Possibility of a zero-temperature metallic phase in granular two-band superconducting films
PPossibility of a zero-temperature metallic phase in granular two-band superconductingfilms
Bojun Yan and Tai-Kai Ng
Department of Physcis, Hong Kong University of Science and Technology, Hong Kong, People’s Republic of China (Dated: Dec. 2011)A variational approach is used to study the superconductor-insulator transition in two-band gran-ular superconducting films using a resistance-shunted Josephson junction array model in this letter.We show that a zero-temperature metallic phase may exist between the superconducting and insu-lator phases which is absent in normal single band granular superconducting films. The metallicphase may be observable in some dirty pnictide superconductor films.
PACS numbers: 74.20.-z,74.78.-w,74.81.-g
Intensive studies had been devoted to the problem ofsuperconductor-insulator (SI) transition in low- T c thinfilms. These systems undergo phase transitions from su-perconductor to insulator as a function of disorder, filmthickness as well as external magnetic fields[1]. The SItransition is usually modelled by a Josephson junctionarray model, expressed in terms of the phases of the su-perconductor order parameter θ i ’s on superconductinggrain i ’s. The Hamiltonian describing the system con-sists of the Josephson coupling between superconductinggrains ∼ J cos( θ i − θ j ) where ( i.j ) are nearest neighborsites, and the charging energy ∼ C ( ˙ θ i − ˙ θ j ) . The sys-tem is in a superconducting phase if the Josephson termdominates, and is in the insulator phase if the charg-ing energy dominates. It has been proposed by differentauthors that a dissipative term arising from coupling be-tween superconducting grains and a dissipative metallicbath may also be important in describing the SI transi-tion (shunted Josephson array model)[2–4]. In particular,a zero-temperature metallic phase between superconduc-tor and insulator phases may be stabilized by dissipation.The physical reason behind the metallic phase is asfollows: Imagine first a state dominated by charging en-ergy. In this case the metallic bath would screen theCoulomb potential, leading to a weakening of chargingenergy and drives the system towards a metallic phaseif the resistance is small enough ( R < R cI )[5]. Alter-natively, the coupling of Cooper pairs in the supercon-ducting phase to a dissipative environment suppressescoherent tunnelling of Cooper pairs between grains ow-ing to the Calderia-Leggett effect[6] and superconduct-ing coherence is destroyed if R > R cS . As a result ametallic phase between the superconducting and insulat-ing phases may exist if R cS < R < R cI . The metal-lic phase, if exist, is a new phase of matter because ofparticipation of incoherent boson (Cooper pairs) in lowtemperature transports which is absent in usual metals.Experimentally the zero-temperature metallic phase insingle band superconducting films has not been found toexist so far in the absence of external magnetic fields,consistent with a theoretical finding that R cI < R cS insingle-band superconductors[7]. More recently, superconductors with more than one or-der parameters, i.e., the multi-band superconductors[8]have raised attention in the physics community. Exam-ples of multi-band superconductors include MgB [9] andthe pnictide superconductors[10]. It is interesting to seewhether a metallic