Charge density wave and superconductivity competition in Lu 5 Ir 4 Si 10 : a proton irradiation study
Maxime Leroux, Vivek Mishra, Christine Opagiste, Pierre Rodière, Asghar Kayani, Wai-Kwong Kwok, Ulrich Welp
CCharge density wave and superconductivity competition inLu Ir Si : a proton irradiation study Maxime Leroux, ∗ Vivek Mishra, † Christine Opagiste, PierreRodi`ere, Asghar Kayani, Wai-Kwong Kwok, and Ulrich Welp Materials Science Division, Argonne National Laboratory, Argonne, USA Institut N ´EEL, CNRS, Univ. Grenoble Alpes, Grenoble 38000, France Department of Physics, Western Michigan University, Kalamazoo, USA (Dated: June 8, 2020)
Abstract
Real-space modulated Charge Density Waves (CDW) are an ubiquituous feature in many familiesof superconductors. In particular, how CDW relates to superconductivity is an active and openquestion that has recently gathered much interest since CDWs have been discovered in manycuprates superconductors. Here we show that disorder induced by proton irradiation is a full-fledgedtuning parameter that can bring essential information to answer this question as it affects CDWand superconductivity with different and unequivocal mechanisms. Specifically, in the model CDWsuperconductor Lu Ir Si that develops a 1D CDW below 77 K and s-wave superconductivitybelow 4 K, we show that disorder enhances the superconducting critical temperature T c and H c2 while it suppresses the CDW. Discussing how disorder affects both superconductivity and the CDW,we make a compelling case that superconductivity and CDW are competing for electronic density ofstates at the Fermi level in Lu Ir Si , and we reconcile the results obtained via the more commontuning parameters of pressure and doping. Owing to its prototypical, 1D, Peierls type CDWand the s-wave, weak-coupling nature of its superconductivity, this irradiation study of Lu Ir Si provides the basis to understand and extend such studies to the more complex cases of densitywaves and superconductivity coexistence in heavy fermions, Fe-based or cuprates superconductors. a r X i v : . [ c ond - m a t . s up r- c on ] J un . INTRODUCTION A charge density wave (CDW) is a spatial modulation of the electronic density of stateswhich opens a gap at the Fermi level. CDW can arise from electronic instabilities such asFermi surface nesting in low-dimension metals or a peak in electron-phonon coupling .This charge modulation is usually accompanied by a periodic lattice distorsion, via theelectron-phonon coupling. An analog modulation known as a spin density wave (SDW) alsoexists for the electronic spin . The presence of real-space modulated CDW or SDW is afeature of many families of superconductors .Recently, CDWs have been found to be ubiquitous in many cuprates superconductors,whether in hole-doped La − x Ba x CuO , YBa Cu O − δ , Bi Sr CaCu O x , HgBa CuO δ or in electron doped Nd − x Ce x CuO . Other examples include: Fe-based superconductorsin which superconductivity seems to compete with spin density waves ; heavy fermioncompounds where the SDW appears linked to d-wave superconductivity ; transition metaldichalcogenides where the CDW is well-known to compete with superconductivity in 2H-TaS and 2H-TaSe but 1T-TiSe has been proposed as an excitonic superconductorenhanced by the CDW ; finally organic superconductors also exhibit superconductivityin coexistence with density waves (e.g. (TMTSF) PF ) . Thus, whether CDW competeswith or on the contrary are a key ingredient in explaining the origin of cuprates’ hightemperature superconductivity , the relation between density waves and superconduc-tivity is an active and open question. Among superconductors with CDW, Lu Ir Si is a well established case of s-wave super-conductivity coexisting with a standard Peierls-type CDW . This compound possessesa first order CDW transition below T CDW = 77 K and it also becomes superconductingbelow T c = 4 K . The CDW develops on 1D Lutetium atom chains along the c-axis, fol-lowing the nesting mechanism , with clear signatures in electrical transport , x-ray ,specific heat or elastic constants . The CDW gaps an estimated 36% of the densityof states at the Fermi level as determined from resistivity and specific heat measurements ,with more recent optical estimates ranging from 16% to 30% . The effect of pressurepoints to a competition scenario: from 0 to 2 GPa T CDW decreases continuously and T c is constant, but above 2 GPa the CDW suddenly vanishes and T c jumps from 4 to 9 K .Chemical doping also points to a competition scenario: the CDW state is suppressed and2 c continuously increases up to at least 6 K for increasing doping . We note that theincrease of T c is binary in the former case, but progressive in the latter. To explain thisdiffering behavior of T c between pressure and doping, it has been proposed that Lu Ir Si