Tc suppression and resistivity in cuprates with out of plane defects

0.19 MB·English

by  S. Graser

T suppression and resistivity in cuprates with out of plane defects
c
S. Graser, P. J. Hirschfeld, and L.-Y. Zhu
Department of Physics, University of Florida, Gainesville, FL 32611, USA
T. Dahm
8
Institut fu¨r Theoretische Physik, Universit¨at Tu¨bingen, Tu¨bingen, Germany and
0
Institute for Solid State Physics, University of Tokyo, Kashiwanoha, Kashiwa, Chiba 277-8581, Japan
0
(Dated: February 4, 2008)
2

n Recentexperimentsintroducingcontrolleddisorderintooptimallydopedcupratesuperconductors
a by both electron irradiation and chemical substitution have found unusual behavior in the rate of
J suppression of the critical temperature Tc vs. increase in residual resistivity. We show here that

7 the unexpected discovery that the rate of Tc suppression vs. resistivity is stronger for out-of-plane
1 thanforin-planeimpuritiesmaybeexplainedbyconsistent calculation of bothTc andresistivity if
thepotentialscattering is assumed tobenearly forward innature. Forrealistic models ofimpurity

] potentials, we further show that significant deviations from the universal Abrikosov-Gor’kov Tc
n
suppression behaviormay beexpected for out of plane impurities.
o
c
PACSnumbers: 74.72.-h,74.25.Jb,74.20.Fg
-
r
p
u The destruction of superconductivity by disorder has comparison to resistivity measurements.
s been traditionally used to probe the nature of the su-
. A further paradox was reported recently by Fujita et
at perconducting state. In classic superconductors, pair- al.13, who showed that the rate of suppression dTc/dρ is
m breakingiscausedonlybymagneticimpurities1,andthe significantly higher for out-of-plane cation substituents
functional form of the T suppression, when plotted vs.
- c inBi2Sr2CuO6 (Bi-2201)and(LSCO)thanforZninthe
d impurity concentration or change in normal state resis- same system. Since Zn is thought to be a near-unitary
n tivity, is known to follow the universal curve predicted
scattererin these materials,this is somewhatmysterious
o by Abrikosovand Gor’kov(AG)2. In unconventionalsu-
atfirstsight. Ifoneaccountsforthefactthattheout-of-
c
perconductors,ordinarynonmagnetic impurities arealso
[ planedefects arepoorlyscreened,andmayactprimarily
expected to break pairs, and in the simplest approxima- as forwardscatterers,however,the increase of resistivity
2 tionwheretheimpuritiesaretreatedaspoint-like(delta-
with defect concentrationwill be slowedandone may be
v function)potentialscatterers,theT suppressionalsofol-
c able to understand this discrepancy. It is the primary
5
lows the AG form.
3 purpose ofthis paperto correlate,within simple models,
6 As in so many other respects, the experimental situa- the slower rates of both Tc suppression and resistivity
3 increase in the case of near-forward scatterers to see if
tioninthecupratesagreesqualitativelywiththesimplest
0 light can be shed on this puzzle.
