Enhanced Thermoelectric Efficiency of Porous Silicene Nanoribbons

There is a critical need to attain new sustainable materials for direct upgrade of waste heat to electrical energy via the thermoelectric effect. Here we demonstrate that the thermoelectric performance of silicene nanoribbons can be improved dramatically by introducing nanopores and tuning the Fermi energy. We predict that values of electronic thermoelectric figure of merit ZTe up to 160 are achievable, provided the Fermi energy is located approximately 100 meV above the charge neutrality point. Including the effect of phonons yields a value for the full figure of merit of ZT = 3.5. Furthermore the sign of the thermopower S can be varied with achievable values as high as S = +/− 500 μV/K. As a method of tuning the Fermi energy, we analyse the effect of doping the silicene with either a strong electron donor (TTF) or a strong electron acceptor (TCNQ) and demonstrate that adsorbed layers of the former increases ZTe to a value of 3.1, which is insensitive to temperature over the range 100 K – 400 K. This combination of a high, temperature-insensitive ZTe, and the ability to choose the sign of the thermopower identifies nanoporous silicene as an ideal thermoelectric material with the potential for unprecedented performance.

I n the recent years, the challenge of removing heat from nanoelectronic devices 1 and thermoelectrically converting waste heat into electricity has attracted huge scientific interest 2 , not only due to the questions posed for fundamental science, but also because thermoelectric energy conversion is an essential requirement for the next generation of nanoscale electronic, optoelectronic and photonic devices 2 . The efficiency of a thermoelectric device and material is determined by its thermoelectric figure of merit (ZT) defined as: where S is the Seebeck coefficient (thermopower), G is the electrical conductance, T the temperature and k the thermal conductance given by k 5 k e 1 k p , where k e (k p ) is the electronic (phononic) contribution to k 3 . Clearly ZT could be enhanced by increasing the power factor (S 2 GT) or decreasing the thermal conductance and therefore a high-performance thermoelectric material should possess a large Seebeck coefficient and electrical conductance and simultaneously a low thermal conductance. The search for new materials with enhanced thermal properties continues to intensify, because these factors are correlated and increasing ZT to values greater than unity requires a delicate optimisation of several material properties. One promising approach has been to reduce the contribution k p of parasitic phonons by nanostructuring materials 4 . Although acoustic phonons are the main heat carriers in bulk crystals, nanostructures can exhibit significantly lower phonon thermal conductances k p , due to increased phonon boundary scattering, changes in the phonon density of states and modified phonon dispersion in low-dimensional materials 2 . The nanostructuring of materials is also a promising route to increasing the power factor, because it can lead to sharp features in the electronic density of states and the transmission coefficient T(E) describing the propagation of electrons of energy E through a device. It is well known that the thermopower S is controlled by the asymmetry of T(E) at the Fermi energy E F and therefore if E F is close to such an asymmetry, both S and ZT will be enhanced. Unfortunately, asymmetries, even if they occur, are not necessarily located near E F and therefore a method of tuning them is required. In a nanoscale device, one could of course consider introducing a third gate electrode to control such features, but in practice this costs energy and does not solve the problem of designing a new material.
The above considerations suggest that an effective strategy for enhancing thermoelectric properties should 1) start from a parent material with low intrinsic thermal conductance, 2) nanostructure the material to further reduce the thermal conductance, 3) implement additional nanostructuring to introduce sharp features in T(E) and 4) chemically modify the material to move these sharp features towards the Fermi energy. In this paper our aim is to demonstrate that silicene, a new counterpart of graphene is an ideal material for implementing the four key elements of this strategy, because it is not only CMOS-compatible, but also it possesses a low intrinsic thermal conductance, which can be further reduced by nanostructuring the material to form nanoribbons. Since silicon is the most common material used in the electronics industry, it is highly desirable to utilise silicon-compatible materials for thermoelectric energy conversion. Bulk silicon has a very low ZT (<0.01) and therefore as a first step in a strategy for enhancing thermoelectric performance it is natural to focus on silicene, which is a recently-observed one-atom-thick crystalline form of silicon atoms arranged in a slightly buckled honeycomb lattice structure [5][6][7][8][9][10][11][12] . Silicene nanoribbons have been synthesised on silver (111) 7,13-25 , gold (110) 26 , iridium (111) 27 and the zirconium diboride (0001) 28,29 substrates and are predicted to be stable on non-metallic substrates 30 . Calculations of thermoelectric properties of armchair and zigzag silicene nanoribbons with and without hydrogen-passivated edges suggest that this material may be attractive for thermoelectric devices [31][32][33][34] , because the thermal conductivity of 2D silicene is predicted to be much smaller than bulk silicon and its counterpart graphene 35 , with only ,10% of the total phononic thermal conductivity being due to the out-of-plane acoustic phonons 36,37 . Both of these desirable features arise from the presence of small buckling, which breaks the reflection symmetry of the structure 35 44 and SiGe 45 . Moreover, it is been shown that placing nanopores in bulk silicon greatly reduces the thermal conductivity and enhances ZT by the factor of two 46 .
