Modulation of pairing symmetry with bond disorder in unconventionalsuperconductors

Yao-Tai Kang,1 Wei-Feng Tsai,2, 1, ∗ and Dao-Xin Yao1, †

1State Key Laboratory of Optoelectronic Materials and Technologies,School of Physics, Sun Yat-Sen University, Guangzhou 510275, China

2Department of Physics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan

We study a two-orbital t-J1-J2 model, originally developed to describe iron-based superconductorsat low energies, in the presence of bond disorder (via next-nearest-neighbor J2-bond dilution). Byusing the Bogoliubov–de Gennes approach, we self-consistently calculate the local pairing amplitudesand the corresponding density of states, which demonstrate a change of dominant pairing symmetryfrom s± wave to d wave when increasing disorder strength as long as J1 . J2. Moreover, thecombined pairing interaction and strong bond disorder lead to the formation of s± wave “islands”with length scale of the superconducting coherence length embedded in a d wave “sea.” This pictureis further complemented by the disorder-averaged pair-pair correlation functions, distinct from thecase with potential disorder, where the “sea” is insulating. Due to this inevitable formation ofspatial inhomogeneity, the superconducting Tc determined by the superfluid density ρs(T ) obviouslydeviates from the value predicted by the conventional Abrikosov-Gorkov theory, where the pairingamplitudes are viewed as uniformly suppressed as the disorder increases.

I. INTRODUCTION

Studying disorder effects in superconductors (SCs) isusually beneficial, though not in a direct manner, for un-veiling their underlying pairing mechanism. An economicearly indicator of the unconventional nature of a super-conductor, for instance, can be the sensitivity of the su-perconducting transition temperature (Tc) to an amountof finite disorder.1–5 In addition, most unconventionalSCs, such as cuprates6 and iron-based SCs7–9, becomesuperconducting after doping, which naturally introducescertain types of disorder in the materials. These mate-rials are often complicated in composition and are inter-mediately or strongly coupled systems, causing a rathercomplex phase diagram with intertwined orders in whichdisorder might play a role.10 With substantial amountsof disorder, a SC may even undergo a zero-temperaturequantum phase transition to another superconductingphase with distinct pairing symmetry11–16 or to a non-superconducting one.17,18

From a modeling point of view, a real disorder environ-ment in a system may be simulated by adding either ran-dom one-body or random many-body interactions. Bothtypes of interactions include the variations of either am-plitude or phase and can be further classified by the inter-action range, i.e., short range or long range. For instance,in Zn-doped cuprates like YBa2(Cu1−xZnx)3O6.9 such adisorder effect could be represented by a random set ofscalar impurity potentials with a finite range.19 In fact,disorder effects resulting from random one-body poten-tials in SCs are widely discussed,19–22 while, in contrast,those from random many-body interactions are relativelyless studied in a systematic way.23–27

The discovery of iron-based SCs has enriched the un-conventional SC physics in several aspects. One partic-ularly interesting aspect is that the isovalent doping inboth 1111 and 122 materials, which is theoretically pre-

dicted not to change electron density significantly,28–30

can suppress the stripe antiferromagnetic (AFM) or-der in the parent compounds.31,32 Moreover, it even in-duces superconductivity after intermediate doping in 122materials;33,34 in some cases, there have been observednodal structures in the superconducting gap,35–39 dis-tinct from the fully gapped one upon charge doping.These findings may provide a possible playground tostudy random many-body interactions in SCs throughthe following intuition, provided the strong-coupling pic-ture is used.40 For instance, a common consensus inBaFe2(As1−xPx)2 is that both magnetism and supercon-ductivity should occur in the FeAs plane, at which Featoms themselves form a square lattice with As atoms sit-ting above and below each plaquette center of the latticealternately. Assuming that the stripe AFM order arisesfrom the competition between nearest-neighbor (NN) J1

and next-nearest-neighbor (NNN) J2 exchange interac-tions on the square lattice, the (random) substitution ofAs by P could result in two leading effects to disorderthe system: one is to mainly suppress NNN J2 exchangeinteractions41,42 and the other is to introduce a scalarpotential at each plaquette center.

Thus, motivated by the experiments done in isovalent-doping iron-based SCs, we study the effects of purelyexchange-interaction disorder (random J2-bond dilution;see Sec. II for the definition), from “weak” to “strong”(i.e., x from 0 to 1), in a two-dimensional (2D), two-orbital t-J1-J2 model.40,43 Although this model was orig-inally developed to describe the low-energy physics ofiron-based SCs, we simply consider a relatively ideal sit-uation, to make our study be a topic of broad interest,but do not intend to claim it directly applicable to a cer-tain real material.

When the bond disorder is “strong” with 0 � x < 1in a SC with short coherence length ξ, the standardtheories for dirty SCs, valid as long as the electronmean free path l � ξ > k−1

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Abrikosov and Gorkov2 (AG) are expected to be insuf-ficient to account for the disorder effects. Therefore, weaddress these issues by using self-consistent Bogoliubov–de Gennes (BdG) formulation with the emphasis on thespatial inhomogeneity for the pairing amplitudes. Signif-icantly, we find the following: (1) As J1 . J2, the pair-ing symmetry of our model at zero temperature (T = 0)is modulated from sx2y2-wave to dx2−y2-wave symmetrywhen the “disorder strength” x becomes greater than xc;this phenomenon is further confirmed by showing theelectron density of states as a function of x. (2) TheT = 0 spectral gap Egap decreases following the sametrend of the position/disorder averaged pairing ampli-tudes, ∆sx2y2 and ∆dx2−y2 , as x grows. (3) When x is

large, the combined pairing interaction and the J2-bonddilution disorder lead to the formation of sx2y2-wave “is-lands” of length scale O(ξ) embedded in a dx2−y2 -wave“sea”; this picture is complemented by the disorder-averaged pair-pair correlation functions. (4) Due to thisinevitable formation of spatial inhomogeneity, the super-conducting Tc determined by the superfluid density ρs(T )obviously deviates from the value predicted by the AGtheory, where the pairing amplitudes are viewed as uni-formly suppressed as the disorder increases.

The rest of the paper is organized as follows. In Sec.II, we describe our model Hamiltonian and briefly sketchthe numerical method we used. We then demonstrateour numerical results in Sec. III to show the modula-tion of the pairing symmetry with bond disorder. Sev-eral disorder-dependent physical quantities are presentedto assist the understanding of this type of disorder, suchas local pairing amplitudes, density of states, spectralgap, and superfluid density. In Sec. IV, we repeat all thecalculations in Sec. III by taking away all J1 exchangeinteractions to sharpen the effects due to J2-bond dilu-tion. Finally, we conclude our findings in terms of a fewremarks and a summary.

