Table Of ContentIntra-valley Spin-triplet p+ip Superconducting Pairing in Lightly Doped Graphene
Jianhui Zhou and Tao Qin
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
Junren Shi
International Center for Quantum Materials, Peking University, Beijing 100871, China
Weanalyzevariouspossiblesuperconductingpairingstatesandtheirrelativestabilities inlightly
3 doped graphene. Weshow that, when inter-sublattice electron-electron attractive interaction dom-
1 inates and Fermi level is close to Dirac points, the system will favor intra-valley spin-triplet p+ip
0 pairing state. Based on the novel pairing state, we further propose a scheme for doing topological
2 quantumcomputation in graphene by engineering local strain fields and external magnetic fields.
n
PACSnumbers: 74.20.-z,73.22.Pr,03.67.Lx
a
J
1
Graphene has attracted great experimental and theo- E E
reticalinterestssinceitssuccessfulfabrication.[1,2]The
]
n uniqueelectronicstructureofgraphene,characterizedby
o massless relativistic particle-like dispersion near two in- Q Q Q Q
0 0 0 0
c − −
- equivalent corner points of Brillouin zone, gives rise to O ky O ky
r exotic physical properties. [3] One of interesting ques-
p
u tions would be: how electrons could be paired in such a kx kx
s system? There had been many theoretical studies that
t. explore the possibility of the superconductivity in sin- (a) (b)
a
gle/multilayergraphenes. Inparticular,Ref.4predictsa
m
spinsingletp+ippairingstateinthepresenceofeffective Figure 1: Two scenarios of pairings: (a) intervalley pairing,
- attractive e-e interaction between the nearest neighbor- (b) intravalley pairing. Q0=(0,4π/(3√3a)).
d
n ing sites. More exotic possibilities such as chiral s-wave,
o d-waveandf-wavepairingsarealsopredicted.[5–9]First
e e interaction has the form:
c principlescalculationhassuggestedthatsuperconductiv- −
[
2 icteyntcloyu,lsdubpeericnodnudcuecdtibvyitdyohpainsgbeaelknarlienaelizaeddationmpso.t[a1s0s]iuRme-- Hint = g20 Xha†isaisa†i−sai−s+b†isbisb†i−sbi−si
v doped few-layer graphene. [11] i,s
36 +g1 X Xa†isaisb†js′bjs′, (1)
<i,j>ss′
9 In this Brief Report, we investigate possibility of ob-
2 taining intra-valley spin-triplet p + ip superconducting whereg0 andg1 denotethe strengthofonsiteinteraction
.
5 state (see below) in lightly doped graphene. Unlike the and interaction between the nearest neighboring (NN)
20 spin-singlet p+ip pairing state, [4] the intra-valley spin sites, respectively. a†is (b†is) and ais (bis) are the creation
tripletp+ipstatemaysupporttopologicalexcitationsof andannihilationoperatorsontheunitcelliwiththespin
1
: Majorana fermions, and could be useful for implement- index s = , , and in the sublattice A(B). i,j denotes
v ing topological quantum computation. We carry out a all pairs o↑f N↓N sites. h i
i
X systematic investigation on the possible superconductiv- We rewrite Eq. (1) in the basis of non-interacting
r ity pairing states and their relative stabilities in lightly eigenstates: ηl(k) = (1/√2)[1, lexp( iϕk)]T, where
a doped graphene, taking account of multiple degree of l = 1 ( 1) for conduction (valan−ce) band−, exp[iϕ(k)]
− ≡
freedoms of orbit, spin and valley. We find that, when γ /γ , γ eik·δi for δ = a 3,√3 /2, δ =
k | k| k ≡ Pδi 1 (cid:0) (cid:1) 2
effective e e attractive interaction between the nearest a 3, √3 /2, δ = 0, and a is the C-C bond length.
