Dear Xing Zhang,

Silicon is not a direct bang-gap semiconductor. Your approximation is only valid for direct band gap semiconductors, because in this case you can approximate the wave functions of electrons and holes by plane waves of free particles. This is not the case for silicon.

In the Phys.Rev.Lett. you cite, there is the following paper cited:

Exciton energy levels in germanium and silicon, TP McLean, R Loudon - Journal of Physics and Chemistry of Solids, 1960

There is given an explicit explanation . The band structure of silicon causes additional geometrical, i.e. symmetry related restraints which are solved there using a variational approach. I highly recommend to read this paper.

Dear Xing Zhang,

Silicon is not a direct bang-gap semiconductor. Your approximation is only valid for direct band gap semiconductors, because in this case you can approximate the wave functions of electrons and holes by plane waves of free particles. This is not the case for silicon.

In the Phys.Rev.Lett. you cite, there is the following paper cited:

Exciton energy levels in germanium and silicon, TP McLean, R Loudon - Journal of Physics and Chemistry of Solids, 1960

There is given an explicit explanation . The band structure of silicon causes additional geometrical, i.e. symmetry related restraints which are solved there using a variational approach. I highly recommend to read this paper.

Dear Xing,

This is a classical problem which was solved long time ago.

An understandable description is given in our recent textbook.

See Chapter 7 on Wannier excitons.

Luminescence Spectroscopy of Semiconductors, by I. Pelant and J. Valenta, Oxford University Press, 2012.

http://ukcatalogue.oup.com/product/9780199588336.do#.UkWKMtJA9og.

In case of having trouble to find the book, write me an email.

Regards

Jan