Hidden Order as a Source of Interface Superconductivity

0.54 MB·English

by  Andreas Moor

HiddenOrderasaSourceofInterfaceSuperconductivity
Andreas Moor,1 Anatoly F. Volkov,1 and Konstantin B. Efetov1,2
1TheoretischePhysikIII,Ruhr-UniversitätBochum,D-44780Bochum,Germany
2NationalUniversityofScienceandTechnology“MISiS”,Moscow,119049,Russia
(Dated:February4,2015)
Interfacialsuperconductivityisobservedinavarietyofheterostructurescomposedofdifferentmaterials
includingsuperconductingandnonsuperconducting(atappropriatedopingandtemperatures)cuprates
andiron-basedpnictides. Theoriginofthissuperconductivityremainsinmanycasesunclear. Here,we
proposeageneralmechanismofinterfacialsuperconductivityforsystemswithcompetingorderparame-
5 ters. Weassumethatparameterscharacterizingthematerialallowformationofanotherorderlikecharge-
1 orspin-densitywavecompetingandprevailingsuperconductivityinthebulk(hiddensuperconductivity).
0 Diffusiveelectronscatteringontheinterfaceresultsinasuppressionofthisorderandreleasingthesuper-
2 conductivity. OurtheoryisbasedontheuseofGinzburg–Landauequationsapplicabletoabroadclass
ofsystems. Wedemonstratethatthelocalsuperconductivityappearsinthevicinityoftheinterfaceand
b
thespatialdependenceofthesuperconductingorderparameter∆(x)isdescribedbytheGross–Pitaevskii
e
F equation.Solvingthisequationweobtainquantizedvaluesoftemperatureanddopinglevelsatwhich∆(x)
appears. Remarkably,thelocalsuperconductivityshowsupeveninthecasewhentherivalorderisonly

3 slightlysuppressedandmayarisealsoonthesurfaceofthesample(surfacesuperconductivity).

]
PACSnumbers:74.70.Xa,74.62.Dh,74.20.De,74.78.Fk
n
o
c
- I. INTRODUCTION the superconducting OP may arise in these materials (see
r reviewsRefs.23–27)withtheamplitudesW and∆depend-
p
ingontemperatureanddoping. Theinterfacialsupercon-
u Interesting phenomena have been discovered few years
s ago in the study of superconductivity in different ma- ductivity in one of the Fe-based pnictides (CaFeAs doped
. withLa,Ce,ProrNd)wasobservedbyWeietal.28Whereas
t terials, especially in high-T superconductors—cuprates
a c the bulk critical temperature was equal to about 30 K, a
m and Fe-based pnictides. It turned out that the crit- small fraction of samples had a T (cid:39)49K. Even higher
ical temperature of the superconducting transition T c
- c T (cid:39)77K was achieved by a Chinese experimental group
d in heterostructures, e.g., in bilayers, is higher than the c
inanotherFe-basedmaterial(singleunit-cellFeSefilmson
n critical temperature T of bare films that can be even
o nonsuperconducting.1–c13 The authors of Ref. 14 (see also SrTiO3).29
c Ref. 9) studied bilayers consisting of two cuprates— Very encouraging for understanding the very nature of
[
overdoped (La2−xSrxCuO4 with x=0.35) and underdoped high-Tc superconductivityseemstobetheeffectofappar-

2 (0.1<x<0.12) cuprate films. The largest increase of T ent enhancement of superconducting transition tempera-
c
v wasabout11K—fromT =21Kinbareunderdopedfilms ture at the interface between an iron-based chalcogenide
c
5 to Tc=32K in bilayers. Moreover, superconductivity has superconductor (FeSe) and SrTiO3 used as substrate.30,31
0
beenobservedinbilayerscomposedofnonsuperconduct- This discovery questioned the role of phonons in bulk
5
2 ing materials, La2−xSrxCuO4 with x=0 (an insulator) and iron-based superconductors30 and confirmed the pres-
0 La2−xSrxCuO4 with x=0.45 (normal metal).2 The critical ence of the magnetic order (spin-density wave) as a key
. temperature reached 50 K. Further experimental studies ingredient for high-Tc superconductivity in iron-based
1 showedtheindependenceofthecriticaltransitiontemper- superconductors.31
0
5 atureinbilayersLa2−xSrxCuO4/La2SrCuO4onx inarather Rather actively the interface superconductivity is stud-
1 widedopinginterval(0.15<x<0.47).15 iedinheterostructuresLaAlO /SrTiO .1 Itisassumedthat
3 3
v: A slight increase of Tc has been measured also in an- atwo-dimensionalelectrongasisformedattheinterface.
i other high-Tc superconductor—YBa2Cu3O7 films covered Effectofanelectricfieldhasbeenemployedtoexplorethe
X byathinAgfilm.16 Afewdecadesago,asimilareffecthas phasediagramofLaAlO /SrTiO interface.3,4Asidefromsu-
3 3
r beenobservedinbilayerscomposedofconventionallow-Tc perconductivity, the phenomenon of ferromagnetism in-
a
superconductors and normal metals.17–19 These latter ex- ducedattheinterfaceofanoxideheterostructurehasbeen
periments were motivated by original Ginzburg’s ideas on observed recently.32,33 The review in Ref. 34 provides an
thepossibilitytogetasurfacesuperconductivitywithahigh excellent overview over the possible symmetries and de-
transitiontemperature.20,21 greesoffreedomofcorrelatedelectronsthatcanevolveat
Another high-T superconductors where an enhance- oxide interfaces. It includes, inter alia, superconductivity,
c
ment of superconductivity at the interface has been es- magnetism,ferroelectricity,andcharge-andspinordersas
tablishedaretheso-callediron-basedpnictidesdiscovered well. Moreover,ithasbeenfound,thatattheinterfacesbe-
in2008.22Inthistypeofsuperconductorscompetingorder tweenLaAlO andSrTiO , superconductivitycoexistswith
3 3
parameters(OP),magneticandsuperconducting, mayco- ferromagnetism,33,35–37asurprisingresultofferingapoten-
exist. Tobemoreexact, thespindensitywave(SDW)and tial for exotic superconducting phenomena due to highly
2
brokeninversionsymmetryoftheinterfaceandaferromag- tentialwellwhichalwayshasdiscreteenergylevels(oronly
neticbackground.36 onelevel).Inone-dimensionalcase(flatinterface)thelocal
Severaltheorieshavebeensuggestedtoexplainthephe- superconductivityarisesevenatarathersmallsuppression
nomenon of interface superconductivity. Some of them ofW.
