Bandgap nanoengineering of graphene tunnel diodes and tunnel transistors to control the negative differential resistance Viet Hung Nguyen · Jérôme Saint-Martin · Damien Querlioz · Fulvio Mazzamuto · Arnaud Bournel · Yann-Michel Niquet · Philippe Dollfus

Published online: 31 January 2013 © Springer Science+Business Media New York 2013

Abstract By means of numerical simulation based on the Green’s function formalism on a tight binding Hamiltonian, we investigate different possibilities of achieving a strong effect of negative differential resistance in graphene tunnel diodes, the operation of which is controlled by the interband tunneling between both sides of the PN junction. We emphasize on different approaches of bandgap nanoengineering, in the form of nanoribbons (GNRs) or nanomeshes (GNMs), which can improve the device behaviour. In particular, by inserting a small or even zero bandgap section in the transition region separating the doped sides of the junction, the peak current and the peak-to-valley ratio (PVR) are shown to be strongly enhanced and weakly sensitive to the length fluctuations of the transition region, which is an important point regarding applications. The study is extended to the tunneling FET which offers the additional possibility of modulating the interband tunneling and the PVR. The overall work suggests the high potential of GNM lattices for designing high performance devices for either analog or digital applications.

V. Hung Nguyen · J. Saint-Martin · D. Querlioz · F. Mazzamuto · A. Bournel · P. Dollfus () Institute of Fundamental Electronics (IEF), CNRS, UMR 8622, Univ. Paris-Sud, Orsay, France e-mail: [email protected] V. Hung Nguyen Center for Computational Physics, Institute of Physics, Vietnam Academy of Science and Technology, Hanoi, Vietnam V. Hung Nguyen · Y.-M. Niquet L_Sim, SP2M, UMR-E CEA/UJF-Grenoble 1, INAC, Grenoble, France

Keywords Graphene device · Dirac fermions · Green’s function · Quantum transport · Negative differential resistance · Tunnel diode · Tunnel transistor

1 Introduction The effect of negative differential resistance (NDR) has been widely investigated in devices based on conventional semiconductors, e.g. the Esaki tunnel diode or the double-barrier resonant tunnelling diode. This effect is suitable for a wide range of high-frequency applications [1]. Given the peculiar tunnelling properties of chiral particles in graphene [2] and the potential of this material for high speed electronics [3], it sounds appealing to investigate the possibility to generate and control an NDR in graphene devices. Actually, different suggestions of graphene structures and devices exhibiting an NDR behaviour have been proposed, based on various physical mechanisms. It has been predicted that the parity selective rule [4] existing in perfect zigzag (Z) GNRs with an even number of zigzag lines may generate a negative differential resistance [5]. The mismatch of modes in the left and right sides of a P+ P ZGNR junction may also induce an NDR regardless of the width of the GNR [6, 7]. The bandgap opened in a double-gate bilayer graphene structure has been used too to generate an NDR effect [8]. A similar behaviour has been observed in different kinds of armchair GNRs [9, 10] or double-barrier graphene structures [11] working in resonant tunnelling regime. The same effect has been predicted in GNRs made of armchair and zigzag sections of different widths [12–14] and in graphene/BN heterostructures [15]. The NDR has been predicted to occur also in GNR superlattices with different ballistic transport regimes, including the resonant tunnelling through the minibands and the Wannier-Stark ladder regime