II. MODEL AND SELF-CONSISTENT BDGTHEORY

Our adopted model Hamiltonian to capture the low-energy physics in clean iron-based superconductors isthe so-called t-J1-J2 model developed in Ref. 40, whichhas been further justified by functional renormalizationgroup study.44 Explicitly, H = H0 +Hint and the nonin-teracting part reads

H0 =∑

rr′

∑

αβ

∑

σ

(tαβrr′c

†rασcr′βσ +H.c.

)− µ

∑

rα

nrα,

(1)

where c†rασ (crασ) creates (annihilates) an electron of theα-orbital with spin σ (α = 1, 2 for two degenerate dxzand dyz orbitals, respectively) at site r. nrα is the localelectron density operators of the α-orbital.

The normal-state Fermi surfaces in the unfolded Bril-

louin zone (one iron per unit cell) can be reasonablyproduced by setting t11

r,r+x = t22r,r+y = −1.0, t22

r,r+x =

t11r,r+y = 1.3, t

11/22r,r±x±y = −0.85, t

12/21r,r+x−y = −0.85,

t12/21r,r+x+y = 0.85, and other hopping parameters as zero,

where x and y are unit vectors along the axes. For sim-plicity, we will take |t11

r,r+x| = 1 as our energy units, lat-tice constant a ≡ 1, and set chemical potential µ = 1.8,corresponding to electron density ne ≈ 2.18.

The interacting part includes several terms as follows,

Hint =∑

〈rr′〉

∑

α

J1(r, r′)(Srα · Sr′α − nrαnr′α)

+∑

〈〈rr′〉〉

∑

α

J2(r, r′)(Srα · Sr′α − nrαnr′α)

+ · · · , (2)

where Srα = c†r,α,σ~σσσ′cr,α,σ′ and nrα are the local spinand density operators with orbital index α = 1, 2. 〈rr′〉and 〈〈rr′〉〉 denote NN and NNN pairs of sites, respec-tively, and thus the first two terms represent intraorbitalexchange interactions. In addition, “· · · ” represents ourignored interorbital exchange and Hund’s coupling terms,which are shown to be unimportant in determining thepairing symmetry of the SC state in this model.

On a square lattice with N = Nx × Ny sites, the ex-change couplings J2(r, r′) are taken to be zero when the(diagonal) bonds (rr′) cross the randomly selected xNplaquettes. In this way we introduce the “disorder” intoour otherwise clean system. This kind of bond disorder isa quantum analog of bond-dilute Ising models45,46 whileit is rarely considered in superconducting systems. Phys-ically, these selected plaquettes might represent the situ-ation where the central atoms As are isovalently replacedby the atoms P, causing strong suppression of J2 bondsin plaquettes. For simplicity, we further assume that theexchange couplings J1(r, r′) = J1 are unaffected. A moredelicate choice for J1(r, r′) will be discussed later.

Following Ref. 40, we assume that the superconduc-tivity of the system can be reliably captured by mean-field approximation as long as the exchange interac-tions are small compared to the bandwidth (∼ 12|t1|).The most dominant SC order parameters in our studywould generally have the following symmetry form fac-tor: ∆α = aα cos kx cos ky−(−1)αbα(cos kx−cos ky). Therelative sign between a1(b1) and a2(b2) determines whichirreducible representation the pairing symmetry belongsto, namely, A1g for the plus sign and B1g for the minussign, given D4h point group symmetry of our model. Inthe presence of disorder, we define the (local) real-spacepairing amplitude for each orbital α as

∆α(r, r + δ) = −Jl(r, r + δ)〈cr,α,↓cr+δ,α,↑〉 (3)

with δ = ±x,±y for NN pairing (l = 1) and δ = ±x± yfor NNN pairing (l = 2). The mean-field Hamiltonian of

3

H is then written as

HMF = H0 +∑

r,δ,α

∆∗α(r, r + δ)cr,α,↓cr+δ,α,↑ +H.c.(4)

Within the BdG formalism, we diagonalize thequadratic, mean-field Hamiltonian (4) through the BdGequation,

(Krαr′β ∆αβ

∆∗αβ −K∗rαr′β

)(unr′βvnr′β

)= En

(unrαvnrα

), (5)

with Krαr′β = −trαr′β − µδrr′δαβ . The relation betweenBogoliubov quasiparticle operators γ and electron oper-

ators is crασ =∑n(unrαγnσ − σvn∗rαγ†nσ), and hence com-

bining with the definition, for instance, of the s-wavecos kx cos ky SC order parameter, this gives rise to thefollowing self-consistent conditions,

∆α(r, r + δ) =1

2

∑

n

J2(r, r + δ) tanhEn

2kBT

×(unr,αvn∗r+δ,α + vn∗r,αu

nr+δ,α), (6)

and

〈nα,r〉 = 〈c†rαcrα〉=∑

n

|vnrα|2[1− f(En)] +∑

n

|unrα|2f(En), (7)

where f(E) is the Fermi distribution function. We havestudied the model for a few sets of J1, J2 (values in a cleansystem), and a wide range of isovalent doping percentage0 < x < 1 on square lattices of sizes up to N = 32× 32.We always perform our computations on finite latticesites with periodic boundary conditions. For a given(quenched) disorder configuration, we obtain the resul-tant quasiparticle spectrum by repeatedly diagonalizingthe BdG equation (5) after each iteration of the pairingamplitudes according to self-consistency conditions (6)and (7) until sufficient accuracy is achieved (e.g., the rel-ative error of the pairing amplitudes is less than 0.01%).

III. MODULATION OF PAIRING SYMMETRYWITH BOND DISORDER

A. Pairing symmetry and local pairing amplitudes

The zero-temperature phase diagram of the t-J1-J2

model in the context of iron-based superconductors hasbeen well studied in the clean limit, x = 0.40,47 Thereare four spatial symmetries for the singlet pairing by de-coupling J1 and J2 interactions: sx2y2 , sx2+y2 , dx2−y2 ,and dxy. Given the two-orbital nature of our model andD4h point group symmetry (when taking one iron perunit cell), every spatial symmetry may even belong todifferent irreducible representations. For instance, one

can have sA1g

x2y2 (also called s± wave) and sB1g

x2y2 , or dA1g

x2−y2

and dB1g

x2−y2 .

In the presence of disorder (x 6= 0), the pairing am-plitudes could be no longer uniform and are likely to beinhomogeneous in a self-consistent manner. Therefore,we define the intraorbital spin-singlet pairing amplitudeon an NN bond as (similarly for an NNN bond pairing)

∆α (r, r + δ) = −J1

2〈cr,α,↓cr+δ,α,↑ − cr,α,↑cr+δ,α,↓〉,(8)

and three dominant local pairing amplitudes with appro-priate pairing symmetries all in A1g irreducible represen-tation,

∆sx2y2 (r) =1

8

∑

αδ′

∆α (r, r + δ′) ,

∆sx2+y2 (r) =1

8

∑

αδ

∆α (r, r + δ) ,

∆dx2−y2 (r) =1

8

∑

α

(−1)α[∆α (r, r + x)−∆α (r, r + y)

+ ∆α (r, r− x)−∆α (r, r− y)], (9)

where δ = ±x,±y and δ′ = ±x ± y. When determiningwhich pairing symmetry is most dominant for a givendisorder proportion x, we take an average over the wholelattice positions and disorder configurations for each localpairing amplitude shown in Eq. (9).

