Superconductivity in intercalated group-IV honeycomb structures
We present a theoretical investigation on electron-phonon superconductivity of honeycomb MX layered structures. Where X is one element of the group-IV (C, Si or Ge) and M an alkali or an alkaline-earth metal. Among the studied composition we predict a T of 7 K in RbGe, 9 K in RbSi and 11 K in SrC. All these compounds feature a strongly anisotropic superconducting gap. Our results show that despite the different doping level and structural properties, the three families of materials fall into a similar description of its superconducting behavior. This allows us to estimate an upper critical temperature of about 20 K for the class of intercalated group-IV structures, including intercalated graphite and doped graphene.
A large research effort has been lately focused on atomic-thin layered materials and their properties Geim and Novoselov (2007); Geim (2009); Novoselov et al. (2012). This was triggered by the creation of graphene from graphite Novoselov et al. (2004) and also motivated by the belief in many potential applications since thin systems can be significantly modified in their electronic properties simply by acting on parameters as stacking, chemical and physical doping Fedorov et al. (2014); Yang et al. (2014). In fact this versatility is an extraordinary playground for searching for new superconductors (SC) Klemm (2012). Many (low temperature) SC are already known in the class of graphite intercalated compounds GICs Belash et al. (1989); Nalimova et al. (1995); Avdeeva et al. (1990); Weller et al. (2005); Emery et al. (2005), graphene itself has been predicted to superconduct with a critical temperature (T) of 18 K upon Li doping Profeta et al. (2012).
Among all possible compounds, those chemically and structurally closer to graphite are the honeycomb lattices of silicon Sanfilippo et al. (2000); Bordet et al. (2000); Imai et al. (1998); Imai and Kikegawa (2003) and germanium Tobash and Bobev (2007); Bobev et al. (2004). For which suplerconductivity upon intercalation was also reported Sanfilippo et al. (2000); M. Imai and Hirano (1995); Flores-Livas et al. (2011); Evers and Weiss (1974); Demchyna et al. (2006); Yamanaka (2009). Hence GICs and doped graphene are not unique systems, having a Si and Ge counterpart can be seen as members of a generalized family of group-IV intercalated honeycomb lattices (gIV-ICs).
So far the highest T reported on gIV-ICs is 11.5 K in CaC Weller et al. (2005); Emery et al. (2005). This system is also the most studied among the family and its superconducting properties are rather well understood Calandra and Mauri (2005); Boeri et al. (2007); Kim et al. (2006); Sanna et al. (2007); Yang et al. (2014). It is particularly clear that an important role is played by the existence at the Fermi level of 2D electron like bands as well as anti-bonding C- states. It is also know that a sufficiently large intercalation is therefore a necessary condition to obtain high critical temperatures. But what is the highest conceivable T in an intercalated graphite-like system? Could Si and Ge iso-morphs be better candidates than GICs? We will address these questions by focusing our investigation on the high doping limit, with one intercalating atom per two honeycomb atoms. We will indicate this family of compounds as MX where M stands for a metal of the I and II column of the periodic table and X is carbon, silicon or germanium. This composition is known to occur Demchyna et al. (2006) in several silicides Evers and Weiss (1974); Bordet et al. (2000); Flores-Livas et al. (2011) and germanides Tobash and Bobev (2007); Bobev et al. (2004); Yamanaka (2009).
We will show by means of theoretical ab-initio methods, that finding high temperature superconductivity in these families is a false hope. On the other hand breaking the record critical temperature of CaC is likely to be possible.