neighborin−g sites (inter-sublattice) dominates and Fermi W(cid:0)e o−btain(cid:1): 3
level is close to the Dirac points, the intra-valley spin-
triplet p+ip pairing state will be favored. Based upon H = 1 Vss′(k )δ
the novel pairing state, we propose a scheme for doing int 2N X i k1+k2−k3−k4−K
kiKss′
quantum computation in graphene by engineering local
strain fields and external magnetic fields. ×c†s(k1)c†s′(k2)cs′(k3)cs(k4), (2)
where c†(k) and c (k) are creation and annihilation op-
s s
We start our investigation from a simplified lattice eratorsof conduction (or valance) band with wavevector
model introduced in Ref. 4, which assumes that effective k and spin s, respectively, N is the total number of unit
2
cells in the system, K is a reciprocal unit vector, and We can then determine all possible pairing states and
the summation of k is over the Brillouin zone. We as- their relativestabilities. First, we consider the case g <
i 0
sume that inter-band contribution can be ignored, and 0 and g >0. For the inter-valley pairing, we define the
1
the band index l is dropped for brevity. The interaction superconducting gap ∆ = (g /2N) c c ,
1 − | 0| Pkh −k−↓ k+↑i
matrix elements have the form: and ∆ = (g /2N) c c . It is easy to
2 − | 0| Pkh −k+↓ k−↑i
Vss′(ki)= g0 1+eiϕk1+iϕk2−iϕk3−iϕk4 δs,−s′ s∆how=th(agt /se2l)f-con∆sisttaennht(βgaEp/e2q)u/a(2tiEon)s, rweahde:re ∆E1 ==
4 (cid:2) (cid:3) 2 | 0| Pk k k k
+ g41 (cid:2)γk4−k1eiϕk1−iϕk4 +γk3−k2eiϕk2−iϕk3(cid:3) . (3) tphe(ǫfkre−eµel)e2c+tro∆n2,diaspnedrs∆ion=of∆gr1ap+he∆n2e,, µǫkis=the±vch|ke|miis-
We focus on the case of lightly doped graphenes with cal potential of the system. The equations yield usual
kFa 1, where kF is the Fermi momentums of the two s-wave pairing with superconducting gap ∆ = ∆0 =
Ferm≪i pockets around the corners of the Brillouin zone 2~ω0exp( 1/g0N0)atthezerotemperature,where~ω0
− | |
at Q , as shown in Fig. 1. We introduce valley index isthecut-offenergyoftheattractiveinteraction(e.g.,De-
0
±
λ = to denote which valley the electron is in, and bye energy in the case of phonon-mediated interaction)
±
the corresponding valley electron annihilation operator and N0 is the density of state per unit cell per valley
is defined as c c (λQ + k), where k is electron at the Fermi level. We note that the pairing correlation
kλs s 0
≡
momentum relative to the center of the corresponding function has even parity when exchanging the valley in-
valley. dexes,andoddparitywhenexchangingthespinindexes,
It is easy to see that there could exist two forms of respectively,correspondingtoaspin-singletpairingstate
superconducting pairings in graphene, i.e., inter-valley ( + + + )( ). On the other hand, there
| −i |− i |↑↓i−|↓↑i
and intra-valley pairings, as shown in Fig. 1. In all the is no pairing in the spin triplet channel.
previous studies, only the inter-valley pairing, i.e., two For the intra-valley pairing, the relevant interaction
electronsinapairhavetheoppositemomentumsandre- component is Ukλk′ ≡ −(|g0|/2)cos(θk−θk′)exp[iλ(θk −
sideatthedifferentvalleys(Fig.1(a)),isconsidered. On θk′)] in Eq. (5). We define the superconducting gap
tthroensotihnerahpaanidr,rtehsiedeinitnrat-vhaellseaympeaivrainllge,y,i.ae.n,dtwthoeetlewco- aTshe∆λs(eklf)-c=ons(i1s/tNen)tPgka′pUkλekq′uhca−tiko′nλ↓crke′aλd↑is,: an∆dλλ(k=) ±=.