To consider the effects of the bond disorder, we firstplot the zero-temperature x vs J1 phase diagram inFig. 1(a) with corresponding averaged pairing amplitudesin Fig. 1(b), where J2 is fixed to be |t1|. One of the mostimportant observations is that as long as J1 . J2, thepairing symmetry of the superconducting ground statewould be modulated from an s±-wave to a d-wave pair-ing symmetry when the disorder x grows greater thancertain critical xc; on the contrary, as J1 & J2 the d-wave symmetry would dominate no matter what x is.

Note that all the phases appear in the phase diagrambelong to the A1g irreducible representation. Thus, in astrict sense, there should be no sharp phase transitions inour disordered system. The crossover boundary betweenphases, the green line, represents the degeneracy of the

averaged pairing amplitudes between the dominant sA1g

x2y2

and dA1g

x2−y2 symmetries, while the dashed line represents

the subdominant line between sA1g

x2y2 and sA1g

x2+y2 symme-

tries, as shown in Fig. 1(b). The pairing amplitudes be-longing to the B1g irreducible representation always havea smaller weight than those of A1g’s. So, every pairingsymmetry in this work will refer to the A1g irreduciblerepresentation and we will omit the superscript hereafter.

When x = 0 (1), the pairing amplitudes for variouspairing symmetry channels are uniform with ∆sx2y2 (r) =

0.1585 > ∆dx2−y2 (r) = 0.0991 [∆dx2−y2 (r) = 0.0194 >

∆sx2+y2 (r) = 0.0049], given J1 = 0.7 and J2 = 1 in the

system. But, as we have mentioned earlier in this subsec-tion, the pairing amplitudes are not necessary so when

4 4

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

J1

x

( )1 1

2 2 2 2

g gA A

x y x yd s

-

( )1 1

2 2 2 2

g gA A

x y x yd s

- +

1

2 2

gA

x ys

+

1

2 2

gA

x ys

( )1

2 2

gA

x yd

-

(a) (b)

FIG. 1. (a) Schematic zero-temperature phase diagram as a function of J1 and disorder x, given J2 = |t1| on a 32 × 32 square

lattice. The green line marks the boundary between the dominant dA1g

x2−y2 -wave pairing region and sA1g

x2y2 -wave pairing one,

while the dashed line marks the boundary between subdominant sA1g

x2y2 -wave pairing region and sA1g

x2+y2 -wave pairing one; the

subdominant pairing symmetries are shown within the brackets. (b) The position and disorder averaged pairing amplitudes foreach pairing symmetry as a function of J1 and x.

the uniformity of the system. As x grows from 0.1, themajor peak becomes lower and broader (|∆| are spread ina wider range of values), indicating the non-uniformity ofthe system which reaches its maximum at x = 0.5. Fur-ther increasing x, the distribution is basically reversedand the amplitudes become more uniform again.

The behavior we found here as x increases is qual-itatively similar to that seen in a usual s-wave super-conductor from weak to strong site-impurity disorder.20

(AG theory breaks down!) However, there exists a sharpfeature never seen in the simple site-impurity case: Forsx2y2-wave pairing symmetry [see Fig. 2(b)], the distribu-tion function shows a few extra sub-peaks except for themajor one for every given x. This is a unique propertywhich distinguishes sx2y2 -wave pairing from the other twopairing channels [see Figs. 2(a) and 2(c)]. In fact, this fea-ture may be physically understood as follows. First, thesx2y2-wave pairing originates from J2 exchange term, notfrom J1 term. Second, since the disorder is introduced byrandomly taking away the NNN (J2) bonds, there shouldbe five different local disorder configurations, as illustrat-ed in Fig. 2(d). Therefore, in a macroscopic system, allthe five configurations should in principle be reflected onthe distribution function in terms of the peaks observedin Fig. 2(b). (Sometimes the distribution peak corre-sponding to either the configuration I or V is too smallto be seen.)

B. Density of states and energy gap

One simple way to demonstrate a possible modulationof pairing symmetry as x increases is to calculate theelectron density of states (DOS), which is expressed as

N (ω) =1

N

∑

rαn

[|un

αr|2δ (En − ω) + |vnαr|2δ (En + ω)

],

(10)

where the delta function has the form of Cauchy-Lorenzdistribution function, δ(x) = γ/π(x2 + γ2), with an in-finitesimal scale parameter γ = 0.008.

Fig. 3 shows the DOS for different x at zero tempera-ture in a 56 × 56 lattice. Each data curve is obtained byaveraging over the whole lattice positions as well as 30disorder configurations at any given x. With increasingdisorder x, the superconducting gap around ω = 0 de-creases and evolves from U-shape like to more V-shapelike, indicating a modulation of the pairing symmetryfrom sx2y2-wave to dx2−y2 -wave. In addition, the coher-ence peaks are slowly smeared out, behaving similarly tothe case with impurity disorder.20

The evolution of the gap in the DOS result may becomplemented by the quasi-particle energy gap Egap,which is defined as the smallest positive eigenvalue ofthe BdG Hamiltonian in Eq. (5). As shown in Fig. 4,Egap decreases when x increases and it also evolves in the

same trend as (position and disorder averaged) ∆sx2y2 ;

otherwise, if there were no ∆sx2y2 component in the SC

state, Egap should be ideally zero due to nodal quasi-

FIG. 1. (a) Schematic zero-temperature phase diagram as a function of J1 and disorder x, given J2 = |t1| on a 32 × 32

square lattice. The green line marks the boundary between the dominant dA1g

x2−y2 -wave pairing region and sA1g

x2y2 -wave pairing

one, while the dashed line marks the boundary between the subdominant sA1g

x2y2 -wave pairing region and sA1g

x2+y2 -wave pairing

one; the subdominant pairing symmetries are shown within the parentheses. (b) The position and disorder averaged pairingamplitudes for each pairing symmetry as a function of J1 and x.

0 < x < 1. In Figs. 2(a)-(c), we provide a statisticaldistribution of the local pairing amplitudes P (|∆(r)|) forseveral disorders x in different pairing symmetry channelsat zero temperature. When x = 0.1 or x = 0.9, each dis-tribution function shows a major sharp peak, indicatingthe uniformity of the system. As x grows from 0.1, themajor peak becomes lower and broader (|∆| are spread ina wider range of values), indicating the nonuniformity ofthe system which reaches its maximum at x = 0.5. Fur-ther increasing x, the distribution is basically reversedand the amplitudes become more uniform again.

