To fully confirm the qualitative explanation of the previous paragraph, we have performed some simulations using the ATLAS code framework . As in this code it is not possible to define a semiconductor with an IB, we have defined the TIL sheet as a “new” semiconductor with the following characteristics:
• Conduction band: Is the same than Si, having the same equivalent density of states, mobility, affinity and so on.
• The gap of this “new” semiconductor is the EC — EIB energy, avoiding the VB at the TIL.
• Density of States (DOS) and hole concentration in the IB: as stated before, RBS measurements showed that Ti is located mostly at interstitial positions. As Ti electronic configuration is 1s2 2s2 2p6 3s2 3p6 4s2 3d2, each atom is able to give one to four electrons to the IB. If we assume as a first approach that each Ti atom gives an electron to the IB, the DOS should be 2x [Ti] (where [Ti] is the Ti concentration), in agreement with standard semiconductor theory of band formation, and the hole density will be the Ti concentration [Ti]. Later in this chapter, we have some considerations about the possibility of having more electrons per Ti atom. These assumptions are in accordance with the theoretical calculations of Wahnon and coworkers  that have shown through ab initio calculations that the Ti interstitial atoms in a host Si lattice should produce a half filled intermediate band.
• From the statistical point of view, the IB is supposed to be narrow enough to be considered as a single level.
• It is worth noting that, for an energy band narrow enough, NV (equivalent density of states) and DOS should be roughly the same. Hence, we will use for NV the DOS quoted above, i. e., NV = 2 x [Ti] cm-3.
• Hole mobility at the IB (дщ) has to be guessed but it is supposed to have a very low value as corresponding to a narrow energy band. Careful Hall measurements at low temperatures limits this variation between about 0.1-0.6cm2 V-1 s-1 as we will see further on.
• TIL thickness will be obtained from the ToF-SIMS measurements at the point where the concentration becomes equal to the Mott limit. For easier calculation, we assume a squared-box Ti concentration obtained as: [Ti] = 0.6 x D/t where D is the implanted dose, t is the thickness obtained through ToF-SIMS measurement and the 0.6 figure takes into account the Ti losses during the PLM annealing as explained before.
The band diagram, resulting from ATLAS code, is depicted in Fig. 13.9 for two temperatures, 90 and 300 K. Notice that the TIL “normal” VB has disappeared in this model in relation to the model proposed in Fig. 13.8 because, as stated before, holes in both “normal” valence bands do not have any significant role in the model due to its very low carrier density.
We have simulated the bilayer sheet resistance for the 1015, 5-1015 and 1016 cm-2 doses, all of them with 0.8 J cm-2 annealing. For this simulation, we have used
a tri-dimensional structure like the one represented in Fig. 13.1. A comparison between sheet resistance experimental data and the ATLAS simulation values is presented in Fig. 13.7a. For the fitting, only two parameters have been scanned: the IB energetic position and the IB hole mobility. Best fit is obtained with EC — EIB = 0.36 eV for the three samples and hole mobility = 0.4 cm2 V-1 s-1 for doses of 1015 cm-2 and 5-1015 cm-2 and 0.6 cm2 V-1s-1 for dose of 1016 cm-2. No fit is done neither in the IB hole concentration which remains equal to Ti volume concentration nor in the NV value at the IB that is 2x [Ti]. The temperature of the sheet resistance minimum is very sensitive to the pseudo-gap chosen, i. e. the energetic position of the IB, being the sheet resistance plateau at low temperatures dependent on the hole mobility times the IB hole density. For this reason, we cannot know exactly how many carriers are given for each Ti atom; it goes from one electron per Ti atom with a mobility of 0.4 cm2 Vs-1 for the corresponding hole to four electrons with 0.1 cm2 Vs-1 hole mobility.
The electron mobility at the conduction band of the implanted layer is a point of concern because it is well known that it should be very dependent on the crystallinity. Nevertheless, there is not appreciable variation on the fitting results if we scan its value at room temperature from 0 to 1,500 cm2 V-1 s-1. That simply means that the product at the intermediate band is always higher than ид„ at the TIL conduction band for all the temperatures. p is the hole concentration at the IB, n is the electron concentration at the conduction band of the TIL and and the mobilities of these carriers.
At high temperatures the fitting is always excellent and the position of the minimum and its shape is very well modelled. For temperatures below the one corresponding to the resistance minimum, the simulated sheet resistance increases faster than the experimental results. The difference could be related with a possible uncontrolled current leak through the TIL-substrate interface. As this is similar to the behaviour of a reverse polarized P-N junction without special technology to avoid surface leaks, we suppose that here the most probable region to have superficial states is the rim of the sample, which is a highly damaged zone. This extra conduction path has not been simulated by the ATLAS code.
Fig. 13.10 ATLAS simulated equipotential lines in a cross section of a sample implanted with 1015 cm-2 dose. (a) is the whole sample at 90 K and (b) at 300 K, (c) and (d) are enlargements of the TIL area at 90 K and 300 K, respectively (Reprinted with permission from . Copyright 2009. Institute of Physics)
Figure 13.10 shows the simulated equipotential lines in a cross section from the injection current electrode at left (electrode 1 in Fig. 13.1) to ground at right (electrode 2) at 90 (Fig. 13.10a, c) and 300K (Fig. 13.10b, d). In both cases, the external potential is 10V positive at the left electrode. Figure 13.10b presents, for 300 K, the equipotential lines in the whole sample showing that they are perpendicular to the surface.
Figure 13.10d is an enlargement of the TIL area showing that there are no differences between the TIL and the substrate because the current is carried by both layers in parallel. At low temperature almost all the substrate has the potential of the positive electrode, as is depicted in Fig. 13.10a. All this behaviour is congruent with the idea that the junction between the TIL and the substrate is blocking if TIL
is negative and not blocking with the opposite polarity. Figure 13.10c depicts the enlarged TIL area for low temperature showing perpendicular equipotential lines meaning that the current is supported only by this part of the bilayer. The potential differences between TIL and substrate causes a depletion zone at the substrate with a triangular shape which become thicker as we approach to the right (ground electrode) because the potential difference is becoming higher. The depletion zone, which is clearly shown in Fig. 13.10a, is essentially developed at the substrate and not in the TIL layer due to the very different carrier concentration between both layers.
Unfortunately, we were unable to simulate the Hall mobility with the ATLAS code because convergence problems.