Silicene nanomesh

Similar to graphene, zero band gap limits the application of silicene in nanoelectronics despite of its high carrier mobility. By using first-principles calculations, we reveal that a band gap is opened in silicene nanomesh (SNM) when the width W of the wall between the neighboring holes is even. The size of the band gap increases with the reduced W and has a simple relation with the ratio of the removed Si atom and the total Si atom numbers of silicene. Quantum transport simulation reveals that the sub-10 nm single-gated SNM field effect transistors show excellent performance at zero temperature but such a performance is greatly degraded at room temperature.

holes are passivated by hydrogen atoms. Each type of SNM is designated by the notation [R, W], where the R index reflects the radius of the hole calculated by N removed 5 6R 2 (N removed is the number of the removed Si atoms from one lattice cell) and the W index is the width of the wall between the nearest-neighboring holes. Fig. 1(a) shows an example of [R, W] 5 [1,4] SNM.
After relaxation, the unit cell size of the SNM structures is nearly unchangeable compared with the corresponding silicene supercell. However, constructing periodic holes will slightly affect the buckling distance (D) of the edge silicon atoms, which is larger than that of pristine silicene (0.46 Å) and increases with the increase of W given the same R 5 1 (Fig. S1(a)). For the silicon atoms far away from the holes, the D tends to be the value of pristine silicene, as shown in Fig.  S1(b).
To investigate the stability of SNMs, the cohesive energy E coh and Gibbs free energy dG, are calculated. The per-atom cohesive energy E coh is calculated according to the follow equation: where E(Si) and E(H) are the energies of the free silicon and hydrogen atoms, respectively, E(SNM) the total energy of a SNM in one supercell, N Si and N H the numbers of silicon and hydrogen atoms in a SNM supercell, respectively. The calculated E coh for [R 5 1, W] SNM is positive (3.64 , 4.71 eV) and increases with the increasing W as shown in Fig. 2a. We define the per-atom Gibbs free energy (dG) of formation for SNMs as where n Si and n H are the mole fraction of Si and H atoms, respectively, for a given structure, and m Si and m H are the per-atom chemical potentials of Si and H, respectively, at a given state. We chose m Si and m H as the binding energies per atom of bulk Si and H 2 molecule, respectively. As given in Fig. 2b, the calculated dG values for W 5 1 , 3 are 0.012 , 0.075 eV, and therefore a small amount of additional energy is required to make these reaction processes happen. However, SNMs have a negative dG when W . 3, indicating a higher thermodynamical stability relative to their elemental reservoirs. Based on the width dependence of E coh and dG, the stability of SNMs monotonically increases with the increasing W given the same R 5 1. To further study its stability, a molecule dynamic simulation of the [R 5 1, W 5 2] SNM is performed at temperature of 1000 K. As shown in the Supplementary Movie S1 we added, the structure is well kept, suggesting SNMs are stable enough against the high temperature.  The calculations done by Liu et al. 26 show that the band gap of graphene hexagonal nanomeshes is only opened when W is even. While W is odd, the GNMs behave semimetallically like pristine graphene. Our calculations show that SNMs have similar properties. The band structures of [1, W] SNMs with W 5 1 , 10 are provided in Fig. 1(b) and 1(c). Similar to pristine silicene, the SNMs with odd W show semimetallic behavior, with a pair of linear bands crossing at the K point ( Fig. 1(b)). By contrast, a direct band gap (E g ) is opened at the G point when W is even ( Fig. 1(c)). The band structures of [2, W] SNMs with even W are provided in Fig. S2 and the band gaps are also opened in them. The opened band gap when W is even monotonically decreases with the increasing W given the same R. A maximum band gap of about 0.68 eV is observed in both R 5 1 and 2 cases ( Fig. 1(d)).