The behavior we found here as x increases is quali-tatively similar to that seen in a usual s-wave supercon-ductor from weak to strong site-impurity disorder.21 (AGtheory breaks down.) However, there exists a sharp fea-ture not seen in the simple site-impurity case: For sx2y2-wave pairing symmetry [see Fig. 2(b)], the distributionfunction shows a few extra subpeaks except for the majorone for every given x. This is a unique property whichdistinguishes sx2y2-wave pairing from the other two pair-ing channels [see Figs. 2(a) and 2(c)]. In fact, this featuremay be physically understood as follows. First, the sx2y2-wave pairing originates from the J2 exchange term, notfrom the J1 term. Second, since the disorder is intro-duced by randomly taking away the NNN (J2) bonds,there should be five different local disorder configura-tions, as illustrated in Fig. 2(d). Therefore, in a macro-scopic system, all five configurations should in principlebe reflected on the distribution function in terms of thepeaks observed in Fig. 2(b). (Sometimes the distributionpeak corresponding to either the configuration I or V istoo small to be seen.)

B. Density of states and energy gap

One simple way to demonstrate a possible modulationof pairing symmetry as x increases is to calculate theelectron density of states (DOS), which is expressed as

N (ω) =1

N

∑

rαn

[|unαr|2δ (En − ω) + |vnαr|2δ (En + ω)

],

(10)

where the δ function has the form of Cauchy-Lorenz dis-tribution function, δ(x) = γ/π(x2+γ2), with an infinites-imal scale parameter γ = 0.008.

Fig. 3 shows the DOS for different x at zero tempera-ture in a 56× 56 lattice. Each data curve is obtained byaveraging over the whole lattice positions as well as 30disorder configurations at any given x. With increasingdisorder x, the superconducting gap around ω = 0 de-creases and evolves from U-shape like to more V-shapelike, indicating a modulation of the pairing symmetryfrom sx2y2 wave to dx2−y2 wave. In addition, the coher-ence peaks are slowly smeared out, behaving similarly tothe case with impurity disorder.21

The evolution of the gap in the DOS result may becomplemented by the quasiparticle energy gap Egap,which is defined as the smallest positive eigenvalue of theBdG Hamiltonian in Eq. (5). As shown in Fig. 4, Egapdecreases when x increases and it also evolves in the sametrend as (position and disorder averaged) ∆sx2y2 ; other-

wise, if there were no ∆sx2y2 component in the SC state,

Egap should be ideally zero due to nodal quasiparticle ex-

5 5

(a) (b)

(c)

Ⅴ

Ⅰ

Ⅳ

Ⅲ

Ⅱ

(d)

FIG. 2. Distribution of the local pairing amplitudes P (|∆(r)|)for various disorder x at zero temperature, with J1 = 0.7and J2 = 1. (a), (b) and (c) correspond to sx2+y2 -, sx2y2 -,and dx2−y2 -wave pairing amplitudes, respectively. (d) Fivepossible disordered configurations for a local lattice site. Thedashed lines denote the NNN bond-defect (J2 = 0).

particle excitations from d-wave pairing. Note that whenx & 0.75, Egap is slightly greater than ∆sx2y2 , which is

due to the presence of ∆sx2+y2 in the system (not shown

in Fig. 4). Therefore, Egap appears to be dictated mainly

by ∆sx2y2 in a certain way. The subtle feature of it could

be further revealed by the physics we will discuss next.

C. Formation of superconducting clusters

In our study, local pairing amplitudes are self-consistently calculated at every lattice site by using BdGtheory for a given disorder realization. In particular, asmentioned in subsection IIIA, the distribution of sx2y2 -wave pairing amplitudes seems to be correlated with localdisorder configurations. Usually this should be contrast-ed with the electron density distribution. Therefore, inFigs. 5(a), 5(b), and 5(c), we plot the electron densityn(r) =

∑ασ⟨nα,r,σ⟩ and spatial variation of sx2y2 -wave

and dx2−y2 -wave pairing amplitudes, respectively, for agiven disorder realization with x = 0.8 at zero tempera-ture. Obviously, the electron density n(r) is nearly homo-geneous and uncorrelated with the given bond disorder.However, the sx2y2 -wave pairing amplitudes are inhomo-geneous and tend to form some s-wave SC “islands” on

FIG. 3. Density of states for various disorder proportions xat zero temperature, with J1 = 0.7 and J2 = 1. Each DOSfor a given x has been averaged over position and 30 disorderconfigurations.

FIG. 4. Quasi-particle gap Egap and position/disorder aver-aged pairing amplitudes ∆s

x2+y2 , ∆sx2y2 and ∆d

x2−y2 as a

function of x with J1 = 0.7 and J2 = 1.

top of a relatively less inhomogeneous d-wave SC “sea”,even though there is no intrinsic correlation between dis-ordered bonds. Note that there is also a small clusteringtendency for dx2−y2 -wave pairing amplitudes, which islikely to be induced by s-wave piece.

To further confirm the formation of s-wave SC “is-lands” without a strong correlation with a certain dis-order configuration, we display the disorder-averagedcorrelation functions ∆(r1)∆(r2) for sx2y2- and dx2−y2-wave pairing amplitudes as a function of the distancerij = |r1 − r2| at x = 0.8 in Fig. 6(a). The s-wave com-ponent shows a clear structure on a scale of two to threelattice constants, while the d-wave counterpart shows al-most no structure. Note that the coherence length in a

FIG. 2. Distribution of the local pairing amplitudes P (|∆(r)|)for various disorder x at zero temperature, with J1 = 0.7and J2 = 1. (a), (b), and (c) correspond to sx2+y2 -, sx2y2 -,and dx2−y2 -wave pairing amplitudes, respectively. (d) Fivepossible disordered configurations for a local lattice site. Thedashed lines denote the NNN bonddefect (J2 = 0).

citations from d-wave pairing. Note that when x & 0.75,Egap is slightly greater than ∆sx2y2 , which is due to the

presence of ∆sx2+y2 in the system (not shown in Fig. 4).

Therefore, Egap appears to be dictated mainly by ∆sx2y2

in a certain way. The subtle feature of it could be furtherrevealed by the physics we will discuss next.

C. Formation of superconducting islands

In our study, local pairing amplitudes are self-consistently calculated at every lattice site by using BdGtheory for a given disorder realization. In particular, asmentioned in Sec. III A, the distribution of sx2y2 -wavepairing amplitudes seems to be correlated with local dis-order configurations. Usually this should be contrastedwith the electron density distribution. Therefore, inFigs. 5(a), 5(b), and 5(c), we plot the electron densityn(r) =

∑ασ〈nα,r,σ〉 and spatial variation of sx2y2-wave

and dx2−y2 -wave pairing amplitudes, respectively, for agiven disorder realization with x = 0.8 at zero tempera-ture. Obviously, the electron density n(r) is nearly homo-geneous and uncorrelated with the given bond disorder.However, the sx2y2-wave pairing amplitudes are inhomo-geneous and tend to form some s-wave SC “islands” on

FIG. 3. Density of states for various disorder proportions xat zero temperature, with J1 = 0.7 and J2 = 1. Each DOSfor a given x has been averaged over position and 30 disorderconfigurations.

