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Pseudosaturation and Negative Differential Conductance in Graphene Field-Effect Transistors Alfonso Alarcón, Viet-Hung Nguyen, Salim Berrada, Damien Querlioz, Member, IEEE, Jérôme Saint-Martin, Arnaud Bournel, and Philippe Dollfus, Member, IEEE

Abstract—We study theoretically the different transport behaviors and the electrical characteristics of a top-gated graphene field-effect transistor where boron nitride is used as the substrate and gate insulator material, which makes the ballistic transport realistic. Our simulation model is based on the Green’s function approach to solving a tight-binding Hamiltonian for graphene, self-consistently coupled with Poisson’s equation. The analysis emphasizes the effects of the chiral character of carriers in graphene in the different transport regimes including the Klein and band-to-band tunneling processes. In particular, the Klein tunneling is shown to have an important role on the onset of the current saturation which is analyzed in detail as a function of the device parameters. Additionally, we predict the possible emergence of negative differential conductance and investigate its dependence on the BN-induced bandgap, the temperature, and the gate insulator thickness. Short-channel effects are evaluated from the analysis of transfer characteristics as a function of gate length and gate insulator thickness. They manifest through the shift of the Dirac point and the appearance of current oscillations at short gate length. Index Terms—Dirac point, graphene field-effect transistor (GFET), negative differential conductance (NDC), short-channel effect.

I. I NTRODUCTION

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HE ISOLATION of one-atom-thick layer of graphene triggered a revolution in solid-state physics and gave rise to a lot of expectations toward electronic applications [1], [2], particularly high-frequency electronics [3]. Recently, graphene transistors operating in the radio-frequency range have been demonstrated with graphene reported on sapphire substrate [4] or with CVD-grown graphene reported on diamondlike carbon substrate [5], [6]. However, the benefits of the extraordinary intrinsic transport properties of graphene [7] are usually hindered by the defects of the supporting insulating substrate. Recently,

it has been shown that graphene reported on hexagonal boron nitride (h-BN) has higher mobility than that on any other substrate [8]. Indeed, such a substrate is flat, with a low density of charged impurities, does not have dangling bonds, and is relatively inert [8]–[10]. A mobility of 275 000 cm2 /V · s at low temperature, as high as that for suspended graphene, has even been reported [11]. It should allow reaching ballistic transport at room temperature and exploiting the peculiarities of graphene properties inherent in the massless and chiral character of charge carriers (chiral band-to-band and Klein tunneling) [12]. Additionally, h-BN has the same atomic structure as graphene, with a lattice constant 1.8% higher. It offers the possibility of assembling graphene/BN multilayers. In the case of Bernal stacking, it has been shown that the interaction of BN with graphene may open a bandgap of 53 meV [13] or 100 meV [14] in graphene, which could improve the poor pinchoff characteristic of graphene field-effect transistors (GFETs). The onset of the current saturation is also an important issue of GFET operation [15], [16]. The physical mechanisms involved in this saturation are very specific of graphene and of the chiral nature of particles. The conditions to get it and its dependence on device parameters will be analyzed and discussed in detail. Devices exhibiting a negative differential conductance (NDC) are also of strong interest for high-frequency applications. With different physical origins, this effect has been predicted in different kinds of graphene nanoribbon or nanomesh junctions [17]–[23], in graphene superlattices [24], in resonant tunneling diodes [25], [26], in FETs [27], and in tunnel FETs [28]. It has been even observed experimentally in a graphene FET as a consequence of ambipolar transport behavior [29]. An NDC effect of different origin will be analyzed in this paper as a particular case of the saturation of current. II. S IMULATED D EVICE AND T RANSPORT M ODEL

Manuscript received November 5, 2012; revised December 31, 2012; accepted January 11, 2013. Date of current version February 20, 2013. This work was supported in part by the French ANR through the projects NANOSIMGRAPHENE (ANR-09-NANO-016) and MIGRAQUEL (ANR-10-BLAN0304). The review of this paper was arranged by Editor A. C. Seabaugh. A. Alarcón, S. Berrada, D. Querlioz, J. Saint-Martin, A. Bournel, and P. Dollfus are with the Institute of Fundamental Electronics, Centre National de la Recherche Scientifique, University of Paris-Sud, UMR 8622 Orsay, France (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; arnaud. [email protected]; [email protected]). V.-H. Nguyen is with the L_Sim, SP2M, UMR-E CEA/UJF-Grenoble 1, INAC, 38054 Grenoble, France, and also with the Institute of Physics, Hanoi, Vietnam (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TED.2013.2241766

