Doped graphane: a prototype high-Tc electron-phonon superconductor
aa r X i v : . [ c ond - m a t . m t r l - s c i ] F e b Doped graphane: a prototype high- T c electron-phonon superconductor G. Savini,
1, 2
A. C. Ferrari, ∗ and F. Giustino Department of Engineering, University of Cambridge, Cambridge, CB3 0FA, UK Institute of Protein Research, University of Osaka, Osaka, 565-0871, Japan Department of Materials, University of Oxford, Oxford, OX1 3PH, UK
We show by first-principles calculations that p -doped graphane is a conventional superconductorwith a critical temperature ( T c ) above the boiling point of liquid nitrogen. The unique strength ofthe chemical bonds between carbon atoms and the large density of electronic states at the Fermienergy arising from the reduced dimensionality synergetically push T c above 90K, and give riseto large Kohn anomalies in the optical phonon dispersions. As evidence of graphane was recentlyreported, and doping of related materials such as graphene, diamond and carbon nanostructures iswell established, superconducting graphane may be feasible. The discovery of superconductors such as magnesiumdiboride[1] and iron pnictides[2, 3] opened new hori-zons in the landscape of superconductivity research, fu-eling renewed interest in the quest for high-temperaturesuperconductivity in materials other than the copperoxides[4, 5]. The critical temperature, T c , reflects the en-ergy scale of the quantum-mechanical interactions driv-ing the electron condensation into the superconductingstate[6]. In high- T c copper oxides[7] the nature of theinteraction leading to superconductivity is still underdebate[8], yet it is generally accepted that Coulomb ex-change and correlation effects, with energy scales aroundfew hundred meVs, play an important role[9, 10]. Incontrast, in conventional superconductors the pairing isknown to be driven by the interaction between electronsand lattice vibrations, with an associated energy scale ofonly a few ten meVs[11]. Due to the order-of-magnitudedifference between such energy scales, it is generally as-sumed that conventional superconductors cannot exhibit T c as high as copper oxides[4, 5]. Here, we report first-principles calculations showing that p -doped graphanewould make a conventional superconductor with T c wellabove the boiling point of liquid nitrogen.The Bardeen-Cooper-Schrieffer (BCS) theory[11] de-fines the basic theoretical framework to understand con-ventional superconductivity. Its generalization, known asthe Migdal-Eliashberg theory[12], incorporating the lat-tice dynamics, provides a predictive computational tool.Within BCS, T c is given by[11]: k B T c = 1 . ~ ω exp (cid:18) − N F V (cid:19) (1)where k B is the Boltzmann constant, ~ ω a character-istic phonon energy, N F the electronic density of states(EDOS) at the Fermi Energy, E F , V an effective pairingpotential resulting from the net balance between the at-tractive electron-phonon coupling (EPC) and the repul-sive electron-electron interaction[11]. Even though theoriginal BCS formula for T c is now replaced by morerefined expressions such as, e.g., the modified McMil-lan equation[13], Eq.1 still proves useful for discussing trends. Eq.1 indicates that one could maximize T c by in-creasing the materials parameters ω , N F , V . However,these are strongly intertwined, making such optimizationcomplex[13, 14]. Here, we propose a simple procedure,based on Eq.1, to design a high- T c superconductor.Let us first consider the conventional superconductorwith the highest T c , MgB ( T c = 39K)[1]. For simplicity,we neglect multi-band and anisotropy effects, which werethe object of detailed investigations[15–19]. In MgB theEPC contribution to V is large ( ∼ λ = N F V ,using N F = 0 . λ ∼ E F (those which conden-sate in the superconducting state[11]) are of σ character,i.e. derive from bonding combinations of planar B sp hy-brids localised around the middle of B-B bonds[15–19].These electronic states are significantly affected by the B-B bond length variation associated with bond-stretching E g phonons[16, 20], resulting into a large EPC contri-bution to V . At the same time, the E g phonon energyis large ( ∼ ω in Eq.1. Furthermore, MgB is a metalwith a significant EDOS at E F ( ∼ T c = 39K[15–19]. However,many attempts to improve upon MgB , by investigatingrelated materials, only met limited success[21], with theexperimental T c never higher than MgB .We thus search for an alternative material having atleast some of the desirable features of MgB , i.e.