Silicene and Germanene as prospective playgrounds forRoom Temperature Superconductivity

G. Baskaran

The Institute of Mathematical Sciences, Chennai 600 113, India

Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5

Combining theory and certain striking phenomenology we suggest that silicene and germanene areelemental Mott insulators and abode of doping induced high Tc superconductivity. In our theory, a3 fold reduction in silicene π-π∗ band width, in comparison to graphene, and short range coulombinteractions enable Mott localization. Recent experimental results are invoked to provide supportfor our Mott insulator model: i) a significant π-band narrowing, in silicene on ZrB2 seen in ARPES,ii) a superconducting gap appearing below 35 K with a large 2∆

kBTc∼ 20 in silicene on Ag, iii)

emergence of electron like pockets at M points, on electron doping by Na adsorbent, iv) certaincoherent quantum oscillation like features exhibited by silicene transistor at room temperatures andv) absence of Landau level splitting upto 7 Tesla and vi) superstructures, not common in graphenebut, ubiquitous in silicene. A synthesis of the above results using theory of Mott insulator, with andwithout doping, is attempted. We surmise that if competing orders are taken care of and optimaldoping achieved, superconductivity in silicene and germanene could reach room temperature scales;our estimates of model parameters, t and J ∼ 1 eV, are encouragingly high, compared to cuprates.

PACS numbers:

INTRODUCTION

A wealth of activity in the field of graphene, followingthe seminal work of Novosolev and Geim [1–3], has pavedway for silicene [4–7], a silicon analogue of graphene. Thisnew entrant might have a potential to begin another fer-tile direction in condensed matter science and technol-ogy. Replicating a rich graphene physics has been a partof recent efforts. Stable silicene layer has been createdon a few metallic substrates, Ag, ZrB2 and Ir. However,synthesis of free standing silicene remains a challenge.Interesting ARPES and STM results are available [8–15]. Silicene based field effect transistor has been alsofabricated [16]. There have been successful attempts tosynthesize germanene and related systems [17] on certainmetallic substrates.

Aim of the present article is to provide a low energymodel, a rather unexpected one, for electrical and mag-netic properties of silicene. Finding a suitable low energymodel for strongly interacting quantum matter continuesto be a challenge. The very experimental results we wishto understand guide us to correct theoretical modelling.Theory in turn guides experiments. The synergy con-tinues. This is true from Standard Model building inelementary particle physics to Standard Model buildingfor cuprate. Silicene is no exception.

The currently prevalent view is that neutral siliceneis a Dirac Metal, a semimetal qualitatively similar tographene [7]. Purpose of the present paper is to offer adifferent view point, that Silicene is not a Carbon Copyof Graphene - it is an elemental Mott insulator. If provedcorrect, our provocative proposal will make silicene differ-ent from semi metallic graphene in a fundamental fashionand open new avenues for physics and technology, arising

from strong electron correlation effects.

It is well appreciated now, thanks to the path breakingdiscovery of high Tc superconducting cuprates by Bed-norz and Muller [18] and subsequent resonating valencebond theory by Anderson and collaborators [19–23], thatMott insulators and strong electron correlations are seatsof a variety of rich physics and phenomena. In addi-tion to superconductivity it includes, quantum spin liq-uids, emergent fermions with Fermi surfaces, gauge fields,quantum order, topological order and so on. Further,inspired by certain recent theoretical development [24]there is a debate and search for quantum spin liquids inhoneycomb lattice Hubbard model [25–27]. We believethat silicene and germanene will fit well into the discus-sion as real candidate materials, albeit with added novelfeatures.

Several ab-initio calculations[4–6], many body theory[28], quantum chemical calculations and insights [29–34]are available for silicene. Interestingly there is a differingview, which has not been well appreciated. Existing solidstate manybody calculations predict stable free standingsemi metallic silicene that is a similar to graphene. How-ever, quantum chemical methods and insights doubt exis-tence of a stable free standing silicene [29, 30, 33] becauseof radicalization/reactivity and reduced aromaticity, aris-ing from a weakened p-π bond.

It is clear that theory of silicene is challenging and lessunderstood compared to graphene, because of a growingimportance of electron electron interaction and a soft c-axis displacement (puckering) degree of freedom, arisingfrom an easy sp3 mixing. Our model and theory is aimedto initiate new discussion, theoretical and experimentalstudies.

The present article is organized as follows. We inter-

arX

iv:1

309.

