Effects of geometrical fluctuations on the transition temperature of disordered quasi-one-dimensional superconductors

0.16 MB·English

by  Dror Orgad

Effects of geometrical fluctuations on the transition temperature of disordered
quasi-one-dimensional superconductors
Dror Orgad
Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel
(Dated: January 28, 2009)
We study the superconducting transition temperature, Tc, of an array of fluctuating, maximally
gapped interacting ladders, embedded in a disordered plane. Renormalization group analysis indi-
9 cates that geometrical fluctuationsmitigate thesuppression of theintra-ladderpairing correlations
0 bythedisorder. Inaddition,thefluctuationsenhancetheJosephsontunnelingbetweentheladders.
0 BotheffectsleadtoanincreaseofTcintheJosephsoncoupledarray. Thesefindingsmayberelevant
2 totheunderstandingofthe1/8anomalyinthelanthanumbasedhigh-temperaturesuperconductors.

n
PACSnumbers: 74.81.-g,74.62.Dh,74.40.+k
a
J

8
I. INTRODUCTION insights outlined above, it is tempting to associate the
2
suppression of T with enhanced disorder effects due to
c
the slowing down of the stripe fluctuations.
] Both perturbative renormalization group (RG)
n Motivated by such questions we study in this paper
o studies1 and strong coupling numerical calculations2
a model of a fluctuating disordered array of maximally
c have demonstrated that doped t J and Hubbard
- ladders typically exhibit a maxima−lly gapped phase, gapped n-leg ladders. We find that the fluctuations re-
r ducethedisorderinduceddegradationofthepairingcor-
p with a single remaining gapless charge mode protected
relationson eachone-dimensionalsystem, althoughthey
u by symmetry3. Such a phase possesses substantial
typicallydonotchangethecriticalvalueoftheLuttinger
s superconducting pairing correlations with approximate
. interactionparameter,K ,abovewhichinfinitesimaldis-
t d-wave symmetry. By coupling many ladders together ∗
a order is irrelevant. The latter is a decreasing function
m via Josephson tunneling, the phases of the different
of the width n. Moreover, the fluctuations enhance the
Cooper pairs may establish coherence, in which case
- Josephson tunneling between the ladders. Both of these
d global superconductivity ensues. It has been suggested
effects lead to a higher T for the array.
n that the underlying mechanism of superconductivity in c
o the cuprates is essentially the one described here for the
c quasi-one-dimensionalsystem4,5.
[ II. THE SINGLE LADDER MODEL
Previous studies have shown that Josephson coupled
2
arraysofspin-gappedchainsandtwo-legladdersarepar-
v We begin by considering a single fluctuating n-leg
ticularly prone to the deleterious effects of disorder on
7 ladder. We assume that the fluctuations occur on
T 6,7. Ontheotherhand,wehavedemonstratedthatge-
5 c length scales which are long compared to the electronic
3 ometrical fluctuations of a spin-gapless one-dimensional
wavelength and thus neglect backscattering from them.
3 systemtendtoreducetheeffectsofdisorder8. Itisthere-
Specifically,wetreatthelimitwheretheladderoscillates
. fore interesting to try andmarry these two observations,
2 rigidly inside a parabolic potential. Its state is given by
1 and study the interplay between disorder and fluctua- the deviation u(τ) from the bottom of the well (taken in
7 tionsinaquasi-one-dimensionalsuperconductor. Beyond
the following as the x-axis), and its action in imaginary
0 its purely theoretical appeal this question may also be
time is
: relevantto a distinctive feature in the phenomenologyof
v
i the lanthanum based cuprates. M Mω2
X S = dτ (∂ u)2+ 0u2 , (1)
Among the high-temperature superconductors, the u 2 τ 2
r Z (cid:20) (cid:21)
a lanthanum compounds stand out as the ones with the
mostextensiveexperimentalevidencefortheexistenceof where M is the ladder’s effective mass. The fluctuations
quasi-one-dimensionalchargeandspinstripeorder,both arecharacterizedby the oscillationfrequencyω0, andby
intheir”normal”andsuperconductingstates9,10. Exper- their typical amplitude λ=1/√Mω0.
