supplementary information - … the format provided by the authors and unedited. charge density wave...

In the format provided by the authors and unedited.

Charge density wave quantum critical point with strongenhancement of superconductivity

Thomas Gruner1, Dongjin Jang1, Zita Huesges1, Raul Cardoso-Gil1, Gerhard H. Fecher1, Michael M. Koza2,

Oliver Stockert1, Andrew P. Mackenzie1, Manuel Brando1 and Christoph Geibel1

Supplementary Note 1 | Analysis of the diamagnetic susceptibility

The diamagnetic contribution is essentially an atomic contribution which can be estimated from atomic calcula-tions. Taking the results obtained within a relativistic Hartree-Fock calculation [1], we obtain a total diamagneticcontribution of -2.6 · 10-9m3mol-1 and -2.2 · 10-9m3mol-1 for LuPt2In and LuPd2In, respectively. The Pauli suscepti-bility can be estimated from the density of states (DOS) obtained within DFT calculations, 1.81 states eV-1f.u.-1 and1.90 states eV-1f.u.-1 for the Pt- and the Pd-based compound, respectively, which are in good agreement with the mea-sured Sommerfeld coefficients (see Supplementary Note 6). In the absence of magnetic correlations this corresponds toa Pauli susceptibility χP of about +0.8 · 10-9m3mol-1, thus a factor of three smaller than the diamagnetic contribution.The orbital (Van-Vleck) susceptibility provides a further positive contribution, which is however difficult to calculate.Therefore, the small negative susceptibility observed in the experiments can be explained by the dominant Langevindiamagnetic contribution. We further note that the increase in the high T (T > TCDW) susceptibility observed betweenLuPt2In and LuPd2In, about 0.3·10-9m3mol-1, agrees reasonably well with the difference in the calculated diamagneticcontribution, 0.4 · 10-9m3mol-1.

Supplementary Note 2 | Estimation of decrease in N(εF) induced by the CDW transition

The small negative value of χ(T ) proves the absence of magnetic correlations in Lu(Pt1-xPdx)2In (the Curie-likeincrease below 50K is due to a tiny amount of paramagnetic impurities) (see Fig. 1b). Hence, one can assume theWilson ratio between χP and Sommerfeld coefficient γ0 to be 1 and thus use the drop in χ(T ) to estimate the reductionin N(εF):

∆γ = ∆χ · (πkB)2/(µ0µ2eff) .

Supplementary Note 3 | Low temperature structure

A comparison of room temperature x-ray powder diffraction patterns for different Pd-contents (see SupplementaryFig. 2) reveals additional Bragg peaks present in Pt-rich samples. However, the interpretation of these data is morecomplicated because microprobe studies indicated an increase of foreign phases with increasing Pt-content. Theissue was solved by T dependent x-ray and neutron scattering studies on pure LuPt2In, which shows some of theadditional Bragg peaks to be T dependent and to disappear above TCDW (see Supplementary Fig. 3), while othersare T independent. This allows to discriminate between foreign phase peaks and superstructure peaks induced by theCDW transition. The latter ones are marked by grey lines in Supplementary Fig. 2 and 3. At 300K they disappearfor Pd-content x larger than 0.2, in agreement with TCDW dropping below 300K for x � 0.2 as deduced from ρ(T )and χ(T ) results.We observe that changes related to TCDW in the powder diffraction patterns are restricted to the appearance ofadditional superstructure reflections. We detect no splitting of the peaks which would correspond to the high T cubicstructure. This indicates that the low T structure retains its cubic symmetry, but with a lattice parameter aLT thatis a multiple of that of the high T structure aHT. Preliminary results from single crystal x-ray diffraction suggest adoubling of the unit cell along all three directions. The collected intensity data indeed lead to a preliminary structuremodel based on a body centered cubic structure with aLT = 2 · aHT. The dominant difference in this structure

1Max Planck Institute for Chemical Physics of Solids, Nothnitzer Straße 40, 01187 Dresden, Germany. 2Institut Laue Langevin, 6 Rue JulesHorowitz, B.P. 156, 38042 Grenoble, Cedex 9, France.

© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

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respective to the high T phase is a rotation of the Pt8 cubes (see Supplementary Fig. 4 and Supplementary Video).However, the poor quality of the collected intensity due to the insufficient quality of the single crystals resulted inunsatisfactory reliability factors of the refinement, which prevents a definitive statement on the low T structure.Visualizing the continuous structural change | The position of each atom in the tentative low T structure canbe described as position of the same atom in the high T cubic Heusler type structure plus a small displacement ∆R.Expressing ∆R as a function of TCDW − T allows simulation of the continuous deformation of the structure belowTCDW. This simulation was used to visualize the continuous structural transition in the Supplementary Video.

Supplementary Note 4 | Properties of the superconducting state

In order to confirm the bulk nature of the SC state, we measured a magnetization loop at 0.5K on the sample withthe highest Tc (Supplementary Fig. 5a). The observed hysteresis corresponds to the expected behaviour for a type-II superconductor. The initial slope of the virgin curve corresponds to a susceptibility of -0.8 in SI units, yieldinga Meissner effect of at least 80%. First departure from this linear behaviour is observed at a lower critical fieldBc1 = (2.4± 0.2)mT, while SC is completely suppressed at the upper critical field Bc2 = (74± 2)mT.Furthermore, in order to shed more light on the nature of the SC state, we studied the specific heat of the x = 0.54sample in the millikelvin regime. The experimental data are plotted as C/T versus T in Supplementary Fig. 5b.Unfortunately, below 350mK C(T ) is dominated by a nuclear contribution Cn(T ), even at B = 0. A significantnuclear contribution at these temperatures is not expected for a non-magnetic system where all atoms are located onsites with local cubic symmetry. Therefore, we suspect that the disorder connected with statistical Pd/Pt occupancysufficiently disturbs the local cubic symmetry, for example on the Lu site. Then the large nuclear quadrupole momentof 175Lu would result in a nuclear quadrupole splitting and thus in an observable Cn(T ). Since the unknown electronicspecific heat in the superconducting state prevented a precise direct determination of Cn(T ) at B = 0, Cn(T ) wasdetermined at field B > Bc2 by fitting the data with C = An(B)/T 2 + γT + βT 3. Extrapolating An(B) to B = 0yields the Cn(0) shown as black curve in Supplementary Fig. 5b. The electronic specific heat Ces in the SC state isthen obtained by subtracting Cn(0) and βT 3 from the experimental data. This is shown as Ces(T )/T (green circles)in Supplementary Fig. 5c. Below 0.2K, Ces(T )/T could be nicely extrapolated to the origin using an exponentialfunction, which suggests that the order parameter is fully gapped. With this extrapolation, the entropy expectedin the superconducting and in the normal state match reasonably well at Tc, as indicated by the green areas inSupplementary Fig. 5c. This further supports our method for the determination of Ces(T ). We note that the stepin Ces(T )/T at Tc is significantly smaller than the BCS value. Accordingly, the decrease of Ces(T )/T below Tc ismuch weaker than for BCS SC. Both features indicate the presence of two different gaps, one being smaller than theexpected BCS value.

Supplementary Note 5 | Influence of the phonon softening on the transition temperature Tc

In the BCS theory of SC the transition temperature Tc is proportional to ωD exp[-1/(N(εF)Vep)], where ωD is acharacteristic phonon frequency and Vep is the BCS pairing interaction. In the standard model of electron-phononmediated SC, the pairing interaction originates from the exchange of phonons. Therefore, the contribution of a phononto Vep is proportional to g2/ω, with g an electron-phonon scattering amplitude and ω the energy of the phonon. Takingan appropriate average of the contribution of all phonons results in the electron-phonon coupling parameter λep, whichfor example enters more sophisticated equations for Tc. If a phonon softens to ω = 0, g2/ω diverges and therefore oneexpects a significant increase in λep. This suggests an easy way to get a peak in Tc(x) at a CDW QCP. However, ωalso enters the prefactor ωD in the BCS equation for Tc, where it will result in a decrease of Tc. Replacing in a firstapproximation the whole phonon spectrum by an Einstein mode representing the dominant soft phonon mode, onewould obtain Tc ∝ ω exp(-ω) in a very simple approximation. This function has a maximum at finite ω but vanisheswhen ω drops to zero. Thus, Tc would present a maximum away from the QCP and disappear at the QCP. On amore elaborate level, the relation between Tc, ω and λep for an Einstein mode has been studied for example by P.B.Allen and R.C. Dynes using numerical solution of the Eliashberg equations [2]. In the limit of large coupling λ, Tc/ωis proportional to the square root of λep. With λep ∝ 1/ω, this results in Tc ∝

