Abstract
Recent development of trilayer graphene nanoribbon Schottkybarrier fieldeffect transistors (FETs) will be governed by transistor electrostatics and quantum effects that impose scaling limits like those of Si metaloxidesemiconductor fieldeffect transistors. The current–voltage characteristic of a Schottkybarrier FET has been studied as a function of physical parameters such as effective mass, graphene nanoribbon length, gate insulator thickness, and electrical parameters such as Schottky barrier height and applied bias voltage. In this paper, the scaling behaviors of a Schottkybarrier FET using trilayer graphene nanoribbon are studied and analytically modeled. A novel analytical method is also presented for describing a switch in a Schottkycontact doublegate trilayer graphene nanoribbon FET. In the proposed model, different stacking arrangements of trilayer graphene nanoribbon are assumed as metal and semiconductor contacts to form a Schottky transistor. Based on this assumption, an analytical model and numerical solution of the junction current–voltage are presented in which the applied bias voltage and channel length dependence characteristics are highlighted. The model is then compared with other types of transistors. The developed model can assist in comprehending experiments involving graphene nanoribbon Schottkybarrier FETs. It is demonstrated that the proposed structure exhibits negligible shortchannel effects, an improved oncurrent, realistic threshold voltage, and opposite subthreshold slope and meets the International Technology Roadmap for Semiconductors nearterm guidelines. Finally, the results showed that there is a fast transient between onoff states. In other words, the suggested model can be used as a highspeed switch where the value of subthreshold slope is small and thus leads to less power consumption.
Keywords:
Trilayer graphene nanoribbon (TGN); ABA and ABC stacking; TGN Schottkybarrier FET; Highspeed switchBackground
Graphene, as a single layer of carbon atoms with hexagonal symmetry and different types such as monolayer, bilayer, trilayer, and multilayers, has attracted new research attention. Very high carrier mobility can be achieved from graphenebased materials which makes them a promising candidate for nanoelectronic devices [1,2]. Recently, electron and hole mobilities of a suspended graphene have reached as high as 2 × 10^{5} cm^{2}/V·s [3]. Also, ballistic transport has been observed at room temperature in these materials [3]. Layers of graphene can be stacked differently depending on the horizontal shift of graphene planes [4,5]. Every individual multilayer graphene sequence behaves like a new material, and different stacking of graphene sheet lead to different electronic properties [3,6,7]. In addition, the configuration of graphene layers plays a significant role to realize either metallic or semiconducting electronic behavior [4,8,9].
Trilayer graphene nanoribbon (TGN), as a onedimensional (1D) material, is the focus of this study. The quantum confinement effect will be assumed in two directions. In other words, only one Cartesian direction is greater than the de Broglie wavelength (10 nm). As shown in Figure 1a, because of the quantum confinement effect, a digital energy is taken in the y and z directions, while an analog type in the x direction. It is also remarkable that the electrical property of TGN is a strong function of interlayer stacking sequences [10]. Two wellknown forms of TGN with different stacking manners are understood as ABA (Bernal) and ABC (rhombohedral) [11]. The simplest crystallographic structure is hexagonal or AA stacking, where each layer is placed directly on top of another; however, it is unstable. AB (Bernal) stacking is the distinct stacking structure for bilayers. For trilayers, it can be formed as either ABA, as shown in Figure 1, or ABC (rhombohedral) stacking [1,12]. Bernal stacking (ABA) is a common hexagonal structure which has been found in graphite. However, some parts of graphite can also have a rhombohedral structure (the ABC stacking) [6,13]. The band structure of ABAstacked TGNs can be assumed as a hybrid of monolayer and bilayer graphene band structures. The perpendicular external applied electric or magnetic fields are expected to induce band crossing variation in Bernalstacked TGNs [1416]. Figure 1 indicates that the graphene plane being a twodimensional (2D) honeycomb lattice is the origin of the stacking order in multilayer graphene with A and B and two nonequivalent sublattices.
As shown in Figure 1, a TGN with ABA stacking has been modeled in the form of three honeycomb lattices with pairs of equivalent sites as {A_{1},B_{1}}, {A_{2},B_{2}}, and {A_{3},B_{3}} which are located in the top, center, and bottom layers, respectively [11]. An effectivemass model utilizing the SlonczewskiWeissMcClure parameterization [17] has been adopted, where every parameter can be compared with a relevant parameter in the tightbinding model. The stacking order is related to the electronic lowenergy structure of 3D graphitebased materials [18,19]. Interlayer coupling has been found to also affect the device performance, which can be decreased as a result of mismatching the AB stacking of the graphene layers or rising the interlayer distance. A weaker interlayer coupling may lead to reduced energy spacing between the subbands and increased availability of more subbands for transfer in the lowenergy array. Graphene nanoribbon (GNR) has been incorporated in different nanoscale devices such as interconnects, electromechanical switches, Schottky diodes, tunnel transistors, and fieldeffect transistors (FETs) [2024]. The characteristics of the electron and hole energy spectra in graphene create unique features of graphenebased Schottky transistors. Recently, the fabrication and experimental studies as well as a hypothetical model of GSchottky transistors have been presented [25]. The studies have focused towards the properties of TGN, and a tunable threelayer graphene singleelectron transistor was experimentally realized [6,26].
In this paper, a model for TGN Schottkybarrier (SB) FET is analyzed which can be assumed as a 1D device with width and thickness less than the de Broglie wavelength. The presented analytical model involves a range of nanoribbons placed between a highly conducting substrate with the back gate and the top gate controlling the sourcedrain current. The Schottky barrier is defined as an electron or hole barrier which is caused by an electric dipole charge distribution related to the contact and difference created between a metal and semiconductor under an equilibrium condition. The barrier is found to be very abrupt at the top of the metal due to the charge being mostly on the surface [2731]. TGN with different stacking sequences (ABA and ABC) indicates different electrical properties, which can be used in the SB structure. This means that by engineering the stack of TGN, Schottky contacts can be designed, as shown in Figure 2. Between two different arrangements of TGN, the semiconducting behavior of the ABA stacking structure has turned it into a useful and competent channel material to be used in Schottky transistors [32].