2242

v3 [

cond

-mat

.str

-el]

25

Sep

2016

2

pret certain existing theoretical results, Coulomb screen-ing argument for Mott transition and quantum chemicalinsights as providing support for our proposal of a nar-row gap Mott insulating state for neutral silicene. Wealso identify and discuss a set of about 6 anomalous ex-perimental results that point to a Mott insulating state.

A Heisenberg model, containing additional multispininteractions, is introduced to describe spin dynamics ina small gap Mott insulator, and a tJ model for the spin-charge dynamics of doped Mott insulator. Then we dis-cuss the aforementioned anomalous experimental resultsin the light of our model.

Prospects for high Tc superconductivity, within ourmodel, is discussed next. In view of larger and morefavourable tJ parameters, in comparison to layeredcuprates, there is a distinct possibility of a Tc approach-ing room temperature scale provided competing interac-tions are taken care of. Using existing theoretical workswe come to the conclusion that superconductivity is likelyto be a chiral spin singlet d + id superconductivity.

Superstructures, not common in graphene but ubiq-uitous in silicene grown on substrates, are interpretedas arising from a strong response of c-axis deformableMott localized p-π electrons, to substrate perturbations,through site dependent sp3 mixing. In a recent ab-initiocalculation with Vidya [38] we have we have found indi-rect evidence for Mott localization, through presence ofsizable Kekule (valence bond) order and a weak antiferro-magnetism. Further we have interpreted experimentallyseen reconstructions as Kekule or Valence Bond order.

The present paper assumes that tJ model in 2d de-scribes Kosterlitz-Thouless superconductivity for a rangeof doping. While there is no rigorous proof for this,it is well certified by a body of analytical, numericaland experimental efforts available for the square latticecuprates, ever since Bednorz Mullers discovery and thebeginning of RVB theory. Existing studies on honeycomblattice tJ model indicate that spin singlet superconduc-tivity continues to be a dominating phase over a range ofdoping, but with d + id order parameter symmetry.

Our present work is a natural extension of our earlierwork on graphene [39–41]. It has been our view thatgraphite and graphene should show interesting electroncorrelation effects, even though electron-electron interac-tion strengths are moderate compared to the band width.According to us, reduced 2-dimensionality of graphenecauses an amplification of electron correlation effects.Our earlier prediction of spin-1 collective mode in neu-tral graphene [40] and very high Tc superconductivity indoped graphene [39, 41] are based on use of a moderateelectron repulsion strength. Our main message in thepresent article is that 2d silicene and germanene, havinga third of graphene band width, but two thirds of on sitecoulomb interaction strength of graphene should exhibitmore pronounced electron correlation effects.

In a recent article we have suggested a Five fold way

to new high Tc superconductors [42] one of the ways isgraphene route. Silicene and germanene are most likelyto lie in the graphene route and help us achieve roomtemperature superconductivity.

IS FREE STANDING SILICENE STABLE ?

Beginning with hexasilabenzene, Si6H6, a silicon ana-logue of benzene, chemists have wondered about the ex-istence of stable planar p-π bonded Si based molecules[31, 43]. Their concern is that an increased Si-bondlength, a simple consequence of a 60 % increase in atomicradius, will weaken the p−π bond, leading to reducedaromaticity and increased chemically reactivity. It is alsoan experimental fact that free Si6H6 molecule has notbeen synthesized so far.

Silicene is an infinitely extended planar p-π bondednetwork of Si atoms. DFT calculations and beyond ar-gue for a stable free standing graphene. On the otherhand, Sheka in 2009 [29, 30] and Hoffman [33] in 2013,have questioned stability and very existence of free stand-ing silicene because of reduced aromaticity and enhancedradicalization. This, according to them, will make sil-icene react with any molecular dirt.

Radicalization in quantum chemistry is creation of un-paired lone electron in certain molecular orbitals in theground state; the loners are generically weakly coupledto other loners, if present. In the context of periodic sys-tems such as a crystalline solid we interpret it to mean anextreme Mott localization and formation of nearly decou-pled spins. In this Mott insulating state quantum fluctu-ations lead to residual (superexchange) couplings amongspins quantum magnetism.

Free standing silicene has not synthesized experimen-tally so far. A key stabilizing factor, namely metallicsubstrates, Ag, Ir and ZrB2 is needed. Further, a stronghybridization between Ag bands and Si orbitals at thefermi level leads to significant modification of electronicproperties of free standing silicene.

In what follows, by stable Mott insulating single layersilicene we mean the following. We assume that (metal-lic) substrate stabilizes a single layer silicene and at thesame time does not significantly modify the Mott ordoped Mott insulator character, that we are after.

