On the mechanism of room temperature superconductivityin substitutionally doped graphene

K.P. Sinha b,c, Apoorv Jindal a

a Birla Institute of Technology & Science, Pilani, Indiab Department of Physics, Indian Institute of Science, Bangalore-560012, Indiac Indian National Science Academy, New Delhi, India

a r t i c l e i n f o

Article history:Received 4 July 2013Received in revised form30 October 2013Accepted 7 November 2013by Y.E. LozovikAvailable online 22 November 2013

Keywords:A. GrapheneB. Boron dopingC. Phonon and bond-polarizationmechanismD. High-temperature superconductivity

a b s t r a c t

A combined mechanism involving phononic and electronic processes is suggested for superconductivityin substitutionally doped graphene. The electronic mechanism is similar to the one used for dopedfullerene system, MxC60 (M¼K, Rb, etc.) and triggered by bond polarization due to doped impurities suchas B or Al. It is found that on increasing the doping, the superconducting critical temperature can beraised to room temperature.

The details of the combined model are given along with the predicted values of TC.& 2013 Elsevier Ltd. All rights reserved.

The discovery of graphene (in 2004) has sparked tremendousresearch activity in the field of condensed matter physics [1,2]. It isa single atom thick layer of hexagonal rings of carbon atoms with ahoneycomb like structure. The interesting property of this materialis that it combines features of semiconductors and metals. Themobility of charge carriers is much higher than those in silicon.There is an expectation of producing high temperature super-conductivity in substitutionally doped graphene. This is motivatedfrom the situation in fullerenes (carbon cluster balls, molecularstructure C60) which become superconducting at a critical tem-perature TC equal to 18 K and 28 K on doping with alkali metals,M¼K or Rb, respectively [3]. These molecular systems becomesimilar to inorganic cuprates as high-temperature superconduc-tors (HTS). A combined phononic and electronic mechanism hasbeen suggested for the pairing in MxC60 and cuprates [3–5]. In fact,very high TC superconductors observed in Pd–[H(D)]x systemshave been accounted for by similar mechanisms [6].

In this paper, we explore the possibility of producing very high(almost room temperature) TC superconductors in substitutionallydoped graphene. To achieve this, some carbon atoms in grapheneshould be substituted by B, Al or N, in order to create a band-gap in

an otherwise zero band-gap structure. Doping with B or Al ispreferred so as to have holes as carriers.

As done in earlier papers, a combined phononic and electronicmechanism is suggested for pairing in graphene substitutionallydoped with boron. The s C–C bonds are less prone to polarizationas compared to the π C–C bonds. Accordingly, we consider thepolarization of π bonds. Although the C¼C π bond is energeticallymore favorable, the ionic Cþ–C�þC�–Cþ bonds will also haveappreciable weightage on doping. As the spin of the pairs in C¼C,Heitler–London bond and the onsite local pairs (Cþ–C� , C�–Cþ)remain in the singlet state, we have the two-level Bose system ateach such bond [7,8].

The Hamiltonian of the system for the above model can bewritten as

H¼∑Ekcþkscksþ∑ℏΩpb

þj bjþ∑Pjðcþk1s

ck2sbjþh:c:ÞþHphþ He�phþ Hc

ð1Þwhere, Ek is the single particle energy of electrons in the conduc-tion band with cþks; cks denoting the usual creation and annihila-tion operators in the state |ks⟩, k is the wave vector and s is thespin index. The second term in the Hamiltonian is the energyoperator for the two-level Bose system with bþ

j ; bj representingthe creation and annihilation operators for the bond j and

ℏΩp ¼ EðCþ–C� ;QjnÞ�E C¼ C;Qj

� �, is the energy difference

between the ionic and the covalent energy configuration. Qj* and

Qj denote the lattice configuration of the two states. It is estimated

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/ssc

Solid State Communications

0038-1098/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ssc.2013.11.009

E-mail addresses: kritpsinha@gmail.com (K.P. Sinha),jindal.apoorv0810@gmail.com (A. Jindal).

Solid State Communications 180 (2014) 44–45

to range between 0.5 and 1.8 eV. For our calculations we will takeit to be 1 eV.

The third term in H denotes the interaction between theconduction electrons and the two-level Bose system with Pj beingthe coupling constant. A rough estimate of |Pj| is obtained fromthe expression e2rb/R2, where e is the elementary charge,rb�1.42 Å (bond length in hexagon) and R is the distance fromthe center to the end point of hexagon. For graphene, R�1.42 Å.With this choice of parameters, the value of Pj turns out to be10.14 eV. The terms Hph and He–ph denote the phonon andelectron–phonon interaction Hamiltonian, respectively. Theirexplicit forms are not written here as they are well known [9].