valleys in the vicinity of ±Q0 act like two independent −(1/N)Pk′Ukk′∆λ(k′)tanh(βEλk′/2)/(2Eλk′)
sub-systems(Fig.1(b)),isalsopossible. Toconsiderboth
possibilities,weenumerateallpossiblecombinationsofki and Eλk = q(ǫk−µ)2+|∆λ(k)|2. The equa-
in the proximityof Q0 or Q0 in Eq. (3), andcollect all tion has an s-wave solution ∆λ(k) = ∆0 and
thetermsrelevanttothes−uperconductivitypairings,i.e., a d + id solution ∆λ(k) = ∆0exp(2iλθk), with
those terms satisfying either k1+k2 =k3+k4 = 0 (for ∆0 =2~ω0exp(−4/|g0|N0). [23]
theinter-valleypairing)ork1+k2 =k3+k4 = 2Q0(for Second, we consider the case g0 > 0 and g1 < 0.
±
the intra-valley pairing). The effective electron-electron For the inter-valley pairing, the relevant interac-
interaction can then be written as, tion component is Ukk′ (3g1 /2)cos(θk θk′)
≡ − | | −
in Eq. (6). We define the superconducting gap
Hisnct ≈ 21N kk′λX=±,ss′Vinλt,sras′(k,k′)c†k′λsc†−k′λs′c−kλs′ckλs aTshe ∆seslsf′-(ckon)siste=nt gap−(1eq/Nua)tPionk′Urekakd′sh,c−∆k′s−ss′′(ckk)′+s=i.
+ 21N kk′λX=±,ss′hVinsste′r(k,k′)c†k′λsc†−k′λ¯s′c−kλ¯s′ckλs −Ek(1/N=)Pkq′U(ǫkkk′−∆µss)′2(k+′)|t∆asnsh′((kβ)E|2k.′/2)/(T2Ehek′)equatainodn
yields two stable zero temperature solutions in
+Vin′,tsesr′(k,k′)c†k′λsc†−k′λ¯s′c−kλs′ckλ¯si, (4) p-wave channel: ∆ss′(k) = ∆0exp( iθk) and
∆ = 2~ω exp[ 4/3g N ]. The pairi±ng can be
and to the first order of ka, k′ a, the matrix elements 0 0 − | 1| 0
| | | | either spin singlet or triplet, with the corresponding
can be approximated as:
valley states ( + + ) and ( + + + ),
Vinλt,sras′(k,k′)≈ g20 cos(θk−θk′)eiλ(θk−θk′)δs,−s′ respectively. [2|4] −i − | − i | −i | − i
For the intra-valley pairing, the relevant interac-
3g
+ 21eiλ(θk−θk′), (5) tion component is Ukλk′ = −(3|g1|/2)exp[iλ(θk − θk′)]
in Eq. (5). We define the superconducting gap
VViin′sn,tsstee′srr′((kk,,kk′′))≈≈ gg2200δδss,,−−ss′′,+ 32g1 cos(θk−θk′), ((76)) −saes(lf1-/∆cNoλsns)s′P(iskt)ke′nUt=kλk′g∆a(1pλss/′N(ke)′qP)utaaktn′iUho(nkλβkE′hkrce′−/ak2d′)sλ/s(′2cE∆k′kλλs′ss)′i(.k) Tanh=de
where we have made use of approximations: γk ≈ 3, Ek = q(ǫk−µ)2+(cid:12)∆λss′(k)(cid:12)2. The equation yields the
tγo±Qth0e+kfir=stγo±rd2Qer0+okf k≈a,0a,nedxpθ(iϕis±Qth0e+ka)zi≈mu−thieoxfpk(∓.iθk), s∆pλin =tri∆pletexpp(+iλθip)(cid:12)paanidrin∆g (cid:12)s=ta2t~eωatexzpe(ro-2t/e3mgperNatu).re:
k ss′ 0 k 0 0 − | 1| 0
3
The above considerations assume that the Fermi level Intervalley Intravalley (µ<2 ~v∆)
q a
is close enough to the Dirac points, such that the quasi-
particle dispersionhas perfect rotationsymmetry within (k) 1 exp(iθk) 1 exp(iλθk)
each valley. However, real graphene systems only have △ exp( iθk) exp(2iλθk)
−
three-fold rotation symmetry. The asymmetry, i.e., valley + +
± ±
ǫ = ǫ , becomes more pronounced when spin +
F±erQm0+iklev6el is±Qre0m−koved from the Dirac points, leading V |g˜0(0)±∓g˜0(2Q0)| |g˜1±(0)| |g˜0−(0)| |g˜1(0)|
0 2 4 4 2
to pair breaking that destabilizes the intra-valley pair-
