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The question didn't state what type of matter or what minimum depth, so I'll give a couple of answers.
Matter from the Earth's interior can be released into the atmosphere when it evaporates. Water that is stored in underground aquifers could eventually make its way to Earth's surface via a spring. From there thermal energy will cause water molecules to evaporate into the air. At that point it is in Earth's atmosphere. Underground water could also be pulled up through plant roots. From there it will be released into the atmosphere by a process called transpiration.
Matter from the Earth's interior can be released into the atmosphere when a volcano erupts explosively. A non-explosive eruption produces a lot of lava. It might get shot up a hundred feet or so, but it doesn't become part of the atmosphere. An explosive eruption, on the other hand, produces large of amounts of pyroclastic material such as volcanic ash. Volcanic ash is light, and volcanoes can be strong. That means volcanic ash can be thrown quite high into the atmosphere. The ash column of Mt. St. Helens reached a height of 80,000 ft in 15 minutes. Mt. Everest is less than 30,000 ft. It took that ash 15 days to circle the planet. Some of that ash was held aloft in the atmosphere for years.
Matter from the Earth's interior can be released into the atmosphere through the carbon cycle. Generally that carbon dioxide is outgassed through plate tectonic activities (volcanoes/earthquakes).
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