phase may exist more easily betweenthe SI-transition in these materials. This is the purposeof this letter.Using a variational approach, we consider in this let-ter the superconductor-insulator transition in two-band s (and s ± )-wave superconducting films where the possi-bility of an intermediate metallic phase is investigated.We show that contrary to the case of single-band su-perconductors, a physically realizable condition for thezero-temperature intermediate metallic phase is foundfor these systems. We propose that the metallic phasemay be observable in some recently discovered disorderedpnictide superconductors[11]. Figure 1: A qualitative sketch of our two band Josephsonjunction model. The two bands on different grains are con-nected by a capacitor, a resistor and Josephson coupling (notshown in the sketch). The two bands on the same grain areconnected by in-grain inter-band Josephson coupling J I We start with the phase-action which is a generaliza-tion of the phase action used to study superconductor-insulator transition in one-band systems[2, 3, 7]. Thesystem is schematically sketched in Fig.1. The actiondescribes a resistance-shunted Josephson network of two-band superconductor grains and is given in imaginarytime by S = S θ + S diss , where a r X i v : . [ c ond - m a t . s up r- c on ] D ec S θ = (cid:88) i,ν,a,b (cid:90) β dτ [ 12 C ab (∆ ν ˙ θ abi ) − J ab cos(∆ ν θ abi )]+ J I (cid:88) i (cid:90) β dτ cos( θ i − θ i ) (1)is the phase action without the dissipative term. a, b =1 , θ ai is the phase of band a superconducting order parame-ter in grain i . ∆ ν θ abi = θ ai − θ bi + ν represents the phasedifference between band a and b superconducting orderparameters in neighboring grains i , i + ν , respectivelyand J ab > C ab (∆ ν ˙ θ abi ) represents the charging energy aris-ing from charge imbalance between band a and band b electrons on grain i and i + ν respectively, where C ab is thecorresponding capacitance. J I is the in-grain inter-bandJosephson coupling which favors θ i = θ i + π for J I > s ± superconductor and favors θ i = θ i for J I < s -wave superconductor). S diss = Q (cid:88) i,ν,a,b (cid:90) β dτ (cid:90) β dτ (cid:48) × α o ab ( τ − τ (cid:48) ) sin [ D ν θ abi ( τ ) − D ν θ abi ( τ (cid:48) )2 Q ] (2)where α o ab ( τ ) = ( h/ e R o ab )[ T / sin( πT τ )] . R o ab is the re-sistance between band a and band b electrons on grains i and i + ν , respectively (see Fig.(1)) and Q = 2 is thecharge of a Cooper pair. D ν θ i = D ν θ i = [∆ ν θ i +∆ ν θ i ], D ν θ aai = [ ∆ ν θ aai + ∆ ν θ ¯ a ¯ ai ], where ¯1(¯2) = 2(1). S diss is derived phenomenologically from a multi-bandresistance network model represented by Fig.1. The de-tails of the derivation can be found in the supplementarymaterials.To simplify calculation we shall consider the grainsforming a two-dimensional square lattice with J = 0in our following analysis. With the later condition the s and s ± superconductors can be transformed to eachother by simply shifting θ i → θ i + π . The main effectof J is to renormalize J I → J I − zJ where z is thelattice co-ordination number and is not going to affectour conclusion in renormalization-group sense.Due to the compactness of the phase field ( e i ( θ +2 nπ ) = e iθ ), the phase variables θ ai ( τ ) can be decomposed into aperiodic part and a winding number contribution, θ ai ( τ ) = 2 πn ai τβ + θ ai ( τ )where θ ai ( β ) = θ ai (0) and