Materials in proximity to quantum critical points (QCPs) experience strong fluctuations in the order parameter associated with the transition and often, as a result, display interesting properties. In this dissertation, we have used a variety of experimental probes such as Shubnikov-de Haas quantum oscillations, thermal conductivity and heat capacity, to better understand two such materials — $A3T4$Sn$\mathrm{<mrow\; data-mjx-texclass="ORD">13</mrow>}$ and YFe$2$Ge$2$.

$A3T4$Sn$\mathrm{<mrow\; data-mjx-texclass="ORD">13</mrow>}$ ($A$ = Ca, Sr; $T$ = Ir, Rh) is a family of quasi-skutterudite superconductors with moderate $Tc$’s between 4 and 8 K. Although the superconductivity is believed to be phonon-mediated with s-wave pairing symmetry, an unusual second-order structural transition makes this material family fascinating to study. Whether this structural transition is a result of three distortions with perpendicular wavevectors resulting in a cubic-to-cubic transformation, or each wavevector acting independently giving rise to cubic-to-tetragonal transformations and formation of twinned domains is a disputed issue. We have measured quantum oscillations in the resistivity of Sr3Ir4Sn13 and compared it to density functional theory (DFT) calculations for both scenarios. Our results strongly suggest that the former interpretation is correct.

The structural transition temperature $T\ast$ in $A3T4$Sn$\mathrm{<mrow\; data-mjx-texclass="ORD">13</mrow>}$ can be suppressed to zero by tuning with physical or chemical pressure. In (CaSr)$3$Rh$4$Sn$13$, the quantum critical point can be accessed purely by chemical substitution at x ~ 0.9. In the vicinity of the QCP, we expect large fluctuations of the order parameter at low temperatures, which for a structural transition could manifest as a structural disorder. We have measured thermal conductivity at temperatures much lower than $Tc$ and found that it is well described by a single power law with suppressed exponents near the QCP. The heat capacity, however, remains ~ $T3$. After excluding conventional phonon scattering mechanisms, we propose the possibility of intrinsic quasi-static spatial disorder that is related to the structural QCP.

YFe$2$Ge$2$ is closely linked to the “122” family of iron-based superconductors like KFe$2$As$2$, although it has a significantly lower $Tc$ ~ 1 K. It has a rather three-dimensional Fermi surface which closely resembles that of KFe$2$As$2$ in the pressure-induced collapsed tetragonal phase. YFe$2$Ge$2$ is in proximity to several types of magnetic order which are predicted by DFT calculations to have lower energy than the non-spin polarised case. Even though YFe$2$Ge$2$ is non-magnetic, its superconductivity could be strongly affected by magnetic fluctuations. Through a collaboration with researchers at the University of Waterloo, we have measured the thermal conductivity of YFe$2$Ge$2$ down to millikelvin temperatures and up to 2.5 T in field. Our results suggest that YFe$2$Ge$2$ is a nodal superconductor. This result could assist in the explanation of the unconventional superconductivity in iron-based superconductors.