ing. On the other hand, the inter-valley pairing is not Table I: The possible superconducting pairing states. The
momentum dependence of gap function ∆(k), the parity in
affected because the symmetry between ǫ and ǫ is
k −k exchangingthevalley(“valley”)andspin(“spin”)index,aswell
guaranteed by time-reversal symmetry. To see asym-
astheeffectivepairingstrengthV areshown. Theamplitude
0
metry effect to the intravalley pairings, we expand the
of thezero temperaturesuperconductinggap isrelated to V
0
tight-binding dispersion to the second order, and for the by∆ =2~ω exp( 1/N V ).
0 0 0 0
valley centering Q , we have ǫv ǫ ~vk +h , −
and h = (~k2va0/4)sin3θ . k T≡heQg0a+pk ≈equation fokr
k k
the intra-valley pairing when g < 0 is modified to the Fermi level is close to the Dirac points such that
1
∆λss′(k) = −(1/N)Pk′Ukλk′∆λss′(k′)[1−f(Ek′ +hk′)− µ ≤ 2p(~v/a)∆. It is important to note that the spin-
f(Ek′ + h−k′)]/(2Ek′). Solving the equation, we find triplet intra-valley p + ip pairing state does not break
thattheintravalleypairingstatesbecomeunstablewhen the time reversal symmetry (TRS), because the order
µ>2 (~v/a)∆ 130 ∆/1meVmeV for graphene. parametersat the two valleys are related by the time re-
p ≈ p
The discussion can be further extended to the more versal, i.e., the order parameter at one valley is p+ip,
general form of the interaction. We introduce gij as while that at the other valley is p ip. The origin of the
αβ −
e-e interaction between two electrons resided in the intra-valley chiral p+ip pairing can be traced back to
unit cell i sublattice α and the unit cell j sublat- π-Berry phase associating with the Dirac point in each
tice β with α, β A, B . The Fourier transforma- of the valleys. [12, 13] The topological origin of the un-
tionsareg˜ (q)=(1∈/N{) gi}j exp[ iq (R R )]and conventional pairing, which is dictated by a band singu-
0 P α=β − · i− j
ij larityburieddeeplyinsidetheFermisea,isdifferentfrom
g˜1(q) = (1/N)Pgαij6=βexp[−iq·(Ri−Rj)], correspond- usual mechanisms derived from strong correlations. The
ij
intra-valleychiralp+ipsuperconductingstateisspatially
ing to intra-sublattice and inter-sublattice interactions,
non-uniform,due to the finite centerofmassmomentum
respectively. In the limit of k a 1, only Fourier com-
F
≪ (CMM) of Cooper pairs. It is important to note that
ponents at q =0 and q = 2Q are relevant. Moreover,
0
± the pairing with the finite CMM does not incur extra
we can show that g˜ ( 2Q ) always vanishes because of
1 0
± kinetic energy in graphene, because the electron disper-
C rotationalsymmetry. Theinteractionmatrixelements
3
sion does have minimums at Q (see Fig. 1). This is
in Eq. (4) are then modified to: ± 0
distinctfromtheanotherwellknownnon-uniformsuper-
Viλnt,sras′(k,k′)≈ g˜02(0)cos(θk−θk′)eiλ(θk−θk′) cstoantdeu,c[t1i4n,g1s5t]atwehinichcoinnvceunrtsiolnaarglesykstinemetsic,ie.en.e,rtghyeaLnOdFiFs
g˜ (0) very fragile.
