Our natural tendency is to think that to see the effects of Einstein's special theory of relativity you have to be travelling at some significant fraction of the speed of light. However, this is not the case. In solid state physics I am aware of three concrete phenomena that are purely due to relativistic effects.
1. Gold metal is the colour "gold".
According to Wikipedia, "non-relativistic gold would be white. The relativistic effects are raising the 5d orbital and lowering the 6s orbital."
2. Mercury is a liquid at room temperature.
This is nicely discussed in a recent blog post by Henry Rzepa concerning a recent paper that shows that relativistic effects shift the melting temperature by about 100 K.
3. Magnetic anisotropy and hysteresis in ferromagnets.
This results from spin-orbit coupling which is a consequence of relativity.