Device structure

The advantages of graphene bilayer for the steep current switching can be fully realized in the structure of the TFET shown schematically in Fig. 2A. A heavily doped silicon substrate acts as a bottom gate used to create the transverse electric field and thus open and manipulate the band gap in GBL39. The oxidation of the substrate results in formation of SiO 2 layer playing the role of back gate oxide and substrate for graphene bilayer. Alternatively, the SiO 2 layer can be replaced with hexagonal boron nitride (hBN) possessing a small (~1010 cm−2) density of residual charged impurities40. A nanometre – thin layer of high-κ dielectric (e.g., zirconium oxide) covers the graphene channel and the top metal gates are formed above. The side gates near the source and drain contacts induce large densities of holes and electrons, respectively, which also leads to the formation of an abrupt tunnel junction and energy barriers (see Fig. 2B) for the thermally activated electrons and holes contributing to the OFF-state leakage current.

The operation of a normally open TFET switched off by a negative top gate voltage is illustrated in the band diagram, Fig. 2B. Application of positive voltage to the bottom gate, V B > 0, induces the band gap and provides an excess electron density in bilayer. The p+ doping of source emerges upon application of negative voltage U S < 0 to the source doping gate. An additional increase in the barrier height for the holes injected from the drain is achieved by applying positive voltage U D > 0 to the drain doping gate. It is instructive that application of high voltage to the doping gates does not result in increased power consumption as this voltage is not changed during the device operation. At zero top gate voltage, the valence band in p+–source overlaps with the conduction band in the n–type channel, which corresponds to the ON state (red band profiles in Fig. 2B). Upon application of negative voltage to the top gate, the transistor is switched off (dashed blue band profiles in Fig. 2B).

The optimization of the device dimensions aiming at the increase in the ON-state and reduction in the OFF-state currents is quite straightforward: both the effective thickness of the gate dielectric and the distance between the source doping gate and the control gate should be small. These distances are limited just by the possible gate leakage current (see below), we choose them to be d t = 2 nm and d g = 5 nm. The doping gate at the drain is used just to induce high barrier for thermally activated holes; the distance between this gate and the control gate should be large to reduce the transparency of tunnel junction at the drain and get rid of ambipolar leakage.

The fabrication of the device structure in Fig. 2A is technologically feasible with recent advances in the growth of graphene on hBN41. The most challenging operation is the formation of the gates at sub-10 nm distance, which is, however, achievable with the combination of self-assembled molecular and electron beam lithographic techniques42.

Model of the interband tunneling enhanced by van Hove singularities

Our modeling of graphene bilayer TFET relies on a self-consistent determination of the carrier density and band structure22 followed by the calculation of tunnel current under assumption of ballistic transport (see supplementary material, sections I–III). However, the principal dependence of the tunnel current on the gate voltage can be derived in a very simple fashion. The current is proportional to the number of electrons capable of tunneling between the valence band of source and the conduction band of channel. Once these band overlap by dE in the energy scale, the electrons available for tunneling in graphene bilayer occupy a ring in the momentum space (Fig. 1B, left panel). Their number is proportional to p min dE, where p min is the momentum corresponding to the bottom of the ‘Mexican hat’. One thus concludes that the tunnel current is a linear function of the band overlap which, in turn, is a linear function of the gate voltage. This contrasts with the 2d materials having parabolic bands where the number of electrons available for tunneling is proportional to (Fig. 1B, right panel). As a result, the current in TFETs based on these materials is proportional to the gate voltage raised to the power 3/2.

A rigorous expression for the tunnel current density involves an integral of the single-particle velocity v || = dE/dp || timed by the barrier transparency and the difference of occupation functions in the valence and conduction bands f v (E) − f c (E) over the momentum space d2p = 2dp ⊥ dp || 10:

Here, g s g v = 4 is the spin-valley degeneracy factor in graphene, p max (E) is the maximum transverse momentum of electron at a given energy E, p max (E) = min{p c (E), p v (E)}, where p c (E) and p v (E) are the inverse functions to the electron dispersion in the conduction and valence bands. The limits of integration over energy are the conduction band edge in the channel, E c and the valence band edge in the source, E v . The factor of two before the quasi-classical barrier transparency comes from the presence of two turning points with zero group velocity in the GBL dispersion at which an electron attempts to tunnel.

The effect of ‘Mexican hat’ on the current switching steepness can be traced analytically from Eq. 1 by assuming that the conduction band states are empty, valence band states are occupied and the barrier transparency weakly depends on the energy and transverse momentum. At small band overlap, the momenta of the tunneling electrons in graphene bilayer are close to p min (Fig. 1B, left panel), which results in

This linear dependence is in agreement with the above qualitative considerations. Previously, such a dependence of the tunnel current on the band overlap was attributed just to the 1D semiconductor structures which proved to be among the best candidates for the TFETs5,7,43.

