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Theoretical calculations predict transition frequencies in the terahertz range for the field-effect transistors based on carbon nanotubes, and this shows their suitability for being used in high frequency applications. In this paper, we have designed a field-effect transistor based on carbon nanotube with high transition frequency suitable for ultra-wide band applications. We did this by optimizing nanotube diameter, gate insulator thickness and dielectric constant. As a result, we achieved the transition frequency about 7.45 THz. The environment of open source software FETToy is used to simulate the device. Also a suitable model for calculating the transition frequency is presented.

There is a tangible demand in the semiconductor industry for progressively diminution in the dimensions of electronic circuits, and consequent increase in the circuits speed due to RC delay reduction. But as the dimensions of circuits shrink, fabrication processes become more complicated. And silicon-based manufacturing technology encounters problems such as inefficiency of the Dennard’s method in design [

carriers (electrons and holes), the symmetry of the valence and conduction band, high electric conductivity, ability to withstand high current density and resistive to electron immigration, one-dimensional conduction (only forward and back scattering), nearly ballistic transport over less than 100 nm lengths, the adjustable energy gap seem to be a good candidate to replace silicon as the channel material [

Theoretical calculations predict transition frequencies in the terahertz range for the field-effect transistors based on carbon nanotubes. And this shows their suitability for being used in high frequency applications. It should be noted that measurements from manufactured samples show only a few tens of gigahertz transition frequency. These differences may be result from the relatively ideal model, weakness of the manufacturing process and measurement problems.

Several structures, for geometry of transistors based on carbon nanotubes, are suggested that each has its own advantages and disadvantages. One of the interesting structures is transistor with coaxial gate structure. With Coaxial gate structure, more gate control on the channel will be obtained [

Here, FETToy software was used for simulations. FETToy has been created to simulate carbon nanotube transistor with coaxial gate structure. An example of coaxial structure is shown in

In FETToy software, carrier transport is considered as ballistic, and circuit model has two capacitors, one is an electrostatic capacitor (C_{∑}), and the other a quantum capacitor (C_{Q}). According to the ballisticity of transistor model, capacitors are in farad per meter. An electrostatic capacitor is composed by a sum of the three capacitors, the gate capacitor (C_{G}), source capacitor (C_{S}) and drain capacitor (C_{D}) that are represented in _{scf} (self- consistent potential), is controlled by the gate, drain, and source potentials through the three capacitors shown. The mobile charge at the top of the barrier is determined by U_{scf} and by the location of the two Fermi levels. The nonlinear semiconductor (or quantum) capacitance is not shown explicitly but is implicit in the treatment of band filling [

Gate capacitor is a cylindrical capacitor and is calculated from Equation (1.a). Equation (1.b) shows relations between source capacitor, drain capacitor, gate capacitor and electrostatic capacitor in the FETToy model. We present _{G} is calculated regarding the nanotube diameter. The source capacitor and drain capacitor and electrostatic capacitor can be calculated from Equation (1.b).

FETToy software, alone, does not have the ability to calculate the transition frequency. Also, this software performs the calculations for a single point of in-

put parameters. The inputs of this software comprise insulator thickness, dielectric constant, nanotube diameter, device temperature, gate bias and drain bias and the number of the calculation points, the Fermi level of the source, effectiveness coefficient of the gate and drain capacitors on the electrostatic capacitor. To calculate the transition frequency, Equation (2.a) was used. In this equation

To evaluate the effect of nanotube diameter on the frequency behavior of the transistor in FETToy model, a transistor with length of 32 nm, insulator thickness of 1.5 nm, dielectric constant of 3.9, the source Fermi level of −0.32 eV and gate effectiveness coefficient of 0.88, at the temperature of 300˚K and at the bias of V_{GS} = 1 V and V_{DS} = 1 V is contemplated. Carbon nanotube diameter swept from 0.4 nm to 3.6 nm and the curves of the energy band gap, capacitors, transconductance and transition frequency have been achieved.

Changing the diameter of carbon nanotubes, affects two parts, one is the energy gap of the nanotube, and the other is electrostatic capacitor of the circuit, in cylindrical gate structure. Band gap of carbon nanotubes is driven by imposing boundary conditions on the graphene equation of the energy-momentum. Equation (3) displays, the approximate value of the energy gap for semiconducting carbon nanotubes, where d is the diameter of the nanotube [

In some geometrical structures of carbon nanotube transistor, changing the diameter of carbon nanotubes may lead to changes in the electrostatic capacitor of the circuit, too. In FETToy model, where gate capacitor is cylindrical in shape, with an increase in the diameter of carbon nanotube the surface area of the gate capacitor will increase. Equation (1.a) shows the formula of the gate capacitor for the cylindrical structure of the gate. In denominator of the Equation (1.a), expression