imentshavealsofoundclearcorrelationbetweenthepres- The non-interacting spectrum of the ladder consists
ence of robust static stripe order and the suppression of of n energy bands. We assume that the filling is such
superconductivity in these materials10. For example, in that all of them cross the Fermi level. If this is not the
La Nd Sr CuO the ordering temperature, T , of case, then, in the following, n should be taken as the
1.6 x 0.4 x 4 m
the in−commensurate magnetic structure reaches a max- number of bands that do cross EF. Each of those do
imum around x = 1/8, while the superconducting tem- so attwo Fermi points k(b), one containing left-moving
± F
perature, T , exhibits a substantial dip there. This 1/8 (r = 1), and the other right-moving (r = 1) particles.
c
anomaly was first identified in La Ba CuO 11, where The a−nnihilation operator of an electron in band b, with
2 x x 4
it is especially pronounced. In vi−ew of the theoretical chirality r and spin projection σ = 1 can be expressed
±
2
using bosonization language as where n is the average electronic density in band a.
a
Assuming a short-range Gaussian disorder, one finds
ψb,r,σ(x)= √Fb2,rπ,σaeirkF(b)x−√i2[θc(b)−rφc(b)+σθs(b)−σrφs(b)](x), that its components obey Vaη,b,r(x,yj)[Vaη′,b′,r′(x,yj′)]∗ =
DBj,j′[δ δ δ +δ δ δ ]δ(x x)δ(y y ),
(2) a,b r,r′ a,a′ b,b′ r,−r′ a,b′ b,a′ − ′ j− j′
wcohmermeutthaetioKnlebinetwfaecetnortsheFdb,irff,σereennstupraerttihcelecsoprerceicets,aanntid- andVaξ,b,r(x,yj)[Vaξ′,b′,r′(x,yj′)]∗ =DBaj,,jb′δr,r′[δa,a′δb,b′+
δ δ ]δ(x x)δ(y y ). Here, D is the disorder
a is a short distance cutoff of the order of kF−1. Phys- star,be′ngb,tah′ and−Bj,′j′ arej n−umjb′ers of order unity which de-
ically, φ(b) and φ(b) are, respectively, the phases of the a,b
c s pend on the details of the band structure.
2k(b) charge and spin-density wave fluctuations in band
F
b, while θ(b) is the superconducting phase of the band. In the following we focus on the physics at energies of
c the order of the inter-ladder couplings,which we assume
When the electronic interactions are turned on, one
lie wellbelow the scale ∆, setby the gapsof the massive
finds for generic densities away from half filling that the
modes. Under suchconditions the lattercanbe replaced
system ends up in a phase where all the spin fields φ(b),
s by their non-vanishing expectation values, thus yielding
and all the relative charge modes θ(a) θ(b), acquire
c c
a gap1,2. Consequently, the pronounced−fluctuations of
their conjugated fields θs(b), and φ(ca) −φ(cb), cause their χη,ξ(a,b,r,σ) = r(φ(ca)∓φc(b))−σ(θs(a)−θs(b))
correlations to decay exponentially to zero. In this so
+ c (a,b,r,σ), (8)
η,ξ
called C1S0 phase the only gapless excitations are cre-
ated by the total charge fields
1 n 1 n where cη,ξ(a,b,r,σ) are constants. All the χη,ξ contain
φ = φ(b), θ = θ(b). (3) fields conjugated to the massive modes, whose correla-
c+ √n c c+ √n c
tion functions decay exponentially. As a result, except
b=1 b=1
X X
for the intra-band forwardscattering part (whose effects
Note that if the interaction strength is larger than the
wediscussinthenextsection)thebaredisorderpotential
band-structure gaps, the number of occupied bands may
coupling,Eq. (4),isirrelevantintheRGsense. However,
change relative to the occupancy of the bare spectrum,
higherorderterms,generatedinthecourseoftheRGby
and n should be adjusted correspondingly. integrating overthe massive modes, may be relevant7,12.
