Germanium is a semiconductor because:

The band gap of each germanium atomic energy level is small.

The correct answer is option 3. The band gap of each germanium atomic energy level is small.

Semiconductors like germanium have a small energy gap (band gap) between their valence band and conduction band. This small band gap allows electrons in the valence band to be easily excited to the conduction band, where they can conduct electricity. However, the band gap is still large enough to prevent complete electron conductivity like in metals. Germanium has a band gap of approximately 0.67 eV, which is relatively small compared to insulators but larger than that of conductors.

The other options do not correctly explain why germanium is a semiconductor:

(1) The band gap of each germanium atomic energy level is large. - This statement is incorrect because semiconductors have small band gaps, not large ones.

(2) There is overlapping of bands in germanium atoms. - This statement is incorrect because overlapping of bands is a characteristic of metals, not semiconductors.

(4) Germanium conductivity ranges between \(10^{-20}\) to \(10^{-10}\) \(ohm^{-1}m^{-1}\). - This statement describes the conductivity range of germanium but does not explain why it is a semiconductor.