Exponential Traps

The transient current, derivable from equation 1, is given in equations 2 and 3 where T is the transit time and I is the absorbed photon flux. The parameter a can be further derived as equation 4 (4), where Tis the absolute temperature and is the distribution width (in units of kT) of a series of exponential traps. In this context, the carrier mobdity is governed by trapping and detrapping processes at these sites. 

Figure 9-27. Experimental (dots) and theoretical (solid line) t/V characteristics of. a Ca/PPV/Ca electron-only device with a thickness, L, of 310 nm. The theoretical curve is obtained assuming an exponential trap distribution with a trap density of Nt=5-I()17 cm 1, a trap distribution parameter Tt 1500 K, and an equilibrium electron density n = L5-I011 cm"1. The dashed line gives the hole SLC according to Eq. (9.13). Reproduced from Ref. 85J.

The lifetime T and diffusion coefficient D of photoinjected electrons in DSC measured over five orders of magnitude of illumination intensity using IMVS and IMPS.56) fis proportional to the r m, indicating that the back reaction of electrons with I3 tnay be second order in electron density. On the other hand, D varied with C0 68, attributed to an exponential trap density distribution of the form Nt(E) <=< exp[ P(E - Ec)l(kBT) with 0.6. Since T and D vary with intensity in opposite senses, the calculated electron diffusion length L = (JD-z)m does not change linearly with the irradiance. 

Assuming exponential trap distribution, combining continuity and Poisson equations and integrating the resultant equation, we obtain,... 

In principle the situation seems to be similar for energetically distributed traps with Gaussian or exponential density of states. But especially in the case of an exponential trap density of states in the gap with the maximum at a distance Eq above the valence band edge... 

First, we consider the exponential trap distribution Gexp (Eq. (8.89)). For this distribution, as mentioned, there is an analytic approximate solution [58]. It yields... 

Mathematically equivalent with the MT model is the Continuous Tune Random Walk (CTRW) model [47-49], where the exponential trap distribution density g(s) in the MT model. 

Figure 12-11. Thickness dependence of the electron only j(V) characteristics at L=0.22, 0.31, and 0.37 pm. Solid lines have been calculated for an exponential distribution of electron traps of the total density 101 cnTJ and a characteristic temperature T,.= 1500 K (Ref. [41[).

It is obvious, and verified by experiment [73], that above a critical trap concentration the mobility increases with concentration. This is due to the onset of intertrap transfer that alleviates thermal detrapping of a carrier as a necessary step for charge transport. The simulation results presented in Figure 12-22 are in accord with this notion. The data for p(c) at ,=0.195 eV, i.e. EJa—T), pass through a minimum at a trap concentration c—10. Location of the minimum on a concentration scale depends, of course, on , since the competition between thermal detrapping and inter-trap transport scales exponentially with ,. The field dependence of the mobility in a trap containing system characterized by an effective width aeff is similar to that of a trap-free system with the same width of the DOS. 

To study the electrical transport properties of this double-barrier system Pd nanoclusters have been trapped in this gap. Figure 14 shows a typical l(U) curve. The most pronounced feature at 4.2 K is the Coulomb gap at a voltage of about 55 mV, which disappears at 295 K. Above the gap voltage, the l(U) curve is not linear, but increases exponentially, which was explained by a suppression of the effective tunnel barrier by the applied voltage. 

Another conceivable limiting case, though one less likely to be approached in practical cases, is that where the total hydrogen concentration always remains far below that of the traps, which continue to capture hydrogen irreversibly. For this case, as Corbett et al. (1986) have pointed out, the concentration of free monatomic hydrogen will approach a quasisteady-state profile that decays exponentially with the depth x. The concentration of trapped hydrogen, of course, will at any point of space approach a linear increase with time. 

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