The plane containing the ladder is taken to include a
To obtain the structure of these terms, which are the
weak random potential which couples to the electrons,
ones responsible for the degradation of the pairing cor-
and through them to u. This choice is motivated by the
relations on the ladder, we need to consider all possible
situationinthecuprateswherethe”ladders”aredefined
linear combinations of the χ with integer coefficients.
η,ξ
by the sites occupied by holes, and as such interact with
The most relevant coupling is that which contains the
the disorder only indirectly via the electronic degrees of
gapless field φ with the smallest coefficient in front of
freedom. Consequently, the forward, Vη(r), and back- c+
it, and no gapped fields. Eq. (8) indicates that this cou-
ward scattering, Vξ(r), disorder components couple as plingisgenerically13generatedby n χ (b,b,r,σ),and
b=1 ξ
Hdis = dx n hηa,b(x,yj)+hξa,b(x,yj) +H.c., (4) stacaktetsetrhinegfobrymth2e enbff=e1cφtic(vbe) =dis2o√rdneφrPci+s.dCesocnrsibeqeduebnytly,the
P
Z a,Xb,j=1h i
where y =jd+u, with d the inter-leg distance, and
j V (x,u)
H = dx eff ei√2nφc+(x)+H.c., (9)
eff dis
hηa,,bξ(r) = eir(kFa∓kFb)xVaη,,bξ,r(r)ψa†,r,σ(x)ψb, r,σ(x) − Z 2πα
±
r,σ
X
Vη,ξ (r) where α is of the order of the correlation length v/∆ of
a,b,r e−i√π2χη,ξ(a,b,r,σ)(x). (5) the gapped fields, with v being their typical velocity14.
∼ 2πa
Xr,σ Theleadingcontributiontotheeffectivedisordersatisfies
V (x,y)V (x,y )=D δ(x x)δ(y y ), with
The exponents in Eq. (5) are given by eff e∗ff ′ ′ eff − ′ − ′
χ (a,b,r,σ) = r(φ(a) φ(b)) (θ(a) θ(b))
η,ξ c ∓ c − c − c D D(D/aλ∆2)n 1(∆/ω )(n 1)/2(α/a)n+1. (10)
+ rσ(φ(a) φ(b)) σ(θ(a) θ(b)). (6) eff ∼ − 0 −
s ∓ s − s − s
They vanish for the intra-band forward scattering pro-
cesses and instead these terms take the form
Hence,afteraveragingoverthedisorderandtheladder
dynamics,asdescribedbyEq. (1),weareledtoconsider
√2
hη (r) Vη (r) n + ∂ φ(a) , (7) thefollowingactionfortheC1S0phaseofthen-legladder
a,a ∼ a,a a π x c !
3
1 v K v
S = dxdτ i∂ θ ∂ φ + c+ c+(∂ θ )2+ c+ (∂ φ )2
x c+ τ c+ x c+ x c+
π − 2 2K
Z (cid:20) c+ (cid:21)
dxdτdτ
D ′F(τ τ )cos √2n[φ (x,τ) φ (x,τ )] , (11)
− eff (2πα)2 − ′ c+ − c+ ′
Z n o
where the fluctuations kernel is given by8
1 1
F(τ τ′)= δ[u(τ) u(τ′)] = . (12)
− h − i √2πλ√1−e−w0|τ−τ′|
III. RG ANALYSIS Few comments are appropriate at this point. First,
note that formally the static ladder limit is reached in
our model by taking λ 0, ω , in order to keep
TheRGanalysisofthisactionismostnaturallyformu- 0
→ → ∞
theladderinitsgroundstateinsidethewell. Secondly,as
lated in terms of the following dimensionless parameters
mentionedintheprevioussection,theintra-bandforward
nD α αω scattering terms, Eq. (7), survive in the gapped phase.
eff 0
= , ̟ = ,
D √2π3λvc2+ vc+ Tsthreenygtahres orfeltehveanptaritns wthheicRhGcousepnleset:o tthhee adviemreangseiobnalnedss-
K =K D 4 +K2 1ds 1 . (13) density and to its long-wavelength fluctuations scale as
c+− 4 (cid:18)n2 c+(cid:19)Z0 √1−e−̟s dDf,1/dℓ = 3Df,1 and dDf,2/dℓ = Df,2, respectively.