consideranonuniformchargedistributionneartheinter- Note that stimulation of the bulk superconductivity by
face and use a phenomenological relation of T to this impurities in materials with two OPs was considered by
c
distribution.7,10,38 Different ideas have been used in other oneoftheauthorsalongtimeago.61Recently,theeffectof
theories,39 where bilayers composed of superconductors superconductivitystimulationbyimpuritiesinthebulkof
withdifferentratiooftheT andpairingstrengthwerecon- Fe-basedpnictideswithtwoOPshasbeenanalysedbyFern-
c
sideredanditwasassumedthatT issuppressedbyphase nadesetal. inRef.62. However,tothebestofourknowl-
c
fluctuations.Inthevicinityoftheinterfacetheroleofthese edge,therehasbeennostudyofthemechanismofthein-
fluctuationsisnotimportantascomparedtothebulkdue terfacesuperconductivitypresentedinthispaper.
tosuppressionoffluctuationsbytheproximityeffect.How- In Section II, the general nonhomogeneous Ginzburg–
ever,thissuggestioncannotexplaintheobservedindepen- Landauequationsdescribingtwocoupledorderparameters
denceofT onthedopinglevelx.Perhaps,thereisnosingle areintroduced. Inthehomogeneouscase, wederivecon-
c
mechanismresponsiblefortheinterfacesuperconductivity ditions for the coefficients in the Ginzburg–Landau equa-
becauseitwasobservedinquitedifferentmaterialsunder tionsthatshouldbefulfilledforobtainingoneofthepossi-
variousconditions. Theimportantingredientofthiseffect blethreephases: 1)puresuperconducting,2)purecharge-
is the presence of an interface or some sort of nonhomo- orspin-densitywave,3)amixedstate. Inthenonhomoge-
geneity. neous case, a solution of the Ginzburg–Landau equations
fortheorderparametersisobtained,wherebydetailedcal-
Inthispaperweproposeanewmechanismfortheinter-
culationsareshiftedtotheAppendix.InSectionIII,wedis-
face superconductivity. The proposed mechanism is very
cuss applicability of the theory to the case when the con-
robust and general being independent of the microscopic
stituentsoftheheterostructurearehigh-T cupratesoriron-
detailsofconsideredmaterials. Itisapplicabletoanyma- c
based pnictides. We propose also an experimental setup
terialswhere,alongsidewiththesuperconductingorderpa-
suitablefortestingourpredictions.Concluding,wediscuss
rameter(OP),anotherOPexists.Wedonotpretendtoapply
ourresultsinSectionIV.
ourtheorytoanysystemwheretheinterfacesuperconduc-
tivityoccurs, butweshowthatitcanbeusedtomaterials
inwhichtwoOPsmaypotentiallyexist. Asiswellknown,
II. GENERALCONSIDERATIONS
inhigh-T superconductors,animportantroleisplayedby
c
thecharge-orspin-ordering.23–27Incuprates,achargeden-
sitywave(CDW)hasbeenobservedrecentlyinnumerous A. Freeenergyandself-consistencyequations
experiments40–50 and discussed in Refs. 51–55. It may ex-
ist alongside with superconductivity, whereas in Fe-based WestartwiththeexpressionforthefreeenergyF fora
superconductors, the spin density wave is more impor- systemwithtwoorderparameters(OPs),∆andW. Inone-
tant.ThepresenceofthenonsuperconductingOPW (CDW dimensionalcase,i.e.,incaseofapreferreddirectionpro-
or SDW) changes such characteristics of superconducting videdbytheinterface,thefreeenergyisgivenby
stateasLondonpenetrationdepth,56–58 heatcapacity,59,60
etc. F=1(cid:90) dx(cid:183)ξ2(∆(cid:48))2−a ∆2+bs∆4+γ∆2W2+ (1)
We show that the presence of a ‘hidden’ OP ∆ in a het- 2 s s 2
epreorsctornudctuucrteivwitiythatWth(cid:54)=ei0nmtearfyalceeaadtttoemapppeeraartaunrecseTof>loTcca(lWsu)-, +ξ2w(W(cid:48))2−awW2+b2wW4(cid:184),
where T (W) is the critical temperature of the supercon-
c
ductingtransitionofbarefilmscomposingtheheterostruc- where ∆(cid:48) and W(cid:48) denote the spatial derivatives of cor-
ture which depends on the amplitude W. We consider a responding order parameters, and the coefficients ξ ,
s,w
heterostructure with an interface where W is locally sup- a ,b , andγareingeneralnotindependentandshow
s,w s,w
pressed. Thissuppressionmaybecausedbyanenhanced acomplicateddependenceondopingand/ortemperature,
impurityscatteringanddopinglevelinthevicinityofthein- andonthemeanfreepath.Thesecoefficientsarepresented
terface.Theenhancedimpurityscatteringcanbecausedby inSectionIIIforthecaseofcupratesandiron-basedpnic-
the interdiffusion of atoms and/or roughness of the inter- tides.
face. BothfactorssuppresstheCDWorSDW.Inthiscase, ThefreeenergyiswrittenintheGinzburg–Landauform
in a vicinity of the interface where W is suppressed, lo- of Eq. (1) in the vicinity of a critical temperature. How-
calsuperconductivityariseswith∆(x)decayingonachar- ever, the critical temperatures T of the transitions into
dw,s
acteristicscaleoftheorderofthesuperconductingcoher- astatewithafiniteW,respectively,∆maybequitediffer-
ence length ξ . Interestingly, the local superconductivity ent. Weassumethatthedopingleveldescribedbythepa-
s
occursat‘quantized’temperaturesbecausetheOP∆(x)is rameterµischoseninsuchaway(µ=µ ),thatthecritical
c
describedbythelinearizedGross–Pitaevskiiequation, i.e., temperatureT forthenon-superconductingOPW coin-
dw
by the Schrödinger equation with a one-dimensional po- cideswiththecriticaltemperatureofthesuperconducting
3
transition Ts. This is possible becauseTdw depends on µ, C. Nonhomogeneouscase
whereasT doesnot.Thus,thecoefficientsa andb de-
s s,w s,w
pendonthedifferencesη=(1−T/T ),δ[µ2]≡µ2−µ2,and
s c IncaseofheterostructureswheretheOPsdependonthe
onimpurityconcentrationn .