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[16]. A possible NDR effect has been shown to occur also in single-gate graphene sheets though the occurrence of Klein tunnelling and band-to-band tunnelling in gapless graphene should be an obstacle to the NDR effect [17]. It is worth noting that a small NDR behaviour has been observed experimentally in graphene transistors [18–20] and explained theoretically [21]. Actually, the stronger NDR effect has been predicted to occur in Esaki-like PN junctions (tunnel diodes), the operation of which is controlled by the interband tunnelling between the conduction band of the N-doped side and the valence band in the P-doped side. Though small in gapless mono-layer and bi-layer graphene sheets [22, 23], this effect may increase significantly if a bandgap can be generated in graphene. In particular, it has been shown that the bandgap engineering in GNRs [24] and in graphene nanomeshes (GNMs) [25] made of lattices of nanoholes may not only enhance the peak-to-valley ratio but can also make the I –V characteristics weakly sensitive to the doping profile and to the atomic disorder. Additionally, the tunnel field-effect transistor (TFET), which is the transistor counterpart of the tunnel diode and works also in interband tunnelling regime, has been predicted to be a good option to make the NDR effect strong and tunable by the gate voltage [26]. The objective of the present article is to review and synthesize the operation and performance of these different types of tunnel diodes and of TFETs. To achieve such results, the bandgap engineering of graphene is a key-point. The zero band gap in graphene is a consequence of the identical potential felt by the two carbon atoms of the graphene unit cell. To open a bandgap, one has to make the potential on the two atoms different. It is possible for instance in the case of Bernal stacking of graphene on hexagonal BN sheet, the atomic structure of which being very close to that of graphene. Bandgaps of 53 meV and 100 meV have been predicted theoretically [27, 28] and may even increase up to 200 meV and more under external pressure [29, 30]. Additionally, it has been shown that when graphene is reported on h-BN, the interface is clean enough to make the mobility almost as high as in suspended graphene [31], which 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. However, though it is relevant to investigate theoretically the possibilities offered by the control of Bernal stacking of h-BN and graphene layers, such stacking remains challenging and not demonstrated experimentally yet. Other techniques of bandgap opening by nanostructuring graphene have been previously investigated and demonstrated experimentally. The most popular approach consists in reducing the dimensionality of graphene by cutting 2D graphene into 1D narrow ribbons, i.e. in graphene nanoribbons. However, for the bandgap to be significant, the GNR

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width should be smaller than about 3 nm, which is difficult to achieve, gives rise to problems of reproducibility and edge roughness control, and provides limited driving current. Alternatively, it has been proposed to open up a band gap in a large sheet of graphene by punching a high-density array of periodic nanoholes [32]. Depending on the neck width, the meshing orientation and the shape of holes, bandgaps higher than 0.5 eV could be achieved [33–35]. All these approaches of bandgap engineering are considered in this article to design devices with their own advantages and drawbacks. We especially emphasize on the possibilities offered by GNRs and GNMs for designing high performance tunnel diodes and on the tunability of the peak-tovalley ratio (PVR) in TFET. On this basis we suggest some possible new directions for future investigations in the conclusive part of the paper.

2 Model and numerical issues The Hamiltonian of the honeycomb graphene lattice, with ac = 0.142 nm as carbon-carbon distance, is considered via a nearest-neighbor tight-binding approximation [36], i.e. εn |nn| − t |nm| + |mn|, (1) Htb = n

n,m

where n and m refer to the atomic sites of the 2D lattice, εn is the on-site energy and t = 2.7 eV is the next-neighbor hopping energy [37]. The sum over atomic sites is restricted to the nearest-neighbor atoms. The elementary unit cell of the honeycomb lattice consists of two carbon atoms, each of them belonging to the triangular sublattices A and B, respectively [38]. The difference of onsite energies Δ = εA − εB between A and B atoms determines if a bandgap is opened. In pristine graphene this difference vanishes, which yields a gapless band structure with linear conduction and valence bands that meet at the K and K (Dirac) points of the Brillouin zone. If Δ = 0, as e.g. in the case of graphene on h-BN in a Bernal stacking arrangement, the inversion symmetry is broken and a finite bandgap EG = 2Δ is opened. The energy dispersion close to the Dirac points is well approximated by the simple relation (2) E(k) = ± 2 vF2 kx2 + ky2 + Δ2 , where vF = 3ac t/2 ≈ 106 m/s is the Fermi velocity, k = (kx , ky ) is the 2D wave vector and the sign + (−) stands for the conduction (valence) band. In the case of GNRs with armchair edges (AGNRs) the bond relaxation at the edges is taken into account by using a different hopping energy te = 3.02 eV for the edge bonds instead of t = 2.7 eV for the other bonds [39]. In the case of large graphene sheet where we can assume the lateral width of the device to be much larger than the

