.0362 Å^{2}; ζ = 1.9214 Å- T
_{C0}^{calc}= 92.16 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011).

**HgBa _{2}CaCu_{2}O_{6.22}** (

*P4/mmm*, T

_{C0}

^{meas}= 127 K)

We know of at least two optimal compounds for which σ > σ_{0}, HgBa_{2}CuO_{4.15} and HgBa_{2}CaCu_{2}O_{6.22}, T_{C0} = 95 K and 127 K, where the two electrons per excess oxygen (0.15 and 0.22, respectively) are equally distributed among the (3+η) layers (this is equivalent to dividing the carriers between the two reservoirs, and then by the average number of layers per reservoir). For HgBa_{2}CaCu_{2}O_{6.22}, one then finds that σ = σ_{0} + 2(0.22)/(3+1) = 0.3160. This enhancement in T_{C0}^{meas} is attributed to the unique structure of the Hg compounds which provides vacancy locations for excess oxygen and the ≤ 2 valence of Hg. Thus one has [1]:

- Type I reservoir: BaO-HgO-BaO
- Type II reservoir: CuO
_{2}-Ca-CuO_{2} - σ = and σ = σ
_{0}+ 2(0.22)/(3+2) = σ_{0}+ 0.088 = 0.3160, - η = 2; ν=2
*A*= 14.9375 Å^{2}; ζ = 2.0390 Å- T
_{C0}^{calc}= 125.84 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011).

**La _{1.837}Sr_{0.163}CuO_{4–δ}** (

*I4/mmm*, 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 [rule (1b)], 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 such that [1]:

- Type I reservoir: (La
_{1.837}Sr_{0.163})O-(La_{1.837}Sr_{0.163})O - Type II reservoir: CuO
_{2} - 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

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011).

**La _{1.8}Sr_{0.2}CaCu_{2}O_{6±δ}** (

*I4/mmm*, T

_{C0}

^{meas}= 58 K)

Since x_{0} = 0, the total charge doping is 0.2. 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 such that [1]:

- Type I reservoir: (La
_{1.8}Sr_{0.2})O-(La_{1.8}Sr_{0.2})O - Type II reservoir: CuO
_{2}-Ca-CuO_{2} - σ = (0.2)(1/2)(1/2) = 0.05
- η = 2 ; ν = 2
*A*= 14.3761 Å^{2}; ζ = 1.7829 Å- T
_{C0}^{calc}= 58.35 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011).

**(Sr _{0.9}La_{0.1})CuO_{2}** (

*P4/mmm*, T

_{C0}

^{meas}= 43 K)

For this material, superconductivity occurs with La doping (with x_{0} = 0). Given that the total charge doping is 0.1, which is distributed equally between the type I and type II reservoir, rule (1b) introduces a factor of 1/2 in γ. Since ν = 1, there are no additional γ-factors such that [1]:

- Type I reservoir: (Sr
_{0.9}La_{0.1}) - Type II reservoir: CuO
_{2} - σ = (0.1)(1/2) = 0.05
- η = 1; ν = 1
*A*= 15.6058 Å^{2}; ζ = 1.7051 Å- T
_{C0}^{calc}= 41.41 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011).

**Ba _{2}Y(Ru_{0.9}Cu_{0.1})O_{6}** (

*Fm3m*, T

_{C0}

^{meas}= 35 K)

The ruthenate compounds A_{2}Y(Ru_{0.9}Cu_{0.1})O_{6} (with A = Ba or Sr; x = 0.05–0.15) are double-perovskites containing no cuprate planes and with η = ν = 1 (reference [82] in original paper [1]). The determination of γ is done using charge allocation, wherein rule (1b) introduces the factor 1/2. In the lower limit, one expects a minimum of ∼ 2 charges per Cu dopant, which are shared between two charge reservoirs of each layer type [AO and 1/2 (YRu_{1–x}Cu_{x}O_{4})], producing a net factor of unity. Thus, for Ba_{2}YRu_{0.9}Cu_{0.1}O_{6} (with T_{C0}^{meas} ∼ 30–40 K), we have γ = (1/2)(1) = 1/2, as stated by equation in the original paper of [1]. The corrigendum also corrects a typographical correction in the definition of the corresponding ruthenate type II reservoir in the last column of table 1 in the original paper [1], which should read 1/2 (YRu_{0.9}Cu_{0.1}O_{4}). Thus one has [1]:

- Type I reservoir: BaO
- Type II reservoir: ½(YRu
_{0.9}Cu_{0.1}O_{4}) - σ = (0.1)(1/2) = 0.05
- η = 1; ν = 2
*A*= 17.3208 Å^{2}; ζ = 2.0809 Å- T
_{C0}^{calc}= 32.21 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011).

**(Pb _{0.5}Cu_{0.5})Sr_{2}(Y/Ca)Cu_{2}O_{7–δ}** (P4/mmm; T

_{C0}

^{meas}= 67 K)

As the description reported in the original work [1] was a bit misleading, so a correct discussion is given here [2]. The (Pb_{0.5}Cu_{0.5.})Sr_{2} structure makes up the type I reservoir in this material, with both Pb and Cu possessing a +3 charge state (primarily, e.g., from Pb-O bond length measurements). Using Stoichiometric (valency) scaling with respect to YBa_{2}Cu_{3}O_{6.92}, rule (2a) introduces a factor of 1/2 for both. However, the large electronegativity (see discussion of Bi-2212 in Ref. [2]) for Pb (χ = 2.33) when compared with Cu (χ = 1.90), an additional factor of 1/2 must accompany the Pb component, such that, γ = (1/2)(0.5_{Pb}/2 + 0.5_{Cu}) = (1/2)(3/4). Thus one has [1-3]:

- Type I reservoir: SrO-(Pb
_{0.5}Cu_{0.5})O-SrO - Type II reservoir: CuO
_{2}-(Y/Ca)-CuO_{2} - σ = (1/2)(3/4) σ
_{0}= 0.375 σ_{0} - η = 2; ν=2
*A*= 14.5771 Å^{2}; ζ = 1.9967 Å- T
_{C0}^{calc}= 67.76 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011). - D. R. Harshman and A. T. Fiory, J. Phys. Chem. Solids
**85**, 106 (2015). - Unpublished (2014).

