220.127.116.11 Sheet Resistance and Hall Mobility
Figure 13.7a shows the sheet resistance as a function of measured temperature for a reference Si substrate and for samples with implanted doses of 1014, 1015, 5-1015 and 1016 cm-2. All the implanted samples were PLM annealed with 0.8 Jcm-2 energy density. Hollow symbols are the experimental measurements while lines with symbols correspond to a simulation we will explain later. We also checked that unimplanted substrates have the same electrical characteristics before and after a PLM process. We did this experiment to confirm that the annealing process has not any influence on the substrate sheet resistance or mobility. Figure 13.7b shows the mobility modulus versus temperature for the same samples quoted in the previous figure. In any case, the samples were n type, that is to say, they have negative mobility. Both the sheet resistance and the mobility for samples implanted with a 1014 cm-2 dose and their corresponding for the substrate are almost indistinguishable.
For samples from 1015 to 1016 cm-2 doses, sheet resistance shows an uncommon feature. For temperatures lower than 180-190K, the sheet resistance is higher than the substrate resistance while for higher temperatures is lower. As we assume that we are dealing with a bilayer (implanted layer and substrate), this implies these layers are in parallel at high temperature and isolated at low temperature. For a parallel scheme, there is no way to have a sheet resistance higher than one of the branches, and for this reason we have to think in a decoupling at low temperatures. That means that for low temperatures we are measuring just the implanted layer while at high temperatures we are measuring both the implanted layer and the substrate in parallel.
The decoupling would be straightforward if we had opposite majority carriers in each layer at low temperature (We mean a rectifying behaviour) and the same type at high temperature, but there is no way to change from p type at low temperatures
to n type at high temperatures for an usual semiconductor. We have to note that the substrate remains n type for all the temperature range.
In the following and in order to explain the results, we will assume that a half – filled IB has been formed at the implanted layer and there is a delocalization of the impurity electron wavefunctions in the thickness, where Ti concentration is overcoming the Mott limit as it is shown in Fig.13.2; that is to say, 20 nm for 1015 cm-2 dose, 60 nm for 5-1015 cm-2 dose and 80 nm for 1016 cm-2 dose.
This assumption is congruent with the different behaviour observed for the sample with 1014 cm-2 dose where it is supposed that the Mott limit has not been reached yet and there is not enough Ti to obtain the overlapping of the wavefunctions.
According to the IB theory, the band diagram of the TIL/n-Si double sheet should be like the one showed in Fig. 13.8. In the TIL side, the Fermi level (FL) is pinned at the IB energy (EC-EIB) due to the high density of states at this energy and is almost constant with the temperature. Electrons at the conduction band and holes
Fig. 13.7 (a) Sheet resistance as a function of measured temperature for a n-Si substrate (-) and for double sheet TIL/n-Si substrate for different implantation doses:
1014 cm-2 (plus experimental); 1015 cm-2 (square experimental; filled square ATLAS simulation); 5-1015 cm-2 (o experimental; bullet ATLAS simulation) and 1016 cm-2 (triangle experimental; filled triangle ATLAS simulation). (b) Mobility absolute value as a function of the measured temperature for a n-Si substrate (-) and for double sheet TIL/n-Si substrate for different implantation doses: 1014 cm-2 (plus); 1015 cm-2 (square); 5-1015 cm-2 (circle) and 1016 cm-2 (triangle) (Reprinted with permission from . Copyright 2009. Institute of Physics)
Fig. 13.8 Generic band diagram of the TIL/n-Si double sheet. This drawing is not at scale and correspond to the expected band diagram at T = 90 K (Reprinted with permission from . Copyright 2009. Institute of Physics)
at the valence band are determined through standard Maxwell-Boltzman statistics and their densities are strongly dependent on the temperature in spite of the constant position of the Fermi level (n = NCexp(—(EC — EIB)/kT) for electrons and p = NV exp(—(EIB—EV)/kT) for holes). Nevertheless, the carrier concentration at the IB will not agree with this statistics and they should be calculated as in a degenerated band, in agreement with the theory for semiconductors with IB .
