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i.e., it is exactly in the middle of band gap, this integral is numerically equal 1 / 2 and there is no temperature dependence, i.e., 

A simple particular case is if we observe the evolution of the occupation probability of a single trap, with time constants τ$_{c}$ and τ$_{e}$ , is numerically equal to lim$_{N}$$_{tr}$ →∞ 〈 n ( t ) 〉$_{uniform}$ /N$_{tr}$ when all traps have the same τ$_{e}$ and τ$_{c}$ , which is given by: 

<img src='content_image/370151.jpg'>

<img src='content_image/370152.jpg'>

where β = τ$_{c}$/τ$_{e}$ is a important ratio considered in this context. Naturally Pr( σ$_{i}$ ( t ) = 1) → 1 1+ β = τ$_{e}$ τ$_{e}$ + τ$_{e}$ when t → ∞ . This leads to a simple but important conclusion: if β > 1 (time capture is greater than time emission) we have Pr( σ$_{i}$ = 1) < 1 / 2 and otherwise – i.e., β < 1(time emission greater than time capture), Pr( σ$_{i}$ = 1) > 1 / 2 when t → ∞ . 

But what is the behavior of 〈 n ( t ) 〉 in a realistic case(i.e., when τ$_{c}$ and τ$_{e}$ are randomly distributed)? In this case we solve numerically the exponential integral from 2. We adopt usual values found in the literature for this problem ( p$_{min}$ = 0 and p$_{max}$ = 7) what means a frequency ranging from 1 to 10 7 Hz. 

The continuous curve in figure 3 shows the time evolving 〈 n ( t ) 〉 theoretically obtained (i.e. numerical integration of equation 2). 

The points corresponds to our MC simulations. For these MC simulations, we start from a given number of empty traps. For each time, each empty trap i = 1 , ..., N$_{tr}$ becomes occupied with probability p$_{i}$ (0 → 1) = 10 − $^{p$_{i}$}$(1 + e − $^{q$_{i}$}$) − 1 and similarly an occupied trap becomes empty again with probability p$_{i}$ (1 → 0) = 10 − $^{p$_{i}$}$(1 + e $^{q$_{i}$}$) − $^{1}$, where p$_{i}$ is uniformly drawn in [ p$_{min}$, p$_{max}$ ], and here the same values ( p$_{min}$ = 0 and p$_{max}$ = 7) were used. Similarly E ( i ) t is uniformly drawn in [ E$_{v}$, E$_{c}$ ]. For real devices E$_{t}$ should change from ˜ 0 . 2 eV to ˜ 1 . 0 eV. First of all, E$_{F}$ corresponds to middle of band-gap ( E$_{F}$ ˜ 0 . 6 eV), k$_{B}$ = 8 . 617385 × 10 − $^{5}$eV K − $^{1}$and T = 300 K, which leads to q$_{i}$ ∈ [ − 15 . 473 , 15 . 473]. We can observe a excellent agreement between the MC simulations and our theoretical equations, showing that in semi- log plot in figure 3 the relaxation dynamics follows a logarithm law: 

<img src='content_image/370153.jpg'>

