  Superconductivity in the Tl-based cuprates encompasses a notably broad range
of measured optimal transition temperatures Tc0, ranging from lowest in the
charge-depleted Tl-1201 compounds
(Tl$_{1-x}$(Ba/Sr)$_{1+y}$La$_{1-y}$CuO$_{5-{\delta}}$), such as
Tl$_{0.7}$LaSrCuO$_5$ (37 K) and TlBa$_{1.2}$La$_{0.8}$CuO$_5$ (45.4 K), to
highest in the Tl-1223 compound TlBa$_2$Ca$_2$Cu$_3$O$_{9{\pm}{\delta}}$ (133.5
K). Seven Tl-based cuprates are considered and compared using the model of
superconductive pairing via electronic interactions between two physically
separated charge reservoirs, where Tc0 $\propto$
({\sigma}{\eta}/A)$^{1/2}${\zeta}$^{-1}$ is determined by the superconducting
interaction charge fraction {\sigma} the number {\eta} of CuO$_2$ layers, and
the basal-plane area A, each per formula unit, and the transverse distance
{\zeta} between interacting layers. Herein it is demonstrated that {\sigma}
follows from the elemental electronegativity and the oxidation state of Tl, and
other structurally analogous cations. The comparatively lower elemental
electronegativity of Tl, in conjunction with its oxidation state, explains the
higher {\sigma} and Tc0 values in the Tl-based compounds relative to their
Bi-based cuprate homologues. A derivation of {\sigma} is introduced for the
optimal Tl$_2$Ba$_2$Ca$_{{\eta}-1}$Cu$_{\eta}$O$_{2{\eta}+4}$ (for {\eta} = 1,
2, 3) compounds, which exhibit a Tl oxidation state at or near +3, obtaining
the fundamental value {\sigma}$_0$ = 0.228 previously established for
YBa$_2$Cu$_3$O$_{6.92}$. Also reported is the marked enhancement in {\sigma}
associated with Tl$^{+1}$ and analogous inner-layer cations relative to
higher-valence cations. For a model proposition of {\sigma} = {\sigma}$_0$, the
fractional Tl$^{+1}$ content of the mixed-valence compound,
TlBa$_2$Ca$_2$Cu$_3$O$_{9{\pm}{\delta}}$, is predicted to be 1/3 at
optimization, in agreement ...