Carriers at the IB could behave as electrons or holes depending on the sign of its effective mass, that is to say, the concavity or convexity of the band at the energy where it is crossed by the Fermi level. Of course the statistics now should be the Fermi Dirac one. Anyway, and independently of its type, carriers at the IB could not cross to the substrate because of the lack of continuity between the IB and the substrate. Tunneling is avoided by the low doping of the substrate that implies a wide depletion thickness, mostly developed in this substrate. Therefore at every temperature, IB carriers are electrically confined in the TIL plane, with low mobility as corresponding to a narrow band.
The sheet resistance at low temperatures only comes from the TIL because of the decoupling effect. To compute it, we have to take into account the electrons at the conduction band, holes at the valence band and carriers at the IB, irrespective of its sign. The only way to discriminate between electrons and holes at this band are the mobility measurements that determine that these carriers are holes as we will see later. Anyway, for the decoupling explanation its nature is irrelevant.
If we assume that the Fermi level (or the IB position which is the same at the TIL) is closer to the conduction band than to the valence band, the only carriers to worry about are electrons at the conduction band and, of course, holes at IB. At low temperature, if the IB energetic position is deep enough, the electron density should be very low at the TIL conduction band. In the substrate, the electron concentration is almost temperature independent (from 90 to 300 K) as corresponding to a semiconductor with a n-type shallow doping of 2.2-1013 cm-3. The substrate Fermi level position can be easily calculated using the Maxwell- Boltzmann statistics. At 90 K the FL in the substrate is about 0.1eV below the conduction band. In the Fig. 13.8, we have assumed that the IB is deeper than this energy and as a consequence, the electrical connection between the two layers
becomes unidirectional; the substrate can inject electrons to the TIL if the net applied voltage is high enough for the electrons to surmount the barrier energy but the TIL cannot inject electrons to the substrate because the electron density at TIL is negligible at this temperature. Due to the Fermi level position represented in Fig. 13.8, holes at TIL valence band or substrate valence band are also negligible. In a parallel double sheet, the fact of having an unidirectional contact between the layers means an effective decoupling of them because any current line coming and returning from TIL to the substrate has to cross twice the junction in opposite senses.
When the temperature goes up, the electron density at the TIL conduction band increases as it corresponds to the thermal generation from a level with an activation energy of EC — EIB. This activation energy is smaller than the Si intrinsic one and electrons density increases relatively fast. There should be a temperature at which the electron density in the TIL becomes similar to the substrate electron density, which remains constant. Electrons can now flow freely back and forth from one layer to another and both layers are now electrically coupled. At this temperature both conduction bands at the TIL and the substrate should be leveled and there is not any current limitation mechanism between them.
According to the previous paragraph, our bilayer limits the electron flux going from the implanted layer to the substrate but only at low temperatures. As holes flux does not cross from TIL to the substrate at the IB, this border behaves as a junction having the P side at the implanted layer and the N side at the substrate. This unidirectional behaviour disappears at high temperature. In this sense, and only in this sense, the implanted layer behaves as a P semiconductor at low temperatures. It is important to highlight that for a semiconductor with an IB the issue of the definition of the type is not as clear as for a normal semiconductor.
As the carrier density at the IB have only influence on the sheet resistance of the implanted layer and they are not involved in the coupling/decoupling mechanism, their nature is irrelevant to determine the current limitation mechanism at the TIL/substrate junction.
Assuming the previous hypothesis, the densities of carriers are easy to calculate using the classical semiconductor equations. Let us suppose that the IB is located 0.36 eV below the conduction band. As we will see in the next paragraph, this is the value that best fits the experimental results, it is not temperature dependent and it is worth to say that it is close to the Ti level referred in the literature . With this hypothesis, at low temperature (90 K) the electrons carrier density in the TIL should be extremely low and the barrier height (AE, see Fig. 13.8) with the substrate is 0.26 eV. Conversely at 300 K the electron density in the TIL increases up to values similar to the electron concentration in the substrate. In that case, there is no appreciable barrier for electrons and they can flow freely in both directions, depending on the external voltage polarity.