10. Appendix

634 Appendix B
of energy density ρ(˘) and frequency ˘ is given by:
P (1/˘ 2 ) 〈i|HXR|f 〉 2 ‰(Ei Ef ˘)ρ(˘)
HXR represents the exciton-photon interaction, |i〉 the initial state with the
exciton in the ground state and |f 〉 the final state of the exciton. Ei and Ef
are, respectively, the exciton initial state and final state energies. For absorption
to the continuum state of the exciton, the exciton wave functions |f 〉 are
expressed in terms of the hypergeometric functions within the hydrogenic approximation.
These continuum wave functions can be decomposed into the
product of a radial wave function R(r) and the spherical harmonic. R(r) is
indexed by the exciton kinetic energy E and wave vector K and has the form:
R(r) exp(apple·/2)/V 1/2 (2l 1)! |°(l 1 i·)|(2Kr) l
· exp(iKr)F(i· l 1; 2l 2; 2iKr)
In this expression l is the usual angular momentum quantum number of the
exciton. · is a dimensionless quantity related to the exciton kinetic energy
E(K) (K) 2 /2M (M is the exciton mass) and the R∗ exciton Rydberg constant
(defined in (6.81)) by
· (R ∗ /E) 1/2 .
° is the Gamma function and F is the confluent hypergeometric function.
Some references in physics textbooks on the hypergeometric functions are:
L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Nonrelativistic Theory
(Addison-Wesley, Reading, MA 1958), Mathematical Appendices.
N.F. Mott and H.S.W. Massey, Theory of Atomic Collisions (Clarendon Press,
Oxford, 1949), second edition, p. 52.
They are defined by an infinite series of the form:
F(a; b; z) 1 a z a(a 1) z
b 1! b(b 1)
2
...
2!
From this definition it is clear that b cannot be 0 or a negative integer. For
b 0 the series will converge for all finite z provided both a and b are real.
We are, however, only interested in direct and allowed excitonic optical transitions.
As shown in (6.86) the transition matrix element depends on the magnitude
of the final exciton wave function |f 〉 at r 0. For direct and allowed
transitions l 0. The radial wave function of |f 〉 simplifies to:
R(0) exp(apple·/2)/V 1/2 |°(1 i·)|F(i· 1; 2; 0)
exp(apple·/2)/V 1/2 |°(1 i·)|
The magnitude of this radial function is therefore:
|R(0)| 2 [exp(apple·)/V] |°(1 i·)| 2
Our problem now is to calculate the magnitude of the Gamma function with a
complex argument. First, we will recall the properties of the Gamma function