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Ab initio perspective of ultra-scaled CMOS from 2D-material fundamentals to dynamically doped transistors

Abstract

Using accurate dissipative DFT-NEGF atomistic-simulation techniques within the Wannier-Function formalism, we give a fresh look at the possibility of sub-10-nm scaling for high-performance complementary metal oxide semiconductor (CMOS) applications. We show that a combination of good electrostatic control together with high mobility is paramount to meet the stringent roadmap targets. Such requirements typically play against each other at sub-10-nm gate length for MOS transistors made of conventional semiconductor materials like Si, Ge, or III–V and dimensional scaling is expected to end ~12 nm gate-length (pitch of 40 nm). We demonstrate that using alternative 2D channel materials, such as the less-explored HfS2 or ZrS2, high-drive current down to ~6 nm is, however, achievable. We also propose a dynamically doped field-effect transistor concept, that scales better than its MOSFET counterpart. Used in combination with a high-mobility material such as HfS2, it allows for keeping the stringent high-performance CMOS on current and competitive energy-delay performance, when scaling down to virtually 0 nm gate length using a single-gate architecture and an ultra-compact design (pitch of 22 nm). The dynamically doped field-effect transistor further addresses the grand-challenge of doping in ultra-scaled devices and 2D materials in particular.

Introduction

Scaling and Moore’s law, that sets the footprint area of a transistor to scale by a factor 2, that is the transistor gate length L to scale by a factor √2, every 2 years, have been the driving force of the electronic industry1. Today L has been scaled well below 20 nm and further scaling has become increasingly difficult owing to short-channel effects (SCE) that degrade the subthreshold slope (SS) of a transistor (i.e., the efficiency with which the current is switched from off to on state by changing the gate bias). SCE leads to an increased off-state leakage current, IOFF. To mitigate SCE, i.e., to keep good electrostatic control of the gate over the transistor channel, its thickness, ts, has to be scaled as well1,2. Also, transistors have evolved from planar single-gate transistors to 3D multi-gate devices such as FINFETs3, nanowires4,5, and nanosheets6. As a rule of thumb, in a multi-gate device, the channel thickness ts has to be of the order of ½ L in order to keep the electrostatic integrity leading to ts of a few nm only in modern advanced nanoscale technologies7. At such value of ts, conventional 3D semiconductors, like Si, or possible high-mobility channel-replacement materials like Ge3 or III–V5, suffer from quantum-confinement effects that strongly deteriorate their performance (e.g., current drive, gate coupling, mobilities…)8,9,10, as well as, lead to increased variability (e.g., strong threshold-voltage variations with surface roughness for instance)2,10,11. It is commonly accepted that conventional dimensional scaling will stop for L of the order of 10 nm. The current international roadmap for device and system (IRDS) predictions have actually forecast that gate-length scaling will stop for a L of 12 nm (https://irds.ieee.org/editions/2018).

As an attempt to further push the scaling, transistors made of novel 2D materials12, i.e., an atomistically thin layer of material that does not create strong atomic bonds in the 3rd dimension, such as transition-metal dichalcogenide (TMD)13,14,15 or black phosphorus (P4)13,16, are being actively investigated as future replacement of Si as channel material. These materials would offer the ultimate electrostatic control and are free from quantum confinement due to their 2D nature. In principle also, their thicknesses could be well controlled, which would remove the variability issue. The research development on 2D material is still at an early stage today. Despite the ever-growing list, including several thousands of newly discovered such materials17, an ideal complementary metal oxide semiconductor (CMOS) candidate for the sub-10-nm channel has not yet emerged. Experimentally, only a subset of 2D-material transistors, such as those using MoS2, WS2, WSe2, and P413,14,16, have been explored and the current drive is typically too low for high-performance (HP)-CMOS applications (https://irds.ieee.org/editions/2018). Although the low drive-current is, at least in part, related to the immaturity of the technology, the fundamental physics and performance of these transistors are not yet fully elucidated. Even using 2D materials, scaling L below 10 nm is further complicated by an additional quantum-mechanical SCE. This effect, called source-to-drain tunneling (SDT), which is the ability of the electronic quantum-mechanical waves to evanescently leak through the channel barrier, further degrades SS and IOFF.

Here, using our state-of-the-art density functional theory–non-equilibrium Green’s function formalism (DFT-NEGF) ATOmistic modeling Solver (ATOMOS)18, we offer unique insights on several 2D-material physics and performance, including those of the less-explored HfS2 and ZrS2 that feature appealing performance for ultra-scaled CMOS. We demonstrate the possibility of L = 6 nm high-performance devices, providing that high doping can be achieved. Finally, we demonstrate that further geometrical scaling, down to a virtually 0 nm gate length footprint, is possible using the dynamically doped field-effect transistor (D2-FET). The D2-FET concept further addresses the difficult challenge of doping in nanoscale devices (https://irds.ieee.org/editions/2018) and 2D materials in particular19, and the need for chemical doping could be suppressed.

Results

Device structure and methodology

The schematic of the studied monolayer (1 ML) double-gate (DG) MOSFETs is shown in Fig. 1. An intrinsic channel of length L is used. The source- and drain- (S&D) extension regions are doped with a concentration NSD. An abrupt junction profile is assumed. The 2 nm thick HfO2 gate oxide has a relative permittivity εR = 15.6 and an equivalent oxide thickness (EOT) = 0.5 nm. The work function of the gate-voltage bias, VG, is typically adjusted to achieve a fixed IOFF value at VG = 0 V. A low-K spacer oxide with εR = 4 surrounds the S&D extensions. Ohmic contacts are assumed with S&D bias VS = 0 V and VD, respectively.

Fig. 1: Schematic view of a double-gate MOSFET, where the channel is made of one of the monolayer 2D materials studied here.
figure 1

The doped contact, oxides (gate and spacer regions), as well as the main device parameters, are shown on the figure. The atomic structure that is depicted in this figure is that of a TMD, here it is HfS2, with the metallic atom (Hf) in the center, sandwiched between the two chalcogen (S) atoms at the top and bottom.

The 1st step toward transport simulations of a given material is a first-principle geometry relaxation of its primitive unit cell, followed by an electronic structure calculation. We used the DFT package Quantum ESPRESSO20 and the generalized gradient approximation with the optB86b exchange-correlation functional21. The Bloch wavefunctions are then transformed into maximally-localized Wannier functions (MLWF) typically centered on the ions using the wannier90 package22. Figure 2 demonstrates the validity of our MLWF representation for the case of 3 of the 2D materials studied here. The resulting supercell information, including atoms and MLWF positions, lattice vectors, as well as the localized Hamiltonian-matrix elements, are used by ATOMOS as building blocks to create the full-device atomic structure and Hamiltonian-matrix. Transport calculations are then performed using a real-space NEGF23,24 formalism including electron–phonon (e-ph) scattering within the self-consistent Born approximation25. More details can be found in the method section.

Fig. 2: Band structure computed with QUANTUM ESSPRESSO using plane-wave DFT and with ATOMOS using the Wannierized Hamiltonian.
figure 2

Band structure for a a monolayer WS2 (2H-phase) b a monolayer HfS2 (1T-phase) and c a monolayer ZrS2 (1T-phase). The insets also show the atomic structure and chosen cartesian-axes directions for the various supercells.

