THE III–V MOSFET
The previously discussed HEMT takes advantage of the high mobility of the III-V semiconductors but suffers from a disadvantage compared to the Si MOSFET: the Schottky gate with which it operates exhibits a high leakage current as shown in figure 8.10. The remedy for this disadvantage is the introduction of a dielectric under the gate, thus making a complete analog to the Si MOSFET. However, this proved to be a formidable task. Existing dielectrics НЮ2 and A1203 are not easily deposited without creating surface defects which scatter the electrons and slow down their response. Fortunately, ways of passivating the defects have been devised and nowadays III-V MOSFETs with both НЮ2 and A120, under the gate exist. Figure 8.11 portrays two versions of III-V
FIGURE 8.11 Typical III-V MOSFET devices.
MOSFETs, one with НЮ2 as the oxide layer and the other with A1203. Observe the complexity of the devices, in particular the many layers required for stability and to contain the current in the channel layer.
The theoretical tools to analyze III—V MOSFETs are essentially no different than what has been presented for both Si MOSFETs and III—V FIEMTs. However, due to the complexity of the devices on the one hand and the nanometric length of the channel on the other hand a combination of both quantum (i.e. Landauer) methods and classical equations together with the Schroedinger equation are needed to fully describe the device. Furthermore, some additional modifications of the classical equations are necessary to fully account for the effects of the nanometric size of the channel that we have discussed in the previous chapter.
In the closing paragraphs of the previous chapter we emphasized that the saturation regime of the Si MOSFET is not dictated by the saturation velocity occurring at the pinch- off point but instead by the injection velocity over the top of the barrier that is near the source-end of the channel. The latter has nothing to do with the term vsJ? that we have used in the analysis of the long Si MOSFET. Guided by these considerations, we can allow the saturation velocity entering the mobility expression in the continuity equation to attain higher values than those usually quoted in the literature, essentially treating this quantity as a parameter.
One may ask why we need the continuity equation when the Natori formula will directly give the current? The answer is that to use the Natori formula, equation 7.48, requires the eigenvalues £„. of equation 7.40 and this, in turn, requires the construction of the corresponding potential energy U'. If the latter is to be calculated self-consistently the system of equations 8.27-8.34 needs to be solved. We note that if the depth of the quantum well were of the order of 2-3eV as in Si MOSFET we could assume that all electrons reside in the lowest sublevel of the well. However, as explained in the case of the HEMT, the conduction band-offset for III—V semiconductors is of the order of a few tenths of an eV (between 0.2-0.4eV). Hence the employment of the full set of equations 8.27-8.34 is necessary.
A further point that needs to be clarified is that the potential energy V(x,z) of equation 8.28 and the potential energy U(x,z) of equation 7.40 in chapter 7 are not the same. Equation 7.40 is essentially a 1-dimensional equation, the variable x simply denotes the position along the channel. Equation 7.40 is solved only along the depth of the channel. On the other hand, equation 8.27 is a 2-dimensional equation and so is the corresponding potential energy V(x,z). Of course, at any point x, U(x,z) is the corresponding column of V(x,z). To use the Natori formalism we first solve the system of equations 8.27-8.34 and obtain V(x,z). Then extract the column I/(x(0p,z) corresponding to the position of the top of the barrier xtop, and solve equation 7.40. We finally evaluate the current using 7.49. Accordingly, flowchart 4 corresponding to these actions is the same as flowchart 3 until convergence of V(x,z) and then it proceeds to the evaluation of the current by the Natori method.
In the above flowchart one should not confuse the symbol Etop which refers to the top energy value of the 2-dimensional potential well with £„. (xtop) which are the eigenvalues of the 1-dimensional well along z located at x,op which is used in the Natori formula.
Some calculations by the author  for a typical device portrayed in figure 8.12 which show the influence of the many layers in III—V as opposed to Si MOSFETs, are shown in Figure 8.13. The figure gives the longitudinal current density Jx (z)as a function of the depth г of the device for several values of the gate voltage Vc. The channel layer (In0 53Ga0 47As) is located between г = 70nm and z = 80nm. We observe that the current density Jx below threshold (V0ff <0) is located near the doped area and gradually, as the gate voltage is increased above threshold, it shifts in the channel layer. This shifting of the current density
FIGURE 8.12 The device for which the author has performed calculations.
FIGURE 8.13 Variation with depth of the longitudinal current density as the gate voltage increases: the subthreshold current occurs below the channel and gradually moves to the channel area.
FIGURE 8.14 Current as a function of gate voltage for the device shown in figure 8.12. When the quantum well in which the current is calculated by the Natori method is identical to the channel, the calculated subthreshold current deviates from the experimental values. When the quantum well is enlarged to include lower layers, agreement with experiments is restored; see text for more details.
can also be seen if one looks at the current as a function of VG. This is shown in figure 8.14 together with the experimental results. The current is calculated in two different ways: the circles represent a calculation where the quantum box (the domain where the Schroedinger equation is solved) coincides with exactly the channel (InGaAs) area while the triangles represent a calculation where the quantum box includes the layers above and below the channel. In both cases the Natori formula is used. Consequently the 1 dimensional well, where the £„. are calculated, includes in the latter case the barrier layers also. The former calculation coincides with Natori’s original assumption that the wavefunctions are contained mainly within the channel layer with only an exponentially decreasing tale outside this layer. It can be seen that the calculation with the larger quantum box (the one including the barriers) is very near the experimental results, whereas the calculation with the quantum box containing only the channel layer misses all the current that exists below the channel at sub-threshold. Above threshold the two calculations agree because all the current is contained within the In0 53Ga047As layer. Then if the quantum box is bigger than the In053Ga047As layer makes no difference. A third calculation, denoted by asterisks, is shown in which the current is calculated in the InGaAs layer by the Natori method and in the layers below by the Poisson-Schroedinger-Continuity (PSC) equations. This also agrees with experiment. Therefore great care should be exercised when using the Natori formalism to many layer III—V MOSFETs.