⇦Chinese Physics Letters, 2019, Vol. 36, No. 9, Article code 097401Express Letter Superconducting Single-Layer T-Graphene and Novel Synthesis Routes * Qinyan Gu (顾琴燕), Dingyu Xing (邢定钰), Jian Sun (孙建)** Affiliations National Laboratory of Solid State Microstructures, School of Physics and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093 Received 31 July 2019, online 03 August 2019; Erratum Chin. Phys. Lett. 36 (2019) 109901 *Supported by the National Key Research and Development Program of China under Grant No 2016YFA0300404, the National Basic Research Program of China under Grant No 2015CB921202, the National Nature Science Foundation of China under Grant Nos 11574133 and 11834006, the Nature Science Foundation of Jiangsu Province under Grant No BK20150012, the Fundamental Research Funds for the Central Universities, the Science Challenge Project (No TZ2016001).
**Corresponding author. Email: jiansun@nju.edu.cn Citation Text: Gu Q Y, Xing D Y and Sun J 2019 Chin. Phys. Lett. 36 097401    Abstract Single-layer superconductors are ideal materials for fabricating superconducting nano devices. However, up to date, very few single-layer elemental superconductors have been predicted and especially no one has been successfully synthesized yet. Here, using crystal structure search techniques and ab initio calculations, we predict that a single-layer planar carbon sheet with 4- and 8-membered rings called T-graphene is a new intrinsic elemental superconductor with superconducting critical temperature ($T_{\rm c}$) up to around 20.8 K. More importantly, we propose a synthesis route to obtain such a single-layer T-graphene, that is, a T-graphene potassium intercalation compound (C$_4$K with $P4/mmm$ symmetry) is firstly synthesized at high pressure ($>$11.5 GPa) and then quenched to ambient condition; and finally, the single-layer T-graphene can be either exfoliated using the electrochemical method from the bulk C$_4$K, or peeled off from bulk T-graphite C$_4$, where C$_4$ can be obtained from C$_4$K by evaporating the K atoms. Interestingly, we find that the calculated $T_{\rm c}$ of C$_4$K is about 30.4 K at 0 GPa, which sets a new record for layered carbon-based superconductors. The present findings add a new class of carbon-based superconductors. In particular, once the single-layer T-graphene is synthesized, it can pave the way for fabricating superconducting devices together with other 2D materials using the layer-by-layer growth techniques. DOI:10.1088/0256-307X/36/9/097401 PACS:74.62.Bf, 74.62.Fj, 74.70.Wz © 2019 Chinese Physics Society Article Text Superconductivity in single layer materials,[1] such as FeSe,[2] MoS$_2$,[3] NbSe$_2$[4] has attracted tremendous attention recently due to their inspiration in fundamental science and potentials in future applications. However, without charge doping or other adjustments, very few examples of intrinsic single layer superconductor have been found. For instance, graphene is not a superconductor due to the vanishing density of states at the Dirac point, although it can become a 2D single layer superconductor with charge doping or tensile strain.[5,6] Recently reported superconductivity in magic-angle graphene[7] caused a sensation, but a magic angle between two adjacent graphene layers is required, which is difficult to be controlled. From what has been proposed so far, the 2D boronphene seems to be the only intrinsic single layer superconductor without external strain or charge doping.[8,9] However, the synthesis of such a kind of 2D elemental superconductor remains a challenge. Carbon is one of the most versatile elements and can constitute various types of molecules and crystals due to its rich electronic hybridization configurations: $sp$, $sp^2$ and $sp^3$. In addition to the well-known diamond, graphite, C$_{60}$ fullerene,[10] carbon nanotube[11] and graphene,[12] many other types of carbon allotropes have been studied, including M-carbon,[13] bct-carbon,[14] graphdiyne,[15] nanotwin diamond,[16] penta-graphene,[17] V-carbon,[18] haeckelites,[19] etc., as an incomplete list. Recently, a buckled T-graphene was predicted to have Dirac-like fermions and a high Fermi velocity similar to graphene.[20] This new form of carbon sheet was predicted to present good mechanical properties[21] and could be used in hydrogen storage.[22] Although there is no superconductivity in pure graphite, many carbon-allotrope related materials were reported to be superconducting, such as graphite intercalation compounds (GICs),[23] fullerene alkali metal compounds,[24] nanotubes,[25] boron-doped diamond,[26] boron carbide[27] and magic-angle graphene superlattices.[7] Due to the similarity of the graphene-like boron layers in MgB$_2$,[28] GICs draw extensive attention, in which metallic atoms intercalate between the graphene sheets. Alkali metal carbon compounds (C$_8$A, A=K, Rb, Cs) were the first type of GIC superconductors[23] studied, with $T_{\rm c}$ being usually less than 1 K. Since C$_8$K is one of the easiest to fabricate in alkali carbon compounds,[23] intense efforts have been taken to the study on its superconducting properties.[29,30] With pressure, the $T_{\rm c}$ of C$_8$K increases to 1.7 K at 1.5 GPa.[31] Up to now, among the GICs theoretically proposed, C$_6$Yb (with $T_{\rm c}$ = 6.5 K)[32] and C$_6$Ca (with $T_{\rm c}$ = 11.5 K)[33] have the highest $T_{\rm c}$ at ambient pressure. And C$_6$Ca exhibits $T_{\rm c}$ = 15.1 K at high pressure of 7.5 GPa.[34] In addition to the bulk GICs, superconductivity in doped graphene with lithium,[35] potassium[36] and Calcium,[37] etc. has been studied theoretically and experimentally. On the other hand, pressure is widely applied to explore new materials with unexpected stoichiometries as well as structures.[38,39] Considering the abundant carbon allotropes under normal or pressurized conditions, one would ask the questions: Is there any new 2D carbon intercalation compound with better superconducting properties? Can we find a 2D pure carbon allotrope with intrinsic $T_{\rm c}$? In this Letter, using crystal structure search methods and first-principles calculations, we predict a very interesting C$_4$K compound with $P4/mmm$ symmetry. It has 2D carbon layers with the structure as T-graphene, where potassium atoms intercalate in between, on the analogy of GICs. We predict that this $P4/mmm$ C$_4$K can be synthesized above 11.5 GPa and is quenchable to ambient pressure, exhibiting a high $T_{\rm c}$ up to around 30.4 K at 0 GPa, which is more than one order of magnitude higher than that of C$_8$K. We also propose that T-graphene can be peeled off from the bulk C$_4$K, once it is synthesized by high pressure method. Most surprisingly, T-graphene itself is found to be an intrinsic 2D elemental superconductor with $T_{\rm c}$ of about 20.8 K at ambient pressure.
cpl-36-9-097401-fig1.png
Fig. 1. The crystal structure and stability of C$_4$K. (a) The crystal structures of C$_4$K, orange and purple balls denoting the C and K atoms, respectively. (b) Calculated formation enthalpies for C + K or C$_8$K + K relative to C$_4$K under high pressure, with optB88-vdw[40] and optPBE-vdw[41] functionals. (c) The mean square displacement (MSD) of the K and C atoms from the AIMD simulations for C$_4$K at ambient pressure and temperatures of 300 (upper panel) and 1000 K (lower panel). (d) Calculated exfoliation energies of single-layer T-graphene from C$_4$K and bulk T-graphite (C$_4$) versus separation distance $d-d_0$, compared with that of graphene.
