Specific-heat studies of the spin-orbit interaction in noncentrosymmetric Li 2 (Pd 1-x Pt x ) 3 B (x=0,0.5,1) superconductors

The temperature- and field-dependent specific heat $C(T,H)$ measurement on the noncentrosymmetric superconductors ${\mathrm{Li}}_{2}{\mathrm{Pd}}_{3}\mathrm{B}$, ${\mathrm{Li}}_{2}{\mathrm{Pt}}_{3}\mathrm{B}$, and ${\mathrm{Li}}_{2}{({\mathrm{Pd}}_{0.5}{\mathrm{Pt}}_{0.5})}_{3}\mathrm{B}$ were carried out over a wide range of $T$ and $H$. $C(T,H)$ of the limit compounds is found to be distinctly different: within the superconducting phase, the electronic specific heat of the Pd-based compound follows ${C}_{e}^{S}(Tl{T}_{c}=6.95\phantom{\rule{0.3em}{0ex}}\mathrm{K},0\phantom{\rule{0.3em}{0ex}}\mathrm{T})=3\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[\ensuremath{-}16.4∕T]\phantom{\rule{0.3em}{0ex}}\mathrm{J}∕\mathrm{mol}\phantom{\rule{0.2em}{0ex}}\mathrm{K}$ while for the Pt-based compound ${C}_{e}^{S}(Tl{T}_{c}=2.56\phantom{\rule{0.3em}{0ex}}\mathrm{K},0\phantom{\rule{0.3em}{0ex}}\mathrm{T})=\ensuremath{\alpha}{T}^{2}\phantom{\rule{0.3em}{0ex}}\mathrm{mJ}∕\mathrm{mol}\phantom{\rule{0.2em}{0ex}}\mathrm{K}$, where $\ensuremath{\alpha}=7.3\phantom{\rule{0.3em}{0ex}}\mathrm{mJ}∕\mathrm{mol}\phantom{\rule{0.2em}{0ex}}{\mathrm{K}}^{3}$. An increase in $H$ tends to reduce weakly the exponential argument and the coefficient of the quadratic term. The above-mentioned findings are in agreement with the description that, due to the difference in the strength of the spin-orbit coupling, the pairing symmetry in the Pd- (Pt-) based compound is predominately a spin-singlet (spin-triplet) state. Within the very low-temperature range, the field-induced normal-state of the Pd-based limit manifests ${C}^{N}(T,7Tg{H}_{c2})=9.3T+1.1{T}^{3}\phantom{\rule{0.3em}{0ex}}\mathrm{mJ}∕\mathrm{mol}\phantom{\rule{0.2em}{0ex}}\mathrm{K}$ while that of the Pt-based isomorph shows ${C}^{N}(T,7Tg{H}_{c2})=9.3T+0.92{T}^{3}\phantom{\rule{0.3em}{0ex}}\mathrm{mJ}∕\mathrm{mol}\phantom{\rule{0.2em}{0ex}}\mathrm{K}$. For the intermediate alloy ${\mathrm{Li}}_{2}{({\mathrm{Pt}}_{0.5}{\mathrm{Pd}}_{0.5})}_{3}\mathrm{B}$, $C(T,H)$ mimics that of the Pt-based compound; specifically, ${C}_{e}^{S}(Tl{T}_{c}=3.9\phantom{\rule{0.3em}{0ex}}\mathrm{K},0\phantom{\rule{0.3em}{0ex}}\mathrm{T})=\ensuremath{\alpha}{T}^{2}\phantom{\rule{0.3em}{0ex}}\mathrm{mJ}∕\mathrm{mol}\phantom{\rule{0.2em}{0ex}}\mathrm{K}$ where $\ensuremath{\alpha}=4.6(2)\phantom{\rule{0.3em}{0ex}}\mathrm{mJ}∕\mathrm{mol}\phantom{\rule{0.2em}{0ex}}{\mathrm{K}}^{3}$. The ratio of the coefficients of the quadratic terms in ${\mathrm{Li}}_{2}{({\mathrm{Pt}}_{0.5}{\mathrm{Pd}}_{0.5})}_{3}\mathrm{B}$ and ${\mathrm{Li}}_{2}{\mathrm{Pt}}_{3}\mathrm{B}$, ${\ensuremath{\alpha}}_{{\mathrm{Pt}}_{0.5}{\mathrm{Pd}}_{0.5}}∕{\ensuremath{\alpha}}_{\mathrm{Pt}}$, is equal to ${Z}_{{\mathrm{Pt}}_{0.5}{\mathrm{Pd}}_{0.5}}^{2}∕{Z}_{\mathrm{Pt}}^{2}$, confirming that the line nodes are driven by the spin-orbit interaction. The finding that these nodes can survive even in a 50% Pd-diluted material gives extra support to the reported claim that the $x$-dependent evolution of ${T}_{c}$, ${H}_{c1}$, ${H}_{c2}$, and ${(\ensuremath{\partial}{H}_{c2}∕\ensuremath{\partial}T)}_{{T}_{c}}$ is almost linear.