Non-Radiative Multi-Phonon Theory

8.5.3 Non-Radiative Multi-Phonon Theory

An alternative process excludes the absorption and emission of a photon, which is actually the use condition of a MOSFET. This makes it a non-radiative transition (NMP) [161, 111, 162, 130], like depicted in the right of Fig. 8.6. In a classical transition the defect can only surmount the barrier $ϵ12$ or $ϵ21$ between the intersection point of the parabolic potentials and its initial ground state. For linear electron-phonon coupling, i.e. $ω1 = ω2 = ω$ , these forward and reverse barrier energies are derived in Appendix D.2 to be

(Sℏ-ω +-E21)2 ϵ12 = 4Sℏω (8.20) (Sℏω − E )2 ϵ21 = ---------21-. (8.21) 4Sℏω

When shifting the defect level by applying a bias, the defect system in state $1$ is shifted with respect to the defect system in state $2$ . Since the intersection point is changed hereby, the transition rates are directly affected. This approach was already used for the permanent component of the two-stage model depicted in Fig. 8.5 and is schematically shown in Fig. 8.7 for two different bias conditions.

Figure 8.7: The total energy $Etot$ as a function of the reaction coordinate $q$ for non-radiative multi-phonon emission assuming strong ( $Sℏω > E21$ ) and linear coupling ( $ω1 = ω2$ ). Left: Without applied bias $ϵ21 < ϵ12$ and state $1$ is preferred due to its lower total energy. Right: Applying a bias induces an oxide electric field which shifts the defect state $1$ with respect to state $2$ . At the same time the intersection point is changed and consequently the barriers $ϵ12$ and $ϵ21$ . For the case shown here the barriers were changed by $Fox$ , i.e. $ϵ21 < ϵ12$ , making the state $2$ more likely to be occupied.

When comparing Fig. 8.6 (right) with Fig. 8.7 (left), a strong difference in $q2 − q1 = q21$ can be observed. While for small $q21$ the relaxation energy $Sℏω$ is much smaller compared to $E21$ , it is exactly the opposite for large $q21$ . Depending on which case to deal with, (8.20) and (8.21) can be further approximated. In the first case this yields

2 ϵ12 ≈ -E21-+ E21- (8.22) 4Sℏω 2 -E221 E21- ϵ21 ≈ 4S ℏω − 2 . (8.23)

Since the barriers mainly depend on the difference in electronic energy $E21$ , even quadratically, and not as much on the phonon part contained in the relaxation energy, this case is called weak coupling. Usually one deals with the other case, termed strong coupling, where the barriers are

S-ℏω E21- ϵ12 ≈ 4 + 2 (8.24) Sℏ ω E21 ϵ21 ≈ ---- − ---. (8.25) 4 2

The barriers here are dominated by the relaxation energy and only linearly depend on $E21$ . This approximation is also visible in Fig. 8.6 (right) for weak and in Fig. 8.7 (left) for strong coupling. When comparing the barriers (8.24) and (8.25) with those within the SRH model (8.13) and (8.14), it can be seen that it is no longer necessary to distinguish whether the trap level is below or above the reservoir level. Furthermore, in the NMP model both barriers of the capture and emission process ( $ϵ 12$ and $ϵ 21$ ) depend on the applied field to the same degree with only opposite sign, as can be easily seen in Fig. 8.7. As such the same amount one barrier is lowered is added to the reverse barrier. The resulting field depence of $τc$ and $τe$ is hence symmetric for linear coupling. By considering also quadratic electron-phonon coupling terms [163, 111], this symmetry is lifted so that one barrier is increased at the expense of the reverse barrier after [130]

ϵ12 ≈ --Sℏω1-2 + RE21-- (8.26) (1 + R) 1 + R --Sℏω1-- -E21-- ϵ21 ≈ (1+ R)2 − 1+ R . (8.27)

with $R$ as $ω1∕ω2$ . Unfortunately, this does not solve the undesired correlation of $τc$ and $τe$ , stated at the end of Chapter 8.1.

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