To find a possible route to synthesize T-graphene, we first use our newly developed machine learning accelerated crystal structure search method,[42] to explore possible stable phases of C–K system up to 20 GPa. More details about the crystal structure search and ab initio calculations can be found in the method section and the Supplementary Material (SM). We find that the known C$_8$K compound becomes unstable at 20 GPa and tends to decompose to more stable C$_4$K and other compounds. A C$_4$K compound with $P4/mmm$ structure becomes energetically favorable at this pressure, in which carbon atoms construct a 2D sheet with 4- and 8-membered rings, named as T-graphene. While the K atoms in C$_4$K occupy the interlayer site above the centers of the octagonal ring (shown in Fig. 1(a)). Due to the unique layered structure of C$_4$K and its similarity to GICs, we believe that it belongs to a new type of intercalation compounds, and we call them the T-graphene intercalation compounds (TGICs). To further confirm the stability of $P4/mmm$ C$_4$K, we calculate its formation enthalpy in two different possible synthesis routes, 4C + K $\rightarrow$ C$_{4}$K and C$_{8}$K + K $\rightarrow$ 2C$_{4}$K, ranging from 0 to 20 GPa. Under different pressures, the most stable phases of the elements (graphite or diamond for carbon, BCC or FCC for potassium) are taken as the references to calculate the formation enthalpy. A hard version pseudopotential with small core radium for carbon and a potassium pseudopotential with 9 valence electrons (including all the 3$s$, 3$p$ and 4$s$ electrons), together with an extremely high cutoff energy (1050 eV) and very dense $k$-mesh are used. More details can be found from the method section and the SM. As shown in Table S1 in the SM, the optB88-vdw[40] and optPBE-vdw[41] functionals yield the best agreement with the experimental lattice constants for graphite and potassium, as well as the correct ground state of potassium. Therefore, we use them to calculate the formation enthalpy of C$_4$K. As shown in Fig. 1(b), although the transition pressures from different functionals are slightly different, the conclusions are consistent that C$_4$K is more stable relative to both C + K and C$_{8}$K + K under high pressure. The highest transition pressure is around 11.5 GPa. We calculate the phonon spectra of $P4/mmm$ C$_4$K at 0 GPa and 20 GPa, see Fig. S3 in the SM, and find that both of them are stable. This indicates that $P4/mmm$ C$_4$K is possible to be synthesized at high pressure and quenchable to ambient pressure. To check the stability of C$_4$K at finite temperature, we perform ab initio molecular dynamics simulations. As shown by the mean square displacement (MSD) results in Fig. 1(c), C$_4$K is also stable at ambient pressure and at finite temperature (300 K and 1000 K). Nevertheless, the synthesis of C$_4$K may still be challenging and it requires cutting-edge techniques that provide high pressure and high temperature condition, such as laser-heated diamond anvil cells[43] or large-volume multianvil apparatus. These techniques have been used to successfully synthesize many new materials, including carbon-based materials.[16,18,43] As shown by convex hulls in Fig. S1(a) in the SM, we find that there are three new stable/metastable candidates of C–K compounds at 20 GPa: the C$_4$K with $P4/mmm$ structure, the C$_3$K$_4$ with $C2/m$ structure, and the C$_2$K$_3$ with $C2/m$ structure. The C$_8$K becomes energetically unfavorable and tends to decompose to C$_4$K and other compounds when pressure increases to 20 GPa. We note that the carbon atoms in C$_3$K$_4$ and C$_2$K$_3$ form chain-like structures (see Fig. S2 in the SM). As we know, liquid exfoliation and mechanical cleavage haven been widely used to get 2D monolayers from their laminated bulk crystals with weakly coupled layers.[44,45] Exfoliation energy is the energy cost which is used to peel off the topmost single layer from the surface of bulk crystal. To explore the feasibility of peeling, we calculate the exfoliation energy of T-graphene from bulk $P4/mmm$ C$_4$K and T-graphite C$_4$, respectively, where bulk T-graphite C$_4$ can be obtained from C$_4$K by evaporating the potassium atoms, possibly using similar procedure in the synthesis of Si$_{24}$ from Na$_{4}$Si$_{24}$.[46] As shown in Fig. S5 in the SM, the potential energy barrier of moving the K atom in C$_4$K is very similar to that of the Na atom in Na$_{4}$Si$_{24} $, which shows the feasibility of obtaining bulk C$_4$ from C$_4$K. We take the graphene peeled off from bulk graphite as a reference system and the exfoliation energy of graphene is calculated to be around 24.1 meV/Å$^2$, which is in good agreement with the experimental results (20$\pm$1.88 meV/Å$^2$).[47,48] Here, we use a vacuum layer of 20 Å to make sure that the charge density of the two surfaces do not overlap. We then used the same parameters to calculate the exfoliation energy of T-graphene. With different parent materials of bulk $P4/mmm$ C$_4$K and T-graphite C$_4$, the resulted exfoliation energy are 85.8 meV/Å$^2$ and 21.6 meV/Å$^2$, respectively, as shown in Fig. 1(d).
cpl-36-9-097401-fig2.png
Fig. 2. A sketch map of the proposed synthesis routes for the superconducting single-layer T-graphene. Path I represents the electrochemical exfoliation route from C$_4$K and Path II represents the mechanical exfoliation route from the bulk T-graphite, where the bulk T-graphene (C$_4$) can be obtained from C$_4$K by evaporating the potassium atoms with a similar method used in the synthesis of Si$_{24}$ from Na$_{4}$Si$_{24}$.[46]
According to the literature,[49] a 2D material is potentially exfoliable (or easily to be exfoliated) when the exfoliation energy is $ < $130 meV/Å$^2$ ($ < $30-35 meV/Å$^2$). Therefore, T-graphene is potentially exfoliable from C$_4$K and much easier to be exfoliated from bulk T-graphite C$_4$. As shown in Table. 1, we find that C$_4$K has similar charge transfer as that in many other graphite intercalation compounds, such as C$_8$Li, C$_8$Na, C$_8$K and C$_6$Li. Although having ionic-like interactions, it was reported that graphene sheets can be exfoliated from C$_6$Li, C$_8$Na and C$_8$K in experiments using the electrochemical exfoliation method.[50–52] Thus due to the structural similarity and also the similar charge transfer, T-graphene can very likely be exfoliated from bulk C$_4$K with similar electrochemical method as used in the graphite intercalation compounds mentioned above. The routes we proposed to synthesize monolayer T-graphite are summarized in Fig. 2.
Table 1. The amount of charge transfer from the intercalated metal atoms to the carbon sheets in C$_4$K and some typical graphite intercalation compounds.
System C$_4$K C$_6$Li C$_8$Li C$_8$Na C$_8$K
Charge transfer 0.8$e$ 0.9$e$ 0.8$e$ 0.9$e$ 0.8$e$
cpl-36-9-097401-fig3.png
Fig. 3. Calculated electronic structures of C$_4$K and single-layer T-graphene at 0 GPa. The orbital-resolved band structures (left panel) and electron DOS (right panel) are shown in (a) for C$_4$K and (c) for T-graphene. The size of the dots in the band structures stands for the weights of different orbitals. The Fermi level is denoted by the dashed horizontal line. (b) Front view (left) and the top view (right) of the Fermi surface of C$_4$K in the extended Brillouin zone. (d) Fermi contour of T-graphene in the extended BZ. The bold rectangle represents the conventional BZ and the arrows point to the nesting vectors.
In what follows, we investigate the electronic properties of bulk C$_4$K and single-layer planar T-graphene. The calculated electronic structures of C$_4$K and T-graphene are plotted in Fig. 3. We also calculate the band structures of the C$_4$ layer without potassium atoms (C$_4$K$_0$), which is obtained by removing the K atoms from the C$_4$K structure and keeping the lattice constants at 0 GPa (see Fig. S6 in the SM). Compared with C$_4$K$_0$, the intercalation of K atoms pushes up the Fermi level and increases the occupancy of electrons in $\pi$ bands. From Fig. 3, one can see that either in C$_4$K or T-graphene, the electronic density of states at the Fermi level, $N(\epsilon _{F})$, is mostly dominated by the $\pi$ bands. In C$_4$K, there is only one band crossing the Fermi level, which results in a relatively smooth cylinder-like Fermi surface around the $M$ point, as it can be seen in Fig. 3(b). While in the T-graphene, the absence of K atoms lowers the Fermi level and adds a new hole pocket around the ${\it\Gamma}$ point. As shown in Fig. 3(b) and 3(d), both fermi surfaces of C$_4$K and T-graphene have 2D features and perfect nesting vectors. Since the Fermi contours of C$_4$K and T-graphene have similar cylinder shapes as those in the cuprates[53] and iron-based superconductors,[54] it is natural to guess if they are also superconducting. Therefore, we investigate their superconducting properties by electron-phonon coupling calculations and the Allen-Dynes modified McMillian equation.[55] We estimate that the $T_{\rm c}$ of T-graphene is around 20.8 K at ambient pressure with a commonly used screened Coulomb potential $\mu^*$ = 0.1. And in C$_4$K, its $T_{\rm c}$ is around 30.4 K at 0 GPa, which is the highest among all known graphite intercalation compounds. One can consider that the number of doping atoms in C$_4$K is more than that in C$_8$K, thus it seems that a higher doping level is a positive factor to the superconductivity in these carbon-sheet based systems.
cpl-36-9-097401-fig4.png
Fig. 4. Electron-phonon coupling and electronic structures with/without perturbation of C$_4$K at 0 GPa. (a) Phonon dispersion curves, Eliashberg spectral functions $\alpha^2F$($\omega$) together with the electron-phonon integral $\lambda$($\omega$) and phonon density of states (PHDOS). (b) the $E_{\rm g}$ vibrational mode. (c) The electronic band structures of C$_4$K in the presence (red) and absence (blue) of the perturbation from the $E_{\rm g}$ mode. (d) Fermi surface in the $k_z\,=\,0$ plane after perturbation and the arrows point to the nesting vectors.