These processes can be shown not to affect the pairing
Note that the parameter K is no longer a pure measure correlationsin the static limit16, or more generallywhen
of the electronic interactions,but is ratheran admixture ω . However, for finite ω they induce a retarded
0 0
of the interactions and the disorder15. However, it is attr→act∞ionbetweenthe electronsviaexchangeofladder’s
K which controls the asymptotic decay of the various oscillation quanta. This attraction tends to renormalize
correlationfunctions16. TheresultingRGflowequations, K towards higher values and enhance pairing, thus re-
with respect to the running cutoff α(ℓ)=αeℓ are inforcing the effects outlined above. The intra-band for-
ward scattering processes do contribute to the localiza-
dK 1 d
= D K2, D =(3 nK) , tion of the ladder since they provide additional channels
dℓ −2√1 e ̟ dℓ − D foritscouplingtothedisorder[beyondtheonecontained
− −
d̟ 4 (cosh̟ 1) inEq. (11)]. Consequently,inthelasttwoflowequations
=̟ LD − , (14)
of set (14) one should substitute + + .
dℓ − n (1 e ̟)3/2 f,1 f,2
− − Due to its large scaling dimensioDn → D isDparticuDlarly
dλ 2 (1 cosh̟+̟sinh̟) Df,1
= LD − λ, effective in localizing the ladder.
dℓ − n ̟(1 e ̟)3/2
− − Finally, one should also take into account the renor-
malization of K at energy scales which span the range
winigthasthde /ddimℓe=nsionl.esTshleadddeetraillesnogfththLeir=deLri/v2aπtiαonscaarle- betweenEF and∆,i.e.,duringthegenerationofthegaps
L −L which define the C1S0 phase. Perturbative RG analysis
presented in Ref. 8.
of the single chain problem8,16 indicates that K renor-
The RG equations (14) imply that infinitesimal disor-
malizes by an amount of the order of D/v λ∆ during
F
derisirrelevantforK >K =3/n,independently ofthe
∗ this stage. In the following we assume that D v λ∆,
F
fluctuations. Moreimportant, however,arethe effects of ≪
and ignore this ”high energy” renormalization. More-
fluctuationsinthepresenceoffinitedisorder. First,since
over, such a condition is necessary in order to ensure
D ∝ λ−nω0−(n−1)/2, see Eqs. (10) and (13), large ampli- thatdisorderdoesnotdestroythe gaps. Since thesizeof
tude and frequency oscillations tend to decrease the ini- the gaps is known to decrease rapidly with the number
tialstrengthofthe effective disorder. Secondly,Eq. (14) of legs4,5,17, the region of applicability of our treatment
showsthatthedisorderismoreefficientinrenormalizing shrinks correspondingly with n.
