Upper critical field of (Bi,Pb${)}_2$${\mathrm{Sr}}_2$${\mathrm{CaCu}}_2$${\mathrm{O}}_8$ single crystals

Reversible temperature-dependent magnetization measurements are conducted in fields up to 5 T on two (Bi,Pb${)}_2$ ${\mathrm{Sr}}_2$ ${\mathrm{CaCu}}_2$ ${\mathrm{O}}_8$ single crystals for fields along the c axis. A linear temperature dependence of the reversible magnetization has been obtained over a wide temperature range. The value of the magnetization slope dM/dT in the linear region at a fixed field decreases at higher field. Both the wide range of rounding of the magnetization near ${\mathit{T}}_{\mathit{c}}$ and the field-dependent dM/dT in the (Bi,Pb${)}_2$ ${\mathrm{Sr}}_2$ ${\mathrm{CaCu}}_2$ ${\mathrm{O}}_8$ system indicate that the traditional extrapolation method based on a linear Abrikosov formula near ${\mathit{H}}_{\mathit{c}2}$ may not be applied. A theoretical model is employed to determine the Ginzburg-Landau parameter κ and upper critical field ${\mathit{H}}_{\mathit{c}2}$ versus temperature. For these two crystals, we find κ=94 and 89, and ${\mathit{H}}_{\mathit{c}2}$(0)=95 and 83 T, respectively, for fields parallel to the c axis.