In Fig. 4, we plot phonon dispersions together with phonon linewidth, Eliashberg spectral function $\alpha^2F$($\omega$), electron-phonon coupling strength $\lambda$($\omega$) and phonon density of states (PHDOS) of C$_4$K at ambient pressure. According to the electronic structures discussed above, there is only one electron pocket in the Fermi surface of C$_4$K, which is very similar to the case of single-layer FeSe on SrTiO$_3$.[56] According to $\lambda =2\int \frac{\alpha ^{2}F\left ( \omega \right )}{\omega }d\omega$, there is a negative correlation between $\lambda$ and $\omega$, so that the low frequency vibrations usually can enhance the electron-phonon coupling and thus the superconductivity. In C$_4$K, most contributions to the electron-phonon coupling constant arise from the low-frequency modes, especially those related to the vibration of the K atoms and the out-of-plane $E_{\rm g}$ mode of the carbon sheet, as shown in Fig. 4(b). This also corresponds to the two predominant peaks in the Eliashberg spectral functions $\alpha^2F$ at about 300 and 450 cm$^{-1} $. To provide more insights into the electron-phonon coupling, we also compare the electronic bands without and with the perturbation induced by the $E_{\rm g}$ phonon mode with $\Delta Q$ = 0.1Å. The calculated band structures and Fermi surfaces with the perturbation are plotted in Figs. 4(c) and 4(d), respectively. Compared to the unperturbed case, the perturbated interlayer band near ${\it\Gamma}$ point is dramatically lowered by 0.3 eV leading to a decrease in electronic occupation of the $\pi$ bands. Particularly, the unperturbated interlayer bands have a gap near the Fermi level around the ${\it\Gamma}$ point, while the perturbated bands get in touch together and produce a new electron pocket at the ${\it\Gamma}$ point. These results clearly show that the low-energy carbon out-of-plane vibrations are critical to the electron-phonon pairing, similar to the case in GICs. Under pressure, the reduced interlayer distance between the carbon sheet and K atoms will push up the bands of K atoms (see Fig. S9 in the SM) and decrease the electron-phonon coupling. This argument is consistent with the present calculated results, where $T_{\rm c}$ of C$_4$K decreases under pressure (see Fig. S10 in the SM).
cpl-36-9-097401-fig5.png
Fig. 5. Electron-phonon coupling and electronic structures of single-layer T-graphene at 0 GPa. (a) Phonon dispersion curves, Eliashberg spectral functions $\alpha^2F$($\omega$) together with the electron-phonon integral $\lambda$($\omega$) and phonon density of states (PHDOS). (b) The vibrational $A_{1}$ mode. (c) The electronic band structures in the presence (red) and absence (blue) of the perturbation from the $A_{1}$ mode.
Figure 5 shows the electron-phonon coupling, electronic structures as well as the representative phonon mode of the single-layer T-graphene. From Fig. 5(a), one can see that three acoustic modes in the low-frequency range ($\omega \le 175$ cm$^{-1}$) couple strongly with electrons on the Fermi surface, which makes a great contribution to $\lambda$ and yields $\lambda \approx 1.23$, almost 78.3% of the total electron-phonon coupling constant. Among these three acoustic modes, the softest out-of-plane mode has the largest phonon linewidth, which gives rise to the highest peak in the Eliashberg spectral functions. We plot the vibration of the softest acoustic mode at the $X$ point ($A_1$ mode with $\omega \sim 160$ cm$^{-1}$) in Fig. 5(b). To reveal its effect on the electrons on the Fermi surface, in Fig. 5(c), we show the calculated electronic bands with and without the perturbation induced by the $A_1$ phonon mode with $\Delta Q$ = 0.1 Å. With the perturbation, the band dispersions and Fermi velocities near the Fermi level have obvious changes, indicating the coupling between the electrons and this softest phonon mode. In Fig. 5(a), although the high frequency phonon modes above 1000 cm$^{-1}$ have a large linewidth, they seem to have less contribution to the Eliashberg spectral functions and also the total electron-phonon coupling constant $\lambda$ compared to the low frequency modes discussed above. As shown in Fig. S7 in the SM, the band dispersions in the presence of the perturbation induced by the $B_3u$ phonon at $X$ point are almost consistent with those in the absence of perturbation. Therefore, the influence of the high-frequency carbon in-plane vibrations on the Fermi-surface electrons is weaker and they do not make a significant contribution to the superconducting pairing. Although the out-of-plane modes in single-layer T-graphene have similar frequencies as those in bulk C$_4$K, there have not been too much electronic states to couple with. In summary, using ab initio calculations, we have found that single-layer T-graphene with 4- and 8-membered rings is an intrinsic elemental 2D superconductor with a $T_{\rm c}$ of around 20.8 K, in which the low frequency out-of-plane vibrational acoustic modes play a key role in superconducting pairing. We have also proposed a novel route to synthesize the single-layer T-graphene, that is, first synthesize the T-graphene intercalation compounds by high pressure method, and then exfoliate the single-layer T-graphene using electrochemical or other methods. As an example, we have searched carefully the C–K system using our machine learning accelerated crystal structure search method and find a $P4/mmm$ C$_4$K, which is exactly the T-graphene intercalation compound we want. This C$_4$K compound can be synthesized when the pressure is higher than 11.5 GPa, and can be quenched to ambient pressure. Our calculation results show that the T-graphene should be feasible to be exfoliated from C$_4$K using the electrochemical exfoliation method once C$_4$K is synthesized by high pressure method. Or be peeled off from bulk T-graphite C$_4$, where C$_4$ can be obtained from C$_4$K by evaporating the potassium atoms. Interestingly, it is found that the calculated $T_{\rm c}$ of $P4/mmm$ C$_4$K is about 30.4 K at 0 GPa, which sets a new record for the layered carbon-based superconductors. The coupling strength between the interlayer electronic states and carbon out-of-plane vibrations has the decisive effect on the superconducting properties of layered carbon intercalation compounds. From the strong electron-phonon coupling, it follows that both T-graphene and C$_4$K exhibit conventional phonon-mediated superconductivity. Comparing the electronic structures and superconducting properties between T-graphene and C$_4$K, we can see the importance of doping effect. Therefore, a further enhancement on the $T_{\rm c}$ for the single layer T-graphene may be possible by charge doping, for instance, by doping metallic atoms or tuning gate voltage. Defects may affect the superconducting properties of T-graphene and C$_4$K, which is interestingly investigated in future studies. As one of the very few examples of intrinsic elemental single-layer superconductors, the T-graphene is an ideal 2D material that can be used to fabricate superconductor/semiconductor heterojunctions with other 2D materials using the so-called "vertical" techniques. This will greatly promote the development of the field. Method We use the machine learning accelerated crystal structure search method[42] to investigate the stable structures of C–K system at 20 GPa, with system sizes up to 18 atoms per simulation cell. Structural optimizations, electronic band structure calculations and strength calculations are performed by the projector augmented wave (PAW) method implemented in the Vienna ab initio simulation package (vasp).