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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 ID - VD-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 ID - VG-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 ID - VD-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 εF 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 ID - VG-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.
In Refs. [13,14], the transport problem in a nano-FET was reduced to a one-dimensional effective problem invoking a "single-mode abrupt transition" approximation. Here, the electrons move along the transport direction in an effective potential given
where Ek = 1 is the bottom of the lowest two-dimensional subband resulting in the z-confinement potential of the electron channel at zero drain voltage (see Figure Figure4b4b of Ref. ). The parameter W is the width of the transistor. Finally, VD = eUD is the drain potential at drain voltage UD which is assumed to fall off linearly.
Experimentally, one measures in a wide transistor the current density J, which is the current per width of the transistor that we express as
was derived. Here, m = μ/εF is the normalized chemical potential in the source contact, vD = VD/εF is the normalized drain voltage, and vG = VG/εF is the normalized gate voltage. As illustrated in Figure 1(b) the gate voltage is defined as the energy difference μ - V0 = VG, i.e., for VG > 0 the transistor operates in the ON-state regime of classically allowed transport and for VG < 0 in the source-drain tunneling regime. The control variable VG is used to eliminate the unknown variable V0. For the chemical potential in the source contact one finds (see next section)
where u = kBT/εF is the normalized thermal energy. Equation 4 has the form of a Tsu-Esaki formula with the normalized supply function
Here, F-1/2 is the Fermi-Dirac integral of order -1/2 and is the inverse function of F1/2. 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.
For a wide enough transistor and a sufficient junction depth a (see Figure Figure1)1) the electrons in the contacts can be treated as a three-dimensional non-interacting electron gas. Furthermore, we assume that all donor impurities of density Ni are ionized. From charge neutrality it is then obtained that the electron density n0 is independent of the temperature and given by
Here me is the effective mass and NV is the valley-degeneracy factor in the contacts, respectively. In the zero temperature limit a Sommerfeld expansion of the Fermi-Dirac integral leads to
Equating 7 and 8 results in
which is identical with (5) and plotted in Figure Figure2.2. As well-known, with increasing temperature the chemical potential falls off because the high-energy tail of the Fermi-distribution reaches up to ever higher energies.
As shown in Ref.  the supply function for a wide transistor can be written as
This expression can be interpreted as the partition function (loosely speaking the "number of occupied states") in the grand canonic ensemble of a non-interacting homogeneous three-dimensional electron gas in the subsystem of electrons with a given lateral wave vector (ky, kz) yielding the energy in the y-z-direction. Formally equivalent it can be interpreted as the full partition function in the grand canonic ensemble of a one-dimensional electron gas at the chemical potential μ - ε. Performing the limit the Riemann sum in the variable can be replaced by the Fermi-Dirac integral F-1/2. It results that
with the normalized transistor width w = W/λ. For the scaling of the supply function in Equation 11 we define (see Ref. )
where and we use the identity V0= εF = m - vG. For the source contact we write
leading to the first factor in the square bracket of the Tsu-Esaki equation 4. In the drain contact, the chemical potential is lower by the factor VD. Replacing μ → μ - VD yields
Below we will show that for transistor operation the low temperature limit is relevant (see Figure Figure2).2). Here, one may apply in leading order (resulting from a Sommerfeld expansion) and F-1/2(-x → ∞) → exp (x). Since V0 > 0 the factor vG - m is negative and we obtain from (12)
From Figure Figure33 it is seen that for ε below the chemical potential the supply function is well described by the square-root dependence in the limit. If ε lies above the chemical chemical one obtains the limit which is a small exponential tail due to thermal activation.
The effective current transmission in Equation 16 is given y
It is calculated from the scattering solutions of the scaled one-dimensional Schrödinger equation
with β = 2m*V0L2/ħ2 = l2(m - vG), and ŷ = y/L. The scaled effective potential is given by , , and ,where . As usual, the scattering functions emitted from the source contact obey the asymptotic conditions and
with and .
As can be seen from Figure Figure4,4, around the current transmission changes from around zero to around one. For weak barriers there is a relatively large current transmission below one leading to drain leakage currents. For strong barriers this remnant transmission vanishes and we can approximate the current transmission by an ideal one.