Pedersen et al. 25 found that the band gap E g in GNM is determined by the relation where N total and N removed are the numbers of the total Si atoms before digging the holes and the removed hole atoms in a unit cell, respectively, and g is a fitting factor. For GNM, one has g 5 25 eV. Fig. 1(e) shows the band gap of SNM against N 1/2 removed /N total . The linear relation remains with g 5 7.246 eV, which is much smaller than that for GNM. Therefore, given the same N total and N removed , i.e. with the same notation [R, W], the band gap in SNM is much smaller than that in GNM. The band gap opening in GNM is proved not directly caused by quantum confinement as in graphene nanoribbions (GNRs); instead, it has a geometric symmetry origin 32,37 . When the two reciprocal lattice vectors of a GNM overlap with Dirac points of the pristine graphene, degeneracy at the Dirac points is lifted and a sizable band gap appears; otherwise, it's semimetal like graphene. SNM shares similar mechanism of band gap opening with GNM. In [R, W] SNMs, when W is even its two reciprocal lattice vectors overlap with Dirac points of the pristine silicene, i.e. the K and K' points of pristine silicene are folded into the G points of SNM. Due to the intervalley scattering, a band gap is opened in SNMs when W is even.
can be explained if we approximately treat SNM as a periodic potential perturbation U(r) to pristine silicene. If K 2 K9 5 G, where G is one reciprocal lattice vector of the SNM supercell, there is interaction between the two degenerate Dirac points. As a result, a band gap is opened and is expressed in terms of degeneracy perturbation theory as, where S is the square of the supercell of SNM, y K Ã A (r) and y K 0 A (r) are the Bloch function of A sublattice at the K and K9 points, with periodic part of m K A (r) and m K 0 A (r), respectively. The external periodical potential U(r) induced by the hole is assumed to be rather localized. Given the same hole of different SNMs, ð dr is approximately independent of the size of the supercell, and we therefore have E g ! 1 N total due to N total / S.
The effective mass m* monotonically decreases from 0.093 to 0.022 m 0 for R 5 1 and from 0.151 to 0.034 m 0 for R 5 2 with the increasing W because of the reduced band gap, where m 0 is the free electron mass. m e CK is approximately equal to m e CM except for W 5 2 case. At the same W, the m* values with R 5 2 are slightly larger than their respective m* values with R 5 1.
Transport properties of SNM FETs. The schematic model of a single-gated FET based on the [1,2] SNM is presented in Fig. 4(a). The electrodes are composed of semi-infinite silicene. To avoid the interaction between SNM and SiO 2 dielectric, a hBN buffer layer is inserted between the SNM and SiO 2 substrate 2 , and the thickness of SiO 2 dielectric plus hBN buffer region is d i 5 11 Å . To start with, we calculated the transmission spectrum of a 6.5 nm-gate-length SNM FET by using the DFT method with single-f (SZ) basis set to benchmark our SE extended Hückel result (Fig. S3). The transmission spectra calculated between the two methods are similar, except that the size of the transmission gap generated by the SE method (0.9 eV) is a litter larger than that by the DFT method (0.7 eV) and the transmission coefficients generated by the SE method near the Fermi level are unsmooth and generally larger than those by the DFT method. The larger transmission gap generated by the SE method can cause the decrease of on/off ratio compared with that by the DFT method given the same gate voltage window. The on/off ratio may be further slightly decreased when using the SE method due to the relative larger conductance in the off-state contributed by the larger transmission coefficient near E f . However these won't affect much the results. Then we focus on the transport properties of the SNM FET with a larger gate length  www.nature.com/scientificreports SCIENTIFIC REPORTS | 5 : 9075 | DOI: 10.1038/srep09075 L gate 5 9.1 nm. The conductance in SNMs can be modulated by applying a gate voltage to the channel, and an on/off switch is expected.
The transmission spectra of the 9.1 nm-gate-length SNM FET at V g 5 0, and 0.5 V with V bias 5 0.2 V are presented in Fig. 4(b). When V g 5 0 V, there is a transport gap of 0.9 eV centered at the Fermi level (E f ). The transmission coefficient nearly vanishes within the bias window, indicating an off state. By applying a positive gate voltage, the transport gap can be shifted towards low energy direction. At V g 5 0.5 V, relatively large transmission coefficients are moved inside the bias window. According to Eq. (6) in the Method section, the drain current is calculated and then normalized by the channel width to obtain the current density I ds (Fig. 4(c)). Clear on/ off current modulation is achieved. If we set V dd 5 V on 2V off 5 0.5 V and V g 5 0.5 V is chosen as the on-state, the on/off ratio can reach 5.1 3 10 4 , which is about three orders of magnitude larger than the maximum on/off ratios obtained in dual-gated silicene FET 2 and already meets the requirement of 10 4 , 10 7 for the high-speed logic applications. The subthreshold swing (SS, here is defined as dV gate / d(logI)) is 68 mV/dec, which approaches the 60 mV/dec switching limit of the classical transistors. To provide an insight into the switch capability, we investigate the transmission eigenchannels of the off-state (V g 5 0 V) and on-state (V g 5 0.5 V) at E 5 0.05 eV and k 5 (0, 0), as shown in Fig. 4(d). The transmission eigenvalue of the off-state is merely 6.71 3 10 27 , and the corresponding incoming wave function is obviously scattered and unable to reach to the other lead. On the contrary, the transmission eigenvalue of the on-state is 0.78; as a result, the scattering is weak, and the most of the incoming wave is able to reach to the other lead.