All systems are structurally relaxed within Kohn-Sham density-functional theory. 111We used the two plane-wave based code abinit Gonze et al. (2009), and espresso Giannozzi et al. (2009) within the Perdew-Burke-Ernzerhof (PBE) Perdew et al. (1996) exchange correlation functional and the core states were accounted for by norm-conserving Troullier-Martins pseudopotentials Fuchs and Scheffler (1999). The pseudopotential accuracy has been checked against all-electron (LAPW+lo) method as implemented in the elk code (http://elk.sourceforge.net/). Upon relaxation sup () all carbon compounds, apart from CaC, converged to the AlB crystal structure (space group , number 191), while all silicides and germanides as well as CaC converged to the EuGe crystal structure (space group , number 164). In both, M occupies the Wyckoff position (0,0,0) and X the the positions (1/3, 2/3, ) and (2/3, 1/3, ). In the AlB structural prototype the parameter is fix to , while in the EuGe structure it is related to a buckling () of the honeycomb lattice: . The EuGe structural prototype and the values of are shown in Fig. 1. This figure shows clearly that intercalating lighter ions (Li, Be, Na) induce high buckled honeycomb plans, while heavier ions (Rb,Cs,Ba) tend to induce low-buckled plans. CaC deviates from the general trend, this structure has a mixture of (75 respectively) bonding and therefore at ambient pressure it present a finite buckling (energetically more favorable than in a flat AlB structure). In this respect, it has been recently predicted by Li and coworkers Li et al. (2013) that the flat-layered phase could be stabilized at high-pressures.
As many of the compounds discussed in this work are not experimentally known, in order to assert on their potential synthesis we calculated their thermodynamic stability, this is derived from the total DFT energy of the system (MX ) and of its elemental ground state solid (see supplemental material sup () for details). This analysis leads to the conclusion that all graphite compounds in the MX layered phase are unstable towards this elemental decomposition. While most of silicides and germanides are stable towards decomposition. Nevertheless, since a positive formation energy does not completely exclude these materials from their possible synthesis, we will also investigate their dynamical stability (phonons).
For all systems under investigation we computed phonons and only for those systems dynamically stable, the electron-phonon coupling was calculated by means of density-functional perturbation theory. 222The phonon spectrum and the electron-phonon matrix elements were obtained employing density-functional perturbation theory. Baroni et al. (2001); Gonze and Vigneron (1989); Savrasov and Savrasov (1996), within the pseudopotential approximation. A cutoff energy of 60 Ry was used in the plane-wave expansion. A Monkhorst-Pack -grid and a -grid was used for all the materials under consideration. With the only exception of SrC, RbSi and RbGe, where we have increased the sampling grid to in order to achieve an accurate description of anisotropic properties. We found most of the intercalated carbon compounds to be dynamically unstable, with the only exception of Sr and Ca intercalation. This suggests that the 1 to 2 intercalation is too large for this family and is evidenced experimentally by the reported challenging synthesis of LiC Belash et al. (1989), that turns out to be metastable, partially loosing its Li content and converting in LiC Nalimova et al. (1995); Avdeeva et al. (1990). On the other hand, with the exception of light-ion intercalants, most of the disilicides and digermanides are dynamically stable.
Eliashberg spectral functions Carbotte (1990); Allen and Mitrović (1983) for all the dynamically stable systems are reported in Fig. 2. From now on we will only consider this subset of materials. In this figure we can clearly observe that the spectral functions are scaled in their frequency by the mass of the atom forming the honeycomb layer. And this extends not only, obviously, to the high energy modes that originate from strong in-layer bonds, but also to the low frequency modes that are dominated by the intercalant motion in the weak interlayer potential. Thus, indicating a chemical effect. We also observe that alkali metals (as compared with alkaline earths) lead to systematically lower phonon branches, therefore to an enhanced coupling strengths Carbotte (1990)
at the same time this lowers the average frequency, that we conventionally express as
From an electronic point of view, all the materials share a qualitative similar structure. As in the case of CaC or doped graphene, there are two type of electronic states located at the Fermi energy: anti-bonding states provided by the honeycomb layer (C, Si, Ge) and 2D interlayer states with contributions from the M -orbitals. These electronic states hybridize differently along the alkali or the alkaline-earth column and lead to different effective doping and band alignment. This affects the density of states at the Fermi energy () and whit it the occurrence of superconductivity, as we will show below.