n ai can be any arbitrary integer(winding number). With this decomposition the phase action becomes S θ → π β (cid:88) i,ν,a,b C ab ∆ ν n abi + (cid:88) i,ν,a,b C ab (cid:90) β dτ (∆ ν ˙ θ abi ) − (cid:88) i,ν,a J aa (cid:90) β dτ cos[∆ ν θ aai + 2 πτβ ∆ ν n aai ]+ J I (cid:88) i (cid:90) β dτ (cid:88) ν cos[∆ θ i + 2 πτβ ∆ n i ] . (3)where ∆ ν n abi = n ai − n bi + ν , ∆ θ i = θ i ( τ ) − θ i ( τ ) and∆ n i = n i − n i and S diss → (cid:88) i,ν,a,b Qπ R ab | D ν n abi | + 18 (cid:88) i,ν,a,b (cid:90) β dτ (cid:90) β dτ (cid:48) α o ab ( τ − τ (cid:48) ) cos[ 2 π ( τ − τ (cid:48) ) Qβ D ν n abi ] × [ D ν θ abi ( τ ) − D ν θ abi ( τ (cid:48) )] . (4)where D ν n abi is defined in the same way as D ν θ abi with∆ ν θ abi → ∆ ν n abi = n ai − n bi + ν . We have assumed strongdissipation and keep only to second order terms of ∆ ν θ aa i in S diss for simplicity[7].To proceed further we employ a varia-tional approach[7]. We consider a trial action S trial = S P trial + S n trial where the periodic and thewinding number contributions to θ are decoupled. S P trial = (cid:88) i,ν,a,b (cid:90) β dτ [ C ab ν ˙ θ abi ) + J eff aa ν θ aai ) ]+ J eff I (cid:88) i (cid:90) β dτ (∆ θ i ) +18 (cid:88) i,ν,a,b (cid:90) β dτ (cid:90) β dτ (cid:48) α eff , o ab ( τ − τ (cid:48) ) × [ D ν θ abi ( τ ) − D ν θ abi ( τ (cid:48) )] (5)is an effective action describing Gaussian fluctuations ofthe periodic phases around the saddle point θ i ( τ ) = 0, θ i ( τ ) = 0( π ) for J I < ( > )0 and S n trial = 2 π β (cid:88) i,ν,a,b C ab ∆ ν n abi − βJ MSI (cid:88) i δ ( n i − n i )+ (cid:88) i,ν,a,b (cid:34) Qπ R eff , o ab | D ν n abi | − βJ MSaa δ (∆ ν n aai ) (cid:35) (6)in an effective action for the winding number field. S n trial is a generalized absolute solid-on-solid model (ASOS) fortwo species of winding numbers with additional βJ MS terms originating from superconductivity. J eff aa , J eff I , J MSaa , J MSI , α eff , o a ( τ ) /α ( τ ) = R o a /R eff , o a are variational param-eters to be determined by minimizing the free energy ofthe system given approximately by F = F + (cid:104) S − S trial (cid:105) ,where F is the free energy computed using S trial and (cid:104) ... (cid:105) denotes averages taken with respect to S trial .The different phases can be identified in our trial ac-tion as follows: First we note that S P trial describes a stablesuperconducting phase as long as the phase stiffness’ssatisfy J eff aa >
0. The nature of the J eff aa = 0 (non-superconducting) state is determined by S n trial which de-scribes two different possibilities. For small R eff , o ab ’s, thesystem is in a “smooth” phase where fluctuations in n ai ’sare suppressed and charges become mobile. The systemis in a metallic phase. For large R eff , o ab ’s n ai ’s at differentsites fluctuate violently (rough phase) and charge fluc-tuations are suppressed. The system is in the insulatorphase[5].Minimizing the free energy we obtain after somelengthy algebra the mean-field equations R eff , o ab = R oab (7) J MSaa = J aa e −(cid:104)| ∆ ν θ aai | (cid:105) J MSI = | J I | e −(cid:104)| ∆ θ i | (cid:105) J eff aa = J MSaa P aan (0) J eff I = J MSI P In (0) , where P aan ( m ) = (cid:104) δ ( m − | ∆ ν n aai | ) (cid:105) S n and P In ( m ) = (cid:104) δ ( m − | n i − n i | ) (cid:105) S n are the probabilities that the inte-ger differences | ∆ ν n abi | = m and | n i − n i | = m in S n trial ,respectively. (cid:104)| ∆ ν θ aai | (cid:105) = 1 βN d (cid:88) iω n ,(cid:126)k γ ( (cid:126)k )2 a ¯ a ¯ a a a − a I , (8a)where ¯1(¯2) = 2(1) and (cid:104)| ∆ θ i | (cid:105) = 12 βN d (cid:88) iω n ,(cid:126)k a + a + 2 a I a a − a I , (8b)where a bb = (cid:18) J eff bb + 12 ( α bb + α ) | ω n | (cid:19) γ ( (cid:126)k ) + J eff I (9a)( b = 1 ,
