Impact of Charge Transitions at Atomic Defect Sites on Electronic Device Performance

Chapter B Crossing points of PECs from relaxation energies

As discussed in Section 2.1.4, the classical energy barrier for a NMP transition is given by the energy difference between the equilibrium energy and the crossing point of the two PECs in different charge states. In the following, it will be emphasized that the barrier height is determined by the CTL and the relaxation energies only and does not depend on the configuration coordinate change $Δ Q$ .

Consider a defect system in two charge states, which CTL and $E_{Relax}^{q_{1} / q_{2}}$ are obtained from DFT calculations. For better readability, $E_{Relax}^{q_{2} / q_{1}} = R_{1}$ and $E_{Relax}^{q_{1} / q_{2}} = R_{2}$ will be used. Following Eq. (2.20), $E_{T}$ is defined as the energy difference between the CTL and the respective charge reservoir $E_{T} = ε (q_{1} / q_{2}) - E_{Res}$ . Consequently, $E_{T}$ gives the energy alignment between the minimum energy configurations of the two PECs. For arbitrary $E_{Res} = 0$  eV, the PECs can simply be expressed as

$\begin{aligned} (B.1a) & V_{1} (Q) & = \frac{R_{1}}{Δ Q^{2}} Q^{2} \\ (B.1b) & V_{2} (Q) & = \frac{R_{2}}{Δ Q^{2}} (Q - Δ Q)^{2} + E_{T} . \end{aligned}$ Consequently, the CPs (math image) of the two parabolas are located at

$\begin{matrix} (B.2) & Q_{1, 2} = Δ Q \frac{- R_{2} ± \sqrt{R_{2} R_{1} + R_{1} E_{T} - R_{2} E_{T}}}{R_{1} - R_{2}} . \end{matrix}$

When $Q_{1, 2}$ is inserted into Eq. (B.1), the $Δ Q$ dependence cancels out and the energy of the crossing point is given by

$\begin{matrix} (B.3) & V_{1} (Q_{1, 2}) = V_{2} (Q_{1, 2}) = \frac{R_{1}}{(R_{1} - R_{2})^{2}} {(- R_{2} ± \sqrt{R_{1} R_{2} + R_{1} E_{T} - R_{2} E_{T}})}^{2} . \end{matrix}$

Even though two CPs exist for two parabolas with different curvatures of the same sign, typically only the lower barrier has to be considered, as transitions are dominated by the minimal activation energy.

From Eq. (B.2b) it is obvious that the classical energy barrier is independent of the configurational change upon charge trapping. This is visualized in Fig. B.1, where the PECs of a model defect in the neutral ( $V_{1} (Q)$ ) and negative ( $V_{2} (Q)$ ) charge states are plotted for (a) $Δ Q = 1.25$ and (b) $Δ Q = 10$ for the same $R_{1}$ , $R_{2}$ and $E_{T}$ . While the positions and shapes of the according PECs are determined by $R_{1}$ , $R_{2}$ , $Δ Q$ and $E_{T}$ , the energy of the crossing point is independent of the structural changes as encoded in $Δ Q$ .

Therefore, $Δ Q$ can be arbitrarily set to e.g. 1 for extraction of the CP as done in Sections 4.1.8 and 5.1.4.

Still, calculating $Δ Q$ is necessary for evaluations of the vibrational wave functions and the spectral function of Eq. (2.9c). An increased accuracy of the PECs curvatures can be obtained by additionally calculating the linearly interpolated intermediate images between the minimum energy configurations and subsequently fitting parabolic functions as discussed in Section 2.1.2.