Transport model requirement for 2D-material screening

We first focus on evaluating intrinsic monolayer (1 ML) 2D-material device physics and performance using our DFT-NEGF model. The goal is to find a meaningful upper limit to identify the trends and screen the most promising candidates for scaled CMOS applications. Looking at the list of existing TMDs and other 2D materials, we have pre-selected five TMDs13,15,17, as well as P416,17, owing to their relevant electronic and transport properties (band structure, phonon properties, material stability) and/or experimental relevance. For the TMDs, we focus here on MoS2, WS2, and WSe2 in the trigonal prismatic (2H) phase, as these materials are among the most studied and mature experimentally. We also focus on the less-explored HfS2 and ZrS2 TMD’s (in their most stable octahedral, 1T, phase), as their band structures hint for better transport properties (higher drive current), whereas retaining a sufficiently high bandgap and balanced properties15,17 to expect good off-state currents at scaled gate lengths. Finally, current-voltage characteristics for P4 will be investigated here as well, owing to the strong attention and expectation this material has stirred in the recent literature16.

Non-atomistic models, such as effective-masses (EM) and derived simplified two-bands NEGF models, although widely used26,27,28 owing to their wider availability and strongly reduced computational cost, are typically inaccurate to model 2D materials. In Fig. 3, it can be seen that although a reasonable fitting of the lowest part of the band structure can be obtained for WS2 using an effective-mass Hamiltonian model (Fig. 3b), the current is strongly over-estimated (Fig. 3a). This is owing to the combination of two facts. First, owing to their specific band structures, e.g., many TMD’s (in particular in the conduction band of that made of W or Mo atoms) have narrow energy valleys with discontinuous density-of-state, DoS, profiles that are not captured with simplified band models (e.g., see Fig. 3b, where the 1st WS2 conduction-band valley is a narrow valley with an energy extend that is <0.6 eV in the KΓ direction)29. Second, in an extremely thin material, a full atomistic treatment of how the charge is distributed within the 2D layers is required to accurately capture the charge-centroid position. For the case shown in Fig. 3a, as typically the case in TMDs, ~90% of the charge is located on the metallic (W) atom, which is in the middle, not on the surface chalcogenide (S) atoms (Fig. 1). This information is lost in a non-atomistic model, and a homogeneous charge distribution with a centroid closer to the surface is obtained. The effective-mass model is also not able to predict accurately SDT, hence the subthreshold characteristics of the device (Fig. 3a), a crucial effect for the sub-10 nm gate-length regime where 2D materials are envisioned to be used for CMOS. Finally, using these approximate EM and derived 2-bands NEGF models, it was assessed that a bi-layer DG device could deliver more drive-current than that of a 1 ML device for the sub-10 nm HP -CMOS application27,28. Our DFT-NEGF results, however, show that a 1 ML material, which is the main focus of this paper, is preferred (more details can be found in Supplementary Note 1). For accurate results, full-band atomistic-transport simulations, such as the DFT-NEGF results presented here, are therefore needed.

Fig. 3: DFT vs fitted effective-mass (EM) model.
figure 3

a Drain-current–gate-voltage, ID(VG), characteristics of a L = 5 nm 1ML-WS2-DG nMOSFET computed with DFT and a fitted EM model (including the two first valleys seen in Fig. 3b). VD = 0.6 V. b DFT-computed band structure around EC (blue line) and that fitted with the 1st valley (centered at K-point) isotropic effective mass = 0.4 m0 (red stars), m0 being the free electron mass. Despite that a good agreement between the EM- and the DFT-band structure model is achieved in the vicinity of EC, the EM-NEGF model strongly overestimates the current drive in the device. The second valley (located between K and Γ) was also fitted with an EM of 0.9 m0 and included in the effective-mass NEGF model.

It, however, turns out that for 2D materials, ballistic full-band transport simulations are not accurate enough, even at a gate length as short as 5 nm. The argument often used of very-short L to justify ballistic transport in conventional 3D-material transistors does not hold true here. In 2D-materials with narrow valleys, a valley-filtering effects typically strongly reduce the ballistic current and inelastic-scattering effects need to be included to recover the current in the device29 as shown in Fig. 4a for a bi-layer WS2 transistor. In other cases, like for black phosphorus, strong optical-phonon modes can significantly reduce the current compared with the ballistic case (Fig. 4b). Thus, a full-band dissipative atomistic treatment, as presented here is needed to get an accurate and meaningful upper limit of 2D-material devices. This upper limit may still be far from today’s reality, as we are neglecting interactions with the environment (e.g., contact resistance, surrounding oxides…), and defects that are usually strongly present in nowadays experimental devices13,16,19. It, however, gives an insight of what is to be the fundamental potential of such a technology, as it matures19,30.

Fig. 4: Impact of electron–phonon scattering in 2D-material transistors.
figure 4

a ID(VG) characteristics of a L = 14 nm bi-layer WS2-DG nMOSFET computed with DFT-NEGF with and without e-ph. VD = 0.7 V. As q.VD is larger than the width of the bi-layer first valley, the 1st valley electrons cannot travel ballistically from the channel to the drain and the ballistic current is reduced compared with the case with scattering. b ID(VG) characteristics of a L = 5 nm, 1ML-P4-DG nMOSFET computed with DFT-NEGF with and without e-ph. VD = 0.6 V. Owing to the strong optical-phonon coupling (DOP = 170 meV/nm, ħω0 = 32 meV)43, the drive current drops significantly when e-ph is included, despite the very-short-channel length. IOFF = 10 nA/μm. ΓY (zigzag) channel orientation.

DFT-computed material parameters and properties

Supplementary Table 1 (in Supplementary Note 2) gives the relaxed unit-cell dimensions and bandgaps we obtained for the TMDs studied here. They are in good agreement with other DFT calculations in the literature15,17 and experimental results (http://www.hqgraphene.com/All-Semiconductors.php).

From our NEGF simulations, we have also extracted the electron or hole concentrations vs the Fermi level, EF, position with regards to conduction- or valence-band edges, EC or EV, respectively, i.e., EF − EC or EV − EF. By fitting those to an analytical 2D-DoS model, the conduction- or valence-band DoS, N2D, as well as an equivalent DoS mass, mDoS, can be computed. Both values are reported in Supplementary Table 1 for the TMDs studied here. This mDoS folds the DFT-computed non-parabolicity of the occupied bands close to the conduction- or valence-band edges into a simplified, equivalent, single-band, parabolic effective-mass model (the details are in the method section). Supplementary Fig. 4 shows, for a representative sample, the good level of agreement that can be achieved between the analytical charge model and the DFT-NEGF-simulated data. N2D, or equivalently mDoS, as well as the mobilities, that we will extract next, are useful quantities for developing simplified TCAD or compact models and benchmarking 2D-material performance31. Note that, in this paper, densities, as well as doping concentrations, are given per unit of volume. Those can be converted to the per unit of area, often used for 2D materials, by multiplying by the 2D-film thickness, tS, ~0.6 nm for a monolayer TMD (the exact value used for each studied monolayer can be found in Supplementary Table 1).