In Fig. 1, we display the schematic view of the simulated GFET wherein the channel is formed by a monolayer graphene sheet in the Oxy plane. The source and drain extensions are N-type doped to 1013 cm−2 , and their length is LS = LD = 20 nm. The thickness of the substrate is WS = 100 nm. Different gate lengths (between 10 and 100 nm) and gate insulator thicknesses (2 and 10 nm), including the possibility of a BNinduced bandgap opening (53 and 100 meV), were considered. We have used a quite standard model based on a nearestneighbor tight-binding Hamiltonian εn |nn| − t [|nm| + |mn|] (1) Htb =

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n

n,m

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Fig. 1. Schematic of the simulated GFET. h-BN is used as a substrate and insulator material with a dielectric constant εr = 3.5.

where n and m refer to the nearest atomic sites of the 2-D lattice, εn is the on-site energy, and t = 2.7 eV is the nextneighbor hopping energy [30], [31]. Assuming the lateral width of the device to be much larger than the channel length, the y-direction can be considered through Bloch periodic boundary conditions [32], [33]. The lattice is split into elementary cells, and by Fourier transform of the operators in (1) (along the Oy-direction), the Hamiltonian (1) is rewritten in the form of decoupled quasi-1-D Hamiltonians for each ky value [23], [26]. The retarded Green’s functions of these Hamiltonians are computed in the ballistic approximation. Electron-phonon scattering has been shown to affect the I–V characteristics in long-channel GFETs [34] but can be neglected here, where the gate length is smaller than the electron mean free path of about 1 μm for graphene sandwiched between two h-BN crystals [35]. The coupling of graphene with the h-BN layers is not explicitly taken into account in the model, but it reinforces the assumption of ballistic transport. The graphene/metal contact is modeled in the weak coupling limit, i.e., the broadening of Dirac states due to the contact-induced states being neglected [36], [37]. The local charge density is computed as in [33], and the Green’s functions are solved self-consistently with the Poisson equation [26]. The method of moments [38] is used to solve the latter equation. Since the potential is assumed to be y independent, the Poisson equation is solved in the 2-D-space Oxz. The self-consistence is implemented through the Newton–Raphson method [38]. The current density is computed from the Landauer formula. III. R ESULTS AND D ISCUSSION A. Carrier Transport Regimes in GFET In Fig. 2(a), we plot the self-consistent results of the local density of states (LDOS) and potential profile Ug (black solid line) obtained for the gate length LG = 15 nm at gate and drain voltages VGS = −0.5 V and VDS = 0.2 V, respectively. The corresponding transmission coefficient is displayed in Fig. 2(b) while the local charge density profile is shown in Fig. 2(c). Here, the chirality of carriers appears in the density profile which exhibits positive (electrons) and negative (holes) values outside and inside the gated region, respectively. In Fig. 2(a), LDOS is plotted for a transversal momentum ky = (2/3 + √ 0.005)π/ac 3, where ac = 0.142 nm is the lattice constant. Although the graphene bandgap is actually zero, we observe in the LDOS an apparent energy gap in the regions of flat

Fig. 2. (a) Potential profile Ug (Solid line) for a device with LG = 15 nm and LDOS for the transverse momentum ky = Ky + 6.4 × 107 m−1 . EFS and EFD are the Fermi levels in source and drain contacts, respectively. (b) Corresponding transmission coefficient. (c) Local charge density (negative values correspond to holes). Horizontal and vertical dotted lines (electron/hole and gate boundaries, respectively) serve as a guide to the eye. (d) Transmission coefficient for LG = 50 nm. Other parameters: WIN = 2 nm, VDS = 0.2 V, VGS = 0.5 V, EGAP = 0 meV, and T = 300 K. LG = 15 nm in (a), (b), and (c), and LG = 50 nm in (d).

