The question is very simple.Why silicon band gap energy is more than germanium at Ok? I know the equation $$EG(Si) = 1.21−(3.6×10-4 )⋅T$$ and $$EG(Ge) = 0.77−(3.6×10-4 )⋅T$$ But what is the physical (atomic) phenomenon behind these features? Thanks for your time
For most common elemental and compound semiconductor materials, there is a trend that larger lattice constants coincide with smaller bandgaps.
In a strongly simplified picture, the crystal lattice resembles a one dimensional superlattice, where the nuclei represent electronic barriers and the "empty space" in between, where the electron orbitals are located, corresponds to quantum wells. Now, if you increase the lattice-constant and therefore the superlattice period in a gedankenexperiment, the energy difference between the subbands of the superlattice will decrease.
As already mentioned by @JohnRennie, reality is much more complex. Take e.g. Ge and GaAs as an example. They both have approximately the same lattice-constant, but strongly differing bandgaps (0.77 eV for Ge and 1.42 eV for GaAs). Furthermore, Ge is an indirect and GaAs is a direct semiconductor. There are, to my knowledge, no simple rules on how to construct the band diagram from just a few material parameters.
The silicon atom electrons are more tightly bound to the nucleus than the germanium atom electrons due to its small size. So energy gap is more in that case.
If one looks at the band structure from a tight-binding point of view, the overlap between the germanium $4sp$ orbitals on neighbouring sites is larger than for silicon $3sp$. So bandwidts are increased, leaving less room for gaps.
Further down group IV in the periodic table, there is grey tin in a diamond-like structure, with only a marginal gap. And higher up there is diamond with a large band gap.
In a different picture with electron waves throughout the crystal, gaps at zone boundaries are due to the amplitude of the relevant Fourier component of the potential from the ion cores. This is large in diamond and goes down when the ions are larger and have more electrons.