To determine the scaling effect of the gate length L gate on the device performance, we calculate the transfer characteristics of the SNM FET with different gate lengths (3.8 , 9.1 nm) at a fixed bias voltage of V bias 5 0.2 V as shown in Fig. 5. The maximum current I max is insensitive to L gate . By contrast, the minimum current I min increases with the decreasing L gate . Such a scaling behavior is attributed to the increasing off-state leakage current with the decreased L gate . Therefore, the maximum and minimum current ratio I max /I min decreases significantly from 5.8 3 10 5 at L gate 5 9.1 nm to 1.9 3 10 2 at L gate 5 3.8 nm (Fig. 6(a)). The on/off current ratio I on /I off (the gate voltage window is limited to a supply voltage) is a more important parameter than I max /I min to characterize switching effect of an electronic device. We limit the gate voltage window to 0.5 V and show the L gate dependent I on /I off in Fig. 6(b). It also monotonously decreases from 5.1 3 10 4 at L gate 5 9.1 nm to 17 at L gate 5 3.8 nm.
The subthreshold swing SS 5 dV gate /d(logI) is another important parameter of FET and determines how effectively the transistor can be turned off by changing the gate voltage. The SS of the SNM FETs monotonously increases from to 68 to 336 mV/dec when L gate scales down from 9.1 to 3.8 nm. Transconductance g m is another important parameter to characterize switching effect of an electronic device, which is computed from g m 5 hI ds /hV g . The g m value decreases from 555 mS/mm at L gate 5 3.8 nm to 351 mS/mm at L gate 5 9.1 nm ( Fig. 6(d)). Another key parameter the intrinsic gate capacitance C g is calculated in Fig. 6(e). C g is defined as C g 5 hQ ch /hV g , where Q ch is the total charge of the channel. The relationship between the C g and L gate is the following equation 39 : C g 5 e 0 e r W gate L gate /t ox , where e 0 and e r are the dielectric constant of vacuum and the relative dielectric constant of the gate dielectric, W gate is the width of the gate, and t ox is the thickness of the gate dielectric. As shown in Fig. 6(e), C g indeed increases almost linearly with L gate from 159 aF/mm at L gate 5 3.8 nm to 258 aF/mm at L gate 5 9.1 nm. According to the charge control model at low bias, m~g m V bias : L 2 gate C g , since C g / L gate , if we assume the mobility is a constant, the transconductance g m tends to vary inversely with L gate , and our results is consistent with this tendency. Fig. 6(f) shows L gate dependence of charge carrier transit time t based on the calculated C g and g m , i.e. t 5 C g /g m . t increases from 0.29 to 0.73 ps when L gate increases from 3.8 to 9.1 nm. The intrinsic cut-off frequency f T indicates how fast the channel current is modulated by the gate and is described as f T 5 1/(2pt) [40][41][42] . f T decreases monotonically with L gate from 557 GHz at L gate 5 3.8 nm to 217 GHz at L gate 5 9.1 nm (Fig. 6(g)), which is much smaller  compare with that in the sub-10 nm graphene FETs (4 , 22 THZ) 43 . The drift velocity of a transistor can be derived by v drift 5 L gate /t. As shown in Fig. 6(h), v drift is insensitive to the gate length and is 12.4 3 10 5 , 13.3 3 10 5 cm/s when L gate 5 3.8 , 9.1 nm.