potential, which is characterized by blue areas in Fig. 2(a). This feature can be explained in terms of the energy band structure of graphene giving rise to a ky -dependent pseudoenergy gap which writes [39] √ ˆg (ky ) = 2t 1 − 2 cos ac ky 3/2 . E (2) This √pseudoenergy gap is truly zero for ky = Ky = 2π/3ac 3 (transversal momentum at the Dirac point) but finite ˆg (ky ) ≈ 77 meV for ky = Ky + for ky = Ky . For instance, E 6.4 × 107 m−1 . This energy gap plays an important role since it governs the off-current, the current saturation, and the NDC effect, which will be discussed later. We can separate the transport regimes in the energy regions (I), (II), and (III), as schematized in the LDOS of Fig. 2(a) and the corresponding transmission function in Fig. 2(b). The first is the thermionic regime (I) for energies above the potential barrier. The second is the chiral tunneling regime through the gate-induced barrier (II) that results from the good matching of the hole states in the gated region with the electron states in the source [40], [41]. It appears in Fig. 2(b) where the transmission coefficient exhibits resonant peaks reaching the value of 1 in the energy range [−0.3 eV, 0 eV]. The transport regime (III) corresponds to the chiral band-to-band tunneling between hole states in the source and electron states in the drain. We will see in this work that, because of the zero (or small) bandgap, these chiral transport regimes play an important role in the electrical characteristics. It is also important to note in Fig. 2(b) that the “transmission valley” separating the regions (I) and (II) at the top of the barrier (around the Dirac point in the channel) ˆg is a consequence of the ky -dependent pseudoenergy gap E

ALARCÓN et al.: PSEUDOSATURATION AND NDC IN GFETs

Fig. 3. ID –VDS characteristics for different values of VGS and for the gate length LG = 15 nm. (a) VGS = −0.5 V. (b) VGS = 0 V. (c) VGS = 1 V. (d) Transfer characteristics for VDS = 0.1 V. The vertical dashed lines are guides to see the bias correspondences between the different figures. Other device and simulation parameters in all cases: WIN = 2 nm, EGAP = 0 meV, and T = 300 K.

commented before. Indeed, while the normal incident particles (with ky = Ky ) are fully transmitted according to the Klein paradox, the particles with ky = Ky may be reflected, which generates this transmission valley. Note also that, due to the influence of electron states on both sides of the gated region, the deepness of the transmission valley is dependent on the gate length, as shown in Fig. 2(b) and (d) where the transmission is plotted for the two gate lengths LG = 15 nm and LG = 50 nm, respectively. Another transmission valley separates the regions (II) and (III). It corresponds to the pseudobandgap which is formed around the Dirac point in the source region and is gate length independent. It plays also an important role in the device characteristics, as will be shown later. Now, we analyze the contribution of the different transmission regimes in the drain current–drain voltage characteristics plotted in Fig. 3 for a gate length LG = 15 nm and for three values of VGS . The total current density (black solid line) is the sum of the three contributions (I), (II), and (III). For VGS = −0.5 V [see Fig. 3(a)], the chiral tunneling is the dominant process at low drain bias but reduces at high VDS where the bandto-band and the thermionic currents increase monotonically which makes the saturation of current impossible. For VGS = 0 V [see Fig. 3(b)] the chiral and the thermionic currents are dominant and comparable at low drain bias. When increasing the drain bias, the most important contributions become the thermionic current and, on a smaller extent, the band-to-band transmission (at high bias). Finally, in Fig. 3(c), it is clear that, for VGS = 1 V, the thermionic current strongly dominates, with a small contribution of the band-to-band tunneling at very high drain voltage. The vertical dashed lines in Fig. 3(a)–(c) correspond to the three values of VGS marked with vertical dashed lines in the transfer characteristic of Fig. 3(d).

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Fig. 4. (a) ID –VDS characteristics of a device with gate length LG = 50 nm for values of VGS between 1 and −1 V (by steps of 0.5 V). Other device and simulation parameters: WIN = 2 nm, EGAP = 0 meV, and T = 300 K. (b) ID –VDS characteristics for different bandgap values EGAP = 0 meV, EGAP = 53 meV, and EGAP = 100 meV and for different values of temperature (solid lines) T = 77 K and (dashed lines) T = 300 K.

