Upward curvature of the upper critical field and the V-shaped pressure dependence of $T_c$ in the noncentrosymmetric superconductor PbTaSe$_2$

The temperature evolution of the upper critical field $H_{c2}(T)$ in the noncentrosymmetric superconductor PbTaSe$_2$ was determined via resistivity measurements down to 0.5 K. A pronounced positive curvature in the $H_{c2}$-$T$ phase diagram was observed in the whole temperature range below $T_c$. The Seebeck coefficient $S(T)$ in the temperature range 5K$\leq$$T$$\leq$350K was found to be negative in sign, modest in magnitude and non-linear in temperature. In addition, the superconducting transition temperature $T_c$ under hydrostatic pressure shows a marked non-monotonic variation, decreasing initially with the applied pressure up to $P_c$$\sim$5-10 kbar but then rising with further pressurization. The underlying physical mechanisms of all these findings have been discussed.