presents a sharp feature in the electronic density of states just above the Fermi level. In this article, we establish disorder induced by irradiation as a full-fledged axis inthe phase diagrams of superconductors, via an extensive study of this model compoundLu Ir Si . In particular, we evidence the mechanism through which proton irradiation actsas a tuning parameter suppressing the CDW in favor of superconductivity and we show howthis tuning parameter brings its own set of unique information on superconductivity-CDWcompetition. In sharp contrast with the expected effect of disorder on superconductivity, weobserve an increase of T c after irradiation. This irradiation induced disorder also stronglysuppresses the CDW and broadens its transition, thus revealing the precise mechanism ofCDW suppression. Moreover the increase of H c, with disorder reveals that the channelfor competition between CDW and superconductivity is the electronic density of states atthe Fermi level. These results make a compelling case that reconciles how CDW and su-perconductivity competes in Lu Ir Si with pressure, doping and disorder. This extensiveset of results in a BCS s-wave compound with a prototypical 1D CDW of the Peierls typeprovides the basis to pursue such irradiation studies in the more complex cases of densitywave coexistence in heavy fermions, Fe-based or cuprates superconductors .The article is organized as follows: Part. II presents the materials and methods. Part. IIIpresents how irradiation induced disorder raises T c and reduces T CDW . Part. IV presents theevolution of H c , as a function of disorder. Part. V reconciles the different evolutions of T c with pressure, doping and disorder by discussing how they relate to the CDW suppressionmechanisms. II. MATERIALS AND METHODS Lu Ir Si has a tetragonal unit cell with lattice parameters a = 12 . c =4 . and space group symmetry P4/mbm. The CDW forms along the c-axis onquasi 1D chains of lutetium atoms. The samples are high quality single crystals that growin needle shape along the c-axis, and have been characterized previously . We usedthe tandem van de Graaf accelerator at Western Michigan University to irradiate a sample3everal times with 4 MeV protons. This sample has dimensions 10 µ m × µ m × µ m(a × b × c). The 10 µ m thickness of the sample ensures uniform irradiation damage andnegligible proton implantation, as SRIM calculations show the projected range of protonsis 67 µ m in these conditions.For the irradiations, the sample is mounted onto an aluminum sample holder that allowsfor linear and rotational motion. In order to avoid heat damage of the sample we use arelatively low beam current of 500 nA and a cooling stage that maintains the sample at-10 ◦ C during irradiation. The incident proton beam of 4.7 mm diameter is homogenized bypassing through a 1 µ m gold foil placed at 240 mm upstream from the sample. The beamis defined through a 7.8 mm aperture placed at 40 mm upstream. This set-up is calibratedwith the help of a Faraday cup placed down-stream from the sample which captures allprotons passing the aperture while the sample is moved out of the beam path. The sampleis electrically connected to the sample holder and the irradiation chamber which, in turn,is isolated from all other electronics and from the beam pipe through plastic rings. Thisapproach allows to accurately determine the irradiation dose by integrating the current fromthe chamber, not affected by spurious effects due to the emission of secondary electrons.The sample was irradiated in four sessions at Western Michigan University to a ratherhigh cumulative dose of 12x10 p/cm (protons per cm ). Such a high dose is known tostart to degrade some superconducting properties in several families of superconductors .After each irradiation we measured the resistivity using a Keithley 2182 voltmeter and 6221current source, with currents ranging from 50 µ A to 1 mA, in an helium 4 cryostat with a7 T magnet. Contacts were made with sputtered platinum and silver epoxy Epotek H20E.Typical contact values are < ∼ µ m apart along thec-axis. III. DISORDER INCREASES T c AND DECREASES T CDW
Fig. 1 shows the evolution of the superconducting (panel a) and of the CDW transition(panel b) with increasing irradiation dose. We define T c as the point where the resistivitydropped to 50% (midpoint). Fig. 1a reveals a clear increase of T c with irradiation, from4.15 K in the pristine state to 5.25 K at the highest irradiation dose. Even though thereappear shoulders of unknown origin near the top / bottom of the superconducting transitions4 .5 4.0 4.5 5.0 5.5 6.0 Temperature (K) R e s i s t i v i t y ( . c m ) Lu Ir Si J = 77 A/cm p r i s t i n e p / c m . p / c m . p / c m . p / c m a
10 100
Temperature (K) R e s i s t i v i t y ( . c m ) pristine10 p/cm p/cm p/cm p/cm b FIG. 1.