notions of what should happen to d-wave superconduc-
7
torsinthepresenceofdisorder,butdiffersinsomeimpor- OthertypesofdeviationsfromtraditionalAGbehavior
0
/ tantdetails. Forexample, whenZnis substituted forCu have been observedand require explanation. In the past
t
a in the Cu-O planes, Tc is suppressedrapidly as expected several years, electron irradiation studies have appeared
m forad-wavesuperconductor. Here,“rapidly”meansthat which are able to suppress superconductivity to zero in
the disorder-induced scattering rate required to destroy a controlled fashion in contrast to early studies which
-
d superconductivity is on the order of the gap scale rather studied only the initial slope of the T suppression by
c
n thantheFermienergyE ,aswouldbeexpectedinans- disorder9. Rullier-Albenque et al.10 used 2.5 MeV elec-
F
o
wave system. Nevertheless, the initial slope of the T vs. tronirradiationtocreatedefectsthroughoutYBa Cu O
c c 2 3 7
∆ρ curve found in experiment is a factor of 2-3 smaller (YBCO-123)andYBa Cu O samples. Inneithersam-
: 2 3 6.6
v thantheuniversalAGcurve3. Thisdiscrepancyhasbeen ple was an AG-type behavior for T observed. In the
c
i
X attributed to scattering in higher angular momentum optimally doped sample, on the contrary, a remarkable
channels by several authors, who modelled the scatter- linear behavior in T vs. ρ was observed down to and
r c
a ing potential with a separableformdescribingscattering including samples of vanishingly small T . Because the
c
inboths-andasinglehigherℓ-wavechannel4,5. Thisisa nearly ”normal” samples are expected to have a small
simple and tractable way of including the finite range of superfluid stiffness, this effect was interpreted provision-
the scatterers qualitatively, but is neither consistent by ally in terms of a phase fluctuation model proposed by
the microscopics of screened impurities in the cuprates, Emery and Kivelson11. Deviations from AG theory at
norcapableoftreatingthelimitofextremeforwardscat- small superfluid stiffness are to be expected even within
tering,claimedtobeofrelevanceinthecupratecase6,7,8. the framework of mean field theory, however, as pointed
Furthermore, in these studies T is calculated as a func- out by Franz et al.12, who studied the problem numeri-
c
tion of the single-particle normal state scattering rate cally and included the self-consistent suppression of the
1/τ , as opposedto the transportrate1/τ relevantfor order parameter around each impurity site. These au-
N tr
2
thorsreportedpositivecurvaturetailsinT vs. impurity
c 4
concentration n for large n , in contrast to the negative
i i
curvature in the AG plot. 3.5
3
I. TOY MODEL LΦc
HQ 2.5
To understand physically how some of these effects 2
might arise, we consider a simple model of forwardscat-
tering by impurities in which all effects can be calcu- 1.5
lated analytically. We will assume that in-plane impu-
1
rities to be described by δ-function like isotropic poten-
0 0.5 1 1.5 2 2.5 3
tials, and out-of-plane scatterers to be described by an Φ
c
extendedpotentialintheplane;thissimplyassumesthat
the screened Coulomb potential created by out-of-plane FIG.1: ThefactorQ(φc)asafunctionofthemaximumscat-
impurities produces a “footprint” sensed by quasiparti- tering angle φc. It reflects the decrease of the initial slope of
cles moving in the CuO2 planes. Tc suppression as a function of resistivity if one approaches
As proposed by Kee6, we interpolate between these theforward scattering limit (φc →0).
cases by considering a weak scattering potential V
~k~k′
which exists only at the Fermi surface |~k| = k , and
F solvingtheresultingsetofequationsfortheratio∆˜ /ω˜