imp coordinatex,themostinterestingnontrivialsolutionofthe
ThevariationofFwithrespectto∆andW yieldstheself-
systemofequations(2)and(3)correspondstothosewhere
consistency(ortheGinzburg–Lanadau)equations the OPW goes to a finite valueW∞=W−∞ (an asymmet-
−ξ2∆(cid:48)(cid:48)+∆(cid:163)−a +b ∆2+γW2(cid:164)=0, (2) riccaseW∞(cid:54)=W−∞canbeconsideredanalogously)while∆
s s s vanishes at distances from the interface exceeding ξ . In
w
−ξ2wW(cid:48)(cid:48)+W(cid:163)−aw+bwW2+γ∆2(cid:164)=0, (3) otherwords,weconsiderthecasewhenthesystemisatthe
pointΓ farawayfromtheinterface. Weassumethatthe
W
whichrepresentthefoundationofourconsiderations. OPW(x)issuppressedneartheinterface,e.g.,duetoanen-
hancedimpurityscatteringinthevicinityoftheinterfaceor
diffusivescatteringontheinterface.Thediffusivescattering
B. Homogeneouscase maybecausedeitherbyinterdiffusionorinterfacerough-
nesses. Asisknown(seeRef.63andreferenceswithin),the
Equations(2)and(3)withoutthespatialderivativesyield criticaltemperatureT issuppressedbyimpurityscatter-
dw
different uniform solutions, i.e., three different points on ingwhilethecriticaltemperatureTsofthesuperconducting
the plane of order parameters (∆,W). We denote these transition is only weakly affected. The most essential de-
pointsby,respectively,Γ∆(where∆(cid:54)=0,W =0),ΓW (where pendenceofthecoefficientsas,wandbs,wonimpurityscat-
∆=0,W (cid:54)=0),andΓW∆ (where∆(cid:54)=0,W (cid:54)=0,i.e.,bothOPs teringistheoneofthecoefficientaw.
coexist),eachcorrespondingtoanextremumofthefreeen- Dopinglevelneartheinterfacealsomaybechangeddue
ergyfunctionalF(∆,W). Analyzingthesepointswedeter- tointerdiffusionofatoms. Anotherreasonforachangeof
minetheconditionsforaparticularpointtocorrespondto thecoefficientsintheGinzburg–Landauequationsisadif-
aminimum: ferent crystal symmetry at the interface. Such a mecha-
1. Γ∆,where∆=(cid:112)as/bs,correspondstoaminimumif nbeisemnocfoennshidaenrceeddfosurpceorncvoenndtuiocntiavlitlyowat-tTwinsubpoeurcnodnadriuecsthoarss
thesecondderivativesofF withrespectto∆andW c
inRef.64.Achangeintheenergyspectrumofcupratesnear
arepositive,implying
thesurfacehasbeencalculatedinRef.65. Thischangecan
alsoleadtoamodificationofcoefficientsintheGinzburg–
b >0, γa −a b >0. (4)
s s w s Landau equations or even to a surface superconductivity.
Itisworthnotingthatthepropertiesofsurfacesupercon-
Inparticular,thecoefficienta mustbepositive.
s
ductivityinsystemswithone(superconducting)OPinmag-
2. Γ ,whereW =(cid:112)a /b ,correspondstoaminimum neticfieldhavebeenstudiedtheoreticallyinRefs.66–69.
W w w
iftheconditions TheinterfacebehaviorofW canbemodeledviathespa-
tialdependenceofthecoefficienta ,i.e.,
w
b >0, γa −a b >0, (5)
w w s w
(cid:40)
−a , |x|<L,
arefulfilled. Inparticular,thecoefficientawmustbe aw(x)= a ,0 |x|>L, (8)
positive. w
3. ΓW∆, where ∆=(cid:112)(asbw−γaw)/D and whereL isacharacteristicwidthoftheregionwhereW is
W =(cid:112)(a b −γa )/D with D=b b −γ2, corre- suppressed.Theexpressionfora ispresentedinSectionIII
w s s s w 0
spondstoaminimumprovidedtheconditions for a particular case. Formula for a (x) at |x|>L implies
w
thattheamplitudeW±∞=(cid:112)aw/bwisthesameatx→±∞,
bs>0, bw>0, bsbw−γ2>0, (6) havingalowervalueattheinterfacex=0,seeFig.1(a).We
willshowthat,underthesecircumstances,asuperconduct-
aresatisfied. ItfollowsfromthedefinitionofD and ingOP∆arisesattheinterfacedecayingtozeroasx→∞.
theexpressionsfor∆andW thattheconditions Notethatinourpreviouspublication70weanalyzednon-
homogeneous solutions for the Ginzburg–Landau equa-
a b −γa >0, a b −γa >0, (7)
s w w w s s tions(topologicaldefects)assumingthatallthecoefficients
intheseequationsareconstant. Thefoundsolutionsmay
shouldbefulfilled.
correspondtometastablestateswithenergieshigherthan
Clearly, the latter inequalities (7) are incompatible with thatforanuniformsolution. Inthecaseconsideredhere,a
those in (4) and (5). This fact is evident from topolog- nonuniformsolutionfor∆(x)isenforcedbybuilt-indefect
ical arguments, namely, if the point ΓW∆ is a minimum describedbytheanon-constantcoefficientas(x).
ofF(∆,W),thenthepointsΓ∆andΓW canonlycorrespond InordertofindthespatialdependenceofW(x)and∆(x),
to a maximum or a saddle point of the free energy func- weassumethatthesuperconductingOP∆issmall. Then,
tional. inthemainapproximationtheequationforW acquiresthe
4
where ξ˜ =ξ (cid:112)b , E =a b −γa , U =γa , and
s s w s w w w
g=b b . Equation 12 has a form of the Gross–Pitaevskii
s w
equation.71,72 Solving this equation one can determine
the spatial dependence of the superconducting OP ∆.
