1 Introduction

Since the discovery of carbon nanotubes in 1991 by Iljama, fundamental and science technology have devoted much attention to the investigation of the electronic transport through single-wall carbon nanotubes (SWCNTs) [1]. Experiments and theory have predicted that this material can be either metallic or semiconducting (SC). The extraordinary electrical properties of such individual CNTs can rival or even exceed the best performance of usual metallic or semiconducting materials [2,3,4,5]. Semiconductor nanotubes can be used as active elements in field-effect transistors (CNTFETs) and CMOS-like logic circuits using n- or p-type MOS-like FETs with Schottky or ohmic contacts [6, 7].

There are two main types of CNTFETs that are being currently studied, differing by their current injection methods. CNTFETs can be fabricated with ohmic or Schottky contacts. The type of the contact determines the dominant mechanism of current transport and device output characteristics. CNTFETs are mainly divided into Schottky barrier CNTFETs (SB-CNTFETs) with metallic electrodes, which form Schottky contacts [8,9,10,11] and MOSFET-like CNTFETs with doped CNT electrodes, which form ohmic contacts [11, 12]. In SB-CNTFETs, tunneling of electrons and holes from the potential barriers at the source and drain junctions constitutes the current. The barrier width is modulated by the application of gate voltage, and thus the transconductance of the device is dependent on the gate voltage [12,13,14].

MOSFET-like CNTFETs take advantage of the n-doped CNT as the contact. Potassium doped source and drain regions have been demonstrated, and the behavior of MOSFETs-like CNTFETs have been experimentally verified [12]. In this type of transistors, since the depletion length, in case of carbon nanotube, is a function of diameter of carbon nanotube and thickness of gate insulators, a potential barrier is formed at the middle of the channel for small values of these parameters [4]. Consequently, for these devices it is mainly the potential barrier between the contacts that controls the current [4].

One of the important aspects of nanotube transistors is the ambipolarity or unipolarity of their current–voltage characteristics. SB-CNTFETs exhibit strong ambipolar behavior; they show electron conduction at high gate voltages and hole conduction at low gate voltages. Recently, CNTFETs with zero or slightly negative Schottky barriers were achieved by attaching an intrinsic nanotube channel to the high work-function metal contacts [7]. When the gate oxide is thick, reducing the Schottky barrier height to zero suppresses the ambipolar conduction, but for the thin gate oxide, the transistor is still ambipolar, even if the Schottky barrier height for electrons/holes is zero [15]. Ambipolar conduction leads to a large leakage current that exponentially increases with the power supply voltage, especially when the tube diameter is large. Figure 1 shows a coaxial CNTFETs with a (13, 0) zigzag CNT as the channel which corresponds to a bandgap of about 0.83 eV and a diameter of 1 nm [16]. The gate is 10 nm long and wrapped around the channel. The gate thickness is assumed 6 nm. SB-CNTFET employs an intrinsic CNT and 20 nm long metallic contacts as the source and drain.

The energy bands for low and high gate voltages and the Schottky barriers are shown in Fig. 1b.

Fig. 1
figure 1

Schottky barrier CNTFET, a 2-D cross section of the coaxial structure with intrinsic CNT as the channel and metal source/drain contacts, b energy band diagram obtained from Poisson equation. The metal Fermi level is taken to be at the midgap of the CNT [16]

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However, for MOSFET-like CNTFETs, only a positive gate voltage increases the current, because of the lowering of the barrier in the channel [16]. The energy bands for low and high gate voltages and the potential barrier in the channel are shown in Fig. 2.

Fig. 2
figure 2

MOSFET-like CNTFET, a 2-D cross section of the coaxial structure with intrinsic CNT as the channel and doped CNT sections as source/drain contacts, b energy band diagram obtained from Poisson equation [16]

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It is clear from the figures why the characteristics of these two transistors differ. The current–voltage characteristics of the devices have been compared. Ambipolar behavior of the SB-CNTFETs constraints the use of these transistors in conventional CMOS logic families [17]. Methods have been proposed to reduce or eliminate the ambipolarity in CNTFETs. One of them uses asymmetrical contact types to introduce Schottky-ohmic CNTFET, which has a Schottky barrier at the CNT-metal interface at the source and an ohmic contact at the drain at the channel-doped CNT interface [18,19,20].