In order to perform a fast screen of our MX set, the superconducting critical temperatures were estimated within McMillan-Allen-Dynes parametrization of the Eliashberg equations McMillan (1968); Allen and Dynes (1975); Carbotte (1990); Eliashberg (1960)
where is the Boltzmann constant. This formula depends on three parameters: the Coulomb pseudopotential (here fixed to by comparison with SCDFT results, see below); the logarithmic average of the phonon frequency ; and the coupling constant . The computed T couplings and are shown in Fig. 3.
In the limit of an homogeneous coupling in k-space, is proportional to . Within BCS theory, this parameter splits as , where is the BCS coupling strength. In Fig. 3 b) we observe a remarkable proportionality between and . Leading to the conclusion that is approximately the same on this MX class of systems, with the sole exception of few systems characterized by strong softening. Eventually this softening will leads to a phononic instability and to a structural phase transition. Perhaps under different thermodynamic conditions of pressure and temperature. Although not belonging to this MX family we observe that CaC lies perfectly in this regime Sanna et al. (2007, 2012). And similarly does MgB, however, this is accidental as we have ignored its multi-band nature Liu et al. (2001); Floris et al. (2005, 2007a).
These calculations predict several interesting superconductors and in particular RbSi, RbGe and SrC. RbGe has the highest density of states and, as discussed above, also presents the highest , even though it shows a modest T of 7 K. In fact T (see Eq. 3) depends also on the phonon energy, which is larger for systems having lower mass, for instance SrC. Also in this Fig. 3 (in panel a), we included the iso-mass lines as a reference to indicate how the T in a material would be affected by (on ) or by . The outcome of this analysis suggests the existence of an upper critical temperature for each family. And this is imposed by the electronic structure, as hardly would exceed the value of 0.7 states/eV/spin (of RbGe). Following the iso-mass lines in this figure for each subfamily, leads to the conclusion that an upper critical temperature of about 10 K, 15 K and 20 K exists respectively for intercalations in Ge, Si and carbon honeycombs. We firmly believe that this conclusion can be extended beyond the MX class, since different intercalation density will not plausibly affect the coupling strength. However, the coupling strength could be significantly affected if states were involved (as in MgB), but this would require an unphysical doping level.
We will now focus our investigation on three selected systems SrC, RbSi and RbGe as the most interesting representative of each sub-family. As discussed in the introduction both RbSi and RbGe are stable towards elemental decomposition. In addition they are also more stable than their RbSi and RbGe rhombohedral counterparts sup (). Therefore, we believe, these two systems are likely to be accessible to the experimental synthesis. On the other hand, SrC is not stable with respect to elemental decomposition and turn to be less energetically competitive than its rhombohedral SrC configuration that, in fact, has been synthesized Kim et al. (2007). Nevertheless, since the system is dynamically stable, it may still be possible to find a way to its synthesis, perhaps by means of a non-equilibrium process or by high temperature and high pressure, as often used to synthesize clathrates, Toulemonde et al. (2005) carbon borides, Shah and Kolmogorov (2013) and layered disilicides Evers and Weiss (1974); Imai and Kikegawa (2003); Bordet et al. (2000); Flores-Livas et al. (2011) and germanides Tobash and Bobev (2007).
The electronic band structures of these three selected materials are shown in Fig. 4. The bands of SrC essentially differ from those of RbSi and RbGe, due to the effect of symmetry breaking (both have buckling) and as well the doping level of the honeycomb lattice (charge projection shows that divalent strontium donates electrons - while monovalent Rb donates electrons for both RbSi and RbGe). The Fermi surfaces (FS) shown in Fig. 5 a, b and c, present multiple Fermi sheets with different orbital character. In SrC the inner FS comes from interlayer states, while the outer surface is formed by carbon states. In RbSi and RbGe the hybridization between interlayer and honeycomb states is much stronger. The outer FS is mostly due to Si/Ge states, while the inner FS has an interlayer character, however with a relatively large overlap (25%) to Si/Ge states.
