Ballistic Performance of Quasi-One-Dimensional Hafnium Disulfide Field-Effect Transistors

Abstract

Hafnium disulfide (HfS₂) monolayer is one of the most promising two-dimensional (2D) materials for future nanoscale electronic devices, and patterning it into quasi-one-dimensional HfS2 nanoribbons (HfS₂NRs) enables multi-channel architectures for field-effect transistors (FETs). Electronic, transport and ballistic device characteristics are studied for sub-7 nm-wide and ~15 nm-long zigzag HfS₂NR FETs using non-equilibrium Green’s functions (NEGF) formalism with density functional theory (DFT) and maximally localized Wannier functions (MLWFs). We provide an in-depth analysis of quantum confinement effects on ON-state performance. We show that bandgap and hole transport mass are immune to downscaling effects, while the ON-state performance is boosted by up to 53% but only in n-type devices. Finally, we demonstrate that HfS₂NR FETs can fulfill the industry requirements for future technology nodes, which makes them a promising solution for FET architectures based on multiple nanosheets or nanowires.

1. Introduction

Atomically thin two-dimensional (2D) materials (2DMs) have arisen as potential candidates for future transistor channel materials [1,2,3] since the discovery of graphene in 2004 [4]. The 2DMs exhibit near-ballistic transport properties and show potential for future high-performance electronic devices due to atomic thickness and dangling-bond-free surfaces [5]. Despite their promising characteristics, the 2DM devices are severely limited by the high contact resistance that degrades the device performance [6,7,8]. After graphene, more than 1800 2DMs have been predicted to be stable [9], and many of them have shown promise as channel material with transition metal dichalcogenides (TMDs), such as MoS₂, MoTe₂, WS₂, WSe₂, SnS₂, etc. [10,11,12], being among the most promising for future FETs. One hundred 2DMs were studied recently in [13] for future logic devices where monolayer hafnium disulfide (HfS₂) is reported to be one of the best-performing 2DMs for future ultra-scaled FETs due to a combination of electronic and transport properties such as low effective mass and high injection velocity that result in high ON-state current for both n- and p-FETs. The electronic, transport, and device properties of 2D HfS₂ were studied in detail in [13,14,15] by advanced theoretical calculations, whereas experimental work on few-layer HfS₂ FETs was reported in [16,17] where integration compatibility of monolayer HfS₂ with HfO₂ high-k dielectric was emphasized as one the biggest strengths of realization of devices with HfS₂.

With current state-of-the-art multi-bridge channel (MBC) FETs with silicon nanosheets or nanowires gaining momentum in the industry, 2DMs patterned into quasi-one-dimensional (quasi-1D) structures such as nanoribbons could replace silicon and extend Moore’s law by offering higher integration density [18,19,20]. Additionally, quantum confinement effects provide avenues for tuning the material properties, which could benefit the performance of such electronic nanodevices. Recently, various nanoribbons with ultimately downscaled widths were experimentally reported, including graphene nanoribbons (GNRs) [21,22] and phosphorene nanoribbons (PNRs) [23,24], which affirms the possibility of constructing MBC FETs, at least in principle, with quasi-1D 2DM nanostructures. Therefore, given the promising performance of 2D HfS₂ FETs, it is imperative to systematically explore the electronic, transport, and device properties of quasi-1D nanoribbons of HfS₂ (HfS₂NRs) about which nothing is currently known in the literature.

In this paper, we present an ab initio study on HfS₂NRs using density functional theory (DFT) to obtain the electronic structure, and maximally localized Wannier functions (MLWFs) to transform DFT Hamiltonians into localized basis suitable for quantum transport simulation. The MLWF Hamiltonians are then coupled to our in-house non-equilibrium Green’s function (NEGF)-based quantum transport [25,26] solver to obtain the relevant properties of HfS₂NRs and HfS₂NR FETs. Most importantly, we report the degradation of pFET performance while scaling down the nanoribbon width; however, the driving current in HfS₂NR nFETs is surprisingly boosted in ~2 nm-wide transistors by the quantum confinement effects. These findings are further explained by investigating the bandgap, density of states and carriers, current density, and injection velocity. Finally, we compare HfS₂NR n- and p-type FETs to other 2DMs and to the requirements set in the International Roadmap for Devices and Systems (IRDS). We show that both n- and p-type HfS₂NR FETs with nanoribbon widths in the range from ≈2 nm to ≈5 nm can fulfil the goals set for future logic devices.