2) and a I = − J eff I + α | ω n | γ ( (cid:126)k ) (9b)where α ab = h/ (4 πe R ab ). The resistance R ab ’s are givenby R − aa = (3 R o − aa − R o − a ¯ a ), and R − = R o − + ( R o − + R o − ) where α oab = h/ (4 πe R oab ). This rather compli-cated form of resistance is a result of appearance of D ν θ terms in S diss . γ ( (cid:126)k ) = 4(sin ( k x /
2) + sin ( k y / J eff aa = 0 is deter-mined by S P trial only and is independent of S n trial as longas P aan (0) and P In (0) are nonzero. Similarly, the metal toinsulator transition is determined by S n trial only (rough orsmooth phase) when J eff a = 0. In Fig.2 we present the re-sulting phase diagram for the symmetric case J = J , C = C and R = R for two different values of J I .We note that a metallic phase is found in a narrow regionof parameter space when J I is small enough, contrary tothe single-band case where no metallic phase is found. Figure 2: Phase diagram of the two band system for differentvalues of α + α versus R for two values of J I with J eff I (cid:54) = 0in the upper panel and J eff I = 0 in the lower panel. A metallicphase is found in a narrow region of parameter space when J eff I = 0 and R < . h/Qe To understand the phase diagram, we observe first thatthe phase diagram is divided into two regimes, (i) J eff I (cid:54) = 0(upper panel) and (ii) J eff I = 0 (lower panel). The effec-tive Josephson coupling between the two bands is nonzeroin the first regime and it is easy to show from Eqs.(7-9) that the two-band superconductor becomes effec-tively like a single band superconductor at low energy ω << J eff I in S P trial . Correspondingly, S n trial becomes aneffective one band model with n i ≡ n i = n i at temper-ature T → βJ MSI (cid:80) i δ ( n i − n i ) term,i.e. S n trial → π C t β (cid:88) i ∆ ν n i + Qπ R eff (cid:88) i,ν | ∆ ν n i | when superconductivity is destroyed, where C t = C + C and R − = R − + R − + 2 R − . The system isat a roughening (insulator) phase when R eff > R cI =0 . h/Qe )[5] and a metallic phase exist only if J eff aa = 0at a finite region of resistances R cI > R eff > R cS . Forsingle-band superconductors, R cS = ( h/Qe ) > R cI [7],and an intermediate metallic phase cannot exist in thiscase.We next consider regime (ii) where J eff I = 0. Firstit is straightforward to show that J eff I (cid:54) = 0 as long as J eff aa (cid:54) = 0, indicating that the system behaves always likean effective one-band system at low enough energy inthe superconducting state. The situation is different ifsuperconductivity is destroyed. Substituting J eff aa = 0into Eq. (8b), we obtain a self-consistent equation for J eff I , J eff I = | J I | P In (0) exp − βN d (cid:88) iω n ,(cid:126)k J eff I + ˜ α I | ω n | γ ( (cid:126)k ) , (10)where ˜ α I = α α + α ( α + α ) α + α +4 α .Equation (10) is solved numerically where we findthat the equation has a non-zero solution only when | J I | > J cI (˜ α I , J ), which is a number depending on ˜ α I and roughly proportional to max( J , J ), the transi-tion from the J eff I (cid:54) = 0 to J eff I = 0 