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channel length, the y direction can be considered through Bloch periodic boundary conditions [40, 41]. The lattice is then split into unit 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-1D Hamiltonians as [25] Htb = ky H˜ 1D (ky ) (3) H˜ 1D (ky ) = n H˜ n− (ky ) + H˜ n (ky ) + H˜ n+ (ky ),

where T (E, ky ) = Trace[ΓS G ΓD G† ] is the transmission function and the integral over ky is performed in the first Brillouin zone. For graphene nanoribbon devices, the Green’s function is determined using the Hamiltonian (1) and the formula (6) for current becomes

2e ∞ dET (E) fS (E) − fD (E)]. (7) I= h −∞

where H˜ n (ky ) is the Hamiltonian of cell {n} and H˜ n± (ky ) denotes the coupling of cell {n} to cell {n ± 1} along the transport direction Ox. Using the Hamiltonian (3) the retarded Green’s function is computed in the ballistic approximation for a given momentum ky as −1 G(E, ky ) = (E − i0+ )I − H1D (ky ) − Σ(ky ) , (4)

3 Results

where the self-energy Σ(ky ) = ΣS (ky ) + ΣD (ky ) describes the coupling between the graphene channel and the semiinfinite source and drain contacts. It can be expressed for the lead α as Σα (ky ) = τD,α gα (ky )τα,D , where τ is the hopping matrix that couples the device to the lead and gα (ky ) is the surface Green’s function of the uncoupled lead. The surface Green’s function and the Green’s function (4) are calculated using the fast iterative scheme described in [42] and the standard recursive algorithm [43], respectively. The local charge density is computed as [41]

1 ∞ dE dky ρ(x) = π −∞ BZ × DS (E, x) fS (E) + sgn(E − EN ) − 1 /2 + DD (E, x) fD (E) + sgn(E − EN ) − 1 /2 , (5) where EN is the charge neutrality level, sgn(E) is the sign function, fS(D) is the source and drain Fermi distributions with the Fermi levels EF S(F D) , respectively, DS(D) = GΓS(D) G† is the local density of states (LDOS) resulting from the source (drain) states, and ΓS(D) = i(ΣS(D) − † ) is the injection rate at source (drain) contact. The ΣS(D) Green’s functions (4) are solved self-consistently with the Poisson equation. The method of moments [44], known to be computationally efficient, is used to solve the latter equation. Since the potential is assumed to be y-independent, the Poisson equation is solved in the 2D-space Oxz. The selfconsistence is implemented through the Newton-Raphson method. The updated values of potential resulting from Poisson’s equation are reintroduced as on-site energies εn in (1). In the results presented below, the self-consistence was activated only in the case of the TFET. In the case of PN junctions, the self-consistence has been shown to not change significantly the results [26] and frozen field simulations were performed. Finally, the current is computed as

∞ e dE dky T (E, ky ) fS (E) − fD (E) , (6) I= πh −∞ BZ

A large part of this work is devoted to the transport in PN junctions, the band structure of which is schematized in Fig. 1 by assuming a finite bandgap in the whole structure, as in any conventional PN junction. The P-type and N-type doping can be generated in graphene either by electrostatic doping [45] or by chemical doping [46]. In the former case, a back gate is formed on the substrate to control the density by tuning the applied voltage Vsub . The junction is characterized mainly by the potential barrier U0 and the length L of the transition region across which the charge density changes monotonically from N-type to P-type. Though expected to be short, this length is difficult to control and will be considered as a parameter in this work. As in the conventional Esaki diode, the peak current in this structure is governed by the interband tunnelling through the transition region at low bias (Fig. 1). The valley current appears at high bias when the filled states in one side of the junction do not see any free state available for tunnelling in the other side. If the bias voltage is further increased the thermionic current above the potential barrier leads to the re-increase of current. In what follows, all simulations were performed at room temperature.