cuts off unless~k and~k′ are sufficiently close 0 n
that enters Eq. (4), we can write
′
V(φ,φ′)= v0 if |φ−φ|<φc (1) ωc 1
(cid:26) 0 otherwise 1=VdN0πTc (5)
ω +2Γφ −Γsin(2φ )
ωXn>0 n c c
The diagonaland off-diagonalself-energies in this model
are then It is worthwhile noting at this point that an expansion
in powers of φ applied to (5) as performed in Ref. 6
c
Σ (φ,ω ) = −Γ φ+φcdφ′ i˜ωn (2) is appropriate except when Tc → 0. Following this ap-
0 n Zφ−φc ω˜2 +∆˜2f(φ′)2 proach would lead to the incorrect conclusion that the
q n 0 Tc suppression is linear all the way to Tc = 0 in the for-
φ+φc ′ ∆˜0f(φ′) ward scattering φc → 0 limit. Instead, it is clear that
Σ (φ,ω ) = Γ dφ (3)
1 n the expression (5) may be summed exactly, leading to a
Zφ−φc ω˜n2 +∆˜20f(φ′)2 modified AG result of the form
q
where Γ = πniN0v02, with N0 the density of states at ln Tc =ψ 1 −ψ 1 + Γ(φc−sinφccosφc) (6)
the Fermi level. In Eqs. (2) and (3), the renormalized Tc0 (cid:18)2(cid:19) (cid:18)2 πTc (cid:19)
Matsubara frequencies and gap magnitudes are iω˜ =
n
iω −Σ and∆˜ =∆ +Σ ,andweassumead-waveform where the pair breaking parameter in the forward scat-
n 0 0 0 1
of the unrenormalized order parameter, f(φ) = cos2φ. tering limit (φc ≪1) is of the order of φ3c:
NoteweworkinunitswhereBoltzmann’sconstantk =
B Γ(φ −sinφ cosφ ) 2Γφ3
1. The critical temperature is determined as usual from c c c ≈ c (7)
the linearized gap equation πTc 3πTc
ωc 1 ∆˜ Thus we see that the effective pairbreaking rate will de-
∆ =V N 2πT dφ′f(φ′)2 0 (4) crease dramatically as the scattering becomes more for-
0 d 0 c
ωXn>02π Z ω˜n wardinnature. Thisisindeedconsistentwiththeconclu-
sions of earlier works4, which concluded that anisotropic
where the d-wave pairing interaction is given by scattering slows T suppression for fixed impurity con-
(VdN0)−1 =1/2ln(2eγωc/πTc0), and Tc0 is Tc in the ab- centration. c
senceofimpurities. Ifwenowtakeφc ≪1,andTc .Tc0 Itremains,however,tocalculatethedependenceofthe
we find to leading order ∆˜0/ω˜n = ∆0/ωn, i.e. there is normalstateresistivityontheimpurityscatteringwithin
nopairbreakingbypureforwardscattering,analogousto the same framework if one wishes to compare directly
Anderson’s theorem in an s-wave superconductor with to experiments where pairbreaking is measured by the
isotropic nonmagnetic impurities. increaseinresidualresistivity. Foranisotropic2DFermi
Since we are only interested in calculating the criti- surface this may be written as
cal temperature we can neglect quadratic variations of
the order parameter and therefore perform the angular ρ= 2m2 dφdφ′(1−cos(φ′−φ))n |V(φ,φ′)|2 (8)
integrations in the definition of the self-energies. After e2p2 Z i
F
3
1 Φc=0.1 1 Κ=0.05
Φc=1.0 Κ=0.1
0.8 Φc=2.0 0.8 Κ=0.5
Φc=2.5 Κ=1.0
Tc0 0.6 Φc=Π Tc0 0.6 Κ=2.0
T(cid:144)c T(cid:144)c
0.4 0.4
0.2 0.2
0 0.25 0.5 0.75 1 1.25 1.5 1.75 0 0.0005 0.001 0.0015 0.002
Ρ(cid:144)€E€Ñ€€€FT€€€€ €ec€€€0€2€€€€ Ρ(cid:144)€e€€Ñ€€2€€€
FIG. 2: Tc suppression as a function of resistivity calculated FIG. 3: Tc suppression as a function of resistivity calculated
withinthetoymodel. Wefindadecreaseoftheinitialslopeof within a tight binding model. 1/κ gives the range of the
the Tc suppression as a function of resistivity if we approach scatteringpotential,whereκ≪1correspondstotheforward
theforward scattering limit. scattering and κ ≫ 1 to the isotropic scattering case. The
non-monotonic behavior of the curves for different κ results
from the competing effects of a reduction of the downward
leading us to the expression curvaturewithasimultaneousdecreaseoftheinitialslopefor
decreasing κ. The critical temperature of the pure sample is
~ 2Γ chosen tobe Tc0 =0.01t.