B. Drain Current–Drain Voltage Characteristics: Current Saturation and NDC Effect In Fig. 4(a), the drain current–drain voltage characteristics obtained at room temperature for the 50-nm-gate-length device with WIN = 2 nm are shown for different values of VGS between 1 and −1 V (by steps of 0.5 V). At high VDS , a rapid and quasi-linear increase of the current density is observed in almost all cases, which is essentially due to the contribution of thermionic transport. At lower VDS , we observe a pseudosaturation behavior only in the case of negative gate voltage. A small NDC effect may even be detected, particularly for VGS = −1 V. This effect has been evidenced experimentally in [29] in the case of long-channel FETs under diffusive transport. It has been explained as a consequence of the distribution of high local resistance between the n-doped and p-doped regions of the channel. In the case of ballistic transport in the short channel, the NDC effect may be explained differently. Both the pseudosaturation and the NDC effects observed here have the same physical origin, the latter being a particular case of the former. Their physical origin may be understood basically from the analysis of the current contributions in Fig. 3 and from the transmissions shown in Fig. 2. To observe a current saturation when increasing VDS , at least one of the three current contributions should saturate or even reduce. It is clear, particularly from Fig. 3, that only the chiral tunneling current is likely to reduce at high VDS . As soon as they are significant, the two other components, i.e., the thermionic and the band-to-band ones, can only increase. Hence, the saturation of current can only be observed when the chiral tunneling is the dominant transport process, i.e., at negative gate voltage when the gate-induced barrier is significant. It is the basic but is not the only condition. Additionally, one can imagine that, if it is possible to increase the separation between the drain bias of the maximum chiral tunneling from that of the onset

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of thermionic/band-to-band currents, an NDC effect may be achieved. To fully understand the parameters which play a major role in the saturation/NDC effects, we have first to summarize, in terms of characteristic energies, why, when, and how the chiral tunneling may reduce and the thermionic and band-toband currents increase. The chiral tunneling is maximum when the transmission window of current transmission [EFD , EFS ] is within the gate-induced source-channel barrier, i.e., when the Dirac point in the channel EKch is higher than EFS and the Dirac point in the source EKS is lower than EFD . It is important to have in mind that, whatever the gate length, the source-channel barrier in the GFET is very sensitive to VDS : For instance, for the gate length LG = 50 nm, the top of the barrier reduces with a rate of about 0.51 eV/V when increasing VDS in the range [0–0.6 V] (not shown). Hence, EKch may go below EFS , and/or EFD may go below EKS , which both reduce the contribution of chiral tunneling. In the hypothetical case where there is no transmission valley around EKch , there would be a continuous switching between the chiral tunneling and the thermionic current, and the total current would increase monotonically. Since there is a transmission valley resulting ˆg , there is a VDS range in from the ky -dependent pseudogap E which the chiral tunneling decreases, and the thermionic current is still limited. Similarly, due to the transmission valley around the Dirac point EKS in the source, the increase of the bandto-band tunneling is limited when EFD approaches EKS . It is the origin of the onset of saturation. The best condition to have a good saturation or even NDC behavior is met in the case where, when increasing VDS , EKch approaches EFS at the same time as EFD approaches EKS . Indeed, it follows that the chiral tunneling through the barrier tends to reduce while the increase of both thermionic and band-to-band currents is limited. We can now examine the influence of the different parameters on the saturation/NDC: bandgap, temperature, oxide thickness, and gate length. As previously stated, it is theoretically possible to open a bandgap in graphene in the case of Bernal stacking on h-BN. Accordingly, we plot in Fig. 4(b) the ID –VDS characteristics of a 50-nm-gate-length device for the different EGAP values of 0 meV, 53 meV [13], and 100 meV [14] and also for two different values of the temperature, i.e., T = 77 K (solid lines) and T = 300 K (dashed lines). The NDC effect appears to be strongly dependent on the bandgap in graphene and, at a smaller extent, on the temperature. For EGAP = 100 meV, the peak-tovalley ratio (PVR) reaches the value of 3 at T = 77 K and 2 at T = 300 K. To understand this behavior, let us consider the latter simulation scenario, i.e., EGAP = 100 meV and T = 300 K to explain the origin of the NDC. In Fig. 5(a), we present the potential profile superimposed on the LDOS obtained for the momentum ky = Ky + 6.4 × 107 m−1 for the peak-current state in Fig. 4(b), i.e., for VDS = 0.2 V. The same quantity is plotted in Fig. 5(c) for the valley-current state VDS = 0.4 V. On the right side, in Fig. 5(b) and (d), we display the corresponding current spectra JSpec . Here, with EGAP = 100 meV, the transmission valley between regions (I) and (II) results from the addition of the actual bandgap EGAP and the ky -dependent transmission

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Fig. 5. In the particular case of LG = 50 nm, T = 300 K, and EGAP = 100 meV in Fig. 4(b). In (a) and (c) are shown the potential profile and LDOS for the transverse momentum ky = Ky + 6.4 × 107 m−1 at the current peak bias VDS = 0.2 V and the current valley bias VDS = 0.4 V, respectively. In (b) and (d) are shown the corresponding current spectra.