Temperature dependence of the c-axis resistivity of Lu Ir Si for increasingirradiation doses. (a) The superconducting transition shifts to higher temperature after irradi-ation, in contrast with expected behavior. The transition width (15 - 85 %) surprisingly decreases with irradiation, evidencing uniform irradiation damage (see text), even though there appear shoul-ders of unknown origin near the top / bottom of the superconducting transitions at high doses. (b)
The large increase in resistivity below 80 K is caused by a CDW which gaps density of states at theFermi level and increases electronic scattering. Contrary to T c , the CDW transition temperatureshifts to lower temperature, the transition width increases , and the amplitude of the increase isreduced for increasing irradiation dose. at high doses, the width of the main part of the transition (15 – 85%) surprisingly decreasesupon irradiation.Fig. 1.b, shows an overview of the c-axis resistivity of Lu Ir Si in semi-log scale. Thetransition to the CDW phase at low temperature appears as a large increase of resistivity5elow T CDW ≈
77 K, as previously observed . We define T CDW as the midpoint (50%)of this increase in resistivity. As the irradiation dose increases, T CDW shifts toward lowertemperature and the amplitude of the increase in resistivity is reduced. Contrary to thesuperconducting transition, the width of the CDW transition strongly increases with irradi-ation dose.The CDW transition also has an hysteresis of approximately 1 K, as was previouslyobserved and recently studied in details . We find that this hysteresis survives upto the highest irradiation dose. However, after irradiation the hysteresis occurs only below T CDW (midpoint), whereas, in the pristine state, the hysteresis extends up to the onset ofthe transition (83.5 K).The simultaneous variations of T CDW and T c are summarized in Fig. 2 as a functionof irradiation dose. T c increases almost linearly at a rate of +0.14 K/10 p/cm (or0.093 K/ µ Ω.cm) and starts saturating after the last irradiation. T CDW decreases linearly inthe whole range of irradiation doses at a rate of -1.85 K/10 p/cm (or -1.18 K/ µ Ω.cm).According to Anderson’s theorem , in an isotropic s-wave superconductor small con-centration of non-magnetic defects should not affect T c while magnetic defects should bepair-breaking and reduce T c . Generally, the effect of pair-breaking scattering is described byAbrikosov-Gorkov theory . In this theory, T c is found to always decrease . We thusconclude that the increase of T c we observed, cannot be explained by the standard effects ofdisorder on a superconductor.Rather, such an increase of T c with irradiation dose arises naturally from a competi-tion scenario betwen the CDW and superconductivity, if irradiation suppresses the densitywave more than superconductivity . We recently demonstrated such an increaseof T c via competition with CDW using irradiations in the d-wave cuprate superconductorLa . Ba . CuO . This has also been evidenced in the dichalcogenides superconductorsusing irradiation induced disorder and substitution disorder . Both types of disorderstrongly suppress CDW, either via real-space phase fluctuations (domains) or by pair-breaking . A competition scenario was also proposed for Lu Ir Si based on pressure and doping studies. The increase of T c that we observe upon irradiation, is thereforedefinitive evidence that the CDW is competing with superconductivity in Lu Ir Si .Quantitatively, Lu Ir Si is a weak coupling limit s-wave superconductor (∆ C/γT c =1.41 , close to 1.43), so that electronic density of states released by the CDW should yield6 .84.04.24.44.64.85.05.25.4 T c ( K ) T c T CDW a Dose (10 p/cm ) S C t r a n s i t i o n w i d t h ( K ) Lu Ir Si J = 77 A/cm SCCDW b T C D W ( K ) C D W t r a n s i t i o n w i d t h ( K ) FIG. 2.