Since the low phonon thermal conductance of silicene is well established and therefore elements 1) and 2) of the above strategy are satisfied, in this paper we focus steps 3) and 4) and demonstrate that they can be achieved by inserting nanopores into silicene nanoribbons, whose edges are terminated by hydrogen or oxygen and by introducing adsorbates to tune the position of features in T(E) relative to E F . In view of the low value of k p , we will focus primarily on the electron contribution in thermoelectric figure of merit ZT e (obtained by setting k p 5 0 in equation 1) and show that for ribbons containing nanopores with hydrogen-passivated edges, ZT e could be enhanced to values as high as 160 at room temperature by creating sharp features in the T(E) and controlling the Fermi energy or by introducing adsorbates such as Tetrathiafulvalene (TTF) and Tetracyanoquinodimethane (TCNQ) onto the silicene surface. We will then show that by including the k p of silicene nanoribbons, a high value of the full ZT of order 3.5 is achievable.

Results and Discussion
We have calculated the electronic contribution to the thermal conductance, the Seebeck and Peltier coefficients and ZT e for the struc-tures shown in Figure 1c-h. The optimized lattice constant (a 0 ) and buckling (d) of the silicene nanoribbon shown in Figure 1a are found to be 3.6 Å and 0.53 Å , respectively, similar to that reported elsewhere 10,32,47 . The engineered silicene nanoribbons shown in Figure 1 include: a zigzag monolayer silicene nanoribbon with hydrogen terminated edges (ZSiNR-H, Figure 1c), a zigzag monolayer silicene nanoribbon containing a nanopore with all edges terminated by hydrogen (ZSiNR-P, Figure 1d), a zigzag monolayer silicene nanor-   Figure 1f). The nanoribbons length and width in all cases are almost equal (L < 6 nm (scattering region < 3.48 nm), W < 3 nm) and the pore diameter is <1.3 nm. We shall find that a key strategy for improving the thermoelectric performance of these structures involves tuning the Fermi energy (E F ). As an example of how this could be achieved, we investigated the effect of introducing adsorbates onto the surface of the ZSiNR-P hydrogen-terminated nanoribbon, as shown in Figures 1f and 1g. The latter show examples of such functionalised ribbons containing adsorbed TTF (a strong electron donor) and TCNQ (a strong electron acceptor), which form charge-transfer complexes with the silicene.
