The design of next generation electrical and thermal transport materials is of far-reaching importance for myriad applications from thermoelectrics to dynamic transport switching. To that end, axis-dependent conduction polarity and thermal-switching materials hold significant promise. Axis-dependent conduction polarity (ADCP) is a phenomenon in which the electrons (n-type carriers) and holes (p-type carriers) are preferentially conducted along orthogonal directions in a crystal. The driving force for this phenomenon is a large (> 10x) anisotropy in the electron and hole mobilities between orthogonal directions. Herein is discussed the development of the first air-stable, wide bandgap (> 0.4 eV) semiconductor that displays ADCP, orthorhombic PdSe2. The anisotropy in the hole mobilities between the cross-plane and in-plane directions is > 100x, with holes preferentially conducting along the cross-plane direction. Additionally, the onset temperature of ADCP can be controlled via extrinsic doping with Ir and Sb as p-type and n-type dopants, respectively. When the chemical potential is near the valance band (Ir doping), ADCP is not observed up to 400 K. When it is mid-gap the onset temperature is about 350 K. But when it is near the conduction band (Sb doping), the onset temperature can be as low as 100 K. The dopant dependent onset temperature indicates the necessity for both the conduction and valance bands to be populated sufficiently to observe ADCP. Studies in this model system pave the way for further ADCP studies in semiconductors.Solid-state thermal switching is the rapid and reversible control over the thermal conductivity of a material between some low and high value without the need for physical phase changes or moving parts. Topologically non-trivial materials are promising candidates for solid-state thermal switching on account of their anomalous transport properties. Therefore, EuCd2As2 and MnBi2Te4 were studied for their thermal switching potential. EuCd2As2, initially thought to be a topological semimetal, was confirmed to be an antiferromagnetic semiconductor (Eg ~0.8 eV) via ARPES. Additionally, p-type (intrinsic) and n-type (La) doping in this interesting material was developed, and the effects on transport, optical, and magnetic properties are discussed. Though EuCd2As2 is a trivial magnetic semiconductor, solid-state thermal switching is still accessible via mechanisms beyond topologically non-trivial transport. Magnon based thermal transport is hypothesized to be an important factor for understanding the solid-state switching mechanism in EuCd2As2. EuCd2Sb2 is isostructural with EuCd2As2 and was also predicted to be a topological semimetal, but through substitutional alloying between the two compounds to afford EuCd2As2-xSbx from x = 0 to 2 strong evidence is provided to support recent claims that it too is a semiconductor. Particularly, the fact that a continuous alloy series between the two shows a continuously tunable band gap from 0.72 eV (EuCd2As2) to 0.49 ev (EuCd2Sb2) supports both as semiconductors. Additionally, the same antiferromagnetic ordering present in EuCd2As2 (TN) is also present in EuCd2Sb2 (TN). This alloy series has not been reported before and offers an interesting platform for studying the interplay between magnetic, electrical, thermal, and optical properties. This alloy series certainly merits further study.It was also found that the thermal conductivity of the magnetic topological insulator, MnBi2Te4, could be switched by applying a magnetic field to alter magnon-phonon scattering. While not a topological effect, this observation highlights an important avenue in the quest for solid-state thermal switches.Finally, in chapter 6 the unique properties of a highly correlated system known as an excitonic insulator are discussed and a study on excitonic insulator candidate Ta2NiSe5 is presented. This exotic phase holds much promise for thermal switching applications. The Bose-Einstein condensate of excitons predicted to exist therein could offer a mechanism to switch between very high and much lower thermal conductivities in a magnetic or electric field. While no solid evidence is reported for the existence of such a condensate in Ta2NiSe5, the data evidences the possibility of a large exciton density. Furthermore, comparisons with a sister material, Ta2NiS5, as a control does show anomalous thermal transport. A quantitative model is proposed to explain the anomalous thermal conductivity and new insights into this fascinating material are offered.