Fig. 1 Schematic view of the band structure in a PN junction with finite bandgap. The transition region of length L separates the N-doped side from the P-doped side of the structure. The interband tunnelling (symbolised by the arrow) between the conduction band of the N-doped side and the valence band of the P-doped side is the dominant contribution to the current. The quantity EF S , EF D , EC EV and EN stand for the Fermi level in the source (N) side, the Fermi level in the drain side, the bottom of conduction band, the top of valence band and the neutrality point, respectively. U0 is the potential barrier height in the unbiased structure

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Fig. 2 I –V characteristics of the PN junction for different values of energy gap EG . Other parameters: L = 10 nm, EF S = 0.26 eV, U0 = 0.52 eV

Fig. 3 I –V characteristics of the PN junction for different values of transition length L. Other parameters: EG = 0.26 eV, EF S = 0.26 eV, U0 = 0.52 eV

3.1 2D graphene PN junctions

This behavior is explained as follows in the gapless structure. At the current peak, when increasing further the bias voltage, the neutral point in the P-type region approaches the Fermi level EF S in the source. Hence, the ky -dependent transmission gap reduces the interband tunneling for nonnormal incident particles and delays slightly the onset of the thermionic current. It is the origin of the small NDR effect. This effect is obviously reinforced in the case of finite bandgap EG . The interband transmission is reduced due to the finite bandgap to cross through the transition region, leading to reduced peak current. In return, the enlarged transmission gap separates more strongly the regimes of interband tunneling and thermionic current, which makes the valley current much smaller than in the gapless case and the PVR higher. It should be noted that in the region of interband tunneling, the good matching of electron and hole states due to the chiral nature of particles in graphene makes the transmission, and thus the peak current, much higher than in conventional semiconductor structures, even with finite bandgap [22]. It has been shown that all other things being equal, the PVR may be optimized by adjusting the bandgap and the doping levels in the device [22, 23]. However, it is more important to have a look at the effect of the length L of the transition region which is a critical parameter in the sense that it plays a major role on the device performance and is difficult to control. We plot in Fig. 3 the I –V characteristics obtained for three different values of L, from 5 nm to 20 nm in the case of the high bandgap EG = 260 meV. Increasing the transition length does not change the valley current but strongly reduces the peak current. It is due to the large distance to be tunneled through the bandgap which reduces exponentially the transmission in the interband tunneling regime. The PVR falls from 123 for L = 5 nm to 8.5 for L = 20 nm and even 2.5 for L = 30 nm (not shown). This sensitivity to the transition length is obviously an issue

First, we consider the case of 2D diodes formed on wide graphene sheets. In this case, the most probable is to have a gapless material, which is not expected to provide an NDR effect. Indeed, without bandgap the current should continuously switch from the interband tunneling regime to the thermionic regime. However, the detailed analysis of the energy-dependence of the transmission reveals that the situation is actually more subtle. As shown in the case of a single-barrier structure [17], the transmission around the neutrality point exhibits a valley, the depth and energy width of which depend on the barrier length and the transverse energy Ey = vF ky , respectively. This valley separates the region of band-to-band tunneling from that of thermionic current. It is similar here where the P-type (N-type) region may be seen by electron (hole) states of the source (drain) as an infinitely long potential barrier. Let’s consider the case of electron states, keeping in mind that the situation is symmetric for hole states [22]. Around the neutrality point of the P-type region a transmission gap is formed due to the fact that the longitudinal momentum kx becomes imaginary, making the carrier states evanescent. It results in a transmission gap of width ET G = 2Δ = 2 Δ2 + Ey2 . When Δ = 0, as in pristine graphene, the transmission gap is just a function of ky and vanishes for ky = 0. When Δ is finite, the actual bandgap of graphene is EG = 2Δ and the transmission gap takes the minimum value ET G = EG for ky = 0. This transmission gap is the origin of the NDR effect which may be observed even in gapless graphene structures. We plot in Fig. 2 the I –V characteristics obtained for EG = 0 and for EG = 130 meV in the case of a transition length L = 10 nm. The NDR effect is very small in the gapless structure but significantly enhanced when EG = 130 meV, with a peak-to-valley ratio (PVR) reaching 2.6.