_{G}. In

diameter in FETToy is displayed. With increasing diameter from 0.4 nm to 3.6 nm, energy gap is reduced from 2.2 eV to 0.25 eV, as expected from Equation (3), and for the metallic cases energy gap is zero [

effect of changing the diameter. This graph is calculated for a transistor with a channel length of 32 nm. Results of FETToy Model indicate a maximum for f_{T} in diameter of 2.3 nm, with a value of 2.4 THz respectively. According to the Equa-

tion (2.a), for the FETToy, transition frequency depends directly to the transconductance, and inversely proportional to the effective capacitor of the circuit; as

For evaluating the effect of gate insulator on transistor frequency behavior in FETToy model, a transistor with the length of 32 nm, diameter of 1 nm, the source Fermi level of −0.32 eV, and gate effectiveness coefficient of 0.88, at temperature of 300˚K, and in bias of V_{GS} = 1 V and V_{DS} = 1 V is contemplated. in this simulation, dielectric constant and insulator thickness have been swept from 1 to 40 and from 1 nm to 5.5 nm respectively.

Equation (1.a) shows how to calculate the electrostatic gate capacitors per unit length for a cylindrical gate structure , that

The graphic on

the change of the gate insulator thickness for different dielectric constants, in FETToy model; where the quantum capacitors increases with increasing thickness of the insulator, this increase is more for smaller dielectric constants.

ness change for different dielectric constants in the FETToy model. transconductance decreases with increasing thickness of the insulator. This increase is lesser for smaller dielectric constants.

Property | Effects on property | |
---|---|---|

Dielectric constant increases | Insulator thickness increases | |

Electrostatic capacitor Quantum capacitor Effective capacitor Mutual conductance | Increase ( | Reduction ( |

to insulator thickness and dielectric constant in FETToy model. In

For designing a transistor with high transition frequency was necessary to calculate proper values of circuit parameters. For this purpose, in the simulation, each parameter is swept until the point, where the transition frequency is maximum, can be found [

In the FETToy code that has been developed in MATLAB, thickness and dielectric constant of the gate insulator, the nanotube diameter, bias, temperature, source Fermi level and gate effectiveness coefficient parameters are investigable. Each of these parameters has been swept and the appropriate values of them for achieving maximum transition frequency are selected. And a transistor has been designed based on these values. As from Equations (2.a), (2.b) and (2.c), it is clear that for assessing the effect of a parameter on the transition frequency it is appropriate that the effect of that on the transconductance and the capacitors should also be investigated.

The value of source Fermi level has been chosen to be −0.32 eV, that gives a threshold voltage of 0.25 V. In the case of gate effectiveness coefficient, its value can change from 0 to 1. Indeed the higher values of gate effectiveness coefficient show more gate control on channel, but in practice, it may be impossible to achieve the value 1. The value of gate effectiveness coefficient is assumed to be 0.88.

We investigated the effect of various parameters on the transition frequency of transistor.

Maximum Transition Frequency | Points | |
---|---|---|

Dielectric Thickness | Dielectric Constant | |

1.18 THz | 1.1 × 10^{−9} m 1.3 × 10^{−9} m 1.8 × 10^{−9} m 2.7 × 10^{−9} m 3 × 10^{−9} m 3.5 × 10^{−9} m 4.1 × 10^{−9} m 4.6 × 10^{−9} m 5 × 10^{−9} m | 8 9 10 12 13 14 15 16 17 |

The number of points with the maximum transition frequency in the FETToy model.

Parameter (change manner) | Effect of parameter on transition frequency | Proper value in CNTFETToy model |
---|---|---|

Bias (increase) Nanotube’s diameter Nanotube’s length (increase) Insulator thickness Dielectric constant Temperature (increase) α_{G} coefficient (increase) Source Fermi level | Increase Has a maximum in one point Decrease Has maximums Has maximums Decrease Increase Affects the threshold voltage | 1 V 2.3 nm 17 nm 4.1 nm 15 (Y_{2}O_{3}) 300˚K 0.95 −0.32 |

Summary of results.

Transistor | Transition frequency |
---|---|

This design Array CNTFET [ | 7.45 THz 5.5 GHz 110 GHz. μm (for lengh of 17 nm 6.47 THz) 8 GHz |

Comparison between performances of several transistors.

proper transistor can be designed using them. If based on the parameter values in

One issue, which should be noted about difference between practical and theoretical results, is that it is possible grow closer together these results with increasing the accuracy of models. For example, in FETToy model carrier scattering effect and the effect of chirality of the nanotubes, metallic nanotubes must be added to model and also model can be developed for array of nanotubes and multi-walled nanotubes. Also, defects in lattice structure, which depends on the manufacturing process are created and can also be added to the model for making results closer to the actual values [

Nouri-Bayat, R. and Kashani-Nia, A.R. (2017) Designing a Carbon Nanotube Field-Effect Transistor with High Transition Frequency for Ultra- Wideband Application. Engineering, 9, 22- 35. http://dx.doi.org/10.4236/eng.2017.91003