(i) σ electrons at the Fermi surface,(ii)large bond-stretchingphonon frequencies, and (iii)large EDOS at E F . Wenote that the first two requirements are both met byB-doped diamond, a conventional BCS superconductorwith T c = 4K[22], where a small hole-like Fermi surfaceappears around the top of the valence band[23]. Theelectronic states at E F have σ character deriving fromthe bonding combination of tetrahedral C sp hybrids,bearing some analogy to MgB . As these σ states are lo-calized in the middle of the C-C bonds, they couple con-siderably to bond-stretching phonons[23, 24], resulting ina large EPC contribution to V , even superior to MgB ( ∼ λ = N F V , using N F = 0 . -2 -1.5 -1.0 -0.5 Energy (eV) E D O S ( s t a t e s / e V / C a t o m ) GrapheneNanotubeDiamondNanowire 1d1d2d3d (sp )(sp )(sp )(sp )
Graphane 2d (sp ) K Γ M K -3-2-10 E ne r g y ( e V ) -3 -2 -1 0 1 2 3 4 Energy (eV) E D O S ( s t a t e s / e V / c e ll ) Pristine graphaneHole-doped graphane (c)(b)(a)
FIG. 1: (Color online)(a)EDOS per carbon atom of 1d (nan-otube; diamond nanowire),2d (graphene; graphane) and 3d(diamond) systems. With the exception of graphene, withlinear dispersions, the EDOS is proportional to E − / in 1d,a step-like function in 2d, and E / in 3d. The step-likeEDOS in graphane implies that N F is large even at low dop-ing.(b)EDOS of pristine (solid black line) and 12.5% p-dopedgraphane (dashed red line). The top of the valence bandis set as zero, and E F =-0.96eV (green line). The EDOS at E F is similar in the two models (0.26 states/eV/cell in rigid-band and 0.27 states/eV/cell in supercell).(c)Band structureof pristine (solid black line) and 12.5% p-doped graphane(dashed red line). (inset) Ball-and-stick 2 × and λ ∼ . ∼ E F is rather small( ∼ T c . Thus, while B-doped diamond shares some of the desirable features of MgB , its 3-dimensional (3d)nature implies that the EDOS in proximity of the valenceband scales as ∼ E / (with E measured from the valenceband edge)[29], Fig.1(a). Then, the number of carriersavailable for the superconducting state remains relativelysmall even for large doping. Superconducting diamondis thus a 3d analogue of MgB [23, 24].This leads to the question of what would happen ina hypothetical B-doped diamond structure with reduceddimensionality, such as a thin film or a nanowire, wherethe EDOS can be significantly enhanced by quantum con-finement. Indeed, the EDOS of a two-dimensional semi-conductor goes as ∼ θ ( E ) ( θ being the step function)[29],hence the number of available carriers can be large, evenat low doping. In order to estimate the expected EDOSincrease in a diamond thin film it is helpful to considera simple parabolic band model. For 2% B doping, bulkdiamond has N F =0.1 states/eV/cell at E F . A 0.5nmthick diamond film with the same doping would have N F ∼ T c . Using the electron-phonon po-tential and the phonon frequency of bulk diamond, Eq.1gives that a 0.5nm film would superconduct at T c ∼ sp bonded, the latter is sp , as diamond[37].Recently, some experimental evidence of graphane wasreported[38]. Since graphane is the 2d counterpart ofdiamond, our scaling arguments immediately point todoped graphane as a potential high- T c superconduc-tor. Doping could be achieved by gating, including us-ing an electrolyte gate, or by charge-transfer, as donein graphene[31–34, 39, 40]. Substitutional doping ofgraphene was also reported, up to ∼ cm − [35, 41].We thus perform density functional perturbation the-ory (DFPT) calculations of EPC and superconductivityin doped graphane within the framework of the Migdal-Eliashberg theory[12] and the local density approxima-tion (LDA)[42, 43]. By analogy with B doped dia-mond, we consider p -doping. This is simulated usingthe rigid-band approximation[44]. Fig.1(b) shows thatthe calculated EDOS in p -doped graphane close to thevalence band maximum follows a step-like behavior, asexpected for a 2d system. At 3% doping the EDOS is0.22 states/eV/cell, compared to 0.13 states/eV/cell inbulk diamond, with a factor 1.7 enhancement. Fig.1(c)indicates that the dispersions close to E F are essentiallyidentical for a supercell containing B and for a rigid-bandmodel of doped graphane. We expect this to hold also Γ M K Γ P honon F r equen cy ( c m ) - P honon E ne r g y ( m e V ) (a) P honon E ne r g y ( m e V ) Κ Γ M P honon F r equen cy ( c m ) - - (b)(b) Pristine1% doping FF FIG. 2: (Color online)(a)Phonon dispersion of pristine (solidblack line) and 1% p -doped graphane (dashed blue lines).The C-H stretching modes have higher frequencies (2655-2711cm − ) and are not shown.