Figure 5 shows the long-channel low-field intrinsic electron and hole mobilities. Each mobility curve was extracted from 4 to 7 DFT-NEGF simulations using devices with various channel lengths, typically ranging from 100 nm to 1 μm, using the dR/dL method32 to correct for the ballistic resistance. The intrinsic mobility is a convoluted value, resulting from band structure (intrinsic transport properties) and electron–phonon (e-ph) scattering. It is a key-performance metrics for long-channel devices. For electron-phonons, we used the DFT-computed values of ref. 15 and included the dominant acoustic, and optical modes. For the 1 T TMDs that are polar materials, we also include the polar-optical phonons (POP) using a Fröhlich model33,34 that considers the electronic screening and the long-range interaction up to 3 nm (more details can be found in Supplementary Note 3). We note that the long-range LO polar component can be directly computed using DFT calculations, as it was done in ref. 35 and as can be seen in Supplementary Fig. 14b. However, a Fröhlich model provides a direct way to consider the screening, which is important and was neglected in ref. 35. 1T TMD material mobilities could also suffer, in principle, from a non-vanishing ZA phonon term owing to the lack of horizontal mirror symmetry36. In order to verify this, we have computed the HfS2 electron–phonon matrix elements from DFT, using a similar approach that the one used in ref. 35. We have found that ZA phonons do contribute in HfS2, but that their contribution is small compared with that of the other acoustic phonon modes (TA and LA, see Supplementary Fig. 14a). A similar conclusion was found for both the mobilities of HfS2 and ZrS2 in ref. 35.

Fig. 5: Electron–phonon-limited mobility.
figure 5

Intrinsic electron–phonon-limited mobility vs carrier concentration in a n- and b p-type TMD transistors. The mobilities were extracted from our DFT-NEGF simulations using long-channel devices (for several channel L ranging from 100 nm to 1 μm typically) at VD = 50 mV. The L-independent ballistic resistance was removed from the extraction by using the dR/dL method32. HfS2 and ZrS2 results are shown along the ΓK channel orientation and include polar-optical phonons using a Fröhlich model.

For the 2H TMDs, the mobility curves typically present a plateau region dominated by intra-band low-energy acoustic e-ph-scattering. At higher carrier concentrations, when the position of the energy band with regards to EF is sufficiently degenerated so that satellite energy valleys start to be populated, higher-energy intervalley e-ph scattering mechanisms further degrade the mobilities. The carrier concentration at which this degradation happens depends on the energy separation between the 1st and the satellite valleys and the mDoS (a larger mDoS leads to less degeneracy at a given concentration as illustrated in Supplementary Fig. 4). The NEGF-computed mobility values for the 2H TMDs are in qualitative agreement with mobilities calculated in the literature with various methods35,37,38, showing the same order of magnitude and ordering. WS2 has the highest mobility. For p devices, WSe2 also features an interesting value.

For the 1T TMDs, the plateau region is not observed in the mobility curves. Their mobility rather increases for increasing carrier concentration. Supplementary Fig. 7 compares the n- and p-type HfS2 total mobilities including POP and screening to that without POP and that with POP but neglecting the screening. The total mobility value is limited by the strong, and nearly unscreened, POP interaction at low carrier concentration. As the carrier concentration increases, however, screening renders POP scattering less efficient and the mobility increases toward the limit without POP. It is to be noted that the n-type mobility value of 1896 cm2/V s that we obtain in the plateau region for the case without POP well agrees with the ~1800 cm2/V s acoustic phonon-limited value computed in refs 35,37. The n-type low mobility value of ~60 cm2/V s obtained for the case that included POP, but neglecting the screening, is consistent with the results and hypothesis presented in ref. 35.

Finally, as presented in Supplementary Fig. 8 for HfS2, for scaled nanoscale devices, the impact of POP scattering is not significant, owing to the short-channel lengths and the strong screening related to the high carrier concentration in on-state. The impact of high-energy optical phonons is typically rather limited in the subthreshold regime of a scaled transistor. Despite the weak screening in the channel, related to the low subthreshold-carrier concentration, most of the electrons are injected at an energy in thermal equilibrium with the top-of-the-channel barrier (e.g., see Fig. 6). The majority of empty states, in which electrons could scatter to, are, however, localized at the same energy. Hence, low-energy acoustic phonons are rather the dominant scattering mechanism in the subthreshold regime. In on-regime, the situation is different (e.g., see Fig. 7a) but POP is effectively screened. From the above discussion and results, one concludes that for nanoscale devices with strong polar interactions, the high-density screened mobility is likely to be the relevant one. The rather high mobilities obtained for HfS2 and ZrS2, at high carrier concentration (in on-state carrier densities of several 1020 cm−3 are typical), highlights their interesting transport properties.

Fig. 6: Impact of source and drain doping on internal properties of a 2D MOSFET in off-state.
figure 6

Current spectrum J(E, x) (surface plot), as well as top conduction-band (EC) (-) edge along the channel direction, x, of a HfS2 nMOS with L = 5 nm, in off-state. a For NSD = 1 × 1020 cm−3. In this case, the lower NSD value is not sufficient to ensure degeneracy of the extension as can be seen at the source and drain-sides, where the conduction band is above the Fermi levels, EFS and EFD, respectively. Both Fermi levels are indicated by a red dashed line. b For NSD = 3 × 1020 cm−3. The resulting narrower channel barrier allows for a larger part of the current spectrum to tunnel under the channel barrier (SDT). This effect is enhanced for larger NSD values as the effective-channel length is reduced. VD = 0.5 V.

Fig. 7: Impact of source and drain doping on internal properties of a 2D MOSFET in on-state.
figure 7

Current spectrum J(E, x) (surface plot), as well as top conduction-band (EC) (-) edge along the channel direction, x, of the L = 5 nm HfS2 nMOS in on-state. a For NSD = 1 × 1020 cm−3. In this case, due to the lower NSD value, the current flow is limited by the source conduction band and source starvation is observed in the ID(VG) characteristics. b For NSD = 3 × 1020 cm−3. In this case, the current is still limited by the channel barrier that is well controlled by the gate. Current saturation is not yet observed in the characteristics at this gate voltage. VD = 0.5 V. VG = 0.6 V.

By definition, the current is the product of the number of mobile charge carriers, that is proportional to mDoS, times their velocity, that is proportional to the mobility. Hence the mDoS × mobility product of a given material is an indication of its MOSFET drive-current potential. This product, normalized so that it is equal to 1 for the nWS2-case, is reported in Supplementary Table 1 for n- and p-type conduction and allows a relative comparison between the different TMDs reported here. Again, the drive potential of HfS2 and ZrS2 stands out, while WS2 comes in 3rd position.

Sub-10 nm fundamentals

2D materials are, however, envisioned to be used in the sub-10-nm gate-length regime as potential replacement for Si. At such L, the mobility × mDoS product alone is not a sufficient metrics to compare performance. Other effects such as SDT, that become important owing to the narrow channel barrier, typically penalize more high-mobility materials (as the ability of quantum-mechanical tunneling is enhanced in low-effective mass materials) and a trade-off exists. The case of P4 that we will further discuss below is a good example. Similarly, a recent publication has used DFT-NEGF simulations to screen 100 2D materials and found 13 potential candidates with very high-drive current potential at L = 15 nm (pending a detailed study on the impact of e-ph scattering)39. When scaling L down to 5 nm, however, the SS of these devices were all degraded and ranging in the 110–275 mV/decade using a DG architecture. It is to be noted that the dynamically doped-transistor concept, that will be studied in the last part of this manuscript, might be a way to utilize the strong drive potential of such materials at further scaled dimensions.