[57] In the structure searching, the generalized gradient approximation (GGA), and the Perdew-Burke-Ernzerhof (PBE) functional[58] are employed. In the formation enthalpy calculations, a hard version pseudopotentials with very small core radium for carbon and a 9-valence-electron (including all the $3s$, $3p$ and $4s$ electrons) pseudopotential for potassium are used. Together with an extremely high cutoff energy (1050 eV) and very dense Monkhorst-Pack[59] $k$-sampling using a small $k$-spacing of $0.02\times{2\pi}$Å$^{-1}$. Due to the 2D features of graphite, C$_4$K and other related intercalated compounds, it is important to include the van der Waals (vdW) interaction to reproduce the correct lattice constants and phase order. We compare the results from different vdW corrections, including the Grimme's DFT-D2,[60] DFT-D3,[61] optB88-vdW,[40] optPBE-vdW,[41] vdW-DF[62,63] and vdW-DF2,[64] with the experimental measured lattice constants of graphite and bcc potassium. As listed in Table S1 in the SM, optB88-vdW and optPBE-vdW give relatively better agreement and also give the correct ground state of potassium at ambient pressure, which should be bcc but not fcc. To check the stability of C$_4$K at finite temperature, we perform ab initio molecular dynamics (AIMD) simulations at ambient pressure and temperatures of 300 and 1000 K for 15 picoseconds with a time step of 1 fs using the $NVT$ ensemble with Nose-Hoover thermostat[65] within a supercell containing 135 atoms. Electron-phonon coupling (EPC) calculations are performed in the framework of Density functional perturbation theory (DFPT), as implemented in the quantum-espresso code.[66] For C$_4$K, we adopt a $16\times16\times16$ $k$-point mesh for the charge self-consistent calculation, and a $32\times32\times32$ $k$-point mesh for EPC linewidth integration and a $8\times8\times8$ $q$-point mesh for dynamical matrix. For T-graphene, we adopt a $16\times16\times1$ $k$-point mesh for the charge self-consistent calculation, and a $32\times32\times2$ $k$-point mesh for EPC linewidth integration and a $8\times8\times1$ $q$-point mesh for Dynamical matrix. Norm-conserving pseudopotentials are used with the energy cutoffs of 160 Ry for the wave functions and 640 Ry for the charge density to ensure that the converge criteria of total energy is less than 10$^{-6} $ Ry. We thank Tong Chen, Pengchao Lu, Kang Xia, Cong Liu, Yong Wang, Dexi Shao for the fruitful discussions. Calculations were performed on the computing facilities in the High Performance Computing Center of Collaborative Innovation Center of Advanced Microstructures, the High Performance Computing Center of Nanjing University and "Tianhe-2" at NSCC-Guangzhou. References Highly crystalline 2D superconductorsInterface-Induced High-Temperature Superconductivity in Single Unit-Cell FeSe Films on SrTiO 3Evidence for two-dimensional Ising superconductivity in gated MoS2Ising pairing in superconducting NbSe2 atomic layersFirst-Principles Calculations on the Effect of Doping and Biaxial Tensile Strain on Electron-Phonon Coupling in GrapheneSuperconducting graphene sheets in CaC6 enabled by phonon-mediated interband interactionsUnconventional superconductivity in magic-angle graphene superlatticesCan Two-Dimensional Boron Superconduct?Superconductivity in two-dimensional boron allotropesC60: BuckminsterfullereneHelical microtubules of graphitic carbonTwo-dimensional gas of massless Dirac fermions in grapheneSuperhard Monoclinic Polymorph of CarbonAb initio study of the formation of transparent carbon under pressureGraphdiyne and graphyne: from theoretical predictions to practical constructionNanotwinned diamond with unprecedented hardness and stabilityPentaheptite modifications of the graphite sheetNovel Superhard s p 3 Carbon Allotrope from Cold-Compressed C 70 PeapodsNew Metallic Allotropes of Planar and Tubular CarbonStructural and Electronic Properties of T Graphene: A Two-Dimensional Carbon Allotrope with TetraringsDensity functional theory study on structural and mechanical properties of graphene, T-graphene, and R-graphyneOctagraphene as a versatile carbon atomic sheet for novel nanotubes, unconventional fullerenes, and hydrogen storageSuperconductivity in Graphitic CompoundsPolymorphism control of superconductivity and magnetism in Cs3C60 close to the Mott transitionSuperconductivity in 4 Angstrom Single-Walled Carbon NanotubesSuperconductivity in diamondSuperhard and superconducting B6CSuperconductivity at 39 K in magnesium diborideElectronic structure and electron-phonon coupling of doped graphene layers in KC 8 Electronic Structure of Superconducting KC 8 and Nonsuperconducting LiC 6 Graphite Intercalation Compounds: Evidence for a Graphene-Sheet-Driven Superconducting StatePressure dependence of the superconducting transition temperature in C 6 Yb and C 6 Ca Superconductivity in the intercalated graphite compounds C6Yb and C6CaSuperconductivity of Bulk CaC 6 Enhancement of Superconductivity and Evidence of Structural Instability in Intercalated Graphite CaC 6 under High PressurePhonon-mediated superconductivity in graphene by lithium depositionSuperconductivity in Potassium-Doped Few-Layer GrapheneSuperconductivity in Ca-doped graphene laminatesUnexpected Stable Stoichiometries of Sodium ChloridesMaterials discovery at high pressuresChemical accuracy for the van der Waals density functionalVan der Waals density functionals applied to solidsA novel superhard tungsten nitride predicted by machine-learning accelerated crystal structure searchIn situ synchrotron X-ray diffraction in the laser-heated diamond anvil cell: Melting phenomena and synthesis of new materialsElectric Field Effect in Atomically Thin Carbon FilmsPhosphorene: An Unexplored 2D Semiconductor with a High Hole MobilitySynthesis of an open-framework allotrope of siliconInterlayer cohesive energy of graphite from thermal desorption of polyaromatic hydrocarbonsPotassium intercalation in graphite: A van der Waals density-functional studyTwo-dimensional materials from high-throughput computational exfoliation of experimentally known compoundsFabrication of high quality graphene nanosheets via a spontaneous electrochemical reaction processDevelopment of adsorption and electrosorption techniques for removal of organic and inorganic pollutants from wastewater using novel magnetite/porous graphene-based nanocompositesGraphene as a flexible electrode: review of fabrication approachesAngle-resolved photoemission studies of the cuprate superconductorsUnconventional Pairing Originating from the Disconnected Fermi Surfaces of Superconducting LaFeAsO 1 − x F x Transition temperature of strong-coupled superconductors reanalyzedPhase diagram and electronic indication of high-temperature superconductivity at 65 K in single-layer FeSe filmsEfficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis setGeneralized Gradient Approximation Made SimpleSpecial points for Brillouin-zone integrationsSemiempirical GGA-type density functional constructed with a long-range dispersion correctionA consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-PuVan der Waals Density Functional for General GeometriesEfficient Implementation of a van der Waals Density Functional: Application to Double-Wall Carbon NanotubesHigher-accuracy van der Waals density functionalCanonical dynamics: Equilibrium phase-space distributionsQUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials
[1] Saito Y, Nojima T and Iwasa Y 2017 Nat. Rev. Mater. 2 16094
[2] Wang Q Y, Li Z, Zhang W H, Zhang Z C, Zhang J S, Li W, Ding H, Ou Y B, Deng P, Chang K, Wen J, Song C L, He K, Jia J F, Ji S H, Wang Y Y, Wang L L, Chen X, Ma X C and Xue Q K 2012 Chin. Phys. Lett. 29 037402
[3] Lu J M, Zheliuk O, Leermakers I, Yuan N F Q, Zeitler U, Law K T and Ye J T 2015 Science 350 1353
[4] Xi X, Wang Z, Zhao W, Park J H, Law K T, Berger H, Forró L, Shan J and Mak K F 2016 Nat. Phys. 12 139
[5] Si C, Liu Z, Duan W and Liu F 2013 Phys. Rev. Lett. 111 196802
[6] Ludbrook B M, Levy G, Nigge P, Zonno M, Schneider M, Dvorak D J, Veenstra C N, Zhdanovich S, Wong D, Dosanjh P 2015 Proc. Natl. Acad. Sci. USA 112 11795
[7] Cao Y, Fatemi V, Fang S, Watanabe K, Taniguchi T, Kaxiras E and Jarillo-Herrero P 2018 Nature 556 43
[8] Penev E S, Kutana A and Yakobson B I 2016 Nano Lett. 16 2522
[9] Zhao Y, Zeng S and Ni J 2016 Phys. Rev. B 93 014502
[10] Kroto H W, Heath J R, O'Brien S C, Curl R F and Smalley R E 1985 Nature 318 162
[11] Iijima S 1991 Nature 354 56
[12] Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V and Firsov A A 2005 Nature 438 197
[13] Li Q, Ma Y, Oganov A R, Wang H, Wang H, Xu Y, Cui T, Mao H K and Zou G 2009 Phys. Rev. Lett. 102 175506
[14] Zhou X F, Qian G R, Dong X, Zhang L, Tian Y and Wang H T 2010 Phys. Rev. B 82 134126
[15] Li Y, Xu L, Liu H and Li Y 2014 Chem. Soc. Rev. 43 2572
[16] Huang Q, Yu D, Xu B, Hu W, Ma Y, Wang Y, Zhao Z, Wen B, He J, Liu Z 2014 Nature 510 250
[17] Zhang S, Zhou J, Wang Q, Chen X, Kawazoe Y and Jena P 2015 Proc. Natl. Acad. Sci. USA 112 2372
[18] Yang X, Yao M, Wu X, Liu S, Chen S, Yang K, Liu R, Cui T, Sundqvist B and Liu B 2017 Phys. Rev. Lett. 118 245701
[19] Terrones H, Terrones M, Hernández E, Grobert N, Charlier J C and Ajayan P M 2000 Phys. Rev. Lett. 84 1716
[20] Liu Y, Wang G, Huang Q, Guo L and Chen X 2012 Phys. Rev. Lett. 108 225505
[21] Majidi R 2017 Theor. Chem. Acc. 136 109
[22] Sheng X L, Cui H J, Ye F, Yan Q B, Zheng Q R and Su G 2012 J. Appl. Phys. 112 074315
[23] Hannay N B, Geballe T H, Matthias B T, Andres K, Schmidt P and MacNair D 1965 Phys. Rev. Lett. 14 225
[24] Ganin A Y, Takabayashi Y, Jeglic P, Arcon D, Potocnik A, Baker P J, Ohishi Y, McDonald M T, Tzirakis M D, McLennan A 2010 Nature 466 221
[25] Tang Z K, Zhang L, Wang N, Zhang X X, Wen G H, Li G D, Wang J N, Chan C T and Sheng P 2001 Science 292 2462
[26] Ekimov E A, Sidorov V A, Bauer E D, Mel'nik N N, Curro N J, Thompson J D and Stishov S M 2004 Nature 428 542
[27] Xia K, Ma M, Liu C, Gao H, Chen Q, He J, Sun J, Wang H T, Tian Y and Xing D 2017 Mater. Today Phys. 3 76
[28] Nagamatsu J, Nakagawa N, Muranaka T, Zenitani Y and Akimitsu J 2001 Nature 410 63
[29] Grüneis A, Attaccalite C, Rubio A, Vyalikh D V, Molodtsov S L, Fink J, Follath R, Eberhardt W, Büchner B and Pichler T 2009 Phys. Rev. B 79 205106
[30] Pan Z H, Camacho J, Upton M H, Fedorov A V, Howard C A, Ellerby M and Valla T 2011 Phys. Rev. Lett. 106 187002
[31] Smith R P, Kusmartseva A, Ko Y T C, Saxena S S, Akrap A, Forró L, Laad M, Weller T E, Ellerby M and Skipper N T 2006 Phys. Rev. B 74 024505
[32] Weller T E, Ellerby M, Saxena S S, Smith R P and Skipper N T 2005 Nat. Phys. 1 39
[33] Emery N, Hérold C, d'Astuto M, Garcia V, Bellin C, Marêché J F, Lagrange P and Loupias G 2005 Phys. Rev. Lett. 95 087003
[34] Gauzzi A, Takashima S, Takeshita N, Terakura C, Takagi H, Emery N, Hérold C, Lagrange P and Loupias G 2007 Phys. Rev. Lett. 98 067002
[35] Profeta G, Calandra M and Mauri F 2012 Nat. Phys. 8 131
[36] Xue M, Chen G, Yang H, Zhu Y, Wang D, He J and Cao T 2012 J. Am. Chem. Soc. 134 6536
[37] Chapman J, Su Y, Howard C A, Kundys D, Grigorenko A N, Guinea F, Geim A K, Grigorieva I V and Nair R R 2016 Sci. Rep. 6 23254
[38] Zhang W, Oganov A R, Goncharov A F, Zhu Q, Boulfelfel S E, Lyakhov A O, Stavrou E, Somayazulu M, Prakapenka V B and Konôpková Z 2013 Science 342 1502
[39] Zhang L, Wang Y, Lv J and Ma Y 2017 Nat. Rev. Mater. 2 17005
[40] Klimeš J, Bowler D R and Michaelides A 2010 J. Phys.: Condens. Matter 22 022201
[41] Klimeš J, Bowler D R and Michaelides A 2011 Phys. Rev. B 83 195131
[42] Xia K, Gao H, Liu C, Yuan J, Sun J, Wang H T and Xing D 2018 Sci. Bull. 63 817
[43] Salamat A, Fischer R A, Briggs R, McMahon M I and Petitgirard S 2014 Coord. Chem. Rev. 277 15
[44] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V and Firsov A A 2004 Science 306 666
[45] Liu H, Neal A T, Zhu Z, Luo Z, Xu X, Tománek D and Ye P D 2014 ACS Nano 8 4033
[46] Kim D Y, Stefanoski S, Kurakevych O O and Strobel T A 2015 Nat. Mater. 14 169
[47] Zacharia R, Ulbricht H and Hertel T 2004 Phys. Rev. B 69 155406
[48] Ziambaras E, Kleis J, Schröder E and Hyldgaard P 2007 Phys. Rev. B 76 155425
[49] Mounet N, Gibertini M, Schwaller P, Campi D, Merkys A, Marrazzo A, Sohier T, Castelli I E, Cepellotti A, Pizzi G 2018 Nat. Nanotechnol. 13 246
[50] Lee J W, Kim M, Na W, Hong S M and Koo C M 2015 Carbon 91 527
[51] Bharath G, Alhseinat E, Ponpandian N, Khan M A, Siddiqui M R, Ahmed F and Alsharaeh E H 2017 Sep. Purif. Technol. 188 206
[52] Tan R K L, Reeves S P, Hashemi N, Thomas D G, Kavak E, Montazami R and Hashemi N N 2017 J. Mater. Chem. A 5 17777
[53] Damascelli A, Hussain Z and Shen Z X 2003 Rev. Mod. Phys. 75 473
[54] Kuroki K, Onari S, Arita R, Usui H, Tanaka Y, Kontani H and Aoki H 2008 Phys. Rev. Lett. 101 087004
[55] Allen P B and Dynes R C 1975 Phys. Rev. B 12 905
[56] He S, He J, Zhang W, Zhao L, Liu D, Liu X, Mou D, Ou Y B, Wang Q Y, Li Z 2013 Nat. Mater. 12 605
[57] Kresse G and Furthmüller J 1996 Comput. Mater. Sci. 6 15
[58] Perdew J P, Burke K and Ernzerhof M 1996 Phys. Rev. Lett. 77 3865
[59] Monkhorst H J and Pack J D 1976 Phys. Rev. B 13 5188
[60] Grimme S 2006 J. Comput. Chem. 27 1787
[61] Grimme S, Antony J, Ehrlich S and Krieg H 2010 J. Chem. Phys. 132 154104
[62] Dion M, Rydberg H, Schröder E, Langreth D C and Lundqvist B I 2004 Phys. Rev. Lett. 92 246401
[63] Román-Pérez G and Soler J M 2009 Phys. Rev. Lett. 103 096102
[64] Lee K, Murray E D, Kong L, Lundqvist B I and Langreth D C 2010 Phys. Rev. B 82 081101
[65] Hoover W G 1985 Phys. Rev. A 31 1695
[66] Giannozzi P, Baroni S, Bonini N, Calandra M, Car R, Cavazzoni C, Ceresoli D, Chiarotti G L, Cococcioni M, Dabo I 2009 J. Phys.: Condens. Matter 21 395502
⇦Chinese Physics Letters, 2019, Vol. 36, No. 9, Article code 097301 The 2D InSe/WS$_2$ Heterostructure with Enhanced Optoelectronic Performance in the Visible Region * Lu-Lu Yang (杨露露)1, Jun-Jie Shi (史俊杰)2, Min Zhang (张敏)3, Zhong-Ming Wei (魏钟鸣)4, Yi-Min Ding (丁一民)2, Meng Wu (吴蒙)2, Yong He (贺勇)3, Yu-Lang Cen (岑育朗)2, Wen-Hui Guo (郭文惠)2, Shu-Hang Pan (潘书航)2, Yao-Hui Zhu (朱耀辉)1** Affiliations 1Physics Department, Beijing Technology and Business University, Beijing 100048 2State Key Laboratory for Artificial Microstructures and Mesoscopic Physics, School of Physics, Peking University, Beijing 100871 3College of Physics and Electronic Information, Inner Mongolia Normal University, Hohhot 010022 4State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences & College of Materials Science and Opto-Electronic Technology, University of Chinese Academy of Sciences, Beijing 100083 Received 12 January 2019, online 23 August 2019 *Supported by the National Natural Science Foundation of China under Grant Nos 11404013, 11474012, 11364030, 61622406, 61571415, 51502283 and 11605003, the National Key Research and Development Program of China under Grant No 2017YFA0206303, the MOST of China, and the 2018 Graduate Research Program of Beijing Technology and Business University.