The phonon dispersion for the three systems is shown in Fig. 4. The overall structure of the phonon modes is the same for the three systems. Low frequency modes present a strong intercalant component, fundamentally due to the weak force constants that binds the atoms to their position in the lattice, but also because of their relatively large mass. To the scope of this work, the most interesting feature of the phononic dispersion is the behavior of the buckling modes. In the unbuckled (flat) SrC compound this mode has 50 meV in the zone center and and cannot falls below 40 meV. While in the buckled RbSi and RbGe compounds it becomes “soft” moving from ( at 30 meV in RbSi and 23 meV in RbGe) to (3.5 meV). This mode is strongly coupled in both RbSi and RbGe, and anharmonic effects (not considered in the present work) may also affect the strength of its coupling.
We will now reconsider the superconducting properties of these selected systems by means of a more accurate superconductivity theory than the McMillan formula used so far. We will adopt density-functional theory for superconductors (SCDFT), as it is completely parameter free Oliveira et al. (1988); Lüders et al. (2005); Marques et al. (2005) and allows for a full k-resolved description 333The phononic functional we use is an improved version with respect to Ref. Lüders et al., 2005; Marques et al., 2005 and is discussed in Ref. Sanna and Gross, 2014. Coulomb interactions are included within static RPA Sanna et al. (2007).
It should be observed (see Fig. 5 on panels a, b and c) that the electron phonon coupling in all these systems is rather anisotropic, meaning strongly k-dependent on the FS. SrC has a continuous distribution, while the two FS of RbSi and RbGe, have remarkably different coupling strength: stronger on the small FS around the point and weaker in the outer FS (at large |k|). The distribution of superconducting gaps on the Fermi energy (not shown) follows the anisotropy in , similarly to the behavior observed in bulk lead Floris et al. (2007b). The gap distribution function at (i.e. the energy distribution of the SC gaps: ), as well as the temperature dependence (in a two-band and single band model) are plotted in Fig. 5 d,e,f. Both RbSi and RbGe show two distinct gaps (like in MgB or bulk lead Floris et al. (2007b, 2005, a)), while SrC has an anisotropic gap continuously distributed. This gap distribution reminds that of CaC Sanna et al. (2007); Gonnelli et al. (2008); Nagel et al. (2008). This anisotropy will affect the specific heat and the thermodynamical properties. However, unlike in MgB, the critical temperature is not much affected by it (less than 1 K). The role of coupling anisotropy on the superconducting behavior can be clearly understood within the qualitative model of Suhl, Mattias and Walker Suhl et al. (1959). The observed combination of a large anisotropy in the gap with a small enhancement in T is a consequence of the strong inter-band coupling between -states (having a smaller gap) and the interlayer states (that dominate on the larger gap). The gap distribution of SrC is even broader and clearly cannot be completely captured within a two-band model. The system is in fact almost gapless, since the small part of the FS shows a negligible superconducting pairing as a consequence of the weak phononic coupling.
In summary, we presented a theoretical study on honeycomb layered binary carbides, silicides and germanides intercalated by alkali or alkaline-earth metals. Our superconductivity analysis has shown that in this class of materials are many compounds with a relatively high critical temperature ( 10 K) as well as a quite complex superconducting state. In addition, the stability investigation has shown that several compounds should be accessible to their experimental synthesis. Finally, we demonstrate an intrinsic physical similarity among the group, which can be traced back to their characteristic +interlayer character of states at the Fermi surface. From this feature we estimate an upper limit for the transition critical temperatures: K, K and K respectively for carbon, silicon and germanium intercalated honeycombs. This limit could be broken only in the unlikely case in which the doping level would be able to drive -states at the Fermi level. Nevertheless this study indicates that superconductivity in doped graphite and similar systems is a rather general behavior and many more superconductors may still be discovered.
J.A.F.L. acknowledge financial support from EU’s 7th Framework Marie-Curie scholarship Program within the “ExMaMa” Project (329386).
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