Figure 2. Schematic of TGN SB contacts.
In fact, the TGN with ABC stacking shows a semimetallic behavior, while the ABAstacked TGN shows a semiconducting property [32]. A schematic view of TGN SB FET is illustrated in Figure 3, in which ABAstacked TGN forms the channel between the source and drain contacts. The contact size has a smaller effect on the doublegate (DG) GNR FET compared to the singlegate (SG) FET.
Figure 3. Schematic representation of TGN SB FET.
Due to the fact that the GNR channel is sandwiched or wrapped through by the gate, the field lines from the source and drain contacts were seen to be properly screened by the gate electrodes, and therefore, the source and drain contact geometry has a lower impact. The operation of TGN SB FET is followed by the creation of the lateral semimetalsemiconductorsemimetal junction under the controlling top gate and relevant energy barrier.
Methods
TGN SB FET model
The scaling behaviors of TGN SB FET are studied by selfconsistently solving the energy band structure equation in an atomistic basis set. In order to calculate the energy band structure of ABAstacked TGN, the spectrum of full tightbinding Hamiltonian technique has been adopted [3337]. The presence of electrostatic fields breaks the symmetry between the three layers. Using perturbation theory [38] in the limit of υ_{F}k « V « t_{⊥} gives the electronic band structure of TGN as [35,39]
where k is the wave vector in the x direction, , t_{⊥} is the hopping energy, ν_{f} is the Fermi velocity, and V is the applied voltage. The response of ABAstacked TGN to an external electric field is different from that of mono or bilayer graphene. Rather than opening a gap in bilayer graphene, this tuned the magnitude of overlap in TGN. Based on the energy dispersion of biased TGN, wave vector relation with the energy (Ek relation) shows overlap between the conduction and valence band structures, which can be controlled by a perpendicular external electric field [6,39]. The band overlap increases with increasing external electric field which is independent of the electric field polarity. Moreover, it is shown that the effective mass remains constant when the external electric field is increased [3,33]. As an essential parameter of TGNs, density of states (DOS) reveals the availability of energy states, which is defined as in [40,41]. To obtain this amount, derivation of energy over the wave vector is required. Since DOS shows the number of available states at each energy level which can be occupied, therefore, DOS, as a function of wave vector, can be modeled as [39]
where E is the energy band structure and A, B, C, D, and F are defined as A = −6.2832α, B = 14.3849α^{2}β, , D = −9β^{2}, and . As shown in Figure 4, the DOS for ABAstacked TGN at room temperature is plotted. As illustrated, the lowDOS spectrum exposes two prominent peaks around the Fermi energy [39].
Figure 4. The DOS of the TGN with ABA stacking.
The electron concentration is calculated by integrating the Fermi probability distribution function over the energy as in [42]. Biased ABAstacked TGN carrier concentration is modified as [43]
where , the normalized Fermi energy is , and M and N are defined as and . Based on this model, ABAstacked TGN carrier concentration is a function of normalized Fermi energy (η). The conductance of graphene at the Dirac point indicates minimum conductance at a charge neutrality point which depends on temperature. For a 1D TGN FET, the GNR channel is assumed to be ballistic. The current from source to drain can be given by the Boltzmann transport equation in which the Landauer formula has been adopted [44,45]. The number of modes in corporation with the Landauer formula indicates conductance of TGN that can be written as [32]
where the momentum (k) can be derived by using Cardano's solution for cubic equations [46]. Equation 4 can be assumed in the form G = N_{1}G_{1}+ N_{2}G_{2}, where N_{1} = 2αq^{2}/lh and N_{2} = −6βq^{2}/lh. Since G_{1} is an odd function, its value is equivalent to zero. Therefore, G = N_{2}G_{2}[32], where
This equation can be numerically solved by employing the partial integration method and using the simplification form, where x = (E − Δ)/k_{B}T and η = (E_{F} − Δ)/k_{B}T. Thus, the general conductance model of TGN will be obtained [32] as
It can be seen that the conductivity of TGN increases by raising the magnitude of gate voltage. In the Schottky contact, electrons can be injected directly from the metal into the empty space in the semiconductor. When electrons flow from the valence band of the semiconductor into the metal, there would be a result similar to that for holes injected into the semiconductor. So, the establishment of an excess minority carrier hole in the vicinity is observed [28]. The current moves mainly from the drain to the source which consists of both drift and diffusion currents. The created 2D anticipated framework is expected to cause an explicit analytical current equation in the subthreshold system. Considering the weak inversion region, the diffusion current is mainly dominated and relative to the electron absorption at the virtual cathode [47]. A GNR FET is a voltagecontrolled tunnel barrier device for both the Schottky and doped contacts.
The drain current through the barrier consists of thermal and tunneling components [48]. The effect of quantum tunneling and electrostatic short channel is not treated, which makes it difficult to study scaling behaviors and ultimate scaling limits of GNR SB FET where the tunneling effect cannot be ignored [20]. The tunneling current is the main component of the whole current which requires the use of the quantum transport. Close to the source within the band gap, carriers are injected into the channel from the source [49]. In fact, the tunneling current plays a very important role in a Schottky contact device.
The proposed model includes tunneling current through the SB at the contact interfaces, appropriately capturing the impact of arbitrary electrical and physical factors. The behavior of the proposed transistor over the threshold region is obtained by modulating the tunneling current through the SBs at the two ends of the channel [20]. The effect of charges close to the source for a SB FET is more severe because they have a significant effect on the SB and the tunneling possibility. When the charge impurity is situated at the center of the channel of a SB FET, the electrons are trapped by the positive charge and the sourcedrain current is decreased. If the charges are situated close to the drain, the electrons will collect near the drain. In this situation, low charge density near the source decreases the potential barrier at the beginning of the channel, which opens up the energy gap more for the flow of electrons from the source to the channel [50].