2. Methods

The unit cell of monolayer HfS₂ (Figure 1a) is obtained from the Materials Cloud [27] and scaled to construct the supercells of zigzag HfS₂ nanoribbons with hydrogen-passivated edges. The nanoribbon structure from the top and side views is shown in Figure 1b and Figure 1c, respectively. The zigzag direction exhibits the highest curvature of the dominant subbands in the valence and conduction bands, which hints at excellent carrier transport in these devices. The HfS₂NR supercells are constructed with respect to the number of HfS₂ unit cells repeated along the nanoribbon width (W). This number ranges from 2 to 10 unit cells, which corresponds to W from 1.36 nm to 6.39 nm. After defining the initial structure, we employ an ab initio plane-wave DFT, implemented in the Quantum Espresso (QE) program package (v.6.8) [28], to relax the nanoribbon structure and obtain the Hamiltonians. A 20 Å vacuum is added in confined directions to exclude any interactions between layers. The DFT calculations use Perdew–Burke–Ernzerhof generalized gradient approximation (PBE-GGA) [29] as the exchange–correlation functional with plane augmented wave (PAW) pseudopotentials. The Brillouin zone is sampled with 1 × 15 × 1 Monkhorst–Pack k-point grid [30] where 15 k-points are in transport direction, and the cutoff energy is set to 1360 eV. Convergence thresholds for ionic forces are fixed to 10^–3 eV/Å, while 10^–4 eV is used for total energy.

Energy-localized plane-wave Hamiltonians from QE are then converted into a space-localized basis using MLWFs [31] implemented in the Wannier90 (v.3.1.0) program package [32,33]. The main input into the Wannier90 tool are the trial orbitals that are used for Wannier transformation, and here we use d orbitals for Hf atoms and p orbitals for S atoms. The band structure calculated with MLWF and DFT Hamiltonians shows good agreement in the relevant energy region around the bandgap, as seen in Figure 1d that reports the dispersion of the 3.25 nm wide HfS₂NR. Finally, the supercell MLWF Hamiltonian matrices are upscaled to construct the total HfS₂NR Hamiltonian, where the ~15 nm-long nanoribbon represents the channel of the HfS₂NR FET.

Being primarily interested in the ON-state performance, we use the top-of-the-barrier (ToB) ballistic FET model [34] to simulate single gate n- and p-type FETs with HfS₂NR channel. Within the ToB model, only the thermionic current is calculated, which is reasonable because the assumed 15 nm long channel is long enough for negligible tunnelling. The main inputs of the ToB FET model are transmission and density of states (DOS) calculated using the NEGF formalism implemented in our in-house code [35,36]. Within NEGF, the channel is described with the total HfS₂NR Hamiltonian and source/drain (S/D) contacts are described with S/D self-energy matrices calculated using the numerically efficient Sancho–Rubio method [37,38]. The n- and p-type FETs with zigzag HfS₂NR channels are simulated with ideal electrostatic control by the gate, resulting in a perfect subthreshold slope, equivalent oxide thickness (EOT) of 1 nm, S/D doping set at 0.01 areal molar fraction or ≈2.7 × 10¹³ cm⁻², and supply voltage (V_DD) set to 0.7 V. To provide a meaningful comparison between various HfS₂NR devices, simulations are performed with a common OFF-state current (I_OFF) set to 10 nA/μm as defined by IRDS for high performance (HP) devices [39].The setting of I_OFF is performed by adjusting the gate work function for each device automatically within our code. The ballistic ON-state current (I_ON) and the ON-state charge density at ToB are extracted when both gate and drain are biased at the supply voltage. Finally, we set EOT to 0.6 nm and compare I_ON and injection velocity to our previous work on GeS nanoribbon devices, in addition to one hundred 2DMs reported in [13].