**Bi _{2}Sr_{2}CaCu_{2}O_{8+δ}** (

*N Bbmb/111*, T

_{C0}

^{meas}= 89 K)

In Bi_{2}Sr_{2}(Ca,Y)_{x}Cu_{2}O_{8+δ} (Bi-2212), where excess oxygen located in the Bi_{2}O_{2} block, as well as cation stoichiometry and substitutional defects, are correlated with a strong incommensurate modulation along the *b*-axis of the basal plane [1,2]. This extra oxygen content, which appears to be enhanced (along with T_{C}) by increasing the Ca vacancies (for x < 1, with zero Y content) or the Y/Ca substitution ratio (x ≈ 1), introduces additional carriers for which the cation scaling rules alone do not account (accurate knowledge of the excess oxygen content in the Bi_{2}O_{2} block would be required in order to apply anion scaling). Variation of ζ for non-stoichiometric compositions would also come into play. Consequently, cation scaling to YBa_{2}Cu_{3}O_{6.92} is applicable only the near stoichiometric Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}, compound (as-grown and unannealed), assuming minimal extra-oxygen enhancement in carrier density, and having a measured transition of 89 K [2]. Given that there are two BiO layers compared to the one CuO chain in YBa_{2}Cu_{3}O_{6.92}, rule (2b) introduces a γ-factor of 2, and since Bi^{+3} is mapped onto the chain Cu^{+2} cation, rule (2a) provides another factor of 1/2. Finally, because the electronegativity [2] of Bi (χ = 2.02) is significantly greater than that of Cu (χ = 1.90), charge transfer along the hard axis is suppressed, generating another factor of 1/2. Utilizing more recent structural refinement data specific to a near-stoichiometric single-crystal sample at 12 K (see Ref. [2]), one then has [1,2]:

- Type I reservoir: SrO-BiO-BiO-SrO
- Type II reservoir: CuO
_{2}-Ca-CuO_{2} - σ = (2)(1/2)(1/2) σ
_{0}= 0.5 σ_{0} - η = 2; ν=2
*A*= 14.5201 Å^{2}; ζ = 1.7500 Å^{[2]}- T
_{C0}^{calc}= 89.32 K^{[2]}

Note that the value of 86.65 K originally published in Ref. [1] was obtained using structural data for a nonstoichiometric powder sample.

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011). - D. R. Harshman and A. T. Fiory, J. Phys. Chem. Solids
**85**, 106 (2015).

**(Bi/Pb) _{2}Sr_{2}Ca_{2}Cu_{3}O_{10+δ}** (

*Fmmm*, T

_{C0}

^{meas}= 112 K)

Given that there are two BiO layers compared to the one CuO chain in YBa_{2}Cu_{3}O_{6.92}, rule (2b) introduces a γ-factor of 2, and since Bi^{+3} is mapped onto the chain Cu^{+2} cation, rule (2a) provides another factor of 1/2. Finally, because the electronegativity (see discussion of Bi-2212 in Ref. [2]) of Bi (χ = 2.02) is significantly greater than that of Cu (χ = 1.90), charge transfer along the hard axis is suppressed, generating another factor of 1/2. Thus, one has [1,2]:

- Type I reservoir: SrO-(Bi/Pb)-(Bi/Pb)-SrO
- Type II reservoir: CuO
_{2}-Ca-CuO_{2}-Ca-CuO_{2} - σ = (2)(1/2)(1/2) σ
_{0}= 0.5 σ_{0} - η = 3; ν=2
*A*= 14.6340 Å^{2}; ζ = 1.6872 Å- T
_{C0}^{calc}= 113.02 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011). - D. R. Harshman and A. T. Fiory, J. Phys. Chem. Solids
**85**, 106 (2015).

**Pb _{2}Sr_{2}(Y/Ca)Cu_{3}O_{8}** (T

_{C0}

^{meas}= 75 K)

Given that there are two PbO layers compared to the one CuO chain in YBa_{2}Cu_{3}O_{6.92}, rule (2b) introduces a γ-factor of 2. Assuming a +3 average charge state for Pb, and since Pb^{+3}O is mapped onto the chain Cu^{+2}O, rule (2a) provides another factor of 1/2. Finally, because the relative electronegativity (see discussion of Bi-2212 in Ref. [2]) of Pb (χ = 2.33) is significantly greater than that of Cu (χ = 1.90), charge transfer along the hard axis is suppressed, generating another factor of 1/2. Thus, one has [1,2]:

- Type I reservoir: SrO-PbO-Cu-PbO-SrO
- Type II reservoir: CuO
_{2}-(Y/Ca)-CuO_{2} - σ = (2)(1/2)(1/2) σ
_{0}= 0.5 σ_{0} - η = 2; ν=2
*A*= 14.6458 Å^{2}; ζ = 2.0280 Å- T
_{C0}^{calc}= 76.74 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011). - D. R. Harshman and A. T. Fiory, J. Phys. Chem. Solids
**85**, 106 (2015).

**Bi _{2}(Sr_{1.6}La_{0.4})CuO_{6+δ}** (

*Cmmm*, T

_{C0}

^{meas}= 34 K)

For the single-layer material, Bi_{2}(Sr_{1.6}La_{0.4})CuO_{6+δ}, the 1/2 γ-factor (relative to YBa_{2}Cu_{3}O_{6.92}) in rule (2a) arising from the presence of a double BiO layer structure (in place of a single CuO chain layer), would naturally apply. However, since YBa_{2}Cu_{3}O_{6.92} has two corresponding Ba^{+2} ions, the partial Sr^{+2} doping (x = 1.6) of the outer layers introduces a relative doping factor of ( 1.6 − 1.16 ) = 0.44 according to rule (2b), where x_{0} = 1.16. Finally, the comparatively large electronegativity (see discussion of Bi-2212 in Ref. [2]) of Bi (χ = 2.02) to that of Cu (χ = 1.90) yields [1,2]:

- Type I reservoir: ½(Sr
_{1.6}La_{0.4})-BiO-BiO-½(Sr_{1.6}La_{0.4}) - Type II reservoir: CuO
_{2} - σ = (1/2)(0.44)(1/2) σ
_{0}= 0.11 σ_{0} - η = 1; ν=2
*A*= 14.5422 Å^{2}; ζ = 1.4880 Å- T
_{C0}^{calc}= 34.81 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011). - D. R. Harshman and A. T. Fiory, J. Phys. Chem. Solids
**85**, 106 (2015).