ˆg at the top of the barrier. Hence, the transmission energy gap E reaches truly 0 (not shown here) in contrast to the cases of Fig. 2 for gapless pristine graphene. In the peak-current state [see Fig. 5(a)], the transmission gap at the top of the barrier is outside of the energy window [EFD , EFS ]. The energy gap in the source is also outside of the energy window. Hence, the chiral tunneling is possible in the full range of this window, while both thermionic current and bandto-band tunneling are negligible. When increasing the drain bias, both transmission gaps enter in the transmission window [EFD , EFS ], which reduces the possibility of chiral tunneling and delays the onset of thermionic and band-to-band currents to higher VDS . As shown in the current spectra, in the valleycurrent state [see Fig. 5(d)], the contribution of the chiral tunneling is strongly reduced while both contributions of the thermionic and the band-to-band transmission are still limited, which results in a total current smaller than that in in the previous case [see Fig. 5(b)]. It is the origin of the NDC effect, as already predicted from non-self-consistent simulations [41]. Now, we discuss the influence of the oxide thickness WIN . For the gate length LG = 15 nm, we consider two oxide thicknesses, i.e., WIN = 2 nm (as in previous devices) and WIN = 10 nm. To evaluate correctly the impact of WIN , the results obtained for the two values should be compared in the same electrostatic conditions: Given the WIN dependence of shortchannel effects, we cannot make the comparison at the same gate voltage but at the same gate-induced barrier height. We first considered VGS = −0.5 V for WIN = 2 nm. Then, for WIN = 10 nm, we have adjusted VGS so that, at VDS = 0, the barrier height is the same as that for WIN = 2 nm and VGS = −0.5 V. In this way, it was necessary to reduce VGS down to −2.16 V. In Fig. 6(a), we plot the ID –VDS characteristics in both cases. The NDC effect is more pronounced for the thinner oxide. It can

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Fig. 7. ID –VGS characteristics for different gate lengths from 10 to 100 nm. Device and simulation parameters: WIN = 2 nm, VDS = 0.1 V, EGAP = 0 meV, and T = 300 K. Inset: The same parameters except T = 77 K.

Fig. 6. (a) ID –VDS characteristics and (inset) potential profiles at VDS = 0 for a device of gate length LG = 15 nm and for two sets of WIN /VGS values: (Solid lines) WIN = 2 nm/VGS = −0.5 V and WIN = 10 nm/VGS = −2.16 V. EGAP = 0. (b) ID –VDS characteristics for different values of LG . Device and simulation parameters: VGS = −0.5 V, WIN = 2 nm, EGAP = 0 meV, and T = 300 K.

be explained by the shape of the potential profile shown in the inset. Due to the more effective influence of the gate, the barrier is more flat for WIN = 2 nm than for WIN = 10 nm. Hence, the transmission valley at the Dirac point EKch in the channel is deeper in the former case, which makes the NDC effect stronger. It appears from the aforementioned discussion that the VDS range of saturation is strongly correlated to the width and deepness of the transmission valley around the K point, EKch , in the channel (top of the barrier) which controls the onset of thermionic current. As seen in Fig. 2 and as previously commented, these characteristics of this transmission valley are gate length dependent. It explains why the saturation range in the ID –VDS characteristics is smaller at short gate length than for long LG . This is shown in Fig. 6(b) where we plot the ID –VDS curves at VGS = −0.5 V for different gate lengths ranging from 10 to 100 nm. Hence, the longer the gate, the better the saturation. Of course, if a bandgap was generated in graphene, the onset of both the thermionic and band-to-band currents would be shifted to higher VDS , and the saturation could be better, with even the possibility of strong NDC effect, as described earlier. Regarding the current saturation/NDC behavior in 2-D graphene devices as discussed here, owing to its possibility of gate-induced bandgap, the bilayer graphene channel could be alternatively used. Indeed, this idea has been evidenced experimentally in [16] where a good current saturation due to the gate-induced bandgap is observed. C. Transfer Characteristics: Current Oscillations and Shift of the Dirac Point In Fig. 7, we show the transfer characteristic for different gate lengths LG at room temperature and, in the inset, at low temperature (T = 77 K). The asymmetry between n- and p-branches due to the doping of the contact regions is clearly