Figure 8 compares the ID(VG) characteristics of L = 5 nm DG n- and p-MOSFETs made of six different 2D materials (Fig. 1), the five TMDs previously studied as well as P4, at a typical HP off-state leakage of IOFF = 10 nA/μm (https://irds.ieee.org/editions/2018). We further benchmark their performance against that of an optimized Si n-type gate-all-around nanowire (GAA) with a square cross-section, but with a relaxed gate length (L = 12 nm)40. The optimized L = 5 nm Si-GAA ID(VG) is shown as well. The GAA were simulated using cleaned mode-space sp3d5s* tight-binding NEGF models34,40.

Fig. 8: ID(VG) characteristics and SS(VG) (inset) of the optimized L = 5 nm 2D materials DG-MOSFETs, and the L = 5 and 12 nm Si optimized GAA.
figure 8

a n- and b pMOS transistors. |VD | = 0.6 V. IOFF = 10 nA/μm. e-ph scattering is included. The devices are optimized in terms of source- and drain-doping concentration (NSD), channel orientation (if anisotropic) and thickness (tS) for the GAA. HfS2 and ZrS2 IV’s are shown along the ΓK channel orientation. The current is normalized by the gate perimeter.

For Si, scaling L below 10 nm typically results in SS and ION degradation. This is due to electrostatic control losses, quantum confinement, and SDT. SDT and QC become significant for L < 10 nm and tS < 4 nm, respectively. It was observed that e-ph scattering was significantly enhanced, even at short L, for Si-GAA with tS < 4 nm8,10,25, one reason being the increase of the electron–phonon wave-function overlap in strongly confined wires due to volume inversion10. This is one of the fundamental reasons behind the strong mobility reduction observed in thin-film 3D materials. For the 5 nm long Si device, the narrow tS of 3.5 nm used in the simulations will further result in a strong threshold-voltage variability related to surface roughness induced bandgap change with QC2,10,11.

For all the 2D materials shown on Fig. 8, excepted for the p-type P4-device case that will be discussed below, we observed less ION and SS degradation than for Si, when scaling L down to 5 nm. This is related to their excellent electrostatic control (a better electrostatic control enables a larger effective-channel length at same nominal L, hence less SDT) and QC-free characteristics stemming from their 2D nature.

The outstanding performance of the two group-IV TMDs, i.e., of HfS2 and ZrS2, that feature, by far, the highest on-current levels is also highlighted on the plot. Besides the afore-mentioned excellent electrostatic control and confinement-free characteristics, this is related to HfS2 and ZrS2 well-balanced transport properties that allow for high ION with limited SDT. The closely matched characteristics of both HfS2 and ZrS2 can be understood in light of their similar band structure (Fig. 2b, c) and the mitigation of the mobility × mDoS product (that is better for HfS2) at very-short L. It is to be noted that HfS2 and ZrS2 have anisotropic band structure properties. Figure 9 shows the impact of crystal orientation on the performance of the HfS2 and ZrS2 n-type devices. Their performance is not varying much along the principal-axis directions shown here. Overall, group-IV TMDs show promise for scaled HP CMOS.

Fig. 9: ID(VG) characteristics of group-IV TMD along principal crystallographic axes.
figure 9

ID(VG) characteristics of group-IV TMD nMOSFETs for three different channel orientations along principal crystallographic axes (ΓK, ΓM, and KM), a for HfS2 and b for ZrS2. VD = 0.6 V. L = 5 nm. IOFF = 10 nA/μm. e-ph scattering is included.

From the more studied group-VI TMD family, WS2 emerges as the best candidate for n and p on average, i.e., second best and close to MoS2 for n, and best with WSe2 for p. MoS2 performs poorly for p-type, whereas WSe2 performs poorly for n-type conduction. This can be correlated to the intrinsic transport properties (e.g., see mobility × mDoS in Supplementary Table 1) of these materials for p-type. For n-type, an additional factor has to be considered. Excepted for MoS2 that has very poor p-type performance, group-VI TMDs have a markedly stronger p-type drive-current than that of the n-type. This is related, at least to a great extent, to the narrow valleys that are present in the conduction band of the Mo- or W-based TMDs, as discussed above. These prevent, at least partially, direct ballistic current from the channel to the drain at VDS ≥ 0.6 V, so that a less-efficient phonon-assisted transport is required. 1ML-WS2 or MoS2 feature a relatively wide 1st conduction-band valley (for WS2, for instance, its width is actually not isotropic and can be especially large in certain orientation such as the KM orientation shown on Figs. 2a and 3b). The combination of this fact with its higher electron mobility × mDoS product explains WS2 good position for n-type transport in this group.

The P4 device shows the worst performance, which might come as a surprise. P4 is a strongly anisotropic material with a low-effective mass in the ΓX direction (i.e., high mobility) and a high effective mass in the ΓY direction (i.e., low mobility). This is true both for n- and p-type conduction (more information can be found in Supplementary Note 4). Hence, this material has a very strong drive current for longer channel in the ballistic regime and the ΓX transport direction. In the sub-10 nm regime, however, the armchair (ΓX) oriented P4 transistor, strongly suffers from SDT and is difficult to switch-off due to its very-low transport effective mass. For L < 10 nm, the strongly anisotropic P4 material was shown to perform best in the zigzag (ΓY) orientation for the n-type transistor18. Still, the ΓY P4 n-device shows a strong drive current in the ballistic regime (Fig. 4b). Most theoretical studies on scaled P4 devices have looked either at ballistic performance and simplified band models41, or have neglected the optical-phonon coupling42. The ΓY P4 nMOSFET ION is strongly degraded by its optical-phonon (OP) coupling (De-ph,OP = 170 eV/nm for a single monolayer)18,43 (Fig. 4b). Concerning the pMOS, the ΓX transistor still performs the best at L = 5 nm but its drive-current is severely degraded by SDT (Fig. 8b). The ΓY drive-current is indeed even lower than in the n-case, while the SDT-related sub-10 nm SS degradation in the ΓX direction is not as strong as for the n-case (more details are available in the Supplementary Note 4).

For all the 2D materials studied here, we found that a high-doping density in the source and drain extensions, NSD, (NSD = 2 to 4 × 1020 cm−3) had to be used to reach their fundamental current level. Figure 10 shows the impact of NSD on the current for the L = 5 nm HfS2 and WS2 nMOS devices. Figures 6 and 7 give details on the current spectrum flow and the conduction-band position in the HfS2 transistor for NSD = 1 and 3 × 1020 cm−3 in off- and on-state, respectively. Owing to the typically high density of states in these materials (~30× (10×) that of Si for HfS2 (WS2, respectively)) (see Supplementary Table 1), a high NSD value is required to fully degenerate S&D extensions (Fig. 6) and avoid a saturation in the ID(VG) characteristics in on-regime related to a source-starvation effect44. In the case of source starvation, as the gate voltage is increased in on-regime, the current is limited by the availability of carrier in the source. On Fig. 7a, it can be seen that, when the source-starvation regime is reached, the current is not limited by the channel barrier. It is rather limited by the energy band at source side. The latter is only indirectly and weakly affected, when switching on VG, by the increase of the non-equilibrium transport charge through the device. This leads to a weak increase and eventually a saturation of the current in the ID(VG) characteristics. By increasing the source- and drain-extension doping to ensure good degeneracy at the source side, however, this effect is delayed to higher gate-overdrive values (Figs. 7b and 10). As can also be seen on Fig. 10, increasing NSD has a detrimental effect on SS at such a scaled gate length, so that an optimal value exists. This is related to a reduction of the effective-channel length and an increase of SDT for higher NSD values (Fig. 6). A similar trend is observed for all the n- and p-type devices studied here and an optimal value between NSD = 2–4 × 1020 cm−3 is observed for L = 5 nm and VDD = 0.6 V in all cases.