FIG. 4. Quasiparticle gap Egap and position/disorder aver-aged pairing amplitudes ∆s

x2y2 and ∆dx2−y2 as a function of

x with J1 = 0.7 and J2 = 1.

top of a relatively less inhomogeneous d-wave SC “sea,”even though there is no intrinsic correlation between dis-ordered bonds. Note that there is also a small clusteringtendency for dx2−y2 -wave pairing amplitudes, which islikely to be induced by an s-wave piece.

To further confirm the formation of s-wave SC “is-lands” without a strong correlation with a certain dis-order configuration, we display the disorder-averagedcorrelation functions ∆(r1)∆(r2) for sx2y2- and dx2−y2 -wave pairing amplitudes as a function of the distancerij = |r1 − r2| at x = 0.8 in Fig. 6(a). The s-wave com-ponent shows a clear structure on a scale of two to threelattice constants, while the d-wave counterpart shows al-most no structure. Note that the coherence length in a

66

(a)

(b) (c)

FIG. 5. (a) Electron density on a 32 × 32 lattice for a givendisorder realization with x = 0.8 at zero temperature. (b)and (c) correspond to two dominant sx2y2 - and dx2−y2 -wavepairing channels. The darker the larger density/amplitudewould be.

(a) (b)

FIG. 6. (a) The disorder-averaged correlation functions

∆(r1)∆(r2) for sx2y2 - and dx2−y2 -wave pairing amplitudesas a function of the distance rij = |r1 − r2| with x = 0.8. (b)

∆(r1)∆(r2) of sx2y2 -wave pairing for various x. Both (a) and(b) are averaged over 15 random disorder configurations and

both are scaled by ∆2(r1).

clean sx2y2 -wave SC in this model can be estimated byvF /∆ ∼ 4. Moreover, the clustering tendency for thes-wave component persists as x increases as can be seenin Fig. 6(b).

Additionally, it is worth noticing that for the first fewlowest quasi-particle excited states, each correspondingwave function distribution, |Ψ(r)|2, appears to be a prop-agating mode with a definite momentum along the diag-onal direction; along each wave front (e.g., of the largestamplitude), the probability distributes alomost uniform-ly but suppresses around s-wave SC “islands”. Such vari-

ation in |Ψ(r)|2 suggests its nature as a gapped mode.Thus, this feature might well explain why Egap > 0, asfound in Fig. 4, is dictated by the s-wave pairing channelin our inhomogeneous SC system.

D. Superfluid density, Critical temperature, andPhase diagram

The temperature dependence of the superfluid densi-ty ρs(T ) is usually viewed as a good indicator to de-termine if a superconductor is “unconventional” or not.For instance, ρs(T ), which is measurable through themagnetic penetration depth λ−2(T ), exhibits an expo-nential behavior in T in a clean (hole-doped) 122 com-pound, Ba1−xKxFe2As2, suggesting a fully gapped SC;45

while ρs(T ) behaves like a power law in an electron-dopedBa(Fe1−xCox)2As2, indicating the presence of nodes orsomething unconventional.46–48

Moreover, the qualitative behavior of ρs(T ) gives a use-ful hint to understand 1) how disorder of certain typemay affect SC (phase rigidity) at low temperatures18,20

and 2) what type of fluctuations is essential near criticaltemperature Tc.

49 Although the answer to question 2) isbeyond our mean-field study, we can still use it to ob-tain Tc in a self-consistent manner when inhomogeneouspairing is inevitable and not negligible.

We generalize the linear-response approach in Refs. 50and 51 for a multi-orbital case to obtain the superflu-id density. Considering a weak vector potential Ax(r, t)along x direction, the hopping terms in Eq. (1) are mod-ified by the Peierls phase factors −eieAx(r). Expandingthe phase factors up to second order, we obtain

HAt = Ht −

∑

r

[ejp

x(r)Ax(r) +1

2e2kx(r)A2

x(r)

],(11)

with the paramagnetic current

jpx(r) = i

∑

sσ

∑

αβ

tαβr,r+s

(c†r+s,βσcrασ − c†

rασcr+s,βσ

),

and the kinetic energy associated with x direction

kx(r) = −∑

sσ

∑

αβ

tαβr,r+s

(c†r+s,βσcrασ + c†

rασcr+s,βσ

).

According to the linear response theory,

ρs = ⟨−kx⟩ − Λxx (qx = 0, qy → 0, iω = 0) , (12)

where ⟨−kx⟩ is the kinetic energy along x direction, and

Λxx (q, iωm) =1

N2

∫ β

0

dτeiωmτ ⟨jpx (q, τ) jp

x (−q, 0)⟩.

(13)

Note that in our disordered model, we calculate the

FIG. 5. (a) Electron density on a 32 × 32 lattice for a givendisorder realization with x = 0.8 at zero temperature. (b)and (c) correspond to two dominant sx2y2 - and dx2−y2 -wavepairing channels. The darker the larger density/amplitudewould be.

6

(a)

(b) (c)

FIG. 5. (a) Electron density on a 32 × 32 lattice for a givendisorder realization with x = 0.8 at zero temperature. (b)and (c) correspond to two dominant sx2y2 - and dx2−y2 -wavepairing channels. The darker the larger density/amplitudewould be.

(a) (b)

FIG. 6. (a) The disorder-averaged correlation functions

∆(r1)∆(r2) for sx2y2 - and dx2−y2 -wave pairing amplitudesas a function of the distance rij = |r1 − r2| with x = 0.8. (b)

∆(r1)∆(r2) of sx2y2 -wave pairing for various x. Both (a) and(b) are averaged over 15 random disorder configurations and

both are scaled by ∆2(r1).

clean sx2y2 -wave SC in this model can be estimated byvF /∆ ∼ 4. Moreover, the clustering tendency for thes-wave component persists as x increases as can be seenin Fig. 6(b).

Additionally, it is worth noticing that for the first fewlowest quasi-particle excited states, each correspondingwave function distribution, |Ψ(r)|2, appears to be a prop-agating mode with a definite momentum along the diag-onal direction; along each wave front (e.g., of the largestamplitude), the probability distributes alomost uniform-ly but suppresses around s-wave SC “islands”. Such vari-

ation in |Ψ(r)|2 suggests its nature as a gapped mode.Thus, this feature might well explain why Egap > 0, asfound in Fig. 4, is dictated by the s-wave pairing channelin our inhomogeneous SC system.

