Modulation of pairing symmetry with bond disorder in unconventional superconductors

3.2 MB·

by  Yao-Tai Kang

Modulation of Pairing Symmetry with Bond Disorder in Unconventional
Superconductors
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
Westudyatwo-orbitalt-J -J model,originallydevelopedtodescribeiron-basedsuperconductors
1 2
at low energies, in the presence of bond disorder (via next nearest-neighbor J -bond dilution). By
2
using Bogoliubov-de Gennes approach, we self-consistently calculate the local pairing amplitudes
7 and the corresponding density of states, which demonstrate a change of dominant pairing symme-
1
try from s -wave to d-wave when increasing disorder strength as long as J1 (cid:46) J2. Moreover, the
0 ±
combined pairing interaction and strong bond disorder lead to the formation of s -wave “islands”
2 ±
with length scale of the superconducting coherence length embedded in a d-wave “sea”. This pic-

n tureisfurthercomplementedbythedisorder-averagedpair-paircorrelationfunctions,distinctfrom
a the case with potential disorder, where the “sea” is insulating. Due to this inevitable formation
J of spatial inhomogeneity, the superconducting T determined by the superfluid density ρ (T) obvi-
c s
3 ously deviates the predicted value by the conventional Abrikosov-Gorkov theory, where the pairing
1 amplitudes are viewed as uniformly suppressed as the disorder increases.

]
n I. INTRODUCTION dicted not to change electron density significantly,27–29
o
can suppress the stripe antiferromagnetic (AFM) or-
c
- der in the parent compounds.30,31 Moreover, it even in-
r Studying disorder effects in superconductors (SCs) is ducessuperconductivityafterintermediatedopingin122
p usuallybeneficial,thoughnotinadirectmanner,forun-
u materials;32,33 in some cases, there have been observed
veilingtheirunderlyingpairingmechanism. Aneconomic
s nodal structures in the superconducting gap,34–38 dis-
t. early indicator of the unconventional nature for a super- tinct from the fully gapped one upon charge doping.
a conductor, for instance, can be the sensitivity of the su-
These findings may provide a possible playground to
m perconductingtransitiontemperature(T )toanamount
c study random many-body interactions in SCs through
of finite disorder.1–5 In addition, most unconventional
- thefollowingintuition, providedthe strongcoupling pic-
d SCs, such as cuprates6 and iron-based SCs7–9, become
ture is used.39 For instance, a common consensus in
n superconductingafterdoping,whichnaturallyintroduces
co certain types of disorder in the materials. These mate- BduacFtei2v(itAys1s−hoxuPlxd)2ociscutrhaintbFoeAthsmplaagnnee,taistmwahnicdhsFuepeartcoomns-
rials are often complicated in composition and are inter-
[ themselves form a square lattice with As atoms sitting
mediately or strongly coupled systems, causing a rather
above and below each plaquette center of the lattice al-
1 complexphasediagramwithintertwinedordersinwhich
v ternately. Assuming that the stripe AFM order arises
disorder might play a role.10 With substantial amounts
4 from the competition between nearest-neighbor (NN) J
of disorder, a SC may even undergo a zero-temperature 1
4 and next nearest-neighbor (NNN) J exchange interac-
5 quantum phase transition to another superconducting tions on the square lattice, the (rand2om) substitution of
3 phase with distinct pairing symmetry11–15 or to an non-
AsbyPcouldresultintwoleadingeffectstodisorderthe
0 superconducting one.16,17
system: one is to mainly suppress NNN J exchange in-
. 2
1 From modeling point of view, a real disorder environ- teractionsandtheotheristointroduceascalarpotential
0 mentinasystemmaybesimulatedbyeitheraddingran- at each plaquette center.
7 dom one-body or many-body interactions. Both types Thus, motivatedbytheexperimentsdoneinisovalent-
1
: of interactions include the variations of either amplitude doping iron-based SCs, we study the effects of purely
v or phase and can be further classified by the interac- exchange-interactiondisorder(randomJ -bonddilution,
2
Xi tion range, i.e., short range or long range. For instance, see Sec. II for the definition), from “weak” to “strong”
in Zn-doped cuprates like YBa2(Cu1 xZnx)3O6.9 such a (i.e., x from 0 to 1), in a two-dimensional (2D), two-
ar disorder effect could be represented−by a random set of orbital t-J -J model.39,40 Although this model is orig-
1 2
scalar impurity potentials with a finite range.18 In fact, inally developed to describe the low-energy physics of
disorder effects resulting from random one-body poten- iron-based SCs, we simply consider a relatively ideal sit-
tials in SCs are widely discussed,18–21 while, in contrast, uation, make our study to be a topic of broad interest,
thosefromrandommany-bodyinteractionsarerelatively butdonotintendtoclaimitdirectlyapplicabletoacer-
less studied in a systematic way.22–26 tain real material.
The discovery of iron-based SCs has enriched the un- When the bond disorder is “strong” with 0 x < 1
(cid:28)
conventional SC physics in several aspects. One partic- in a SC with short coherence length ξ, the standard the-
ularly interesting aspect is that the isovalent doping in ories for dirty SCs, valid as long as the electron mean
both 1111 and 122 materials, which is theoretically pre- free path l (cid:29) ξ > kF−1, by Anderson1 and by Abrikosov
2
and Gorkov2 (AG) are expected to be insufficient to ac- produced by setting t11 = t22 = 1.0, t22 =
r,r+xˆ r,r+yˆ − r,r+xˆ
countforthedisordereffects. Therefore,weaddressthese t11 = 1.3, t11/22 = 0.85, t12/21 = 0.85,
issuesbyusingself-consistentBogoliubov-deGennesfor- tr1,2r/+2y1ˆ = 0.85,ra,rn±dxˆ±oyˆther h−opping pra,rr+amxˆ−eyˆters as−zero,
mulationwiththeemphasisofthespatialinhomogeneity r,r+xˆ+yˆ
where xˆ and yˆ are unit vectors along the axes. For sim-
for the pairing amplitudes. Significantly, we find that:
plicity, we will take t11 =1 as our energy units, lat-
1) As J1 (cid:46) J2, the pairing symmetry of our model at | r,r+xˆ|
tice constant a 1, and set chemical potential µ = 1.8,
zero-temperature (T = 0) is modulated from s -wave
x2y2 ≡
corresponding to electron density n 2.18.
to dx2 y2-wave symmetry when the “disorder strength” e ≈
x beco−mes greater than x ; this phenomenon is further
c
The interacting part includes several terms as follows,
confirmed by showing the electron density of states as a
function of x. 2) The T =0 spectral gap E decreases
gap H = J (r,r)(S S n n )
following the same trend of the position/disorder aver- int 1 (cid:48) rα· r(cid:48)α− rα r(cid:48)α
agedpairingamplitudes,∆sx2y2 and∆dx2 y2,asxgrows. (cid:104)(cid:88)rr(cid:48)(cid:105)(cid:88)α
3)Whenxislarge,thecombinedpairing−interactionand + J2(r,r(cid:48))(Srα·Sr(cid:48)α−nrαnr(cid:48)α)
the J2-bond dilution disorder lead to the formation of (cid:104)(cid:104)(cid:88)rr(cid:48)(cid:105)(cid:105)(cid:88)α
sx2y2-wave “islands” of length scale (ξ) embedded in a + , (2)
d -wave “sea”; this picture is coOmplemented by the ···
x2 y2
diso−rder-averagedpair-paircorrelationfunctions. 4)Due where Srα = c†r,α,σ(cid:126)σσσ(cid:48)cr,α,σ(cid:48) and nrα are the local
tothisinevitableformationofspatialinhomogeneity,the spin and density operators with orbital index α = 1,2.
superconductingTc determinedbythesuperfluiddensity (cid:104)rr(cid:48)(cid:105) and (cid:104)(cid:104)rr(cid:48)(cid:105)(cid:105) denote NN and NNN pairs of sites, re-
ρ (T) obviously deviates the predicted value by the AG spectively, and thus the first two terms represent intra-
s
theory, where the pairing amplitudes are viewed as uni- orbital exchange interactions. In addition, “ ” repre-
···
formly suppressed as the disorder increases. sent our ignored inter-orbital exchange and Hund’s cou-
The rest of the paper is organized as follows. In Sec. pling terms, which are shown to be unimportant on de-
II,wedescribeourmodelHamiltonianandbrieflysketch termining the pairing symmetry of the SC state in this
the numerical method we used. We then demonstrate model.