MOTT INSULATING SILICENE -THEORETICAL SUPPORT

In this section, using recent estimates of Hubbard U,nearest neighbour repulsion V and Mott’s argument formetal insulator transition invoking screened long rangecoulomb, we will argue a Mott insulating ground statefor silicene.

3

FIG. 1: Figure 1. ARPES data of silicene on ZrB2 (reproduced from reference [11] with permission). A hole like narrow bandof width (marked X2), below a gap of 0.3 eV at K point is seen in figure 1a . This band is only 1 eV wide, compared to 6 eVπ − π∗ band width given by LDA. It has appreciable spectral weight only in one third of the Brilluoin zone. We interpret thisband as a strongly renormalized hole band of a Mott insulator. Interestingly all silicene bands X1, X2 and X2’ lie within anenergy interval of 0.5 eV. S’s refer to substrate bands.

We begin with a summary of basic quantum chemistryand band theory results for silicene. Silicon, located justbelow carbon in the periodic table has a larger atomic ra-dius ∼ 1.17 AA, in contrast to a smaller value ∼ 0.77 Afor carbon. This leads to a significant 3p-π bond stretch-ing : Si-Si distance is ∼ 2.3 A, while C-C distance is only1.4 A in graphene.

Unlike graphene, Si allows a small sp3 admixture to sp2

bonding, resulting in puckered sigma bonds. That is, Siatoms of the two triangular sublattices of the honeycombnet undergo a small and opposite displacements, leadingto a net c-axis stretching∼ 0.4 Au. A reduced 3s-3p levelseparation in Si atom, compared to 2s-2p level separationin C atom encourages a small sp3 hybridization.

Electronic structure calculations [4–6] predict asemimetal band structure qualitatively similar tographene, containing two Dirac cones at K and K’ points.A major differences from graphene is a 3 fold reductionin the π − π∗ band width.

Following a pioneering work of Sorella and Tosatti [25]a recent estimate [24] gives an accurate value of the crit-ical value U

t ∼ 3.8, for metal to Mott transition in Hub-bard model on Honeycomb lattice. In a very recent manybody theory Schuler et al. [28] estimate a value U

t ∼ 4.1for silicene. This puts silicene on the Mott insulator side,close to the phase transition point. However, Schuler etal. argue that inclusion of nearest neighbour repulsionVt ∼ 2.31, will reduce U to an effective U ≈ U - V and

bring silicene back to a semi metallic state like grapheme.In what follows we argue, on the contrary, that presenceof finite V will hasten Mott localization and reinforce aMott state through a first order phase transition.

We start with Mott’s argument for a first order Motttransition. Mott begins by asking whether a screenedcoulomb interaction present in the metallic state is suf-

ficient to form a quantum mechanical bound state of anelectron and the hole it left behind at its home site, at thefermi level. Within a Hubbard model idealization thishappens when band width becomes comparable to onsiteU; it also means that at very large U every site bindsa lone daughter electron. In Mott’s argument the tran-sition is pre-empted by reduced screening of long rangeinteraction (as we loose free carriers by bound state for-mation), through a feed back resulting in a first orderphase transition. This aspect is not contained in thesimple Hubbard model.

Let us assume that in addition to U we have a non-zero nearest neighbour coulomb interaction V, which is aleading term in the long range part of screened coulombinteraction. According to Mott’s arguments, V will addto the already existing on site attraction U between anelectron and the hole it left behind. This additional at-traction (widening of the potential well) decreases kineticenergy of bound state. Resulting increase in binding en-ergy hasten bound state formation. Roughly, the onsiteHubbard U gets enhanced: U ≈ U + αV, where α ∼ 1.

Thus we conclude, by what we believe to be a correctuse of important estimates of Schuler et al. [28] that sil-icene is in the Mott insulating side of the metal insulatortransition point. We hope to address this point in somedetail in a separate paper.

MOTT INSULATING SILICENE -PHENOMENOLOGICAL SUPPORT

In this section we briefly review experimental resultsin silicene which are anomalous, from the point of view ofa Dirac metal. But, as we will discuss latter, they seemnormal from Mott insulator point of view.Doped Hole and a Narrow Band: In a work that

4

has received a wide attention, using ARPES, Vogt et al.[9] show presence of a Dirac cone dispersion over a wideenergy range below Fermi level, with a vF comparable tothat in graphene. There is an ongoing debate [44–46] onwhether this is primarily a silicene p-π band or primarilya Ag metallic bands or a strong hybrid. Recent ARPESresult shows [47] multiple (more than the expected dou-ble) Dirac cones in the silicene on Ag system. This hasbeen attributed to a strong hybridization of silicene andsilver states at the fermi level.