**Corresponding author. Email: yaohuizhu@gmail.com Citation Text: Yang L L, Shi J J, Zhang M, Wei Z M and Ding Y M et al 2019 Chin. Phys. Lett. 36 097301    Abstract Two-dimensional (2D) InSe and WS$_2$ exhibit promising characteristics for optoelectronic applications. However, they both have poor absorption of visible light due to wide bandgaps: 2D InSe has high electron mobility but low hole mobility, while 2D WS$_2$ is on the contrary. We propose a 2D heterostructure composed of their monolayers as a solution to both problems. Our first-principles calculations show that the heterostructure has a type-II band alignment as expected. Consequently, the bandgap of the heterostructure is reduced to 2.19 eV, which is much smaller than those of the monolayers. The reduction in bandgap leads to a considerable enhancement of the visible-light absorption, such as about fivefold (threefold) increase in comparison to monolayer InSe (WS$_2$) at the wavelength of 490 nm. Meanwhile, the type-II band alignment also facilitates the spatial separation of photogenerated electron-hole pairs; i.e., electrons (holes) reside preferably in the InSe (WS$_2$) layer. As a result, the two layers complement each other in carrier mobilities of the heterostructure: the photogenerated electrons and holes inherit the large mobilities from the InSe and WS$_2$ monolayers, respectively. DOI:10.1088/0256-307X/36/9/097301 PACS:73.22.-f, 73.63.-b, 78.67.-n © 2019 Chinese Physics Society Article Text In the past decade, many researchers have turned their attention to 2D semiconductors, such as transition-metal dichalcogenides ($MX_2$; $M$=Mo, W; $X$=S, Se, Te)[1,2] and group-III monochalcogenides ($MX$; $M$=Ga, In; $X$=S, Se, Te).[3,4] Studies on 2D semiconductors have demonstrated their advantages over the corresponding bulk materials in various optoelectronic applications.[5] Recently, 2D InSe has exhibited great potential for optoelectronic applications due to its high electron mobility, good metal contacts, and wide bandgap range.[6,7] The bandgap increases from 1.26 eV to 2.6 eV as bulk InSe is thinned to a monolayer.[7–9] The electron mobility of few-layer InSe exceeds $10^3$ cm$^2$V$^{-1}$s$^{-1}$ and $10^4$ cm$^2$V$^{-1}$s$^{-1}$ at room temperature and at liquid-helium temperature, respectively.[7] However, the optoelectronic properties of monolayer InSe need to be improved before it can be exploited widely in practice. First, its monolayer has a poor absorption of visible light due to the wide bandgap.[10] As a result, the performance of optoelectronic devices based on it would be reduced significantly in the visible region. Second, its hole mobility is rather low, although it has a large electron mobility. This severely impedes its applications in some types of optoelectronic devices; e.g., photoconductors and photodiodes. Fortunately, it has been shown that there is a unifying strategy to overcome both the difficulties; i.e., the construction of van der Waals (vdW) heterostructures with type-II band alignment.[11–14] The present work aims to enhance the optoelectronic performance of 2D InSe by combining it with another suitable 2D material to build a heterostructure. One of its possible counterparts is 2D WS$_2$, which also has a hexagonal structure.[15–18] Researchers have investigated its various applications, such as photocatalysis and optoelectronics.[19–24] Several heterostructures of WS$_2$ and other 2D materials have been fabricated and characterized experimentally.[15,24] In this Letter, we study the optoelectronic properties of the 2D InSe/WS$_2$ heterostructure by first-principles calculations before it is fabricated in experiments. The computational methods are outlined as follows: our first-principles calculations were performed with the Vienna ab initio simulation package.[25] Within the framework of the density functional theory, we use the generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) functional to describe the exchange and correlation interactions between the valence electrons.[26] The electron-ion interactions are described by the projector augmented-wave method.[27] Then, the GGA-1/2 scheme is used to make corrections to the band structures and optical properties, since the usual GGA underestimates bandgaps.[28] The GGA-1/2 scheme can yield the results in good agreement with experiments and with other theoretical methods such as the GW and HSE06 schemes. Meanwhile, this scheme is much less expensive than other computational methods. Within the GGA-1/2 scheme, the atomic self-energy potential is expressed as the difference between the all-electron potentials of the atom and those of the half-ion, $$ V_{\rm s}\approx{V}(0,r)-V(-1/2,r),~~ \tag {1} $$ where $r$ is the radial coordinate. The potential $V_{\rm s}$ has a long-range Coulomb tail that must be trimmed by $$ {\it \Theta}(r)=\begin{cases}\!\! \left[1-\left(\dfrac{r}{r_{\rm CUT}}\right)^m\right]^3,&~~r\leq{r_{\rm CUT}}; \\\!\! 0,&~~r>r_{\rm CUT}. \end{cases}~~ \tag {2} $$ The values of the dimensionless parameter $m$ and the trimming radius $r_{\rm CUT}$ are chosen to ensure that the result of the bandgap reaches its extreme. In optimizing geometric structures, the vdW interaction is accounted for by the optB88 vdW exchange functional.[29] We determine the positions of the conduction band minimum (CBM) and the valence band maximum (VBM) using the equation proposed by Toroker et al., $E_{\rm CBM/VBM}=E_{\rm BGC}\pm{E}_{\rm g}^{\rm QP}/2$, where $E_{\rm BGC}$ denotes the bandgap center calculated with the PBE functional and it is insensitive to different exchange functionals.[30] The quasiparticle bandgap $E_{\rm g}^{\rm QP}$ can be calculated by the GGA-1/2 method. We calculate the optical absorption spectrum of the 2D InSe/WS$_2$ heterostructure, as well as those of monolayer InSe and WS$_2$ by writing the absorption coefficient $I(\omega)$ in terms of the frequency-dependent dielectric function.[31] Furthermore, the electron and hole mobilities at temperature $T$ are computed by the widely used formula[11,32] $$ \mu_{\rm 2D}=\frac{e\hbar^3C_{\rm 2D}}{k_{\rm B}Tm^\ast{m}_{\rm d}E_1^2},~~ \tag {3} $$ where $C_{\rm 2D}$ denotes the in-plane stiffness and $E_1$ is the deformation potential constant. Various effective masses are used in Eq. (3): $m^\ast$ is the effective mass in the transport direction; i.e., $m_x^\ast$ or $m_y^\ast$, and $m_{\rm d}=\sqrt{m_x^\ast{m}_y^\ast}$ gives the average one over the $x$ and $y$ directions.
cpl-36-9-097301-fig1.png
Fig. 1. The top and side views of the crystal structures.[33] (a) Monolayer InSe has a honeycomb lattice connected by Se–In–In–Se sequence. (b) Monolayer WS$_2$ also has a hexagonal configuration, with each W atom anchored by three pairs of S atoms by S–W–S sequence. One of the lattice vectors (thick solid-black arrows) of the InSe (WS$_2$) supercell makes an angle of 30$^{\circ}$ (23.4$^{\circ}$) to the horizontal lattice vector (parallel to the thin dashed-black line) of its primitive cell. (c) One (optimized) primitive cell of the 2D InSe/WS$_2$ heterostructure is composed of the supercells of the InSe and WS$_2$ monolayers shown in (a) and (b). One monolayer is stacked on the other by matching the rhombuses of their supercells until they coincide with each other. The optimized interlayer distance is $d=3.40$ Å in the heterostructure.
Table 1. The equilibrium lattice constants $a$ (in units of Å) and bandgaps $E_{\rm g}$ (in units of eV) of monolayer InSe and WS$_2$.
MLs $a$ $E_{\rm g}$ (PBE) $E_{\rm g}$ (GGA-1/2) $E_{\rm g}$ (GW) $E_{\rm g}$ (Exp.)
InSe 4.09 1.47 2.75 2.83$^{\rm a}$ 2.6$^{\rm c}$
WS$_2$ 3.19 1.77 2.59 2.64$^{\rm b}$ 2.38$^{\rm d}$
$^{\rm a}$From Ref.  [9], $^{\rm b}$From Ref.  [21], $^{\rm c}$From Ref.  [7], $^{\rm d}$From Ref.  [34].
The crystal structures of 2D InSe, WS$_2$, and InSe/WS$_2$ heterostructure are shown in Fig. 1. Our structural optimizations yield the lattice constants for InSe and WS$_2$ monolayers (listed in Table 1) and are in good agreement with the values in previous reports.[4,9,20,21,35–40] Because of the huge lattice mismatch (over $20$%) between the two pristine monolayers, we build a large cell for the 2D InSe/WS$_2$ heterostructure in such a way that the lattice mismatch is made as small as possible and the computational cost is still acceptable at the same time. As a reasonable compromise, we choose a primitive cell of the heterostructure consisting of two hexagonal supercells: a $\sqrt{12}\times \sqrt{12}$ supercell of InSe (24 In and 24 Se atoms) and a $\sqrt{19}\times \sqrt{19}$ supercell of WS$_2$ (19 W and 38 S atoms) as shown in Fig. 1. Then, one can find a lattice mismatch smaller than 1.7% by comparing $\sqrt{12}a_{\rm InSe}$ and $\sqrt{19}a_{\rm WS_2}$ in Fig. 1 and using the lattice constants listed in Table 1. Our structural optimization yields a lattice constant of 14.09 Å (the side length of the rhombus in Fig. 1(c)) for the heterostructure, which induces a compressive strain of 0.6% in the InSe monolayer and a tensile strain of 1.3% in the WS$_2$ monolayer. During the geometry optimization, the cell shape, cell volume, and ion positions are allowed to change automatically (ISIF=3) with the lattice vector perpendicular to the layer plane fixed (negligible stress due to the thick vacuum layer). We begin the optimizations with a series of initial interlayer distances (from 3.20 Å to 4.00 Å in increments of 0.10 Å) for the heterostructure, and all of them end up with the same equilibrium interlayer distance $d=3.40$ Å. The key computational parameters are outlined as follows: two $k$-point meshes, $5\times5\times1$ and $7\times7\times1$, are employed for the geometry optimizations and self-consistent calculations of the supercell, respectively. The thickness of the vacuum layer is over 20 Å so as to ensure decoupling between periodically repeated layers, and the energy cutoff is set to 450 eV. The structures are fully optimized until the residual atomic forces are smaller than 0.01 eV/Å. Moreover, we use the Gaussian smearing in combination with a reasonable smearing parameter, i.e., 0.05 and 0.1 for the monolayers and the heterostructure, respectively.
cpl-36-9-097301-fig2.png
Fig. 2. Band structures and projected DOS. (a) Monolayer InSe has an indirect bandgap (2.75 eV) with the CBM located at ${\it \Gamma}$ point and the VBM located between ${\it \Gamma}$ and $K$ points. Its CBM (VBM) is mainly composed of Se 4$p$ and In 5$s$ (5$p$) orbitals. (b) Monolayer WS$_2$ has a direct bandgap of 2.59 eV with both its CBM and VBM located at $K$ points.[35] Its W 5$d$ orbitals play the dominant role in both the CBM and VBM, while its S 3$p$ orbitals make a minor contribution. (c) The weighted band structure and DOS of the InSe/WS$_2$ heterostructure. Its CBM is dominated by the InSe layer as shown by the inset.