However, the device still suffers from band to band tunneling. That is, in low or negative voltages the electrons from the valence band can tunnel to the conduction band and contribute to the total current of the transistor and cause the increase of the current in negative voltages [21]. On the other hand, an advantage of ambipolar SB-CNTFETs is that they can be used as either an n-type or p-type FET in a CMOS application [22]. In MOSFET-like CNTFETs with heavily doped source and drain regions, when applying a negative gate voltage, the band to band tunneling may lead to ambipolarity. For suppressing this effect in MOSFET-like CNTFETs, it has been proposed [16] a non-uniform doping profile as shown in Fig. 3. This reduces the gradient of the channel potential barrier at each interface between the intrinsic and doped sections of the CNT and suppresses the band to band tunneling and ambipolar conduction.

Fig. 3
figure 3

a Cross section of the CNTFET. b Step and linear doping profiles for the MOSFET-like structure versus position

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In this paper, we have studied the effect of physical and geometrical parameters on the performance of the SB-CNTFET. The relationship between the pass state current and the blocked state current is an essential factor for improved static performance of SB-CNTFET. The limitation of the use of this transistor is due generally to the effects of the height of the Schottky barrier, nanotube diameter and the capacity of the oxide layer. For this reason, we study the effect of the height of Schottky barrier on the pass state current. For dynamic system, we study the effect of the gate–source voltage, the chirality and the CNT diameter on the transition frequency.

2 Mathematical formulation

A carbon nanotube transistor is formed by two metal contacts, metal/nanotube in drain and source sides. The nature of the metal may be either aluminum (Al), titanium (Ti) [23], palladium (Pd) [14, 24], or scandium (Sc). Platinum (Pt) is often used since it has poor wettability with the carbon nanotubes [14]. Because it has low adhesion energy, at the interfaces formed by the metal and the semiconductor nanotube contact, barriers of potential are formed and opposed to the carriers transition between source and drain through the nanotube channel. Thus, these barriers have an important role in determining the current since they determine the number of carriers present on the metal side and transmitted in the channel [25].

A polarization change of the gate or the drain affects the channel potential, and affects also the dimensions (height and width) of the two Schottky barriers. Generally, the profile of the band gap close to the contact depends strongly on the polarizations \({V}_{\mathrm{GS}}\) and \({V}_{\mathrm{DS}}\). Figure 4 shows a SB-CNTFET structure in “back-gate” configuration for different polarization schemes \({V}_{\mathrm{GS}}\) and \({V}_{\mathrm{DS}}\). In this example, only the first energy band is shown. Let’s examine the variation of the energy band profile and shape of the Schottky barriers depending on the range bias \({V}_{\mathrm{GS}}\) and \({V}_{\mathrm{DS}}\).

Fig. 4
figure 4

Schematic representation of bands energy between source and drain of a SB-CNTFET in back gate (top) configuration for different regimes of polarization bands (Case A: \({V}_{\mathrm{DS}}= {V}_{\mathrm{GS}} = 0\), Case B: \({V}_{\mathrm{GS}}> 0\, {V}_{\mathrm{DS}} = 0\), Case C: \({V}_{\mathrm{GS}}> {V}_{\mathrm{DS}}>0\) and Case D: \({V}_{\mathrm{DS}}> {V}_{\mathrm{GS}}> 0\)), only the 1st sub-band is shown [26]

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Fig. 5
figure 5

Schematic of the equivalent model of the carbon nanotube transistor with modulated height barrier

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3 Equivalent electrical circuit