ρ= (2φ −2sinφ ) (9)
e2E c c
F
so that T may be expressed directly in terms of the II. REALISTIC MODEL
c
impurity-induced resistivity as
To confirm the above intuition, and get a sense of the
Tc 1 1 e2EF range of behavior possible in real systems, we now con-
ln =ψ −ψ +Q(φ )ρ (10)
T (cid:18)2(cid:19) (cid:18)2 c 4π~T (cid:19) sider a more realistic model consisting of randomly dis-
c0 c
tributed out-of-plane impurities with Yukawa potentials
where the factor Q(φ ) is given as V = V exp(−κr )/r , where r is the distance from a
c i 0 i i i
dopant atom to the lattice site i in the plane. 1/κ gives
Q(φc)=(φc−sinφccosφc)/(φc−sinφc) (11) the screening length of the impurity potential and the
forward scattering case (κ → 0) as well as the isotropic
ItrangesfromQ(0)=4intheforwardscatteringlimitto scattering case (κ → ∞) are included with this specific
Q(π)=1inthecaseofisotropicscatteringanditsdepen- choiceoftheimpuritypotential. TheFouriercomponents
dence on φc is shown in Fig. 1. The factor Q(φc) is also of the screened Coulomb potential can be written as
directly related to the initial slope of the T suppression
c
that can be derived from Eq. (10) as 2 |V |2
V(~k,~k′) = 0 (13)
2
Tc π EF e2 (cid:12)(cid:12) (cid:12)(cid:12) ~k−~k′ +κ2
T ≈1− 8Q(φc)ρ(cid:18)T ~ (cid:19) (12) (cid:12) (cid:12) (cid:12)(cid:12) (cid:12)(cid:12)
c0 c0 (cid:12) (cid:12)
We use a square lattice to mimic the copper-oxide
We see that although the rate of T suppression (or plane and include up to next-nearest hopping terms in
c
resistivity increase) is certainly much slower for forward numericalcomputations. Theelectronicbandistakento
′
scatterersthanintheisotropicscatteringcase,thedepen- be ǫk =−2t(coskx+cosky)−4t coskxcosky −µ, with
′
denceonthe resistivityis strongerforthe formerclassof t = −0.3t and µ = −1.1t in order to mimic the known
defects – within our model about a factor of four. This shape ofhole-dopedcuprateFermisurfacesnearoptimal
may indeed explain the results of Fujita et al.13, since doping. The Tc suppression is calculated by solving the
in these experiments out of plane (more forward scat- gap equation self-consistently:
tering) and in-plane (isotropic) impurities are studied in
separate samples. In Fig.2 we show the suppression of ∆ = 1 Vd ∆˜k′ (14)
T as a function of resistivity for differentvalues ofφ as ~k Λβ ~k,~k′ω˜2 +ǫ2
gicven by Eq. (10). While the initial slope is indeed sceen ωXn,~k′ n k′
to increase as the scattering is made more anisotropic,
with an effective electron-electron pairing interaction
approaching the value of 4 as φc → 0, we see also that Vd = Vdf(~k)f(~k′). The choice of the symmetry func-
withinthe toymodelthereis nodeviationfromtheform ~k,~k′
of the AG curve, as exhibited explicitly in Eq. (6). tion f(~k) = 1/2(coskx−cosky) leads to a dx2−y2 sym-
4
suppressionduetoforwardscatteringprocesses. However
1 Κ=0.05 Tc0=0.005t the particularshapeoftheT suppressioncurvedepends
c
Κ=0.1 Tc0=0.005t on the details of the considered model, e.g. the band
0.8 Κ=0.05 Tc0=0.01t structure, the doping strength or the functional form of
Κ=0.1 Tc0=0.01t the scattering potential. In Fig.4, it is shown that by
TT(cid:144)cc0 0.6 approaching the weak coupling limit Tc0 ≪EF due to a
lowering of T , we find a more linear dependence of the
0.4 c0
criticaltemperature onthe resistivity, reminiscentof the
linearity found in the electron irradiation experiments.
0.2
At present, we do not have an analytical understanding
of the origins of this quasilinearity.
0 0.00025 0.0005 0.00075 0.001 0.00125 0.0015
Ñ
Ρ(cid:144)€€€€€€€€
e2
III. CONCLUSIONS
FIG. 4: Tc suppression as a function of resistivity in the for-
ward scattering limit for two different values of Tc0. The We have investigated the disorder-induced reduction
upwardcurvaturethatisvisibleespeciallyforverylowvalues
of the critical temperature in high T compounds due to
ofκdiminishesforlowervaluesofTc0 andtheTc suppression c
out-of-planedisorder,assumingthatthe out-of-planede-
as a function of resistivity becomes more linear.