K forsmalleroscillationfrequencies,andthatitdrives̟
towards this regime. Due to both mechanisms, stronger
and faster fluctuations cause K to flow more slowly to
IV. COUPLED LADDERS
smallervalues. Consequently,theladder’ssuperconduct-
ing pairing correlations,which decay as x 1/nK, are less
−
degradedby the disorder. As demonstratedbelow, these Next, we consider an array of coupled C1S0 n-leg lad-
strongerpairingcorrelationstranslateintoanincreaseof ders. The arguments leading to Eq. (9) imply that the
T for the Josephson-coupled array. most relevant inter-ladder charge-density wave (CDW)
c
4
1
coupling, , is the 2nk CDW coupling. Its scaling
F
V
dimension, 2 nK, makes it more relevant than the
−
JosephsontunnelingonlyonceK <1/n. Moreover,hη , 0.8
a,a
which we neglected in this work, is known to suppress
V
exponentially16. Inter-ladder q 0 couplings also pro-
mote over 18. Owing to thes∼e reasons and our inter- 0.6
J V Tc
est in applications to the superconducting cuprates, we Tc0
assume that the Josephson tunneling between adjacent 0.4
ladders dominates over their CDW coupling. Hence we
analyze the effect of
0.2
n
HJ =−Z dxhXi,jiaX,b=1Jab(ui−uj)∆ia†∆jb+H.c., (15) 0 10−5 10−4 10−3̟ 10−2 10−1 100
with ∆i = ψi ψi + ψi ψi the singlet su-
a a,+, a, , a, , a,+, FIG. 1: Tc of an array of 2-leg ladders as obtained by inte-
perconducting ord↓er p−a↑rameter−↓in ban↑d a of the i-th gratingtheRGequations(14)and(17)(solidline),andfrom
ladder. The Josephson tunneling amplitude depends, the mean field theory, Eq. (20), (broken line). The results
exponentially, on the inter-ladder separation, J are for a system with bare K = 0.7, J = 10−2, D = 10−4,
ab
J0e−|s+ui−uj|/γ,wheresisthemeanspacingofthearra∼y λ/γ =10−3, and L=10.
andγdependsontheenvironmentbetweentheladders19.
Assuming λ s, in order to remain in the quasi-one-
dimensional l≪imit, we obtain for H in the C1S0 phase W is vanishingly small, this condition reads
J
1 1 1/Tc
HJ =−Z dx i Je((u2i−πuαi)+21)/γ cos"rn2 θci++1−θci+ #, J˜ = (2πα)2 Z dxZ0 dτP(x,τ), (19)
X (cid:0) ((cid:1)16) with P(x,τ) = cos[ 2/nθ (x,τ)]cos[ 2/nθ (0,0)]
c+ c+
h i
where14 J = 8 n J (0) cos(√2φ(a) cos(√2φ(b) . calculated using the action (11). Approximating the ef-
Its contribution toa,bt=h1e aRbG hflow, in steirhms of s =i fectoftemperaturebypacutofflengthlT =pvc+/T beyond
J/2π√nv , andPderived assuming20 ,λ/γ 1, iJs whichallcorrelatorsdecayexponentiallytozero,andus-
c+
J ≪ ing the scaling properties of P , Eq. (19) becomes6,7
∂K λ2 ∂ 1
= 2 1+ (1+e−̟) , J = 2 , 1 vc+/Tc RdR ln(R/α) dℓ
∂ℓ J γ2 ∂ℓ − nK J = exp , (20)
∂̟ (cid:20) λ2 (cid:21) (cid:18) (cid:19) J˜ Zα 4πvc+α2 "−Z0 nK(ℓ)#
=̟ 2nπ2 2 [1+I (̟)], (17)
∂ℓ − LJ γ2 0
where K(ℓ) is calculated from Eq. (14).
∂λ =nπ2 2λ31+I0(̟)−̟I1(̟), Results for the ratio of Tc to the transition temper-
∂ℓ LJ γ2 ̟ ature of the pure system, in the case n = 2, are pre-
sented in Fig. 1. The effect of the fluctuations on T is
c
where I0,1(x) are modified Bessel functions. substantial22 and is particularly important whenever
By integrating Eqs. (14) and (17) one can calcu- ismorerelevantthan ,i.e. forK <(1+√5)/2n. Fig.D1
J
late the scale, ℓ , at which grows to be of order was generated for fixed bare , ignoring the dependence
∗
J D
1, and obtain an estimate for the transition tempera- oftheeffectivedisorderonω ,asgivenbyEq. (10). Tak-
0
ture of the array Tc ∆e−ℓ∗. Since fluctuations tend ingthisdependenceintoaccountwillonlymakethemiti-
∼
to increase K, via its renormalization by both and gatingconsequencesofthefluctuationsmorepronounced.