our numerical results in Sec. III to show the modulation
of the pairing symmetry with bond disorder. Several On a square lattice with N = Nx Ny sites, the ex-
×
disorder-dependent physical quantities are presented to change couplings J2(r,r(cid:48)) are taken to be zero when the
assist the understanding for this type of disorder, such (diagonal) bonds (rr(cid:48)) cross the randomly selected xN
as local pairing amplitudes, density of states, spectral plaquettes. In this way we introduce the “disorder” into
gap, superfluid density and so on. In Sec. IV, we repeat ourotherwisecleansystem. Thiskindofbonddisorderis
all the calculations in Sec. III by taking away all J ex- a quantum analog of bond-dilute Ising models42,43 while
1
changeinteractionstosharpentheeffectsduetoJ -bond is rarely considered in superconducting systems. Physi-
2
dilution. Finally, we conclude our findings in terms of a cally,theseselectedplaquettesmightrepresentthesitua-
few remarks and a summary. tion where the central atoms As are isovalently replaced
by the atoms P, causing strong suppression of J bonds
2
in plaquettes. For simplicity, we further assume that the
exchange couplings J (r,r) = J are unaffected. More
II. MODEL AND SELF-CONSISTENT BDG 1 (cid:48) 1
delicate choice for J (r,r) will be discussed later.
THEORY 1 (cid:48)
Following Ref. 39, we assume that the superconduc-
Our adopted model Hamiltonian to capture the low- tivity of the system can be reliably captured by mean-
energy physics in clean iron-based superconductors is field approximation as long as the exchange interac-
the so-called t-J1-J2 model developed in Ref. 39, which tions are small compared to the bandwidth ( 12t1 ).
has been further justified by functional renormalization The most dominant SC order parameters in o∼ur s|tud|y
group study.41 Explicitly, H = H0 +Hint and the non- would generally have the following symmetry form fac-
interacting part reads tor: ∆ =a cosk cosk ( 1)αb (cosk cosk ). The
α α x y α x y
− − −
relativesignbetweena (b )anda (b )determineswhich
H = tαβc c +h.c. µ n , 1 1 2 2
0 rr(cid:48) †rασ r(cid:48)βσ − rα irreducible representation the pairing symmetry belongs
(cid:88)rr(cid:48) (cid:88)αβ (cid:88)σ (cid:16) (cid:17) (cid:88)rα to, namely, A1g for the plus sign and B1g for the minus
(1) sign, given D point group symmetry of our model. In
4h
the presence of disorder, we define the (local) real-space
where c (c ) creates (annihilates) an electron of α-
†rασ rασ pairing amplitude for each orbital α as,
orbital with spin σ (α = 1,2 for two degenerate “d ”
xz
and “dyz” orbitals, respectively) at site r. nrα is the ∆α(r,r+δ)= Jl(r,r+δ) cr,α, cr+δ,α, (3)
localelectrondensityoperatorswithspinpolarizationα. − (cid:104) ↓ ↑(cid:105)
The normal-state Fermi surfaces in the unfolded Bril- with δ = xˆ, yˆfor NN pairing (l=1), and δ = xˆ yˆ
± ± ± ±
louin zone (one-iron per unit cell) can be reasonably for NNN pairing (l=2). The mean-field Hamiltonian of
3
H is then written as plitudes could be no longer uniform and are likely to be
inhomogeneous in a self-consistent manner. Therefore,
HMF =H + ∆ (r,r+δ)c c +h.c..(4)
0 ∗α r,α, r+δ,α, wedefinetheintra-orbitalspin-singletpairingamplitude
↓ ↑
r(cid:88),δ,α on an NN bond as (similar for an NNN bond pairing)
Within BdG formalism, we diagonalize the quadratic, J
1
mean-field Hamiltonian (4) through the BdG equation, ∆α(r,r+δ)= cr,α, cr+δ,α, cr,α, cr+δ,α, ,(8)
− 2 (cid:104) ↓ ↑− ↑ ↓(cid:105)
Kˆ ∆ˆ un un andthreedominantlocalpairingamplitudeswithappro-
(cid:18) ∆ˆrα∗αrβ(cid:48)β −Kˆr∗ααβr(cid:48)β (cid:19)(cid:18) vrnr(cid:48)(cid:48)ββ (cid:19)=En(cid:18) vrnrαα (cid:19), (5) ptartiaiotne,pairingsymmetriesallinA1g irreduciblerepresen-
with Kˆ = t µδ δ . The relation between
rαr(cid:48)β − rαr(cid:48)β − rr(cid:48) αβ 1
Bogoliubov quasi-particle operators γ and electron op- ∆sx2y2(r)= 8 ∆α(r,r+δ(cid:48)),
erators is crασ = n(unrαγnσ − σvrnα∗γn†σ¯), and hence (cid:88)αδ(cid:48)
combining with the definition, for instance, of s-wave 1
coskx cosky SC or(cid:80)der parameter, this gives rise to the ∆sx2+y2(r)= 8 ∆α(r,r+δ),
· αδ
following self-consistent conditions, (cid:88)
1
∆ (r)= [∆ (r,r+xˆ) ∆ (r,r+yˆ)
∆α(r,r+δ)=21 J2(r,r+δ)tanh2kEnT dx2−y2 8(cid:88)α α − α
((cid:88)unn vn +vn un B), (6) +∆α(r,r−xˆ)−∆α(r,r−yˆ)], (9)
× r,α r+∗δ,α r,∗α r+δ,α where δ = xˆ, yˆ and δ = xˆ yˆ. When determining
(cid:48)
± ± ± ±
and which pairing symmetry is most dominant for a given
disorderproportionx,wetakeanaverageoverthewhole
n = c c
(cid:104) α,r(cid:105) (cid:104) †rα rα(cid:105) latticepositionsanddisorderconfigurationsforeachlocal
= vn 2[1 f(E )]+ un 2f(E ), (7) pairing amplitudes shown in Eq. (9).
| rα| − n | rα| n To consider the effects of the bond disorder, we first
n n
(cid:88) (cid:88)
plot the zero-temperature x vs. J phase diagram in
1
where f(E) is the Fermi distribution function. We have
Fig.1(a)withcorrespondingaveragedpairingamplitudes
studiedthemodelforafewsetsofJ ,J (valuesinaclean
1 2 inFig.1(b), whereJ isfixedtobe t . Oneofthemost
2 1
system),andawiderangeofisovalentdopingpercentage | |
important observations is that as long as J1 (cid:46) J2, the
0<x<1 on square lattices of sizes up to N =32 32.
pairing symmetry of the superconducting ground state
×
We always perform our computations on a finite lattice
would be modulated from a s -wave to a d-wave pairing
sites with periodic boundary conditions. For a given symmetry when the disorder±x grows greater than cer-
(quenched) disorder configuration, we obtain the resul-
tain critical xc; on the contrary, as J1 (cid:38) J2 the d-wave
tant quasi-particle spectrum by repeatedly diagonalizing
symmetry would dominate no matter what x is.
BdGEq.(5)aftereachiterationofthepairingamplitudes
Note that all the phases appear in the phase diagram
according to self-consistency conditions (6) and (7) until
belong to A irreducible representation. Thus, in a
1g
sufficient accuracy is achieved (e.g., the relative error of
strict sense, there should be no sharp phase transitions
the pairing amplitudes is less than 0.01%).
in our disordered system. The crossover boundary be-
tween phases, the green line, represents the degeneracy
oftheaveragedpairingamplitudesbetweenthedominant
III. MODULATION OF PAIRING SYMMETRY sA1g and dA1g symmetries, while the dashed line rep-
x2y2 x2 y2
WITH BOND DISORDER resents the su−bdominant line between sA1g and sA1g
x2y2 x2+y2
symmetries, as shown in Fig. 1(b). The pairing ampli-
A. Pairing symmetry and local pairing amplitudes
tudes belonging to B irreducible representation always
1g
haveasmallerweightthanthoseofA ’s. So,everypair-
1g
The zero-temperature phase diagram of t-J -J model
1 2 ing symmetry in this work will refer to A irreducible
1g
in the context of iron-based superconductors has been
representationandwewillomitthesuperscripthereafter.