The calculations in this case formally coincide with those
carried out in Ref. 70 if the interchange ∆↔W is done.
For completeness we repeat the steps one has to perform
seeking for a solution ∆(x). Assuming that ∆ is small, the
right-hand side of Eq. (12) can be neglected and we are
left with the linearized Gross–Pitaevskii equation, i.e., the
‘Schrödinger’ equation, which can be solved considering
theeigenvalueproblem
Lˆ∆ =E ∆ , (13)
n n n
where the operator Lˆ =−ξ˜2∂2 −Ucosh−2(κ x), and ∆
s xx w n
andE aretheeigenfunctionsandeigenvaluesofLˆ. The
n
solutions∆ correspondingtoadiscretespectrumofE can
n n
beexpressedintermsofhypergeometricfunctionsandthe
‘energy’levels(E <0)beinggivenby73
ξ˜2κ2 (cid:183) (cid:115) 4U (cid:184)2
E =− s w −(1+2n)+ 1+ . (14)
n 4 ξ˜2κ2
s w
Provided the inequality 4U/ξ˜2κ2 <8 is fulfilled, there is
s w
FIG.1. (Coloronline.) (a)Sketchofconsideredsystem. TheCDW onlyone‘energy’levelwithn=0andthecorrespondingso-
(orSDW)orderparameterW issuppressednearaninterfacebe- lution ∆ (x) has a form of a soliton. Otherwise there are
tweentwomaterialsinwhichsuperconductingorderparameter∆ 0
severalsolutionsdecayingfarawayfromtheinterfaceand
mayexistalongsideW. Thissuppressionleadstotheappearance correspondingtoE .
of interfacial superconductivity, which can be thus regarded as n
RepresentingtheOP∆closetoacertain‘energy’levelE
‘hidden’. (b)ThecaseofstrongsuppressionofW withsolutions n
of∆n(x)givenbyhypergeometricfunctions. Notethat∆0(x)has oasrt∆ho(xg)o=naclnt∆on∆(x()x+),δo∆nne(xo)bwtaiitnhsathsemcaolelfcfiocrierencttsiocn,δ∆n(x)
theshapeofasoliton,whereasothersolutionshavenodes. (c)In n n
thecaseofaweaksuppressionofW, onehasashallow‘poten- E−E
tial’inthe‘Schrödinger’equationandthereexistsonlyone‘en- c2= n , (15)
n g〈〈∆4(x)〉〉
ergy’levelgivenbyEq.(16). n
where 〈〈f(x)〉〉 = (cid:82)−∞∞dxf(x) (double angle brackets are
used to distinguish the notation from the averaging over
form
momentadirectionsintroducedinAppendixA).
−ξ2W(cid:48)(cid:48)+W(cid:163)−a (x)+b W2(cid:164)=0, (9) Note an important point. The condition E <0 that de-
w w w
terminestheappearanceoftheinterfacesuperconductivity
where aw(x) is given by Eq. (8). This equation can be coincideswiththecondition(5)thatprovidesthestability
solved exactly, but for simplicity we restrict ourselves to ofthestatewithW (cid:54)=0and∆=0inthebulk. Thismeans
the simp(cid:112)lest case of a narrow region of suppression, i.e., thatifanonsuperconductingstateinthebulkischaracter-
L(cid:191)ξw/ a0,thusobtainingthesolution izedbyanonzeroOPW,anysuppressionofW leadstothe
W(x)=W∞tanh(cid:163)κw(|x|+x0)(cid:164), (10) acopnpseiadrearnincegothfeloccaaslesuopfearscmonadlluscutpivpitrye.sWsioendeomfWon.stratethis
wheretheintegrationconstantx obeystheequation Weaksuppressio(cid:112)n,r (cid:192)1. Inthiscase,oneobtainsfrom
0
Eq.(11) x κ (cid:39)ln 2r andthespatialdependenceof∆(x)
0 w
sinh(2x0κw)=4ξ2wκw/a0L≡r, (11) isdeterminedbythe‘Schrödinger’equation.(12)withthe
withκ =ξ−1(cid:112)a /2. ‘potential’ Ucosh−2(κwx)→V(x)=1−tanh2(cid:163)κw(|x|+x0)(cid:164).
w w w In other words, the function ∆(x) is determined by the
Next,weconsiderseparatelythecasesofastrong[r (cid:191)1,
‘Schrödinger’equationthatprovidesthe‘energy’levelsina
cf. Fig. 1 (b)] and, respectively, weak [r (cid:192)1, cf. Fig. 1 (c)]
shallowpotentialwellV(x).Asiswellknown,73therealways
suppressionofW attheinterface.
existsasingle‘energy’level
Strong suppression, r (cid:191)1. In this case, the product
2x0κw(cid:39)r issmallandthequantityx0inEq.(10)canbene- E0=−J2/(2ξ˜s)2, (16)
glected.SubstitutingW(x)inthisapproximationintoEq.(2)
whereJ=〈〈V(x)〉〉=4κ−1exp(−2κ x ).Theamplitudec is
oneobtains w w 0 0
givenagainbyEq.(15)withn=0. Thismeansthatasuper-
ξ˜2∆(cid:48)(cid:48)+(cid:163)E+Ucosh−2(κ x)(cid:164)∆=g∆3, (12) conducting condensate with a small amplitude ∆(cid:39)T E ,
s w s 0
5
theequation(seeAppendixA)
〈2µ2s (µ )〉=ln(T /T ), (17)
c 1m c dw s
where the critical temperature T depends, generally
dw
speaking, on impurity concentration which is assumed to
besmallfarfromtheinterface.