Rewriting the Hamiltonian in pseudo-spin formalism withspin¼1/2, we have

H ¼HoþHe�plþ… ð2Þwhere

Ho ¼∑Ekcþkscksþ ∑ℏΩpS

zj ð3Þ

and

He�pl ¼∑Pjðcþk1sck2sS

þj þh:c:Þ ð4Þ

The consolidated form of the Hamiltonian turns out to be

H ¼∑Ekcþkscks� ∑ðVphþ Vbp� VscÞcþk↑cþ�k↓c�k↓0 ck↑0 ð5Þ

where Vph is the usual phonon-induced interaction and Vsc is thescreened Coulomb repulsion between conduction electrons.

Vbp ¼ xjPjj2ℏΩpx

* +ð6Þ

is the interaction involving exchange of polarization modes ofexcitation and de-excitation of the two level system and x is thenumber of bonds polarized.

The expression for the superconducting critical temperature, TCtakes the consolidated form,

TC ¼ 1:14Wn1phW

n2bpexp½�ðλphþλnbpÞ�1� ð7Þ

where, n1¼λph (λphþλnbp)�1, n2¼λnbp (λphþλnbp)�1, λnbp¼λbp�μn, μn

denotes the screened Coulomb repulsion. Further,λbp¼N(0)Vbp, λph¼N(0)Vph; N(0) is the density of states at the

Fermi energy. The density of states is related reciprocally to theconduction band width which is of the order of 0.5 eV for thesesystems. Thus, N(0) will be in the range in 1–2 states eV�1.

In temperature units, Wph¼80 K. Wbp (half band-gap energy intemperature units) changes according to the doping concentrationin graphene [10] and is given in Table 1.

Further μn¼0.1 and λph¼0.25. The value of λbp is dependentupon Vbp which in turn depends upon x (Eq. (6)). Therefore, λbpvaries linearly with x,

λbp¼102.8196x; taking N(0)¼1 eV�1.For each doping concentration, there would be an inherent

bond polarization present. Taking this value, we get our λbp andhence, we calculate the TC (Table 2).

It is seen that critical temperatures for superconductivity reachas high as 342.71 K (�70 1C) for a doping concentration of 2%.

The above model shows that as in the case of cuprate super-conductors, a combined phononic and electronic mechanism [5,6,11]can lead to very high TC superconductivity and is capable to explainother features of the system. There is considerable interest in produ-cing room-temperature superconductivity at present [12]. The modelpresented in this paper seems capable of fulfilling the goal.

Acknowledgments

K.P. Sinha would like to thank Indian National Science Academyfor their support during this work. A. Jindal extends his gratitudeto the IASc-INSA-NASI Summer Research Fellowship Program forproviding him the opportunity to work at IISc, Bangalore andproviding him a comfortable stay.

References

[1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos,I.V. Gregoreiva, A.A. Firsov, Science 306 (2004) 666.

[2] A.K. Geim, K.S. Novoselov, Nat. Mater. 6 (2007) 183.[3] K.P. Sinha, Solid State Comm. 83 (1992) 291. (and references therein).[4] K.P. Sinha, Solid State Comm 79 (1991) 735.[5] K.P. Sinha, Mod. Phys. Lett B12 (1998) 805.[6] K.P. Sinha, Natl. Acad. Sci. Lett. 29 (2006) 125.[7] K.P. Sinha, Physica B163 (1990) 664.[8] R. Micnas, J. Rannninger, S. Robaszkieuicz, Rev. Mod. Phys. 62 (1990) 113.[9] J.R. Schreiffer, Theory of superconductors, Benjamin/Cummings, Reading, MA,

1964.[10] Pooja Rani, V.K. Jindal, Designing band gap of graphene by B and N dopant

atoms, arXiv 1209.5228, 2012.[11] A.F. Andreev, JETP Lett. 79 (2004) 88.[12] Marouchkine Andrei, Room temperature superconductivity, Cambridge Inter-

national Science Publishing, 2004.

Table 1Variation of Wbp with substitutional doping concentration of Boron.

Boron dopingconcentration in graphene (%)

Wbp (K)

0.5 2031 4061.5 6092 812

Table 2Variation of critical temperatures for substitutionally doped graphene according todoping concentrations.

Doping concentration of Boron (%) Inherent bond polarization (x) TC (K)

0.5 0.004 25.731 0.007 91.901.5 0.011 214.022 0.014 342.71

A. Jindal, K.P. Sinha / Solid State Communications 180 (2014) 44–45 45