well studied in the clean limit, x=0.39,44 There are four
When x = 0 (1), the pairing amplitudes for various
spatial symmetries for the singlet pairing by decoupling
pairingsymmetrychannelsareuniformwith∆ (r)=
J and J interactions: s , s , d , and d . sx2y2
1 2 x2y2 x2+y2 x2 y2 xy 0.1585 > ∆ (r) = 0.0991 (∆ (r) = 0.0194 >
Giventhetwo-orbitalnatureofourmodela−ndD4h point ∆ (r) =dx20−.0y2049), given J = 0d.x72−ayn2d J = 1 in the
groupsymmetry(whentakingoneironperunitcell),ev- sx2+y2 1 2
ery spatial symmetry may even belong to different irre- system. But,aswehavementionedearlierinthissubsec-
duciblerepresentations. Forinstance,onecanhavesA1g tion, the pairing amplitudes are not necessary to be so
x2y2 when0<x<1. InFigs.2(a)-(c),weprovideastatistical
(also called s -wave) and sBx21yg2, or dAx21gy2 and dxB21gy2. distributionofthelocalpairingamplitudesP(∆(r))for
± − − | |
In the presence of disorder (x = 0), the pairing am- severaldisorderxindifferentpairingsymmetrychannels
(cid:54)
44
1.0
sA1g
0.8 x2+y2 dxA21-gy2(sxA21g+y2)
0.6
x
0.4 sA1g dA1g (sA1g )
x2y2 x2-y2 x2y2
( )
0.2 dxA21-gy2
0.0 (cid:1876)
0.0 0.5 1.0 1.5 2.0
J1 (cid:1836)(cid:2869)
(a) (b)
wFlaIhtFlwGtailiIht.ecGtie1ilt.ce..he1t.e(T.hadeh()TaaedhS)saecghShsrecgheedhreemeednlemainnltlaiieinnctlieimenczmeeamzrreamoakrr-orsaktk-resttskmehtsmhtephtepebhereboearbouatbuountouunrduerndaenadrpdyrpahyarhabryabsyeseebtebtwdewdeteiteaiweawegnegnereraesnasnmumubtbthdahdaesoesommadadioiofnfnumumaannninicncnttttaaiissnonoAxAxtntn2211yyddggoo22xAAxff--2211wwJJ−−gg11aayyvv22aaee--nnwwdpdpaaaavdvdiiereirisisinopnopgragraddiirrerreeireinrgngxgixgio,o,nrngregeigavigavineoinedonndnnsJasJAx2anxA221n2d=+g1d=+gysy|2tsAx|2-1txAw2-11|yw2g1a|2oygva-2onvew-newaapavapa3evia23rei2ip×rnaip×gni3arg2ioi3rnn2oisgneqns;gueoqt;anuohretaneeh,reee,
susbudbodmoimniannatnptapiariinrigngsysmymmmetertireisesaarereshshoowwnnwwitithhinintthheebbrraacckkeettss.. ((bb))TThheeppoossiittiioonnaannddddiissoorrddeerraavveerraaggeeddppaairiirninggamamplpitliutduedsesfofror
each pairing symmetry as a function of J and x.
each pairing symmetry as a function of J 1and x.
1
at zero temperature. When x=0.1 or x=0.9, each dis- B. Density of states and energy gap
the uniformity of the system. As x grows from 0.1, the B. Density of states and energy gap
tribution function shows a major sharp peak, indicating
majorpeakbecomeslowerandbroader(∆ arespreadin
awthideeurnriafonrgmeiotyfvoafluthese),syinstdeimca.tiAngstxhegrno|owns|-ufrnoimfor0m.1it,ythoef One simple way to demonstrate a possible modulation
thmeasyjosrtepmeawkhbiecchomreeasclhoewseirtasnmdabxriomaudemr(a|t∆x|a=re0s.p5r.eaFdurin- ofOpnaeirisnimg psylemwmaeytrtyo dasemxoninsctrreaatseeas pisostsoibclaelcmuoladtuelatthieon
thaerwiindcerreraasninggeoxf,vathlueesd)i,sitnrdibicuattiionngitshebnaosinc-aulnlyiforremveirtyseodf oelfecptariornindgensysimtymoeftsrtyataess (xDiOnScr)e,awshesichisistoexcparlecsusleadteasthe
the system which reaches its maximum at x=0.5. Fur- electron density of states (DOS), which is expressed as
and the amplitudes become more uniform again.
ther increasing x, the distribution is basically reversed 1
N(ω)= un 2δ(E ω)+ vn 2δ(E +ω) ,
and the amplitudes become more uniform again. N(ω)= N1 |uαnr|2δ(En− ω)+| vαnr| 2δ(En +ω) ,
N (cid:88)rαn(cid:2)| αr| n− | αr| n (cid:3)
rαn (10)
∑[ ]
(10)
where the delta function has the form of Cauchy-Lorenz
The behavior we found here as x increases is qual- wdihsterriebuthtieondeflutnacftuionnc,tiδo(nx)ha=s tγh/eπ(fxor2m+oγf2C),awuicthhy-aLnorinen-z
iTtahteivebleyhsaivmioilrarwteofothuantdseheenreinasaxusiunaclresa-sweasveissquupaelr-- dfiinsittreisbiumtaiolnscfaulencptaioranm, eδt(exr)γ==γ0/.π00(x8.2 +γ2), with an in-
conductor from weak to strong site-impurity disorder.20
itatively similar to that seen in a usual s-wave super- finFitiegs.im3ashloswcaslethpearDaOmSetfeorrγdi=ffe0re.0n0t8x. at zero tempera-
co(nAdGuctthoerofrryombrewaekaskdotowns!t)roHnogwseivtee-ri,mthpeurreiteyxidsitssoardsehra.2r0p turFeigin. 3as5h6ows56thlaetDticOeS. Efoarchdidffaetraencturxveatiszoebrotatienmedpberya-
(AfsGeattuh-reweoarnvyeevbperraeiasrekinesngdsioynwmtnmh!)eetHsriymow[pseleeveesFri,tiget-.hi2me(rbpe)ue]r,xittiyhstescdaaissest:hriaFbroupr- tauvreeraigninag5o6v×er5t6helatwthicoel.eElaatcthicedaptoasictuiornvseaissowbetlalinased30by
featxu2yr2e never seen in the simple site-impurity case: For disorder confi×gurations at any given x. With increasing
tion function shows a few extra sub-peaks except for the averaging over the whole lattice positions as well as 30
sxm2ya2-jowravoenepafoirrinegvesryymgmiveetnryx[.seTehFisigi.s2a(bu)]n,iqthueedpirsotpriebruty- ddiissoorrddeerr xco,ntfihgeusruaptieorncsonadtuacntiynggigvaepn axr.ouWndithωin=cr0eadsein-g
tion function shows a few extra sub-peaks except for the creases and evolves from U-shape like to more V-shape
whichdistinguishess -wavepairingfromtheothertwo disorder x, the superconducting gap around ω = 0 de-
major one for every gxiv2ey2n x. This is a unique property like, indicating a modulation of the pairing symmetry
pairingchannels[seeFigs.2(a)and2(c)]. Infact,thisfea- creases and evolves from U-shape like to more V-shape
whichdistinguishess -wavepairingfromtheothertwo from s -wave to d -wave. In addition, the coher-
ture may be physicax2llyy2understood as follows. First, the like, inxd2yic2ating a mxo2duyl2ation of the pairing symmetry
pairingchannels[seeFigs.2(a)and2(c)]. Infact,thisfea- encepeaksareslowlysm−earedout, behavingsimilarlyto
tusfrrxeo2mmy2a-Jwyabvteeerppmah.iyrsiSnicegacololrnyidgui,nnsaditneecrsesftrtoohomeddJai2ssoefroxdlclehorawnissg.einFttierrrosmtd,,utcnheoedt fetrhnoecmecapssexea2ykw2si-twahraeivmeslpotuworlidytxys2m−diyes2oa-rrwdeadevr.oe2.u0tI,nbaedhdavitiinogn,sitmheilacrolhyetro-
1
sxb2yy2-rwaanvdeompalyiritnagkionrgigianwaatyestfhroemNJN2Nex(cJha)nbgeontdersm, t,hneorte theTchaeseevwoliuthtioimnpoufrtithyedgiaspordinert.2h0e DOS result may be
2
fromJ term. Second,sincethedisorderisintroducedby complemented by the quasi-particle energy gap E ,
shoul1d be five different local disorder configurations, as gap
The evolution of the gap in the DOS result may be
randomlytakingawaytheNNN(J )bonds,thereshould which is defined as the smallest positive eigenvalue of
illustrated in Fig. 2(d). Therefore2, in a macroscopic sys-
complemented by the quasi-particle energy gap E ,
befivedifferentlocaldisorderconfigurations,asillustrat- the BdG Hamiltonian in Eq. (5). As shown in Fig.g4a,p
tem, all the five configurations should in principle be re-
which is defined as the smallest positive eigenvalue of
ed in Fig. 2(d). Therefore, in a macroscopic system, all E decreaseswhenxincreasesanditalsoevolvesinthe
flectedonthedistributionfunctionintermsofthepeaks gap
the BdG Hamiltonian in Eq. (5). As shown in Fig. 4,
thoebfisverevceodnifingFuriga.ti2o(nbs).sh(oSuomldeitnimpersintchiepldeisbterirbeuflteiocntedpeoank Esamedtercernedasaess(wphoesnitixoninacnredasdeissoarndderitaavelsroageevdo)lv∆essxin2y2th;e
tinhcsemoFdriargiles.lstprt2ioo(bnbubd)te.iinosegn(eSntfouo.)mneceitttihiomenreistnhtetheecromdnsifisgotrufirbtahutetioiopnneaIkposeraokVbsceiosrrvtreoedo- sosattgamhatepeer,wtErisegena,dpifsathshoe(urpledowsibeteiroenidneoaanl∆ldysxdz2ieys2roorcdodemurepaotvnoeernnatgoedidna)lthq∆eusaxSs2Ciy-2;
sponding to either the configuration I or V is too small otherwise, if there were no ∆sx2y2 component in the SC
to be seen.) state, E should be ideally zero due to nodal quasi-
gap
5 5
(a) (b)
Ⅰ Ⅱ
Ⅲ
FIG. 3. Density of states for various disorder proportions x
FIG. 3. Density of states for various disorder proportions x
at zero temperature, with J = 0.7 and J = 1. Each DOS
Ⅳ Ⅴ foartazegriovetnemxphearsabtueeren,awveitrhag1Je1d=ove0r.7poasnitdio2Jn2a=nd13.0EdaiscohrdDerOS
foragivenxhasbeenaveragedoverpositionand30disorder
configurations.