Electrons moving from the metal into the semiconductor can be defined by the electron current density J_{m→s}, whereas the electron current density J_{s→m} refers to the movement of electrons from the semiconductor into the metal. What determines the direction of electron flow depends on the subscripts of the current. In other words, the conventional current direction is opposite to the electron flow. J_{s→m} is related to the concentration of electrons with velocity in the
 x
where e is the magnitude of the electronic charge and ν_{x} is the carrier velocity in the direction of transport:
High carrier mobility reported from experiments on graphene leads to assume a complete ballistic carrier transport in the TGN, which means that the average probability of injected electron at one end that will transmit to the other end is approximately equal to 1:
Kinetic energy, as a main parameter, is considered over the Fermi level, and the current densityvoltage response of the TGN SB FET device is determined with respect to the carrier density and its kinetic energy as
where (V_{A} is the applied bias voltage and V_{T} is the thermal voltage) [51]. The dependence of the drain current on the drainsource voltage is associated with the dependence of η on this voltage given by
where V_{GT} = V_{GS} − V_{T} and V(y) is the voltage of channel in the y direction. By solving Equation 11, the normalized Fermi energy can be defined as
In order to obtain an analytical relation for the contact current, an explicit analytical equation for the electric potential distribution along the TGN is presented. The channel current is analytically derived as a function of various physical and electrical parameters of the device including effective mass, length, temperature, and applied bias voltage. According to the relationship between a current and its density, the current–voltage response of a TGN SB FET, as a main characteristic, is modeled as
where l is the length of the channel.
Results and discussion
In this section, the performance of the Schottkycontact doublegate TGN FET is studied. A novel analytical method is introduced to achieve a better understanding of the TGN SB switch devices. The results will be applied to identify how various device geometries provide different degrees of controlling transient between onoff states. The numerical solution of the presented analytical model in the preceding section was employed, and rectification current–voltage characteristic of TGN SB FET is plotted as shown in Figure 5.
Figure 5. Simulated I_{D }(μA) versus V_{DS }(V) plots of TGN Schottkybarrier FET (L = 25 nm, V_{GS }= 0.5 V).
It further revealed that the engineering of SB height does not alter the qualitative ambipolar feature of the current–voltage characteristic whenever the gate oxide is thin. The reason is that the gate electrode could perfectly screen the field from the drain and source for a thin gate oxide (less than 10 nm). The SB whose thickness is almost the same as the gate insulator diameter is almost transparent. However, the ambipolar current–voltage (IV) characteristic cannot be concealed by engineering the SB height when the gate insulator is thin. Lowering the gate insulator thickness and the contact size leads to thinner SBs and also greater oncurrent. Since the SB height is half of the band gap, the minimum currents exist at the gate voltage of V_{G,min} = 1/2V_{D}, at which the conduction band that bends at the source extreme of the channel is symmetric to the valence band and also bends at the drain end of the channel, while the electron current is the same as the hole current. The consequence of attaining the least leakage current is the same as TGN SB FET with middlegap SBs [23]. Raising the drain voltage leads to an exponential increase of the minimal leakage current which shows the importance of proper designing of the power supply voltage to ensure small leakage current. As depicted in Figure 6, the proposed model points out strong gatesource voltage dependence of the current–voltage characteristic showing that the V_{GS} increment effect will influence the drain current. In other words, as V_{GS} increases, a greater value of I_{D} results. As the drain voltage rises, the voltage drop through the oxide close to the drain terminal reduces, and this shows that the induced inversion charge density close to the drain also decreases [28]. The slope of the I_{D} versus V_{DS} curve will reduce as a result of the decrease in the incremental conductance of the channel at the drain. This impact is indicated in the I_{D}V_{DS} curve in Figure 6. If V_{DS} increases to the point that the potential drop across the oxide at the drain terminal is equal to V_{T}, the induced inversion charge density is zero at the drain terminal. At that point, the incremental conductance at the drain is nil, meaning that the slope of the I_{D}V_{DS} curve is zero. We can write
where V_{DS} (sat) is the draintosource voltage which is creating zero inversion charge density at the drain terminal. When V_{DS} is more than the V_{DS} (sat) value, the point in the channel where the inversion charge is zero moves closer to the source terminal [28]. In this case, electrons move into the channel at the source and pass through the channel towards the drain, and then at that point when the charge goes to zero, the electrons are infused into the space charge region where they are swept by the Efield to the drain contact. Compared to the original length L, the change in channel length ΔL is small, then the drain current will be regular for V_{DS} > V_{DS} (sat). The region of the I_{D}V_{DS} characteristic is referred to as the saturation region. When V_{GS} is changed, the I_{D}V_{DS} curve will also be changed. It was found that if V_{GS} increases, the initial slope of I_{D}V_{DS} will be raised. We can also infer from Equation 14 that the value of V_{DS} (sat) is a function of V_{GS}. A family of curves is created for this nchannel enhancementmode TGN SB FET, as shown in Figure 6.
Figure 6. I_{D }(μA)V_{DS }(V) characteristic of TGN SB FET at different values of V_{GS }for L = 100 nm.
Also, it can be seen that by increasing V_{GS}, the saturation current increases, showing the fact that a larger voltage drop occurs between the gate and the source contact. Also, there is a bigger energy value for carrier injection from the source contact channel [20]. The impact of power supply upscaling decreases the SB length at the drain side, allowing it to be more transparent and resulting in more turnon current to flow. Therefore, an acceptable performance comparable to the conventional behavior of a Schottky transistor is obtained. The scaling of the channel length improves gate electrostatic control, creating larger transconductance and smaller subthreshold swings. The effect of the channel length scaling on the IV characteristic of TGN SB FET is investigated in Figure 7. It shows a similar trend when the gatesource voltage is changed. It can be seen that the drain current rises substantially as the length of the channel is increased from 5 to 50 nm.