(b)Optical modes around thezone centre, showing the Kohn Anomalies. The horizontal(green) arrows indicate the average Fermi surface diameter. for lower doping, where the perturbation to the pristinedispersions is smaller. The similarity between these twomodels justifies our use of the rigid-band approximation.A supercell calculation with the B dopant explicitly in-cluded does not show impurity states inside the gap.Fig.2(a,b) report the phonon dispersions of pristineand p -doped graphane and Fig.3(a) the correspondingphonon density of states (PDOS). Upon doping, the opti-cal zone-centre modes with in-plane C-C stretching softenas a result of the inception of Kohn Anomalies[45]. Thetwo degenerate TO modes, having planar C-C stretchingand H atoms moving in-phase with the C atoms, down-shift from 1185 to 715cm − (147 to 89meV). This is dueto the large EPC of planar C-C stretching, which signif-icantly affects the sp -like electronic states at the Fermisurface. The two degenerate zone-centre modes, hav-ing in-plane C-C stretching and H atoms moving out-of-phase with respect to the C atoms, downshift from 1348to 1257cm − (167 to 156meV). The LO mode, with out- Phonon Frequency (cm ) F ( ω ) ( s t/ c m / un i t c e ll ) -1 - Pristine graphane8 % doping (a)
Phonon Frequency (cm ) α F ( ω ) / ω ( c m ) -1 - TO mode TO mode (c)
Phonon Frequency (cm ) α F ( ω ) / ω ( c m ) -1 - (b) FIG. 3: (Color online)(a)PDOS of pristine and dopedgraphane.(b) Eliashberg function in p -doped graphane for in-creasing doping. The largest contribution comes from theoptical modes, similar to diamond[24], but also the acous-tic phonons couple to holes at the Fermi surface, similar toSiC[44].(c)Contributions to the Eliashberg function arisingfrom the TO stretching modes (hashed region). (insets) Ball-and-stick representations of two TO modes. The arrows indi-cate the in-plane C-C stretching motions (carbons are shownin grey, hydrogens in white) of-plane C-C stretching, does not couple to the electronsdue to the different parity of potential and wavefunctions,resulting into a vanishing EPC. The two degenerate opti-cal modes corresponding to the shear motion of the C andH planes (at ∼ − ) and the C-H stretching modes(2 modes at 2663 and 2711cm − ) do not undergo soften-ing upon doping. This is consistent with the electronicstates associated with the C-H bonds having little weightat E F , hence a small EPC.The softening of modes with a large C-C stretch-ing component is similar to that reported in B-dopeddiamond[24, 25]. In particular, the region of recip-rocal space where the phonon softening is observedmatches the diameter, 2 k F , of the hole Fermi surface Hole doping (%) T ( K ) c MgBDiamondCaCGraphane (b) Hole doping (%) λ MgBGraphaneDiamondCaC (a) FIG. 4: (Color online)(a)EPC of graphane as a function ofdoping, calculated using the standard DFPT formalism[51]:the Brillouin zone is sampled with an electron grid up to300 × ×
1, smearing from 50 to 270 meV, and phonongrid of 100 × ×
1. For comparison, we plot literaturevalues for MgB (solid red line[24]), CaC (dashed greenline[52]), and diamond (solid black line[24]; triangles[25]).More sophisticated calculations taking explicitly into accounta substitutional dopant, such as B, could slightly changethe EPCs[25, 28]. However, in B-doped diamond a rigid-band model provides a lower EPC and a lower bound to T c [25].(b) T c calculated using the modified McMillan formulaand a Coulomb pseudopotential µ ∗ = 0 . T c [15, 53]. For com-parison we also show T c of MgB (solid red line, T c = 39K[1]),CaC (dashed green line, T c = 11 . T c = 4K at ∼
3% B[22]; 11K at ∼
7% B[55] around the Γ point, this being a typical signature ofthe Kohn effect[45]. The calculated phonon softening ofthe TO C-C stretching modes ( ∼ ∼ − )is significantly larger than in other materials, as typi-cal Kohn anomalies range from ∼ ∼ α F ( ω ) = 12 X q ν ω q ν λ q ν δ ( ω − ω q υ ) (2)where λ q ν is the EPC for a phonon mode ν with momen-tum q and frequency ω q ν , and δ is the Dirac delta (weused a Gaussian of width 2meV for this purpose). Weget that the TO in-plane C-C bond-stretching phononswith C and H atoms moving in-phase (see Fig.3(c)) havethe largest EPC, due to the σ character of the elec-tronic states at E F and the large C displacements asso-ciated with these modes. This is similar to B-doped bulkdiamond[23–25, 28] and validates our hypothesis that p -doped graphane can be regarded as an atomically thindiamond film, exhibiting similar EPC and vibrational fre-quencies, but larger EDOS at E F . We note that the in-plane C-C bond-stretching phonons, with C and H atomsmoving out-of-phase, do not contribute to the EPC. Thishappens because, upon softening, the four C-C planarstretching modes hybridize in such a way that those at715cm − carry an increased weight on the C atoms, whilethe opposite happens for the two modes at 1257cm − .Figure 4(a) plots the EPC as a function of doping, andFig. 