configurations.
(c) (d)
FFIIGG..22..DDiissttrriibbuuttiioonnoofftthheellooccaallppaaiirriinnggaammpplliittuuddeessPP((∆∆((rr))))
ffoorrvvaarriioouuss ddiissoorrddeerr xx aatt zzeerroo tteemmppeerraattuurree,,wwiitthhJJ11||==00.|7.|7
aannddJJ22==11.. ((aa)),, ((bb)) aanndd ((cc))ccoorrrreessppoonnddttoossxx22++yy22--,,ssxx22yy22-,-,
papanonodsdsssiidbdbxlxle2e2−−ddyyii2s2so-o-wwrrddaaeevvrreeeeddppaacciioorrnniinnfifigggguuaarrmmaattppiiololiintntususddffoeoesrsr,,aarrelelososcpcpaeaelclctltlaiaivtvtetetililcycye.e.ss(i(tditde)e.).FTFTihvihveeee
ddaasshheeddlliinneessddeennootteetthheeNNNNNNbboonndd--ddeeffeecctt((JJ22==00))..
particleexcitationsfromd-wavepairing. Notethatwhen
particleexcitationsfromd-wavepairing. Notethatwhen
xdxu&(cid:38)e t00o..77t55h,,eEEpggraaeppseiinsscesslliioggfhh∆ttllyy ggrreeaattienerrtththheaasnnys∆∆tessmxx22yy(22n,,owtwhshihciochhwinsis
due to the presence of ∆sx2+y2 in the system (not shown
inFig.4). Therefore,Egaspx2a+pyp2earstobedictatedmainly
inFig.4). Therefore,E appearstobedictatedmainly
by∆ inacertainwgaayp. Thesubtlefeatureofitcould
by∆sx2y2 inacertainway. Thesubtlefeatureofitcould
be fusrxt2hye2r revealed by the physics we will discuss next.
be further revealed by the physics we will discuss next. FIG. 4. Quasi-particle gap E and position/disorder aver-
gap
aFgeIdG.p4a.iriQnguaasmi-ppalirttuidcleesg∆aspx2E+gya2p, ∆ansdx2yp2osaitnidon∆/ddxis2oryd2erasavaer-
C. Formation of superconducting clusters fuangcetdiopnaoirfinxgwaimthpJli1tu=de0s.7∆asnxd2+Jy22,=∆1s.x2y2 and ∆d−x2 y2 as a
function of x with J =0.7 and J =1. −
C. Formation of superconducting clusters 1 2
In our study, local pairing amplitudes are self-
coInnsistoeunrtlystcuadlcyu,latloedcaaltepvaeirryinlgattaicmepsliitteubdyesusianrgeBsdeGlf- top of a relatively less inhomogeneous d-wave SC “sea”,
ctohnesoirsytefnotrlyacgailvceunladtiesdoradteervreeraylilzaatttiiocne.siItnepbayrtuisciunlgarB,daGs etvoepnothfoaugrehlatthievreelyislensosiinnthrionmsiocgceonreroeluastido-nwbaevtewSeeCn“dsiesa-”,
tmheeonrtyionfoerdaingisvuebnsedcitsioorndeIIrIrAe,altihzaetdioisnt.riIbnutpioanrtiocfuslaxr2,y2a-s oervdeenretdhbouongdhst.hNeroeteistnhoatinthtreirnesiics caolsrorealastmioanllbceltuwsteeernindgis-
wmwcaaealvnvedteiipospaonaireirdrdiinenriggncaaosmmunpfibplgslieiuttcurutadditoeeisnosnssIeseIe.IemmAUs,sstuttoahoblelbyedectioschtroirrsrierbsleauhlattoeituoedlnddwowbitfiehtshcxloo2lcnyoa2---l ltieotkenrenddlydeerentencodycybbfeoofnorinrdddsxdu.2c−Neydo2t-bewy-wtahsva-aevwteaptvapheiaerriipnreiigenicsgeaa.malsmpolpiatluistdmuedsae,lslw,chlwuicshhtiecrhiisnigs
x2 y2
dtirsaosrtdederwciotnhfitghuerealteicotnrso.nUdesunasiltlyydthisitsrisbhuotuioldn.bTehceornetfroarset,- likTeolyfutorthbeerincdonufic−ermd btyhes-wfoarvmeaptiioence.of s-wave SC “is-
eidnFwiigtsh.5th(ae),e5le(cbt)r,oanndde5n(sdi)ty, wdeisptlroibtutthieonel.ecTtrhoenredfoenres,ityin landTso” fwuirtthhoeurtcaonsfitrromngthceorrfoerlamtiaotniownitohf as-wcearvteainSCdis“-is-
Fni(grs).=5(a),α5σ((cid:104)bn)α,,ra,σn(cid:105)da5n(dc)s,pwateiapllvoatritahteioenleocftrsoxn2y2d-ewnasvitey olradnedrs”cownfiitghuoruattioans,trwoengdcisoprlraeylattihoen dwiistohrdaerc-aevrtearaingeddis-
nag(nivrde)nd=xd2i−(cid:80)soyrα2d-σwe⟨rnavαree,ra,plσiaz⟩iaratiinnodgnaswpmiatpthliiatxlu=dvaesr0,i.a8rteiaostpnezceotrfiovsetxle2ymy,2p-fowerraava-e cwooarrvdreeelraptaicoiornninfifgugnuacrmatitpoilnoitnsu,∆dew(sre1a)sd∆isa(rp2lfau)ynfocttrihosenx2dyo2ifs-otarhdneedrd-diasxvt2ea−rnyac2ge-ed
and d -wave pairing amplitudes, respectively, for a correlation functions ∆(r )∆(r ) for s - and d -
x2 y2 1 2 x2y2 x2 y2
gtiuvreen. Odib−∑sovrioduesrlyr,etahliezaetleioctnrownitdhenxsi=ty0n.(8r)atiszneeraorltyemhopmeroa-- rwija=verp1airirn2g aatmxpl=itu0d.8esinaFsiga.f6u(nac).tioTnheofs-twhaevedicsotma−n-ce
| − |
geneous and uncorrelated with the given bond disorder. ponent shows a clear structure on a scale of two to three
ture. Obviously,theelectrondensityn(r)isnearlyhomo- r = r r at x = 0.8 in Fig. 6(a). The s-wave com-
ij 1 2
However, the s -wave pairing amplitudes are inhomo- lattice|con−stant|s, while the d-wave counterpart shows al-
geneous and unxc2oy2rrelated with the given bond disorder. ponent shows a clear structure on a scale of two to three
geneous and tend to form some s-wave SC “islands” on most no structure. Note that the coherence length in a
However, the s -wave pairing amplitudes are inhomo- lattice constants, while the d-wave counterpart shows al-
x2y2
geneous and tend to form some s-wave SC “islands” on most no structure. Note that the coherence length in a
6
66
ation in Ψ(r)2 suggests its nature as a gapped mode.
gaatipopnedinm|Ψod(er.)|T2 hsuusg,gtehstissfietastunraetumreigahst waegllaepxppeldaimnowdhey.