The equivalent electrical circuit for the SB-CNTFET in Fig. 5 follows from the equations of the density of source and drain; \({Q}_{\mathrm{QS}}\) and \({Q}_{\mathrm{QD,}}\) (Eq. 1) and drain current (Eq. 2).

$$\begin{aligned} Q_\mathrm{S,D}= & {} f_\mathrm{SMO} Q_\mathrm{low\,energy} +\left( 1-f_\mathrm{SMO}\right) Q_\mathrm{high\,energy} \end{aligned}$$
(1)
$$\begin{aligned} I_\mathrm{D}= & {} \frac{4ek_{B}}{h}T\sum \limits _{p=1}^{{nb-sbbd}} \left[ \begin{array}{l} \ln \left[ 1+\hbox {exp}\frac{{eV}_{\mathrm{s}}+\Phi _{\mathrm{SB}}^{\mathrm{eff}} -{sbbd}[{p}]}{{k}_{{B}}{T}}\right] \\ -{\ln }\left[ 1+\hbox {exp}\frac{{eV}_{\mathrm{D}}+\Phi _{\mathrm{SB}}^{\mathrm{eff}} -{sbbd}[{p}]}{{k}_{{B}}{T}}\right] \\ \end{array}\right] \nonumber \\ \end{aligned}$$
(2)

\({C}_{\mathrm{INS}}\) is the capacity of the gate oxide, it depends on the geometry and the dielectric constant of the insulator; \({V}_{\mathrm{FB}}\) is the voltage of flat bands, which takes into account the difference between the work function of the metal and the electron affinity of the nanotube; \({R}_{\mathrm{G}}\) is the access gate resistance; \({R}_{\mathrm{S}}\) and \({R}_{\mathrm{D}}\) are the access resistance of source and drain; \({C}_{\mathrm{SE}}\) and \({C}_{\mathrm{DE}}\) are the electrostatic capacity, which represent the change in the load interfaces doped nanotube/intrinsic nanotube [27]. The AC system, which is obtained from the equivalent circuit and the influence of the barrier Schottky is introduced into the calculation of the current \({I}_{\mathrm{DS}}\) and charge densities \({Q}_{\mathrm{QS}}\) and \({Q}_{\mathrm{QD}}\) [28].

3.1 Current calculation

Drain current in a SB-CNTFET is calculated by using the following equation:

$$\begin{aligned} {I}_{\mathrm{DS}}= & {} \frac{2e}{2{\uppi }} {M}\left[ \int \nolimits _{0}^{\infty }{v}\left( {k}\right) {f}_{\mathrm{s}} \left( {k}\right) {T}_{\mathrm{s}} \left( {E}\right) \hbox {d}k\right. \nonumber \\&\left. -\int \nolimits _{-\infty }^{0} {v}\left( {k}\right) {f}_{\mathrm{D}} \left( {k}\right) {T}_{\mathrm{D}} \left( {E}\right) \hbox {d}k\right] \end{aligned}$$
(3)

Ferry [29] assumes a one-dimensional (1D) channel, which is characterized by a ballistic transport between source and drain [30, 31]. Thus, it is given by using the formulation of the effective Schottky barrier:

