2.2 Franck-Condon Theory
Charge trapping models found in literature usually assume elastic electron tunneling,
which ignores the atomic configuration of the defects. In the Franck-Condon
theory [107, 108, 109, 110], however, changes in the defect configuration play a
crucial role in the charge trapping process. For instance, this theory predicts that the
defect levels can be subject to a shift [111, 112, 113, 114] (discussed in
Section 2.2). In order to support the physical understanding of such trapping
processes, one has to deal with the basics of microscopic theories, which
provide the most complete description of a physical process, such as charge
trapping.
Therefore, the Schrödinger equation [92, 91] for a system involving electrons and
nuclei is taken as a starting point. Its Hamiltonian
can be formulated as
where the subscripts
and
refer to nuclei and electrons, respectively.
and
are the coordinates of a certain electron
or a certain nucleus
,
respectively, whereas the curly braces stand for the whole set of coordinates,
such as
,
,
,
for
. The Hamiltonian
contains all
contributions from the electrons (
) and nuclei (
) kinetic energies as well
as from the Coulombic electron-electron
, nucleus-nucleus
, and
electron-nucleus
interactions. Then, the Schrödinger equation reads
where the energy of the whole system of the electrons and the nuclei is denoted as
. It is important to note at this point that the wavefunctions
depend on all positions of the electrons as well as the nuclei and thus the equation
(2.14) cannot be solved due to its mathematical complexity.
In order to obtain an approximate solution, one usually employs the Born-Oppenheimer
approximation, also known as the adiabatic approximation [92, 91], which states
that the electrons move much faster than the nuclei. In this picture, the electrons
instantaneously adapt to each atomic configuration and the motion of the nuclei can
be neglected for the solution of the electron system. Based on this assumption the
wavefunction
can be split into an electron (
) and a
nuclei (
) part.
In a further consequence, this ansatz separates the electron and nuclei system, which
are then characterized by the so-called ‘electronic’ and ‘vibrational’ states,
respectively. The electron system is described by the electronic Schrödinger equation
with a fixed nuclei configuration
.
stands for the many-electron wavefunction, which is a function of
the electron coordinates
. The nuclei have fixed positions
and enter the
equation (2.16) as parameters only. Consequently, their positive charges form an
external potential
to the electron system. The eigenvalues
of equation (2.16) re-enter the Schrödinger equation of the nuclei system,
and act as an external potential, in which the nuclei move. This potential is therefore
referred to as the adiabatic potential or the potential energy surface. The nuclei in
the Schrödinger equation (2.17) are treated as quantum mechanical particles.
Therefore, they cannot be visualized as being to be located at certain points in space
but are described by their wavefunctions
. Furthermore, the energy
spectrum of the nuclei system is quantized and includes discretized excitations known
as lattice vibrations in crystals. In quantum chemistry, the nuclei system is frequently
visualized in so-called configuration coordinate diagrams (see Fig. 2.2). The
ordinate of these diagrams represents the total energy
presuming that
the electron system is in its ground state. The abscissa is the configuration
coordinate
, which summarizes the nuclei configuration
in one
quantity. It is important to note here that the coupling of the two Schrödinger
equations (2.16) and (2.17) results in so-called ‘vibronic’ states, which are
combinations of the electronic and the vibrational states and are defined by the
wavefunctions
and their corresponding energy
.
According to the Franck-Condon principle [107, 108, 109, 110], these vibronic
transitions involve a change in the electronic as well as in the vibrational state. The
corresponding rate can be calculated using Fermi’s golden rule (see Appendix A.1)
with
being the perturbation operator.
and
denote the initial and the
final vibronic state, respectively, where the latin and greek symbols refer
to the electronic and vibrational states. According to the Franck-Condon
approximation, the matrix element
can be separated into two factors:
Using the above equation, the transition rate (2.19) can be rewritten as
Here,
denotes the electronic matrix element and is associated with an
electronic transition, for instance electron or hole tunneling in the case of charge
trapping in defects. The Franck-Condon factor
gives the probability of
the vibrational transition, which is determined by the overlap of the nuclei
wavefunctions. The probability for such a transition is lower than that for the
electron transition. Therefore,
becomes the decisive factor in equation
(2.21).