state is a first orderphase transition. The phase diagram determined by (10)is provided in the supplementary material.This interesting result suggests that although the su-perconducting state behaves always like an effective one-band superconductor at low enough energy, there ex-ists two kinds of non-superconducting states. The non-superconducting state is effectively one-band like when J eff I (cid:54) = 0 and two-band like when J eff I = 0. We find thatan intermediate metallic phase may exist in the two-bandlike non-superconducting state.To see how this can occur we consider Eq. (8a) with J eff I = 0. In this case we obtain J eff aa ∼ J aa P aan (0) exp − βN d (cid:88) iω n ,(cid:126)k J eff aa + ˜ α aa | ω n | , (11)where ˜ α aa = h πe (cid:18) R aa + 1 R ¯ a ¯ a + 2 R (cid:19) , corresponding to a single-band superconductor with ef-fective resistance R − = R − aa + ( R ¯ a ¯ a + 2 R ) − , whichis the effective resistance obtained from the resistancenetwork model shown in Fig.(1). The SI transition isdetermined by Eq. (11) and S n trial . To see the plausi-ble existence of metallic phase, we examine the limit R o12 ∼ R →
0. In this limit, a long-range orderof n , i ’s are built up in the winding number action S n trial because n i ≡ n i + ν ∀ i , and the system is alwaysin the smooth phase. A metallic phase exists as long as R eff → R R / ( R + R ) > h/Qe where J eff aa → R increases as shown in Fig.(2)lower panel. Notice that the winding number field is basically controlled by R when R and R arelarge, so for a metallic phase to occur, we generally re-quire R to be smaller than 0 . h/Qe . For small R ,the superconductor-insulator transition is governed by α + α . For the superconducting stiffness to vanish,we require α + α ≤ . J I and inter-band dissipative resis-tance term R are small enough. Physically, the morecomplicated circuit network structure for two-band su-perconductors (Fig.1) gives rise to the possibility thatthe effective dissipation responsible for screening andquantum dissipation are coming from different resistancechannels which is not possible for single-band supercon-ductors. With the recent advancements of research inIron pnictide and other multi-band superconductors, webelieve that this new metallic phase of matter may bereachable in the near future[11].This work is supported by HKRGC through grantHKUST3/CRF09. [1] M. P. A. Fisher, Phys. Rev. Lett. , 923 (1990).[2] S. Chakravarty, G. Ingold, S. Kivelson, and A. Luther,Phys. Rev. Lett. , 2303 (1986).[3] S. Chakravarty, G. Ingold, S. Kivelson, and G. Zimanyi,Phys. Rev. B , 3283 (1987).[4] A. Kapitulnik, N. Mason, S. Kivelson,and S. Chakravarty, Phys. Rev. B. , 125322 (2001).[5] See for example, R. Fazio and G. Sch¨on, Phys. Rev. B , 5307 (1991) and J.E. Mooij et.al. , Phys. Rev. Lett. , 645 (1990).[6] A.O. Caldeira and A.J. Leggett, Ann. Phys. (N.Y.) ,374 (1983).[7] Tai-Kai Ng, Derek K. K. Lee, Phys. Rev. B , 144509(2001).[8] H. Suhl, B. T. Matthias, L. R.Walker, Phys. Rev. Lett. , 552 (1959).[9] Nagamatsu J. et al., Nature , 63 (2001).[10] Y. Kamihara et al., J. Am. Chem. Soc. , 3296 (2008).[11] Ming-Hu Fang et al , Europhysics Lett. , 27009 (2011)[12] T.K. Ng, Europhysics Lett. , 47004 (2008) SUPPLEMENTARY MATERIALS