Superconducting and CDW transitions as a function of irradiation dose.(a) T c and T CDW vary linearly up to high irradiation doses. T CDW decreases at a constant rateof -1.85 K/10 p/cm , whereas T c increases at a rate of +0.14 K/10 p/cm , which appears tosaturate at the highest irradiation dose. (b) For both transitions we define the width using a 15% –85% criterion (for the CDW: between the min/max resistivity above/below T CDW , respectively).As irradiation dose increases, the superconducting transition width is reduced whereas the CDWtransition width increases, evidencing the different mechanisms through which disorder affects them(see text). an exponential increase of T c following the standard formula for a BCS superconductor: T c = α θ D exp (cid:16) − N ( E F ) V (cid:17) where α ≈ .
14 in the weak coupling limit, θ D = [315 − , N ( E F ) is the density of states at the Fermi level involved inCooper pairs and V is the attractive potential between the electrons of the pair.The evolution of the residual resistivity as a function of irradiation dose is presented in7 m a x m i n ( . c m ) Dose (10 p/cm ) ( . c m ) max ( T < T CDW ) min ( T > T CDW ) FIG. 3.
Irradiation dose dependence of the residual resistivity ρ and the resistivityjump at T CDW . We define the amplitude of the jump as the difference ρ max − ρ min between themaximum and minimum resistivity below and above T CDW , respectively (see Fig. 1.b). ρ increaseslinearly with dose at a rate of 1.57 µ Ω.cm/10 p/cm . No saturation of defects creation is observedup to 12 × p/cm . Fig. 3. Before irradiation, the residual resistivity is ρ = 49 . µ Ω.cm and the residual resistiv-ity ratio (RRR) is ρ
295 K /ρ = 1 .
3, in-line with previous studies . After irradiation, wefind that the residual resistivity increases linearly at a rate of 1.57 µ Ω.cm/10 p/cm with-out saturation up to 12 × p/cm . Such a linear increase is what is typically expectedin metals following the ”unitary limit” , but this was not a priori obvious in Lu Ir Si because of the CDW. Indeed, on the one hand, irradiation suppresses the CDW, which in-creases the density of states at the Fermi level and should reduce the residual resistivity.But on the other hand, irradiation increases the number of defects and reduces the size ofCDW domains, both of which should raise electronic scattering and increase the residualresistivity. Here, as the residual resistivity increases overall, we can at least conclude thatthe latter (increased scattering) more than compensates the former (increased density ofstates). We also find that ρ max − ρ min , the amplitude of the jump in resistivity at T CDW , isreduced after irradiation (see Fig. 3). Again, a natural explanation for this reduction would8
Temperature (K) H c , ( T ) J = 77 A/cm pristine10 p/cm p/cm p/cm p/cm a
10 15 20 25 30 T c (K ) H c , ( ) ( T ) b FIG. 4. H c , vs irradiation dose. (a) Superconductivity second critical field H c , for H in-planeas a function of temperature and irradiation dose. Lines are theoretical curves from Werthamer-Helfand-Hohenberg-Maki (WHHM) theory with α = 0 .
21 and λ SO = 0. Data in the pristinestate is taken from Ref. 26. (b) Superconductivity second critical field at zero temperature H c , (0),extrapolated from the WHHM curves in a), appears proportionnal to T with a slope of 0.12 T/K (see text). be that after the CDW is suppressed by disorder, it does not gap as much electronic densityof states. IV. INCREASE OF H c , WITH DISORDER
As expected from the increase of T c we find an increase of the in-plane upper criticalfield ( H c , ) for increasing irradiation dose. Our measurements of H c , are reported Fig.4a,9s a function of temperature for several irradiation doses, and where we define H c , as thepoint where the resistivity drops 10% below the residual resistivity value ρ . Our datais in good agreement with published H c , ( T ) data in non irradiated Lu Ir Si . Wefind that H c , ( T ) follows the Werthamer-Helfand-Hohenberg-Maki (WHHM) scaling atall irradiation doses. Using the WHHM scaling, Ref. 26 found best fit parameters values α = 0 .