Figure 7. Impact of the channel length scaling on the transfer characteristic for V_{GS }= 0.5 V.
To get a greater insight into the effect of increasing channel length on the increment of the drain current, two important factors, which are the transparency of SB and the extension of the energy window for carrier concentration, play a significant role [49,50]. For the first parameter, as the SB height and tunneling current are affected significantly by the charges close to the source of SB FET, the channel length effect on the drain current through the SB contact is taken into account in our proposed model. Moreover, when the center of the channel of the SB FET is unoccupied with the charge impurities, the drainsource current increases because of the fact that free electrons are not affected by positive charges [49]. The effect of the latter parameter appears at the beginning of the channel where the barrier potential decreases as a result of low charge density near the source. This phenomenon leads to widening the energy window and ease of electron flow from the source to the channel [50]. Furthermore, due to the long mean free path of GNR [5255], the scattering effect is not dominant; therefore, increasing the channel length will result in a larger drain current.
For a channel length of 5 nm, direct tunneling from the source to drain results in a larger leakage current, and the gate voltage may rarely adjust the current. The transistor is too permeable to have a considerable disparity among onoff states. For a channel length of 10 nm, the drain current has improved to about 1.3 mA. The rise in the drain current is found to be more significant for channel lengths higher than 20 nm. That is, by increasing the channel length, there is a dramatic rise in the initial slope of I_{D} versus V_{DS}. Also, based on the subthreshold slope model and the following simulated results, a faster device with opposite subthreshold slope or high on/off current ratio is expected. In other words, it can be concluded that there is a fast transient between onoff states. Increasing the channel length to 50 nm resulted in the drain current to increase by about 6.6 mA. The operation of the stateoftheart shortchannel TGN SB FET is found to be near the ballistic limit. Increasing further the channel length hardly changes neither the oncurrent or offcurrent nor the on/off current ratio. However, for a conventional metaloxidesemiconductor fieldeffect transistor (MOSFET), raising the channel length may result in the channel resistance to proportionally increase. Therefore, in this case, downscaling the channel length will result in significant loss of the on/off current ratio as compared to the SG device.
Figure 8 shows a comparative study of the presented model and the typical IV characteristics of other types of transistors [49,50]. As depicted in Figure 8, the proposed model has a larger drain current than those transistors for some value of the drainsource voltages. The resultant characteristics of the presented model shown in Figure 8 are in close agreement with published results [49,50]. In Figure 8, DG geometry is assumed for the simulations instead of the SG geometry type.
Figure 8. Comparison between proposed model and typical IV characteristics of other types of transistors. (a) MOSFET with SiO_{2} gate insulator [50] (V_{GS} = 0.5V), (b) TGN MOSFET with an ionic liquid gate, C_{ins} >> C_{q}[49] (V_{GS} = 0.5 V), (c) TGN MOSFET with a 3nm ZrO_{2} wrap around gate, C_{ins}~ C_{q}[49] (V_{GS} = 0.37 V), (d) TGN MOSFET with a 3nm ZrO_{2} wrap around gate, C_{ins} ~ C_{q}[49] (V_{GS} = 0.38 V).
In order to have a deep quantitative understanding of experiments involving GNR FETs, the proposed model is intended to aid in the design of such devices. The SiO_{2} gate insulator is 1.5 nm thick with a relative dielectric constant K = 3.9 [50] (Figure 8a). Furthermore, the gatetochannel capacitance C_{g} is a serial arrangement of insulator capacitance C_{ins} and quantum capacitance C_{q} (equivalent to the semiconductor capacitance in conventional MOSFETs). Figure 8b shows a comparative study of the presented model and the typical IV characteristic of a TGN MOSFET with an ionic liquid gate. The availability of the ionic liquid gating [49] that can be modeled as a wraparound gate of a corresponding oxide thickness of 1 nm and a dielectric constant ε_{r} = 80 results in C_{ins} >> C_{q}, and MOSFETs function close to the quantum capacitance limit, i.e., C_{g} ≈ C_{q}[49]. As depicted in Figure 8c,d, the comparison study of the proposed model with a TGN MOSFET with a 3nm ZrO_{2} wraparound gate for two different values of V_{GS} is notable. A 3nm ZrO_{2} (ε_{r} = 25) wraparound gate has C_{ins} comparable to C_{q} for solidstate highκ gating, and this is an intermediate regime among the MOSFET limit and C_{q} limit.
Recently, a performance comparison between the GNR SB FETs and the MOSFETlikedoped sourcedrain contacts has been carried out using selfconsistent atomistic simulations [20,21,4850,56,57]. The MOSFET demonstrates improved performance in terms of bigger oncurrent, larger on/off current ratio, larger cutoff frequency, smaller intrinsic delay, and better saturation behavior [21,50]. Disorders such as edge roughness, lattice vacancies, and ionized impurities have an important effect on device performance and unpredictability. This is because the sensitivity to channel atomistic structure and electrostatic environment is strong [50]. However, the intrinsic switching speed of the GNR SB FET is several times faster than that of the Si MOSFETs. This could lead to promising highspeed electronics applications, where the large leakage of the GNR SB FET is of fewer concerns [20]. An efficient functionality of the transistor with a doped nanoribbon has been noticed in terms of on/off current ratio, intrinsic switching delay, and intrinsic cutoff frequency [48].
Based on the presented model, comparable with the other experimental and analytical models, the onstate current of the MOSFETlike GNR FET is 1 order of magnitude higher than that of the TGN SB FET. This is because the gate voltage ahead of the sourcechannel flat band condition modulates both the thermal and tunnel components in the onstate of MOSFETlike GNR FET, while it modulates the tunnel barrier only of the metal Schottkycontact TGN FET that limits the onstate current. Furthermore, TGN SB FET device performance can be affected by interlayer coupling, which can be decreased by raising the interlayer distance or mismatching the AB stacking of the graphene layers.