4(b) the corresponding T c . We find that T c exceedsthe boiling point of liquid nitrogen, and falls within thesame T c range as copper oxides[50]. Due to the rela-tively constant EDOS below the top of the valence band[cf. Fig.1(b)], T c is rather insensitive to doping. This isimportant for the practical realization of superconduct-ing graphane. Our results should be valid throughoutthe entire doping range considered here in the case ofgate- or charge transfer-induced doping, since in thesecases the holes are delocalized and doped graphane isin the metallic regime. On the other hand, for substi-tutional doping we expect our results to be valid onlybeyond the Mott metal-to-insulator transition (MIT). Inabsence of experimental MIT measurements in graphane,we estimate the critical doping concentration, n c , usingthe following argument. In 3d the MIT occurs when theimpurity wavefunctions are close enough that their over-lap is significant[56]. For many materials a H n / c ∼ . a H being the radius of the ground-state wavefunction ofan hydrogenic donor[56]. The radius can be calculated as a H = ǫ/m ⋆ a / a being the Bohr radius, ǫ the dielec-tric constant, and m ⋆ the effective mass[56]. In diamond a H ∼ n c ∼ · − [56], therefore the averageseparation between nearest neighbor B atoms is ∼ a dH = ǫ/m ⋆ a / ǫ = 5 . m ⋆ = 0 .
74) wefind a dH = a H / ∼ ∼ . · holes · cm − . Thiscould be feasible, considering that substitutional dopingin graphene was reported up to 5%[41].The calculated high- T c for p -doped graphane bearsconsequences both for fundamental science and appli-cations. One could envision hybrid superconducting-semiconducting circuits directly patterned through litho-graphic techniques, graphane-based Josephson junctionsfor nanoscale magnetic sensing, and ultimately an idealworkbench for exploring the physics of the superconduct-ing state in two dimensions[58]. The superconductingphase transition in graphane could also be controlledby gating[34, 59]. A high- T c superconductor with gate-controllable T c could lead to novel switching mechanismsin nanoscale field-effect transistors. Furthermore, thediscovery of an electron-phonon superconductor with T c above liquid nitrogen would mean that(i) there are nofundamental reasons to believe that BCS superconduc-tors cannot have T c >
40K (MgB ), and (ii)high- T c su-perconductivity does not take place exclusively in thecopper oxides. In particular, our calculations indicatethat at least one material could exist where a very strongEPC leads to T c in the copper oxide range without trig-gering a lattice instability. The superconducting phasetransition in systems with reduced dimensionality hasbeen the subject of numerous theoretical studies[60, 61].Quantum fluctuations could destroy the superconductingorder in 2d[62]. However, recent experimental evidencesuggests that this is not necessarily the case[58, 59, 63].In particular, for thin Pb it was reported that the super-conducting state is robust down to two atomic layers[58].Since our proposed mechanism of superconductivity indoped graphane is BCS-like, as in Pb, there should be nofundamental limits to prevent the realization of high- T c superconductivity in graphane.It is immediate to extend the present study to diamondnanowires, which have been the subject of intense inves-tigations in the past few years[64]. For a 1d system theEDOS near a band edge has a van Hove singularity goingas ∼ E − / [29]. We can assume phonon energies and EPCto be similar to bulk diamond and graphane. Then Eq.1would yield T c as high as ∼ T c higherthan copper oxides by exploiting dimensionality deserves further investigation. Our work suggests that p -dopeddiamond nanostructures have an intriguing potential forhigh- T c BCS-like superconductivity.
Acknowledgments
Calculations were performed atHPCF (Cambridge) and the Research Center for Scien-tific Simulations of the University of Ioannina (Greece)using
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