√ω. Thus, also in this more elaborate

approach, Tc drops to zero when ω goes to zero.In real systems, not only the soft mode, but the whole phonon spectra is contributing to the electron-phonon couplingand to Tc. Therefore, assessing the effect of one particular mode is difficult. However, there is a well-establishedknowledge about which part of the phonon spectrum is the more effective in enhancing Tc, and how the enhancementdepends on the phonon frequency. In a study based on numerical solution of the Eliashberg equations, G. Bergmannand D. Rainer [3] showed that the most effective phonon frequency range for enhancing Tc is slightly above 2π Tc, and

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that this effectiveness, that is the derivative dTc/d(α2 F (ω)) is decreasing linearly to zero when ω goes to zero. This

result implies that when a phonon becomes soft, it may increase Tc as long as his frequency is well above 2π Tc, butthis increase shall vanish upon further softening, and may even turn into a decrease. Bergmann and Rainer explicitlynote in their paper: “changes of α2 F (ω) in the very low frequency region have no essential influence on the transitiontemperature. This is in contrast to the influence of the low frequency part of α2 F (ω) on the electron-phonon couplingparameter λ. . . . λ depends sensitively on the low frequency behaviour of α2 F (ω)”. These sentences provide in shortthe reason why a phonon which softens to ω = 0 at a CDW QCP may result in a peak in λep at the QCP, but thatshall not result in a peak in Tc at the QCP. This does not imply that within the Eliashberg theory it is impossibleto get a peak in Tc. By combining a multiband system with peculiar electron-phonon couplings, likely any type of Tc

dependence might be reproduced. However, as discussed in the main part of the paper, the observed nice correlationbetween the composition dependence of Tc, of the thermal exponent n and of the coefficient β indicates that criticalfluctuations are an important ingredient.In this context, we note that a peak in Tc near a CDW QCP was reported for pressure or composition tuned TiSe2[4]. However, a detailed study based on a combined pressure/doping study revealed than the dependence of Tc on thetuning parameter is linear on both sides of the QCP [5], in contrast to the concave curvature observed for x < xc inLu(Pt1−xPdx)2In. A more recent study indicates the CDW QCP in TiSe2 to be at a much higher pressure and themaximum in Tc to be connected with a transition from a commensurate to an incommensurate CDW [6].

