An electrical equivalent circuit for our bilayer is depicted in Fig. 13.12. We will assume that both layers are connected with four non-ideal diodes (one on each corner) that represent the electrical TIL-substrate junction. The characteristics of
these diodes have been calculated in reference , where we showed that the reverse current depends exponentially with the temperature with an activation energy which is the difference between the IB energetic position and EC. Some experimental data of the rectifying behaviour of these diodes could be found in .
This model is, strictly speaking, non linear but we will assume that the reverse characteristics of the non-ideal diode could be represented with a temperature depending resistor Rt. We have to bear in mind that the limiting diodes are the ones inversely polarized and that the main variation is not the voltage but the temperature. Direct polarized diodes will be substituted by short circuits.
The limiting resistor has to have a temperature dependence, which is the inverse of the junction saturation current, that is to say:
where the pre-exponential factor A includes all the parameters quoted in reference  and also the contact area.
According to the model, the TIL will be layer 1 and the substrate layer 2. Electron density at the conduction band of TIL will be n and its mobility дП1. Hole density at the IB will be p1 and its mobility p, p1. Substrate electron parameters are n2 and M„2. As explained before, we will not take into account holes in the valence band neither at the TIL nor at the substrate. We will define the following magnitudes:
• Gsi = 1/Rsi = q(n1/in1 C p1^p1)t1, TIL sheet conductance
• GS2 = 1/RS2 = qn2^.„2t2, substrate sheet conductance,
where t1 and t2 are the TIL and substrate thicknesses, respectively.
• GCi = 1/RCi = GSi/a, TIL conductance between two consecutive electrodes
• GC2 = 1/RC2 = GS2/a, substrate conductance between two consecutive electrodes
According to (i3.6), the sheet resistance is:
r _ Gci + Gc2 F2
sheet = a(Gci C GC2F)2 •
With F = Gt/(Gt C GC2) being Gt defined in (i3.8).
According to the model and having in mind that we are dealing with electrons at the CB and holes at IB in the TIL layer and only electrons at the substrate, the effective mobility will be:
-n1M„12t1 C p1Mp12t! – n2Mn22t2F2
П1Мп1,1 C p1^p1h C n2M„2t2F2
where F has the same meaning than in the previous equations.
Table 13.3 IB energetic position, IB hole mobility and diode pre-exponential factor for the three doses
Now, remembering the meaning of all the variables we have in the previous equations and its dependence with the temperature we can adjust our model to the experimental results. As with the ATLAS code, we consider that the hole concentration at the IB in the TIL is the Ti density. To fit the model to the experimental measurements, we have to scan just three parameters: the IB energy (we assume is the same for all the samples), hole mobility at the IB and the A parameter of the diode saturation current. All the other parameters are determined by direct measurements (TIL thickness) or are a consequence of the model (hole concentration at the IB). As with the ATLAS simulation, the electron mobility at the conduction band of the TIL has no influence on the fitting when its value ranged from 0 to 1,500 cm2 V_1 s_1.
Table 13.3 shows the best fit parameters and Fig. 13.13a, b give us the fitted plots for the sheet resistance and the mobility for the same samples represented on Fig. 13.7a, b. Parameters are remarkably similar to the ones used for ATLAS fits, except a difference of 20 mV in the IB energetic position. In Table 13.3, we give also the parameters used in the ATLAS fitting for comparative purposes.