**RuSr _{2}GdCu_{2}O_{8}** (

*P4/mmm*, T

_{C0}

^{meas}= 50 K)

In the case of the ruthenocuprate compound, RuSr_{2}GdCu_{2}O_{8}, the structure contains a type I reservoir SrO–RuO_{2}–SrO, where the Cu^{+2}O chain layer is replaced by a Ru^{+5}O_{2} layer, and Y^{+3} is replaced by Gd^{+3}. Taking the Ru charge state to be +5 in this material, and given the charge equivalence of Bi^{+3}O^{−2} and Ru^{+5}(O^{–2})_{2}, one can draw an analogy with the Bi/Pb compounds, and approximate σ. In this case, however, there is only one layer that is charge equivalent. Thus, from rule (2a), one factor of 1/2 is introduced arising from the valency scaling between Ru^{+5}(O^{–2})_{2} and Cu^{+2}O, Given that the Pauling electronegativity (see discussion of Bi-2212 in Ref. [2]) of Ru is 2.20, which is significantly greater than that of Cu (χ = 1.90), charge transfer aling the hard axis is suppressed, invoking an additional factor of 1/2, such that [1,2]

- Type I reservoir: SrO-RuO
_{2}-SrO - Type II reservoir: CuO
_{2}-Gd-CuO_{2} - σ = (1/2)(1/2) σ
_{0}= 0.25 σ_{0} - η = 2; ν=2
*A*= 14.7372 Å^{2}; ζ = 2.1820 Å- T
_{C0}^{calc}= 50.28 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011). - D. R. Harshman and A. T. Fiory, J. Phys. Chem. Solids
**85**, 106 (2015).

**(Ln/Th)(O _{x–y}F_{1–x})FeAs** (

*P4/nmm*, T

_{C0}

^{meas}= 26 K – 55 K)

In the (n-type) Ln–O/F–Fe–As (Ln-1111) iron pnictides, e.g., La(O_{0.92−y}F_{0.08})FeAs (T_{C0}^{meas} = 26 K), Ce(O_{0.84−y}F_{0.16})FeAs (T_{C0}^{meas} = 35 K), Tb(O_{0.80−y}F_{0.20})FeAs (T_{C0}^{meas} = 45 K) and Sm(O_{0.65 −y}F_{0.35})FeAs (T_{C0}^{meas} = 55 K), where y accounts for the actual O-site occupancy; we assume [x − x_{0}] is given by F stoichiometry, the Coulombic interaction is assumed to occur between the Ln(O/F) and AsFe layers, which defines ζ (Ln–As distance along the *c*-axis), and sets η = ν = 1 (i.e. within the periodicity *d*). Applying rules (1b) (equally shared doping between the hole and electron reservoirs) and (1a) (where the doping is further divided between component layers of the two reservoirs, [(O/F)Ln and (AsFe)], one obtains γ = (1/2)(1/2) = (1/4). For the Th^{+4} doped n-type 1111 iron pnictide compounds, such as (Sm_{0.7}Th_{0.3})OFeAs (T_{C0}^{meas} = 51.5 K), the value of σ is given similarly, but with [x − x_{0}] determined by the Th doping content. Thus one has [1]: [Need to recheck structure parameters]

- Type I reservoir: ½(Ln-2O/F-Ln)
- Type II reservoir: ½(As-2Fe-As)
- σ = (1/2)(1/2) [x – x
_{0}] - η = 1; ν = 1
**La(O**_{0.92−y}F_{0.08})FeAs:*A*= 16.1620 Å^{2}; ζ = 1.7677 Å; T_{C0}^{calc}= 24.82 K**Ce(O**_{0.84−y}F_{0.16})FeAs:*A*= 15.8778 Å^{2}; ζ = 1.6819 Å; T_{C0}^{calc}= 37.23 K**Tb(O**_{0.80−y}F_{0.20})FeAs:*A*= 14.8996 Å^{2}; ζ = 1.5822 Å; T_{C0}^{calc}= 45.67 K**Sm(O**_{0.65−y}F_{0.35})FeAs:*A*= 15.4535 Å^{2}; ζ = 1.6670 Å; T_{C0}^{calc}= 56.31 K**(Sm**_{0.7}Th_{0.3})OFeAs:*A*= 15.4897 Å^{2}; ζ = 1.6710 Å; T_{C0}^{calc}= 51.94 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011).

**(Ba/K)(Fe/Co) _{2}As_{2}** (

*I4/mmm*, T

_{C0}

^{meas}= 37 K & 22 K)

As in the case of the 1111 structures, rules (1a) and (1b) also apply in determining γ for the 122 compounds, e.g., (Ba_{0.6}K_{0.4})Fe_{2}As_{2} and Ba(Fe_{1.84}Co_{0.16})As_{2}, except that the doping is also shared between the two FeAs structures [rule (1a)] yielding an additional factor of 1/2 such that γ = (1/2)(1/2)(1/2) = 0.125. For (Ba_{0.6}K_{0.4})Fe_{2}As_{2} (T_{C0}^{meas} = 37 K, x = 0.4 and x_{0} = 0), the Coulombic interaction occurs between the Ba/K layer and the two adjacent Fe^{+2}As^{–3} structures, where we designate the Ba(K) layers as type II ( η = 1), which interact with the As in the FeAs type I (ν = 2) structures. The γ factor for the n-type homologue to the above p-type compound, Ba(Fe_{1.84}Co_{0.16})As_{2} (T_{C0}^{meas} = 22 K, x = 0.16 and x_{0} = 0), is similarly calculated, but with [x − x_{0}] corresponding to Co doping. Therefore [1]:

- Type I reservoir: As-2(Fe/Co)-As
- Type II reservoir: Ba/K
- σ = (1/2)(1/2)(1/2) [x − x
_{0}] - η = 1; ν = 2
**(Ba**_{0.6}K_{0.4})Fe_{2}As_{2}:*A*= 15.2803 Å^{2}; ζ = 1.932 Å, T_{C0}^{calc}= 36.93 K**Ba(Fe**_{1.84}Co_{0.16})As_{2}:*A*= 15.6848 Å^{2}; ζ = 1.8920 Å, T_{C0}^{calc}= 23.54 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011).