It is also noteworthy that MOSFETs operate in the region of subthreshold (weak inversion) as the magnitude of V_{GS} is smaller than that of the threshold voltage. In the weak inversion mode, the subthreshold leakage current is principally as a result of carriers' diffusion [58,59]. The offstate current of the transistor (I_{OFF}) is the drain current when V_{GS} = 0. The offstate current is affected by some parameters such as channel length, channel width, depletion width of the channel, gate oxide thickness, threshold voltage, channelsource doping profiles, drainsource junction depths, supply voltage, and junction temperature [59].
Shortchannel effects are defined as the results of scaling the channel length on the subthreshold leakage current and threshold voltage. The threshold voltage is decreased by reducing the channel length and drainsource voltage [5861]. In the subthreshold region, the gate voltage is approximately linear [58,59]. It has been studied that the decrease of channel length and drainsource voltage results in shifting the characteristics to the left, and it is obvious that as the channel length gets less than 10 nm, the subthreshold current increases dramatically [62]. Based on the International Technology Roadmap for Semiconductors (ITRS) nearterm guideline for lowstandbypower technology, the value of the threshold voltage is close to 0.3 V [59]. Figure 9 illustrates the subthreshold regime of TGN SB FET at different values of drainsource voltage. As shown in this figure, for lower values of drainsource voltage, the threshold voltage is decreased and meets the guidelines of ITRS.
Figure 9. Subthreshold regime of TGN SB FET at different values of V_{DS }(V) for L = 25 nm.
The subthreshold slope, S (mV/decade), is evaluated by selecting two points in the subthreshold region of an I_{D}V_{GS} graph as the subthreshold leakage current is adjusted by a factor of 10. It has been noted that selfconsistent electrostatics and the gate biasdependent electronic structure have an essential role in determining the intrinsic limits of the subthreshold slope in a TGN SB FET, which stays well over the Boltzmann limit of the ideal value of 60 mV/decade or less than 85mV/decade [58,63].The subthreshold slope, as one of the key issues of deepsubmicrometer devices, is defined as [59]
where V_{t} is the threshold voltage, V_{off} is the off voltage of the device, I_{vt} is the drain current at threshold, and I_{off} is the current at which the device is off. In other words, the subthreshold slope delineates the inverse slope of the log (I_{D}) versus V_{GS} plotted graph as illustrated in Figure 10.
Figure 10. I_{D }(μA)V_{GS }(V) characteristic of TGN SB FET at different values of V_{DS}.
Average subthreshold swing is a fundamental parameter that influences the performance of the device as a switch. According to Figure 10, the subthreshold slope for (l = 100 nm) is obtained as shown in Table 1.
Table 1. Subthreshold slope of TGN SB FET at different values of V_{DS}
Based on data from [64], for the effective channel lengths down to 100 nm, the calculated and simulated subthreshold slope values are near to the classical value of approximately 60 mV/decade. The subthreshold slope can be enhanced by decreasing the value of the buried oxide capacitance C_{BOX} or by increasing the value of the gate oxide capacitance C_{GOX}[64]. Based on the simulated results, it can be concluded that when the channel material is replaced by TGN, the subthreshold swing improves further.
The comparison study between the presented model with data from [62,64] showed that due to the quantum confinement effect [39,43], the value of the subthreshold slope in the case of TGN SB FET is less than those of DG metal oxide semiconductor and vertical silicononnothing FETs [62,64] for some values of drainsource voltage. A nanoelectronic device characterized by a steep subthreshold slope displays a faster transient between onoff states. A small value of S denotes a small change in the input bias which can modulate the output current and thus leads to less power consumption. In other words, a transistor can be used as a highspeed switch when the value of S is small. As a result, the proposed model can be applied as a useful tool to optimize the TGN SB FETbased device performance. It showed that the shortening of the top gate may lead to a considerable modification of the TGN SB FET current–voltage properties. In fact, it also paves a path for future design of the TGN SB devices.
Conclusions
TGN with different stacking arrangements is used as metal and semiconductor contacts in a Schottky transistor junction. The ABAstacked TGN in the presence of an external electric field is also considered. Based on this configuration, an analytical model of junction current–voltage characteristic of TGN SB FET is presented. The dependence of the drain current versus the drainsource voltage of TGN SB FET as well as the backgate and topgate voltages for different values of gatesource voltage and geometric parameters such as channel length are calculated. In particular, we conclude that by increasing the applied voltage and also channel length, the drain current increases, which showed better performance in comparison with the typical behavior of other kinds of transistors. Finally, a comparative study of the presented model with MOSFET with a SiO_{2} gate insulator, a TGN MOSFET with an ionic liquid gate, and a TGN MOSFET with a ZrO_{2} wraparound gate was presented. The proposed model is also characterized by a steep subthreshold slope, which clearly gives an illustration of the fact that the TGN SB FET shows a better performance in terms of transient between offon states. The obtained results showed that due to the superior electrical properties of TGN such as high mobility, quantum transport, 1D behaviors, and easy fabrication, the suggested model can give better performance as a highspeed switch with a low value of subthreshold slope.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
MR wrote the manuscript, contributed to the design of the study, performed all the data analysis, and participated in the MATLAB simulation of the proposed device. Prof. RI and Dr. MTA participated in the conception of the project, improved the manuscript, and coordinated between all the participants. HK, MS, and EA organized the final version of the cover letter. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to acknowledge the financial support from a Research University grant of the Ministry of Higher Education (MOHE), Malaysia, under Projects Q.J130000.7123.02H24, PY/2012/00168, and Q.J130000.7123.02H04. Also, thanks to the Research Management Center (RMC) of Universiti Teknologi Malaysia (UTM) for providing excellent research environment in which to complete this work.
References