Supplementary Note 6 | Calculations of electronic and phononic properties

The electronic band structure, Fermi surface, nesting properties as well as phonon dispersion were calculated in thelocal (spin) density approximation for a large number of rare earth containing Heusler compounds. An extendedaccount of these calculations and their outcome shall be given in a forthcoming paper. Here, we focus on the resultsfor LuPd2In and LuPt2In. For the latter one all calculations refer to the undistorted high T structure, since thelow T structure is presently not well-established. We note that the calculations for LuPt2In in the high T structureresulted in a stable solution, without negative phonon energies, indicating that the structural instability is only a weakone. Not surprisingly both compounds present very similar band structures and electronic density of states N(ε).Therefore, we show in Supplementary Fig. 8a and b only the results for LuPt2In. At first sight they do not providean obvious reason for a CDW transition. N(ε) is small and almost independent on energy in the vicinity of εF. Thus,a sharp peak with an enhanced N(εF), which for example leads to the martensitic transition in Nb3Sn, is missing [7].On the other hand, the Γ-X-direction shows one “flat” band which stays close to εF, ending in a Van Hove singularityat the X-point just 0.23 eV below εF. Interestingly, the strongest differences in the states near εF between the Pt- andthe Pd-compounds are observed just along the Γ-X-direction. In LuPd2In the “flat” band is pushed above εF for allmomenta along this direction, with the corresponding Van Hove singularity at the X-point shifted to +0.32 eV. Thesestates have mainly Pd/Pt character, and the changes for example at the Γ-point are directly connected to the largerspin-orbit coupling of Pt. Accordingly, four bands cross εF in LuPd2In, instead of only three bands in LuPt2In. Twoof these Fermi surfaces display some parallel parts indicating possible nesting. We therefore calculated the autocorre-lation function f(q) of the Fermi surface, also called the nesting function, for both pure compounds and for the alloywith x = 0.5. They show a rich structure, which is furthermore strongly composition dependent. In LuPd2In we gotfor instance strong maxima at around q = (0.5 , 0.5 , 0), which, however, became weaker in the Pt-compound. On theother hand, it is meanwhile clear that the nesting scenario commonly proposed for CDW is rarely applicable for realsystems, and that states far below and above the Fermi level are equally important [8, 9]. Accordingly, the predictivepower of the nesting function has proven to be quite limited [8, 9]. Instead the CDW propagation vector seems to befrequently determined by maxima of the q dependent electron-phonon coupling [10–12]. Therefore, focusing on thenesting function would be misleading because phonons are highly relevant, too.In Supplementary Fig. 8c and d we plot the calculated phonon-dispersion relation and phonon DOS of LuPd2In andLuPt2In in different colours. The main difference is a strong shift of the lowest set of optical branches to lower energiesfrom the Pd- to the Pt-compound. Since these modes essentially correspond to vibration of Pd or Pt, this shift caneasily be accounted for by the differences in the atomic masses (almost a factor of 2). Accordingly, the correspond-ing phonon DOS moves quite significantly to lower energies, resulting in a pseudo gap at about 11meV. The mostinteresting feature is the pronounced dip in the transverse acoustic (TA) mode at the X-point, which results from ananticrossing with the lowest optical mode. Accordingly, this dip gets very pronounced in the Pt-compound, suggestinga strong phonon softening there.We note that for a hypothetical cubic Heusler LuPt2Sn phase we obtain strongly negative energies for this TA modeand for the lowest optical mode at the X-point, indicating a strong lattice instability towards a corresponding distor-tion. This provides a possible origin for the link between the occurrence of a CDW type instability in these Heuslerphases and the proximity to the region where the hexagonal ZrPt2Al structure type (as in LuPt2Sn) is formed instead

3






of the Heusler phase [13].Thus, both the calculated electronic and phononic properties provide a number of possible origins for this CDW typeof structural transition. Further studies are necessary to determine which is the relevant one. The comparison betweenthe electronic states close to εF in LuPd2In and LuPt2In suggest that the stronger spin-orbit coupling of Pt might bethe root cause.Some properties allow a simple and direct comparison between these DFT based calculations and experimentaldata. The calculated electronic DOS at εF amounts to 1.90 states eV-1f.u.-1 and 1.81 states eV-1f.u.-1 for LuPd2Inand LuPt2In, respectively, which in absence of renormalization effects would correspond to Sommerfeld coefficients γ0of 4.5mJmol-1K-2 and 4.3mJmol-1K-2, respectively. The experimental value for LuPd2In, 6.2mJmol-1K-2, is slightlyhigher, but an electron-phonon coupling constant λ = 0.38 would be sufficient to account for this difference. Forthe Pt-compound a direct comparison is not possible due to the strong effect of the CDW transition, but the smalldifference between the calculated values for the Pt and the Pd-compound is in agreement with the evolution of γshown in Fig. 4a. The low T Debye temperature ΘDebye-LT, which determines the coefficient β in the specific heat, canbe calculated from the initial slope of the acoustic modes. These slopes are quite similar in both compounds despitethe strong shift in the first set of optical modes. Accordingly, the calculated ΘDebye-LT is almost the same in bothcompounds, ΘDebye-LT = 205K and 194K for LuPd2In and LuPt2In, respectively. The experimental values deducedfrom the coefficient β (Fig. 4b) are 255K and 190K, respectively. For the Pt-compound the agreement is perfect,while for the Pd-compound experimental data indicate slightly higher mean sound velocity than predicted.