An additional increase in the graphene bilayer TFET subthreshold steepness occurs due to the dependence of the transparency on the junction field and, hence, gate voltage. The transparency is evaluated by integrating the imaginary part of the electron momentum inside the band gap, which results in (see Supporting information, section III)

where Im p || (E = 0) is the imaginary part of electron momentum evaluated at the midgap and l is the length of the classically forbidden region (tunneling path length). The latter is given by for p ⊥ < p min and l = 2E(p ⊥ )/eF for p ⊥ > p min , where F is the electric field at the junction found from the solution of Poisson’s equation and is the band gap in the GBL. The thermionic leakage currents were evaluated with equations similar to (1) by setting the unity transmission probability and constraining the energy integral to the particles with the energies above the barrier.

Characteristics of the graphene bilayer TFET

The calculated room-temperature J(V G )–characteristics of graphene bilayer TFET at different drain bias V D are shown in Fig. 3A. The current density just above the threshold voltage is a linear function of V G , in agreement with the simple density-of-states arguments and Eq. (2). With increasing the top gate voltage, the slope of J(V G )-curve slightly increases due to the sensitivity of the tunnel barrier transparency to the junction field. The subthreshold slope at V G = V th reaches (20 μV/dec)−1 and is limited by the small thermionic current mA/μm and the gate leakage current mA/μm. About 100 mV below the threshold, the ambipolar leakage at the drain tunnel junction becomes pronounced; this can be, however, minimized by placing the drain doping gate at large distance from the control gate.

Figure 3 Calculated room-temperature gate transfer (left) and current-voltage (right) characteristics of graphene bilayer TFET at fixed bias voltages at auxiliary gates: V B = 3.3 V, U S = −0.6 V, U D = 0.25 V. Top gate dielectric is 2 nm ZrO 2 , κ = 25, back gate dielectric is 10 nm SiO 2 , spacing between the source doping and control gates d g = 5 nm, spacing between drain doping and control gates is 10 nm. The regions highlighted in yellow correspond to the drive voltage swing of 150 mV, in which sufficient ON/OFF ratio and high ON-state current are achieved. Inset: gate transfer characteristic in the log scale. Full size image

The drain characteristics of graphene bilayer TFET shown in Fig. 3B demonstrate a pronounced current saturation typically absent in single graphene layer FETs. This saturation is due to the limited energy range in which the tunneling injection is possible. At very high drain bias ~600 mV, the barrier for thermally activated holes at drain junction is sufficiently lowered, which leads to the further increase in current. At negative drain bias, the transistor current can be viewed as that of p+ − n+ tunnel diode between source and channel. The emerging negative differential resistance is due to the dependence of the band edges in the channel on the amount of injected carriers: at high electron density, the bands in the channel are lifted upwards, which reduces the source-channel band overlap and switches the tunneling off.

For low-power applications, the maximization of the highest subthreshold slope is not as important as minimization of the supply voltage V S required to switch the transistor between the ON- and OFF-states. Considering the current at V G = 0 V as the ON-state current (J ON = 0.8 mA/μm in Fig. 3B at V D = 0.15 V) and the leakage current as the OFF-state current (J ON /J OFF = 3.5 × 104), we have obtained V S = 150 mV. In a conventional MOSFET, the gate voltage swing V S ≥ 285 mV is required to achieve the same current switching ratio. The average subthreshold slope of our TFET over 4.5 decades of current is 33 (mV/dec)−1. With this characteristic, it outperforms all sub-thermal tunnel switches2 based on silicon4, germanium6, III-V hetero junctions5 and carbon nanotubes7 reported to date. Only recently a vertical TFET based on MoS 2 /germanium junction with a similar value of the average subthrehold slope was demonstrated44, however, its ON-state current density of 1 μA/μm leaves much to be desired.

The aggregate quality of the TFET, accounting for both average subthreshold slope and current density, can be characterized by an I 60 –figure of merit45 which is the current density at the point where the subthreshold slope equals (60 mV/dec)−1. While the best I 60 reported to date equals 6 nA/μm (InAs nanowire/Si heterojunction TFET5), in our TFET structure I 60 = 150 μA/μm.

