Approximation of the Vibronic Transition

[next] [tail] [up]

8.5.1 Approximation of the Vibronic Transition

Basically, the calculation of the LSF via DFT is feasible, but since the motion of a polyatomic structure, especially at $T > 0$ , is highly complex to treat, simplifications need to be made. By limiting the movement of the defect system to only one vibrational mode (single-mode coupling), the defect transition can be modeled along its most dominant reaction path or coordinate [153, 151]. The total energy $E tot$ as a function of corresponding reaction coordinates (RC) can be further approximated by parabolic potential energy curves (PEC) [158], like schematically depicted in Fig. 8.6. Though originally used for small distortions around the equilibrium, such an harmonic approach is also able to model strong distortions of the defect system [159, 126].

The two solid parabolic potentials in the left of Fig. 8.6 are now given by

1 V0(q) = --M ω21(q − q1)2 + E0min (8.17) 2 V +(q) = 1M ω22(q − q2)2 + E+min (8.18) 2

with the mass $M$ and the vibrational frequencies $ω1$ , $ω2$ of the defect system. The minimum of $V 0(q)$ corresponds to the initial defect configuration. When for example examining hole capture, the defect system has to change from $V 0(q1)$ into its charged configuration $V+ (q)$ . This can be achieved by applying a bias which shifts the uncharged defect configuration (solid $V0$ ) with respect to the charged configuration upwards (dashed $0 V$ ). When assuming $T = 0$ , i.e. there are no phonons, the tunneling process can only occur when the shifted ground state crosses the positive configuration. Starting from $+ V (q1)$ , structural relaxation to the minimum $V+ (q2)$ takes place. This is accomplished by the emission of phonons. Fowler et al. used this picture to model electron tunneling between semiconductor bands and insulator traps at the interface, i.e. band-to-trap tunneling, followed by structural relaxation [160].

Figure 8.6: The total energy $Etot$ as a function of the reaction coordinate $q$ reveals various transition possibilities of certain defects systems [154, 157, 155, 141, 151]. Left: Band-to-trap tunneling is modeled via a two-stage process of tunneling followed by structural relaxation. The dashed line symbolizes the shift of the initial defect system by an applied bias. Center: By absorbing or emitting a photon of the energy $ϵ 12$ or $ϵ21$ , respectively, the defect state can be changed (multi-phonon emission (MPE)). Subsequent structural relaxation always restores the system to the respective equilibrium in both cases. The emitted energy is called relaxation energy $ER$ . In the case of linear coupling ( $ω1 = ω2$ ), $ER = (ϵ12 − ϵ21)∕2$ . Right: Without optical excitation or emission the same mechanism is called non-radiative multi-phonon (NMP) process. The transition energies $ϵ12$ and $ϵ 21$ required have to be supplied by phonons. Classically, the defect has to overcome the barrier determined by the intersection point of the parabolas with reference to $E1$ and $E2$ .

[next] [front] [up]