fects act primarily as elastic forward scatterers. These
calculations may be important not only to understand
metry of the order parameter. The diagonal and off- experimentswherethistypeofdisorderisvariedsystem-
diagonal parts of the self-energy are given by atically, but also to understand Tc’s in systems which
are intrinsically disordered by the doping process. Cal-
Σ (~k,ω) = n V(~k,~k′) 2 ω˜ (15) culating both the normal state resistivity as well as the
0 iX~k′ (cid:12)(cid:12) (cid:12)(cid:12) ω˜2−ǫ2k′ −∆˜2k′ Trecasliusptipcrsecsasitotneriinngbpotohteantsiiamlspalendtoybamndosd,elwaenfdoufonrdmthoaret
(cid:12) (cid:12)
Σ (~k,ω) = n V(~k,~k′) 2 ∆˜k′ (16) although the effect of forward scattering on both quan-
1 iX~k′ (cid:12)(cid:12) (cid:12)(cid:12) ω˜2−ǫ2k′ −∆˜2k′ tpirteiesssioisnsamsaallfeurntchtaionnfoofrriessoitsrtoivpiitcysicsastttreorningge,rtfhorefTocrwsuarpd-
(cid:12) (cid:12)
wherewecanneglectquadraticvariationsof∆˜ sincewe thanforisotropicscatterers. Inthecaseofthetoymodel,
k
we showed that the suppression is AG-like, with modi-
areonlyinterestedintheregionnearT . Tocomparethe
c
fied pairbreaking parameter. The basic physics of the
suppression of T to the increase of the normal conduct-
c
more rapid initial suppression of T vs ρ was confirmed
ing resistivity we have to calculate the resistance within c
the same model. For the transport rate τ−1(~k) we use numerically using more realistic Yukawa-type impurity
potentials within a tight binding model.
an approximation introduced by J. M. Ziman14 that ex-
Our result stands in apparent contradiction to ear-
pands the quasiparticle scattering rate by an additional
lier work comparing T vs ρ for anisotropic potentials
scattering-in term, c
scatterers5, where only s and d-wave components of the
scattering potential were retained. These authors con-
1 2π dk′dk′ ~v (~k)·~v (~k′)
x y F F 2 cludedthatthe greaterthe ratioofpotentialsV /V , the
τk = ~ niZ (2π/a)2δ(ǫk′)1− ~vF(~k) ~vF(~k′) |Vkk′| weaker would be the suppression of Tc vs ρ. Wdhiles this
 (cid:12) (cid:12)(cid:12) (cid:12) model is mathematically consistent, it is physically un-
(cid:12) (cid:12)(cid:12) (cid:12) (17)
(cid:12) (cid:12)(cid:12) (cid:12) realistic and does not yield generic results. As shown
and that has proven to be very accurate even for highly
in the Appendix, more realistic potentials may be ex-
anisotropic transport rates15. In the expression for the
panded in Fermi surface harmonics, but to obtain the
transportrate we have replaced the full T matrix by the
proper (weaker) enhancement of the resistivity with dis-
singleimpurityscatteringpotentialVkk′. Theconductiv- order it is important to retain the p-wave component as
ity can then be calculated from
well. If this is done for a generic potential, the result of
d2k this paper obtains.
ρ−1 =σxx =e2Z (2π)2τkvF2,xδ(ǫk) (18) Thispointleadsustoanimportantconclusionregard-
ing T suppression experiments making use of Zn to re-
c
assuming a cubic symmetry of the transport tensor. place Cu. It has been recognized for some time that
In Fig. 3, the suppression of T as a function of resis- the T suppression rate vs. residual resistivity for Zn is
c c
tivity is shown for the tight binding model discussed in smallerthanpredictedbyAGtheoryforpointlikescatter-
this section and T = 0.01t. As we have already seen ersinad-wavesuperconductor3byaboutafactorofthree
c0
for the toy model, the initial rate of T suppression vs. inoptimallydopedYBCO,andevensmallerinotherma-
c
resistivityincreaseswithincreasingrangeoftheimpurity terialslikeLSCO.Wehavenowarguedthatthe”conven-
potential1/κ,andseemsto be arobustfeature ofthe T tional” explanation, based on highly anisotropic scatter-
c
5
strongly influenced by correlations.
1 Zn-LSCO
We summarize the situation comparing the current
Nd-LSCO
theory to experiments in Fig. 5. Since we are primar-
0.8 Zn-YBCO
6.93 ily interested in order-of-magnitude physics, we discuss
Irr.YBCO
c0 0.6 7 only the initialslopes ofTc suppressionmeasuredinvar-
T ious experiments, and compare to the toy model result
(cid:144)
c
T Eq. (12). To obtain a fit one must make an assumption
0.4
EF(cid:144)Tc0 =10 about the parameter EF/Tc0 which enters this expres-
sion. A reasonable choice for the high-T materials is
0.2 c
E /T = 10, and this choice puts most of the data on
F c0
Zn-YBCO about a factor of 3 higher in slope, as found
0
0 1 2 3 4 by previous authors. The range of Tc suppression initial
Ρ@kW D slopes within the current approach is then shown in the
ƒ
Figureingray,rangingfromtheisotropicresulttotheex-
1 Zn-LSCO tremeforwardscatteringresult. Forcomparison,asetof
Nd-LSCO
curves is also given for the unphysical case E /T = 4.