$$\begin{aligned} {I}_{\mathrm{s}}= & {} \frac{4e}{h}\sum \limits _{{p}=1}^{{ nb-sbbd}}\left[ \int \limits _{{sbbd[p]}}^{\infty }\frac{1}{1+\hbox {exp}\left( \frac{{E}-{e}({v}_{\mathrm{CNT}}-{v}_{\mathrm{s}}}{{k}_{\mathrm{B}}{T}}\right) }\right. \nonumber \\&\text {exp}\left. \left( -4\frac{\sqrt{2{m}}}{3\hbar {qE}_{\mathrm{doc}}}\right) \left( \Phi _{\mathrm{SB}}-{E}-{qV}_{\mathrm{s}}\right) ^{\frac{3}{2}}\hbox {d}E\right] \end{aligned}$$
(4)
$$\begin{aligned} {I}_{\mathrm{DS}}= & {} \frac{2e}{h}{M}\sum \limits _{p=1}^{nb-sbbd}\left[ \int _{{\Phi _{\mathrm{SB}}^{\mathrm{eff}}}}^{\infty }{f}_{\mathrm{s}}({E})\hbox {d}E-\int _{{\Phi _{\mathrm{SB}}^{\mathrm{eff}}}}^{\infty }{f}_{\mathrm{D}}({E})\hbox {d}E\right] \nonumber \\ \end{aligned}$$
(5)
$$\begin{aligned} {I}_{\mathrm{DS}}= & {} \frac{4e}{h}\sum \limits _{p=1}^{{nb-sbbd}} \left[ \int _{\Phi _{\mathrm{SB}}}^{\infty }\frac{1}{1+\hbox {exp}\left( \frac{{E}-{e} ({v}_{\mathrm{CNT}}-{v}_{\mathrm{S}}}{{k}_{\mathrm{B}}{T}}\right) }\hbox {d}E \right. \nonumber \\&\left. -\int _{\Phi _{\mathrm{SB}}}^{\infty } \frac{1}{1+\hbox {exp}\left( \frac{{E}-{e}({v}_{\mathrm{CNT}}-{v}_{\mathrm{D}}}{{k}_{\mathrm{B}}{T}}\right) \hbox {d}E}\right] \end{aligned}$$
(6)

The sbbd energy is the minimum of a sub-band with respect to the Fermi level of the carbon. We can integrate the current.

$$\begin{aligned} {I}_{\mathrm{D}}= & {} \frac{4ek_{B}}{h}T\sum \limits _{p=1}^{-sbbd} \left[ \begin{array}{l} \hbox {ln}\left[ 1+\hbox {exp}\frac{{eV}_{\mathrm{s}}+\Phi _{\mathrm{SB}}^{\mathrm{eff}} -{sbbd}[{p}]}{{k}_{\mathrm{B}}{T}}\right] \\ -\hbox {ln}\left[ 1+\hbox {exp}\frac{{eV}_{\mathrm{D}}+\Phi _{\mathrm{SB}}^{\mathrm{eff}} -{sbbd}[{p}]}{{k}_{\mathrm{B}}{T}}\right] \\ \end{array}\right] \nonumber \\ \end{aligned}$$
(7)

with \(\Phi _\mathrm{SB-S}^{\mathrm{eff}}\) and \(\Phi _\mathrm{SB-D}^{\mathrm{eff}}\) barrier heights of the source and drain, respectively, and defined as

$$\begin{aligned} \Phi _{\mathrm{SB}_\mathrm{S,D} }^{\mathrm{eff}}= & {} \left( \Phi _{\mathrm{SB}} \!-\! \left( {sbbd}_{[p]}\!-\!{eV}_{\mathrm{CNT}} \!+\!V_\mathrm{S,D}\right) \right) \hbox {exp}\left( \frac{\!-\!d_{\mathrm{tunnel}}}{\lambda _{\mathrm{Schottky}}}\!\right) \nonumber \\&+ \left( {sbbd}_{[p]} -{eV}_{\mathrm{CNT}} +V_\mathrm{S,D}\right) \mathrm{exp}\left( -\frac{d_{\mathrm{tunnel}}}{\lambda _{\mathrm{Schottky}}}\right) \nonumber \\&+ \left( {sbbd}_{[p]} -{eV}_{\mathrm{CNT}} +V_\mathrm{S,D}\right) \end{aligned}$$
(8)

In the case of field effect transistors, the transition frequency \({f}_{\mathrm{T}}\) is expressed as:

$$\begin{aligned} {f}_{\mathrm{T}}=\frac{1}{2{\pi }} \frac{g_{\mathrm{m}}}{c_\mathrm{g}}\bigg |_{{{V}}_{\mathrm{DS}} -{V}_{\mathrm{DD}}} \end{aligned}$$
(9)

\({c}_{\mathrm{g}}\) is the intrinsic capacitance of the gate.