Fig. 10: Impact of source and drain doping on transfer characteristics of 2D MOSFETs.
figure 10

ID(VG) characteristics and SS(VG) (inset) of 2D nMOSFETs vs NSD a for HfS2 and b for WS2. VD = 0.6 V. L = 5 nm. IOFF = 10 nA/μm. e-ph scattering is included. The trade-off between on-state (better for higher NSD values, owing to a reduction of source starvation) vs off-state (better for lower NSD values, owing to a reduction of SDT) is observed for both materials.

The D2-FET

The trade-off between on- and off-state for the optimal doping concentration becomes more stringent as L is reduced, ultimately degrading transistor performance and preventing further downscaling. Even using 2D materials, scaling below 5-nm gate length becomes very challenging. The case of the monolayer HfS2 transistor with L = 3 nm is shown on Fig. 11a. Using NSD = 2 × 1020 cm−3 results in strongly degraded SS due to SCEs and SDT. The optimal NSD = 1 × 1020 cm−3 value, however, has poor performance. It suffers both from source-starvation-related on-current saturation and degraded slope due to SDT and SCE.

Fig. 11: L = 3 nm D2-FETs vs MOSFET transfer characteristics.
figure 11

ID(VG) characteristics and SS(VG) (inset) for a HfS2 double gated nMOSFETs and nD2-FETs with ΔL = 4 nm vs NSD, and b optimized Si-GAA nMOSFETs (NSD = 1 × 1020 cm−3) and DG nD2-FETs (intrinsic NSD) for ΔL = 4 and 11 nm. For the D2-FETs the optimized film thickness tS (also indicated in the figure in nm) is larger than for the MOSFET. VD = 0.6 V. L = 3 nm. IOFF = 10 nA/μm.

In addition, as transistor dimensions are scaled down, it becomes increasingly difficult to dope, activate, and control the location of the source-and-drain-extension doping atoms, using traditional implantation and annealing techniques4 (https://irds.ieee.org/editions/2018)19,45,46. In thin-film technologies, direct implantation usually results in a high defect concentration. It is especially the case for 2D materials, where finding a proper way for doping is still an active topic of research (https://irds.ieee.org/editions/2018)19,47. Typically, more complex techniques, such as epitaxial regrowth of the source and drain extensions, using in situ doping during the epitaxy, are needed (https://irds.ieee.org/editions/2018)19,48. Strict control and positioning of the high-doping concentration to prevent its unwanted diffusion during the high-temperature steps of the fabrication process (e.g., during dopants activation or epitaxial-growth phases) in the channel is also challenging and requires advanced techniques such as Flash and Laser anneal4 (https://irds.ieee.org/editions/2018)45, a solution to this particular problem is to use a uniformly doped, or junctionless transistor4. In any case, the discrete nature and limited numbers of doping atoms, resulting from the scaling of the device dimensions, leads to a strong and unavoidable statistical variability when using doping impurities at very small dimensions46.

Owing to the challenge of chemical doping, electrostatic doping is sometime used in today’s experimental 2D or carbon-nanotube material devices to dope their source and drain extensions49. It consists in using a gate, e.g., the wafer back gate, to electrostatically induce a high carrier concentration and decrease the semiconductor resistivity, which is the desired effect of chemical doping. Using electrostatic doping with a gate is indeed free of all the afore-mentioned problems related to chemical doping. It can dope (i.e., control the carrier concentration in) the entire thickness of the semiconductor film as long as this film is sufficiently thin, typically for tS < 10 nm (the exact value also depends on the residual chemical doping of the film and substantially decrease if this residual doping is larger than 1 × 1020 cm−3). Directly, using the wafer back gate is, however, not a manufacturable solution to individually control billions of transistors with different n- and p-type doping on a chip. A local, dedicated gate for each transistor would rather be required for that purpose. Typically, also these techniques are meant to induce a fixed amount of doping in the source and drain extensions of the device, whereas the dynamic-doping implications of this technique have not yet been studied. It would indeed be advantageous to have no or a low carrier concentration in the off-state to minimize leakage and a high carrier concentration in the on-state to maximize drive current, i.e., we want to dynamically control doping with the gate voltage of the transistor to break free of the NSD optimization trade-off.

We propose here a dynamically doped FET, whose purpose is to turn the challenge of scaling into an opportunity (thin-film and multi-gate architecture technologies, which are the by-product of scaling, enable the manufacturability of such a device). It consists of a transistor that is dynamically doped by one of its own gate(s). This doping gate is located opposite (e.g., at the bottom) the source and drain metal contacts (e.g., located on top). Owing to its opposite position, the doping gate, unlike a conventional gate of length L, can now overlap, by a value ΔL on each side, the source and drain extensions to dynamically induces doping without increasing the footprint of the transistor (Fig. 12). This unconventional gate-positioning scheme would alleviate the need for strict alignment control between the doping-gate and the other gates in a multi-gate technology. We insist here that the D2-FET remains a three-terminal device, like a conventional MOSFET. The doping gate is also the gate of the device for a single-gate device and share the same contact-voltage bias that any other conventional gates, if a multi-gate architecture is used (Fig. 12). It, therefore, does not require an additional contact compared with a MOSFET.

Fig. 12: Schematic view of 1 ML 2D D2-FETs.
figure 12

D2-FET with a a single gate, b a double gate, and c an alternative double-gate design with chemical doping in the source-and-drain-contact regions (NSD2), larger LSEP, and shorter ΔL.

It should be reminded here that, scaling L is a way to scale the total contacted gate pitch, CGP, of a transistor, i.e., the minimal distance between the gate of two subsequent transistors. CGP is composed of the sum of L and the length of the highly-doped source-and-drain extensions, LSD (see Figs. 1 and 12). LSD is the sum of the spacer length, that separate the gate contact from the source and drain metal contact of the three-terminal transistor, as well as the metal-contact length (https://irds.ieee.org/editions/2018). The length of the doping gate is LDG = L + 2* ΔL. Technological requirements impose that ΔL = LSDLSEP, LSEP being a separation distance (typically at least half the spacer length, i.e. a few nm) (https://irds.ieee.org/editions/2018) needed to separate the doping gate from one transistor to that of the next. It is, therefore, longer than L, although it does not require a larger CGP footprint than a traditional top-sided gate of length L. To quantify this, the 2031 IRDS-dimensional targets for the so-called 1-nm-technology node and beyond are L = 12 nm, LSD = 14 nm, and CGP = 40 nm (https://irds.ieee.org/editions/2018). The spacer length is 6 nm so that LSEP ≥ 3 nm, ΔL ≤ 11 nm, and LDG ≤ L + 22 nm. As can be seen, a comfortable margin is available for LDG. This allows for keeping a good electrostatic control, as well as a relaxed tS scaling, as can be seen for the Si case in Fig. 12b, even when using a very aggressive pitch scaling (L = 3 nm).