F c0
0.8 Zn-YBCO
6.93 Whiletheseparametersshouldnotbetakentooseriously
Irr.YBCO
7 in a quantitative sense, it seems clear that a) a hitherto
Tc0 0.6 unaccounted for physical mechanism is required to ex-
(cid:144)
Tc plain the data on Zn-substituted samples; and b) the
0.4
effectofout-of-planescattererswithin the presentmodel
EF(cid:144)Tc0 =4 can only account for a factor of 4 or so increase in the
0.2
magnitude of the slope. Thus our work has explained
qualitatively the paradoxicalresult that the out ofplane
0
scatterers generically reduce T more quickly than in-
0 1 2 3 4 c
plane relative to residualresistivity,there is still a quan-
Ρ@kW D
ƒ
titative question remaining regarding the magnitude of
the suppression in both cases.
FIG. 5: Comparison of experimental data on Tc suppres-
sion in optimally doped cuprates. Diamonds: Nd-LSCO13; The final experimental result we have attempted to
squares: Zn-YBCO17;triangles: electron-irradiatedYBCO10; discuss is the fascinating linear Tc vs ρ suppressionmea-
stars: Zn-LSCO13. Solid lines: toy model result for isotropic sured on electron-irradiated single crystals of optimally
scatterers; dashed lines: for purely forward scatterers. Top doped YBCO by Rullier-Albenque et al10. The devia-
(bottom) panel, EF/Tc0 =10(4). tion from the AG pairbreaking result was attributed by
these authorsto phase fluctuations accordingto a model
put forward in Ref. 11. This is plausible for the under-
doped sample where phase fluctuations are expected to
ing by Zn impurities, is unlikely to be correct. It seems,
be strong, and for the highly irradiatedoptimally doped
therefore, that even in the optimally doped cuprates
sample, where the superfluid density is also small. We
the deviation must be ascribed to effects of strong cor-
note,however,thatwithinthisscenarioitispuzzlingand
relations or strong coupling superconductivity. Within
mustberegardedasaccidentalthattheinitialT slopein
Eliashberg theory,18 the AG pair-breaking parameter is c
the optimally doped sample, due to pairbreaking, is the
renormalizedΓ→Γ/(1+λ),whereλisthedimensionless
same as the final slope before superconductivity disap-
coupling,while the disorder-inducedresistivitychange is
pears. It is therefore equally likely, in our opinion, that
unrenormalized to leading order. The physical origin of
the quasilinearity found by these authors is due to the
this effect is not clear in the cuprate context, however,
effects of out-of-plane defects created by the electron ir-
KulicandOudovenko5arguethatrenormalizationsofthe
radiation, as discussed here, and/or the effects of order
scattering vertex due to strong interactions within the t
parameter suppression as discussed in Ref. 12.
-J model introduce significant suppression of the trans-
port rate induced by impurities. While this cannot ex-
plaintheeffectsofforwardpotentialscatteringforoutof
plane scatterersasdiscussedhere,since the renormaliza-
tionintheirmodelisindependentoftheanisotropyofthe
scattering,itmaybe partofthe solutiontothe Znprob-
lem. Another perspectiveonthe samephysicsto explain Acknowledgments
this effect may involve the low-energy spin fluctuations
known to be induced by disorder in strongly correlated The authors acknowledge valuable conversations with
systems16. No theoretical work which considers both T H. Alloul, J. P. Carbotte, H. Eisaki, M. Kulic, E. Nicol,
c
and residual ρ is available at this writing, however; it is and F. Rullier-Albenque. Partial support for this re-
difficult to predict a priori which quantity will be more search was provided by DOE Grant No. DE-FG02-
6
05ER46236.