ETTfoghhuauunpssd,,>ittnhh0iiF|,ssiagffees.aa4ftt|o,uuuirrsneeddmmiciiintggahhtFtteidgww.bee4yllll,teeihsxxeppdslliaa-cwiitnnaatvwweedhhpyybaEEyiriggtnaahgppec>>sh-aw00n,,anvaaeessl
pfoauirnidnginchFaign.n4e,liisndoiuctraitnehdobmyotgheeneso-wusavSeCpasyirsitnegmc.hannel
in our inhomogeneous SC system.
in our inhomogeneous SC system.
DD.. SSuuppeerrflfluuiidd ddeennssiittyy,, CCrriittiiccaall tteemmppeerraattuurree,, aanndd
D. Superfluid density, Critical temperature, and
(a) PPhhaassee ddiiaaggrraamm
(a) Phase diagram
TThhee tteemmppeerraattuurree ddeeppeennddeenncceeooffththeesusuppereflrfluuididdednesni--
The temperature dependence of the superfluid densi-
sttiyytyρρsρ((sTT(T)))iissisuuussuusuaaalllllyylyvvviieeiewwweeedddaaasssaaagggooooooddd iiinnndddiiicccaaatttooorrr tttooo dddeee---
tteerrmmsiinnee iiff aa ssuuppeerrccoonndduuccttoorr iiss ““uunnccoonnvveennttiioonnaall”” oorr nnoott..
termine if a superconductor is “unconventional” or not.
FFFooorrr iiinnnssstttaaannnccceee,,, ρρρss(((TTT))),,, wwwhhhiiiccchhh iiisss mmmeeeaaasssuuurrraaabbbllleee ttthhhrrrooouuuggghhh ttthhheee
mmmaaagggnnneeetttiiiccc pppeeennneeetttrrrsaaatttiiiooonnn dddeeepppttthhh λλλ−−222(((TTT))),,, eeexxxhhhiiibbbiiitttsss aaannn eeexxxpppooo---
nneennttiiaall bbeehhaavviioorr iinn TT iinn aa cclleeaa−nn ((hhoollee--ddooppeedd)) 112222 ccoomm--
nential behavior in T in a clean (hole-doped) 122 com-
pwppwwooohhhuuuiiilllnnneeedddρρρ,,,ssBBB(((TTTaaa1)11))−−−bbbxxxeeeKKKhhhaaaxxxvvvFFFeeeeeesss222AlAAlliiikkkssseee222,,,aaassspppuuuooogggwwwgggeeeeeesssrrrtttllliiiaaannnwwwgggaaaiiinnnfffuuuaaannnllllllyyyeeelllgggeeeaaacccppptttrrrpppoooeeennnddd---dddSSSoooCCCppp;;;eee444ddd555
((bb)) ((cc)) BsBBssoooaaammm(((FFFeeeeteetthhh111−−−siiinnnxxxgggCCCuuuooonxnnxx)c))cc222oooAAAnnnvvvsss2e22ee,n,,nntttiiiiiinnnooodddnnniiiaaaccclllaaa...t4tt44i6ii66nnn–––4g44gg888ttthhheee ppprrreeessseeennnccceee ooofff nnnooodddeeesss ooorrr
FIG. 5. (a) Electron density on a 32 32 lattice for a given
daFpppFddaaniannIaaiiIsssGdiGiiddoroorrir.iirr.(n((nndddc5cc5gggeee)))..rrrcccccc((hrohhrrooaaeeerarraa))araarrnnneleeEllEninniissszzzlepleeppaeaaelllotcoosttcssin.tiinn..tooorrdddnonnoTTTnntttwwwohoohhddeieeiittettttewhwwhhndnddosooasaaxixxirtrrtdddkykky=o==ooeeemormmorrnn000itiittnnn.h..hha8a88aaaeeennn33aaatltt2ll2tttaaasssr×rr××zzzxgxxggeee2e22ee3r3rryyyrrro2oo2222-d--ddltlttaeaeeeaeeaantnntmnmmnnttsssdiddipippiiccttteeeeeydyyddrrr/x//xxfafaa2oa22oaattt−−−rmrmmuuuyyyrrraappp222eeel-ll--...gigiiwwwtttiiuvuuvaaa(((devbeddbbvvnnee)eeee)) fmafmfmuuunalaallMMMdyyyhhhooo2iiiaaarnrrnn)ffffffeeetttwoooeeevcvvtcctthotooetteearrrtSSS,,,uuuCCCttttnnnhhhydddepee(((eeepppeqrqqrrhhhsssuuuotttaaaaaafaaassslllnnnfleeeiiitttddduaaarrrctittii111gtggiiivvvu)))iiidddeeeahhhiiitbbbtttoooiyyyeoeewww)))hhhnaaasaaadddvvvtttiiiiiiissssooolllooooooerrrrrrwwwsooodddsfffeeeetttnρρρrrreeestssmmmoooi(((aTTTfffpppl)))eeecccnrrreeegggeaaarrriiiattttttvvvraaauuueeeiiirrrssscnnneeeraaasssitttt111uuuyyyi888csss,,,ppp222eeae000eee---l
wwwooouuullldddbbbeee... atattbeeennemmmdydoppp22neee))drrraaawwtotthhuuuuaarrrrteteemttTTTyycccepp...a444een999-ooAfiAAffelllflflltttdhuhhuocoocsttuuutuuguggaahhhdttyiitttoo,hhhnnweeesseaaaiissnnncssseeawwwssnsseeeeerrrsnnttttttioooliilaaqqqlluuuunnseeeeeesssaatttirriiitoooccnnntrroiitt222iio)))ccbaaiiisss-ll
bbeeyyoonndd oouurr mmeeaann--fifieelldd ssttuuddyy,, wwee ccaann ssttiillll uussee iitt ttoo oobb--
tain T in a self-consistent manner when inhomogeneous
ttpaaaiiinnrinTTgccc iiinns iaanessveelilftf--accboolnnessaiissnttdeennnttommtaannnenngeleirrgiwwbhlheee.nn iinnhhoommooggeenneeoouuss
ppaaiirriinnggiissiinneevviittaabbllee aanndd nnoott nneegglliiggiibbllee..