Next,considertheregion|x|<L,wheretheimpurityscat-
teringisassumedtobestronger. Inthiscase,thetemper-
ature T (cid:39)T (cid:163)1−(4πT τ)−1(cid:164), where T is the crit-
dw dw0 dw0 dw0
ical temperature of the transition into a state with W (cid:54)=0
intheabsenceofimpuritiesandsuperconductivity,τ=l/v
FIG. 2. (Color online.) Sketch of temperature dependence of
theamplitudes∆ncorrespondingtodifferenteigenvaluesEn,see is the momentum relaxation time for intraband scatter-
Eq.(14).AtagiventemperatureT,thestatewiththelargest∆n,i.e., ing, and the mean free path l is assumed to be larger
thestate∆0,correspondstoaminimumofthefreeenergy. How- than v/Tdw0. One can easily show that, in this case,
ever,thetransitionsbetweendifferent∆narepossible a0=aw−(4πTdw0τ)−1,whereawisgivenbytheexpression
above.Providedtheconditiona (cid:191)(4πT τ)−1(cid:191)1isful-
w dw0
filled,thenthecoefficienta inEq.(8)ispositive(meaning
0
whereTs isthesuperconductingtransitiontemperaturein that−a0<0)andconsiderablyesceedsaw.
the bulk, necessarily arises at the interface as soon as the
competingOPisarbitrarilyweaklysuppressed.
Note that, in the case of a two-dimensional point de- B. Temperatureanddopingdependence
fectinsteadoftheinterface,|E |isanexponentiallysmall73
0
quantityand,therefore,theradiusofthedecayofthecon- Considering the expression (15) for the coefficients c ,
n
densateisexponentiallylarge. onecanseethatatsmalldifference|E−E |,theOP∆isalso
n
small. It turns to zero at certain temperatures or doping
levels,whereE(T ,µ )=E (T ,µ )holds,withE (T ,µ )
n m n n n n n n
III. APPLICATIONTOCUPRATESANDPNICTIDES given by Eq. (14) with coefficients expressed through the
microscopicparametersandthetemperature. InFig.2we
plotthetemperaturedependenceoftheamplitudes∆ cor-
A. Relationofcoefficientstomicroscopicparameters n
responding to different eigenvalues E . Clearly, when the
n
temperatureT becomeslowerthanthetemperatureT de-
0
Here, we present the expressions for the coefficients in terminedbyEq.(14),thesuperconductingOP∆ (x)arises
0
the Ginzburg–Landau expansion for the case of cuprates
at the interface with the amplitude increasing when T is
andiron-basedpnictides. lowered. ThetemperatureT islowerthanT (η>0), but,
0 s
AshasbeenshowninRef.74,themodelthathasbeende-
undercertainconditions,higherthanthetemperatureT at
b
velopedindetailinRefs.75–77forFe-basedpnictides(gen-
which the superconducting state becomes more favorable
erally, to two-band superconductors with an SDW), is ap- inthebulk.AtT <T ,anewbranch∆ (T)appears,etc.
1 1
plicabletoquasi-one-dimensionalsuperconductorswitha
TheminimaltemperatureT (ormaximaln )atwhich
n max
CDW,and,aftercertainmodification,alsotocuprates. the branch ∆ (T) appears, is determined by the condi-
First, we consider the region |x|>L and assume that (cid:113)nmax
tion 2n ≤ 1+4Uξ˜−2κ−2−1. It can be shown that at
theimpurityconcentrationinthisregionissmall, i.e., the max s w
mean free path l(cid:192)ξ . In the model of Refs. 75–77, the a given temperature T, the state with the largest ∆n, i.e.,
s,w thestate∆ ,correspondstoaminimumofthefreeenergy.
coefficients are related to the microscopic parameters of 0
the model via a =η, b (cid:39)1.05, a =η(1−β )−〈β δ[µ2]〉, However,thetransitionsbetweendifferent∆n arepossible
s s w 1 2
b =s , γ=s , where η=1−T/T and δ[µ2]=µ2−µ2 analogouslytotransitionsbetweenanovercooledstateand
w 3m 2m s c
withµbeingafunctionthatdescribesthecurvatureofthe equilibrium.
Fermisheetsofthequasi-one-dimensionalsuperconductor
oradeviationoftheaverageFermisurfacefromtheperfect
C. Interfacesuperconductingtransitiontemperature
circleiniron-basedsuperconductors,wherepartialnesting
betweentheellipticalelectronbandsandthecircularhole
bandsleadstotheformationofthespin-densitywave. The Ourconsiderationsconcernedthecasewhenthesystem
functions s2m, s3m and β1,2 (see Appendix A) depend on isatthepointΓW farawayfromtheinterface,i.e.,thecon-
thedimensionlesscriticalcurvaturem=µ /πT definedin ditions (5) are valid while the conditions (4) are violated.
c s
suchawaythatthecriticaltemperatureT oftheforma- Whenappliedtothecaseofquasione-dimensionalmate-
dw
tionoftheOPW equalsT ,whereT arethecriticaltem- rialswiththeCDWortomateriallikeiron-basedpnictides
s s,dw
peraturesforthetransitionintothe,correspondingly,super- withanSDW,theseconditionscanbepresentedintheform
conductingandCDWorSDWstateinabsenceofthecom-
petingorderanddoping. Thecriticalµ isdeterminedby A2η+B2δ[µ2]<0, A1η+B1δ[µ2]<0, s3m>0, (18)
c
6
where A1=s3m−s2m(1−β1), B1=s2mβ2, exists again at temperatures T <T∆, but this min-
A =s −s (1−β ), and B =1.05β if expressed via imum is global only at T <T , where the critical
2 2m 3 1 2 2 1pt
the microscopic parameters of the model. The coeffi- temperature T for the first-order phase transition
1pt
cients A1,2 and B1,2 depend on µ(µ0,µϕ) and may attain isdeterminedbytheequationF(aw/bw)=F(as/bs).
positiveornegativevalues(onlyB2 isapositivequantity). At T >Tdw, a uniform solution for W =(cid:112)aw/bw
If these coefficients are all positive, then this conditions arises, but it corresponds to a global minimum at
can be fulfilled provided δ[µ2]<0. Thus, we can rewrite T >T . Inthiscase,noregionofcoexistenceexists,
1pt
them as B2|δ[µ2]|>A2η and B1|δ[µ2]|>A1η. In terms of and at T =T1pt a first-order phase transition from
microscopicparametersthelattercanbewrittenas thesuperconductingstatetoastatewithW (cid:54)=0takes
place with increasing temperature, see Fig. 3 (b).
