Non-Radiative Multi-Phonon Process

D.2 Non-Radiative Multi-Phonon Process

When there are no photons available for the transition, the process is called non-radiative multi-phonon (NMP) process. Now the transition energy from one parabolic minimum into the other has to be provided by phonons. Due to energy conservation, a classical transition at the points is possible, where the binding energy is zero. This is the case only at the intersection point¹ IP of (D.1) and (D.2). The value between $V1(qIP) = V2(qIP )$ and either $E1$ or $E2$ then specifies the classical barrier which has to be crossed. Assuming linear coupling, i.e. $ω1 = ω2 = ω$ , yields

2(E1−E2) 2 2 q = --M-ω2--+--q1-−-q2. IP 2(q1 − q2)

(D.5)

Reinserting (D.5) into (D.2) delivers

( )2 1 2 2(EM1−ωE22)+ q12− q22 V2 (qIP) = 2M ω ----2(q-−--q)---- − q2 + E2 1 2 1 2( 2(E1 − E2) + M ω2(q21 − 2q1q2 + q22))2 = 2M ω ----------2M-ω2(q-−-q-)---------- + E2 ( 1 )22 1- 2 2E12-+-M-ω2(q1-−-q2)2- = 2M ω 2M ω2(q1 − q2) + E2. (D.6 )

With the relaxation energy $1 2 2 Sℏω = 2 M ω (q1 − q2)$ (cf. Fig. D.2), (D.6) can be further evaluated to

1- 2-(2E12 +-2S-ℏω)2 V2(qIP) = 2 M ω 4M 2ω4(q1 − q2)2 + E2 (D.7 ) 2 = (E12-+-Sℏω-)-+ E2. (D.8 ) 4Sℏ ω

By combining (D.5) and (D.1), $V1(qIP)$ can be written as

2 V1(qIP) = (E12 −-S-ℏω)-+ E1. 4S ℏω

(D.9)

The forward and reverse rates then read

2 2 ϵ12 = (E12 −-Sℏω)--= (S-ℏω-+-E21)-, (D.10 ) 4Sℏω 4S ℏω (E12 +-Sℏω)2- (S-ℏω-−-E21)2 ϵ21 = 4Sℏω = 4S ℏω . (D.11 )

In Fig. D.2 all derived quantities are depicted. Again $S$ gives the number of emitted phonons with an energy of $ℏω$ . Their product $Sℏω$ reflects the strength of coupling and hence has a huge impact on the transition rates of the defect states, e.g. a smaller $Sℏω$ yields a smaller barrier.

Figure D.2: Description of defect state transitions by assuming harmonic oscillators with equal vibronic frequency $ω$ in a reaction coordinate diagram for the NMP model. The transition from one defect system into another happens at the intersection of the energy curves. Usually one transition is favored, inducing a preferred defect state. In this case the state $V1(q1)$ will be occupied most of the time.

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