**FeSe _{0.977} (7.5 GPa)** (

*Pnnm*, T

_{C0}

^{meas}= 36.5 K) [Check structure information]

While superconducting (tetragonal) Fe_{1+x}Se_{1−y} is very nearly stoichiometric, some degree of symmetry breaking, arising from either a Se deficiency or an excess in Fe content, must exist to induce the superconducting condensate. It is also clear that the optimal superconducting state is achieved at a hydrostatically applied pressure of 7.5 – 8.5 GPa, which we attribute to pressure-induced charge redistribution, analogous to the response of YBa_{2}Cu_{4}O_{8}. Although assigning reservoir types to the Fe and Se layers is somewhat subjective, it is also not a necessary requirement in determining σ (for convention, we refer to the negative valence Se layer as type I). Given the +2 and −2 valences of Fe and Se, respectively, the charge doping is 2x or 2y (or, typically, the sum thereof), where x and y are small and positive quantities. High resistivity (∼33 mΩcm at T ∼ 9 K) observed confirms that stoichiometric FeSe is dominantly a non-metallic material. One may therefore assume that the FeSe binary compound of precisely 1:1 stoichiometry is effectively an insulating material and that doping is achieved by introducing excess Fe or depleted Se. From rule (1b) the dopant charge populates both charge reservoirs, yielding γ = 1/2. For FeSe_{0.977} (y = 0.023, y_{0} = 0), the doping factor has the single term [2(y−y_{0})] = 2(0.023) and the fractional charge is then calculated to be σ = 2(0.023)/2 = 0.023. The basal-plane lattice parameter at 7.5 GPa is *a* = 3.622 A. The interaction distance ζ is measured between nearest-neighbor Fe and Se ions observed at the same pressure. Thus one has [1]:

- Type I reservoir: Se
_{0.977} - Type II reservoir: Fe
_{1.0} - σ = (1/2) [2(y − y
_{0})] = 0.023 - η = 1; ν = 1
*A*= 13.1189 Å^{2}; ζ = 1.4240 Å- T
_{C0}^{calc}= 36.68 K

- D. R. Harshman and A. T. Fiory, J. Phys.: Condens. Matt.
**24**, 135701 (2012).

**Fe _{1.03}Se_{0.57}Te_{0.43} (2.3 GPa)** (T

_{C0}

^{meas}= 23.3 K) [Check structure information]

To calculate σ for Fe_{1+x}Se_{1–y}Te_{y} system (for y > 0.3), one must consider the type I reservoir structure to be Te_{y}–Fe_{x}–Se_{1−y}, where ν = 2 and the charge doping, supplied by the excess Fe^{+2} ions (two charges per excess Fe cation), is shared equally between the Te_{y} and Se_{1−y} layers, with comparable occupancies since y ∼ 1/2. The result for γ is a factor of 1/2 from rule (1a), multiplied by an additional factor of 1/2 associated with rule (1b), yielding γ = 1/4. Since accurate information on Fe stoichiometry is essential, we consider results reported for a sample with composition Fe_{1.03}Se_{0.57}Te_{0.43}, where x ≈ 0.03 was obtained from pressure- and temperature-dependent Rietveld refinements of synchrotron x-ray powder diffraction data. The sample under study exhibited superconducting transitions (determined from magnetization onsets) of 13.9 K at ambient pressure, reaching a maximum T_{C} = 23.3 K under hydrostatic pressure of 2.3 GPa. Unfortunately, only resistance and zero-field-cooled magnetization data are given, limiting our ability to quantify the sample’s quality. In another work on an FeSe_{0.5}Te_{0.5} sample (in this case Fe stoichiometry of unity was apparently presumed, leaving σ undetermined), the superconducting transitions (onset values) varyd from 13.5 K at ambient pressure, reaching a maximum T_{C} of ~25–26 K (26.2 K reported) at 2 GPa. The sample in this case was judged by the authors to be of high quality, based on a 12% Meissner fraction (the field-cooled magnetization is about 38% of the zero-field-cooled magnetization; normal-state resistivity is 1.4 mΩcm extrapolated to T_{C} at 2 GPa). Since one does not expect the structural parameters to differ greatly between these two samples, it would normally be reasonable to accept 25.5 K as representative of the highest T_{C} attained for this compound. Given that the Fe content is unknown for the latter, results from the former are adopted for doping and T_{C0}^{meas}, with the understanding that the quoted transition temperature may be lower than optimum. To calculate T_{C0} for the Fe_{1.03}Se_{0.57}Te_{0.43} compound, we use data taken at 2.0 GPa; *a* = 3.7317 and *b* = 3.7262 Å, giving a plane area (per formula unit) of *A* = 13.9051 Å^{2}. One further estimates an average interaction distance 〈ζ〉 = 1.597 Å (assuming the measurement reflects an average between ζ_{Se} and ζ_{Te}). This value is corroborated elsewhere, which indicates ζ_{Te} = 1.7638 Å and ζ_{Se} = 1.4162 Å for a sample indicates ζ_{Te} = 1.7638 A under pressure (presumably 2.0 GPa), corresponding to an average value 〈ζ〉 = 1.59 Å. Using 〈ζ〉 takes into account disorder in site occupancy and interactions contributed by the further Te ion, given that (ζ_{Te} − ζ_{Se}) << 〈ζ〉. Above ∼2.5 GPa, the compound undergoes an orthorhombic to monoclinic transition, resulting in small abrupt changes to some of the relevant lattice parameters.

- Type I reservoir: Se
_{0.57}-Fe_{0.03}-Te_{0.43} - Type II reservoir: Fe
_{1.0} - σ = (1/2) (1/2) [2(1.03 – 1)] = 0.015
- η = 1; ν = 2
*A*= 13.9051 Å^{2}; 〈ζ〉 = 1.5970 Å- T
_{C0}^{calc}= 25.65 K

- D. R. Harshman and A. T. Fiory, J. Phys.: Condens. Matt.
**24**, 135701 (2012).