On the other hand, ARPES study of silicene grownon ZrB2 by Fleurence et al. [11] shows a very differ-ent behaviour. ZrB2 is a low carrier density metal withsmall Fermi pockets. It is expected to have less electronicinfluence on silicene. Fleurence et al. find a remarkablynarrow band around around K and K’ points (marked X2

in figure 1b), lying below a finite gap ∼ 0.3 eV. Its spec-tral weight vanishes in two-third of the Brillouin zone.An extrapolation from the shape of the visible part ofthe band gives us a total band width ∼ 1 eV. This isto be contrasted with the total π − π∗ band width of 6eV as given by LDA calculations. Thus there is a bandwidth renormalization by a factor of 10. Further 3 sil-icene bands denoted by X1, X2 and X2’ all lie in an energyinterval of 0.5 eV. This spectrum is reminiscent of holespectrum in Mott insulating La2CuO4 .

A Superconducting Gap Anomaly: Chen et al.have observed [13] a superconductor like gap structure intheir STS study of silicene on Ag and a Tc ∼ 35 to 40 K.They provide arguments in support of a gap arising fromsuperconductivity. A large 2∆

kTc∼ 20 makes it anomalous.

Further experiments are needed to substantiate this im-portant result.

Absence of Landau Level formation: An STMwork [14] in the presence of a magnetic field as large as7 Tesla, do not find an expected Landau level splitting.In our view, a doped hole looses nearly all its quasipar-ticle weight because of the strong electron-electron in-teraction. We find that quasiparticle line broadening,as inferred from experiment is larger than the expectedLandau level splitting, making it invisible in STS mea-surements. In graphene, because of quasi particle coher-ence, one sees Landau level structures for similar mag-netic fields in the STS measurements.

Silicene Lattice Reconstructions : Recent experi-ments on silicene grown on metal surfaces exhibit [35–37]lattice reconstructions,

√3×√

3,√

7×√

7,√

13×√

13,4 × 4 etc. They are temperature dependent and some ofthem exhibit finite temperature phase transitions. Fur-ther, reconstructions are accompanied by space depen-dent sp3 mixing and consequent bond length modulation.Even within band theory there is no Fermi surface. Soa standard route for density wave instabilities namelyFermi surface nesting is absent. This explains why re-constructions or CDW order is not seen in graphene.Graphene is stiff, inview of a ∼ 9 eV wide filled π-band

and an associated large aromaticity. We suggest thatp-π electrons in silicene, because of Mott localization, re-spond strongly to substrate perturbations by making useof the soft sp3 hybridization option and correspondingc-axis displacements.Fermi Arc like Electron Pockets: Electron doping

in silicene deposited on ZnB2 by alkali metal leads to ap-pearance of small Fermi arc or pockets like features atthe M points [15]. This is not easily explained withinLDA band structure result. A doped Mott insulator onthe other hand, does not obey Luttinger theorem in theusual fashion and can have unusual emergent Fermi pock-ets and Fermi arcs - a striking example is the under dopedcuprates.Silicene Transistor and Resistance Oscillations

Using an innovative technique a silicene transistor hasbeen successfully fabricated [16]. It exhibits a desired onoff resistance radio as gate voltage is varied. Further,it exhibits an intriguing resistance oscillation at roomtemperatures, resembling a Fabry-Perot type of quantuminterference.

Thus we have some theoretical arguments and a fewphenomenological facts in support of a Mott insulatorpicture for neutral silicene. In the following we will builda model, address the above experimental observationsfrom the Mott insulator point of view.

SPIN LIQUID STATE IN SILICENE ?

Let us first discuss nature of the hypothesized Mottinsulating state of neutral silicene. From ARPES exper-iment [11] we estimate a Mott Hubbard gap of ∼ 0.6 eV.In transition metal oxide and organic Mott insulators,Mott gap is often comparable to band widths. As theinferred Mott gap, 0,6 eV is only a tenth of the totalπ − π∗ band width ∼ 6 eV silicene is a Mott insulatorwith a small charge gap.

In a Mott insulator low energy degree of freedom arespins and spin-spin interaction arise from super (kinetic)exchange processes. One can estimate J, using the stan-

dard expression, J ≈ 4t2

U2 . The values, t ∼ 1.14 eV and aneffective U ∼ 5 eV, discussed earlier for silicene, gives us aJ ≈ 1 eV. As J is comparable to the Mott Hubbard gap of0.6 eV, a strong virtual charge density and charge currentfluctuations will renormalize J to lower values. Further,higher order cyclic exchanges will be also present.