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for the design of this device in spite of promising performance when the doping levels, the doping profile and the bandgap may be controlled properly. We will see in the next sub-sections that it is possible to make the I –V characteristics weakly sensitive to the length L thanks to appropriate bandgap nanoengineering of graphene, in the form of nanoribbons or nanomeshes, or by the gate controllability of the potential profile in tunnel field effect transistors. 3.2 GNR PN junctions Cutting a graphene sheet into 1D nanoribbons of a few nanometer width is the most natural approach to generate a bandgap in graphene. It has been widely investigated both experimentally and theoretically to improve the behavior of graphene devices, though the total amount of current that can be carried in such ribbon is limited.

Fig. 4 Schematic view of a (a) normal and (b) T-shape AGNR PN junction. MC and MT are the numbers of carbon chains between edges in the contact and the transition regions, respectively

Fig. 5 LDOS in (a) a normal junction (with MC = MT = 21) and (b) a T-shape junction (with MT = 29). (c) Transmission in the same structures as a function of energy. The structures are unbiased here

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Hence, one can think of designing an AGNR PN junction as schematized in Fig. 4(a) to get a high PVR, as suggested by the results obtained in the previous sub-section. However, there is no reason for this structure to make the I –V characteristics, and especially the PVR, less sensitive to the length L of the transition region. Regarding this point, we have to keep in mind that if the current peak is controlled by the interband tunneling, i.e. the bandgap in the transition region, the current valley is mainly controlled by the bandgap in the doped-regions which separates the interband tunneling and the thermionic current. Hence, if we can design a structure where the bandgap is finite in the doped regions and small or even zero in the transition region, we should be able not only to enhance strongly the PVR, but also to make the interband tunneling weakly sensitive to L. That is why we suggest to analyzing the T-shape armchair structure schematized in Fig. 4(b) where the width is larger in the transition region (with a small bandgap) than in the doped region (with a larger bandgap). Such GNRs made of alternate sections of different width have been previously suggested to generate a resonant-like behavior [12], or to enhance the thermoelectric properties of GNRs [13]. The AGNR width is defined by the number M of carbon chains between the two edges. For instance, in the case M = 21, the bandgap reaches 370 meV while it falls to 62 meV for M = 29. In Fig. 5 we compare the local density of states (LDOS) in a normal junction (with MC = MT = 21) and in a T-shape junction (with MT = 29). The evanescent states clearly observed in the transition region of the normal junction are not visible in the latter case, due to the very small bandgap in the central region. It follows that the transmission (Fig. 5(c)) is strongly enhanced in the interband tunneling energy range and is expected to be weakly dependent on the length L. As a consequence, as shown in Fig. 6, for a transition length L = 10.2 nm the peak current is even higher than in

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Fig. 6 I –V characteristics of two normal junction of different transition length L and one T-shape junction of length L = 10.2 nm. Other parameters: U0 = 0.7 eV

Fig. 7 Peak-to-valley ratio in normal and T-shape junctions as a function of the length of the central transition region. Other parameters: U0 = 0.7 eV

a normal junction of transition length L = 5.1 nm. Additionally, the valley current in the T-junction appears to be smaller than in the normal ones. It results in a behavior of the PVR very different in these structures, as seen in Fig. 7. It is not only much higher in the T-shape junctions but it also depends weakly on the length L, in contrast to the case of the normal junctions of uniform width. However, until now we have considered AGNRs with perfect edges, but the unavoidable edge disorder is known to generate localized states which usually degrade strongly the electrical performance of devices [6]. It is thus important to evaluate this effect here. In this order, defected structures have been generated by removing randomly 15 % of edge atoms along the structures. Some resulting I –V characteristics are shown in Fig. 8 and compared to that of the perfect structure. It is remarkable that the valley current is not enhanced but slightly reduced by the disorder and that though the transmission is perturbed the peak current remains high and a large PVR is still achieved. Indeed, the disorder tends to enlarge the bandgap in the doped regions, which is not bad