Wegeneralizethelinear-responseapproachinRefs.50
WWeeggeenneerraalliizzeetthheelliinneeaarr--rreessppoonnsseeaapppprrooaacchhiinnRReeffss..5500
and 51 for a multi-orbital case to obtain the superflu-
aanndd 5511 ffoorraammuullttii--oorrbbiittaallccaasseettoooobbtataininththeesusuppereflrfluuid-
id density. Considering a weak vector potential A (r,t)
didendseitnys.ity.CoCnosnidsiedreinrigngaawweaeakkvveeccttoorrppootteennttiiaall AAx((rr,,tt))
alongxdirection,thehoppingtermsinEq. (1)arexxmod-
aalloonnggxxddiirreeccttiioonn,,tthheehhooppppiinnggtteerrmmssiinnEEqq.. ((11))aarreemmoodd--
ified by the Peierls phase factors eieAx(r). Expanding
iififieedd bbyy tthhee PPeeiieerrllss pphhaassee ffaaccttoorrss −eeiieeAAxx((rr)).. EExxppaannddiinngg
the phase factors up to second ord−−er, we obtain
tthheepphhaasseeffaaccttoorrssuupp ttoo sseeccoonndd oorrddeerr,, wwee oobbttaaiinn
1
((aa)) ((bb)) HHHtttAAA===HHHttt−−− r [eeejjjxxxppp(((rrr)))AAAxxx(((rrr)))+++ 12122eee222kkkxxx(((rrr)))AAA2x2x2x(((rrr)))],,,(((111111)))
FFIIGG.. 66.. ((aa)) TThhee ddiissoorrddeerr--aavveerraaggeedd ccoorrrreellaattiioonn ffuunnccttiioonnss (cid:88)∑∑rr (cid:20)[ (cid:21)]
FIG. 6. (a) The disorder-averaged correlation functions with the paramagnetic current
∆a∆(aa∆∆(∆∆(bsbbss(()(((())arraarrrra11aa1111f))ffr))))rruuu∆∆e∆∆∆∆eennna((aac((((ccrrvrrrrtvvtt22ie2222iieeooo))r))))rrnannaaooofgffggofoooffooeeerfrrsffssdddxxxtttsss2h22ohhooxxxyyyvevvee222222eeeyyy-d--ddrrr2w22wwiii---s1ss11aaattt5v55vvaaaaaaeeennnnrnnrrdaddpcaappccnenneeaaaddddiddiirrrrrrxoxxooiiiiii2jn22mjjnnmm−−−ggg===yyydddf2ff22o|ooi||ii---rsrrssrwrrwwo1oo11vavvraarr−−−advaaddvvreerreeeerrriiirrrooo222pppu|uuc||ccaaaosoosswwwiiinnnrrrxxxiiiifiiififittt.n..nnhhhggggggBBBuuuxxxrorrooaaaaaat=tt==mmmthtthhiiipoppoo000(((nlnnll.a..aaiii888st)sstt))...uuuaaaaaaddd(((nnnnnnbbbeeedddddds)ss)) wwiitjtjjhhxxxppp(((ttrrrhh)))ee===ppiaiiar(cid:88)r∑∑sssaaσσσmm(cid:88)∑∑aaαααggβββnnttteeαrαrαrtt,,,βiββirrrcc+++cscssuu(cid:16)((rrcccrr†r†r†ree+++nnssstt,,,βββσσσcccrrrααασσσ−−−ccc†r†r†rααασσσcccrrr+++sss,,,βββσσσ(cid:17))),,,
bbbooottthhhaaarrreeessscccaaallleeedddbbbyyy∆∆∆222(((rrr11)))... and the kinetic energy associated with x direction
1 aannddtthheekkiinneettiicceenneerrggyy aassssoocciiaatteedd wwiitthh xx ddiirreeccttiioonn
cvscvcvsl-Fll-FFeweew/a//aaaa∆n∆∆nnvveesss∼∼∼xxxcc222ooyyym4m44222.-..--ppwwwooMaMMaannvvveoeeooeennrrretSeetSSoooCCCppvvveeeeeiriirrrrnnnss,,,iissttttttththhhhhssiiieeesssaascsmccmmllluxxuuooosssddditittneneeeeelcllrcrririircnccnneeaaagaggannnssetettbbbeseesnenneeaadddsseeeeeessscncnntttaacicciimmmnnyyyaaabbffftttoooeeeeerrrdddsseetttbebbhehhnynyyeee AAAckcckkcccxxxooo(((rrrrrrddd)))iii=n==nnggg−−−tttooo(cid:88)∑∑ssstσttσσhhhe(cid:88)ee∑∑αααβlββlliiinnnttteeeαrαrαraaa,,,βββrrrrrr+++rrrssseee(cid:16)((ssscpccpp†r†r†rooo+++nnnssssss,,,βeββeeσσσtttccchhhrrreeeαααoooσσσrrryyy+++,,, ccc†r†r†rααασσσcccrrr+++sss,,,βββσσσ(cid:17)))...
isiinnn-wAFFFadiiivgggde...it666ci(o((obbbmn)))a.p..lloyn,enitt ipserwsiosrttshasnxotiicnicnrgeatshesatasfocranthbee sfieresnt ρρρsss===(cid:104)⟨⟨−−−kkkxxx(cid:105)⟩⟩−−−ΛΛΛxxxxxx(((qqqxxx ===000,,,qqqyyy →→→000,,,iiiωωω ===000))),,, (((111222)))
Additionally, it is worth noticing that for the first few where k is the kinetic energy along x direction, and
fleowwAeldsotdwieqtisuotanqsaiu-llpaysa,ir-itptiacislretwiceolxercteihtxecndiotetsitdcaistntegas,ttehesaa,ctehafocchrotcrhroeersrfipesropsntodnfiendwg- wwhheerree ⟨−kkxxx⟩iisstthheekkiinneettiicc eenneerrggyy aalloonngg xx ddiirreeccttiioonn,, aanndd
atilwwaanohggaagwepaavvertteewdoiisnntipffaauuggvaqgnnegmmouccanattfootiiuasiooddinln-nneegpcdddtwwaiimriiroiisstettnttoichhcrrdtiildiaabbeeoiuunsddewttt;xeeiirioofificitabnninnhtlu,,oiiettt||ndaeeΨΨigomms((ndtrre,eooa))afimm||tc|22neΨh,,eesit,(nnaaerwpptte)uuppaa|mmm2veec,ehaaorraamassfcllrpooeottopnnnroonreggttbbeuastteerm(phhseaaoee.tngappddo.ldrr,oiiooiaanbnppoggggef---- ΛΛΛxxxxxx((((cid:104)⟨qqq−−,,,iiiωωω(cid:105)⟩mmm)))=== NNN111222 ∫0βββdddτττeeeiiiωωωmmmτττ(cid:104)⟨⟨jjjxxxppp(((qqq,,,τττ)))jjjxxxppp(((−−−qqq,,,000)))⟩⟩(cid:105)...
toohnneaallladdriigrreeeccstttiiooannm;;paallliootnnuggdeee)aa,cchhthwweaapvvreeoffbrraoobnnittli((teey..ggd..,,isootffrittbhhueetellaasrrggaeelosstt- (cid:90)∫00 (13)
amplitude), the probability distributes alomost uniform- ((1133))
mamosptlituundifeo)r,mthlye pbruotbasbuipliptryesdsiesstriabruotuensdalso-mwoasvteuSnCifo“rmis--
lybutsuppressesarounds-waveSC“islands”. Suchvari- Note that in our disordered model, we calculate the
llaynbdust”.suSpupcrhesvsaersiaatrioounnidnsΨ-w(arv)e2SsCug“gieslsatnsditss”.naStuucrhevaasria- NNoottee tthhaatt iinn oouurr ddiissoorrddeerreedd mmooddeell,, wwee ccaallccuullaattee tthhee
| |
7
7
current-current correlation function in real space, quasi-particle energy E as a function of T depicted
current-current correlation function in real space, T (called T ). This can bgaepcontrasted by the minimum
in Fig. 8(bc) with a similar trend. Such an exponential
ΛΛxxxx((rr11,,rr22,,iiωωmm))== NN11 ∫(cid:90)00ββddττeeiiωωmmττ⟨(cid:104)jjxpxp((rr11,,ττ))jjxpxp((rr22,,0(0)(1)⟩14(cid:105).4.)) qibsnbsyueysehaFstAhsaetiiavemg-vfmip.eoiaowir7srir(stsbfrituscfe)rtullmoelrwlyonlayeinggtrnglhkagyelspayrappgsauypesrsugdeeimEggd.weg.gisleaoatsprsrttshatthsrtemehaneedSnf.CuStniCSogcnutrgiiconorhuongnuadhonnfedsreTtexsa.tptdaeoFteneiporefisncottttf,iehaudtelhne-
likeinthecaseofsite-impuritydisorder,ρ (0)isbasically
A few remarks are worth mentioning herse. First, un-
AAfftteerr ddiissoorrddeerr aavveerraaggiinngg,, tthhee ttrraannssllaattiioonnaall iinnvvaarriiaannccee linkoetinsutpheprceassseedofasistet-hime pduisroitrydedrissotrrdeenrg,tρh(x0)iinscbreaasisceasl,lyin-
mmaayy bbee rreeccoovveerreedd aanndd hheennccee wwee sseett ΛΛxxxx((rr11,,rr22,,iiωω))== ndoitcastuipnpgrtehssaetdthase tphheasdeiscoorhdeerrenstcreenfogrthelxsecitnrocrnesaissesa,lmino-st
ΛΛxxxx((rr11−rr22,,iiωω)),, ffoolllloowweedd bbyy aa FFoouurriieerrttrraannssffoorrmmttooqq-- duicnaatffinegcttehda.tStheceopnhda,seatcoxhe>renxcceefvoerneltehcotruognhstishealdm-owsatve
ssppaaccee,, − ucnoamffpecotneedn.tSisedcoonmdi,naatntx,t>herxeceovuelndbtheonuoghgatphleesds-Bwoagvoeli-
c
ΛΛxxxx((qq,,iiωω))== NN1122 ee−−iiqq··((rr11−−rr22))ΛΛxxxx((rr11,,rr22,,iiωω)).(.(1155)) cuuTobbmhooviprvdoqqn,uuea‘nTastisc-i’ip-sspadardtoreitmtcielciernlsmea,sni,antsa,esidtnhidnebidryceiaccρtaeostdu(eTldidn)ibnFaeniFgndi.og7.Eg(8abg(pa)bplea)(stTasT)tBTao<rge<oTlcinT-.oct.