Silicene nanoribbons and nanopores. For the structures (c) to (f) of Figure 1, Figure 2 shows results for electrical conductance G, the electronic contribution of the thermal conductance k e , the thermopower S and the Peltier coefficient P of the junction as a function of the temperature (T). All of these properties are obtained from the energy (E) dependence of the electron transmission coefficient T(E), shown in Figure 2a. Figure 2 shows that the silicene nanoribbons ZSiNR-H, ZSiNR-HO and ZSiNR-O possess high electronic thermal conductances and low thermopowers and consequently their figures of merit (ZT e ) are low. However, the results for the silicene monolayer nanoribbon containing a nanopore (ZSiNR-P) show that ZT e is significantly improved by placing a hole in the ribbon. This arises because the densities of states of all four structures possess sharp peaks around the Fermi energy E F due to band bending in the corner of the k-space in silicene band structure, which introduces a sharp feature (indicated by an arrow in Figure 2a) in T(E) near E F . Furthermore the T(E) of ZSiNR-P possess the desirable feature that T(E) is almost zero on either side of the peak. This feature is associated with edges states, which is why it is sensitive to the chemical nature of the edge terminations. These edge states are a well-known feature of nanoribbons with zigzag edges [48][49][50] and lead to a significant improvement in the thermopower (Figure 2e) as well as a reduction in the thermal conductance ( Figure 2d). Edge states have been predicted earlier theoretically 51 and observed experimentally [52][53][54] in graphene nanoribbons. By comparing ZSiNR-P with ZSiNR-H or by comparing ZSiNR-HO with either ZSiNR-P or ZSiNR-O, Figure 2 demonstrates the general trend that introducing scattering reduces both the electrical and thermal conductances and simultaneously increases the thermopower. Figure 2 also reveals the attractive property that ZSiNR-P possesses the highest room-temperature ZTe (of order 1.4) and that this high thermoelectric efficiency is preserved over a wide range of temperature from 100 K to 500 K.
To highlight the role of sharp features near E F in the transmission coefficients of Figure 2a, we investigate a simple model for which    . This is mainly due to reduction of electronic thermal conductance (Figure 3c), because the thermopower decreases more slowly (Figure 3d) than the electronic thermal conductance.
The above simple model and the results of Figure 2 demonstrate that the thermoelectric performance of silicene nanoribbons is improved by the introduction of nanopores. To demonstrate that further dramatic improvements are available if the Fermi energy is tuned by an external gate, figure 4 shows results for ZT e , S and k e as a function of E F , obtained from the exact transmission curves T(E) of Figure 1a. This demonstrates that at room temperature, the electronic contribution to ZT e can be hugely enhanced by varying the Fermi energy by as little as 100 meV, with achievable values as high as 160. Furthermore the sign of the thermopower can be selected by such tuning, with achievable values as high as S 5 1/2 500 mV/K. This improvement could be achieved by electrostatically-gating the nanopore-containing ribbon. For the purpose of improving intrinsic material performance, we demonstrate below that a Fermi energy shift can also be realised by introducing donor or acceptors onto the surface of the silicene nanoribbon. However before considering the effect of doping, we examine the effect of phonons, since ZT e is obtained by neglecting the phononic contribution (k p ) to the thermal conductance in equation (1). This is a reasonable approximation to the full figure of merit ZT provided the electronic thermal conductance (k e ) is higher than phononic thermal conductance. However large values of ZT e correspond to the opposite limit of low values of k e , where k p . k e . In this limit, one needs to consider only the phononic contribution and consequently the denominator of equation (1) becomes a constant, independent of s and E 0 .
To estimate the k p for the structure of figure 1d, we note that for a perfect zigzag silicene nanoribbon of width equal to the bridging sections of silicene above and below the pore of Figure 1d is predicted to have a phononic thermal conductance k p of less than 6 3 10 211 W/K 32 . This value contrasts markedly with graphene, whose thermal conductance is predicted to be a factor of at least 50 times higher 36 . As discussed in the introduction, boundary scattering by nanopores can lead to a further order of magnitude reduction in k p and therefore a value of k p on the scale of ,10 212 W/K would seem reasonable for the structure of Figure 1d. For the model T(E) of Figure 3a, Figure 3f shows the full ZT when the thermal conductance is dominated by phonons, with a high thermal conductance of k p 5 2 3 10 211 W/K. This demonstrates that even in the most unfavourable scenario, the full ZT could achieve a value as high as 3.5 for values of s and E 0 lying in the red region of Figure 4f  Adsorbate-functionalized silicene. For the purpose of improving intrinsic material performance, a method of shifting E F by doping the silicene nanoribbon is needed. Since covalent bonding of dopants to the nanoribbons may adversely affect electronic properties, we now consider tuning the Fermi energy by adsorbing planar molecules which interact with the surface only through weak pi-pi interactions. The first adsorbed molecule we study is Tetrathiafulvalene (TTF) which is a strong donor and the second is Tetracyanoquinodimethane (TCNQ), which is a strong acceptor 55 .