Ignoring spin-orbit coupling for the moment, our effec-tive spin Hamiltonian in the Mott insulating state is aspin-half Heisenberg Hamiltonian

Hs = J∑〈ij〉

(Si · Sj −1

4) + 4 & 6 spin terms (1)

If the nearest neighbour J dominates we will have longrange antiferromagnetic order in the ground state. As

5

recent results show, within nearest neighbour repulsiveHubbard model one is unlikely to stabilize a spin liquidphase [26]. In the present case, because of softness ofpuckering, there may be valence bond order, rather thana spin liquid, without doping. However, as in cuprates,this may not be an important practical issue. This isbecause we expect that even a small density of dopantsthrough their dynamics will destroy long range AFM or-der and valence bond order and stabilize some kind ofspin liquid state (pseudo gap phase) containing incoher-ent dopant charges. It is this doping induced spin liq-uid state that will determine nature of superconductivityover a range of doping. It is likely that a variety of spinliquids are around the corner.

DOPPED MOTT INSULATOR

For dopant dynamics in a Mott insulator, upto cer-tain range of doping, a projective constraint arising fromupper and lower Hubbard band formation and survivingsuperexchange are important. Thus the relevant modelfor doped Mott insulator is the tJ model:

HtJ = −t∑〈ij〉

(c†iσcjσ +h.c.) +J∑〈ij〉

(Si ·Sj −1

4ninj) (2)

with the local constraint ni↑ + ni↑ 6= 2 or 0, for hole andelectron doping respectively. The value of J, t ∼ 1 eV.

It is known from the study of single hole dynamics inMott insulating cuprates that a free and coherent prop-agation of a doped hole or electron (carrying its chargeand spin) is frustrated by the strongly quantum entan-gled spin background. This leads to a significant bandnarrowing and loss of spectral weight over a large part ofthe Brillouin zone, as seen in ARPES experiments [48]and tJ model calculations [49]. For cuprates band theorygives a width of 8t ≈ 3 to 4eV, while ARPES results inthe Mott insulating cuprates give a band width of 2J ∼0.3 eV (essentially spin wave band width) for holes. Thisis a substantial, ten fold reduction of hole band width.

As we saw earlier ARPES study on silicene grown onZrB2 gives a parabolic hole band at K point, about 0.3eV below the Fermi level (band X2 in figure 1 b) witha band width ∼ 1 eV. Comparing results with cupratesmentioned above, we get a J ∼ 0.5 eV. This is in the rightball park, as our estimate of J ∼ 1 eV.

DOPED MOTT INSULATOR ANDSUPERCONDUCTIVITY

We start with tJ model for our doped Mott insulator ina honeycomb lattice and discuss superconductivity. In-terestingly, the above model (equation 2) was studied ina mean field approach first in [50] for graphite. Later the

Spin

Density

Wave

d + id

Superconductivity

Pseudogap

Phase

0.25.. .

0.1 0.2

doping x

0.0.

T

Rec

onst

ruct

ions

Str

uct

ura

lIn

sula

tor

Mott

Anomalous

Metal

FIG. 2: A schematic phase diagram for doped silicene andgraphene.

present author independently studied the same model ig-noring onsite constraints [39] as a semi microscopic way ofincorporating Pauling’s spin singlet (resonating valencebond) correlations in planar graphitic systems, within aband theory approach. It leads to a prediction of veryhigh Tc superconductivity around an optimal doping.Our meanfield theory result of high Tc superconductiv-ity for graphite was reanalyzed and confirmed by An-nica Black-Shaefer and Doniach [51]. They further founda remarkable spin singlet chiral superconducting state,namely a state with d + id symmetry as the most stablemean field solution for the same range of doping. A moreconservative repulsive Hubbard model has been studiedfrom superconductivity point of view using variationalMonte Carlo method in reference [41], by us and collab-orators for graphene. There are other important worksaddressing the same issue[52–57] by different methods.They all support the possibility of d + id chiral spin sin-glet superconductivity.

A very recent work [60] studies tJ model on a hon-eycomb lattice using a new and powerful variational ap-proach using Grassman Tensor Network states. It con-firms d + id superconductivity for tJ model on honey-comb lattice for a range of doping. Using available theoryand insights we suggest a phase diagram (figure 1), qual-itatively similar to cuprates, including the pseudo gapphase. A new aspect for the honey comb lattice is pres-ence of a van Hove singularity and an associated nesting(figure 1) at a doping of x = 0.25. It has been sug-gested that this nesting might stabilize a complex mag-netic order [61] or charge density wave order. If our vari-ational calculation [41] performed for graphene, a doped

6

semimetal described by intermediate repulsive U Hub-bard model, is any guidance, maximum superconductingTc occurs already around 15 percent, somewhat similarto cuprates.