D. Superfluid density, Critical temperature, andPhase diagram

The temperature dependence of the superfluid densi-ty ρs(T ) is usually viewed as a good indicator to de-termine if a superconductor is “unconventional” or not.For instance, ρs(T ), which is measurable through themagnetic penetration depth λ−2(T ), exhibits an expo-nential behavior in T in a clean (hole-doped) 122 com-pound, Ba1−xKxFe2As2, suggesting a fully gapped SC;45

while ρs(T ) behaves like a power law in an electron-dopedBa(Fe1−xCox)2As2, indicating the presence of nodes orsomething unconventional.46–48

Moreover, the qualitative behavior of ρs(T ) gives a use-ful hint to understand 1) how disorder of certain typemay affect SC (phase rigidity) at low temperatures18,20

and 2) what type of fluctuations is essential near criticaltemperature Tc.

49 Although the answer to question 2) isbeyond our mean-field study, we can still use it to ob-tain Tc in a self-consistent manner when inhomogeneouspairing is inevitable and not negligible.

We generalize the linear-response approach in Refs. 50and 51 for a multi-orbital case to obtain the superflu-id density. Considering a weak vector potential Ax(r, t)along x direction, the hopping terms in Eq. (1) are mod-ified by the Peierls phase factors −eieAx(r). Expandingthe phase factors up to second order, we obtain

HAt = Ht −

∑

r

[ejp

x(r)Ax(r) +1

2e2kx(r)A2

x(r)

],(11)

with the paramagnetic current

jpx(r) = i

∑

sσ

∑

αβ

tαβr,r+s

(c†r+s,βσcrασ − c†

rασcr+s,βσ

),

and the kinetic energy associated with x direction

kx(r) = −∑

sσ

∑

αβ

tαβr,r+s

(c†r+s,βσcrασ + c†

rασcr+s,βσ

).

According to the linear response theory,

ρs = ⟨−kx⟩ − Λxx (qx = 0, qy → 0, iω = 0) , (12)

where ⟨−kx⟩ is the kinetic energy along x direction, and

Λxx (q, iωm) =1

N2

∫ β

0

dτeiωmτ ⟨jpx (q, τ) jp

x (−q, 0)⟩.

(13)

Note that in our disordered model, we calculate the

FIG. 6. (a) The disorder-averaged correlation functions

∆(r1)∆(r2) for sx2y2 - and dx2−y2 -wave pairing amplitudesas a function of the distance rij = |r1 − r2| with x = 0.8. (b)

∆(r1)∆(r2) of sx2y2 -wave pairing for various x. Both (a) and(b) are averaged over 15 random disorder configurations and

both are scaled by ∆2(r1).

clean sx2y2-wave SC in this model can be estimated byvF /∆ ∼ 4. Moreover, the clustering tendency for thes-wave component persists as x increases as can be seenin Fig. 6(b).

Additionally, it is worth noticing that for the first fewlowest quasiparticle excited states, each correspondingwave function distribution, |Ψ(r)|2, appears to be a prop-agating mode with a definite momentum along the diag-onal direction; along each wave front (e.g., of the largestamplitude), the probability distributes almost uniformlybut suppresses around s-wave SC “islands.” Such vari-ation in |Ψ(r)|2 suggests its nature as a gapped mode.

Thus, this feature might well explain why Egap > 0, asfound in Fig. 4, is dictated by the s-wave pairing channelin our inhomogeneous SC system.

D. Superfluid density and critical temperature

The temperature dependence of the superfluid densityρs(T ) is usually viewed as a good indicator to deter-mine whether a superconductor is “unconventional” ornot. For instance, ρs(T ), which is measurable throughthe magnetic penetration depth λ−2(T ), exhibits an ex-ponential behavior in T in a clean (hole-doped) 122 com-pound, Ba1−xKxFe2As2, suggesting a fully gapped SC;48

while ρs(T ) behaves like a power law in an electron-dopedBa(Fe1−xCox)2As2, indicating the presence of nodes orsomething unconventional.49–51

Moreover, the qualitative behavior of ρs(T ) gives a use-ful hint to understand (1) how disorder of certain typemay affect SC (phase rigidity) at low temperatures19,21

and (2) what type of fluctuation is essential near criticaltemperature Tc.

52 Although the answer to question (2)is beyond our mean-field study, we can still use it to ob-tain Tc in a self-consistent manner when inhomogeneouspairing is inevitable and not negligible.

We generalize the linear-response approach in Refs. 53and 54 for a multiorbital case to obtain the superfluiddensity. Considering a weak vector potential Ax(r, t)along the x direction, the hopping terms in Eq. (1) aremodified by the Peierls phase factors −eieAx(r). Expand-ing the phase factors up to second order, we obtain

HAt = Ht −

∑

r

[ejpx(r)Ax(r) +

1

2e2kx(r)A2

x(r)

],(11)

with the paramagnetic current

jpx(r) = i∑

sσ

∑

αβ

tαβr,r+s

(c†r+s,βσcrασ − c†rασcr+s,βσ

),

and the kinetic energy associated with x direction

kx(r) = −∑

sσ

∑

αβ

tαβr,r+s

(c†r+s,βσcrασ + c†rασcr+s,βσ

).

According to the linear response theory,

ρs = 〈−kx〉 − Λxx (qx = 0, qy → 0, iω = 0) , (12)

where 〈−kx〉 is the kinetic energy along x direction, and

Λxx (q, iωm) =1

N2

∫ β

0

dτeiωmτ 〈jpx (q, τ) jpx (−q, 0)〉.

(13)

Note that in our disordered model, we calculate the

7

current-current correlation function in real space,

Λxx (r1, r2, iωm) =1

N

∫ β

0

dτeiωmτ 〈jpx (r1, τ) jpx (r2, 0)〉.

(14)

After disorder averaging, the translational invariancemay be recovered and hence we set Λxx (r1, r2, iω) =Λxx (r1 − r2, iω), followed by a Fourier transform to qspace,

Λxx (q, iω) =1

N2

∑

r1r2

e−iq·(r1−r2)Λxx (r1, r2, iω) .(15)

7

current-current correlation function in real space,

Λxx (r1, r2, iωm) =1

N

∫ β

0

dτeiωmτ ⟨jpx (r1, τ) jp

x (r2, 0)⟩.

(14)

After disorder averaging, the translational invariancemay be recovered and hence we set Λxx (r1, r2, iω) =Λxx (r1 − r2, iω), followed by a Fourier transform to q-space,

Λxx (q, iω) =1

N2

∑

r1r2

e−iq·(r1−r2)Λxx (r1, r2, iω) .(15)

(a) (b)

FIG. 7. (a) Temperature dependence of superfluid densityρs(T ) and energy gap Egap for various disorder proportion x.

