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APPENDIX C: Impurity States in Semiconductors

One of the historically first applications of the effective mass equation was the calculation of the donor and acceptor states, ED and EA> in semiconductors, see section 2.4. When a Si atom is replaced by an As atom which has 5 electrons in its outer shell (as opposed to 4 in silicon) a perturbing potential energy is created which, to a first level of approximation, can be taken to be that of a point charge in a dielectric medium, that is

The same argument can be repeated for a column VI of the periodic table donor in GaAs.

In order to get a feeling of the magnitude involved in E0 and EA we can, to a first level of approximation, replace the effective mass tensor m,* by an average effective mass m* in silicon (Si) which has a degenerate conduction band. No such approximation is needed for direct band semiconductors as, for example GaAs. Then if the effective mass equation for electrons is used to calculate Ed, with the expression Cl substituted for the perturbing potential, the problem becomes identical with that of the hydrogen atom with the difference that m goes to m* and e, appears in the denominator of the potential energy.

We can immediately deduce the eigenvalues (depth below Ec) from the hydrogen case, equation 1.40,

We can easily see that a) because of the minus sign the levels lie below the conduction band-edge and b) because of the e2r(~ 100) in the denominator of C2 and rn in the numerator, the n = 1 level will be of the order of tens of meV. One may question the use of the effective mass equation for such a rapidly varying potential as that of equation Cl. However the justification for the use of the effective mass equation lies in the extent of the wavefunction obtained for the impurity states: we deduce, using equation 1.38, that the “equivalent” Bohr radius would be enlarged by a factor of me,Jm* (~ 100) so that the region r ~ 0 when V(r) of equation Cl is varying rapidly will not play a role. Similar arguments hold for the acceptor states.

 
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