In the rest of this section, if not specified otherwise, we have, however, assumed a worst-case scenario for the D2-FET with smaller LDG. We used ΔL = L/2 with a minimum value of ΔL = 4 nm for L ≤ 8 nm. This either assumes a very aggressive LSD scaling, or the possibility of choosing a smaller ΔL value (i.e., LSEP » 3 nm) in conjunction with high-doping, NSD2, in the LSEP ungated part of the device (as shown in Fig. 12c) to further reduce the contact resistance or for reducing the intrinsic gate capacitance (see discussion below) for instance. Our simulation results show that both cases of D2-FETs (Fig. 12b, c) achieve similar ION and SS for same L and ΔL in case of ohmic or low Schottky-barrier contacts. The second scheme (Fig. 12c) could be advantageous in case of a high Schottky-barrier contact.

On Fig. 11a, several L = 3 nm DG HFS2 D2-FET characteristics are shown, one with no intentional doping (NSD ≤ 1 × 1019 cm−3, which correspond to typical residual doping concentrations in the 2D films, i.e., lowly doped or intrinsic extensions) and 2 with highly-doped extensions (NSD = 1 and 2 × 1020 cm−3). Our simulations show that for NSD ≤ 1 × 1019 cm−3, the presence of a residual doping in the extensions has no impact on the current-voltage characteristics. The carrier concentration in the extensions is mainly determined by the doping-gate bias. In off-state, the conjunction of a low carrier concentration in the extensions and the extended doping-gate geometry allows for a large Leff (typically ≥ 2 × L, Fig. 13a) and nearly ideal SS and low off-state current is achieved. In on-state, a high carrier concentration allows for a high-drive current. As can be seen, the intrinsic DG-D2-FET, free from any chemical doping, already strongly outperforms the optimized NSD = 1 × 1020 cm−3 DG-MOSFET. On Fig. 13b, however, it can be seen that in on-state, for a large gate overdrive, the current might be limited by the source part of the conduction-band barrier that is mostly controlled by the doping-gate, not by the top-gate.

Fig. 13: Current spectrum and top conduction-band edge along the channel direction of a L = 3 nm HfS2 D2-FET.
figure 13

J(E, x) (surface plot) and EC(x) (−) of the simulated intrinsic (NSD ≤ 1 × 1019 cm−3) L = 3 nm HfS2 nD2-FET of Fig. 11a (with the design of Fig. 12c) a in off-state at VG = 0.1 V, and b in on-state at VG = 0.6 V. ΔL = 4 nm. NSD2 = 4 × 1020 cm−3.

In case, a large additional NSD doping is used as an attempt to further boost the on-state current, the carrier concentration in the extensions is still dynamically controlled by the doping gate, but the dynamic-doping level at a given VG can be enhanced vs the intrinsic case. In Fig. 11a, it is observed that the current drive can be slightly increased for NSD ≥ 1 × 1020 cm−3, owing to the enhanced carrier concentration in the source, whereas SS is only slightly affected as the doping gate still deplete the extension in the off-state. As in the case of the regular MOSFET, an optimal doping of NSD = 1 × 1020 cm−3 is observed for VDD = 0.6 V. On the contrary to the MOSFET case, however, ION and SS sensitivity to doping variations are strongly reduced, and ION remains high, whereas SS remains low for all the simulated NSD values. Finally, Supplementary Fig. 10 compares the ID(VG) characteristics of SG and DG 1ML-HfS2 MOSFETS and D2-FETs for L = 3 nm and for L = 5 nm. It is shown that for the D2-FET case, a simpler-to-fabricate SG architecture is as good or even better in term of drive current than that of a DG- D2-FET. The SG-D2-FET indeed keeps a similar and good electrostatic control (SS), when compared with that of the DG- D2-FET case, hence similar drive-current (per gate). For L ≤ 3 nm, the SG-D2-FET ION typically outperforms that of the DG-D2-FET device as the short top-gate drive only a small amount of additional current compared to the doping gate of the device. In the MOSFET case, a SG architecture is not sufficient to maintain a good electrostatic control for sub-10 nm devices. The SG device SS and ION is hence degraded compared to the DG-MOSFET case.

Figure 11b compare L = 3 nm optimized MOSFETS and D2-FETs for the Si case. For the Si-D2-FETs, the number of gates can be reduced, similarly to what was found for the HfS2 case, and an intrinsic DG device was used instead of a GAA. Furthermore, the tS scaling was relaxed towards L rather than ½ L, assuming ΔL = 4 nm. This strongly reduces QC and boost ION of the D2-FET, as for the square cross-section GAA the confinement is both in the width (y-) and height (z-direction). In the rectangular cross-section DG-case, the width (y direction) is typically large compare to its height tS that is further relaxed compared to the GAA case. In case ΔL = 11 nm is used, the optimal tS for the D2-FET is even further relaxed to 4 nm and the performance is further boosted as SS is strongly improved. For Si, we found that the intrinsic case (i.e., unintentionally doped extensions with NSD ≤ 1 × 1019 cm−3) is always better than the case with a larger NSD. In any case, even in case chemical doping would be used in the D2-FET, the related challenges (e.g., variability) would be reduced, one reason being the relaxed dimensions (LDG, tS) at same CGP.

Scaling perspective

Next, we investigate in Fig. 14 the scaling behavior for both optimized MOSFETs and D2-FETs made of the less-explored HfS2, the material showing the highest mobility and performing the best at L = 5 nm, and that of WS2, the best performing material of the more studied group-VI TMDs. The optimized device results for the more conventional Si technology are shown as well. The evolution of their on-current vs L at a fixed off-state current of 10 nA/μm and a supply-voltage VDD = 0.6 V is shown on the figure.

Fig. 14: Maximum achievable ION vs L for optimized n-type MOSFETs and D2-FETs made of Si, WS2, and HfS2.
figure 14

A DG architecture is assumed for all the 2D MOSFETs. For the Si MOSFETs, a GAA transistor was used. A SG architecture is employed for the 2D D2-FETs, whereas a DG transistor is used for the Si-D2-FETs. EOT = 0.5 nm. VDD = 0.6 V. IOFF = 10 nA/μm. The current is normalized by the gate perimeter. For the D2-FETs, we used ΔL = L/2 with a minimum value of ΔL = 4 nm for L ≤ 8 nm.

For the MOSFET case, all materials show degradation of their performance when L is scaled below 7.5 nm. This is related to SCEs and SDT as discussed before. Comparing the Si-GAA and WS2 DG MOSFETs that achieve similar drive current at L = 10 nm, it can be seen that the GAA ION degrades faster when downscaling L. This is related to the better electrostatic control and QC-free characteristics of the 2D material over the GAA. None of these 2 materials, however, are able to meet the stringent high-performance IRDS ION targeted for year 2031 (https://irds.ieee.org/editions/2018) and a higher-mobility channel material is needed. HfS2 on the other hand, owing to its outstanding transport properties, exceeds the target down to a channel length of ~6 nm.

