# Why Silicon band gap energy is more than germanium?

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

• I doubt there is a simple answer to this. In general the band structure of a solid isn't a simple function of the electronic structure of the atoms in it. – John Rennie Mar 18 '17 at 6:41
• @JohnRennie It would really help if you could tell what are the branches of study required to understand the band structure of a solid.Not in detail but where should someone start – debo.stackoverflow Mar 18 '17 at 6:49
• That's a huge subject. I would look for an introductory book on the band structure of solids and see how you get on with it. I can't recommend a book because this isn't my area and I don't know what the best books are. – John Rennie Mar 18 '17 at 7:03
• @JohnRennie Ok will look for the subject of band structure of solids.Thanks for your time. :) – debo.stackoverflow Mar 18 '17 at 7:06
• @debo.stackoverflow - the most famous introductory book is "Solid State Physics" by Ashcroft and Mermin. If you want a shortcut, however, section 2.7 of "Fundamentals of Semiconductors" by Yu and Cardona would be one of the good places to start. The latter discusses tight-binding approach to computing band structures of Carbon (diamond), Silicon, and Germanium. Hopefully, this will give you (at least partial) insight into the relationship between band gaps of group IV elements. – NanoPhys Mar 18 '17 at 20:35

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.

• No, not in between. The potential wells of the valence electrons are at the ion cores (for example in the Kronig-Penney model and in a muffin-tin potential). And the distance between the atoms is not the main reason. – Pieter Mar 19 '18 at 9:13

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.