Future FET technologies will require operation at voltages at or below 0.5 V to reduce power consumption. To compare the SNM FETs with the Si based and CNT transistors at a supply voltage V dd 5 0.5 V, we summarize the critical performance parameters of the sub-10 nm SNM (9.1 and 7.8 nm), advance Si, and CNT FETs at V bias 5 0.5 V in Table 1. The 9.1 nm SNM FET carries an on-state current of 464 mA/mm, which is larger than those (41 , 300 mA/mm) of the 8 , 10 nm advanced Si devices but slightly smaller than that (630 mA/ mm) of the 9 nm CNT device. The on/off current ratio of the 9.1 nm SNM FET is 7.4 3 10 3 , which is a little smaller than those (1 3 10 4 ) of the 10 nm Si nanowire, 8 nm ETSOI, and 9 nm CNT devices but larger than that (1 3 10 3 ) of 10 nm Si Fin device. The SS value (82 mV/dec) of the 9.1 nm SNM FET is slightly smaller than those (83 , 125 mV/dec) of the 10 nm Si nanowire and Si Fin, 8 nm ETSOI, and 9 nm CNT devices. Taking the three criterions together, the 9.1 nm SNM FET is competitive with the sub-10 nm advanced Si devices but is inferior to the 9 nm CNT device.
Adding the total area of the gates is an effective way to strengthen the gates' control over the channel, the gate control ability of a FET is expected to be improved by using a dual gate configuration. The transfer characteristic of the 9.1 nm dual-gated SNM FET at V bias 5 0.5 V is provided in Fig. 7(a) to compare with that of the singlegated one with the same L gate , and improved gate control is apparent. The performance parameters of the 9.1 nm dual-gated SNM FET are generally better than those of the single-gated counterpart as listed in Table 1. The SS is reduce by 8 meV/dec, the large on/off current ratio is increased by a factor of 2.7, and the on-state current is increased by a factor of 1.6. The 9.1 nm dual-gated SNM FET delivers an on-state current of 870 mA/mm, which is larger than those (41 , 630 mA/mm) of the 8 , 10 nm advanced Si devices and 9 nm CNT device. The   30 10 0.  on/off current ratio of the 9.1 nm dual-gated SNM FET is 1.2 3 10 4 , which is comparable with those of the 10 nm Si nanowire, 8 nm ETSOI, and 9 nm CNT devices and one order of magnitude larger than that of the 10 nm Si Fin device. The SS (74 mV/dec) of the 9.1 nm dual-gated SNM is smaller than those (83 , 125 mV/dec) of the 8 , 10 nm advanced Si devices and 9 nm CNT device. Taking the three criterions together, the 9.1 nm dual-gated SNM has a better performance than the sub-10 nm advanced Si devices and 9 nm CNT device. The excellent performance of the SNM FET is attributed to the depressed short channel effects due to their extremely small thickness and fewer traps on semiconductor-dielectric interface due to the smooth interface (Fig. 8).
The output characteristics for the 9.1 nm SNM FET at different gate voltages are shown in Fig. 7(b). The source-drain ballistic current increases with the applied bias voltage, and no current saturation is observed until V bias 5 0.7 V. The current of the dual-gated SNM FET is much larger than that of the single-gated SNM FET at the same V bias under V g 5 0.5 V, indicating an improved gate controlling.
The transfer and output characteristics of the 7.8 nm SNM FET are provided in Fig. S4. Although the performance of the 7.8 nm singlegated SNM FET is inferior to the sub-10 nm advanced Si devices and the 9 nm CNT device, the 7.8 nm dual-gated SNM FET is sufficiently improved: The SS is reduce by 4 meV/dec to 68 mV/dec, the large on/off current ratio is increased by a factor of 2.7 to 8.9 3 10 3 , and the on-state current is increased by a factor of 3 to 607 mA/mm at a supply voltage of 0.5 V (Table 1). Consequently, the 7.8 nm dualgated SNM FET has a better performance than the sub-10 nm advanced Si devices and is competitive with the 9 nm CNT device.