The issue of scale of superconducting Tc is very excit-ing and it shows some promise. As mentioned earlier wehave an unusually large J ∼ 1 eV and t ∼ 1 eV for ourhoneycomb lattice. This is to be contrasted with a valueof J ∼ 0.15 eV and t ∼ 0.25 eV for the square latticecuprates, known for their record Tc ∼ 90◦ K, for for sin-gle layer cuprates. We have an average 4 fold increase int and J for silicene. Does silicene offer a 4 fold increase inTc ? Taking care of lattice structure difference betweencuprate and silicene crudely, a scaling gives at least 3fold increase of Tc. Thus room temperature scales forTc seems within reach, provided competing orders aretaken care of .

As in cuprates, we expect competing orders, chargeand spin stripes to challenge the high Tc superconductingstates. Further, as we will see, electron lattice interactioncan cause patterns of vertical displacements of Si atoms,arising from sp3 mixing, It is conceivable that there willbe stripes and 2d patterns, corresponding to various va-lence bond orders. These valence bond localization ten-dencies will compete and reduce Tc significantly.

The only available, but remarkable indicator for super-conductivity in silicene is an STS study [13]. They finda Tc in the range 30 - 40 K and a large gap of 35 meV;that is, an anomalous value, 2∆

kBTc≈ 20. If we take the

maximal gap value from this experiment and extract amean field Tc, using weak coupling BCS result, we geta number close to 200 K ! We interpret this anomaly in

2∆kBTc

, as due to a superconducting state with a large lo-cal pairing gap but with Tc reduced by a strong phasefluctuations arising from a static or low frequency localcompeting orders. This needs to be investigated theoret-ically and experimentally further.

We have interpreted [62] a quantum oscillation in resis-tance seen in silicene transistor at room temperatures, asa Fabry-Perot type of interference of preformed bosoniccharge 2e Cooper pairs, and physics similar to the pseudogap phase of cuprates. Briefly, the gate induced carri-ers (holons or doublons) enable delocalization and inter-ference of preformed charge 2e singlets, even before fullphase coherence and superconductivity develops.

There are recent suggestions of superconductivity insilicene based on different mechanisms [63, 64]. We wishto point out that Liu et al. [63] have theoretically studiedpossibility of spin fluctuation induced chiral d + id super-conductivity in neutral bilayer silicene. In their mecha-nism, intralayer hopping between the two Dirac metalliclayers produce small electron and hole pockets aroundK and K’ points. In our theory such a bilayer will re-main insulating with an additional feature of a spin gapinduced by a strong interlayer exchange coupling (spinsinglet bond formation between Si atoms along c-axis).

External doping or gate doping by very strong electricfields applied perpendicular to the layers will be neededto create superconductivity in the Mott insulating bilay-ers.

GERMANENE AND STANENE

Group IV elements in periodic table C, Si, Ge, Snand Pb have increasing atomic radii 0.77, 1.17, 1.22,1.45 and 1.8 A respectively. The p-π bonds in silicene,germanene and stanene get weaker because of increasingatomic radii, in spite of increasing size of the p orbitals.Ab-initio calculations for germanene leads to a few per-cent enhancement of Ge-Ge bond length, but a substan-tial increase of puckering and c-axis stretching from ∼0.4 A in silicene to ∼ 0.6 A in germanene. There is asmall reduction in π and π∗ band widths from 3 to 2.5eV.

All things being quantitatively nearly equal at the levelof model, germanene should be a narrow band Mott in-sulator. Consequences we have discussed so for, includ-ing possibility of quantum spin liquid and high Tc su-perconductivity on doping should be anticipated. Thereare interesting experimental developments with respectto germanene and stanene recently [17].

SUMMARY AND DISCUSSION

To summarise, we have hypothesized that silicene andgermanene are narrow gap Mott Insulators. This chal-lenges the widely held belief that they are a Dirac metal,like graphene. In making our proposal we have reliedheavily on a synthesis of theory and phenomenology. Wehave used ARPES, STM and recent conductivity results.Mott’s arguments, quantum chemical insights and exten-sive theoretical study of graphene and silicene has helpedus in our proposal. Using known results we discussed var-ious solutions for our model, including spectral functionof a hole in a Mott insulator, absence of Landau levelsplitting, superconductivity and so on. Unconventionalsuperconducting order namely d + id and high Tc’s wasalso discussed.