FIG. 8. Critical temperature Tc as a function of x, obtainedby quasi-particle gap minimum Egap, superfluid density ρs,and the AG-type calculation with enforced uniform ∆(ri),respectively.

In Fig. 7(a), we show the temperature dependence ofρs(T ), normalized by ρs(0), at various x. At any given x,ρs(T ) deviates from ρs(0) exponentially at low tempera-tures and then decreases until reaching zero at a critical

T (called Tc). This can be contrasted by the minimumquasi-particle energy Egap as a function of T depictedin Fig. 7(b) with a similar trend. Such an exponentialbehavior strongly suggests the SC ground state of thesystem is fully gapped.

A few remarks are worth mentioning here. First, un-like in the case of site-impurity disorder, ρs(0) is basicallynot suppressed as the disorder strength x increases, in-dicating that the phase coherence for electrons is almostunaffected. Second, at x > xc even though the d-wavecomponent is dominant, there could be no gapless Bogoli-ubov quasi-particles, as indicated in Fig. 7(b) at T < Tc.Third, ‘Tc’s determined by ρs(T ) and Egap(T ) are notnecessarily the same. In fact, Egap usually overestimatesTc slightly in our study (see Fig. 8). This is because asystem may have certain local pairing amplitudes whilecompletely lose the phase coherence among them. There-fore, Tc determined by ρs would be more reliable.

Typically the standard AG theory provides an estima-tion of the suppressed Tc in a disordered superconduc-tor. The suppressed Tc, ∆T , is basically proportional tothe disorder scattering rate, x/N(0), times a constant ofO(1). The essential step in its derivation is to replace∆(r) by the spatially averaged one. Therefore, we alsocalculate the quasi-particle gap with an enforced uniformorder parameter ∆(ri) in the self-consistent mean-field e-quations and summarize all of our obtained Tc in Fig. 8.It is important to observe that Egap-determined Tc viathe inhomogeneous solution in the self-consistent man-ner agrees well with the ρs-determined one, but both areaway from the AG-type Tc when 0 < x < 1. This strong-ly implies the breakdown of the conventional AG-typetheory for the bond-disordered system.

IV. SYSTEM WITHOUT J1

We have studied a J2-bond disordered t-J1-J2 modelin the previous sections. However, the presence of J1

terms may couple with J2 terms in a complicated wayand features purely from the bond disorder could be stillobscure. Since this type of disorder is interesting on itsown right, in this section, we repeat previous calculationsby setting J1 = 0.

Focusing on sx2y2-wave only, Fig. 9(b) shows the dis-tribution of the spatial pairing amplitudes at zero tem-perature with its corresponding disorder realization atx = 0.4 depicted in Fig. 9(a). The trend to form “su-perconducting islands” is clear [see Figs. 9(c)→(b)→(d)]:As x & 0.5, “Superconducting islands” are formed whilegradually become smaller until x approaches 1. These“islands” are instead separated from one another by ametallic “sea” (with negligible ∆(r)). This can be moreaccurately confirmed by the disorder-averaged correla-tion function ∆(r1)∆(r2) for various x, as shown inFig. 10.

We next examine the quasi-particle minimum gap Egap

as a function of the disorder proportion x at zero temper-

FIG. 7. (a) Temperature dependence of superfluid densityρs(T ) and energy gap Egap for various disorder proportionsx.

FIG. 8. Critical temperature Tc as a function of x, obtainedby quasiparticle gap minimum Egap, superfluid density ρs,and the AG-type calculation with enforced uniform ∆(ri),respectively.

In Fig. 7(a), we show the temperature dependence ofρs(T ), normalized by ρs(0), at various x. At any given x,ρs(T ) deviates from ρs(0) exponentially at low tempera-tures and then decreases until reaching zero at a criticalT (called Tc). This can be contrasted by the minimum

quasiparticle energy Egap as a function of T depictedin Fig. 7(b) with a similar trend. Such an exponentialbehavior strongly suggests the SC ground state of thesystem is fully gapped.

A few remarks are worth mentioning here. First, un-like in the case of site-impurity disorder, ρs(0) is basi-cally not suppressed as the disorder strength x increases,indicating that the phase coherence for electrons is al-most unaffected. Second, at x > xc even though thed-wave component is dominant, there could be no gap-less Bogoliubov quasiparticles, as indicated in Fig. 7(b)at T < Tc. Third, the Tc’s determined by ρs(T ) andEgap(T ) are not necessarily the same. In fact, Egap usu-ally overestimates Tc slightly in our study (see Fig. 8).This is because a system may have certain local pairingamplitudes while completely losing the phase coherenceamong them. Therefore, Tc determined by ρs would bemore reliable.

Typically the standard AG theory provides an estima-tion of the suppressed Tc in a disordered superconduc-tor. The suppressed Tc, ∆T , is basically proportional tothe disorder scattering rate, x/N(0), times a constant ofO(1). The essential step in its derivation is to replace∆(r) by the spatially averaged one. Therefore, we alsocalculate the quasiparticle gap with an enforced uniformorder parameter ∆(ri) in the self-consistent mean-fieldequations and summarize all of our obtained Tc in Fig. 8.It is important to observe that Egap-determined Tc viathe inhomogeneous solution in the self-consistent man-ner agrees well with the ρs-determined one, but both areaway from the AG-type Tc when 0 < x < 1. This stronglyimplies the breakdown of the conventional AG-type the-ory for the bond-disordered system.

IV. SYSTEM WITHOUT J1

We have studied a J2-bond disordered t-J1-J2 modelin the previous sections. However, the presence of J1

terms may couple with J2 terms in a complicated wayand features purely from the bond disorder could be stillobscure. Since this type of disorder is interesting in itsown right, in this section, we repeat previous calculationsby setting J1 = 0.

Focusing on sx2y2 wave only, Fig. 9(b) shows the dis-tribution of the spatial pairing amplitudes at zero tem-perature with its corresponding disorder realization atx = 0.4 depicted in Fig. 9(a). The trend to form “su-perconducting islands” is clear [see Figs. 9(c)→(b)→(d)]:As x & 0.5, “superconducting islands” are formed whilegradually becoming smaller until x approaches 1. These“islands” are instead separated from one another by ametallic “sea” (with negligible ∆(r)). This can be moreaccurately confirmed by the disorder-averaged correla-tion function ∆(r1)∆(r2) for various x, as shown inFig. 10.