Mak KF, Shan J, Heinz TF: Electronic structure of fewlayer graphene: experimental demonstration of strong dependence on stacking sequence.
Phys Rev Lett 2010, 104:176404. PubMed Abstract  Publisher Full Text

Rahmani M, Ahmadi MT, Kiani MJ, Ismail R: Monolayer graphene nanoribbon pn junction.

Craciun MF, Russo S, Yamamoto M, Oostinga JB, Morpurgo AF, Tarucha S: Trilayer graphene is a semimetal with a gatetunable band overlap.
Nat Nanotechnol 2009, 4:383388. PubMed Abstract  Publisher Full Text

Berger C, Song Z, Li T, Li X, Ogbazghi AY, Feng R, Dai Z, Marchenkov AN, Conrad EH, First PN, de Heer WA: Ultrathin epitaxial graphite: 2D electron gas properties and a route toward graphenebased nanoelectronics.
J Phys Chem B 2004, 108:1991219916. Publisher Full Text

Nirmalraj PN, Lutz T, Kumar S, Duesberg GS, Boland JJ: Nanoscale mapping of electrical resistivity and connectivity in graphene strips and networks.
Nano Letters 2011, 11:1622. PubMed Abstract  Publisher Full Text

Avetisyan AA, Partoens B, Peeters FM: Stacking order dependent electric field tuning of the band gap in graphene multilayers.

Warner JH: The influence of the number of graphene layers on the atomic resolution images obtained from aberrationcorrected high resolution transmission electron microscopy.
Nanotechnology 2010, 21:255707. PubMed Abstract  Publisher Full Text

Zhu W, Perebeinos V, Freitag M, Avouris P: Carrier scattering, mobilities, and electrostatic potential in monolayer, bilayer, and trilayer graphene.

