While scaling of complementary metal-oxide-semiconductor (CMOS) technology is facing serious issues such as mobility degradation, drain-induced barrier lowering, increasing tunneling current through the gate oxide, and source-to-drain tunneling, two-dimensional (2D) materials are under investigation as promising candidates to overcome these issues [1
]. The high mobility of carriers in graphene [2
] along with the possibility of wafer-scale fabrication [3
] make graphene a very attractive material for electronic applications. However, any material to be used as a channel in a performant CMOS technology should exhibit not only a high carrier mobility but also a sufficiently large bandgap (more than 0.4 eV for Ion
) to realize a sufficiently low off-current [7
]. Therefore, despite having high mobility, associated with its Dirac dispersion, the use of graphene as a channel material is limited because of the absence of a bandgap.
On the other side, silicene, the silicon analog of graphene, can be considered to be the limit of an extremely scaled silicon CMOS technology and may be easier to integrate than graphene. Silicene has attracted a significant amount of attention, especially after the first experimental demonstration of a metal-oxide-semiconductor field-effect transistor (MOSFET) with a silicene channel [9
]. While silicene shows many resemblances to graphene, one key difference is the presence of buckling of the crystal, breaking the horizontal mirror symmetry. For potential CMOS application, silicene suffers from an extremely low phonon-limited mobility, which is almost zero [10
], and its negligible bandgap (~2 meV [11
]), resulting in poor performance in both the on- and off-state of a silicene device.
The poor mobility stems from the strong scattering with the out-of-plane acoustic phonon (ZA) modes, which is permitted at first order by the broken horizontal mirror symmetry. As was shown in the detailed theoretical study by Fischetti and Vandenberghe [10
], the parabolic phonon dispersion relation, combined with the Dirac dispersion, quenches the theoretical phonon-limited mobility in free-standing silicene to values as low as 0.01 cm2
/(Vs). Note that some more primitive calculations in previous papers did not effectively account for the ZA phonon coupling and predicted much higher values [12
]. Silicene–substrate interaction will somewhat alleviate mobility depression by suppressing ZA phonon scattering [15
]. The effect of suppression of ZA phonon scattering can be crudely approximated by introducing a cut-off frequency. However, with realistic cut-offs, the carrier mobility in silicene is unlikely to reach a competitive value [10
The absence of a bandgap must be alleviated by inducing a larger bandgap. Several methods of inducing a bandgap have been investigated, including strain, perpendicular electric fields, and hydrogenation. The use of tensile and compressive strain [18
] has been shown to be ineffective and applying a perpendicular electric field [21
] requires an excessively high electric field when applied externally [23
]. However, studies based on density functional theory (DFT) have shown that hydrogenated silicene, called silicane, has a large bandgap, exceeding 2 eV [24
The silicane crystal structure can come in two major configurations, a chair-like and a boat-like configuration. Stability studies based on total-energy calculations have shown that the chair-like configuration is more stable [27
], while Houssa et al. [29
] have shown both configurations have a similar stability. Unfortunately, silicane, similar to silicene, does not have horizontal mirror symmetry, (as can be seen in Figure 1
) and will also suffer from scattering with out-of-plane acoustic phonons. However, the gapped silicane band structure does not display a Dirac cone and will have less pronounced backscattering compared to the out-of-plane acoustic phonon scattering in silicene [10
], offering the prospect of a better mobility.
The silicane band structure has been studied previously but a precise theoretical study of its carrier mobility has not been conducted. The only work on the silicane mobility that we are aware of was performed by Restrepo et al. [24
], who reported a mobility of 464 cm2
/(Vs) for electrons, based on the linearized Boltzmann transport equation. However, this study did not provide detailed information on the contribution of the different phonon branches, nor did it discuss how the parabolic nature of the out-of-plane phonon modes was considered. A detailed and precise treatment is very important and a lack of an accurate treatment has led to an overestimation for the mobility in studies of several other materials, as discussed by Gaddemane et al. [17
]. Moreover, to the best of our knowledge, no work has been done to estimate the hole mobility in silicane.
In this paper, we studied both the electron and hole mobility in silicane in its chair-like structure. In Section 3.1
, the silicane electronic is determined using the DFT framework. In Section 3.2
, the interaction matrix elements between the carriers and the phonons are calculated and, in Section 3.3
, the matrix elements are used to calculate the phonon scattering rates. In Section 3.4
the Boltzmann transport equation (BTE) is solved using a full-band Monte Carlo scheme to obtain electron and hole mobility. Finally, in Section 3.5
, the velocity-field characteristics are investigated, and the average energy of carriers and their distribution is used to better explain the observed velocity-field behavior. Comparing the calculated silicane mobility with that of other gapped 2D materials, we conclude that silicane, i.e., a hydrogen terminated mono-layer of silicon, is a competitive material candidate for future electronic devices, provided scattering with out-of-plane phonons can be sufficiently suppressed.
The silicane bandgap was determined to be 2.19 eV with a valence band maximum and a conduction band minimum at the Γ and M points, respectively. Phonon calculations, including phonon energies, and the interaction of electrons and holes with phonons were performed using density functional perturbation theory. We calculated scattering rates for both electrons and holes, and showed that the acoustic branches had the largest scattering rates. In particular, the ZA phonon branch, with its parabolic phonon dispersion near the Γ point, had a scattering rate exceeding all other scattering rates by at least two orders of magnitude. Different cut-off wavelengths ranging from 0.58 nm to 16 nm were used to investigate the upper limit of mobility when ZA phonon scattering could be suppressed at different levels.
We determined the silicane mobility and velocity-field characteristics using the Monte Carlo method. The electron and hole mobilities were found to be 5 cm2/(Vs) and 10 cm2/(Vs) for a cut-off wavelength of 16 nm. We showed that providing a cut-off wavelength of 4 nm increased electron mobility to 24 cm2/(Vs) and hole mobility to 101 cm2/(Vs), while complete supersession of ZA resulted in an even higher mobility of 53 cm2/(Vs) for electrons and 109 cm2/(Vs) for holes.
Finally, the impact of the electric field on electron and hole transport was studied, showing that the velocity starts to saturate at a field of 3 × 105 V/cm and features negative differential mobility for fields exceeding 4 × 105 V/cm, for both electrons and holes. This phenomenon was consistent with the average energy and the distribution of the Monte Carlo ensemble.
The mobility values we obtained for silicane are competitive with computed values found in the literature for other 2D materials such as phosphorene and some of the TMDs. Silicane, i.e., hydrogen passivated <111> silicon slab scaled to its monolayer limit, can be thought of as the ultimate limit of silicon scaling. Given the competitive mobilities, we conclude that even at the most extreme scaling, silicon can still be considered as a promising material for future CMOS technology.