With the transconductance \({g}_{\mathrm{m}}\) representing the variation of the current in the channel as a function of gate voltage \({V}_{\mathrm{GS}}\):

$$\begin{aligned} g_{\mathrm{m}} =\frac{\partial {I}_{\mathrm{D}}}{\partial {V}_{\mathrm{GS}}} \end{aligned}$$
(10)

4 Results and discussion

In this section, we will study the effect of physical and geometrical parameters on the performances of the SB-CNTFET. We consider only one sub-band, and we are limited to study the effect of the Schottky barrier and oxide layer as material parameters and the carbon nanotube diameter as the geometrical parameter. For a dynamic system we will study the effect of the gate–source voltage, the chirality, and the CNT diameter on the transition frequency.

Fig. 6
figure 6

The variation of the drain current \({I}_{\mathrm{DS}}\) versus gate voltage \({V}_{\mathrm{GS}}\), for three values of Schottky barrier height \(\Phi _{\mathrm{SB}} = [0.3, 0.5, 0.6]\) eV

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Figure 6 shows the change in leakage current \({I}_{\mathrm{DS}}\) with the gate voltage \({V}_{\mathrm{GS}}\), for three values of Schottky barrier height \(\Phi _{\mathrm{SB}} = 0.3\), 0.5, and 0.6 eV (while keeping the other parameters constant). From this figure, it is clear that for negative values of the gate voltage \(({V}_{\mathrm{GS}} \le 0)\), the current degrades almost linear and tends to zero, which is the dominant effect of \(({V}_{\mathrm{GS}} \le 0)\) represented by a weakening of the intensity of the drain current. On the other hand, when the gate voltage increases \(({V}_{\mathrm{GS}}> 0)\), the drain current of a virtually linear course rises. We find that the conduction current \({I}_{\mathrm{DS}}\) is reduced as the barrier \(\Phi _{\mathrm{SB}}\) is increased, which leads to a low static conduction.

Fig. 7
figure 7

The variation of the current in the on state \({I}_{\mathrm{ON}}\) versus the barrier height \(\Phi _{\mathrm{SB}}\)

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In Fig. 7, we plot the variations of the on state current \({I}_{\mathrm{ON}}\) versus the barrier height \(\Phi _{\mathrm{SB}}\). We note that as the value of the barrier height increases, the on-state current decreases exponentially, which affects directly the static performance of the SB-CNTFET.

Fig. 8
figure 8

The characteristics \({I}_{\mathrm{DS}}-{V}_{\mathrm{GS}}\) for two chirality values (13, 0) and (19, 0), and two values of \({V}_{\mathrm{DS}}\) (0.1 and 0.6 V)

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Figure 8 shows the drain current \(({I}_{\mathrm{DS}})\) versus the gate voltage \(({V}_{\mathrm{GS}})\) for SB-CNTFET with two different drain voltage \(({V}_{\mathrm{DS}})\), i.e. 0.1 V and 0.6 V and two chirality values (13, 0) and (19, 0). Since this type of device works on tunneling from SB, there is great influence of \({V}_{\mathrm{DS}}\) on transfer characteristics. As CNT is throughout intrinsic, it behaves as both n-type and p-type for both positive and negative gate bias (ambipolar effect). When the device works under the subthreshold region, \(({V}_{\mathrm{DS}} = 0.1\;\hbox {V})\) it works n-type as we increase gate bias. But when it works under the saturation region there is great influence of \({V}_{\mathrm{DS}}\). There is a very low \({I}_{\mathrm{OFF}}\) in this case, and the threshold voltage is much greater than the subthreshold region, which changes with increased \({V}_{\mathrm{DS}}\). It is also clear that increasing the diameter of the channel will increase the conduction current, which is expected because the rate of passage of electrons is proportional to the diameter of the carbon nanotube.

