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 $S$ 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 $k$ 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 ${\omega }_k$ can be defined by [120]

$$\begin{array}{}\text{(A.1)}& S_k=\frac{{\omega }_k{q}_k^2}{2\hslash }.\end{array}$$

with $q_k=\sum _{\gamma i}{m}_{\gamma }^{1/2}\mathrm{\Delta }{R}_{\gamma i}{q}_{k;\gamma i}$. Here, $m_{\alpha }$ labels the mass of tom $\gamma$, $i$ the Cartesian coordinates $x,y,z$, $\mathrm{\Delta }{R}_{\gamma i}$ the distortion vector as a difference between initial and final atom configuration and $q_{k;\gamma i}$ an unit vector for each atom $k$ in the direction of the mode. Using $S_k$, a spectral density of the electron-phonon coupling can be written as [272, 114, 273]

$$\begin{array}{}\text{(A.2)}& S(ϵ)=\sum _k{S}_k\delta (ϵ-{ϵ}_k).\end{array}$$

The total Huang-Rhys factor of the transition is consequently given by

$$\begin{array}{}\text{(A.3)}& S={\int }_0^{\mathrm{\infty }}S(ϵ)dϵ.\end{array}$$

As was shown in [272], $S(ϵ)$ 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 $⟨{\chi }_{i\alpha }|{\chi }_{f\beta }⟩$ in Eq. (2.7) exists and is given by [274]

$$\begin{array}{}\text{(A.4)}& {\left|⟨{\chi }_{i0}|{\chi }_{f\beta }⟩\right|}^2={\mathrm{e}}^{-S}\frac{S^{\beta }}{\beta !}.\end{array}$$

This expression can only capture the line shape of the multi-dimensional treatment when the electron-phonon coupling is sufficiently large ($S⪆5$) [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 $S=3.67$ leads to a ZPL weight of the total lineshape of $w={\mathrm{e}}^{-S}=3.2\mathrm{\%}$. For a model defect with $S=0.3$, the fraction of the total light in the ZPL would even increase to $75\mathrm{\%}$ [116]. For larger $S$, 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 $S>15$ 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 $\alpha$-Al${}_2$O${}_3$ as presented in Section 6.2.