The GGA-1/2 scheme is applied firstly to monolayer InSe and WS$_2$ to calibrate computational parameters. The bandgaps calculated with these parameters are quite reasonable as each of them falls between the corresponding GW result and that reported by experimental groups as listed in Table 1. The band structures and the corresponding density of states (DOS) of the two monolayers are shown in Figs. 2(a) and 2(b). The electronic properties of monolayer InSe and WS$_2$ are also consistent with the results of previous calculations.[9,21] The stability of the heterostructure is described by its binding energy $E_{\rm b}$. We write $E_{\rm b}$ as $E_{\rm b}=(E_{\rm InSe/WS_2}-E_{\rm InSe}-E_{\rm WS_2})/n$, where $n$ is the number of all types of atoms. Moreover, $E_{\rm InSe/WS_2}$, $E_{\rm InSe}$, and $E_{\rm WS_2}$ are the total energies of the relaxed heterostructure, monolayer InSe, and monolayer WS$_2$, respectively.[17] The calculated binding energy is $-30$ meV/atom for the heterostructure. This indicates a moderate interaction between the two monolayers and the heterostructure satisfies one of the requirements on a stable structure.
cpl-36-9-097301-fig3.png
Fig. 3. The charge density at the VBM (CBM) depicted by the isosurface ($\rho=1.36\times10^{-4}\,e$Å$^{-3}$) in the heterostructure.
Figure 2(c) shows the weighted band structure and the corresponding DOS of the heterostructure. The bandgap of the InSe/WS$_2$ heterostructure is reduced to 2.19 eV, which is much smaller than those of monolayer InSe and WS$_2$ (see Table 1). This reduction will be interpreted qualitatively by the band edge alignment below. Figure 3 depicts the isosurface charge density at the VBM and the CBM of the heterostructure. The size of the isosurface indicates that an electron at the CBM resides almost entirely in the InSe layer. This is consistent with the band structure of the CBM and its DOS shown in Fig. 2(c), where the InSe layer is definitely dominant over the WS$_2$ layer. On the other hand, a hole at the VBM exhibits more complicated behavior. It has noticeable probability to appear in the InSe layer, although it resides mainly in the WS$_2$ layer. This is also in agreement with the band structure of the VBM and its DOS in Fig. 2(c), where the WS$_2$ layer plays a major role. Therefore, photogenerated electron-hole pairs can be effectively separated: electrons and holes reside preferably in the InSe and WS$_2$ layers, respectively. The InSe/WS$_2$ heterostructure has a type-II band alignment. The CBM and VBM of monolayer InSe are $-3.86$ eV and $-6.61$ eV, respectively, with respect to the vacuum level, which are in agreement with the previous results.[9] As for monolayer WS$_2$, the CBM and VBM are $-3.41$ eV and $-6.00$ eV, respectively, which are roughly consistent with the previous reports.[21] Both the CBM and the VBM of InSe are lower than those of WS$_2$. The conduction band offset (CBO) and the valence band offset (VBO) are 0.45 eV and 0.61 eV, respectively. Thus the heterostructure has a type-II band alignment, which is consistent with the electronic structures in Fig. 2(c). Straightforward calculation shows that the bandgap of the heterostructure is reduced to 2.14 eV, which is close to the more rigorous result (2.19 eV) given by the band structure in Fig. 2(c). The difference is attributed to the strain and interlayer interaction in the heterostructure. Figure 4 presents the optical absorption spectrum of the InSe/WS$_2$ heterostructure together with those of monolayer InSe and WS$_2$. In comparison to monolayer InSe and WS$_2$, the spectrum of the heterostructure has a much wider energy range, in which the absorption coefficient for visible light is on the order of 10$^5$ cm$^{-1}$. Specifically, its absorption coefficient has a nearly fivefold (threefold) enhancement for the photon energy 2.53 eV (the light wavelength of 490 nm) in comparison to monolayer InSe (WS$_2$). The enhanced light absorption is largely owing to the smaller bandgap of the heterostructure as shown in Fig. 2(c). It is worthwhile to point out that the bandgap will be reduced further if the exciton effects are taken into account. The first-principles calculation of the exciton binding energy ($E_{\rm eb}$) is usually very time-consuming and beyond the scope of the current work. Fortunately, it can be estimated using a robust linear scaling law, i.e., $E_{\rm eb}\approx{E}_{\rm g}/4$, for 2D semiconductors.[41] We obtain $E_{\rm eb}\approx0.55$ eV using this method, and then the optical bandgap is roughly 1.64 eV when $E_{\rm eb}$ is subtracted from the fundamental bandgap (2.19 eV). This will give rise to a remarkable exciton absorption peak far below the absorption edge in Fig. 4.
cpl-36-9-097301-fig4.png
Fig. 4. Absorption spectra of visible light. The solid-black, dash-dotted-blue, and dashed-black curves depict the variation of the absorption coefficients with photon energy for the InSe/WS$_2$ heterostructure, monolayer WS$_2$, and monolayer InSe, respectively.
Table 2. The electron and hole mobilities listed together with the associated coefficients for monolayer InSe, monolayer WS$_2$, and the 2D InSe/WS$_2$ heterostructure. Both $x$ and $y$ components are given for the effective masses $m^\ast$, deformation potential constants $E_1$, in-plane stiffness $C_{\rm 2D}$, and mobilities $\mu_{\rm 2D}$. Refer to Eq. (3) for details.
