Superconducting single-layer T-graphene and novel synthesis routes

TL;DR

This paper predicts a new single-layer superconducting material called T-graphene with a Tc of around 20.8 K, proposes a synthesis method involving high-pressure intercalation, and reports a record Tc of 30.4 K for related compounds.

Contribution

It introduces a novel 2D carbon superconductor T-graphene, predicts its properties, and proposes a feasible synthesis route, advancing the field of elemental 2D superconductors.

Findings

T-graphene is predicted to be an intrinsic superconductor with Tc ~20.8 K.

A synthesis route involving high-pressure intercalation and exfoliation is proposed.

The Tc of C4K is calculated to be about 30.4 K, setting a new record for layered carbon superconductors.

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 (Tc) 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 (C4K with P4/mmm symmetry) is firstly synthesized at high pressure (>11.5GPa) and then quenched to ambient condition; and finally, the single-layer T-graphene can be either exfoliated using the electrochemical method from the bulk C4K, or peeled off from bulk T-graphite C4, where C4 can be obtained from C4K by evaporating the K atoms. Interestingly, we find that the calculated Tc of C4K is about 30.4K at 0GPa, 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.

Tables

Table 1. 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	C4K	C6Li	C8Li	C8Na	C8K
Charge transfer	0.8$e$	0.9$e$	0.8$e$	0.9$e$	0.8$e$

Full text

Superconductivity in single layer materials Saito et al. (2016), such as FeSe Qing-Yan et al. (2012), MoS2 Lu et al. (2015), NbSe2 Xi et al. (2015) 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 Si et al. (2013); Ludbrook et al. (2015). Recently reported superconductivity in magic-angle graphene Cao et al. (2018) 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. Penev et al. (2016); Zhao et al. (2016) 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, sp2 and sp3. In addition to the well-known diamond, graphite, C60 fullerene Kroto et al. (1985), carbon nanotube Iijima (1991) and graphene Novoselov et al. (2005), many other types of carbon allotropes have been studied, including M-carbon Li et al. (2009), bct-carbon Zhou et al. (2010), graphdiyne Li et al. (2014), nanotwin diamond Huang et al. (2014), penta-graphene Zhang et al. (2015), V-carbon Yang et al. (2017), haeckelites Terrones et al. (2000), 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 Liu et al. (2012). This new form of carbon sheet was predicted to present good mechanical properties Majidi (2017) and could be used in hydrogen storage Sheng et al. (2012).

Although there is no superconductivity in pure graphite, many carbon-allotrope related materials were reported to be superconducting, such as graphite intercalation compounds (GICs) Hannay et al. (1965), fullerene alkali metal compounds Ganin et al. (2010), nanotubes Tang et al. (2001), boron-doped diamond Ekimov et al. (2004), boron carbide Xia et al. (2017) and magic-angle graphene superlattices Cao et al. (2018). Due to the similarity of the graphene-like boron layers in MgB2 Nagamatsu et al. (2001), GICs draw extensive attention, in which metallic atoms intercalate between the graphene sheets. Alkali metal carbon compounds (C8A, A=K, Rb, Cs) were the first type of GIC superconductors Hannay et al. (1965) studied, with $T_{c}$ being usually less than 1 K. Since C8K is one of the easiest to fabricate in alkali carbon compoundsHannay et al. (1965), intense efforts have been taken to the study on its superconducting properties Grüneis et al. (2009); Pan et al. (2011). With pressure, the $T_{c}$ of C8K increases to 1.7 K at 1.5 GPa Smith et al. (2006). Up to now, among the GICs theoretically proposed, C6Yb (with $T_{c}$ = 6.5 K) Weller et al. (2005) and C6Ca (with $T_{c}$ = 11.5 K) Emery et al. (2005) have the highest $T_{c}$ at ambient pressure. And C6Ca exhibits $T_{c}$ = 15.1 K at high pressure of 7.5 GPa Gauzzi et al. (2007). In addition to the bulk GICs, superconductivity in doped graphene with lithiumProfeta et al. (2012), potassium Xue et al. (2012) and Calcium Chapman et al. (2016), 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. Zhang et al. (2013, 2017) 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 $T_{c}$?

In this Letter, using crystal structure search methods and first-principles calculations, we predict a very interesting C4K 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 $P4/mmm$ C4K can be synthesized above 11.5 GPa and is quenchable to ambient pressure. exhibiting a high $T_{c}$ up to around 30.4 K at 0 GPa, which is more than one order of magnitude higher than that of C8K. We also propose that $T$-graphene can be peeled off from the bulk C4K, once it is synthesized by high pressure method. Most surprisingly, $T$-graphene itself is found to be an intrinsic 2D elemental superconductor with $T_{c}$ of about 20.8 K at ambient pressure.

