Impact of Charge Transitions at Atomic Defect Sites on Electronic Device Performance
Chapter A Limitations of the 1D configuration coordinate diagram
The ability to capture the line shape function of a charge transfer at a defect site with a one-dimensional configuration coordinate diagram (CCD) depends on the strength of the electron-phonon coupling. The Huang-Rhys factor
as defined in Eq. (2.14) provides a measure of this strength, as it gives the total number of phonons emitted during the structural relaxation upon a charge transfer at a
defect site [116, 114, 102].
In general, the coupling of the relaxation to all normal phonon modes in the system has to be included in the full multi-dimensional treatment of the spectral function. The partial Huang-Rhys Factor for each phonon mode
with frequency can be defined by [120]
with . Here, labels the mass of tom , the Cartesian coordinates ,
the distortion vector as a difference between initial and final atom configuration and an unit vector for each atom in the direction of the mode. Using , a spectral density of the electron-phonon
coupling can be written as [272, 114, 273]
The total Huang-Rhys factor of the transition is consequently given by
As was shown in [272], can be used to define a so-called generating function. The Fourier transform of this
generating function then gives the spectral function and subsequently the line shape function in the multi-dimensional treatment including all phonon modes.
In the 1D representation, however, the line shape function is calculated as a sum over the 1D vibrational wave functions of the effective phonon mode as given in eq. 2.15. When the
system is initially in the vibrational ground state and the frequencies of the accepting and promoting modes are equal, an analytical expression of the Franck-Condon overlap integral in Eq. (2.7) exists and is given by [274]
This expression can only capture the line shape of the multi-dimensional treatment when the electron-phonon coupling is sufficiently large () [116]. For weak electron-phonon coupling, the contribution of the zero-phonon line to the total line shape is substantial and additionally the line shape can have several
peaks. This is the case for example for the NV center in diamond [275], where leads to a ZPL weight of the total lineshape of . For a model defect with , the fraction of the total light in the ZPL would even increase to [116]. For larger , both the
broadening and the position of the luminescence line shape from the multi-dimensional calculations is well represented by Eq. (A.4).
The reason for this is that contributions from single phonon modes can no longer be identified for charge transfers with strong electron-phonon coupling and it is thus possible to effectively describe such a transition by a single
effective phonon mode [116]. The resulting line shape for is smooth with a potential small asymmetry and with vanishing contributions from the
zero-phonon line. This is for example the case for the N vacancy in ZnO [102], but also for the O vacancy in -AlO as presented in
Section 6.2.