Radiative Multi-Phonon Emission

8.5.2 Radiative Multi-Phonon Emission

The process described above may be also triggered by an optical excitation, which is usually called multi-phonon emission (MPE), and is illustrated in the center of Fig. 8.6. Following the FC principle, the radiative transition takes place at constant $q$ and moves the defect configuration from the minimum $V1(q1)$ up to the $V 2$ -curve via photon absorption. The necessary energy of this photon can be obtained from the general binding energy $EB (q) = V2(q)− V1(q)$ , which is derived in Appendix D.1 as

1- 2 (( 2 2) 2 2) EB (q) = E21 + Sℏω2 + 2 M ω2 2q1q2 − q1 − 2qq2 + q − R (q − q1) .

(8.19)

Evaluated at $q1$ gives an energy of $ϵ12 = E21 + Sℏω2$ , with $E21 = E2 − E1$ . This photon energy exceeds the energy needed for a simple electronic SRH transition, where the barrier only results from the difference of the corresponding energy levels, $E21$ , by $Sℏω2$ . To obtain defect equilibrium within $V2$ , $Sℏω2$ has to be released via structural relaxation (phonons) afterwards. Therefore $S ℏω 2$ is also called relaxation energy. The Huang-Rhys factor $S$ in it gives the number of photons emitted after the FC transition [153] and determines the strengh of the electron-phonon coupling [157].

Now the loop can be closed by a photon emission of $ϵ21$ and again structural relaxation ( $Sℏω1$ ) back to $V1(q1)$ . Consequently, $ϵ21 = E21 − S ℏω1$ . Note that the energy of the two photons $ϵ12$ and $ϵ21$ differs by the sum of the two relaxation energies which are generally not equal due to non-linear electron-phonon coupling.

When electron-phonon coupling is neglected as done in the SRH model, the defect equilibrium does not change. This can be seen by modifying Fig. 8.6 (Center) such that $q1 = q2$ . Consequently, the photon energies have to be equal now and the relaxation energy $E = 0 R$ . As already known, such a harsh approximation it is not able to explain the experimental results of BTI. However, since the calculation of transition barriers with $q1 ⁄= q2$ and $ω1 ⁄= ω2$ , as would be necessary for a physically more correct model, is quite complex, linear electron-phonon coupling will be used instead⁵ . Therein still $q1 ⁄= q2$ holds, but the vibrational frequencies are not allowed to change anymore, i.e. $ω1 = ω2 = ω$ . Using linear coupling simplifies the model picture a lot. For example, the relaxation energy is now constant too, making the difference of the absorbed and emitted photon exactly $2ER$ [159, 130].

So far the defect system was treated at $T = 0K$ . At higher temperatures ( $T > 0K$ ) not only the ground states at the minimum but also higher energies are occupied. For these states the absorbed and emitted photon energies $ϵ12$ and $ϵ21$ are reduced, which is called thermal broadening of the absorption and emission lines [130].

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