observed, in accordance with all experimental results, as, for instance, in [29] and [39]. Here, the maximum of current density is reached in the n-branch because of the N-type doping of the source/drain extensions. A more peculiar phenomenon is observed at very short gate length in the p-branch, where the current is dominated by the chiral tunneling through the hole states in the channel: the oscillations of current as a function of VGS for small channel length. As shown in the inset of Fig. 7, these oscillations are more pronounced when decreasing the temperature. These oscillations result from the quantization of hole states in the gated region which gives rise to resonant chiral tunneling [41]. Indeed, at short gate length, the energy separation of these states is significant, as seen when comparing Figs. 2(a) and 5(a). Hence, at low drain bias, the transmission window [EFD , EFS ] is small enough to cross separately the quantized hole states of the gated region when tuning VGS , which generates current oscillations. A similar effect has been shown to generate strong oscillations of transconductance between positive and negative values in bilayer graphene structures [42]. This effect tends to be lost when the gate length and/or the temperature increases. A zoom of the ID –VGS characteristics of Fig. 7 around the Dirac point is plotted in Fig. 8 for two values of the gate insulator thickness: WIN = 2 nm and WIN = 10 nm. The neutrality point corresponds to the transition between normal transmission above the barrier and chiral tunneling through the barrier. It thus occurs when the transmission window [EFS , EFD ] crosses the transmission energy gap at the top of the barrier. When reducing LG , a shift of the neutrality point to negative VGS values is observed. This short-channel effect is due to the reduced gate charge control at short gate length: It becomes necessary to apply a more negative VGS to push electrons out of the channel and reach the equilibrium between electrons and holes. This effect is obviously stronger when increasing WIN , as experimentally observed in [43]. Additionally, the off-current increases when reducing LG . Finally, to evaluate possible applications of the considered GFET, its ID –VGS characteristics with different h-BN-induced bandgaps were also analyzed in [44]. It was shown that the on/off-current ratio reaches the value of 20 for EGAP = 100 meV and LG = 15 nm. Although this ratio may be slightly

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Fig. 8. Zoom around the Dirac points of Fig. 7 for different LG and two values of the gate insulator thickness. (a) WIN = 2 nm. (b) WIN = 10 nm.

higher at longer gate length, it cannot reach in this device the values required for digital applications. However, the device can exhibit a high transconductance up to a few millisiemens per micrometer. Additionally, a high intrinsic cutoff frequency in the terahertz range and a high voltage gain may be achieved. Thus, although not good enough for digital applications, the considered GFET offers a good potential for analog HF applications. IV. C ONCLUSION We have presented different features of transport behavior in a top-gated GFET where h-BN was used as both the substrate and gate insulator material. This study was performed by means of self-consistent simulation of the Poisson equation and the nonequilibrium Green’s function approach to solving a tightbinding Hamiltonian. The analysis of the different transport regimes provided an understanding of the transport behaviors and emphasized the important role of chiral (Klein and bandto-band) tunneling processes. We have described in detail the conditions to meet to observe a pseudosaturation of the current as a function of drain bias. In particular, it appears clearly that the saturation is possible only when the current is dominated by the chiral tunneling process at low VDS , i.e., at negative VGS . An NDC effect has even been shown to be possible with a PVR dependent on the bandgap in graphene and on the temperature. It may reach the value of 3 at T = 77 K. The gate insulator thickness was shown to strongly influence the shortchannel effects through the gate length dependence of the Dirac point position in the transfer characteristics. At very small gate length, the chiral tunneling through quantized hole states of the channel was shown to give rise to current oscillations. ACKNOWLEDGMENT The authors would like to thank Y.-M. Niquet and F. Triozon from CEA Grenoble for useful discussion regarding the Newton-Raphson algorithm. R EFERENCES [1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science, vol. 306, no. 5696, pp. 666–669, Oct. 2004.

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