In the presence of adsorbates, calculations were performed by relaxing the adsorbates to find their minimum-energy states on the surface of the silicene (see methods), which yielded an optimized distance between the silicene and TCNQ of 3.1 Å and 3.54 Å for TTF. For the lowest dopant concentrations, we place a monolayer or sub-monolayer of TTF or TCNQ on the surface of the electrodes of ZSiNR-P, as shown in Figures 1g and 1h, whereas for the highest concentration, two layers above and below the silicene are included, as shown in Figure 1b. Figure 5 shows that TTF on silicene simultaneously increases the Seebeck coefficient, decreases the thermal conductance and increases ZT e from an undoped value of 1.4 to 1.7, 1.85 and 3.1 with increasing concentration from 21% to 43% and 87%, where the concentration of adsorbate is defined as the ratio of the number of TTF atoms to the number of silicene nanoribbon atoms.
The inset of Figure 6 shows the variation of ZT e with E F for different concentrations of TTF and reveals that an 87% coverage produces a Fermi energy shift of approximately 3 meV. The transmission just around the Fermi energy is shown in Figure S2b of SI.
In summary, we have developed a new strategy for improving the thermoelectric performance of silicene-based nanoribbons by investigating the effect of introducing nanopores and varying their edge termination. We have demonstrated that the thermopower and electronic thermoelectric figure of merit ZT e can be improved by introducing nanopores and tuning their Fermi energy. By shifting the Fermi energy by approximately 100 meV from the chargeneutrality point, we predict that huge values of ZT e up to 160 are accessible and that the sign of the thermopower can be varied with achievable values as high as S 5 1/2 500 mV/K. As a method of  tuning the Fermi energy, we analysed the effect of doping the silicene with either a strong electron donor (TTF) or a strong electron acceptor (TCNQ) and demonstrated that doping by the former tended to decrease the value of ZT e , because the Fermi energy shift was in an unfavourable direction. On the other hand, we found that doping with TTF increased the room-temperature value to ZT e to 3.1 and that this value is insensitive to temperature over the range 100 K -400 K. This combination of a high temperature-insensitive ZT e , and the fact that the low phonon thermal conductance renders ZT e comparable with the full figure of merit ZT, identifies nanoporous silicene as an ideal thermoelectric material with the potential for unprecedented performance.

Methods
To find the optimized geometry and ground state Hamiltonian of the structures of interest, we employed the SIESTA 56 implementation of Density Functional Theory (DFT) using the generalized gradient approximation (GGA) of the exchange and correlation functional with the Perdew-Burke-Ernzerhof parameterization (PBE) 57 a double zeta polarized basis set, a real-space grid defined with a plane wave cut-off energy of 250 Ry and a maximum force tolerance of 40 meV/Ang. The calculation with VDW-DF exchange and correlation functional with BH parameterization was also carried out to check the GGA-PBE result in some cases. The BH is the same as DRSLL with some modification 58 . From the converged DFT calculation, the underlying mean-field Hamiltonian was combined with our transport code, GOLLUM 59 which is an implementation of the non-equilibrium Green's function (NEGF) method. This yields the transmission coefficient T(E) for electrons of energy E (passing from the source to the drain) via the relation In this expression, C L,R (E) 5 i (S L,R (E) 2 S L,R { (E)) describes the level broadening due to the coupling between left (L) and right (R) electrodes and the central scattering region, S L,R (E) are the retarded self-energies associated with this coupling and G R 5 (ES 2 H 2S L 2 S R ) 21 is the retarded Green's function, where H is the Hamiltonian and S is overlap matrix (both of them obtained from SIESTA). Thermal properties such as the electrical conductance G(T), the electronic contribution of the thermal conductance k e (T), the thermo-power S(T) and the Peltier coefficient P(T) of the junction as a function of the temperature are given by 59 : and T(E) is the transmission coefficient, f(E) is the Fermi-Dirac probability distribution function (f(E) 5 (1 1 exp (E 2 E F /k B T)) 21 ), T is the temperature, E F is the Fermi energy, G 0 5 2e 2 /h is the conductance quantum, e is electron charge and h is the Planck's constant. From these expressions, the electronic contribution to the figure of merit ZT e is: