An attempt to settle the one-component versus two-component debate in Cuprate high- T c superconductors
aa r X i v : . [ c ond - m a t . s up r- c on ] J un An attempt to settle the one-componentversus two-component debate in Cupratehigh- T c superconductors N avinder S ingh ∗ Physical Research Laboratory, Ahmedabad, [email protected]
June 25, 2019
Abstract
Related to the electronic structure of cuprate high-Tc superconductors there are two prevailing views:In one view, Pines and collaborators argue that cuprates consist of two electronic subsystems: one withlocalized spins and the other having itinerant character. Contrary to that, Phil Anderson has argued forone-component or one-band model for cuprates. Thus, a very natural question arrises: What is the actualelectronic system in cuprates? Are these two-component or one-component systems? After a careful consid-eration of both the views, we argue in favor of one-component or Andersonian view. The key lies in the dualcharacter of electrons in narrow d bands, and we put forward a semiclassical model using which we couldquantitatively account for oxygen NMR shift data in La − x Sr x CuO within the single component view. I. I ntroduction
At the foundation of the problem of high- T c superconductivity in Cuprates is the nature oftheir electronic structure. More specifically thequestion is: in which hybrid orbitals the mag-netic degrees of freedom reside and where dothe mobile carriers reside? For this fundamen-tal question, there are two prevailing views inthe literature: In one view Pines and collab-orators argue[1] that cuprates consist of two ∗ Cell Phone: +919662680605 electronic subsystems or two interpenetratingfluids: one with localized spins and the otherhaving itinerant or mobile character. The rea-son for this view is that the NMR shift exper-imental data can be nicely explained withinthis two-fluid or two-component picture, asthe investigations of Haase, Slichter and col-laborators have recently shown[2, 3, 4]. Mi-croscopically it is explained on the basis ofin-complete d-p hybridization of copper 3 d orbitals and oxygen 2 p orbitals, and it leadsto localized copper spins and mobile p holes1ne-component versus two-component debate(i.e., two components). With this division mag-netic paring mechanism conceptually becomespleasing as one can imagine mobile carriersbeing paired up by the magnetic spin fluctu-ations of the localized copper spins (along theconventional BCS tradition).Contrary to that, right from the beginning,Phil Anderson has argued for one-componentor one-band model for cuprates as far as lowenergy physics is concerned[5, 6]. This rele-vant band is build up from hybridization ofcopper d orbitals and oxygen p orbitals, moreprecisely, the anti-bonding Cu d x − y − O p σ narrow band. This view is based on very care-ful analysis of the nature of chemical bondingin cuprates. Thus, a very natural question ar-rises: What is the actual electronic system incuprates? Are these two-component systemsor one-component systems? In this paper weattempt to resolve this issue and put forwarda semiclassical model which quantitatively ac-count for the oxygen NMR shift data in LSCOwithin the single component scenario. Thephysical picture of our model consists of tem-porally localized states in Cu d − orbitals and aband of mobile states separated from the tem-porally localized states with a "spin gap". Thisphysical picture at finite doping is some sortof remanent of Mott physics at zero doping.The paper is organized in the following way.We first review Anderson’s argument in favorof one component view which is based on verycareful analysis of the semi-covalent bondingin Cuprates. We then consider the view-point of Pines and collaborators in favor of two-component view. NMR shift data of Haaseand collaborators is reviewed which supportsthe two-component view. We argue how thedebate can be settled and introduce our modelwhich is based on the one-component viewand compute oxygen NMR shift from it. Oncomparing with experimental data we find areasonably good agreement. We close by sum-marizing the results. II. T he correct electronicstructure of C uprates and whichelectrons are removed on doping ? In what follows we closely follow Ander-son’s explanation of the electronic structure ofCuprates[5]. Let us take the example of thesystem La − x Sr x CuO . The composition of theun-doped system La CuO is: two layers of LaO and one layer of
CuO . Thus Cu mustbe in Cu ++ state, as La generally has the va-lence state La +++ and oxygen has O −− . Cu has electronic configuration [ Ar ] d s and Cu ++ has to be [ Ar ] d . As d orbitals ac-commodate 10 electrons, thus in one of the d orbitals in Cu ++ one electron has to remainsingle or un-paired. There are five d orbitals( xy , yz , zx , x − y , 3 z − r ). The question is:in which orbital does this un-paired or loneelectron reside?We will speak in terms of orbitals not hy-bridized orbitals as d orbitals are compara-tively tightly closed in the interior of the atom.2ne-component versus two-component debateThey are relatively "sequestered" as comparedto s orbitals which hybridize considerably andform fat bands. In the system under consider-ation, copper atoms are in a cages of octahe-drons formed by oxygen atoms (figure 1). Wewant to understand how d-orbitals are filled.With zeroth approximation, crystal fields areof cubic symmetry i and these split the five d orbitals into two-sub-groups: e g and t g . Threeorbitals ( xy , yz , zx ) form t g set and thesehave lower energy as their lobs do not directlypoint towards negatively charged four planneroxygen atoms and two negatively charged api-cal oxygen atoms of the cage. This leads to re-duced Coulomb repulsion between electronsin these orbitals and six negatively chargedoxygen atoms at the corners of the octahedron.Thus they will fill first, accommodating sixelectrons in total. Now we have three elec-trons remaining out of nine. The two orbitals3 d x − y and 3 d z − r of the set e g can accommo-date four electrons and are both "un-happy"as their lobs point directly towards negativelycharged oxygen atoms. Question is which oneis more "un-happy" and which one is less "un-happy". Here come’s Anderson’s crucial in-sight. The most "un-happy" orbital is 3 d x − y asits four lobs point directly towards negativelycharged four planner oxygen atoms (figure 1)whereas 3 d z − r face repulsion only from twonegatively charged apical oxygen atoms along i Remember six oxygen atoms, four planner, and twoapical, are negatively charged with two excess electrons(these are semi-covalent bonds, as discussed in the text). the z-direction. Thus two electrons will gointo 3 d z − r (lower energy orbital) and the lastelectron has to reside in 3 d x − y orbital (higherenergy orbital). And it is this electron whichis responsible for magnetism as it remains un-paired as well as for strange metal behavior ondoping, and also for unconventional supercon-ductivity. CuO
Figure 1:
The octahedron cage
This is the basic picture. There are otheraspects. As four planner negatively chargedoxygen atoms face lesser repulsion from oneun-paired electron in 3 d x − y as compared towhat two apical oxygen atoms face from twoelectrons in 3 d z − r , the four planer oxygensare pulled-in towards copper atom and apicaloxygen atoms are pushed out. Thus the octa-hedron becomes distorted and pointy. This isthe Jahn-Teller (JT) distortion. The presence ofJT distortion is an unequivocal signature thatun-paired electron resides in 3 d x − y orbital.Now the question is how the above pic-3ne-component versus two-component debateture is modified when hybridization is con-sidered. Here it is important to understandthe nature of bonding in Cuprates. There aretwo extreme kinds of chemical bonds: cova-lent bonds and ionic bonds. In cuprates thebonding is intermediate i.e., semi-covalent orpartly ionic. Hybridization leads to bondingand anti-bonding orbitals. The highest hybridorbital to be filled finally remains Cu d x − y −− O p σ anti-bonding orbital. This picture isbased on the fact that in Cuprates bondinghas great strength (very high melting points)which is derived from semi-ionic character[5].What will happen when the system is holedoped i.e., when electrons are removed? Thelone electron in 3 dx − y remains "un-happy"as it still has to face Coulomb repulsionfrom four negatively charged planner oxygenatoms. Thus on hole doping, it is THIS elec-tron which is removed. Therefore we only haveone-component and one-band system.
The othertopic of Mott insulating behaviour of a lat-tice of un-paired electrons localized in 3 d x − y orbitals is well understood[5, 7]. Putting an-other electron into half-filled 3 d x − y orbitalcosts energy and the configurations of the type Cu d − − Cu d are not energetically fa-vored. We conclude based on the nature ofchemical bonding in Cuprates that these areone-component or one-band systems as far aslow energy physics is concerned. III. T wo - component picture David Pines and collaborators very stronglyargue that there are two sub-systems incuprates: one with localized spins and theother having itinerant character (figure 2). Itis argued that d-p hybridization is not com-pete and it leads to localized copper spins andmobile p holes[1]. Hole doping does not re-move the un-paired electrons from Cu 3 d x − y orbitals rather electrons are removed from oxy-gen p orbitals. This leads to two types of elec-tronic subsystems: one localized and the othermobile or itinerant. Further, NMR shift experi-mental data can be nicely explained using two-component model. Thus, in recent times thisview seems to become dominant in the high-Tc community. ii Before we briefly review thisanalysis in the following paragraphs we firstreview the status of the literature in this field.
Figure 2:
The two-component view: Localized d-electrons (arrows) on Cu provide the pairing"glue" to itinerant or mobile p-band electronsfrom oxygen (wavy lines).
One component view was put forward in ii In the first decade after 1986 one-component viewwas more popular, but afterwards two-component viewbecame more popular due to NMR shift experiments. iii
Recently, ex-tensive investigations by Haase and collabora-tors strongly support the two-component view.Some selected references are[2, 3, 4].Next, we consider NMR shift experimentsthat points towards the two component view. iii
However, refer to[12] for a different aspect related tocoherence in hyperfine fields.
IV. E xperimental support of thetwo - component picture NMR shift experiments are based on the"shielding" effect of un-paired electronsaround a selected nucleus in which the nu-cleus will not "see" the externally applied mag-netic field ( H ) rather it "sees" a modifiedfield or an effective field ( H e f f ) . This changesthe magnitude of the Zeeman splitting of theenergy levels of the nucleus thus shifts theNMR resonance frequency. This can be accu-rately measured. iv This "shielding" due to un-paired electrons in technical literature is calledKnight shift due to hyperfine interactions. Fors-electrons it is Fermi contact type and dueto p- or d-electrons, it is dipole-dipole type.Let us write the observed NMR resonance fre-quency as ω NMR . It can be written as ω NMR γ n = H + ∑ k − A n , k h S k i + C n . (1)Here, H is the unscreened external magneticfield. h S k i is the average value of the spin dueto k th un-paired electron. A n , k is the hyperfinecoefficient for n th nuclear site and k th electron. C n is the temperature independent part of theshift that comes from the orbital effects etc.Let us assume, along with the supporters ofthe one component view, that there is one sin-gle temperature dependent susceptibility χ ( T ) which can be written as M e f f = µ B g k h S k i = χ ( T ) H , (2) iv For other technical details of NMR shift experiments,reader is referred to dedicated literature[2, 13, 14, 15]. M e f f is the effective magnetization.With this, one can express the relative fre-quency shift in terms of a single temperaturedependent susceptibility: K s = ω NMR − ω ω = − ∑ k A n , k µ B g k χ ( T ) + K .(3)Here, ω is the NMR resonance frequencywithout any shielding. Thus one concludesthat K s ∝ χ ( T ) . This means that shift mustbe a single function of temperature. But thisis not what is observed in general.
Let us takethe case of
LSCO which we are considering. v An experimental data from one of the publi-cations of Haase and collaborators is given infigure 3. The shift at the Cu site is more or
Figure 3:
Experimental data showing two different tem-perature dependences of shifts at Cu nucleusand O nucleus. This is against the one com-ponent view. Figure courtesy[2]. less temperature independent. It is suddenlyreduced when T c is reached, whereas the shiftat O is temperature dependent and smoothly v Even YBCO under hydrostatic pressure show two tem-perature dependences[2]. decreases with decrease in temperature. Thusit is clear that it cannot be rationalized withina single temperature dependent susceptibilityscenario. It can be understood as arising fromtwo two fluids, one having more or less tem-perature independent susceptibility that is ef-fective at Cu sites, and the other fluid havinga temperature dependent susceptibility that isacting on O sites. This motivates the two com-ponent view (for more details refer to[1, 2]).However, we argue differently. Next sectionpresents our argument.
V. A n attempt to settle thedebate
A one-component system can lead to two-components: electrons in narrow bands withstrong electron correlation can show both thelocalized and the itinerant behavior at thesame time. Thermal activation is the key. vi An electron temporally localized in the rele-vant Cu d-orbital can convert to an itinerantelectron through thermal excitation, and re-verse process also happens where an itiner-ant electron converts to a local one. This ther-mal activation occurs over a "spin gap barrier"of magnetic origin.
The physical picture of ourmodel consists of temporally localized states in Cud − orbitals and a band of mobile states separatedfrom the temporally localized states with a spin gap. So, both types of the carriers can "emerge" in a vi This is not entirely new. Similar thermal activationideas has been put forward in[17]. re-manent of the zero doping Mott physics when thesystem is hole doped (figure 4). We mathe-matically model the situation in the followingway.
UPPER HUBBARD BANDMOTT GAPLOWER HUBBARD BAND MOBILE STATESE iti
SPIN GAPSTATES loc
ON DOPING ETEMPORALLY LOCALIZED
Figure 4:
The semiclassical model at finite doping issome sort of a remanent of the Mott physicsat zero doping.
Let us consider Cu − O lattice at zero dop-ing. We have localized Cu spins (each Cu sitewith one un-paired electron). On hole dop-ing let us say on n v fraction of sites electronsare removed. Now the system starts conduct-ing. vii Let, at any given instant of time, n loc bethe fraction of localized electron in the lattice,and n iti be the fraction of itinerant electrons.These are time dependent quantities and keep vii Electronic conduction itself will have two channels:one through the tight band below the spin gap as thegap (as it turns out in Cuprates) has nodes in specificdirections in momentum space. And the other channelis through excitation of carriers to upper band of mobilestates above the spin gap. This excitation is thermal innature. In our crude model we do not consider momen-tum dependence of the spin gap rather we use "an averagespin gap". on changing. But we have the obvious rela-tion: n loc + n iti + n v =
1. (4)The time evolution of the populations can bewritten as dn loc dt = − P l → i n loc + P i → l n iti ( − n loc ) . (5)Here, P l → i represents the transition rate fromlocal to itinerant behaviour, and P i → l is the re-verse rate. The factor ( − n loc ) multiplyingthe last term represents the fact that an itin-erant electron can become local only if vacantsites are available, whereas from going fromlocal to itinerant no such constraint is there.This is our method of imposing the Hubbardconstraint in a soft way in this model. Thetwo terms on the right hand side are notingbut "loss and gain" of the local electron popu-lation. Similarly dn iti dt = − P i → l n iti ( − n loc ) + P l → i n loc . (6)In the steady state P l → i n loc = P i → l n iti ( − n loc ) . (7)Using the constraint (equation 4) and little al-gebra viii , we get n loc = ( η + − n v ) − r ( η + − n v ) − ( − n v ) ,(8)and n iti = r ( η + − n v ) − ( − n v ) − ( η + n v ) .(9) viii Notice that if n v →
1, that is all sites vacant (100 per-cent hole doping), n loc → η is defined as η = P l → i P i → l . (10)That is, the local to itinerant transition ratedivided by itinerant to local transition rate. Toget the feel for numbers, let us assume that n v = η ≃ ix On plugging these numbersinto equations (8) and (9) we get n loc ≃ n iti ≃ Thus we observe that a two compo-nent system does emerge from a one band and onecomponent system.
To test the model we need to compute someobservable from it and compare that with ex-periment. To this end, we will compute Knightshift from this model. But before we do thatwe need to compute the transition rates. Thisis done using the thermodynamical argument.Let E loc be the energy of temporally localized ix In fact η will be less than one, as the tendency of lo-cal to itinerant transition is suppressed as compared tothe tendency of itinerant electron going local. This is dueto short range antiferromagnetic correlations in localizedelectrons which tries to "hold them up" into the lattice(that is magnetic energy lowering while on localization).This energy is denoted by "an effective spin gap" in ourmodel. electrons and E iti ( n iti ) be the energy of the itin-erant electrons which is a function of the itin-erant electron number density ( n iti ). Let ∆ sg be the spin gap. Populations obey the thermo-dynamic relations (refer to figure 4) n loc n loc + n iti = e − β E loc . n iti n loc + n iti = e − β ( E loc + ∆ sg + E iti ( n iti )) . (11)Their ratio gives n loc n iti = e β ( ∆ sg + E iti ( n iti )) . (12)For a 2D system E iti ( n iti ) = ¯ h π m nn iti . Here n is the number of electrons per unit area and n iti is the fraction of itinerant electrons. If a isthe Cu-Cu bond length, then n = a withone 3 d x − y electron per Cu atom. Collectingall this, and writing n loc in terms of n iti usingthe constraint (equation 4 ) we get ln (cid:18) − n iti − n v n iti (cid:19) = k B T ∆ sg + ¯ h π ma n iti ! (13)This is one of our main result. The tem-perature dependence of n iti can be calculatedfrom a numerical solution of the above im-plicit equation for n iti and it is presented in theAppendix figure (6). n iti increases with tem-perature as expected (thermal activation pop-ulates the itinerant band). The temperature de-pendence of the ratio of transition rates η canbe computed from equations (8) or (9). This isalso given in the appendix figure (6). Here wedeal with the important quantity which is the8ne-component versus two-component debatemagnetic susceptibility of this semi-classical"one-level and one-band" model. For that wecan deduce the temperature dependence ofthe Knight shift and can compare with the ex-periment. ▲▲▲ ▲▲▲▲▲ ▲ ▲ ▲ ▲
50 100 150 200 T0.020.040.060.080.10 K ⊥ Figure 5:
Our theory (solid line) for K ⊥ ( T ) agreesreasonably well with the experimental data(triangle)[2]. The best fit value using leastsquares gives ∆ sg ≃ eV which is in goodagreement with its value found by change inthe slop of electrical resistivity and Nernst ef-fect signal[18]. There, the temperature T ∗ de-duced is about K at x = which cor-responds to k B T ≃ ∆ PG about meV for ∆ PG (figures 5 and 10 in[18]). Consider the magnetic susceptibility of itin-erant electrons. In simple metals it is the Paulisusceptibility which is temperature indepen-dent. Temperature independence of the Paulisusceptibility in metals comes form the wellknown relation µ B H << k B T << E F . In sys-tems where Fermi energy is small and is com-parable of k B T , and/or there is a gap in theexcitation spectrum, the "Pauli susceptibility"will depend on temperature. In the next para-graph we compute the magnetic susceptibility due to the itinerant part in our "one-level andone-band" model, in which number density( n iti ( T ) ), and thus Fermi energy are temper-ature dependent quantities!Magnetic susceptibility can be written as χ iti ( T ) = µ B lim H → Z ∞∆ sg d ǫ g ( ǫ ) × f ( ǫ − µ B H ) − f ( ǫ + µ B H ) µ B H .(14)Where g ( ǫ ) is electron density of states( π m ¯ h ) in 2D. f ( ǫ ) is the Fermi function and H is the external magnetic field. A straightfor-ward calculation leads to χ iti ( T ) = π m µ B ¯ h (cid:18) e β ( ∆ sg − E Fiti ( T )) + (cid:19) . (15)Here E Fiti ( T ) = ¯ h π ma n iti ( T ) . This is ourmain result.For the computation of Knight shift we as-sume that Knight shift at oxygen site is af-fected mainly by this component. There willbe some effect on oxygen shift due to tem-porally localized spins on Cu atoms (trans-ferred interactions). We assume that this ef-fect is sub-dominant and oxygen shifts aremainly affected by the above computed sus-ceptibility due to the itinerant part. Thus weset K ⊥ ( T ) ∝ χ iti ( T ) . Figure (5) shows theleast square fitting of oxygen Knight shift com-puted from our model with that of experimen-tal data of Haase et al[2]. The agreement isquite good. To compare the temperature evo-lutions of the Knight shift, the proportionality9ne-component versus two-component debateconstant in K ⊥ ( T ) ∝ χ iti ( T ) is normalized tothe experimental data at the maximum tem-perature ( T ≃ K ). We used ∆ sg as fit-ting parameter and best fit value gives ∆ sg ≃ meV . This is in reasonable agreementwith that found through other methods[18].The pseudogap temperature T ∗ deduced isabout 125 K at x = ∆ PG ≃ meV through k B T ≃ ∆ PG .Theoretical modeling of the temperature de-pendence of the Cu Knight shift is beyond thescope of this investigation. The contributionin this case comes mainly from temporally lo-calized electrons which have dynamically fluc-tuating antiferromagnetic correlations. As canbe seen from figure (3) it is more or less tem-perature independent in the case of LSCOwhile this shift is temperature dependent inthe case of YBCO at ambient pressure. x Thusthere is system to system variation even in thecase of Cu ( K ⊥ ) shift. Proper understandingof it requires details of the electronic structurevariations from one system to another. Thisproblem remains theoretically open[2].However, we offer a qualitative understand-ing of it in the following way. There are twoopposing tendencies acting at the Cu sites.The spin gap leads to lowering of Cu Knightshift on reducing temperature (as in figure 5). x The shift when magnetic field is parallel to c-axis(that is K ) shows large variations with temperature evenwithin a given system. Thus it is less reliable than K ⊥ (magnetic field perpendicular to the c-axis or in the ab-plane). Whereas temporally localized spins of Cu sitesleads to susceptibility that may scale like T (Langevin-Curie type). This tends to increasewith lowering temperature. Thus these twoopposing tendencies may cancel each otherand may lead to more or less temperature in-dependent Knight shift at Cu site as seen infigure 3 (upper squares). It suddenly reduceswhen the system becomes superconducting.In this way a qualitative understanding can beobtained. However, a quantitative theory ismuch needed. VI. S ummary
We conclude that un-paired electrons in Cu d x − y − − O p hybrid orbitals and a nar-row band formed by them are responsible forboth the localized behaviour and the itinerantbehaviour[16]. An electron localized in thatorbital can convert to an itinerant electron in atimescale of the order of ¯ h ∆ sg ∼ f emto − sec (for ∆ sg ∼ meV ) and reverse process canalso happen on similar timescales. Thus a two-component system evolves from a manifestly one-component system through thermal excitation.
Weput forward a semiclassical model to this endwhich takes into account this conversion pro-cess through thermal activation and deactiva-tion. Magnetic susceptibility and Knight shiftis calculated in such a model. We find a rea-sonable agreement between theory and exper-iment for oxygen shifts data.10ne-component versus two-component debate
VII. A cknowledgment
The author would like to thank B. Sriram Shas-try for helpful comments.
VIII. A ppendix
Thermal evolution of n iti ( T ) is plotted in fig-ure (6) below.
50 100 150 200 250 T0.020.040.060.080.100.12 η ( T )
50 100 150 200 250 T0.050.100.150.20 n iti ( T ) Figure 6:
Thermal evolution of n iti ( T ) . It increaseswith increasing temperature as thermal excita-tion populates the itinerant band. Inset showsthe temperature dependence of the ratio η com-puted from equation (9) after solving equation(13). R eferences [1] V. Barzykin and D. Pines, Adv. in Phys. , 1-65 (2009).[2] J. Haase, M. Jurkutat, and J. Kohlrautz,Cond. Matt. , 16 (2017).[3] J. Haase, C. P. Slichter, and G. V. M.Williams, J. Phys.: Cond. Matt. , 434227(2008). [4] J. Haase, C. P. Slichter, and G. V. M.Williams, J. Phys.: Cond. Matt. , 455702(2009).[5] P. W. Anderson, Int. J. Mod. Phys. B. ,1-39 (2011).[6] P. W. Anderson, Science , 1196 (1987).[7] P. W. Anderson, P. A. Lee, M. Randeria,T. M. Rice, N. Trivedi, and F. C. Zhang, J.Phys: Cond. Matt. , R755 (2004).[8] F. Mila and T. M. Rice, Physica (Amster-dam), , 561 (1989).[9] B. Sriram Shastry, Phys. Rev. Lett. ,1288 (1989).[10] A. J. Millis, H. Monien, D. Pines, Phys.Rev. B. , 167 (1990).[11] R. E. Walstedt, B. S. Shastry, and S.-W.Cheong, Phys. Rev. Lett. , 3610 (1994).[12] A. Uldry and P. F. Meier, Phys. Rev. B. ,094508 (2005).[13] A. Abragam, The principles of nuclear mag-netism , Clarendon Press, London (1961).[14] C. P. Slichter,
Principles of magnetic reso-nance , Springer-Verlag, New-York (1989).[15] H. Alloul,
NMR in strongly correlated mate-rials , Scholarpedia (2015).[16] P. W. Anderson and R. Schrieffer, Phys.Today
6, 54 (1991). 11ne-component versus two-component debate[17] L. P. Gor’kov and G. B. Teitel’baum, Phys.Rev. Lett. , 247003 (2006); ibid, Scien-tific Reports, , 8524 (2015).[18] O. Cyr-Choiniere, R. Daou, F. Laliberte,C. Collignon, S. Badoux, D. LeBoeuf,J. Chang, B. J. Ramshaw, D. A. Bonn,W. N. Hardy, R. Liang, J.-Q. Yan, J.-G.Cheng, J.-S. Zhou, J. B. Goodenough, S.Pyon, T. Takayama, H. Takagi, N. Doiron-Leyraud, and Louis Taillefer, Phys. Rev. B97