It is interesting to examine whether the SNM FETs can meet the requirements for the high-performance FETs from the 2013 edition of the International Technology Roadmap for Semiconductors (ITRS) 44 . The required gate lengths of HP logic of 2022 and 2023 are 8.9 nm and 8.0 nm, and supply voltages are 0.72 and 0.71 V, respectively. The transfer characteristic of the 9.1 nm dual-gated SNM FET at V bias 5 0.72 V is provided in Fig. 6(a). The 9.1 nm dual-gated SNM FET, whose gate length is approximately meet the requirement of HP logic of 2022 (8.9 nm), carries an on-state current of 3122 mA/mm at a supply voltage of V dd 5 0.72 V and greatly satisfies the requirement of I on 5 1350 mA/mm for the HP logic of ITRS of 2022. As shown in Fig. S4(a) We perform a molecular dynamics (MD) simulation of the channel region in a 9.1 nm single-gated SNM FET at room temperature to check how the transport properties change as phonon effect is partially included (only elastic scattering is considered) in the device. Compared with the transmission spectra of 9.1 nm single-gated SNM FET at V bias 5 0.2 V without considering the phonon scattering, the transport gap is increased from 0.9 eV to 1.5 eV, and the transmission coefficients of both the conduction and valence bands are greatly depressed at 300 K after phonon scattering effect is included, as shown in Fig. 9. The off-state current at V g 5 0 V isn't affected much. Whereas the on-state current is decreased significantly to 3.9 3 10 -2 mA/mm, and the on/off current ratio is decreased to 39 at V bias 5 0.2 V when the gate bias window is fixed at 0.5 V. When V bias 5 0.5 V, the on-state current is decreased to 4.0 mA/mm, and the on/off current ratio is decreased to 100. Therefore, SNM FETs still works at room temperature, but its performance is greatly degraded. Phonon scattering plays an import role on accurate assessment of SNM FETs even at a short-gate length below 10 nm.
In summary, a band gap is opened in SNM when the width W of the wall between the neighboring holes is even from the firstprinciples calculations. The size of the band gap increases with the reduced W and is proportional to the ratio of the removed Si atom and the total Si atom numbers of silicene. We simulate the transport of the FETs with a sub-10 nm SNM channel based on quantum transport theory and find that the sub-10 nm SNM FETs have an  2D materials have clean surface with fewer traps in semiconductor-dielectric interface and are extremely thin compared to traditional 3D materials, leading to a better gate control. V(x) and | Y(x) | 2 represent the potential and the probability density of the electronic charges, respectively. www.nature.com/scientificreports SCIENTIFIC REPORTS | 5 : 9075 | DOI: 10.1038/srep09075 excellent performance at zero temperature, characterized by a large on-state current up to 870 mA/mm, a large on/off current ratio up to 1.2 3 10 4 , and a small subthreshold swing low to 68 mV/dec at a supply voltage of 0.5 V. However the performance is greatly degraded when phonon scattering effect is included.

Methods
The geometry optimizations and the band structure calculations are performed using the double numerical basis set plus polarization (DNP), implemented in the DMol 3 package 45 . We chose the generalized gradient approximation (GGA) 46 of the Perdew-Burke-Ernzerhof (PBE) form to the exchange-correlation functional 47 . Both the atomic positions and lattice constant are relaxed without any symmetry constraints until the maximum force is smaller than 0.01 eV/Å . A 16 3 16 3 1 Monkhorst-Pack k-points grid 48 is used in the first Brillouin zone sampling. A vacuum space of 20 Å normal to silicene plane is used to avoid spurious interaction between periodic images. To examine the thermal stability of SNMs, ab initio MD simulation within the NVT ensemble is carried out using the DMol 3 package at 1000 K, and the process lasts for more than 1.0 ps with a time step of 1.0 fs.
A single-gated two-probe model is built to simulate the transport of SNM, and the pristine silicene is used as source and drain electrodes for simplicity. Transport properties are calculated by the semi-empirical (SE) extended Hückel method coupled with NEGF formalism implemented in the Atomistix Tool Kit (ATK) 11.2 package [49][50][51] . The Hoffman basis is used, and the temperature is set at 300 K. The kpoint meshes of the electrodes and central region are set to 1 3 50 3 50 and 1 3 50 3 1, respectively. The current is calculated with the Landauer-Bűttiker formula 52 : where T(E, V g , V bias ) is the transmission probability at a given gate voltage V g and bias voltage V bias , f L/R the Fermi-Dirac distribution function for the left(L)/right(R) electrode, and m L /m R the electrochemical potential of the L/R electrode.
To include the phonon effect in the calculation of transport properties, ab initio MD simulation of the central region of the device within the NVT ensemble is performed by using the Dmol 3 package at 300 K, and the process lasts for 3.0 ps with the electrode extension parts constrained. The time step is 1.5 fs. Then different configurations of the central region are built into two-probe models after every 400 MD steps, and their transport properties are evaluated and finally averaged over 5 configurations using a NEGF approach implemented in the ATK package.
During the production of this paper we would like to add that we are aware that silicene FET operating at room temperature has been fabricated recently 53 , corroborating theoretical expectations regarding its ambipolar Dirac charge transport.