Using the D2-FET concept, it can be seen that the performance degradation with L is delayed to L below 5 nm in all cases. For such small gate lengths, the drive-current potential is strongly enhanced compared with that of the MOSFET case. At L = 3 nm, the Si, WS2 and HfS2 D2-FETs show an ION gain of ~3× compared with their MOSFET counterparts. It is now possible to reach the IRDS target with a L of ~1 nm HfS2 SG-D2-FET transistor (using ΔL = 4 nm). Our simulations finally show that, using this scheme, it is still possible to have a transistor effect using L = 0 nm, and that this device performs as well as a regular chemically doped multi-gate transistor with a 4 nm or longer L values for the case with 2D materials, although using only single-gated intrinsic semiconductor materials. This shows the promise of using the D2-FET concept for sub-10 nm transistors and in particular for ultra-scaled high-mobility material devices that would be required to meet the stringent IRDS HP targets.

It is to be noted that the larger LDG, will however increase the intrinsic gate capacitance at same L and a trade-off between ΔL and the ION gain may exist for the speed performance of the D2-FET. In modern scaled technologies, the total load capacitance of inverters or other digital circuits is often dominated by extrinsic (back-end-of-line) capacitances, such as the metal-line capacitance that is proportional to CGP (not LDG) (https://irds.ieee.org/editions/2018). By enabling further downscaling with strongly improved drive-current, the D2-FET may therefore also enable a power-delay benefit. To investigate the trends, we performed, here, a power-delay performance analysis of the basic building block of a digital circuit, i.e., a CMOS inverter, using scaled D2-FET and MOSFET devices. The details about the process assumptions and layout, which determine the number of stacked-transistors per device and their geometries, as well as their back-end-of-line capacitance load that is considered in this analysis, are detailed in Supplementary Note 5 and Supplementary Figs 11 and 12.

Figure 15 compares the switching energy vs delay (EDP) of back-end-loaded high-performance invertor cells made with HfS2 DG MOSFETs and HfS2 SG-D2-FETs, as well as that made with Si-GAA’s and Si SG-D2-FETs (typically the best device architectures per category of materials for MOSFETs and D2-FETs). A more-detailed analysis to identify the best devices per category is available in Supplementary Fig. 13 and surrounding text. The inverters are loaded with a typical 50 contacted gate pitch-long metal line (https://irds.ieee.org/editions/2018). As CGP is reduced for shorter L, more aggressively scaled devices get a net capacitance reduction. The SG-D2-FET devices that only require one spacer length instead of two (see Supplementary Fig. 11) benefit from a further CGP reduction at same L. The extrinsic capacitances of the cell layout are also included in the load capacitance. Again, the SG-D2-FET devices are free from the extrinsic parasitic components CGSext and CGDext (see Supplementary Fig. 11), as the gate metal contact does not directly face the source and drain metal contacts.

Fig. 15: Switching energy vs delay (EDP) of high-performance MOSFET and D2-FET inverters.
figure 15

EDP of 1ML-HfS2 high-performance inverter cells, at various VDD (0.4 V to 0.7 V), made of L = 5 nm and L = 3 nm stacked DG MOSFETs (5 ribbons/device) and L = 0 nm and L = nm stacked SG-D2-FETs (nine ribbons/device). The EDP performance of Si HP inverter cells made of L = 12 nm stacked Si-GAA MOSFETs (tS = 5 nm, 8 wires/device) and L = 5 nm stacked Si SG-D2-FETs (tS = 3 nm, 7 ribbons/device) are also shown for comparison. The inverters are loaded with a 50 contacted-gate-pitch-long metal line (https://irds.ieee.org/editions/2018). The extrinsic capacitances of the cell layout are also included in the load capacitance. IOFF = 10 nA/μm. ΔL = 4 nm for the D2-FETs.

In Fig. 15, VDD is varied from 0.4 to 0.7 V. The best EDP performance is achieved by the L = 0 nm, HfS2 SG-D2-FET, which uses the simplest (SG) architecture and further yields the largest pitch reduction (CGP = 22 nm). It is closely followed by the L = 5 nm, HfS2 DG MOSFET (CGP = 33 nm) and the L = 3 nm, HfS2 SG-D2-FET (CGP = 25 nm). The L = 3 nm, HfS2 DG MOSFET (CGP = 31 nm) performance is strongly degraded, and typically worst (especially at larger VDD where the speed performance saturates due to on-current saturation) to that of the L = 5 nm Si SG-D2-FET (CGP = 27 nm). The latter two devices still comfortably outperform the L = 12 nm Si-GAA (CGP = 40 nm).

These results further confirm and highlight the promising potential of the D2-FET device, paving the way towards ultimately scaled devices, with reduced process complexity and variability (e.g., reduced number of gates, larger tS, doping free or reduced sensitivity to doping fluctuations…) and improved performance.

Discussions

In summary, we have presented an in-depth study of the essential physics and performance potential of several 2D materials towards sub-10 nm gate length high-performance CMOS. We have argued that using very-advanced atomistic-simulation techniques including e-ph scattering, such as the DFT-NEGF technique we have used here, is required to achieve reliable and accurate results for 2D-materials. We have extracted from our simulations relevant 2D-material parameters such as mDoS and mobilities for 5 TMD n- and p-type materials.

We have benchmarked n- and p-type MOSFETs made of six different 2D materials, including the most used Mo and W-based TMDs, P4, as well as, the less-explored HfS2 and ZrS2 materials, against Si-GAA. Our results demonstrate the interest and better scaling potentials of HfS2, ZrS2, and WS2 for sub-10 nm HP CMOS providing that a high-doping concentration could be used in the source and drain extensions to mitigate source-starvation effects. We have further shown that a high-drive current, meeting the stringent IRDS-2031 target down to ~L = 6 nm, would be achievable using HfS2. We finally proposed the D2-FET concept that scales better than its MOSFET counterpart and seems very attractive for the sub-10-nm gate-length regime. Used in combination with a high-mobility material such as HfS2, it allows for keeping the stringent ITRS HP on current when scaling down to 1 nm gate length. It further shows very attractive power-delay performance and its EDP performance keeps increasing when ultimately scaling down L to 0 nm using a simpler SG architecture with an ultra-compact design (CGP = 22 nm). The D2-FET further addresses the grand-challenge of doping in ultra-scaled devices and 2D material in particular.

Methods

Quantum transport solver

Our quantum transport solver, ATOMOS18, was specifically developed for high-performance computing and the use of computationally heavy DFT Hamiltonians. It is written in C++ and uses multi-threaded MPI with various levels of parallelism. Ultimately, any heavy vector-matrix or matrix-matrix operations are performed using BLAS and LAPACK.

ATOMOS core transport solver is a Real-Space NEGF solver based on the recursive-Green’s function (RGF) algorithm50. For completeness, the equations for retarded (GR), lesser (G<) and greater (G>) Green’s functions read33:

$$G^{\mathrm{R}} = \left( {EI_{\mathrm{N}} - H - {\Sigma}^{\mathrm{R}}} \right)^{ - 1},$$
(1)
$$G^ < = G^{\mathrm{R}}{\Sigma}^ < G^{{\mathrm{R}}\dagger },$$
(2)
$$G^ > = G^{\mathrm{R}} - G^{{\mathrm{R}}\dagger } + G^ < .$$
(3)

E is the scalar energy. IN the identity matrix, H the device Hamiltonian, and ΣR,< the retarded, lesser self-energies that include the interaction terms (e.g. with the semi-infinite leads ΣCR,< and the electron–phonon scattering terms ΣSR,<) are matrices of rank N, the total number of atoms in the device × the number of orbitals/atoms. We efficiently store H and other G matrices using our dedicated sparse block-matrix class, which we specifically customized for the RGF method.