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Fig. 8 I –V characteristics of T-shape PN junctions (MC = 21, MT = 29) with perfect edges or with different disorder configurations defined randomly

for the device operation. The transition region being small here, it is not strongly perturbed by the random disorder. Hence, we have shown that it is possible to design AGNR tunnel diodes able to deliver a very high PVR of a few thousand at room temperature. Additionally, using appropriate edge engineering, the PVR may be weakly sensitive to the transition length and not strongly degraded by edge disorder. The remaining problem is actually the small current of a few micro-Amperes that can be driven in such narrow ribbons. Since the normalized current per unit of width is as high as the order of mA/µm, a good solution could be to arrange a large number of identical ribbons to form a single device capable of delivering high total current [47]. However, the fabrication of such a device with an array of well-defined narrow ribbons is certainly a strong issue. We will show in the next sub-section that this limitation may be ruled out thanks to the use of GNM lattices which offer similar possibilities of bandgap engineering but on large sheets of graphene. 3.3 GNM PN junctions By punching a graphene sheet to form a periodic lattice of nanoholes, the resulting GNM has a bandgap that depends on the neck width, the meshing orientation and the shape of holes. For a given hole shape, due to unavoidable disorder effects, experiments have shown that the bandgap EG tends to follow a universal dependence on the neck width Wn in the form of EG = α/Wn [48]. The bandgap may be typically higher than 0.4 eV for a hole distance of about 3 nm [34]. On this basis, we have considered a particular nanomesh lattice with nanoholes corresponding to the removal of 24 carbon atoms, separated by the distances Wx = 4.8 nm and Wy = 2.46 nm [25]. This configuration leads to the bandgap EG = 0.27 eV. This GNM has been used to generate two types of PN junctions: a normal PN junction fully designed on the GNM lattice, and a GNM heterostructure where the transition region is made of pristine (gapless) graphene and

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Fig. 10 ID –VDS characteristics of the TFET for the bandgap EG = 100 meV, the gate length LG = 20 nm and the BN gate insulator thickness of 4 nm

integration of nanodevices, especially when their operation is based on a tunnelling current. 3.4 Tunnel FET

Fig. 9 I –V characteristics of GNM PN junctions for different values of the length L of the transition region. (a) Normal junctions on GNM lattice. (b) Heterostructure GNM with transition region in pristine graphene

the doped regions are designed on nanostructured parts. The principle is the same as for the T-shape GNR junction: the gapless central region is expected to enhance the interband tunnelling and to make the peak current and the PVR weakly sensitive to the length L of the transition region. It is indeed what is observed in the I –V characteristics plotted in Figs. 9(a) and 9(b) for the normal and the heterostructure devices, respectively. In the former case, the results are shown only for the short length L = 5.5 nm and L = 11.1 nm. For higher L (not shown) the characteristics are strongly degraded. In contrast, the heterostructure PN junctions have excellent I –V characteristics even for the transition length as high as L = 44.3 nm, with a PVR of 105 instead of 197 for L = 11.1 nm, which is a remarkable and promising result with a view to achieving NDR devices on large sheets of graphene, with high peak current. One remaining issue that should be considered when designing GNM devices is related to the atomic edge disorder of holes. In principle, this disorder can affect detrimentally the output current. However, in a recent work [34], we demonstrated that the use of suitable GNM sections of finite length in the two doped regions allows us to avoid this effect to a large extent while good device performance is still obtained. Hence, this device seems to offer reduced problems of reproducibility and variability which are usually considered as a nightmare for the