We next examine the quasiparticle minimum gap Egapas a function of the disorder proportion x at zero temper-

8 8

(a) (b)

(c) (d)

FIG. 9. (a) A disordered pattern with x = 0.4. The purpleregions mark the points with J2 = 0. (b) The spatial distri-bution of local pairing amplitudes corresponding to (a). Thedarker purple color indicates the regions with larger pairingamplitudes. (c) and (d) are another two spatial distributionsof local pairing amplitudes with x = 0.1 and x = 0.7 re-spectively. The darker purple color has the same meaningexplained in (b)

.

ature, as illustrated in Fig. 11. Clearly, Egap follows the

same trend as ∆sx2y2 . Note that even though the system

tends to form the “superconducting islands” as x grows,the superfluid stiffness ρs doesn’t vanish until x = 1 (SeeFig. 12). This is because, within our BdG formalism, weignored the possible phase fluctuations between SC is-lands, which can frustrate Cooper pairs to form a coher-ent superconducting state. Thus, in principle, one maynot rule out the occurrence of a superconductor-metaltransition in this purely academic model. Moreover, inFig. 12 Tc determined by either Egap or superfluid densityρs is consistent with each other; while AG-type calcula-tion is again shown to underestimate Tc, reflecting on theimportance of the inhomogeneous pairing amplitudes.

V. DISCUSSION AND CONCLUSION

Before concluding our work, there are a few addition-al remarks worth mentioning here, which are relevant toisovalent doping in iron-based SCs. Firstly, as we havementioned in Sec. II, in the strong coupling picture theleading effect of the substitution of P for As reduces se-riously the exchange interaction J2 (presumably super-exchange type). However, a number of subleading effectsare also anticipated. For instance, one might consider

FIG. 10. The correlation function ∆(r1)∆(r2) for sx2y2 -pairing amplitudes as a function of the distance rij = |r1−r2|with different x, averaged by 15 configurations and scaled by∆2(r1).

FIG. 11. The energy gap Egap as a function of the disor-der proportion x at zero temperature. With x raising, Egap

reduces gradually and reaches zero at x = 1.0.

to include the weakened NN exchange interaction J1 andthe presence of short-range impurity potentials. Both ofthem are basically pair breakers: The former suppress-es the d-wave superconductivity, while the latter coulderase the phase information or even insulate originallysuperconducting regions. As long as the ratio of the im-purity potential strength to J1 is much less than one, ourresults shown in the previous sections may remain validqualitatively.

Secondly, although our prediction on the modulation ofthe pairing symmetry may be applicable for the isovalent-doping 1111 iron-based SCs at finite temperatures, whereLaOFeAs1−xPx (x = 1) has been reported to have gap

FIG. 9. (a) A disordered pattern with x = 0.4. The blackregions mark the points with J2 = 0. (b) The spatial distri-bution of local pairing amplitudes corresponding to (a). Thedarker purple color indicates the regions with larger pairingamplitudes. (c) and (d) are another two spatial distributionsof local pairing amplitudes with x = 0.1 and x = 0.7, re-spectively. The darker purple color has the same meaningexplained in (b)

.

ature, as illustrated in Fig. 11. Clearly, Egap follows the

same trend as ∆sx2y2 . Note that even though the system

tends to form the “superconducting islands” as x grows,the superfluid stiffness ρs does not vanish until x = 1 (seeFig. 12). This is because, within our BdG formalism, weignored the possible phase fluctuations between SC is-lands, which can frustrate Cooper pairs to form a coher-ent superconducting state. Thus, in principle, one maynot rule out the occurrence of a superconductor-metaltransition in this purely academic model. Moreover, inFig. 12 the Tc’s determined by Egap and by the super-fluid density ρs are consistent with each other, while theAG-type calculation is again shown to underestimate Tc,reflecting the importance of the inhomogeneous pairingamplitudes.

V. DISCUSSION AND CONCLUSION

Before concluding our work, there are a few additionalremarks worth mentioning here, which are relevant toisovalent doping in iron-based SCs. First, as we havementioned in Sec. II, in the strong-coupling picture theleading effect of the substitution of P for As reduces seri-ously the exchange interaction J2 (presumably superex-change type).55 However, a number of subleading effects

FIG. 10. The correlation function ∆(r1)∆(r2) for sx2y2 -pairing amplitudes as a function of the distance rij = |r1−r2|with different x, averaged by 15 configurations and scaled by∆2(r1).

FIG. 11. Quasiparticle gap Egap and position/disorder aver-aged pairing amplitudes ∆s

x2y2 as a function of x with J1 = 0

and J2 = 1.

are also anticipated. For instance, one might considerincluding the weakened NN exchange interaction J1 andthe presence of short-range impurity potentials. Both ofthem are basically pair breakers: The former suppressesthe d-wave superconductivity, while the latter could erasethe phase information or even insulate originally super-conducting regions. As long as the ratio of the impuritypotential strength to J1 is much less than 1, our resultsshown in the previous sections may remain valid qualita-tively.

Second, although our prediction on the modulation ofthe pairing symmetry may be applicable for the isovalent-doping 1111 iron-based SCs at finite temperatures, where

9

FIG. 12. Superconducting critical temperature Tc as a func-tion of x, with J1 = 0, J2 = 1.

LaOFeAs1−xPx (x = 1) has been reported to have gapnodes,35,36 our theory cannot sufficiently describe 122systems. This is because we have so far ignored the mag-netic and orbital fluctuations, which are believed to playimportant roles in exhibiting either an antiferromagneticorder or a nematic state.27 We will refer this more genericconsideration to a future study.

In summary, we studied the bond disorder effects, from“weak” to “strong,” in an unconventional superconduc-tor described by the two-orbital t-J1-J2 model. We usedthe self-consistent Bogoliubov–de Gennes formulation to

emphasize the necessity of the spatial inhomogeneity forthe pairing amplitudes. In particular, we found that aslong as J1 . J2, the pairing symmetry of our model atT = 0 can be modulated from sx2y2-wave to dx2−y2 -wavesymmetry when the “disorder strength” x goes beyondxc. This result was best presented by the electron den-sity of states as a function of x and could be partiallyjustified by any negative evidence in experiments aboutprobing fully gapped s±-wave order.56–58 Moreover, whenx is large, we observed the formation of sx2y2-wave “is-lands” with length scale O(ξ) embedded in a dx2−y2 -wave“sea,” due to the combined pairing interaction and theJ2-bond disorder. As a consequence, Tc determined bythe superfluid density ρs(T ) is found to deviate from thepredicted value by the AG theory, suggesting its insuffi-ciency to describe the bond disorder effects.

ACKNOWLEDGMENTS

We would like to acknowledge Fan Yang, Jiang-ping Hu, E. W. Carlson, and Elbio Dagotto for stim-ulating discussions. W.F.T. is supported in partby MOST in Taiwan under Grant No.103-2112-M-110-008-MY3 and the Thousand-Young-Talent Programof China. Y.T.K. and D.X.Y. acknowledge supportfrom NBRPC-2012CB821400, NSFC-11574404, NSFC-11275279, NSFG-2015A030313176, Special Program forApplied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase), Shanhai Fo-rum, and Fundamental Research Funds for the CentralUniversities of China.

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(2012).9 X. Chen, P. Dai, D. Feng, T. Xiang, and F.-C. Zhang,

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