η<cµB1/A1≡1−T˜W, η<cµB2/A2≡1−T˜∆, (19) Now,ifT0asdeterminedbyEq.(14)correspondingto
n=0(thegroundstate)fallsintotheregionT <T ,
1pt s
wherecµ=|δ[µ2]|andweintroducedthe‘critical’tempera- which is possible if the difference T −T is pos-
s 1pt
turesT˜W,∆=TW,∆/(πTdw,s). itive and sufficiently large, then, at the interface,
Onecandistinguishtwofromthephysicalpointofview superconductivityisinduced.
differentcases:
a) The case T∆<TW is realized if the quantity
D≡A B −A B =s s −s2 >0 provided that A
1 2 2 1 3 3m 2m 1
and A arepositive. Inthiscase,thepuresupercon-
2
ducting state (minimum of the free energy at Γ∆) D. Experiments
existsattemperaturesT <T∆with∆2un=as/bs.Inthe
temperature range T∆<T <TW, a mixed state (the
state of coexistence) with ∆(cid:54)=0 and W (cid:54)=0 given by The obtained appearance of superconductivity (or en-
expressionsjustbeforeEq.(6)takesplace. AtT >T , hancementofthecriticaltransitiontemperature)atanin-
0
apureCDW-stateor,moregenerally,aW-stateoccurs terface between two materials in which, alongside super-
with W2 = a /b . In the interval T <T <T , conductivity, another exists and is energetically more fa-
un w w W 0
one has a surface (or interface superconductiv- vorable, may be realized in two prominent examples of
ity), where T is the temperature determined by such systems. One of them, the cuprates, show a charge-
0
Eq.(14)correspondington=0(thegroundstate). At density wave order alongside superconductivity40–50 In
T <T <T ,thesystemisnonsuperconductingwith thesematerials,anenhancementofsuperconductingtran-
0 dw
the OP W (cid:54)=0. In Fig. 3 (a) we show schematically sitiontemperaturehasbeenfoundinabilayerconstructed
the temperature dependence of ∆ and W and also ofLa1.65Sr0.35CuO4andLa1.875Ba0.125CuO4.9,14
the temperature range in which the local supercon- Superconductivity accompanied by a spin-density wave
ductivityexists. AttemperaturesTW,∆,second-order isknowntoexistintheiron-basedpnictides,wherethein-
phase transitions occur and the OPs ∆ and W arise terfacesuperconductivityisproposedtobethedrivingef-
continuously (see(cid:112)expressions for ∆(cid:112)and W before fectbehindthealmostdoublingofsuperconductingtransi-
Eq.(6)where∆∼ TW−T andW ∼ T−T∆). Note tiontemperatureinCaFe2As2.80,81
an obvious analogy with conventional second-
Unfortunately, there are no data on spatial dependence
type superconductors in a magnetic field H.78,79
oftheorderparameteraccompanyingsuperconductivityin
The quantities T∆,W are analogous to the critical these experiments, neither the spatial dependence of the
fields H [cf. Fig. 3 (c)] so that at T <T one
c1,c2 W superconductingorderparametershasbeeninvestigatedin
has a purely superconducting state (full expulsion
these experiments. Such a measurement would provide a
of the magnetic field), a mixed state in the interval
test of our theory, if charge- or spin-density wave would
TW <T <T∆(correspondingly,theAbrikosov’svortex havebeensuppressedneartheinterface.
state) and a surface or interface superconductivity
in the range T∆<T <T0 (correspondingly, in the Another interesting effect also serving as a test of our
range H <H<H ). At last, at T >T , one has a predictionsisrelatedtothefactthattheremightappeara
c2 c3 0
pureW-statewhichcorrespondstothenormalstate hysteretic behavior stemming from the presence of differ-
in conventional superconductors. Note that same ent‘energy’levels[seeEq.(14)]. Thisresultsinasequence
analysiscanbeperformedif,insteadoftemperature, of temperatures Tn at which ∆ formally vanishes but the
the doping (or curvature) given by cµ in Eq. (19) temperaturedependenceof∆isdeterminedbythehighest
is considered and one obtains the corresponding temperature of them all, i.e., by T0 since the minimum of
situationasdepictedinFig.3(d). thefreeenergyisdeepesthere. Interestingly, atatemper-
atureT alocalminimumofthefreeenergyispresentand
n
b) The case T∆>TW is realized if the quantity adjusting∆bymeans,e.g.,ofanexternalfield,onecanlet
D≡A B −A B =s s −s2 <0 provided that A itfollowthetemperaturedependenceof∆ aftertherelax-
1 2 2 1 3 3m 2m 1 n
and A arepositive. Inthiscase,thepuresupercon- ationofexternalconstrains,so,indeed,“there’salotofroom
2
ducting state (minimum of the free energy at Γ∆) fornewcombinations”.80
7
in a system with two OPs may be responsible for the in-
terface superconductivity observed in many materials in-
cludingcupratesandiron-basedpnictides. Asisfirmlyes-
tablished, in cuprates and iron-based pnictides, CDW (or
quadrupolechargeorder82)orSDWcanexistalongsidewith
superconductivity.
We found that in case of a strong suppression of W at
some point, there are several solutions for the supercon-
ducting OP ∆(x) which are localized on the scale of the
coherence length ξ . These solutions are found from the
s
nonlinearSchrödingerequation(orGross–Pitaevskiiequa-
tion)andcorrespondtodifferent‘energy’levels.Eachsolu-
tionarisesatcertainvaluesoftemperaturesT (ordoping
n
levelµ )andhasadifferentformchangingfromasoliton-
n
likeonetoanoscillatoryfunctiondecayingatinfinity.How-
ever, only the soliton-like solution ∆ (x) corresponds to a
0
minimum of the free energy. Other solutions ∆ (x) (with
n
n(cid:54)=0)withnodeshavehigherenergies.Theycorrespondto
metastablestates.IfthecorrespondingpotentialV(x)inthe
Schrödingerequationisnotdeepenough,thereisonlyone
FIG. 3. (Color online.) (a) In the case T∆<TW, three ranges ‘energy’levelandonlyonesoliton-likesolutionfor∆(x).