+ 1 eiλ(θk−θk′), (8) The intra-valley triplet p+ip superconducting phase
2
Vinsste′r(k,k′)≈ g˜02(0) + g˜12(0)cos(θk−θk′), (9) itno ilmighptlelymdenotpegdragprhaepnhee-nbea,seodnctoeproelaolgiziceadl,qcuoaunldtubmecuosmed-
putation. This could be achieved if one could gener-
V′,ss′(k,k′) g˜0(2Q0). (10) ate half quantum vortices (HQV) in the superconduct-
inter ≈ 2
ing phase. [16] Each vortex core will confine a Majorana
Followingthesameprocedure,wecanextendouranal- fermion. The quantum state of a macroscopic system
ysistothegeneralmodel. Theresultsaresummarizedin with a number of such vortices could be transformed by
TableI.ForthesimplemodelEq.(1),g˜ (0)=g˜ (2Q )= adiabaticallymovingthem,withrobustnesstopologically
0 0 0
g , and g˜ (0)=3g . Compared to the simple model, the protected by their spatial separations.[17]
0 1 1
general model yields extra possibility of forming inter- Forgraphene,thereexistsaveryusefulanduniquetool
valley spin triplet s-wave pairing state. for implementing the novel quantum computing scheme,
From Table I, it is straightforward to determine the i.e., pseudo-magneticfields generatedby strainsor topo-
pairinggroundstatesindifferentparameterregimes. The logical defects. [18–20] Using a strain field or topolog-
phase diagram is shown in Fig. 2. The most interest- ical defect, it is possible to create a spatially localized
ing feature of the phase diagram is the presence of spin- pseudo-magneticflux,whichisnoteasilyachievableusing
triplet intra-valley p+ip pairing state when the inter- a real magnetic field. One can use the pseudo-magnetic
sublattice attractive interaction g˜ (0) dominates, and fluxesasnucleiforcreatingHQVs. Forinstance,one can
1
4
g˜ (0) systemhasitselectronicstructuresignificantlymodified,
1
andcannotbeexpectedtosupportthenovelp+ipphase
wepredict,itdoesgivethe hopethatonemayfindother
g˜0(2Q0)>0 S
˜g p dopants that serve the purpose.
(20 in
Spinsinglets−wave )Q0 trip In summary, we systematically investigate the super-
> le conducting phases for lightly doped graphene. We find
˜g t
g˜0(2Q0)<0 (00 −s that an intra-valley spin triplet p+ip superconducting
) w phasecouldbeachievedinthecertainparameterregime.
a
Spintriplets−wave ve g˜ (0) We further show that the novel superconducting phase
0 could be used to implement topological quantum com-
putationingraphenebyutilizingpseudo-magneticfluxes
Intervalleyp+ip created by strain field or topological defects. These pos-
µ>2q¯hva∆ sibilitiesmakegrapheneapromisingcandidateforimple-
menting topological quantum computation.
Mixed This work is supported by MOST 973 program
Spintriplet
No. 2009CB929101.