An important support for Mott localization comesfrom our translation of the quantum chemical results ofSheka [29, 30] to solid state terminology. Sheka stud-ies nano silicene clusters and finds an extensive radical-ization. Radicalization is extracted using a procedurethat uses spin polarized (antiferromagnetic) ab-initio so-lutions. This interesting quantum chemical approxima-tion is less known in solid state context. As indicatedearlier we equate radicalization in finite systems to Mottlocalization in extended solid state context. Thus Sheka’sfinding of extensive radicalization and alleged unstable

7

silicene is a conservative evidence for narrow gap Mottinsulator formation and strongly exchange coupled spins.

We have ignored spin-orbit coupling in the present pa-per, simply to focus on the key physics of strong corre-lations. Spin orbit coupling will play its own, importantunique role, some what different from standard band in-sulator or semi metal. Within the context of semi metal-lic graphene interesting effects of spin orbit coupling arebeing studied [66–68]

Direct and indirect methods should be used to unravelan underlying Mott insulator. Optical conductivity, σ(ω)measurement is an urgent one. It should focus on findingsignatures on Hubbard band features. It will be inter-esting to confirm the existing claim of superconductinggap in STS measurements [13]. Reconstructions shouldbe studied carefully to distinguish valence bond densitywave and plaquette resonance density waves. Pseudo gapphysics needs to be explored using NMR and NQR mea-surement. Our tantalizing prediction of very high Tc su-perconductivity reaching room temperature scales needsto be explored. This needs ways of understanding andovercoming unavoidable competing instabilities such asspin and charge stripes.

Next major experimental and theoretical challenge isto see whether a free standing silicene exists and if it canbe synthesized and studied.

I am not aware of any real material for which tJ modelparameters are as large as we have suggested for siliceneand germanene,t, J ∼ 1 eV. So we consider silicene andgermanene as forefront materials in the race for roomtemperature superconductivity.

Acknowledgement I thank Prof. Yamada-Takamurafor giving permission to reproduce a figure from [11]; DrAyan Datta for a discussion; Dr Kehui Wu and colleaguesfor informative discussions at the Silicene meet at Bei-jing in 2014. I thank Science and Engineering ResearchBoard (SERB), Government of India for the SERB Dis-tinguished Fellowship. Additional support was providedby the Perimeter Institute for Theoretical Physics. Re-search at Perimeter Institute is supported by the Govern-ment of Canada through the Department of Innovation,Science and Economic Development Canada and by theProvince of Ontario through the Ministry of Research,Innovation and Science.

[1] K. S. Novoselov et al., Nature (London), 438 197 (2005)[2] A. K. Geim and K. S. Novoselov, Nat. Mater., 6 183

(2007)[3] A. H. Castro-Neto et al., Rev. Mod. Phys. 81 109162

(2009)[4] K. Takeda and K. Shiraishi, Phys. Rev. B 50 14916

(1994)[5] G. G. Guzmn-Verri and L. C. Lew Yan Voon, Phys. Rev.

B 76 075131 (2007)

[6] S. Cahangirov et al., Phys. Rev. Lett., 102 236804 (2009)[7] A. Kara et al., Surface Science Reports, 67 1 (2012)[8] P. De Padova et al., Appl. Phys. Lett. 96 261905 (2010)[9] P. Vogtet et al., Phys. Rev. Lett. 108 155501 (2012)

[10] L. Chen et al., Phys. Rev. Lett., 109 056804 (2012)[11] A. Fleurence et al., Phys Rev Lett 108 245501 (2012)[12] L. Meng et al., Nano. Lett., 13 685 (2013)[13] L. Chen et al., App. Phys. Lett., 102 081602 (2013)[14] C.L. Lin et al., Phys. Rev. Lett. 110 076801 (2013)[15] R. Friedlein et al., App. Phys. lett., 102 221603 (2013)[16] L. Tao et al., Nature Nanotechnology, 10 227 (2015); G.

Le Lay, Nature Nanotech. 10 202 (2015)[17] L. Li et al., Adv. Mater. 26, 4820 (2014); P. Bampoulis,

P. et al., J. Phys. Condens. Matter 26 442001 (2014);Derivaz, et al., Nano Lett. 15, 2510 (2015); A. Acun etal., J. Phys.: Condens. Matter 27 443002 (2015); Zhu,F. et al., Nature Mat. 14 1020 (2015); M.E. Davila andG. Le Lay Scientific Reports, DOI: 10.1038/srep20714(2016)

[18] J.G. Bednorz and K.A. Mller, Z. Physik, B 64 189(1986).

