Scaling properties of ballistic nano-transistors
© Wulf et al; licensee Springer. 2011
Received: 5 November 2010
Accepted: 28 April 2011
Published: 28 April 2011
Recently, we have suggested a scale-invariant model for a nano-transistor. In agreement with experiments a close-to-linear thresh-old trace was found in the calculated I D - V D-traces separating the regimes of classically allowed transport and tunneling transport. In this conference contribution, the relevant physical quantities in our model and its range of applicability are discussed in more detail. Extending the temperature range of our studies it is shown that a close-to-linear thresh-old trace results at room temperatures as well. In qualitative agreement with the experiments the I D - V G-traces for small drain voltages show thermally activated transport below the threshold gate voltage. In contrast, at large drain voltages the gate-voltage dependence is weaker. As can be expected in our relatively simple model, the theoretical drain current is larger than the experimental one by a little less than a decade.
In the past years, channel lengths of field-effect transistors in integrated circuits were reduced to arrive at currently about 40 nm . Smaller conventional transistors have been built [2–9] with gate lengths down to 10 nm and below. As well-known with decreasing channel length the desired long-channel behavior of a transistor is degraded by short-channel effects [10–12]. One major source of these short-channel effects is the multi-dimensional nature of the electro-static field which causes a reduction of the gate voltage control over the electron channel. A second source is the advent of quantum transport. The most obvious quantum short-channel effect is the formation of a source-drain tunneling regime below threshold gate voltage. Here, the I D - V D-traces show a positive bending as opposed to the negative bending resulting for classically allowed transport [13, 14]. The source-drain tunneling and the classically allowed transport regime are separated by a close-to linear threshold trace (LTT). Such a behavior is found in numerous MOSFETs with channel lengths in the range of a few tens of nanometers (see, for example, [2–9]).
Starting from a three-dimensional formulation of the transport problem it is possible to construct a one-dimensional effective model  which allows to derive scale-invariant expressions for the drain current [15, 16]. Here, the quantity arises as a natural scaling length for quantum transport where ε F is the Fermi energy in the source contact and m* is the effective mass of the charge carriers. The quantum short-channel effects were studied as a function of the dimensionless characteristic length l = L/λ of the transistor channel, where L is its physical length.
In this conference contribution, we discuss the physics of the major quantities in our scale-invariant model which are the chemical potential, the supply function, and the scale-invariant current transmission. We specify its range of applicability: generally, for a channel length up to a few tens of nanometers a LTT is definable up to room temperature. For higher temperatures, a LTT can only be found below a channel length of 10 nm. An inspection of the I D - V G-traces yields in qualitative agreement with experiments that at low drain voltages transport becomes thermally activated below the threshold gate voltage while it does not for large drain voltages. Though our model reproduces interesting qualitative features of the experiments it fails to provide a quantitative description: the theoretical values are larger than the experimental ones by a little less than a decade. Such a finding is expected for our simple model.
Tsu-Esaki formula for the drain current
where E k = 1is the bottom of the lowest two-dimensional subband resulting in the z-confinement potential of the electron channel at zero drain voltage (see Figure 4b of Ref. ). The parameter W is the width of the transistor. Finally, V D = eU D is the drain potential at drain voltage U D which is assumed to fall off linearly.
Here, F -1/2 is the Fermi-Dirac integral of order -1/2 and is the inverse function of F 1/2. The effective current transmission depends on which is the normalized energy of the electron motion in the y-z-plane while is their energy in the x-direction. In the next sections, we will discuss the occurring quantities in detail.
Chemical potential in source- and drain-contact
with and .
To a large extent the Fowler Nordheim oscillations in the numerical transmission average out performing the integration in Equation 4.
Parameters in experimental nano-FETs
Heavily doped contacts
For n ++-doped Si contacts the valley-degeneracy is N V = 6 and the effective mass is taken as . Here m 1 = 0.19m 0 and m 2 = 0.98m 0 are the effective masses corresponding to the principle axes of the constant energy ellipsoids. In our later numerical calculations we set ε F = 0.35 eV assuming a level of source-doping as high as N i = n 0 = 1021 cm-3.
Here ε F = 0.35 eV was assumed. One then has in Equation 3 I 0 = ~ 27μ A and with λ ~ 1 nm as well as = 2 one obtains J 0 = 5.4 × 104 μ A/μ m.
The experimental and the theoretical drain characteristics in Figure 7 look structurally very similar. For a quantitative comparison we recall from Sect. "Parameters in experimental nano-FETs" the value of J 0 = 5.4 × 104 μ A/μ m. Then the maximum value j = 0.15 in Figure 7b corresponds to a theoretical current per width of 8 × 103 μ A/μ m. To compare with the experimental current per width we assume that in the y-axis labels in Figures 7a and 8a it should read μ A/μ m instead of A/μ m. 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 ). 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.
After a detailed discussion of the physical quantities in our scale-invariant model we show that a LTT is present not only in the low temperature limit but also at room temperatures. In qualitative agreement with the experiments the I D - V G-traces exhibit below the threshold voltage thermally activated transport at small drain voltages. At large drain voltages the gate-voltage dependence of the traces is much weaker. It is found that the theoretical drain current is larger than the experimental one by a little less than a decade. Such a finding is expected for our simple model.
linear threshold trace.
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