Here we show how the dissipative term (2) can bederived from a straightforward generalization of the dis-sipative term for one-band system to two-band system.We assume phenomenologically that a metallic compo-nent exists in the system and the dissipative term can bederived from a Hamiltonian with tunnelling and capaci-tance energy between grains, H = (cid:88) m,a H am + (cid:88) a,b (cid:0) H abT + H abQ (cid:1) (12)where m = L, R and a, b = 1 , H am = (cid:88) σ (cid:90) dx m ˆΨ † aσ ( x m )[ (cid:15) am ( − i ∇ )] ˆΨ aσ ( x m ) (13a)describes non-interacting electrons in grain m , band a where σ is the spin index, and H abT = (cid:88) σ (cid:90) dx L dx R T ab ( x L , x R ) ˆΨ † aσ ( x L ) ˆΨ bσ ( x R ) + h.c. (13b)describes tunneling of electrons between grain L , band a and grain R , band b and H abQ = 18 C ( Q aR − Q bL ) (13c)is the charging energy associated with charge imbalancebetween the grains where Q am = e (cid:88) σ (cid:90) dx m ˆΨ † aσ ( x m ) ˆΨ aσ ( x m ) (14)is the total electric charge in grain m , band a . The cor-responding action at imaginary time is S = (cid:88) a,σ (cid:90) β dτ { (cid:90) dx L ¯Ψ aσ ( x L ) ∂τ Ψ aσ ( x L )+ (cid:90) dx R ¯Ψ aσ ( x R ) ∂τ Ψ aσ ( x R ) } + H (15)To derive S diss we first apply a Stratonovich-Hubbardtransformation on H Q to obtain S → (cid:88) m,a,σ (cid:90) β (cid:90) dx m ¯Ψ aσ ( x m ) { ∂τ + (cid:15) am ( − i ∇ )+( − s m ( ie V aa +( 1 + ( − s m V a ¯ a + ( 1 − ( − s m V ¯ aa ] } Ψ aσ ( x m )+ (cid:88) a,b,σ (cid:90) β dτ (cid:90) dx L dx R × [ T ab ( x L , x R ) ¯Ψ aσ ( x L )Ψ bσ ( x R ) + c.c ] − (cid:88) a,b (cid:90) β dτ [ CV ab m = L, R , s L = 0 and s R = 1 and ¯1(¯2) = 2(1). Writing V ab = ˙ θ aR − ˙ θ bL , where ¯ L ( ¯ R ) = R ( L ), the elec-tric potential V ab ’s can be absorbed by a gauge transfor-mation Ψ aσ ( x m , τ ) = e − iθ am + i ( θ m + θ m ) ˜Ψ aσ ( x m , τ ) . (17)where the tunnelling term becomes S T → (cid:88) a,b,σ (cid:90) β dτ (cid:90) dx L dx R { T ab ( x L , x R ) × e i [ (∆ ν θ ab +∆ ν θ ba )+ (∆ ν θ +∆ ν θ )] × ¯Ψ aσ ( x L )Ψ bσ ( x R ) + c.c } (18)where ∆ ν θ ab = θ aL − θ bR ( a, b = 1 , S eff ( θ ) = (cid:88) a,b (cid:90) β dτ [ C ab ν ˙ θ ab ( τ ) + S ab diss ] (19)where S ab diss = 12 (cid:88) σ | T ab | (cid:90) β dτ (cid:90) β dτ { G aLσ ( τ − τ ) G bRσ ( τ − τ ) e i ( D ν θ ab ( τ ) − D ν θ ab ( τ )) + G bRσ ( τ − τ ) G aLσ ( τ − τ ) e i ( D ν θ ba ( τ ) − D ν θ ba ( τ )) } (20)where D ν θ i = D ν θ i = [∆ ν θ i + ∆ ν θ i ] and D ν θ aai =[ ∆ ν θ aai + ∆ ν θ ¯ a ¯ ai ]. G amσ ( τ ) = 1 βV (cid:88) iω,(cid:126)k e − iω n τ iω n − (cid:15) amσ ( (cid:126)k ) = − D amσ ( E F ) πT sin( πT τ ) (21)is the free electron Green’s function at imaginary time. m = L, R and D amσ ( E F ) is the density of states on theFermi surface. Defining α oab ( τ ) = (cid:88) σ | T ab | D aLσ ( E F ) D bRσ ( E F )( πT sin( πT τ ) ) = (cid:18) h e R oab (cid:19) (cid:18) T sin( πT τ ) (cid:19) (22)and put it back into (20) we obtain S diss ∼ (cid:88) a,b (cid:90) β dτ (cid:90) β dτ { α oab ( τ − τ ) × sin [ D ν θ ab ( τ ) − D ν θ ab ( τ )2 Q ] } (23)which is the dissipation term we use in the main text.We attach here also the phase diagram determined byEq. (10) with J = J . The line separating the J eff I =( (cid:54) =)0 phases is a line of first order phase transition. we seethat J /J I ∼ ˜ α I = α α + α ( α + α ) α + α +4 α at the transition. α = α α + α ( α + α ) α + α + 4 α J J I J effI = 0 J effI > Figure 3: Phase diagram for J effI for different values of J − I versus ˜ α . A first order phase transition separates the J effI = 0and J effI (cid:54)(cid:54)