21 for the Maki parameter and λ SO = 9 . α = 0 .
21, fits are essentially insensitive to the choice of the spin-orbitcoupling λ SO , so that we can adopt in the following λ SO = 0 and extrapolate the value of H c , (0) with little uncertainties (see appendix A for details).In Fig. 4b we find that H c , (0) is in good agreement with a T dependence. In a usuals-wave isotropic superconductor in the dirty limit, H c , (0) should scale with T c . Indeed,the upper critical field is equal to φ πξ and in the dirty limit the coherence length ξ isrenormalized to ξ ,d = q ξ ¯ l , where ¯ l = v F τ and ξ = ¯ hv F . πk B T c in the weak-coupling BCStheory, yielding : H c , (0) = φ πξ ,d ≈ φ . k B hv τ T c (1)So usually, in the dirty limit H c , (0) ∝ T c , however here H c , (0) ∝ T . This unusual scalingcan be easily explained by the fact that 1 /τ is proportionnal to T c due to the competitionwith the CDW, namely: the prefactor φ . k B hv is independent of irradiation dose, whereas1 /τ is usually proportional to the irradiation dose for uniform non-overlapping defects inmetals , and the dose itself is empirically proportional to T c (see Fig.2). V. PRESSURE, DOPING AND IRRADIATION: PROGRESSIVE VERSUS BI-NARY INCREASE OF T c Interestingly, the competition scenario between CDW and superconductivity still requiresclarification in Lu Ir Si . Indeed, pressure studies show a binary effect on T c : below2 GPa, T c is 4 K and constant, whereas above 2 GPa, T c is 9 K and constant; in contrastdoping studies show a progressive increase of T c from 4 K to 6 K. This is all the moresurprising as in both cases the CDW is progressively suppressed. Hence, naively, shouldn’tone expect that T c also increases progressively with pressure ? We argue that this can beexplained by the different mechanisms through which the CDW is suppressed when usingpressure, doping and irradiation. 10he variations as a function of irradiation dose of the width (15 - 85 %) of the super-conducting and CDW transitions are summarized in Fig. 2.b. As can be seen, the widthof the superconducting transition decreases with irradiation dose. In general, for the su-perconducting pairs in a s-wave superconductor, disorder only acts via the pair-breakingmechanism from magnetic defects. The suppression mechanism by phase fluctuations fromdisorder does not apply as the superconducting condensate is not modulated in real space(FFLO or pair-density wave superconducting states do have a superconducting condensatemodulated in real space, but such states have remained elusive experimentally ). So in astandard s-wave superconductor no change of the superconducting transition width is ex-pected with disorder at first. However, in this compound superconductivity competes withthe CDW, so that it is still sensitive to how homogeneously the CDW is suppressed bydisorder. Thus, here the decreased superconducting-transition width after irradiation showsthat the irradiation damage caused by proton irradiation is very uniform, to the point thatdisorder in the sample is actually more uniform after irradiation.Let us now turn to how the disorder influences the CDW. By contrast to the super-conducting transition, as shown in Fig. 2, the width of the CDW transition significantlyincreases with irradiation dose, and both T CDW and the jump in resistivity decreases withincreasing disorder. These are strong indications that the CDW is suppressed by real-spacephase fluctuations.Indeed, two different mechanisms have been proposed for CDW suppression by disorder:(i) a pair-breaking mechanism where disorder increases the scattering rate, which inducesa broadening of the Fermi function and reduces the peak in electronic susceptibility. Thisprocess reduces the jump in resistivity at the CDW transition, but it does not affect themacroscopic coherence of the CDW and produces a uniform global reduction of the CDW.Thus this mechanism cannot account for the broadening of the CDW transition. (ii) a real-space phase fluctuations mechanism where disorder pins the phase of the periodic spatialmodulation of the electronic density and associated lattice distortion. This breaks up theCDW into small domains, which broadens the transition(Ref. 58 § in which Sc (Co) were introduced on the Lu(Ir)-sites, respectively, pronounced broadening and suppression of the CDW was observed,in analogy to the results presented here, suggesting that real-space phase fluctuations are11uppressing the CDW in doping studies as well. However, in doping studies, additionaleffects may arise from doping-induced changes of the Fermi surface.Conversely, in pressure studies, the mechanism by which the CDW is suppressed must bedifferent as there is no change in the number of defects, hence no pair-breaking nor phasefluctuations. This is also quite strikingly evidenced experimentally : even though T CDW decreases by up to a factor of 10, there is no significant change in the CDW transition widthunder pressure. A natural