of temperature. For T<T∆, the system is in a pure supercon- InthecaseofanasymmetricpotentialV(x),thelocalized
ductingstate, whereasinthetemperaturerangeT∆<T<TW, a solution for ∆(x) exists provided that the potential well is
mixedstate(thestateofcoexistence)with∆(cid:54)=0andW (cid:54)=0isreal-
deepenough.73IfforsomereasonstheOPW issuppressed
ized. Thus,thebulksuperconductingtransitiontemperaturecor-
atthesurface,thesolutionsfor∆(x)havethesameformas
riseswpiodnednsedto(eTnWha.nTchemeteenmtpofersautpuerrecoranndguecotifvitthye)ocnoeaxpispteeanrcaenscteaotef incaseofthesymmetricV(x)andsurfacesuperconductiv-
the interface superconductivity with the highest transition tem- ityarisesinthesample. Noteanimportantpoint. Thelo-
peratureT0(theremaybeothertransitiontemperaturesTn). Fi- calsuperconductivitymayariseinone-ortwo-dimensional
nally,forT>T0,thesystemisinapureW-state(CDWorSDW) cases (a flat interface or a point defect) if the suppression
up to its transition into the normal state at Tdw. (b) In case oftheOPW issmall. Thismeansthateveniftheminimal
T∆>TW,thesystemmaybeineitherthepuresuperconducting valueofW(x)correspondstoaminimumofthefreeenergy
orinthepureW-state. Thetransitionfromoneintoanotheris
in an uniform case, in a nonuniform case local supercon-
of the first order at a temperature T1pt. If T0 falls into the re- ductivitywouldariseattheinterfaceoratthesurface.
gionT1pt<Ts,whichispossibleifthedifferenceTs−T1ptispos- Thedevelopedtheoryisabletoexplaintheemergentor
itive and sufficiently large, then, at the interface, superconduc-
enhanced interface superconductivity in some cuprate or
tivityisinduced. (c)Ananalogywithconventionalsecond-type
superconductorsinamagneticfield H issketched. Thequanti- iron-basedpnictideheterostructures. Thepredictionscan
tiesT∆,W andT0areanalogoustothecriticalfieldsHc1,c2andHc3, betestedinfurtherexperiments.
respectively, i.e., up to Hc1 the system is in a purely supercon- Remarkable, the used approach based on the consider-
ducting state (full expulsion of the magnetic field), in a mixed ation of Ginzburg–Landau equations for two coupled or-
stateintheintervalHc1<H<Hc2,andintheAbrikosov’svortex derparametersdoesnotdependonthenatureoforderde-
stateforHc2<H<Hc3loosingthesuperconductingpropertiesfor scribedbythose.Theobtainedresultsarealsoapplicableto
H>Hc3.(d)If,insteadoftemperature,thedoping(orcurvature)µ descriptionofanarbitrary‘hidden’orderevolvingatthein-
isconsidered, thesituationissimilartothetemperaturedepen-
terface,e.g.,theferromagnetisminducedattheinterfaceof
dence.
anoxideheterostructureobservedrecently.32
Notethatlocalizedsuperconductivitymayarisealsoina
homogeneous sample if a nonuniformW(x) state (for ex-
IV. DISCUSSION ample, stripes) is energetically favorable due to an inter-
nalmechanism(e.g.,theLarkin–Ovchinnikov–Fulde–Ferrel
mechanism).83,84Analysisofthisstatewithtwocompeting
WehavestudiedasystemwithtwocompetingOPsoneof
whichisthesuperconductingOP∆andanother,W,canbe orderparametersdeservesaseparateconsideration.85
theamplitudeofthecharge-orspin-densitywave. Onthe
basis of Ginzburg–Landau equations we have shown that
if the temperature and doping are chosen in such a way ACKNOWLEDGMENTS
thatthestatewithW (cid:54)=0and∆=0isfavorable,i.e.,itcor-
responds to a minimum of the free energy in the bulk of WeappreciatethefinancialsupportfromtheDFGviathe
the sample, an arbitrary small suppression ofW at an in- ProjektEF11/8-1;K.B.E.gratefullyacknowledgesthefinan-
terface or a defect leads to the appearance of local super- cialsupportoftheMinistryofEducationandScienceofthe
conductivity. This mechanism of local superconductivity RussianFederationintheframeworkofIncreaseCompet-
8
itivenessProgramofNUST“MISiS”(Nr.K2-2014-015). We with∆(cid:48)andW(cid:48)aswellas∆(cid:48)(cid:48)andW(cid:48)(cid:48)denotingthefirstand
thankProf.Dr.I.Ereminforusefulcommentsandvaluable second derivatives with respect to x, respectively. These
discussions. equationsdetermineextremaofthefreeenergyfunctional
[cf.Eq.(1)]
AppendixA:Dopingdependenceofthecoefficientsin F=1(cid:90) dx(cid:183)ξ2(∆(cid:48))2−a ∆2+bs∆4+γ∆2W2+ (A6)
Ginzburg–Landauequations 2 s s 2
(cid:184)
+ξ2(W(cid:48))2−a W2+bwW4 ,
In the notation of Refs. 75–77, the Ginzburg–Landau w w 2
equationshavetheform
withrespectto∆andW,andthecorrespondingcoefficients
−ξ2∇2∆+∆(cid:163)W2s +∆2s −ln(T /T)(cid:164)=0,
s 2m 3 s of the G–L expansion are related to variables in Eqs. (A1)
(A1) and(A2)viaa =η,b =s (cid:39)1.05,a =η(1−β )−〈β δ[µ2]〉,
s s 3 w 1 2
−ξ2w∇2W+W(cid:163)〈2µ2s1m〉+W2s3m+∆2s2m−ln(Tdw/T)(cid:164)=0, bw=s3m, γ=s2m, whereη=1−T/Ts. Theexpressionsfor
(A2) thecoefficientsintermsofthemicroscopicparametersof