In the tunnel FET, the interband tunneling between the Pdoped side and the N-doped side can be tuned by the gate voltage applied on the central region of the device [26]. It is considered as a promising device for low-power digital applications thanks to the steep subthreshold slope that can be less than the usual limit of 60 mV/decade at room temperature [49]. Additionally, it may exhibit an NDR in the I –V characteristics [50] as in the Esaki tunnel diode. Hence, the TFET has the ability to control the PVR, which may be useful for high-frequency applications [51, 52], especially if designed on graphene that is intrinsically a good material for high-frequency operation. We consider a TFET designed on a monolayer graphene sheet with finite bandgap, as in the case of Bernal stacking of h-BN and graphene layers. The operation and performance of this device has been recently described in details [26]. Here, we summarize the main results in terms of electrical characteristics. The I –V characteristics obtained for a gate length LG = 20 nm and a bandgap EG = 100 meV are plotted in Fig. 10 which shows clearly the modulation of the NDR behaviour, with a PVR reaching here the maximum value of 2.2. Actually, the effect is more pronounced for a bandgap of 200 meV that leads to a PVR of 8, but is smaller for a higher bandgap of 300 meV with the same doping levels (not shown). Indeed, it is difficult to switch off efficiently the valley current due to the remaining contributions of the thermionic current and interband tunnelling. Actually, with a uniform bandgap in the structure, it is not possible to achieve a PVR as high as in the best tunnel diodes studied above that benefit from an optimization of the band structure using position-dependent nanostructuring (Sects. 3.2 and 3.3).

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Fig. 11 ID –VDS characteristics of the TFET at VGS = −0.28 V for the bandgap EG = 200 meV and the gate length LG = 20 nm, compared to the I –V characteristics of the 2D PN diode of transition length L = 30 nm

However, this TFET is much less sensitive to the transition (gate) length than the PN diode designed on 2D graphene and compares very well in terms of PVR. Indeed, in Fig. 11 we compare the ID –VDS characteristics of the TFET with EG = 200 meV and LG = 30 nm for VGS = −0.28 V with the I –V characteristics of the “equivalent” PN diode of same transition length L = 30 nm. A much higher peak current is reached in the TFET while the valley current is even smaller than in the PN diode. It makes the TFET a versatile device with tunable non-linear characteristics that can make this device a good option to achieve circuits operating at very high frequency.

4 Conclusion and final remarks This has shown that beyond the potential of 2D graphene tunnel diodes to provide an NDR effect, the bandgap nanoengineering in the form of GNRs and GNMs with positiondependent band structure may be of great help to design high performance devices. In particular, by inserting a small or even zero bandgap section in between the P-doped and the N-doped sides of the junction, it is possible to enhance strongly the interband tunnelling in the peak current regime, while the valley current is efficiently limited by the finite bandgap in the doped regions. It leads to extremely high PVR that, in addition, is weakly sensitive to the length of the small-bandgap transition region. It is definitely a good point with a view to applications with reduced problems of reproducibility and variability. Thanks to the gate contact, the tunnel FET offers the additional advantage of making the peakcurrent and the PVR tunable if the bandgap of graphene is finite. In comparison with the 2D tunnel diode where the PVR is strongly dependent on the transition length, the gate controllability of the potential profile in the channel makes the PVR in the TFET higher and weakly sensitive to the gate length.

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The GNM bandgap nanoengineering opens the way of promising routes for designing high performance graphene devices for both radio-frequency and digital applications. To have the appropriate bandgap of 200 meV, a GNM lattice could be the basis of a TFET of arbitrary large width able to deliver high peak currents. If the gated region was made of gapless pristine graphene or of GNM of different neck width, the peak current could be further enhanced while the valley current would remain small thanks the finite bandgap in the highly doped access regions, making the PVR higher and/or tunable over a wide range. A GNM lattice could be a good option also for designing a conventional FET likely to deliver a high current and a high on/off ratio suitable for digital applications. Acknowledgements This work was partially supported by the French ANR through projects NANOSIM_GRAPHENE (ANR-09NANO-016) and MIGRAQUEL (ANR-10-BLAN-0304). The work at Hanoi was supported by the Vietnamese National Foundation for Science and Technology Development (NAFOSTED) under Projects No. 103.02.64.09 and 103.02.76.09.

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