The unique characteristics of GBL TFET surpassing the existing TFETs are enabled by the three factors. First of all, it is the small extrinsic band gap (for doping gate voltages used in Fig. 3, eV) that guarantees elevated tunneling probability ( ~0.1) and large current density. It is remarkable that there exists a lower limit of the interband transparency in GBL due to the saturation of the band gap at high transverse field, , where γ 1 ≈ 0.4 eV is the interlayer hopping integral and v 0 ≈ 106 m/s is the band velocity. Such transparency is sufficient to reach appreciable ON/OFF ratio and it still enables pronounced ON-state current. At the same time, most semiconducting monolayers have large intrinsic band gaps (1.9 eV for MoS 2 , 1.3 eV for WS 2 , etc.), while in the 2D structures based on III–V materials being narrow-gap in the bulk, the gap value increases significantly due to the quantum confinement46. Secondly, the singular DoS near the band edges allows an abrupt switching of tunnel current. Even if there existed a parabolic-band 2D material with the same band gap and the same barrier transparency in the TFET structure, its current density would be given by (see supplemental material, section IV)

where m c and m v are the conduction and valence band effective masses and, similar to the derivation of Eq. (2), we have assumed the barrier transparency 0 to be energy- and momentum independent. Last but not least, it is large density of states in GBL growing linearly at high energies that contributes to the high ON-state current. The numerical comparison of current density in graphene bilayer and its equivalent parabolic band counterpart is presented in Fig. 4 for the effective mass values typical for narrow-gap III–V semiconductors (m c = 0.024m 0 , m v = 0.026m 0 for InAs). At 150 mV gate voltage above the threshold, the current density in graphene bilayer exceeds 15 times that in a parabolic-band material. The factor of two is due to the valley degeneracy absent in III-V’s, another factor of two is due to the tunneling at two turning points of the ‘Mexican hat’ dispersion and the remainder of 3.5 is due to the finiteness of electron momentum at the edge of the ‘Mexican hat’.

Figure 4 Comparison of the gate transfer characteristics of GBL TFET and a TFET based on an equivalent 2D semiconductor with the same barrier transparency D 0 , but with different (parabolic) band structure. Numerical values of the effective masses are taken for bulk InAs. The insets show the band diagrams overlaid with electron-hole spectra and the energy dependence of DoS. Full size image

Gate leakage and band tailing: the insulator selection rules

The steep switching of the tunnel current by the gate voltage can be masked by the leakage to the gates, band-tail and trap-assisted tunneling19,20,21. The latter factors might have masked the onset of the interband current in the recent measurements of graphene bilayer tunnel junctions38,47. A careful selection of the gate dielectrics providing high interface quality is required to minimize these effects.

The main reason for the band tailing comes from the fluctuations of electric potential produced by the random charged defects or dopants37. This effect is most pronounced in the TFETs with source and drain intentionally doped chemically. In the TFETs with electrically doped contacts, only residual charged impurities inevitably present in the substrate contribute to the band tailing. To provide a quantitative view on the band tailing in graphene bilayer on different substrates, we have evaluated the quasi-classical DoS ρ(E) in the presence of fluctuating potential by integrating the singular ‘bare’ DoS ρ 0 over the probabilities of voltage fluctuations37

where 〈V2〉 is the root-mean-square amplitude of the voltage fluctuations proportional to the impurity density n i . The calculated energy dependencies of the ‘smeared’ DoS are shown in Fig. 5. For the parameters of chemical doping used in the pioneering proposal of the GBL TFET36, n i = 4 × 1013 cm−2, the conduction and valence bands almost merge together, which would result in a poor OFF-state, nothing to say about high switching steepness. A slight peak in the DoS near the band bottom becomes noticeable already at impurity density of 5 × 1012 cm−2 which corresponds to the low-quality graphene on SiO 2 substrates. In graphene samples on a high-quality SiO 2 48, the smearing of the band edge is order of 10 meV. The ultimate band abruptness of ~5 meV can be achieved in graphene samples encapsulated in boron nitride, providing the residual impurity density of ~5 × 1010 cm−2 40. At this limit, the fluctuation-induced smearing of the bands becomes negligible and the behavior of the DoS near the bottom of the “Mexican hat” is governed by the trigonal warping distortions of electron spectrum due to the next-nearest neighbor interactions22. Using the exact spectrum of GBL with trigonal warping, we estimate the energy scale where the trigonal warping is relevant as δε ≈ 20 meV. Already for relatively small gate voltages, V G − V th > δε/e, these corrections are irrelevant and the linearity of the J(V G )–characteristic holds.

Figure 5 Calculated energy dependencies of the DoS in the conduction band of graphene bilayer at different densities of charged impurities (corresponding to the substrates of different quality). The electron density is held fixed at 4 × 1013 cm−2, the nominal energy gap is 0.3 eV. Full size image

The gate leakage may also limit the minimum achievable OFF-state current, while at the same time small effective gate oxide thickness is required to efficiently control the band structure in the channel by the gate voltage. Among the common high-κ materials, zirconium oxide (κ ≈ 25) looks as an optimal solution for the GBL TEFT due to the large band offset with respect to graphene (U b = 2.9 eV49) and elevated tunneling mass m t ≈ 0.3m 0 50. We have evaluated the leakage current from graphene with electron (hole) density of n e(h) into the metal gate to be (see Supporting information, section VI)