carbon di oxide is heavier than oxygen and nitrogen. but why doesn't it form a lower layer in the atmosphere?

In fact it does.

Co2 is much more prevalent at lower altitudes then it is at higher altitudes, however the highly random nature of gas prevents a clear divisions from forming.

Thanks to a combination of gravity and gas buoyancy, In the atmosphere the average density of atmospheric gas steadily (quadratically i think) decrease as altitude decreases. Therefor the requisite densities of each component gas self adjusts to adhere to that average density.

Carbon Dioxide constitutes only about 0.039% of the atmosphere, which is mostly in the lower parts of the atmosphere so it does create a small layer but its quite insignificant compared to nitrogen and oxygen. If you would build a tunnel which its surface is lower then the outside there could be a lethal build of Carbon Dioxide.

Hope i helped

because of the brownian motion of gas molecules, different molecule structures in gas or liquid don't form layers.