Sutter P, Hybertsen MS, Sadowski JT, Sutter E: Electronic structure of fewlayer epitaxial graphene on Ru(0001).
Nano Letters 2009, 9:26542660. PubMed Abstract  Publisher Full Text

Shengjun Y, Raedt HD, Katsnelson MI: Electronic transport in disordered bilayer and trilayer graphene.

Koshino M: Interlayer screening effect in graphene multilayers with ABA and ABC stacking.

Zhang F, Sahu B, Min H, MacDonald AH: Band structure of ABCstacked graphene trilayers.

Lu CL, Lin HC, Hwang CC, Wang J, Lin MF, Chang CP: Absorption spectra of trilayer rhombohedral graphite.
Appl Phys Lett 2006, 89:221910. Publisher Full Text

Xiao YM, Xu W, Zhang YY, Peeters FM: Optoelectronic properties of ABCstacked trilayer graphene.

Rutter GM, Crain J, Guisinger N, First PN, Stroscio JA: Optoelectronic properties of ABCstacked trilayer graphene.
J Vac Sci Technol A 2008, 26:938943. Publisher Full Text

Russo S, Craciun MF, Yamamoto M, Tarucha S, Morpurgo AF: Doublegated graphenebased devices.

Koshino M, McCann E: Gateinduced interlayer asymmetry in ABAstacked trilayer graphene.

Craciun MF, Russo S, Yamamoto M, Tarucha S: Tuneable electronic properties in graphene.
NanoToday Press 2011, 6:4260. Publisher Full Text

Appenzeller J, Sui Y, Chen Z: Graphene nanostructures for device applications. In Digest of Technical Papers on 2009 Symposium on VLSI Technology: June 16–18 2009; Honolulu. Piscataway: IEEE; 2009:124126.

Ouyang Y, Yoon Y, Guo J: Scaling behaviors of graphene nanoribbon FETs: a threedimensional quantum simulation study.

Yoon Y, Fiori G, Hong S, Lannaccone G, Guo J: Performance comparison of graphene nanoribbon FETs with Schottky contacts and doped reservoirs.

Zhang Q, Fang T, Xing H, Seabaugh A, Jena D: Graphene nanoribbon tunnel transistors.

Naeemi A, Meindl JD: Conductance modeling for graphene nanoribbon (GNR) interconnects.

Liang Q, Dong J: Superconducting switch made of graphene–nanoribbon junctions.
Nanotechnology 2008, 19:355706. PubMed Abstract  Publisher Full Text

Zhu J: A novel graphene channel field effect transistor with Schottky tunneling source and drain. In Proceedings of the ESSDERC 2007: 37th European Solid State Device Research Conference, 2007: September 11–13 2007; Munich. Piscataway: IEEE; 2007:243246.

Guettinger J, Stampfer C, Molitor F, Graf D, Ihn T, Ensslin K: Coulomb oscillations in threelayer graphene nanostructures.
New J Phys 2008, 10:125029. Publisher Full Text

Rahmani M, Ahmadi MT, Ismail R, Ghadiry MH: Performance of bilayer graphene nanoribbon Schottky diode in comparison with conventional diodes.
J Comput Theor Nanosci 2013, 10:15. Publisher Full Text

Neamen DA: Semiconductor Physics and Devices. 3rd edition. New York: McGrawHill; 2003.

Kargar A, Lee C: Graphene nanoribbon schottky diodes using asymmetric contacts. In Proceedings of the IEEENANO2009: 9th Conference on Nanotechnology, 2009: July 26–30 2009; Genoa. Piscataway: IEEE; 2009:243245.

Jimenez D: A current–voltage model for Schottkybarrier graphene based transistors.
Nanotechnology 2008, 19:345204. PubMed Abstract  Publisher Full Text

Ahmadi MT, Rahmani M, Ghadiry MH, Ismail R: Monolayer graphene nanoribbon homojunction characteristics.
Sci Adv Mater 2012, 4:753756. Publisher Full Text

Sadeghi H, Ahmadi MT, Mousavi M, Ismail R: Channel conductance of ABA stacking trilayer graphene field effect transistor.
Mod Phys Lett B 2012, 26:1250047. Publisher Full Text

Avetisyan AA, Partoens B, Peeters FM: Electricfield control of the band gap and Fermi energy in graphene multilayers by top and back gates.

McCann E, Koshino M: Spinorbit coupling and broken spin degeneracy in multilayer graphene.

Guinea F, Castro Neto AH, Peres NMR: Electronic states and Landau levels in graphene stacks.

Latil S, Meunier V, Henrard L: Massless fermions in multilayer graphitic systems with misoriented layers: ab initio calculations and experimental fingerprints.

Castro EV, Novoselov KS, Morozov SV, Peres NMR, Santos JMB L, Nilsson J, Guinea F, Geim AK, Castro AH: Electronic properties of a biased graphene bilayer.
J Phys Condens Matter 2010, 22:175503. PubMed Abstract  Publisher Full Text

Kato T: Perturbation Theory for Linear Operators. Berlin: Springer; 1995:132.