D
, and since K controls the scaling of , T likewise Additionalamplificationofthephysicsdiscussedhereoc-
c
J J
rises. Another estimate for T can be derived using curs when shape deformations of the ladders are taken
c
the inter-ladder mean field theory6,7,21. By replacing into account8.
Je(ui−uj)/γ J˜ = 2J e(ui−uj)/γ = 2Jeλ2/2γ2, and
→ h i
cos[ 2/n(θi θj )] cos( 2/nθ ) cos( 2/nθ )
c+ − c+ → h c+ i c+ V. DISCUSSION
in H , the problem turns into that of a single ladder
J
p p p
described by Eq. (11) and coupled to an effective field
Ourstudywasmotivatedbythephysicsofthecuprate
W high-temperature superconductors, especially the 214
H dx cos 2/nθ , (18)
J →− (2πα)2 c+ compounds, which exhibit both static and fluctuating
Z (cid:16)p (cid:17) charge and spin stripe orders over a substantial portion
induced by the neighboring ladders and determined self- of their phase diagram4,5. In particular, we were inter-
consistently from W = J˜cos( 2/nθ ) . At T , when ested to gain insight into the anomalous suppression of
c+ c
h i
p
5
T at x =1/8, which is manifested to various degrees in stripes. These excitations originate from the intervening
c
these systems. One should be cautious, however, when lightly doped regions between the charged stripes. How-
trying to apply the model studied here to the physical ever,apartfromacting aseffective barriersfor tunneling
materials. To begin with, we have considered the quasi- betweenthe stripes [as characterizedby the phenomeno-
one-dimensional limit of weakly coupled ladders. This logical length γ in Eq. (16)] they have little effect on
limit allows for a controlled theoretical treatment of the the charge response of the system governing its super-
problem,butisanextremecaricatureoftheactualcharge conducting susceptibility, which is the focal point of the
modulationobservedinthecuprates,whoseamplitudeis present study.
difficult to ascertain but which is most likely weak. We Notwithstanding, our model constitutes a controlled
then assumed smooth long-wavelength fluctuations and theoretical laboratory for the study of the interplay be-
ignored lattice commensurability effects, which are be- tween the stripe dynamics and the disorder. Provided
lieved to be important in the slowing down of the stripe that ̟ is empirically associated with the distance from
dynamics at x = 1/8. In a related approximation we x = 1/8 in the phase diagram of the lanthanum based
did not include fluctuations of the stripe order towards cuprates, then Fig. 1 offers a mechanism for the T
c
a similar order oriented along the perpendicular direc- anomaly at x = 1/8. It suggests that the above men-
tion. Recent scanning tunneling spectroscopy23 demon- tioned interplay plays an important role in alleviating,
strated domains of perpendicular stripe orientations in atleastpartially,the disorder-inducedsuppressionofsu-
Bi2Sr2CaCu2O8+δ. It is conceivable, however, that in perconductivity in these compounds.
thelowtemperaturetetragonalphaseofLa Ba CuO ,
2 x x 4
where the in-plane rotational symmetry is−broken and
the x = 1/8 anomaly is observed, such an approxima-
Acknowledgments
tion is not too severe. Finally, the striped model pos-
sessesalargespin-gap,whichimpliestheabsenceofgap-
less nodal quasiparticles in the superconducting phase It is a pleasure to thank E. Arrigoni and T. Gia-
(although it exhibits a ”d-wave-like” order parameter). marchi for useful discussions. This work was supported
The same spin-gap means that the model does not con- by the United States - Israel Binational Science Foun-
tain any of the low-energy incommensurate spin excita- dation (grant No. 2004162) and by the Israel Science
tions, which are the primary experimental evidence for Foundation (grant No. 538/08).
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