We discuss our numerical results on the background of experimental characteristics for a 10 nm gate length transistor [
4,
5] reproduced in Figure . As demonstrated in Sect. "Parameters in experimental nano-FETs" one obtains from Equation 21 a characteristic length of
λ ~ 1 nm under reasonable assumptions. For the experimental 10 nm gate length, we thus obtain
l =
L/
λ = 10. Furthermore, Equation 20 yields the value of
εF = 0.35 eV. The conversion of the experimental drain voltage
V into the theoretical parameter
vD is given by
The maximum experimental drain voltage of 0.75 V then sets the scale for
vD ranging from zero to
vD = 0.75 eV/0.35 eV ~ 2. For the conversion experimental gate voltage
VG to the theoretical parameter
vG we make linear ansatz as
where
is the experimental threshold gate voltage (see Figure ). The constant
β is chosen so that
converts into
. In our example, it is shown from Figure
= 0.15 V and from Figure
= -0.05, so that
β = -0.2 eV. To match the experimental drain characteristic to the theoretical one we first convert the highest experimental value for
VG into the corresponding theoretical one. Inserting in (23)
VG = 0.75 V yields
vG ~ 0.5. Second, we adjust the experimental and the theoretical drain current-scales so that in Figure the curves for the experimental current at
VG = 0.7 and the theoretical curve at
vG = 0.5 agree. It then turns out that the other corresponding experimental and theoretical traces agree as well. This agreement carries over to the range of negative gate voltages with thermally activated transport. This can be gathered from the
ID -
VG traces in Figure . We note that the constant of proportionality in Equation 23 given by 1 eV is more then
εF which one would expect from the theoretical definition
vG =
VG/
εF. Here, we emphasize that the experimental value of
e VG corresponds to the change of the potential at the transistor gate while the parameter
vG describes the position of the bottom of the lowest two-dimensional subband in the electron channel. The linear ansatz in Equation 23 and especially the constant of proportionality 1 eV can thus only be justified in a self-consistent calculation of the subband levels as has been provided, e.g., by Stern[
18].
The experimental and the theoretical drain characteristics in Figure look structurally very similar. For a quantitative comparison we recall from Sect. "Parameters in experimental nano-FETs" the value of
J0 = 5.4 × 10
4μA/
μm. Then the maximum value
j = 0.15 in Figure corresponds to a theoretical current per width of 8 × 10
3μA/
μm. To compare with the experimental current per width we assume that in the
y-axis labels in Figures and it should read
μA/
μm instead of A/
μm. The former unit is the usual one in the literature on comparable nanotransistors (see Refs. [
2-
9]) and with this correction the order of magnitude of the drain current per width agrees with that of the comparable transistors. It is found that the theoretical results are larger than the experimental ones by about a factor of ten. Such a failure has to be expected given the simplicity of our model. First, for an improvement it is necessary to proceed from potentials resulting in a self-consistent calculation. Second, our representation of the transistor by an effectively one-dimensional system probably underestimates the backscattering caused by the relatively abrupt transition between contacts and electron channel. Third, the drain current in a real transistor is reduced by impurity interaction, in particular, by inelastic scattering. As a final remark we note that in transistors with a gate length in the micrometer scale short-channel effects may occur which are structurally similar to the ones discussed in this article (see Sect. 8.4 of [
10]). Therefore, a quantitatively more reliable quantum calculation would be desirable allowing to distinguish between the short-channel effects on micrometer scale and quantum short-channel effects.
1 Marcus Krahlisch,1 and Hans Richter1