3. Results and Discussion

Band structure of HfS₂NRs with various widths is reported in Figure 2a–d. All HfS₂NRs exhibit a direct bandgap at the Γ point and we observe an immunity of bandgap (E_g) to width scaling. The bandgap keeps a constant value of ≈1 eV in all nanoribbons, as reported in Figure 2e, which is slightly smaller than the 1.3 eV reported in [13] for the HfS₂ monolayer. As W is scaled down, the conduction band (CB) exhibits fewer bands near the CB minimum (CBM), while the dominant subband qualitatively remains the same. On the other hand, two degenerate subbands are visible in the valence band (VB) of wide HfS₂NRs with W > 4 nm, and this degeneracy is broken in narrower nanoribbons due to strong quantum confinement effect. Namely, two-hole subbands separate and this separation increases with W downscaling, which is followed by an increase in the curvature of the second subband near the VB maximum (VBM). To further investigate the quantum confinement effects on the band structure of all HfS₂NR widths, we extract the electron and hole–band structure effective mass of the dominant subband, closest to the CBM or VBM, from the band structure by fitting its curvature with a parabolic approximation (Figure 2f). Electron effective mass (m_e*) is immune to scaling with m_e* ≈ 0.2m₀ for all nanoribbon widths. In contrast, hole effective mass (m_h*) experiences a significant width-scaling effect. Namely, m_h* is ≈0.37m₀ for HfS₂NRs with W ≥ 3.25 nm; while scaling down the width linearly decreases m_h* to 0.33m₀ when W = 1.36 nm. Considering only the observed differences in CB and VB, where the m_h* of the dominant (highest) subband is at least 1.6 × lower than the m_e* for all observed HfS₂NRs, we expect a considerable difference in the performance of n- and p-FETs with HfS₂NR channels. However, this metric does not take into account degeneracy in VB or the higher number of subbands near CBM in wider nanoribbons.

To assess the performance of n- and p-type HfS₂NR FETs in Figure 3a, we plot the nanoribbon width dependence of the width-normalized ON-state current. The HfS₂NR pFETs exhibit a monotonic I_ON decrease from 1.5 mA/μm to 1.12 mA/μm when the width is downscaled. The only exception is the narrowest pFET that shows a slight I_ON increase to 1.18 mA/μm. On the other hand, for HfS₂NR nFETs, we surprisingly observe a generally monotonic increase of I_ON when HfS₂NR width decreases. The ON-state performance is enhanced from 1.14 mA/μm to 1.74 mA/μm when the width is downscaled from 6.39 nm to 1.99 nm, whereas the narrowest nanoribbon with W = 1.36 nm exhibits a slight decrease of I_ON to 1.66 mA/μm. Therefore, quantum confinement effects induce a current boosting of 53% for nFETs with the ≈2 nm wide HfS₂NR channel. Since the performance of HfS₂NR FETs shows no correlation to the band structure effective mass, in the following paragraphs, we explore the features of charge density, density of states, and carrier injection velocity.

Charge density at the top-of-the-barrier extracted in the ON-state (Q_CH) for various HfS₂NR widths is plotted in Figure 3b. For HfS₂NR nFETs, we report Q_CH ≈ 8.3 × 10¹² cm⁻² for W ≥ 3.88 nm, while scaling down decreases Q_CH to 7.88 × 10¹² cm⁻² in the 1.36 nm wide device. On the other hand, in pFETs, charge density increases monotonically from Q_CH = 7.87 × 10¹² cm⁻² for W = 6.39 nm to Q_CH = 8.24 × 10¹² cm⁻² for W = 1.36 nm. Charge density at the top-of-the-barrier depends on DOS near the CBM or VBM. Therefore, we plot DOS near the CBM (VBM) for various HfS₂NR widths in Figure 3c,d. The HfS₂NR nFETs exhibit a dense DOS with a high number of Van Hove singularities (VHS) near the CBM. Scaling down, HfS₂NR width reduces the number of available bands in the CB, decreases the overall DOS in the 100 meV window, and, therefore, lowers the charge density in narrower HfS₂NRs. However, VB DOS that is relevant for HfS₂NR pFETs shows an increase from a single VHS to two VHSs near the VBM due to band separation of the two dominant subbands (see Figure 2a) which increases the DOS near the VBM and Q_CH when nanoribbon width is scaled down.