explanation for this is that pressure stiffens the elastic constantsof the crystal, which makes the periodic lattice distortion less favorable energetically andreduces T CDW . This suppression-by-elastic-stiffening mechanism follows from the standardCDW stability criterion of Chan and Heine . As this mechanism is global in essence, itexplains why the CDW transition width remains sharp and constant even though T CDW diminishes. Finally, as T c remains essentially constant up to 2 GPa, it also means that thedensity of states at the Fermi level is essentially constant with pressure in the CDW phase.To first order, the elastic stiffening modifies only the temperature at which the Chan andHeine criterion is met, without affecting the density of electronic states involved.Thus, we can now reconcile the effects of pressure, doping and irradiation on the CDWand superconductivity, by considering the differences between the three mechanisms of CDWsuppression: pair-breaking, real-space phase fluctuations and elastic constants stiffening.The main effect of pressure on the CDW is to reduce T CDW via elastic constants stiffening,but both the CDW-transition width and the amount of density of states gapped by the CDWremains constant. Hence T c increases in a binary way: when pressure reduces T CDW below T c .By contrast, with doping or irradiation there are both phase fluctuations and pair-breakingwhich not only reduces T CDW , but also reduces the amount of density of states gapped bythe CDW. Thus T c increases in a progressive way: doping/irradiation frees density of statesthat was gapped by the CDW, which then raises T c60 (as superconductivity is not subjectto the same strong pair-breaking or phase fluctuations effects of disorder). VI. CONCLUSION
We showed that irradiation induced disorder enhances the superconducting critical tem-perature T c and H c2 while it suppresses the CDW in Lu Ir Si . We showed how this increaseof T c cannot be accounted for by the expected effect of disorder, and instead stems from12he increase of density of states at the Fermi level. Our results thus make a very com-pelling case that superconductivity and CDW are competing for electronic density of statesat the Fermi level in Lu Ir Si . Owing to its prototypical, 1D, Peierls type CDW and thes-wave, weak-coupling nature of its superconductivity, Lu Ir Si thus provides a platformfrom which to understand the more complex cases of density waves and superconductiv-ity coexistence in heavy fermions, Fe-based or cuprates superconductors. Recently it wasshown that, in an unconventional superconductor mediated by spin-fluctuation, very inho-mogeneous conditions may result in an increase of T c . However we do not think this lattercase is relevant to the standard s-wave superconductor studied here. Also very recently,it was found that, while the effect of disorder on the dichalcogenide NbSe in bulk formis explained in terms of CDW-superconductivity competition and synergy , in monolayerNbSe a much larger T c dome was discovered as a function of disorder and this has beenproposed to be due to the wavefunction multifractality in a 2D monolayer system . Thus,disorder as a tuning parameter is finding relevance not only for the study of bulk supercon-ductors with density waves but also for 2D materials such as monolayer transition metaldichalcogenides. Interestingly, from a technological perspective, our results also unlock thepossibility of direct-writing superconducting detectors and Josephson junctions by localizedirradiation via a mask or a helium FIB, which could support the current effort in QuantumInformation Science based on SQUID technology. ACKNOWLEDGMENTS
M.L. acknowledges fruitful discussions with S. Eley, F. Ronning, C. Proust and P. Mon-ceau. The experimental study at Argonne National Laboratory was supported by the USDepartment of Energy (DOE), Office of Science, Materials Sciences and Engineering Divi-sion. V.M. was supported by the Center for Emergent Superconductivity, an Energy FrontierResearch Center, funded by the US DOE, Office of Science, Office of Basic Energy Sciences.Proton irradiation was performed at Western Michigan University. CO and PR work aresupported by the ANR-DFG grant ANR-18-CE92-0014-03 ”Aperiodic”. The samples used13n this study were grown at the N´eel Institute by C. Opagiste. ∗ Present address: Laboratoire National des Champs Magn´etiques Intenses (CNRS, EMFL, INSA,UGA, UPS), Toulouse 31400, France † Present address: Kavli Institute for Theoretical Sciences, University of Chinese Academy ofSciences, Beijing 100190, China G. Gruner,
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VII. APPENDICESAppendix A: WHHM Scaling
To support the relevance of the WHHM theory in Lu Ir Si , despite its complex FermiSurface , we verify that the α values estimated in two independent ways are close, fol-lowing the recommendation of Ref. 55. First, using the normal state values just above T c : α = e ¯ hγρ mπ k ≈ .