To a large extent the Fowler Nordheim oscillations in the numerical transmission average out performing the integration in Equation 4.
In the heavily doped contacts the electrons can be approximated as a three-dimensional non-interacting Fermi gas. Then from (8) the Fermi energy above the bottom of the conduction band is given by
For n++-doped Si contacts the valley-degeneracy is NV = 6 and the effective mass is taken as . Here m1 = 0.19m0 and m2 = 0.98m0 are the effective masses corresponding to the principle axes of the constant energy ellipsoids. In our later numerical calculations we set εF = 0.35 eV assuming a level of source-doping as high as Ni = n0 = 1021 cm-3.
In the electron channel a strong lateral subband quantization exists As well-known  at low temperatures only the two constant energy ellipsoids with the heavy mass m2 perpendicular to the (100)-interface are occupied leading to a valley degeneracy of gv = 2. The in-plane effective mass is therefore the light mass m* = m1 entering the relation
Here εF = 0.35 eV was assumed. One then has in Equation 3 I0 = ~ 27μA and with λ ~ 1 nm as well as = 2 one obtains J0 = 5.4 × 104 μA/μm.
Typical drain characteristics are plotted in Figure Figure55 for a low temperature (u = 0.01) and at room temperature (u = 0.1). It is seen that for both the temperatures a LTT can be identified. We define the LTT as the j - vD trace which can be best fitted with a linear regression j = σthvD in the given interval 0 ≤ vD ≤ 2. The best fit is determined by the minimum relative mean square deviation. The gate voltage associated with the LTT is denoted with . It turns out that at room temperature lies slightly above zero and at low temperatures slightly below (see Figure Figure5c).5c). In general, the temperature dependence of the drain current is small. The most significant temperature effect is the enhancement of the resonant Fowler-Nordheim oscillations found at negative vG at low temperatures. From Figure Figure5d,5d, it can be taken that the slope of the LTT σth decreases with increasing l and increasing temperature. For "hot" transistors (u = 0.2) a LTT can only be defined up to l ~ 10.
The threshold characteristics at room temperature are plotted in Figure Figure66 for a "small" drain voltage (vD = 0.1) and a "large" drain voltage (vD = 2.0). For the largest considered characteristic length l = 60 it is seen that below zero gate voltage the drain current is thermally activated for both considered drain voltages. A comparison with the results for l = 25 and l = 10 yields that for the small drain voltage the ID - VG trace is only weakly effected by the change in the barrier strength. In contrast, at the high drain voltage the drain current below vG = 0 grows strongly with decreasing barrier strength. The drain current does not reach the thermal activation regime any more, it falls of much smoother with increasing negative vG. As can be gathered from Figure Figure88 this effect is seen in experiments as well. We attribute it to the weakening of the tunneling barrier with increasing vD. To confirm this point the threshold characteristics for a still weaker barrier strength (l = 3) is considered. No thermal activation is found in this case even for the small drain voltage.
We discuss our numerical results on the background of experimental characteristics for a 10 nm gate length transistor [4,5] reproduced in Figure Figure7.7. 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 Figure8a).8a). The constant β is chosen so that converts into . In our example, it is shown from Figure Figure8a8a = 0.15 V and from Figure Figure8b8b = -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 Figure77 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 Figure8.8. 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.
The experimental and the theoretical drain characteristics in Figure Figure77 look structurally very similar. For a quantitative comparison we recall from Sect. "Parameters in experimental nano-FETs" the value of J0 = 5.4 × 104μA/μm. Then the maximum value j = 0.15 in Figure Figure7b7b corresponds to a theoretical current per width of 8 × 103μA/μm. To compare with the experimental current per width we assume that in the y-axis labels in Figures Figures7a7a and and8a8a 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 ). 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 ID - VG-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.
LTT: linear threshold trace.
The authors declare that they have no competing interests.
UW worked out the theroretical model, carried out numerical calculations and drafted the manuscript. MK carried out numerical calculations and drafted the manuscript. HR drafted the manuscript. All authors read and approved the final manuscript.