Since the trends in I_ON and Q_CH behavior, reported in Figure 3a,b, are qualitatively opposite, the only reasonable explanation for I_ON trends should be found in the carrier transport properties, i.e., injection velocity. Injection velocity (v_inj) represents the average carrier velocity at the top-of-the-barrier under the ballistic limit and it is shown in Figure 3e,f for n- and p-type HfS₂NR FETs, i.e., electrons and holes, respectively. The plots report a gate voltage (V_GS) dependence of v_inj in the range from the threshold voltage (V_GS ≈ 0.2 V) to the supply voltage (V_GS = 0.7 V). The nFETs with W ≥ 5.14 nm show almost no modulation by V_GS with electron v_inj being ≈0.8 × 10⁷ cm/s. In contrast, v_inj increases with increasing bias in the 1.36 nm and 1.99 nm wide HfS₂NR nFETs. These two devices exhibit similar v_inj at the threshold voltage; however, HfS₂NR nFETs with W = 1.99 nm exhibit a stronger bias modulation and a higher maximum electron v_inj of 1.39 × 10⁷ cm/s. The best observed v_inj coincides with the best reported I_ON in nFETs for the 1.99 nm wide HfS₂NR nFET. We attribute the optimum performance and highest electron v_inj for W ≈ 2 nm to the band structure evolution, especially to the second CB subband visible in Figure 2b. Namely, for this nanoribbon, the subband crowding is not as dense as for wider HfS₂NRs, which means that higher subbands with heavier carriers do not contribute significantly to the current-carrying process. As for pFETs, hole v_inj at the threshold equals ≈ 0.8 × 10⁷ cm/s for all HfS₂NR pFETs. Scaling down, HfS₂NR width decreases the strength of V_GS modulation of the hole v_inj, and devices with W ≤ 2.63 nm show almost no bias modulation. The widest observed HfS₂NR (W = 6.39 nm) shows a 49% increase of hole v_inj to 1.2 × 10⁷ cm/s in the ON-state. As the width is downscaled, the hole v_inj decreases monotonically, which agrees with the I_ON behavior reported in Figure 3a.

We further explore the current and transport properties in ultra-scaled HfS₂NR nanodevices by analyzing the current energy density (J_de) in the ON-state. The J_de results are shown for n- and pFETs in Figure 4a and Figure 4b, respectively. In all devices, the current is mainly contained in the energy window up to ≈100 meV from the CBM/VBM. The nFETs exhibit J_de maximum at the CBM for the narrowest HfS₂NR, but wider HfS₂NRs exhibit a slight shift of the maximum by 20–30 meV, due to the higher number of occupied subbands. Namely, setting a common I_OFF for all devices results in different S/D Fermi levels (E_F) depending on device width, with a greater E_F shift upward generally observed for wider HfS₂NRs. Consequently, there are more current-carrying subbands, and the current density at the CBM is lower in wider HfS₂NR nFETs. The 1.36 nm wide HfS₂NR shows maximum current density near the CBM, with only a single dominant subband. In contrast, for W = 1.99 nm, the second subband carries current as well, with the second-subband current density surpassing that of the first subband above CBM + 90 meV, which leads to the maximum I_ON for the 2 nm wide device among all the studied HfS₂NR nFETs.

In addition to the current density, band decomposed charge density (BDCD) of nanoribbons with the widths of 1.36 nm and 6.39 nm is plotted in Figure 5a and Figure 5b, respectively. The BDCD is shown for subbands at the CBM/VBM, along with the nearest next subbands that are denoted as next-CBM/next-VBM. Figure 5a shows that the electron current in the narrowest nanoribbon, mainly determined by the first CB subband (see Figure 4a), is distributed almost through the whole nanoribbon width, except for edge S atoms. On the other hand, electron current in the widest HfS₂NR is mostly determined by the next-CBM subband (see Figure 4a), so that the current flows away from the edges and somewhat through the middle, as shown in Figure 5b. As for HfS₂NR pFETs with current density reported in Figure 4b, J_de is localized near the VBM due to the dominant first subband in the VB with a considerable contribution of the second subband in some devices. The lowest two VB subbands are slightly separated, i.e., degeneracy is broken, only in the narrowest HfS₂NR (see Figure 2a), and this split results in lower transmission probability near the VBM and, therefore, lower J_de in pFETs with the narrowest 1.36 nm wide channel. As shown in the BDCD plots in Figure 5a,b, now addressing the situation for the VBM subband, we observe that the hole current is expected to flow exclusively along nanoribbon edges. Furthermore, for wide HfS₂NRs where two degenerate subbands are dominant near the VBM, we observe that the current flows through separate edges in VBM and next-VBM, as seen in Figure 5b, due to zero overlap between the edges. Scaling down the nanoribbon width decreases the distance between the edges, which increases the overlap of the edge states and, therefore, causes the separation of degenerate subbands near the VBM, as seen in Figure 2. Increasing the current by increasing the width would be limited due to edge transport; hence, the narrowest HfS₂NR pFETs exhibit a decreasing drive current (Figure 3a) due to higher DOS in S/D regions and related E_F downshift.