16 (SI units), with γ = 119 . /K (the volumic Sommerfeld co-efficient from γ = 23 .
42 mJ / mol / K in Ref. 26) and ρ = 56 . µ Ω.cm (also from Ref.26). This result is close to the value deduced from the slope of H c , near T c (in T/K): α = 0 . × − (cid:16) dH c , dT (cid:17) Tc ≈ .
21 (SI units). In Fig. 5, we show that because of the smallvalue α = 0 .
21, the WHHM scaling is essentially insensitive to the choice of the spin-orbitcoupling λ SO in this compound. Appendix B: Detailed CDW and SC Transition Curves and Peak Effect
In Fig. 6, we show that the point where resistivity drops below the resolution of theinstruments ( ≈ − µ Ω.cm), also shifts to higher temperature with increasing irradiationdose, in the same manner as the midpoint of the transition. We also note that in the pristine(non-irradiated) state, higher current densities shift the transition to lower temperature,whereas after irradiation the curves become almost identical at all current densities. Wefound no effect of the current density on the CDW transition in the range of current densitythat we explored ( ≤
154 A/cm ).In Fig. 7, we show that we observe a clear peak effect in the middle of the superconducting18 = TT c h * WHHM scaling = 0.21, SO = 100= 0.21, SO = 9= 0.21, SO = 0= 0.16, SO = 0= 1.50, SO = 2= 1.50, SO = 0 FIG. 5. H c , and WHHM scaling. Superconductivity reduced second critical field h* = H c , / ( − d H c , / d t ) t =1 as a function of the reduced temperature t = T /T c , for all irradiation doses.No significant changes occur with irradiation in this reduced plot. The lines are theoretical curvesfrom WHHM theory. Data in the pristine state was taken from Ref. 26. transition, at both current densities and for all irradiation doses, which shows this peak effectis robust to disorder. Such a peak effect is usually caused by the softening of the vortexlattice near T c which enables it to better adapt to the distribution of defects, hence the dropin resistivity. To our knowledge, such a peak effect had never been reported in Lu Ir Si .We also find that the irreversibility field ( H irr ) defined as the point where the resistivitydrops below the resolution of our instruments (1 nΩ.cm) increases with irradiation dose, inline with H c , . 19 R e s i s t i v i t y ( . c m ) J = 7.7 A/cm pristine10 p/cm p/cm p/cm p/cm R e s i s t i v i t y ( . c m ) J = 77 A/cm Temperature (K) R e s i s t i v i t y ( . c m ) J = 154 A/cm Temperature (K) FIG. 6.
Detailed temperature dependence of the c-axis resistivity of Lu Ir Si at thesuperconducting transition: for increasing current densities (top to bottom) and in linear (leftcolumn) and semilog scale (right column). p/cm J = 77 A/cm p/cm J = 154 A/cm p/cm p/cm p/cm p/cm Temperature (K) R e s i s t i v i t y ( . c m ) p/cm p/cm FIG. 7.
Detailed magnetic field dependence (H//ab) of the c-axis resistivity ofLu Ir Si at the superconducting transition : for increasing irradiation doses (top to bottom)and using a current density of 77 (left column) and 154 A/cm (right column). A peak effect isvisible in the superconducting transition.(right column). A peak effect isvisible in the superconducting transition.