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c
39, 1781 (1960) [Sov.Phys. JETP 12, 1243 (1961)]. that the integration projects out only the s- and the d-
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(1996); ibid,58,12441(1998);M.L.KulicandO.V.Dol- Σ0 =2πΓV0 (A.2)
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and
375 (1997).
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0
5714 (2000). Σ1(φ,ωn)=πΓV2 (A.3)
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B 70, 214503 (2004).
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0 n
Phys. Rev.B 50, 15967 (1994). quency summation in the gap equation we find
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Lett. 91, 047001 (2003).
11 V.J.EmeryandS.A.Kivelson,Phys.Rev.Lett.74,3253 Tc 1 1 Γ(2V0−V2)
ln =ψ −ψ + (A.4)
(1995). T (cid:18)2(cid:19) (cid:18)2 2T (cid:19)
12 M. Franz, C. Kallin, A. J. Berlinsky, and M. I. Salkola, c0 c
Phys. Rev.B 56, 7882 (1997).
13 K. Fujita, T. Noda, K. M. Kojima, H. Eisaki, and This result is in agreement with the results of Kulic and
S. Uchida,Phys. Rev.Lett. 95 097006 (2005). Oudovenko and their parameters Γs and Γd can now be
14 J. M. Ziman, Adv.Phys. 10, 1 (1961). directly compared to the toy model parameter φ :
c
15 W. E. Lawrence and L. A. Cole, J. Phys. F: Met. Phys.
15, 883 (1985).
16 H. Alloul, J. Bobroff, M. Gabay, and P. J. Hirschfeld, un- Γs =4Γφc, Γd =2Γsin(2φc) (A.5)
published.
17 Y. Fukuzumi,K. Mizuhashi, K. Takenaka, and S.Uchida,
To calculate the normal conducting resistivity one has
Phys. Rev.Lett. 76, 684 (1996).
18 E. Schachinger, J. M. Daams, and J. P. Carbotte, Phys. to weight the quasiparticle scattering probability by its
impactonthe resistivityleadingto Eq.(8). It is obvious
Rev.B 22, 3194 (1980).
that in the case of pure forward scattering the increase
of resistivity with impurity concentrationis much slower
APPENDIX: FOURIER EXPANSION OF THE than for isotropic scattering, a fact that has been taken
′
IMPURITY POTENTIAL into account by the factor 1−cos(φ−φ) that projects
not only the s-wave part but also the p-wave part of the
impurity potential out of the Fourier expansion. The
In previous works the effect of forward scattering on
resistivity can then be written as
the critical temperature has been studied by including
higher angular momenta of the impurity potential when
calculating the self-energies. In this section we show the 4π~ 1
ρ= Γ V − V (A.6)
link between the results of our toy model and an ap- e2E (cid:18) 0 2 1(cid:19)
F
proach, where the impurity potential is decomposed in
itsrelevantFouriercomponents. Particularlywewantto
SolvingthisequationforΓandinsertingitintheexpres-
pointouttheimportanceofthep-wavecomponentofthe
sionfortheT suppressionleadstoEq.(12)whereQ(φ )
impurity potentialforcalculatingthe normalconducting c c
canbeexpressedbythes-,p-andd-wavepartoftheim-
resistivitythathasbeenneglectedinpreviousworksand
purity potential as Q(φ ) =(2V −V )/(2V −V ). The
thatisthekeyinunderstandingthestrongersuppression c 0 2 0 1
dependence of Q(φ ) on the maximum scattering angle
of the critical temperature as a function of resistivity in c
φ is shown in Fig. 1. It shows that the effect of im-
the case where forward scattering is the dominant scat- c
purity scattering on the resistivity is drastically reduced
tering process.
in the forward scattering limit due to the p-wave char-
Startingwiththesquareoftheimpuritypotentialthat
acter of the scattering-in term while the effect on the
is givenin Eq.(1) we canexpandit in a Fourierseriesof
′ T reduction is not as strong, since the only anisotropic
its two arguments φ and φ c
componentwhichplaysaroleinthelattercaseisd-wave.
∞
|V(φ,φ′)|2 = v2V +v2 V (A.1)
0 0 0 k
kX=1
′ ′
× [cos(kφ)cos(kφ)+sin(kφ)sin(kφ)]