rr∑(cid:88)11rr22 Tnheicreds,sa‘Tricl’ystdheetesrammien.edInbfyacρts,(ETg)apanudsuEalglayp(oTve)reasrteimnoattes
nTeccesslsiagrhitlylythinesoaumres.tuIndyfa(cste,eEFgaigp.u8s)u.allTyhoivseirsesbteimcaautsees a
Tsysstliegmhtlmyaiynhoauvrestcuerdtyai(nseleocFailgp.a8i)r.inTghaims pislibtuedcaeusswehaile
c
scyostmemplemtealyylhoasveethceerpthaianseloccoahlepreanirciengamamonpglitthuedmes.wThhielere-
cfoomrep,leTtceldyeltoesremtihneedphbayseρcsowheoruelndcebaemmoonrgetrheelimab.lTe.here-
foreT,yTpicdaeltleyrmthienesdtabnydaρrdwAoGuldthbeeormyoprreorveildiaesblaen. estima-
c s
tiTonypoicfatlhlyetshuepsptraensdseadrdTAcGinthaeodriysoprrdoevrieddessuanpeersctoimndau-c-
ttioonr.oTfhtehesuspupprpersessesdedTT, ∆inT,aisdibsaosridcearlelydpsuroppeorcrotinodnuacl-to
cc
ttohr.eTdihseorsdueprpsrceasstetedriTng, ∆raTte,,ixs/bNas(i0c)a,lltyimpreospaorctoinonstaalnttoof
c
the(1d)i.sorTdheresecsastetnetriianlgsrtaetpe,inx/iNts(0d)e,rtiivmateisona cisontsotarnetploafce
O
∆((1r)). bTyhteheesssepnattiiaalllsyteapveinragitesddoenriev.atTiohneriesfotore,rewpelacaelso
O
∆ca(rlc)ublaytethtehespqautaiasil-lpyaarvtiecrlaeggeadpowneit.hTanheerneffoorrce,edwuenaiflsoorm
coarlcduelratpeatrhaemqeutearsi-∆pa(rrti)cliengtahpewsietlhf-acnonesnisfotercnetdmuneaifno-rfimeld
(a) (b) i
oerqdueratpiaornasmaentdersu∆m(rmi)airniztehaellseolff-ocuornsoibstteanintemdeTacn-ifineFldige.-8.
FFIIGG..77.. ((aa)) TTeemmppeerraattuurree ddeeppeennddeenncceeooffssuuppeerrflfluuiiddddeennssitityy qIutaitsioinmspaonrdtasnutmtmoaorbizseeravlel otfhoautrEogbatpa-idneetderTmciinnedFiTg.c8v.ia
ρρss((TT))aannddeenneerrggyyggaappEEggaappffoorrvvaarriioouussddiissoorrddeerrpprrooppoorrttioionnxx.. Itthies iinmhpoomrtoagnetnteoouosbsseorlvuetiothnaitnEtghaep-sdeeltfe-cromnisniesdtenTtc mviaan-
tnheerianghroemesogweenlelowuisthsotlhuetiρon-dientetrhmeisneeldf-coonnes,isbtuenttbomtahna-re
s
naewraaygrferoesmwtehlelwAiGth-ttyhpeeρTsc-dwehteernm0in<edxo<ne1,.bTuthbisostthroanrgely
aiwmapylifersomthtehberAeaGk-dtoywpenTocfwthheenco0n<venxt<ion1a.lTAhGis-tsytrpoentgh-e-
lyoriymfpolrietshethbeonbdre-adkisdoorwdenreodf styhsetecmon.ventional AG-type
theory for the bond-disordered system.
IV. SYSTEM WITHOUT J
1
IV. SYSTEM WITHOUT J
1
We have studied a J -bond disordered t-J -J model
2 1 2
We have studied a J -bond disordered t-J -J model
in the previous sectio2ns. However, the pr1ese2nce of J
1
in the previous sections. However, the presence of J
terms may couple with J terms in a complicated w1ay
2
terms may couple with J terms in a complicated way
andfeaturespurelyfrom2thebonddisordercouldbestill
andfeaturespurelyfromthebonddisordercouldbestill
obscure. Since this type of disorder is interesting on its
obscure. Since this type of disorder is interesting on its
ownright,inthissection,werepeatpreviouscalculations
ownright,inthissection,werepeatpreviouscalculations
by setting J =0.
by setting J 1=0.
Focusing1on s -wave only, Fig. 9(b) shows the dis-
Focusing on s x2y-2wave only, Fig. 9(b) shows the dis-
tribution of thex2syp2atial pairing amplitudes at zero tem-
tribution of the spatial pairing amplitudes at zero tem-
perature with its corresponding disorder realization at
perature with its corresponding disorder realization at
x = 0.4 depicted in Fig. 9(a). The trend to form “su-
x = 0.4 depicted in Fig. 9(a). The trend to form “su-
perconductingislands”isclear[seeFigs.9(c) (b) (d)]:
FbFbyIyIGGqq..uu8a8a.s.siiC-C-pprraaiitrtrititcciicacalllleettegegmmaappppeemmrraaiinnttuuiimmrreeuumTmTccEEaassggaaaapp,,ffuussnunucpcptteieiororflnflnuuoioidfdfxdxd,e,eononbsbsittitatyayininρeρseds,d, pAAgesrrsacxdoxu&n(cid:38)ad0lul0.yc5.t5,ibn,“eg“ScSuiosupmlapeenrecdrsocsmno”dnaiudsllcuectrclietnauignrng[itssiiellsaelxnaFdnaisdg”pssp.”ar9roa(earcef)co→hfreom→(srbme1)d→e.dw→(Thdwih)lh]ee:islee
aannddtthhee AAGG--ttyyppee ccaallccuullaattiioonn wwiitthh eennffoorrcceedduunniiffoorrmm∆∆((rrii)),, g“riasdlaunadllsy”baerceominestsemadallseerpaurnattiledxfarpomprooanceheasn1o.thTerhebsye a
rreessppeeccttiivveellyy.. “islands” are instead separated from one another by a
metallic “sea” (with negligible ∆(r)). This can be more
metallic “sea” (with negligible ∆(r)). This can be more
accurately confirmed by the disorder-averaged correla-
accurately confirmed by the disorder-averaged correla-
In Fig. 8(a), we show the temperature dependence of tion function ∆(r )∆(r ) for various x, as shown in
In Fig. 7(a), we show the temperature dependence of tion function ∆(r )1∆(r )2 for various x, as shown in
1 2
ρρs((TT)),,nnoorrmmaalliizzeeddbbyyρρs((00)),,aattvvaarriioouussxx..AAttaannyyggiivveennxx,, FFigig..1100..
s s
ρ (T) deviates from ρ (0) exponentially at low tempera- Wenextexaminethequasi-particleminimumgapE
ρs(T) deviates from ρs(0) exponentially at low tempera- Wenextexaminethequasi-particleminimumgapE gap
s s gap
ttuurreessaannddtthheennddeeccrreeaasseessuunnttiillrreeaacchhiinnggzzeerrooaattaaccrriitticicaall aassaafufunncctitoionnooffththeeddisiosrodredrerprporpooprotriotinonxxataztezreortoetmempepr-er-
T (called T ). This can be contrasted by the minimum ature, as illustrated in Fig. 11. Clearly, E follows the
c gap
8 8
(a) (b)
FIG. 10. The correlation function ∆(r )∆(r ) for s -
FIG. 10. The correlation function ∆(1r )∆2(r ) forx2sy2 -
pairingamplitudesasafunctionofthedistan1cer 2= r rx2y2
(c) (d) wwpiatihtirhidndigffiffearemerneptnlitxtux,d,aveasevreaarsgaaegdefudbnybcyt1i5o1n5coocnoffintghfiuegrudartiasiottanionsncasenijardnijsdc=|ascl1|ear−dl1e−bd2y|rb2y|
∆2(r ).
