How exactly was the earth’s atmosphere formed in a chronological sense? why was such a formation unique in that we do not observe other planets with similar atmospheric features?
provide reference would be better, .

Earth was formed some 4.6 billion years ago, due to the mix of hot and cold and no atmosphere such format formations unique to the terrestrial planet like earth.

Explanation:

Earth across the geologic time has been rich with the greenhouse gases and higher in carbon and primitive gases and due to the solar winds the gases were blown away from the surface of the earth and such formation as unique as the repeatedly cooling and heating of the earth crust led to the formation of large oceans basins. The first atmosphere is of gases and solar nebula primary hydrogen. Which is mostly found in less dense and gaseous planet like Jupiter. The second atmosphere was formed due to the striking of the asteroids and early cycles of sun, seen to take place from 2.7 billion years ago and this marks the phase of earth's dynamic oxygenation evolution.  The third atmosphere originated due to plate tectonics and after which the layers of the earth began to be divided into 5 major zones as Troposphere, Stratosphere, Ionosphere and Mesosphere and Thermosphere around the earth. Such formation became unique for the origin of life forms on earth and due to the flow of oxygen in the air and formation of ozone in the stratosphere life could exist on land and water on the planet.