Intravalleyp+ip
µ<2q¯hva∆
[1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Kat-
Figure2: Phasediagramoflightlydopedgraphene. Forgiven
snelson M I, Grigorieva I V, Dubono S V and Firsov A
g˜0(0)andg˜1(0),thepairingstatesalsodependontheparam- A 2005 Nature438 197
eters g˜ (2Q ) and µ, and the corresponding conditions are
0 0 [2] Zhang Y B, Tan Y W, Stormer H L and Kim P 2005
shown for each phase. Nature 438 201
[3] Castro NetoA H, Guinea F, Peres N M R, Novoselov K
S and Geim A K 2009 Rev.Mod. Phys 81 109
generate a total pseudo-magnetic flux of a half quanta [4] Uchoa B and Castro Neto A H 2007 Phys. Rev.Lett 98
within the size of superconducting penetrating length. 146801
Because of the presence of two independent valleys, it [5] Black-SchafferAMandDoniachS2007Phys.Rev.B75
will create a pair of HQVs, one for each of the valleys. 134512
[6] Honerkamp C 2008 Phys. Rev.Lett 100 146404
The two Majorana fermions localized inside the vortex
[7] Hosseini M V and Zareyan M 2012 Phys. Rev.Lett 108
cores will maintain their independence within time scale
147001
of inter-valley scattering. [22] On the other hand, to cre-
[8] Nandkishore R, Levitov L S and Chubukov A V 2012
ateasingleMajoranafermion,onecouldengineeratotal Nature Phys8 158
pseudo-magnetic flux of a quarter quanta. Because the [9] Roy B and HerbutI F 2010 Phys. Rev.B 82 035429
pseudo-magneticfield has the opposite directions for the [10] Profeta G, Calandra M and Mauri F 2012 Nature Phys
two valleys, a perpendicular real external magnetic field 8 131
[11] XueM,ChenG,YangH,ZhuY,WangD,HeJandCao
will enhance the magnetic field experienced by one val-
T B 2012 J. Am.Chem. Soc134 6536
ley while suppressing it in the other. It is then possible
[12] Shi J R and Niu Q 2006 arXiv: 0610531[cond-mat]
to generate a HQV only in one of the two valleys. [25]
[13] MaoL,ShiJR,NiuQandZhangCW2011 Phys.Rev.
The resulting Majorana fermion will be fully topologi- Lett 106 157003
cally protected. Finally, one could carefully design the [14] LarkinAJandOvchinnikovYN1965Sov.Phys.JETP
strainfieldandthematchingexternalrealmagneticfield 20 762
so that the creation of multiple HQVs is energetically [15] Fulde P,and Ferrell R A 1964 Phys. Rev 135 A550
[16] JangJ,FergusonDG,VakaryukV,BudakianR,Chung
favorable,[21] andthe graphene-basedtopologicalquan-
S B, Goldbart PM, andMaeno Y 2011 Science331 186
tum computation can then be achieved by tuning the
[17] Ivanov D A 2001 Phys. Rev.Lett 86 268
strain field.
[18] Guinea F, Katsnelson M I and Geim A K 2009 Nature
To realize the novel superconducting phase predicted Phys 6, 30
in this Report, it is important to find a modification [19] VozmedianoMAH,Katsnelson MIandGuineaF2010
to graphene systems that introduces an appropriate Phys. Rep 496, 109
level of carrier density and a sizable effective electron- [20] Levy N,BurkeS A,Meaker KL, Panlasigui M,ZettlA,
Guinea F, Neto A H C and Crommie M F 2010 Science
electron attractive interaction from such as electron-
329,544
phonon coupling or electron-plasmon coupling. [4] Re-
[21] ChungSB,BluhmHandKimEA2007Phys.Rev.Lett
cently, it has been predicted by the first principles cal- 99, 197002
culation that graphene could be transformed supercon- [22] Wu XS,Li X,SongZ,Berger Cand deHeerW A 2007
ducting by lithium deposition. [10] While the particular Phys. Rev.Lett 98, 136801
5
[23] The other solutions have the form ∆λ(k) = [25] Tominimizeelasticenergy,itisusuallynecessarytocre-
√2∆0cos(θk + ϕ)exp(iλθk), and ∆0 atepseudo-magneticfluxesinpairsofoppositedirections.
1.72~ω exp( 4/g N ). The corresponding state≈s In this case, each pair of the fluxes will create a pair of
0 0 0
− | |
havethehigher energy. Majorana fermions residing in the different valleys, spa-
[24] The degeneracy between the spin singlet and triplet tially separated.
states will be lifted in the order of (kFa)2, in favor of
thesinglet state.