To find a possible route to synthesize $T$-graphene, we first use our newly developed machine learning accelerated crystal structure search method Xia et al. (2018), 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 Supplemental Material (SM). We find that the known C8K compound becomes unstable at 20 GPa and tends to decompose to more stable C4K and other compounds. A C4K 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 C4K occupy the interlayer site above the centers of the octagonal ring (shown in Fig. 1(a)). Due to the unique layered structure of C4K and its similarity to GICs, we believe 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 $P4/mmm$ C4K, 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 Klimeš et al. (2009) and optPBE-vdw Klimeš et al. (2011) 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 C4K. As shown in Fig. 1(b), although the transition pressures from different functionals are slightly different, the conclusions are consistent that C4K 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 $P4/mmm$ C4K at 0 GPa and 20 GPa, see Fig. S3 in the SM, and find that both of them are stable. This indicates that $P4/mmm$ C4K is possible to be synthesized at high pressure and quenchable to ambient pressure. To check the stability of C4K at finite temperature, we perform ab initio molecular dynamics simulations. As shown by the mean square displacement (MSD) results in Fig. 1(c), C4K is also stable at ambient pressure and at finite temperature (300 K and 1000 K). Nevertheless, the synthesis of C4K 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 Salamat et al. (2014) or large-volume multianvil apparatus. These techniques have been used to successfully synthesize many new materials, including carbon-based materials Huang et al. (2014); Yang et al. (2017); Salamat et al. (2014).

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 C4K with $P4/mmm$ structure, the C3K4 with $C2/m$ structure, and the C2K3 with $C2/m$ structure. The C8K becomes energetically unfavorable and tends to decompose to C4K and other compounds when pressure increases to 20 GPa. We note that the carbon atoms in C3K4 and C2K3 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 Novoselov et al. (2004); Liu et al. (2014). 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 $P4/mmm$ C4K and $T$-graphite C4, respectively, where bulk $T$-graphite C4 can be obtained from C4K by evaporating the potassium atoms, possibly using similar procedure in the synthesis of Si24 from Na4Si24Kim et al. (2015). As shown in the Fig. S5 in the SM, the potential energy barrier of moving the K atom in C4K is very similar to that of the Na atom in Na4Si24, which shows the feasibility of obtaining bulk C4 from C4K. 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/$\AA^{2}$, which is in good agreement with the experimental results (20 $\pm$ 1.88 meV/$\AA^{2}$) Zacharia et al. (2004); Ziambaras et al. (2007). 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 $P4/mmm$ C4K and $T$-graphite C4, the resulted exfoliation energy are 85.8 meV/$\AA^{2}$ and 21.6 meV/$\AA^{2}$, respectively, as shown in Fig. 1(d).

According to the literature Mounet et al. (2018), a 2D material is potentially exfoliable (or easily to be exfoliated) when the exfoliation energy is $<$ 130 meV/$\AA^{2}$ ($<$ 30-35 meV/$\AA^{2}$). Therefore, $T$-graphene is potentially exfoliable from C4K and much easier to be exfoliated from bulk $T$-graphite C4. As shown in Table. 1, we find that C4K has similar charge transfer as that in many other graphite intercalation compounds, such as C8Li, C8Na, C8K and C6Li. Although having ionic-like interactions, it was reported that graphene sheets can be exfoliated from C6Li, C8Na and C8K in experiments using the electrochemical exfoliation method. Lee et al. (2015); Bharath et al. (2017); Thomas et al. (2018) Thus due to the structural similarity and also the similar charge transfer, $T$-graphene can very likely be exfoliated from bulk C4K 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.

In what follows we investigate the electronic properties of bulk C4K and single-layer planar $T$-graphene. The calculate electronic structures of C4K and $T$-graphene are plotted in Fig. 3. We also calculated the band structures of the C4 layer without potassium atoms (C${}_{4}K_{0}$), which is obtained by removing the K atoms from the C4K structure and keeping the lattice constants at 0 GPa (see Fig. S6 in the SM). Compared with C4K0, 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 C4K or $T$-graphene, the electronic density of states at the Fermi level, $N\left(\epsilon_{F}\right)$, is mostly dominated by the $\pi$ bands. In C4K, 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 $\Gamma$ point. As shown in Fig. 3(b) and (d), both fermi surfaces of C4K and $T$-graphene have 2D features and perfect nesting vectors.