Fig. 9
figure 9

The characteristics \({I}_{\mathrm{ON}}/{I}_{\mathrm{OFF}}\) versus the EOT thickness

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Now in Fig. 9, we plot the variations of \({I}_{\mathrm{ON}}/{I}_{\mathrm{OFF}}\) ratio with respect to the variation of oxide EOT thickness. \({I}_{\mathrm{ON}}\) and \({I}_{\mathrm{OFF}}\) are calculated for a given \({V}_{\mathrm{GS}}\) and \({V}_{\mathrm{DS}}\). It can be seen from the figure that with decrement of EOT, the \({I}_{\mathrm{ON}}/{I}_{\mathrm{OFF}}\) ratio increases and lead to a high on-state current. This is associated with superior control of the gate voltage over the channel, which helps in reducing the off-state current. As the on-state current largely depends upon the higher energy level and flow of thermionic emission current, it becomes insensitive for device parameter compared to \({I}_{\mathrm{OFF}}\) [32]. We observe also an important linear variation of \({I}_{\mathrm{ON}}/{I}_{\mathrm{OFF}}\) ratio in the range 1 nm < EOT < 10 nm, this linear variation will be lower in the range 10 nm < EOT < 100 nm, which leads to an improvement of the static performance when the EOT thickness is lowered [32].

Fig. 10
figure 10

\({I}_{\mathrm{ON}}/{I}_{\mathrm{OFF}}\) ratio as a function of the power supply voltage for a SB-CNTFET with a CNT of thickness EOT \(=\) 0.1 nm and channel length Lg \(=\) 100 nm

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Figure 10 shows the \({I}_{\mathrm{ON}}/{I}_{\mathrm{OFF}}\) ratio as a function of the power supply voltage for a channel length Lg \(=\) 100 nm and EOT \(=\) 0.1 nm. From this figure we can observe that by increasing the power supply voltage, there is a linear evolution of the ratio \({I}_{\mathrm{ON}}/{I}_{\mathrm{OFF}}\), the slope decreases beyond 0.4 V, which explains the saturation current. When increasing power supply voltage, the off current, \({I}_{\mathrm{OFF}}\) increases. This phenomena is also reported experimentally by [15] and theoretically by atomistic scale simulations [19]. However, by increasing the power supply voltage, there is an increase on \({I}_{\mathrm{ON}}\). High \({I}_{\mathrm{OFF}}\) values are important for switching speed. Also, for low power applications, designers should take care of \({V}_{\mathrm{DD}}\) voltage to keep large \({I}_{\mathrm{ON}}/{I}_{\mathrm{OFF}}\) ratio. The off current increases faster with \({V}_{\mathrm{DD}}\) than the on-current [27].

In this section, we investigate the influence of physical and geometric parameters on the dynamic performance of the SB-CNTFET.

This study is limited to materials parameters (Schottky barrier \(\Phi _{\mathrm{SB}}\), the oxide layer EOT), and geometric parameters (the carbon nanotube diameter).

Fig. 11
figure 11

Transition frequency \({f}_{\mathrm{T}}\) as a function of the gate voltage for two Schottky barrier heights \(\Phi _{\mathrm{SB}}\) of 0.275 and 0 eV and \({V}_{\mathrm{DS}} = 0.3\;\hbox {V}\)