The contact self-energies are computed with the Sancho-Rubio method51. Electron–phonon scattering is considered using the self-consistent Born approximation25. Assuming the phonons stay in equilibrium, the scattering self-energy may be written as33:

$${\Sigma}_S^ < ({\mathbf{r}}_i,{\mathbf{r}}_j,E) = {\int} {\frac{{d{\mathbf{q}}}}{{(2{\uppi})^3}}} e^{{\mathrm{i}}{\mathbf{q}}.({\mathbf{r}}_i - {\mathbf{r}}_j)}\left| {M_q} \right|^2 \times \left( {N_q + \frac{1}{2} \pm \frac{1}{2}} \right)G^ < ({\mathbf{r}}_i,{\mathbf{r}}_j,E \pm \hbar \omega _q)$$
(4)

where q and ωq are the phonon wave vector and the corresponding angular frequency, ħ is the reduced Plank’s constant, Nq is the phonon-occupation number. Mq is the e-ph coupling matrix, which depends on the exact scattering mechanism. For TMDs we used the DFT-computed e-ph parameters from ref. 15, for P4 those from ref. 43. Additional details about the e-ph scattering implementation can also be found in Supplementary Note 3. To ensure efficient load-balancing, a master-slave-approach-based dynamic scheduler is used to distribute the various energy-momentum (e-k) points between the different parallel ranks. For optimally generating the energy points, we rely on a recursive adaptive-grid algorithm52, using a trapezoidal integration rule and a global-error estimator.

Similarly, a parallel Poisson solver with its own sparse class is used. To expedite the self-consistent Poisson-NEGF convergence, we employ a predictor-corrector method using the Newton scheme53. To predict the carrier changes with respect to potential variations, various semi-classical predictor functions have been implemented. For 2D materials, we can well fit the NEGF data using a 2D-DoS model, i.e., using a Fermi-Dirac integral of order 0 (See Charge and DoS Fitting section below and Supplementary Fig. 4). Additional adaptive-damping methods can be used, if the current and charge convergence criteria are not met within a given number of iterations. The anisotropic dielectric permittivity’s are taken from ref. 54.

In this work, we have used the Wannierization technique55 to express the DFT-Hamiltonian in a localized-orbital basis that is compatible with the RGF algorithm. The use of advanced and well-optimized algorithms together with high-performance parallel computing allow for scalable and fast calculations. For the 1 ML 2D devices simulated here, using a real-space DFT-Hamiltonian with longer-range interactions and dissipative transport, a full ID(VG) curve is typically achieved within about an hour on one to a few 100 cores, using the latest generation Intel Xeon CPU.

DFT-Hamiltonian computation

The electronic states in the various TMD and BP monolayers are modeled using the DFT-based ab initio tool QUANTUM ESSPRESSO20. Both the geometry relaxation and the computation of the electronic structure are performed using the generalized gradient approximation and the optB86b exchange-correlation functional21. Spin-orbit coupling was not included. The plane-wave cutoff energy and the Monkhorst-Pack k-point grid for the Brillouin-zone integration, that we used for the relaxation and band structure calculations, were selected so that the total energy was well converged. The convergence criteria are set to <10−3 eV/Å for the forces acting on each ion, and a difference smaller than 10−3 eV for the total energy variation between two subsequent iterations. A vacuum layer of 25 Å was employed in the DFT simulations to cutoff the spurious interactions of the periodic images along the out-of-plane (z-direction, see inset of Fig. 2).

We then transformed the Bloch wavefunctions into MLWF, typically centered on the ions, with the wannier90 package22. Figure 2 and Supplementary Fig. 9 demonstrate the validity of our MLWF representation for the case of WS2, ZrS2, HfS2, and P4. ATOMOS uses the resulting supercell information, i.e., MLWF and atom positions, lattice vectors, and the localized Hamiltonian-matrix elements, as building blocks to create the full-device atomic structure and Hamiltonian-matrix. We kept in the device Hamiltonian, the required Wannier-Hamiltonian longer-range interactions (typically 12–15 Å). ATOMOS can further rotate the device geometry to a preferential channel orientation within the 2D layer. We assumed periodic boundary conditions in the width (y axis) direction. They were modeled with 24 kY points.

Charge and DoS fitting

From our NEGF simulations, we can extract the electron concentration vs EF – EC. As can be seen in Supplementary Fig. 4, this can be well fitted by a 2D-DoS model (using a Fermi-Dirac integral of order 0), from which the conduction-band DoS, N2D, can be extracted by:

$$n = N_{2{\mathrm{D}}}.{\mathrm{ln}}\left( {1 + e^{\frac{{E_{\mathrm{F}} - E_{\mathrm{C}}}}{{k_{\mathrm{B}}T}}}} \right)$$
(5)

In Eq. (5), EF is the Fermi level, EC is the conduction-band edge, kB is the Boltzmann constant and T is the absolute temperature. The NEGF-simulated carrier concentrations vs EF – EC curves are extracted from a L = 14 nm device using the averaged values of a cross-section in the middle of the channel under low VD (typically 1 mV) bias condition, whereas varying VG. Owing to non-equilibrium transport, the NEGF Fermi level is only known and well defined at the source and drain contacts. At very low VD, we can, however, safely assume a quasi-linear and close to equilibrium regime and that the Fermi level value in the middle of the channel is halfway between that of the source, EFS, and drain, EFD. Assuming an equivalent single-band parabolic effective-mass model, we can further define an equivalent DoS mass, mDoS, by:

$$N_{2D} = \frac{{m_{DoS}.k_BT}}{{\pi .\hbar ^2}} \times \frac{1}{t}_S$$
(6)

Note that this DoS mass capture the DFT-computed non-parabolicity of the occupied bands close to the conduction-band edge. In Eq. (6), ħ is the reduced Plank constant and tS is the 2D-film thickness, ~0.6 nm for a monolayer TMD (the exact value we used for each studied monolayer can be found in Supplementary Table 1). tS is used to convert the 2D density from per-area to per-volume unit. Similarly, for a p-type device, we can extract the valence-band DoS, N2D from the DFT-NEGF-simulated hole concentration vs EV – EF using:

$$p = N_{2{\mathrm{D}}}.{\mathrm{ln}}\left( {1 + e^{\frac{{E_{\mathrm{V}} - E_{\mathrm{F}}}}{{k_{\mathrm{B}}T}}}} \right)$$
(7)

where EV is the valence-band edge. Using Eq. (6), it is then possible to extract the equivalent hole DoS mass.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability

Access to the code that is used in this study is restricted by imec legal policy.

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Acknowledgements

Part of the computing resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation–Flanders (FWO) and the Flemish Government. The author acknowledges the support of Dr. G. Gaddemane for the DFTP e-ph coupling calculations.

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A.A. developed the theory, simulation code, and concepts (e.g., the D2-FET). He performed the simulations and analyses and wrote the manuscript.

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Afzalian, A. Ab initio perspective of ultra-scaled CMOS from 2D-material fundamentals to dynamically doped transistors. npj 2D Mater Appl 5, 5 (2021). https://doi.org/10.1038/s41699-020-00181-1

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