Rahmani M, Ahamdi MT, Ghadiry MH, Anwar S, Ismail R: The effect of applied voltage on the carrier effective mass in ABA trilayer graphene nanoribbon.
Comput Theor Nanosci 2012, 9:14. Publisher Full Text

Guinea F, Castro Neto AH, Peres NMR: Interaction effects in single layer and multilayer graphene.
Eur Phys J Spec Top 2007, 148:117125. Publisher Full Text

Krompiewski S: Ab initio studies of NiCuNi trilayers: layerprojected densities of states and spinresolved photoemission spectra.
J Phys Condens Matter 1998, 10:9663. Publisher Full Text

Arora VK: Failure of Ohm's law: its implications on the design of nanoelectronic devices and circuits. In Proceedings of the 2006 25th IEEE International Conference on Microelectronics: May 14–17 2006; Belgrade. Piscataway: IEEE; 2006:1522.

Rahmani M, Ahmadi MT, Ismail R, Ghadiry MH: Quantum confinement effect on trilayer graphene nanoribbon carrier concentration.
J Exp Nanosci
in press

Kumar SB, Guoa J: Chiral tunneling in trilayer graphene.
Appl Phys Lett 2012, 100:163102. Publisher Full Text

Datta S: Electronic Transport in Mesoscopic Systems. Cambridge: Cambridge University Press; 2012.

Cubic equation.
[http://eqworld.ipmnet.ru/en/solutions/ae/ae0103.pdf webcite]

Choi B: Improvement of drain leakage current characteristics in metaloxidesemiconductorfieldeffecttransistor by asymmetric sourcedrain structure. In Proceedings of the 2012 IEEE International Meeting for Future of Electron Devices Kansai (IMFEDK): May 9–12 2012; Osaka. Piscataway: IEEE; 2012:12.

Alam K: Transport and performance of a zeroSchottky barrier and doped contacts graphene nanoribbon transistors.
Semicond Sci Technol 2009, 24:015007. Publisher Full Text

Ouyang Y, Dai H, Guo J: Multilayer graphene nanoribbon for 3D stacking of the transistor channel. In Proceedings of the IEDM 2009: IEEE International Electron Devices Meeting: December 7–9 2009; Baltimore. Piscataway: IEEE; 2009:14.

Fiori G, Yoon Y, Hong S, Jannacconet G, Guo J: Performance comparison of graphene nanoribbon Schottky barrier and MOS FETs. In Proceedings of the IEDM 2007: IEEE International Electron Devices Meeting: December 10–12 2007; Washington, D.C. Piscataway: IEEE; 2007:757760.

Datta S: Quantum Transport: Atom to Transistor. New York: Cambridge University Press; 2005:113114.

Mayorov AS, Gorbachev RV, Morozov SV, Britnell L, Jalil R, Ponomarenko LA, Blake P, Novoselov KS, Watanabe K, Taniguchi T, Geim AK: Micrometerscale ballistic transport in encapsulated graphene at room temperature.
Nano Lett 2011, 11:23962399. PubMed Abstract  Publisher Full Text

Berger C, Song Z, Li X, Wu X, Brown N, Naud C, Mayou D, Li T, Hass J, Marchenkov AN, Conrad EH, First PN, De Heer WA: Electronic confinement and coherence in patterned epitaxial graphene.
Science 2006, 312:11911196. PubMed Abstract  Publisher Full Text

Novoselov KS, Geim AK, Morozov SV, Jiang D, Zhang Y, Dubonos SV, Grigorieva IV, Firsov AA: Electric field effect in atomically thin carbon films.
Science 2004, 306:666669. PubMed Abstract  Publisher Full Text

Gunlycke D, Lawler HM, White CT: Room temperature ballistic transport in narrow graphene strips.

Jiménez D: A current–voltage model for Schottkybarrier graphenebased transistors.
Nanotechnology 2008, 19:345204345208. PubMed Abstract  Publisher Full Text

Liao L, Bai J, Cheng R, Lin Y, Jiang S, Qu Y, Huang Y, Duan X: Sub100 nm channel length graphene transistors.
Nano Letters 2010, 10:39523956. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Thompson S, Packan P, Bohr M: MOS scaling: transistor challenges for the 21st century.

Saurabh S, Kumar MJ: Impact of strain on drain current and threshold voltage of nanoscale double gate tunnel field effect transistor: theoretical investigation and analysis.
Jpn J Appl Phys 2009, 48:064503064510. Publisher Full Text

Jin L, HongXia L, Bin L, Lei C, Bo Y: Study on twodimensional analytical models for symmetrical gate stack dual gate strained silicon MOSFETs.
Chin Phys B 2010, 19:107302. Publisher Full Text

Ray B, Mahapatra S: Modeling of channel potential and subthreshold slope of symmetric doublegate transistor.

Rechem D, Latreche S, Gontrand C: Channel length scaling and the impact of metal gate work function on the performance of double gatemetal oxide semiconductor fieldeffect transistors.

Majumdar K, Murali Kota VRM, Bhat N, Lin YM: Intrinsic limits of subthreshold slop in biased bilayer graphene transistor.
Appl Phys Lett 2010, 96:123504. Publisher Full Text

Sviličić B, Jovanović V, Suligoj T: Vertical silicononnothing FET: subthreshold slope calculation using compact capacitance model.
Inform MIDEM J Microelectron Electron Components Mater 2008, 38:14.