∆2(r1 ).
1
FFIIGG..99.. ((aa))AAddiissoorrddeerreedd ppaatttteerrnnwwiitthhxx==00..44.. TThheeppuurrpplele
rreeggiioonnssmmaarrkktthhee ppooiinnttss wwiitthhJJ22==00.. ((bb))TThheessppaattiaiallddisisttrri-i-
bution of local pairing amplitudes corresponding to (a). The
bution of local pairing amplitudes corresponding to (a). The
darker purple color indicates the regions with larger pairing
darker purple color indicates the regions with larger pairing
amplitudes. (c) and (d) are another two spatial distributions
amplitudes. (c) and (d) are another two spatial distributions
of local pairing amplitudes with x = 0.1 and x = 0.7 re-
of local pairing amplitudes with x = 0.1 and x = 0.7 re-
spectively. The darker purple color has the same meaning
spectively. The darker purple color has the same meaning
explained in (b)
explained in (b)
.
.
sametrendas∆ . Notethateventhoughthesystem
atteunrdes,taosfiollrumsttrhastexe2“dys2uinpeFricgo.n1d1u.ctCinlegairsllya,nEdgs”apafsoxllogwroswtsh,e
sametrendas∆ . Notethateventhoughthesystem
thesuperfluidstisffx2nye2ssρs doesn’tvanishuntilx=1(See
tends to form the “superconducting islands” as x grows,
Fig. 12). This is because, within our BdG formalism, we
thesuperfluid stiffnessρ doesn’tvanishuntilx=1(See
ignored the possible phasse fluctuations between SC is-
Fig. 12). This is because, within our BdG formalism, we
lands, which can frustrate Cooper pairs to form a coher-
iegnntorseudpetrhceonpdouscstiibnlge spthaates.e Tfluhcutsu,aitniopnrsinbcieptlwe,eeonneSCmaiys-
lnaontdsr,uwlehoicuhtctahneforuccsutrrarteencCeooofpearspuapiresrctoondfourcmtoar-mcoehtearl-
etnratnssuitpieorncoinndtuhcistinpgursetlaytea.caTdehmusic, imnopdreiln.ciMploer,eoonveer,miany FIG. 11. The energy gap Egap as a function of the disor-
nFoitg.r1u2leToduettetrhmeinoecdcubryreenitcheerofEa suopresrucpoenrdfluucidtodre-mnseittyal dFerIGp.ro1p1o.rtiTonhexeanterzgeyrogtaepmpEegraaptuarse.aWfuinthctxionraiosfintgh,eEdgiaspor-
c gap reddeurcpersogproardtuioanllyxaantdzreeraochteesmzpeerroatautrxe.=W1.i0t.h x raising, E
tρranissictioonnsisintenthtiwsipthureealychaocathdeerm;iwchmileodAeGl.-tMypoerecoavlceur,lai-n gap
s reduces gradually and reaches zero at x=1.0.
Ftiiogn.1is2aTgcadinetsehromwinnetdobuyndeietrheestriEmgaatpeoTrc,surepfleercfltuinidgdoenntshitey
ρims pisocrtoannsciseteonftthweitihnheoamchogoetnheeoru;swphaiilreinAgGa-mtypplietucdaelcs.ula-
tionisagainshowntounderestimateT ,reflectingonthe
c the presence of short-range impurity potentials. Both of
importance of the inhomogeneous pairing amplitudes.
thtoemincalruedbeatshiceawllyeapkaeinrebdrNeaNkeerxs:chTahnegefoirnmteerracsutipopnreJsseasnd
1
V. DISCUSSION AND CONCLUSION ththeed-pwreasveenscuepoefrcsohnodrut-crtaivnigtey,iwmhpiuleritthyeplaottteenrtciaolusl.dBeroatsheof
ththeepmhaasreeibnfaosrimcaalltyiopnaoirr bevreenakienrssu:laTteheorfiogrimnaelrlysusuppperre-ss-
BefoVre. coDncISluCdUinSgSoIuOrNwoArkN, tDheCreOaNreCaLfUewSIaOddNitional ceosndtuhcetidn-gwraevgeiosnusp.eArcsonlodnugctaisvitthye, rwahtiioleotfhtehelaitmteprurciotyuld
remarks worth mentioning here, which are relevant to peortaesnetitahlestprehnagstehintoforJm1aitsiomnucohr leevsesnthinasnuloantee,ooruigrinrea-lly
isoBveafloernetcdoonpcilnugdiinngiroounr-bwaosrekd,StChesr.eFairrestalyf,eawsawdedihtaiovne- ssuultpserschoonwdnucintintgherepgrioevniso.uAssseloctnigonassmthaeyrraetmioaoinf tvhaelidim-
amlernemtioanrekds winorStehc.mIeIn,tinionthinegsthreorneg, wcohuicphlinagreprieclteuvraentthteo qpuuarliittaytpivoetley.ntialstrengthtoJ1 ismuchlessthanone,our
ilseoavdailnegnteffdeocptinogf tihneisruonb-sbtiatsuetdionSCosf.PFfiorrstAlys,raesduwceeshsaev-e reSseuclotsndslhyo,walnthionutghheopurrepvrieoduisctsieocntioonnsthmeamyodreumlaatiinonvoaflid
mrieonutsiloyntehdeinexScheca.ngIeI,iinntetrhaectsiotrnonJg2 c(opurepsluinmgapbilcytusurepetrh-e thqueaplaitiraitnivgeslyym. metrymaybeapplicablefortheisovalent-
exchangetype). However,anumberofsubleadingeffects doping1111iron-basedSCsatfinitetemperatures,where
leading effect of the substitution of P for As reduces se- Secondly,althoughourpredictiononthemodulationof
are also anticipated. For instance, one might consider LaOFeAs P (x = 1) has been reported to have gap
riously the exchange interaction J (presumably super- thepairi1ngxsyxmmetrymaybeapplicablefortheisovalent-
toincludetheweakenedNNexchan2geinteractionJ and nodes,34,35−our theory cannot sufficiently describe 122
exchangetype). However, anumberofsubleadinge1ffects doping1111iron-basedSCsatfinitetemperatures,where
are also anticipated. For instance, one might consider LaOFeAs P (x = 1) has been reported to have gap
1 x x
−
9
phasizethenecessityofthespatialinhomogeneityforthe
pairing amplitudes. In particular, we found that as long
as J1 (cid:46)J2, the pairing symmetry of our model at T =0
can be modulated from s -wave to d -wave sym-
x2y2 x2 y2
metry when the “disorder strength” x go−es beyond x .
c
This result was best presented by the electron density of
states as a function of x and could be partially justified
by any negative evidence in experiments about probing
fully gapped s -wave order.52–54 Moreover, when x is
large, we observ±ed the formation of s -wave “islands”
x2y2
withlengthscale (ξ)embeddedinad -wave“sea”,
x2 y2
duetothecombinOedpairinginteractiona−ndtheJ -bond
2
disorder. Asaconsequence, T determinedbythesuper-
c
fluiddensityρ (T)isfoundtodeviatefromthepredicted
s
valuebytheAGtheory,suggestingitsinsufficiencytode-
scribe the bond disorder effects.
FIG. 12. Superconducting critical temperature T as a func-
c
tion of x, with J =0,J =1.
1 2 ACKNOWLEDGMENTS
We would like to acknowledge Fan Yang, Jiangping
Hu and Elbio Dagotto for stimulating discussions.
systems. Thisisbecausewehavesofarignoredthemag-
W.F.T. is supported in part by MOST in Taiwan
netic and orbital fluctuations, which are believed to play
under Grant No.103-2112-M-110-008-MY3 and the
importantrolesinexhibitingeitheranantiferromagnetic
Thousand-Young-Talent Program of China. Y.T.K.
orderoranematicstate.26Wewillreferthismoregeneric
and D.X.Y. acknowledge support from NBRPC-
consideration to a future study.
2012CB821400, NSFC-11574404, NSFC-11275279,
Insummary,westudiedthebonddisordereffects,from NSFG-2015A030313176, NSFC-Guangdong Joint Fund
“weak” to “strong”, in an unconventional superconduc- and National Supercomputer Center in Guangzhou,
tor described by the two-orbital t-J -J model. We used Shanhai Forum, and Fundamental Research Funds for
1 2
self-consistent Bogoliubov-de Gennes formulation to em- the Central Universities of China.
∗ [email protected] (2011).
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