Finally, we compare the ON-state performance of HfS₂NR FETs with one hundred 2DMs analyzed in [13], to the armchair and zigzag GeSNRs from our previous work [36,40], and to the IRDS requirements for future logic nodes [39]. For a proper comparison with the literature data, we set the EOT to 0.6 nm (HfO₂ with ε_r = 20, t_ox = 3 nm) as in [13]. The injection velocity in the ON-state for nFET and pFETs is shown in Figure 6a. All HfS₂NR n- and pFETs exhibit v_inj close to, or higher, than the IRDS requirement (0.9 × 10⁷ cm/s), while achieving comparable v_inj values to those in 2D HfS₂ along the zigzag direction. Furthermore, only a few other large-area 2D materials such as Ge₂S₂, As₄ and P₄ exhibit higher v_inj compared to HfS₂NRs. This is noteworthy because HfS₂NRs are quasi-1D nanostructures with additional quantum confinement in comparison to 2DMs that are infinite 2D sheets.

The ON-state current is compared in Figure 6b again for HfS₂NR FETs from this work and one hundred 2DMs reported in [13]. We compare the data against IRDS 2021 [41] because the IRDS 2022 version, which we have used for all previous comparisons, does not provide I_ON requirements with zero series or contact resistance. We observe that HfS₂NR FETs exhibit a relatively high I_ON and all devices meet the minimum IRDS I_ON for both nFET and pFET devices. On the other hand, the maximum IRDS I_ON specification is met only by HfS₂NR FETs with W in the range from 1.99 nm to 5.14 nm. The remaining HfS₂NR FETs are close but are nevertheless beyond the edges of the acceptable I_ON window. Compared to the armchair and zigzag GeSNRs, only armchair GeSNRs with W > 3 nm show better I_ON performance than HfS₂NR devices in both n- and p-FET configurations. Finally, in comparison to 2D HfS₂ and other 2D materials such as WS₂, ZrS₂, Ge₂S₂, arsenene and phosphorene, all HfS₂NR FETs perform relatively poorly. Nevertheless, we stress that HfS₂NRs are quasi-1D nanostructures with strong quantum confinement effects along the width, so a direct comparison to 2D monolayers is not completely fair.

The usage of zigzag HfS₂NR for future FET architectures based on multiple parallel nanosheets or nanowires is plausible, given their fulfilment of the IRDS requirements for the ON-state current and injection velocity and, moreover, given the possibility of a matched performance of n- and p-channel devices. Although several monoelemental 2D materials and TMDs outperform hafnium disulphide nanoribbon devices, the HfS₂NR FETs still outperform most of the previously studied 2DMs when both n- and p-FET devices and their performance are considered. Meeting the IRDS requirements for the ON-state current is obtained here for the ideal single-gate FET structure, whereas the usage of multi-channel architectures such as MBC FETs could provide opportunities for further improvements in performance. Additionally, we note that this study deals with purely ballistic devices so only the upper ballistic performance limits are explored in this work, while taking into account realistic effects like carrier scattering in quasi-ballistic transport and contact resistance would deteriorate the reported HfS₂NR FET performance. While investigating the impact of these issues is important, it is also out of the scope of the current manuscript and will be addressed in future work. Nevertheless, the potential of using ultra-scaled HfS₂NRs as a channel material in future logic devices is evident.

4. Conclusions

We present a comprehensive study on the electronic, transport and ballistic device characteristics for HfS₂ nanoribbons with a length of 15 nm and widths under 7 nm. We show that HfS₂NRs and HfS₂NR FETs exhibit promising transport properties and ON-state performance combined with an immunity of bandgap to nanoribbon width downscaling. Moreover, an unexpected and significant I_ON increase of up to 53% is observed in n-type FETs with HfS₂NR width of ≈2 nm. Concerning the industry goals for future technology nodes, we demonstrate that HfS₂NR FETs with nanoribbon widths in the range from 1.99 nm to 5.14 nm meet all IRDS requirements when EOT is set to 0.6 nm. Therefore, just like 2D HfS₂, quasi-1D nanostructures of monolayer HfS₂ present a promising candidate material system for ultra-scaled logic devices, especially for multi-channel nanosheet or nanowire nFETs, in which the ~2 nm wide HfS₂NRs could provide a significant performance boost due to the strong quantum confinement effect on the band structure.