[19] P.W. Anderson, Science, 235 1196 (1987)[20] G. Baskaran, Z. Zou, P.W. Anderson, Sol. St. Commn.,

63 973 (1987)[21] G. Baskaran and P.W. Andersdon, Phys. Rev. B 37 580

(1988)[22] G. Baskaran, Iranian J. Phys. Res., 6 163 (2006)[23] X.G. Wen, Quantum Field Theory of Many-body Sys-

tems (Oxford Graduate Texts, 2007)[24] Z. Y. Meng et al., Nature 464 847 (2010)[25] S. Sorella and E. Tosatti, Europhys. Lett., 19 699 (1992)[26] S. Sorella, Y. Otsuka and S. Yunoki, Sci. Rep., 2012; 2:

992.[27] S.R. Hassan and D. Senechal, Phys. Rev. Lett., 110

096402 (2013)[28] M. Schuler et al., Phys. Rev. Lett., 111 036601 (2013)[29] E.F.Sheka, arXiv:0901.3663[30] Sheka, E. F., Int. J. Quantum Chem., 113 612 (2013)[31] A. Grassi et al., Chemical Physics, 297 13 (2004)[32] N. H. March and A. Rubio, Journal of Nano materials,

Article ID 932350, doi:10.1155/2011/932350 (2011)[33] R. Hoffmann, Ang. Chem. (Int. Ed.), 52 93 (2013)[34] D. Jose and A. Datta, J. Phys. Chem., C 116 24639

(2012)[35] C. L. Lin et al., Applied Physics Express 5 045802 (2012)[36] Z Majzik et al., J. Phys. Condens. Matter., 25 225301

(2013)[37] L Chen al., Phys. Rev. Lett., 110 085504 (2013)[38] R. Vidya and G. Baskaran, (unpublished)[39] G. Baskaran, Phys. Rev. B 65 212505 (2002)[40] G. Baskaran and S. A. Jafari, Phys. Rev. Lett., 89 016402

(2002)[41] S. Pathak, V. B. Shenoy, and G. Baskaran, Phys. Rev.,

B 81 085431 (2010)[42] G. Baskaran, Pramana - Journal of Physics, 73 61 (2009);

available at:http://www.ias.ac.in/pramana/v73/p61/fulltext.pdf

[43] D. A. Clabo and H. F. Schaefer, J. Chem. Phys. 84 1664(1986)

[44] Y.P. Wang, H.P. Cheng, Phys. Rev. B 87 245430 (2013)[45] Z.X. Guo et al., J. Phys. Soc. Jpn., 82 063714 (2013)[46] S. Cahangirov et al., Phys. Rev. B 88 035432 (2013)[47] Ya Feng et al., arXiv:1503.06278[48] T. Tohyama and S. Maekawa, Supercond. Sci. Technol.,

8

13 R17 (2000)[49] B. O. Wells et al., Phys. Rev. Lett. 74 964967 (1995)[50] T. C. Choy and B. A. McKinnon, Phys. Rev. B 52 14539

(1995)[51] A. M. Black-Schaffer and S. Doniach, Phys. Rev. B 75

134512 (2007)[52] C. Honerkamp, Phys. Rev. Lett.100 146404 (2008)[53] W. S. Wang et al., Phys. Rev. 85 035414 (2012)[54] M.L. Kiesel et al., Phys. Rev. B 86 020507 (2012)[55] R. Nandkishore, L. S. Levitov, and A. V. Chubukov, Na-

ture Physics, 8 158 (2012)[56] M. M. Scherer, S. Uebelacker, and C. Honerkamp, Phys.

Rev., B 85 235408 (2012)[57] J . Vucicevic, M. O. Goerbig and M. V. Milovanovic,

Phys. Rev., B 86 214505 (2012)

[58] T. Ma et al., Phys. Rev. B 84 121410(R) (2011)[59] Roy, B. and Herbut, Phys. Rev.B 82 035429 (2010)[60] Z.G. Gu et al., Phys. Rev. B 88 155112 (2013)[61] Tao Li, Euro phys. Lett., 97 37001 (2011)[62] G. Baskaran (Unpublished)[63] F. Liu et al., Phys. Rev. Lett., 111 066804 (2013)[64] W. Wan et al., Eur. Phys. Lett., 104 36001 (2013)[65] G. Baskaran, Phys. Rev. Lett. 90 197007 (2003)[66] C.C. Liu, H. Jiang and Y. Yao, Phys. Rev., B 84 195430

(2011)[67] M. Ezawa, New J, Phys., 14 033003 (2012)[68] M. Ezawa, Y. Tanaka and N. Nagaosa, Sci. Rep., 3 2790

(2013)