Carriers Materials $m_x^{\ast}$ $m_y^{\ast}$ $E_{1x}$ $E_{1y}$ $C_{{\rm 2D}-x}$ $C_{{\rm 2D}-y}$ $\mu_{{\rm 2D}-x}$ $\mu_{{\rm 2D}-y}$
InSe 0.18 0.19 $-$4.42 $-$4.32 49.47 48.70 1667 1584
Electrons WS$_{2}$ 0.45 0.33 $-$11.20 $-$11.80 145.79 148.51 122 179
InSe/WS$_{2}$ 0.23 0.24 $-$4.00 $-$4.10 182.53 185.21 4503 4168
InSe 1.84 2.01 $-$1.40 $-$1.37 49.47 48.70 152 143
Holes WS$_{2}$ 0.53 0.43 $-$5.19 $-$5.40 145.79 148.51 456 529
InSe/WS$_{2}$ 1.22 1.15 $-$3.48 $-$4.23 182.53 185.21 223 162
Table 2 lists the electron and hole mobilities of the heterostructure together with those of monolayer InSe and WS$_2$. There is a pronounced increase in the electron mobility of the heterostructure; i.e., nearly threefold increase in comparison to that of monolayer InSe. At the same time, the electron mobility of the heterostructure is much larger than that of the monolayer WS$_2$, which indicates that there is no close relationship between the two mobilities. The electron mobility is dominated by the contribution from the InSe layer since the photogenerated electrons almost reside in this layer as shown by the band structure in Fig. 2(c) and the CBM electron density in Fig. 3. This explains why the electron mobility of the WS$_2$ is not related to that of the heterostructure. The pronounced increase in the electron mobility of the heterostructure results from the large increase of the stiffness ($C_{{\rm 2D}-x(y)}$) relative to that of monolayer InSe.[42] The stiffness of the heterostructure is roughly equal to the sum of those of InSe and WS$_2$ monolayers. Moreover, the heterostructure effects are dominant over the correction due to the finite thickness, since the structure studied here is rather thin.[43] The mobility of the holes exhibits more complicated characteristics than that of the electrons. The hole mobility of the heterostructure increases remarkably in comparison to that of monolayer InSe while it decreases significantly as compared with that of monolayer WS$_2$. The photogenerated holes reside mostly in the WS$_2$ layer and partly in the InSe layer as shown by the band structure in Fig. 2(c) and the VBM electron density in Fig. 3. This is consistent with the change in the hole effective mass of the heterostructure listed in Table 2; i.e., the hole effective mass falls between the values of the InSe and the WS$_2$ layers. Meanwhile, $E_{\rm 1x(y)}$ also exhibits the similar variation. As compared with monolayer InSe, $m_{x(y)}^\ast$ of the heterostructure decreases and tends to increase the hole mobility, while $E_{1x(y)}$ increases in magnitude and makes the hole mobility smaller instead. The two variations cancel each other partly in the contribution to the hole mobility. However, the hole mobility has an overall increase in comparison to that of the InSe layer when the large increase in the stiffness is taken into account. In summary, our first-principles calculations show that the 2D InSe/WS$_2$ heterostructure exhibits enhanced optoelectronic performance in the visible region. It has a type-II band alignment and its bandgap is reduced to 2.19 eV. Its CBM and VBM derive mainly from the InSe and WS$_2$ layers, respectively, which promotes effectively the spatial separation of the photogenerated electron-hole pairs and in turn decreases their recombination rate. The visible-light absorption is increased considerably in comparison to monolayer InSe (WS$_2$). The two monolayers complement each other in carrier mobilities: the photo-generated electrons and holes inherit the large mobilities from InSe and WS$_2$ monolayers, respectively. The results presented here may be useful for experimentalists working on this structure. References Electronics and optoelectronics of two-dimensional transition metal dichalcogenidesThe chemistry of two-dimensional layered transition metal dichalcogenide nanosheetsSynthesis, properties and applications of 2D layered M III X VI (M = Ga, In; X = S, Se, Te) materials2D layered group IIIA metal chalcogenides: synthesis, properties and applications in electronics and optoelectronicsGraphene-Like Two-Dimensional MaterialsBack Gated Multilayer InSe Transistors with Enhanced Carrier Mobilities via the Suppression of Carrier Scattering from a Dielectric InterfaceHigh electron mobility, quantum Hall effect and anomalous optical response in atomically thin InSeTuning the Bandgap of Exfoliated InSe Nanosheets by Quantum ConfinementSingle-Layer Group-III Monochalcogenide Photocatalysts for Water SplittingStrain induced new phase and indirect–direct band gap transition of monolayer InSeEnhancement of hole mobility in InSe monolayer via an InSe and black phosphorus heterostructureFast, multicolor photodetection with graphene-contacted p -GaSe/ n -InSe van der Waals heterostructuresElectronic properties of silicene in BN/silicene van der Waals heterostructuresTwo-dimensional g-C 3 N 4 /InSe heterostructure as a novel visible-light photocatalyst for overall water splitting: a first-principles studyStrain engineering in monolayer WS 2 , MoS 2 , and the WS 2 /MoS 2 heterostructureDFT investigation on two-dimensional GeS/WS2 van der Waals heterostructure for direct Z-scheme photocatalytic overall water splittingC2N/WS2 van der Waals type-II heterostructure as a promising water splitting photocatalystHybrid density functional study on the photocatalytic properties of AlN/MoSe 2 , AlN/WS 2 , and AlN/WSe 2 heterostructuresTwo-Dimensional Nanosheets Produced by Liquid Exfoliation of Layered MaterialsPhotosensor Device Based on Few-Layered WS 2 FilmsComputational Search for Single-Layer Transition-Metal Dichalcogenide PhotocatalystsElectrical Transport Properties of Single-Layer WS 2Vertically Aligned WS 2 Nanosheets for Water SplittingMoS 2 /WS 2 Heterojunction for Photoelectrochemical Water OxidationEfficient iterative schemes for ab initio total-energy calculations using a plane-wave basis setGeneralized Gradient Approximation Made SimpleFrom ultrasoft pseudopotentials to the projector augmented-wave methodApproximation to density functional theory for the calculation of band gaps of semiconductorsVan der Waals density functionals applied to solidsFirst principles scheme to evaluate band edge positions in potential transition metal oxide photocatalysts and photoelectrodesElectronic structure, chemical bonding, and optical properties of paraelectric BaTiO 3 High-mobility transport anisotropy and linear dichroism in few-layer black phosphorusVESTA 3 for three-dimensional visualization of crystal, volumetric and morphology dataElectronic band gaps and exciton binding energies in monolayer M o x W 1 − x S 2 transition metal dichalcogenide alloys probed by scanning tunneling and optical spectroscopyThickness and strain effects on electronic structures of transition metal dichalcogenides: 2H- M X 2 semiconductors ( M = Mo, W; X = S, Se, Te)Band offsets and heterostructures of two-dimensional semiconductorsStrain engineering of WS 2 , WSe 2 , and WTe 2Electronic structures and elastic properties of monolayer and bilayer transition metal dichalcogenides MX 2 ( M = Mo, W; X = O, S, Se, Te): A comparative first-principles studyTwo-Dimensional Indium Selenides Compounds: An Ab Initio StudyComputational mining of photocatalysts for water splitting hydrogen production: two-dimensional InSe-family monolayersScaling Universality between Band Gap and Exciton Binding Energy of Two-Dimensional SemiconductorsFew-layer Tellurium: one-dimensional-like layered elementary semiconductor with striking physical properties
[1] Wang Q H et al 2012 Nat. Nanotechnol. 7 699
[2] Chhowalla M et al 2013 Nat. Chem. 5 263
[3] Xu K et al 2016 Nanoscale 8 16802
[4] Huang W et al 2016 CrystEngComm 18 3968
[5] Xu M, Liang T, Shi M and Chen H 2013 Chem. Rev. 113 3766
[6] Feng W, Zheng W, Cao W and Hu P 2014 Adv. Mater. 26 6587
[7] Bandurin D A et al 2017 Nat. Nanotechnol. 12 223
[8] Mudd G W et al 2013 Adv. Mater. 25 5714
[9] Zhuang H L and Hennig R G 2013 Chem. Mater. 25 3232
[10] Hu T, Zhou J and Dong J 2017 Phys. Chem. Chem. Phys. 19 21722
[11] Ding Y M et al 2017 Nanoscale 9 14682
[12] Yan F et al 2017 Nanotechnology 28 27LT01
[13] Wu Z B et al 2018 Chin. Phys. B 27 077302
[14] He Y et al 2019 J. Phys. D 52 015304
[15] He X et al 2016 Appl. Phys. Lett. 109 173105
[16] Ju L et al 2018 Appl. Surf. Sci. 434 365
[17] Kumar R, Das D and Singh A K 2018 J. Catal. 359 143
[18] Wang G et al 2018 J. Phys. D 51 025109
[19] Coleman J N et al 2011 Science 331 568
[20] PereaL ópez N et al 2013 Adv. Funct. Mater. 23 5511
[21] Zhuang H L and Hennig R G 2013 J. Phys. Chem. C 117 20440
[22] Ovchinnikov D et al 2014 ACS Nano 8 8174
[23] Yang Y et al 2015 Adv. Funct. Mater. 25 6199
[24] Pesci F M et al 2017 ACS Catal. 7 4990
[25] Kresse G and Furthmüller J 1996 Phys. Rev. B 54 11169
[26] Perdew J P, Burke K and Ernzerhof M 1996 Phys. Rev. Lett. 77 3865
[27] Kresse G and Joubert D 1999 Phys. Rev. B 59 1758
[28] Ferreira L G, Marques M and Teles L K 2008 Phys. Rev. B 78 125116
[29] Klimeš J, Bowler D R and Michaelides A 2011 Phys. Rev. B 83 195131
[30] Toroker M C et al 2011 Phys. Chem. Chem. Phys. 13 16644
[31] Saha S, Sinha T P and Mookerjee A 2000 Phys. Rev. B 62 8828
[32] Qiao J et al 2014 Nat. Commun. 5 4475
[33] Momma K and Izumi F 2011 J. Appl. Crystallogr. 44 1272
[34] Rigosi A F et al 2016 Phys. Rev. B 94 075440
[35] Yun W S et al 2012 Phys. Rev. B 85 033305
[36] Kang J et al 2013 Appl. Phys. Lett. 102 012111
[37] Amin B, Kaloni T P and Schwingenschlögl U 2014 RSC Adv. 4 34561
[38] Zeng F, Zhang W B and Tang B Y 2015 Chin. Phys. B 24 097103
[39] Debbichi L, Eriksson O and Lebègue S 2015 J. Phys. Chem. Lett. 6 3098
[40] Peng Q et al 2017 Catal. Sci. Technol. 7 2744
[41] Jiang Z et al 2017 Phys. Rev. Lett. 118 266401
[42]Kang P et al 2017 2D Mater. 4 045014
[43] Qiao J et al 2018 Sci. Bull. 63 159