**A _{z}Fe_{2-x}Se_{2}** (T

_{C0}

^{meas}= 28.5 – 31.5 K)

Unlike Fe_{1+x}Se_{1−y}, the ternary A_{z}Fe_{2−x}Se_{2} series of compounds can be optimized at ambient pressure, and are characterized by a rather large normal-state resistivity at T_{C}, signifying a higher scattering rate. For these materials, the type I reservoir structures are identified as the Se–Fe_{2−x}–Se triple layers, with ν = 2, and the alkali metal A_{z} layers are defined as the type II (η = 1) reservoir structures (the exact formula-unit structure that we consider is Fe_{1−x/2}–Se–Az–Se–Fe_{1−x/2}, or equivalently A_{z/2}–Se–Fe_{2−x}–Se–A_{z/2} , the latter corresponding to figure 1(b) in Ref. [1]). There are generally two doping sources; one associated with the type I reservoir and the other with the type II. The two terms in equation (2) are [z_{0} − z] and +2[x_{0} − x] for A_{z} and Fe_{2−x}, respectively, where the prefactor of 2 corresponds to the double valence of Fe (the terms v_{i} (x − x_{0})_{i} for the two reservoir types have opposite signs). There also appear to be (at least) two insulating end materials corresponding to the alkali metal and iron components (determined for A = K): KFe_{2}Se_{2} such that z_{0} = 1, and KFe_{2−x0}Se_{2}, where 0.40 ≤ x_{0} ≤ 0.42. Since x and z depend on growth stoichiometries, these two doping sources generally provide unequal contributions to the fractional charge, which is found by combining the magnitudes of the two charges as per equation (2), and treating doping as if from a single source. Thus, by analogy with (Ba_{1−x}K_{x})Fe_{2}As_{2}, the charge allocation rules (1a) and (1b) apply. Here, one factor of 1/2 arises from rule (1b) requiring that the doping charge be shared between the two reservoirs; the factor of (1/2)(1/2) arises from dividing the charge among the four individual layers of the two Fe–Se structures of the type I reservoir (rule (1a) for N = 4, or applied twice with N = 2). From this we can write γ = (1/2)(1/2)(1/2) = 0.125. Thus one has [1]:

- Type I reservoir: Se-Fe
_{2-x}-Se - Type II reservoir: A
_{z} - σ = (1/2) (1/2) (1/2) [(1 – z) + 2(0.40 – x)]
- η = 1; ν = 2
**K**_{0.83}Fe_{1.66}Se_{2}:*A*= 15.2432 Å^{2}; 〈ζ〉 = 2.0241 Å; T_{C0}^{meas}= 29.5 K; T_{C0}^{calc}= 30.07 K**Rb**_{0.83}Fe_{1.70}Se_{2}:*A*= 15.4867 Å^{2}; 〈ζ〉 = 2.1463 Å; T_{C0}^{meas}= 31.5 K; T_{C0}^{calc}= 31.78 K**Cs**_{0.83}Fe_{1.71}Se_{2}:*A*= 16.1419 Å^{2}; 〈ζ〉 = 2.3298 Å; T_{C0}^{meas}= 28.5 K; T_{C0}^{calc}= 29.44 K

- D. R. Harshman and A. T. Fiory, J. Phys.: Condens. Matt.
**24**, 135701 (2012).

**Na _{0.16}(S)_{y}TiNCl** (

*S*= PC, BC ) (T

_{C0}

^{meas}~ 6 – 7 K)

High-T_{C} superconductivity in this model occurs in layered structures forming adjacent type-I and type-II charge reservoir layers containing the superconducting and mediating charges, respectively, repeating alternately along the transverse axis. The superconducting transition temperature depends on the spatially indirect Coulomb interaction across the transverse distance ζ between the two charge reservoirs, measured between the outer chlorines in the type-I [TiNCl]_{2} layer and the locus of the cations *A*_{x} in the neighboring type-II intercalation layer, assuming co-intercalant (*S*)_{y} is uncharged. The layered structure of *A*_{x}(*S*)_{y}TiNCl is characterized by a thickness *d*_{2} of the [TiNCl]_{2} layers, the transverse spacing d between them, and an intercalant thickness *d* − *d*_{2}. Assuming that the mean cation *A*_{x} locus is at the intercalant-layer midplane, the interaction distance is ζ = (*d* − *d*_{2})/2. Since *d*_{2} is approximately the same as for pristine α-TiNCl (see [1] of Ref. [1]), the observed functional dependence of T_{C} on d is expected to correlate with an analogous dependence on ζ . However, one notes that the interlayer interaction length is the shorter distance ζ, rather than the spacing d. Structural and superconductivity data are presented in Table I, listing directly measured values of *d*_{2} where available. The Coulomb energy e^{2}/ζ lies within 1.8 – 8.7 eV. For *A*_{x}(*S*)_{y}TiNCl, doping occurs only in the intercalation layer via *A*_{x} such that σ is determined according to the simplified relation, σ = γ |v(x − x_{0})|, where v is the valence and x is the optimal content of the cation dopant species in the type II *A*_{x}(S)_{y} reservoir; x_{0} is the threshold value of x for superconductivity; here, v = 1 for alkali-ion doping and x_{0} = 0 is inferred elsewhere. The factor γ derives from the allocation of the dopant by considering a given compound’s structure. Following the procedure generally applied to high-T_{C} superconductors, the charge introduced by the dopant is shared equally between the two charge reservoirs [rule (1b)]. Additionally, the methodology requires the doped charge to be distributed pairwise between the charge-carrying layer types within each of the charge reservoirs. Assuming the co-intercalant contributes no doping charge, determination of γ for *A*_{x}(S)_{y}TiNCl is comparable to that of (Ba_{0.6}K_{0.4})Fe_{2}As_{2} for which a structural analogy was previously noted. Sharing the charge equally between the two reservoirs contributes a factor of 1/2 to γ . Sharing between the Cl layer and the double-TiN-layered structure and then to the two TiN layers contributes two factors from rule (1a) of 1/2 to γ. Hence, γ = (1/2)(1/2)(1/2) = 1/8, yielding σ generally smaller than x. Thus, one has [1,2]:

- Type I reservoir: TiNCl
- Type II reservoir:
*A*_{x}(*S*)_{y} - σ = (1/2) (1/2) (1/2) [x
_{opt}] - η = 1; ν = 2
**Na**:_{0.16}(PC)_{y}TiNCl*A*= 13.0331 Å^{2}; ζ = 7.6735 Å; T_{C0}^{meas}= 6.3 K; T_{C0}^{calc}= 6.37 K**Na**:_{0.16}(BC)_{y}TiNCl*A*= 13.0331 Å^{2}; ζ = 7.7803 Å; T_{C0}^{meas}= 6.9 K; T_{C0}^{calc}= 6.28 K

- D. R. Harshman and A. T. Fiory, Phys. Rev. B 90, 186501 (2014).
- D. R. Harshman and A. T. Fiory, J. Supercond. Nov. Magn., accepted for publication (2015).

**Li _{x}(S)_{y}ZrNCl** (T

_{C0}

^{meas}= 13 .7 & 15.1 K)

The exemplary compound is β-Li_{x}ZrNCl, for which optimal stoichiometry is identified as occurring at x_{opt} = 0.08 at which point the full superconducting volume is reached and T_{C} (= 15.1 K) is maximized (see [19, 20] of Ref. [1]). Underdoping occurs for x < 0.08, as identified by diminished superconducting volumes that vanish for x < 0.05 [19], indicating x_{0} = 0.05. Muon-spin-depolarization rates σ_{μ}(T → 0) for Li_{x}ZrNCl are linear in (x − x_{0}) with x_{0} = 0.05 obtained by extrapolation (see [51] in Ref. [1]; also, plasma frequency data show that ω_{p}^{2} extrapolates to zero for x ≈ x_{0} (see [46] in Ref. [1]. Signature characteristics of overdoping occur for x > x_{opt} in that T_{C} is a decreasing function of x, falling off precipitously from 15.1 K for x between 0.08 and 0.2 and reaching a minimum of about 11.5 K for 0.2 < x < 0.4 (see [19, 47] in Ref. [1]). This negates the notion of the superfluid density being a dominant factor in determining T_{C} in the overdoped regime, as suggested elsewhere [15]. Continuous incorporation of Li and charge doping have been concluded from an x dependence in certain Raman modes, particularly for in-plane vibrations of the [ZrNCl]_{2} block (mode denoted “A” in (see [19] of Ref. [1]). Within this x variation, the underdoping and overdoping regimes appear to be reflected in the x dependence of the lattice parameters (see [19] of Ref. [1]). The lattice parameter *c* is particularly sensitive to the intercalant thickness and dictated to a significant extent by the Li–Cl bond length that determines ζ, since *d* is given by c/3 while the [ZrNCl]_{2} thickness *d*_{2} tends to show little variation with intercalation. The variation of *c* vs. x is strongest at low x and crosses over to a weaker dependence for x greater than about 0.1, which is rather close to the optimal doping point x_{opt}. This behavior suggests that participating charges are introduced into the interaction layers for x ≤ x_{opt} with the equilibrium charge structure of the Li–Cl interaction layers fully formed at x = x_{opt}. For x > x_{opt}, the excess charges are transferred to the [ZrNCl]_{2} reservoir, yielding weak variation of intercalant thickness with the further increases in x. This non-participating charge fraction, being more localized in the [ZrN]_{2} structures and minimally affecting the Cl sites, is assumed benign with respect to the superconducting pairing interaction. A similar change in slope is revealed in the careful measurements of ω_{p}^{2} vs. x, which nearly coincide with results from first principles calculations of band structure [46]. In addition, the damping τ^{−1} is reported to be greater for x > x_{opt} containing non-participating charge. Also reported in (see [35] in Ref. [1]) is the compound Zn_{0.04}ZrNCl, with T_{C} = 15 K, which is possibly a divalent doping analogue of Li_{0.08}ZrNCl. The corresponding co-intercalated compound, Li_{0.13}(DMF)_{y}ZrNCl (where DMF is (N, N)-dimethyl-formamide, C_{3}H_{7}NO), with T_{C}^{meas} = 13.7 K (see [7] in Ref. [1]), was also found to be optimal [1]. As in the case of the related TiNCl compounds, ζ = (*d* – *d*_{2})/2 [1] and σ = γ |v[x_{opt} – x_{0}]|. The γ factor is similarly obtained as well; a factor of (1/2) is given by rule (1b), and an additional two factors of (1/2) arises from the pairwise sharing between the Cl layer and the double-TiN-layered structure and then to the two TiN layers, such that γ = (1/2)(1/2)(1/2) = 1/8. Thus [1]:

- Type I reservoir: ZrNCl
- Type II reservoir: Li
_{x}(*S*)_{y} - note hexagonal structure
- σ = (1/2) (1/2) (1/2) |v [x
_{opt}– x_{0}]|; v =1 for Li - η = 1; ν = 2
**Li**x_{0.08}ZrNCl:_{0}= 0.05;*A*= 11.3233 Å^{2}; ζ = 1.5817 Å; T_{C0}^{calc}= 14.35 K**Li**x_{0.13}(DMF)_{y}ZrNCl:_{0}= 0;*A*= 11.3233 Å^{2}; ζ = 3.400 Å; T_{C0}^{calc}= 13.90 K

- D. R. Harshman and A. T. Fiory, J. Supercond. Nov. Magn., accepted for publication (2015).

** A_{x}(S)_{y}HfNCl** (T

_{C0}

^{meas}= 20 – 24 K)

Calculation of σ utilizes the same methodology as is applied in the case of Li_{x}(*S*)_{y}ZrNCl.

- Type I reservoir: HfNCl
- Type II reservoir:
*A*_{x}(*S*)_{y} - note hexagonal structure
- σ = (1/2) (1/2) (1/2) |v [x
_{opt}– x_{0}]|; v = 1 - η = 1; ν = 2
**Na**x_{0.25}HfNCl:_{0}= 0.15;*A*= 11.1484 Å^{2}; ζ = 1.6580 Å; T_{C0}^{calc}= 25.19 K**Li**x_{0.2}HfNCl:_{0}= 0.15;*A*= 11.1195 Å^{2}; ζ = 31.5950 Å; T_{C0}^{calc}= 20.31 K**Li**x_{0.2}(NH_{3})_{y}HfNCl:_{0}= 0;*A*= 11.1117 Å^{2}; ζ = 2.7620 Å; T_{C0}^{calc}= 21.42 K**Ca**x_{0.11}(NH_{3})_{y}HfNCl:_{0}= 0;*A*= 11.1251 Å^{2}; ζ = 2.7370 Å; T_{C0}^{calc}= 22.66 K**Eu**x_{0.08}(NH_{3})_{y}HfNCl:_{0}= 0;*A*= 11.1117 Å^{2}; ζ = 2.6690 Å; T_{C0}^{calc}= 24.28 K

- D. R. Harshman and A. T. Fiory, J. Supercond. Nov. Magn., accepted for publication (2015).

**κ–[BEDT-TTF] _{2}Cu[N(CN)_{2}]Br** (

*Pnma*, T

_{C0}

^{meas}= 10.5 K)

For κ–[BEDT–TTF]_{2}Cu[N(CN)_{2}]Br , the hole conduction is in the ac-plane along the sulfur chains of the two BEDT-TTF molecules (each of which is bisected by a centrally located C–C bond). Equating the Cu^{+1}[N(CN)_{2}]^{–1}Br^{–1} anion layer with Cu^{+2}[O^{–2}][O^{–2}] (type II layer) shows a factor of 1/2 between the valences of the cuprate plane ions compared to those comprising the Cu[N(CN)_{2}]Br anion molecule, leading to a base anion layer charge from rule (2c) of σ_{0}/2, which must equal the positive charge available to the two BEDT-TTF (type I layer) molecules. Dividing this charge between the two BEDT-TTF molecules comprising the dimer [rule (1a)], and further distributing the charge among the two halves of the BEDT-TTF molecule [rule (1a)], yields an additional factor of (1/2)(1/2) such that γ = (1/2)(1/2)(1/2) = 0.125. Thus, one has [1]:

- Type I reservoir: S-chains of the [BEDT-TTF]
_{2}molecules - Type II reservoir: Cu[N(CN)
_{2}]Br - σ = (1/2) (1/4) σ
_{0}= 0.125 σ_{0} - η = 1; ν = 2
*A*= 54.4745 Å^{2}; ζ = 2.4579 Å- T
_{C0}^{calc}= 11.61 K

- D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matt.
**23**, 295701 (2011);**23**349501 (2011).

**Cs _{3}C_{60}** (

*Pm3n*, T

_{C0}

^{meas}= 38.3 K (A15, 0.93 GPa); Fm

*3*m, 35.3 K (FCC, 0.73 GPa)) [1]

When viewed from a local perspective the surfaces of the C_{60} molecules of the 3D (*A*,*B*)_{3}C_{60} macrostructure present an important adaptation of Coulombic pairing involving interactions between *interfacial* structures; the surface of the C_{60} molecule and the three alkali-metal cations located on an intervening virtual surface. Given the n-type character of these materials, one associates the superconducting condensate with the C_{60} molecules (type I reservoir), with the pairing mediated by holes on the cations (type II reservoir). Both the A15 and FCC structures of Cs_{3}C_{60} are Mott insulators at ambient-pressure, exhibiting antiferromagnetism with a Néel temperature of ~46 K and ~2 K, respectively [Takabayashi2009] [Ganin2008] [Ganin2010]. Under hydrostatic pressure, both phases become superconducting; optimization of the A15 and FCC phases occur at 0.93 and 0.73 GPa, respectively, with corresponding measured T_{C0} values of 38.36 K and 35.2 K [Takabayashi2009] [Ganin2010].

The C_{60} molecule has a diameter of 7.1 Å (radius *R* = 3.55 Å) [Hedberg1991], and comprises 12 regular pentagons, with a C–C bond length of 1.45 Å, and 20 hexagons, each with a weighted average C–C bond length of (1.45 + 1.40)/2 = 1.425 Å (the first term corresponding to the 6:5 bond and the second to the 6:6 (with equal weighting) [Pennington1996]), such that the total surface area of the C_{60} molecule at STP (assuming regular polygons) is *A* = 20(5.2757 Å^{2}) + 12(3.6173 Å^{2}) = 148.922 Å^{2}. For simplicity, *A* is assumed the same for all materials (and pressures [Kaur_arXiv_2007]) considered herein. Since η = υ = 1 (where υ is the number of interacting type I layers/interfaces per f.u.), the γ factor is given by rule (1b) as, γ = 1/2, σ = (1/2) [x – x_{0}], and Eq. (1) reduced to, T_{C0} = (72.28 K-Å) [x] ζ^{–1}, with x = 3 for the stoichiometric (optimal) compounds [Haddon1993]. Note that all of this assumes an insulating (or near-insulating) end material.

For A15 Cs_{3}C_{60}, the three Cs cations are located at tetrahedral (T) sites facing the C_{60} hexagons [Ganin2008;suppl.] {check} (of average radius 1.425 Å) [Pennington1996]. In the fcc phase, on the other hand, the Cs cations are distributed between two interstitial locations; two Cs ions occupy the (T) sites, while the octahedral (O) site hosts the third [Ganin2010]. Notice that the Cs type II locations face unoccupied type I lattice positions along the radial, and can be related in 2D to defining ζ in, *e*.*g*., the β-form doped metal-hydride-nitrides [Harshman2015]. Designating to be the (weighted) radial interaction distance for each of the *N* nearest neighbor Cs sites, the average interaction distance is then an average of the occupied nearest-neighbor sites [Note].

** A15 (BCC packing) structure Cs_{3}C_{60}** – Measurements conducted on a 77.7(6)% A15 enriched sample with stoichiometry Cs

_{2.85(1)}C

_{60}(space group Pmn) and

*T*

_{C}= 38.3 K (optimized at 0.93 GPa) show V

_{0.93}(T

_{C}) = 1533.6 Å

^{3}and

*a*

_{0.93}(T

_{C}) = 11.5320 Å [Takabayashi2009]. Determining ζ in the A15 compound is straight forward since Rietveld refinement of this structural phase indicates that the Cs cations occupy only the hexagon-coordinated 6

*d*tetrahedral (T) sites; the 6

*c*sites are left empty. Given that the (T) position is located a distance 5

^{1/2}

*a*

_{0}/4 from the center of a C

_{60}molecule, and faces the center of the hexagon of radius Rthe average interaction distance ζ = ζ

_{(T)}= 5

^{1/2}

*a*

_{0.93}/4 – (

*R*

^{2}– 1.425

^{2})

^{1/2}= 6.4463 – 3.2514 = 3.1952 Å. From this, the above equation gives T

_{C0}

^{calc.}= 38.19 K in excellent agreement with experiment. As no octahedral sites are occupied, the average over nearest-neighbor Cs cations is trivial.

** The FCC packing structure Cs_{3}C_{60}** – At

*T*

_{C}= 35.3 K and applied pressure of 0.73 GPa, the volume per unit cell for an 85.88(2)% fcc (Fmm space group) sample with stoichiometry Cs

_{2.901(6)}C

_{60}is measured to be V

_{0.73}(

*T*

_{C}) = 3038.4 Å

^{3}, corresponding to a lattice parameter of

*a*

_{0.73}(

*T*

_{C}) = 14.4838 Å [Ganin2010]. Unlike its A15 counterpart, the C

_{60}molecules of the fcc phase exhibit merohedral disorder [Pennington1996] [Potocnik2014]. The type II surface comprises 8 Cs cations occupying the (T) sites (with four neighboring C

_{60}molecules) and 6 in the octahedral (O) sites, also with six C

_{60}neighbors; the number of Cs cations per C

_{60}is then 8/4 + 6/6 = 3 [Note]. From Ref. [Potocnik2014], the (T) sites are located over the center of the C-hexagons a distance 3

^{1/2}

*a*

_{0}/4 from the C

_{60}center, while the (O) sites lie above the midpoint of the 6:6 hexagon C-C bonds, a distance

*a*

_{0}/2 from the C

_{60}center. Consequently, ζ

_{(T)}= 3

^{1/2}

*a*

_{0.73}/4 – (

*R*

^{2}– 1.425

^{2})

^{1/2}= 3.0203 Å, and ζ

_{(O)}=

*a*

_{0.73}/2 – (

*R*

^{2}– 0.7

^{2})

^{1/2}= 3.7616 Å. Taking the weighted average then gives, ζ = [(8/14)ζ

_{(T)}+ (6/14)ζ

_{(O)}] = 3.3380 Å, which from Eq. (3) gives,

*T*

_{C0}

^{calc.}= 36.88 K. While the agreement with experiment is reasonable, the difference is sufficiently large to suggest that pair breaking due to the merohedral disordering of the C

_{60}molecules may be a factor.

**Note:** There’s a difference between the number of cation sites per unit cell (12 for fcc) and the number of cation sites that are the nearest neighbors to a given C_{60} (14 for fcc); one is interested in the latter, not the former when calculating ζ. Consider the C_{60} situated at coordinates (0,0,0) and its nearest neighboring Cs; the 8 tetrahedrally coordinated Cs sites (T) are located at (*i*,*j*,*k*)(*a*_{0}/4) and the 6 octahedrally coordinated Cs sites (O) are located at (*i*,0,0)(*a*_{0}/2), (0,*j*,0)(*a*_{0}/2), and (0,0,*k*)(*a*_{0}/2) (with *i*,*j*,*k* = +1 or –1). This C_{60} is neighbored by 8 (T) Cs, shared by four C_{60}s, so the number of (T) Cs per this C_{60} = 8/4 = 2. And by 6 (O) Cs, shared by six C_{60} molecular anions, so the number of (O) Cs per this C_{60} is 6/6 = 1. Thus the total number of Cs per C_{60} comes out correctly as 3. The model for ζ is based on the type II shell of Cs surrounding a given C_{60}. The fractions of (T) and (O) Cs sites per C_{60} are inequivalent to the fractions of nearest-neighbor (T) and (O) sites, because of the model geometry and coordination sharing. In summary one has [1]:

- Type I reservoir: C
_{60} - Type II reservoir: A
_{3} - η = 1; ν = 1
- σ = γ [3 –x
_{0}] = (1/2) [3 ] = 1.5 (assuming x_{0}= 0) **[BCC]:***A*= 148.922 Å^{2}; ζ = 3.1952 Å (assuming negligible pressure dependence in*A*); T_{C0}^{mea}^{s}= 38.36 K; T_{C0}^{calc}= 38.19 K (i.e., A15 phase)**[FCC]:***A*= 148.922 Å^{2}; ζ = 3.510 Å (assuming negligible pressure dependence in*A*); T_{C0}^{mea}^{s}= 35.2 K; T_{C0}^{calc}= 36.88 K

- D. R. Harshman and A. T. Fiory, Journal of Physics: Condensed Matter
**29**, 145602 (2017).

- Y. Takabayashi
*et al.*, Science**323**, 1585 (2009). - A. Y. Ganin
*et al.*, Nature Mater.**7**, 367; suppl. (2008). - A. Y. Ganin
*et al.*,**466**, 221; suppl. (2010). - C. H. Pennington and V. A. Stenger, Rev. Mod. Phys.
**68**, 855 (1996). - R. C. Haddon, Pure and Appl. Chem.
**65**, 11 (1993). - A. Potocnik
*et al*., Chem. Sci.**5**, 3008 (2014). - K. Hedberg
*et al*., Science**254**, 410 (1991). - N. Kaur
*et al*., arXiv (2007).

**H _{3}S** (

*Im3m*, T

_{C0}

^{meas}= 200.0 K (155 GPa)) [1]

The H3S system represents the first true 3D material for which the interfacial Coulombic interaction model has been applied. It is also the first material in which the type I and type II charge reservoirs contain the same ionic species.

- Type I reservoir: half-sublattice
- Type II reservoir: half-sublattice
- η = 1; ν = 1
- σ = γ [x] = (1/2) [3.43 + 3.43] = 3.43
*A*= 3*a*_{0}^{2}= 28.5017 Å^{2}(*a*_{0}= 3.0823 Å); ζ =*a*_{0}/2^{1/2}= 2.1795 Å; T_{C0}^{mea}^{s}= 200 K (155 GPa); T_{C0}^{calc}= 198.5 ± 3.0 K

- D. R. Harshman and A. T. Fiory, Journal of Physics: Condensed Matter
**29**, 445702 (2017).

**Twisted Bilayer Graphene (gated)** (T_{C0}^{meas} = 1.83(5) K) [1]

The TBG system is an extended 2D lattice where, like the 3D compound H_{3}S, the two reservoirs are essentially identical, containing both superconducting and mediating charges.

- Type I reservoir: first graphene sheet
- Type II reservoir: second graphene sheet
- η = 1; ν = 1
- σ/
*A*= γ |*n*_{opt}–*n*_{0}| = 1/2|*n*_{opt}–*n*_{0}| - ζ =
*a*_{0}/2^{1/2}= 3.5 Å; T_{C0}^{mea}^{s}= 1.83(5) K; T_{C0}^{calc}= 1.94 K

- D. R. Harshman and A. T. Fiory, Journal of Superconductivity and Novel Magnetism (2019).