Since the Fermi contours of C4K and $T$-graphene have similar cylinder shapes as those in the cuprates Damascelli et al. (2003) and iron-based superconductors Kuroki et al. (2008), 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 Allen and Dynes (1975). We estimate that the $T_{c}$ of $T$-graphene is around 20.8 K at ambient pressure with a commonly used screened Coulomb potential $\mu^{*}$ = 0.1. And in C4K, its $T_{c}$ 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 C4K is more than that in C8K, thus it seems that a higher doping level is a positive factor to the superconductivity in these carbon-sheet based systems.

In Fig. 4, we plot phonon dispersions together with phonon linewidth, Eliashberg spectral function $\alpha^{2}F$($\omega$), electron-phonon coupling strength $\lambda$($\omega$) and phonon density of states (PHDOS) of C4K at ambient pressure. According to the electronic structures discussed above, there is only one electron pocket in the Fermi surface of C4K, which is very similar to the case of single-layer FeSe on SrTiO3 He et al. (2013). 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 C4K, 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 $E_{g}$ mode of the carbon sheet, as shown in Fig. 4(b). This also corresponds to the two predominant peaks in the Eliashberg spectral functions $\alpha^{2}F$ at about 300 and 450 $cm^{-1}$.

To provide more insights into the electron-phonon coupling, we also compared the electronic bands without and with the perturbation induced by the $E_{g}$ phonon mode with $\Delta Q$ = 0.1 $\AA$. The calculated band structures and Fermi surfaces with the perturbation are plotted in Figis. 4 (c) and (d), respectively. Compared to the unperturbed case, the perturbated interlayer band near $\Gamma$ 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 $\Gamma$ point, while the perturbated bands get in touch together and produce a new electron pocket at the $\Gamma$ 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 Tc of C4K decreases under pressure (see Fig. S10 in the SM).

Fig. 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 ($\omega$ $\leq$ 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 $\AA$. 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 $B_{3}u$ 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 C4K, 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 $T_{c}$ 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 $P4/mmm$ C4K, which is exactly the $T$-graphene intercalation compound we want. This C4K 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 C4K using the electrochemical exfoliation method once C4K is synthesized by high pressure method. Or be peeled off from bulk $T$-graphite C4, where C4 can be obtained from C4K by evaporating the potassium atoms. Interestingly, it is found that the calculated $T_{c}$ of $P4/mmm$ C4K 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 C4K exhibit conventional phonon-mediated superconductivity. Comparing the electronic structures and superconducting properties between $T$-graphene and C4K, we can see the importance of doping effect. Therefore, a further enhancement on the $T_{c}$ 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 C4K, which is interesting to be 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 Xia et al. (2018) 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) Kresse and Furthmüller (1996). In the structure searching, the generalized gradient approximation (GGA), and the Perdew-Burke-Ernzerhof (PBE) functional Perdew et al. (1996) 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 Monkhorst and Pack (1976) k-sampling using a small k-spacing of $0.02\times{2\pi}$Å*-1*. Due to the 2D features of graphite, C4K 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-D2Grimme (2006), DFT-D3 Grimme et al. (2010), optB88-vdW Klimeš et al. (2009), optPBE-vdW Klimeš et al. (2011), vdW-DF Dion et al. (2004); Román-Pérez and Soler (2009) and vdW-DF2 Lee et al. (2010), 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 C4K 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 Hoover (1985) 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 Giannozzi et al. (2009). For C4K , we adopt a $16\times 16\times 16$ k-point mesh for the charge self-consistent calculation, and a $32\times 32\times 32$ k-point mesh for EPC linewidth integration and a $8\times 8\times 8$ q-point mesh for dynamical matrix. For $T$-graphene, we adopt a $16\times 16\times 1$ k-point mesh for the charge self-consistent calculation, and a $32\times 32\times 2$ k-point mesh for EPC linewidth integration and a $8\times 8\times 1$ 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.

**Acknowledgments

**We thank for the fruitful discussions with Tong Chen, Pengchao Lu, Kang Xia, Cong Liu, Yong Wang, Dexi Shao. We appreciate the financial support from the MOST of China (Grants No. 2016YFA0300404 and No. 2015CB921202), the NSFC (Grant Nos. 11574133 and 11834006), the NSF of Jiangsu Province (Grant No. BK20150012), the Fundamental Research Funds for the Central Universities, the Science Challenge Project (No. TZ2016001) 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.