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Regarding Fig. 11, which plots the cut-off frequency as a function of gate voltage, we observe that decreasing the Schottky barrier height provides better dynamic performance. For the two barriers and for low values of \({V}_{\mathrm{GS}}\), the two curves of transition frequency show nearly linear characteristics, which verify formula (9) with \({c}_{\mathrm{g}}\) ineffective. For medium values of \({V}_{\mathrm{GS}}\), the \({c}_{\mathrm{g}}\) effect is dominant and the frequency \({f}_{\mathrm{T}}\) varies exponentially. For high values of \({V}_{\mathrm{GS}}\), the transition frequency \({f}_{\mathrm{T}}\) is almost constant. Furthermore, when the SB height is reduced, the better saturation of drain current gives a weaker dependence of \({f}_{\mathrm{T}}\) on \({V}_{\mathrm{DS}}\), and the behavior of \({f}_{\mathrm{T}}\) becomes similar to that of ohmic-contact CNTFE. However, thanks to smaller gate capacitance in Schottky devices, the maximum transition frequency \({f}_{\mathrm{Tmax}}\) is similar in ohmic and SB. In spite of drawbacks related to the ambipolar behavior (poor saturation of the \({I}_{\mathrm{DS}}-{V}_{\mathrm{DS}}\) characteristics and strong \({V}_{\mathrm{DS}}\) dependence of \({f}_{\mathrm{T}}\)), SB-CNTFET may provide performance similar to that of ohmic contact devices [33]. It makes thus possible to envision the design of new multi-function logic circuits taking advantage of ambipolar transport [34].

Fig. 12
figure 12

Maximum transition frequency \({f}_{\mathrm{Tmax}}\) as a function of chirality of 4 nanotube (10, 0), (13, 0), (16, 0) et (19, 0) for \(\Phi _{\mathrm{SB}} = 0.275\;\hbox {eV}\) and \({V}_{\mathrm{DS}} = 0.4\;\hbox {V}\)

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Figure 12 shows the variation of the maximum transition frequency \({f}_{\mathrm{Tmax}}\) as a function of four chirality: (10, 0), (13, 0), (16, 0), and (19, 0) at \({V}_{\mathrm{DS}} = 0,4\;\hbox {V}\). From the figure, it is clear that the transition frequency \({f}_{\mathrm{Tmax}}\) is almost two times higher for a chirality of (19, 0) than the chirality of (10, 0). Thus, the best transition frequencies are established for the large diameters of the nanotube. This variation of dynamic performances with the diameter of the nanotube contradicts with the static performances and the \({I}_{\mathrm{ON}}/{I}_{\mathrm{OFF}}\) ratio which deteriorate for the largest diameters of the nanotube. This shows that it is necessary to establish a compromise for the choice of the diameter of the nanotube [27].

5 Conclusion

In this paper, a study of the influence of the parameters of a compact model of a SB-CNTFET transistor (height of the Schottky barrier, equivalent oxide thickness and nanotube diameter) was carried out in order to show their impact on the static and dynamic performances of the SB-CNTFET transistor. The analysis indicates that the SB-CNTFET static performances are affected by the height of the barrier \(\Phi _{\mathrm{SB}}\). Other results show that the current \({I}_{\mathrm{OFF}}\) is increased with the increase of the diameter of SB-CNTFET. We also demonstrate that the \({I}_{\mathrm{ON}}/{I}_{\mathrm{OFF}}\) ratio is increased with the decrease of the EOT thickness, which enhances the static performances of SB-CNTFET.

Further results showed that the transition frequency \({f}_{\mathrm{T}}\) degrades with the increase of the Schottky barrier height. This is related to the quality of the \({I}_{\mathrm{D}}\) versus \({V}_{\mathrm{GS}}\) that is to say to the transconductance which is increasingly degraded for the heights of Schottky barriers. This shows the great importance of the restriction of this material parameter to keep the static and dynamic performances of SB-CNTFET very high. We also show that the decrease of oxide thickness makes the \({I}_{\mathrm{ON}}/{I}_{\mathrm{OFF}}\) ratio increases and lead to a high on-state current.

We also found that for CNTs with large diameters, the dynamic performances of the SB-CNTFET are enhanced (increase of the transition frequency) but the static performances (decrease of \({I}_{\mathrm{ON}}/{I}_{\mathrm{OFF}}\) ratio) are degraded. Consequently, the choice of the thin oxide and a compromise for the choice of the diameter of the nanotube may lead to high static and dynamic performances of SB-CNTFET.