themodelforcupratesandiron-basedpnictidesaregiven
whereξ arethecoherencelengths(atlowtemperatures)
s,w asfollows:
for ∆ and W, respectively, and T are, respectively, the
s,dw
∞
criticaltemperaturesforthetransitionintothepuresuper- s = (cid:88)(2n+1)−3, (A7)
conducting state or into a state with a CDW or an SDW 3
n=0
othnelyt.ranInsitoiothneirnwtootrhdes,chTadwrgeis-otrhdeercerditsictaatleteinmapbesreantucereoffo∆r s1m= (cid:88)∞ (2n+1)−1(cid:163)(2n+1)2t2+m2(cid:164)−1, (A8)
andµ, whileT isthe superconductingtransitiontemper- n=0
s
∞
ature in absence of W. The angle brackets mean the an- s = (cid:88)(cid:173)(cid:163)(2n+1)2−m2(cid:164)(2n+1)−1(cid:163)(2n+1)2+m2(cid:164)−2(cid:174),
2m
gle averaging (in Fe-based pnictides) or integration along n=0
thesheetsoftheFermisurfacesinquasi-one-dimensional (A9)
superconductors. The functions s1m, s2m, etc. are func- ∞
tions of the normalized curvature m=µ/(πT ), where s = (cid:88)(cid:173)(2n+1)(cid:163)(2n+1)2−3m2(cid:164)(cid:163)(2n+1)2+m2(cid:164)−3(cid:174),
s 3m
µ=µ0+µϕcos(cid:163)(p2y+pz2)1/2a(cid:164) is a curvature in quasi-one- n=0
(A10)
dimensional superconductors with a doping-dependent
valueofµ0.ItisassumedthattheFermisurfaceofthesesu- β = (cid:88)∞ (cid:173)4m2(2n+1)(cid:163)(2n+1)2+m2(cid:164)−2(cid:174), (A11)
perconductorsconsistsoftwoslightlycurvedsheetswhich 1
n=0
are perpendicular to the x axis74. In the case of Fe-based ∞
pnictides,µ=µ0+µϕcos(2ϕ)isaquantitythatdescribesan β2= (cid:88)2(2n+1)−1(cid:163)(2n+1)2+m2(cid:164)−1, (A12)
elliptic(µϕ(cid:54)=0)andcircular(µϕ=0)Fermisurfacesofelec- n=0
tronandholebands.75–77 Allquantities—∆,W andµ—are wheret=T/T ,theanglebrackets〈...〉denotetheangleav-
measured in units of πT . The expressions for the coeffi- s
s eraging (in iron-based pnictides) or integration along the
cientsintheG–Lexpansionwithaccountforimpurityscat-
sheetsoftheFermisurfaces(inquasi-one-dimensionalsu-
teringhavebeencalculatedinRef.86.
perconductorsorcuprates).
Replacing the derivative ∇→∇−i2πA/Φ , one can use
0
Eqs.(A1)and(A2)todescribevorticesinsuperconductors
withaCDW,87whereΦ0isthemagneticfluxquantum. AppendixB:DetailsonsolutionoftheGross–Pitaevskiiequation
As it is seen from Eq. (A2), the critical temperature T
dw
depends on doping, i.e., on the parameter µ. We choose
HerewesketchthesolutionoftheGross-Pitaevskiiequa-
thisparameterµ=µcinsuchawaythatTdw(µc)=Ts. This tion for ∆. In zero-order approximation we obtain for ∆
meansthatatT =T ,thequantities∆=W =0,and,thus,µ 0
s c fromEq.(12)
obeystheequation
〈2µ2s (µ )〉=ln(T /T ), (A3) ξ˜2∆∆(cid:48)0(cid:48)+∆0(cid:163)E+Ucosh−2(κwx)(cid:164)=0. (B1)
c 1m c dw s
where µc is a function of two parameters, Thisequationisintegrableanditssolutionsψncorrespond-
i.e.,µc=µc(µ0,µϕ). ingtoadiscretespectrumofEn areexpressedintermsof
Then, we expand the function s (µ,T) in the de- hypergeometricfunctions73. Inournotations,the‘energy’
1m
viations δ[µ2]=µ2−µ2 and δT =T −T, thus obtaining levelsofdiscretespectrumaregivenby73
c s
s (µ,T)=s (µ ,T )+β δT+〈β δ[µ2]〉,anduseEq.(A3)
to1mobtain equ1matiocnssin a g1eneral s2tandard form (assuming E =−ξ˜2sκ2w(cid:183)−(1+2n)+(cid:115)1+ 4U (cid:184)2. (B2)
thatallthefunctionsdependonlyononecoordinatex), n 4 ξ˜2κ2
s w
−ξ2∆(cid:48)(cid:48)+∆(cid:163)−a +b ∆2+γW2(cid:164)=0, (A4)
s s s and their maximal number nmax is determined by
−ξ2wW(cid:48)(cid:48)+W(cid:163)−aw+bwW2+γ∆2(cid:164)=0, (A5) 2nmax≤(cid:113)1+4Uξ˜−s2κ−w2−1.
9
We expand the correction δ∆ to the zero-order solu- with R=g∆3+(E−E )∆ and represent ∆ as
N
tion∆ intermsofthenormalizedeigenfunctions∆ ≡ψ ∆(x)=c ψ (x)+δ∆ (x), where δ∆ (x)=(cid:80)(cid:48) c ψ (x),
0 n n N N N N n N,n n
oftheoperatorLˆ =−ξ˜2∂2 −Ucosh−2(κ x). Thesefunc- andthesummationrunsoveralln exceptthetermn=N.
s xx w
tionsobeytheequation Wesubstitutethis∆(x)intoEq.(B4)andmultiplythisequa-
tionfirstbyψ andthenbyψ withn(cid:54)=N,thenintegrating
N n
Lˆψ =E ψ . (B3) the obtained result each time over x. Thus, taking into
n n n
account the orthogonality of different eigenfunctions, we
findthecoefficientsc
n
SolutionsofEq.(12)canbewrittenexplicitlyifthequan-
E−E
tityE =E(η,δ[µ2])isclosetoacertain‘energy’levelEn,say c2 = N , (B5)
to E , such that E (cid:39)E =E(η ,δ[µ2 ]) (if considering the N g〈〈ψ4 〉〉
N N N N N
model for cuprates or iron-based pnictides, the ‘tempera- 〈〈ψ3 ψ 〉〉
ture’ηordopingδ[µ2]shouldbechosenproperly).Wewrite c =gc3 N n withn(cid:54)=N, (B6)
N,n N E −E
Eq.(12)intheform n N
where 〈〈f(x)〉〉 = (cid:82)−∞∞dxf(x). Obviously, in Eq. (B6), ψn
Lˆ∆=E ∆+R(∆), (B4) andψ havetohavesameparity(bothevenorbothodd).
N N
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