Supplementary Note 7 | Dependence of the coefficient β of the phononic specific heat on the Pd-content x

The coefficient β is determined by the small energy / small momentum part of the acoustic branches and therefore

proportional to 〈c〉-3, where 〈c〉 is an appropriate average over the sound velocities. In a first approximation, assumingthe changes in 〈c〉 to be due to the changes in the atomic mass and the stiffness to be independent on x, the sound

velocity of these small q modes is inversely proportional to 〈m〉1/2, where 〈m〉 is the average mass of the atoms withina unit cell.

With f =〈m〉LuPt2In

〈m〉LuPd2In

, we get β(x) = βLuPd2In · [f(1− x) + x]3/2

.

We used the Pd-compound as a basis of the calculation because it seems to be the more “normal” system, without astructural phase transition and without soft phonons in the energy range probed by low T specific heat. The resultingbehaviour is plotted as a dotted line in Fig. 4b.

Supplementary References

[1] L. B. Mendelsohn, F. Biggs, and J. B. Mann, Phys. Rev. A 2, 1130 (1970).[2] P. B. Allen and R. C. Dynes, Physical Review B 12, 905 (1975).[3] G. Bergmann and D. Rainer, Zeitschrift fur Physik 263, 59 (1973).[4] E. Morosan, H. W. Zandbergen, B. S. Dennis, J. W. G. Bos, Y. Onose, T. Klimczuk, A. P. Ramirez, N. P. Ong, and R. J.

Cava, Nature Physics 2, 544 (2006).[5] S. L. Bud’ko, P. C. Canfield, E. Morosan, R. J. Cava, and G. M. Schmiedeshoff, Journal of Physics: Condensed Matter 19,

176230 (2007).[6] Y. I. Joe, X. M. Chen, P. Ghaemi, K. D. Finkelstein, G. A. de la Pena, Y. Gan, J. C. T. Lee, S. Yuan, J. Geck, G. J.

MacDougall, et al., Nature Physics 10, 421 (2014).[7] L. F. Mattheiss and W. Weber, Physical Review B 25, 2248 (1982).[8] M. D. Johannes and I. I. Mazin, Physical Review B 77, 165135 (2008).[9] J. Laverock, D. Newby, E. Abreu, R. Averitt, K. E. Smith, R. P. Singh, G. Balakrishnan, J. Adell, and T. Balasubramanian,

Physical Review B 88, 035108 (2013).[10] F. Weber, S. Rosenkranz, J.-P. Castellan, R. Osborn, R. Hott, R. Heid, K.-P. Bohnen, T. Egami, A. H. Said, and D. Reznik,

Physical Review Letters 107, 107403 (2011).[11] L. P. Gor’kov, Physical Review B 85, 165142 (2012).[12] H.-M. Eiter, M. Lavagnini, R. Hackl, E. A. Nowadnick, A. F. Kemper, T. P. Devereaux, J.-H. Chu, J. G. Analytis, I. R.

Fisher, and L. Degiorgi, Proceedings of the National Academy of Sciences 110, 64 (2013).[13] T. Gruner, D. Jang, A. Steppke, M. Brando, F. Ritter, C. Krellner, and C. Geibel, Journal of Physics: Condensed Matter

26, 485002 (2014).

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1.1

1

0.54

Lu(Pt1-xPdx)2In0.4

0.460.50.52x = 0.56

/5K

a

0 50 100 150

0

b

∂/∂

T(a

rb.u

nits

)

Temperature T (K)

Supplementary Figure 1 | Determination of TCDW in samples close to the quantum critical point. a, The crossing point oftwo straight lines drawn through the ρ(T ) data above and below the kink, respectively, provides a first estimation of TCDW.b, A second estimation of TCDW is given by the location of the largest curvature in ρ(T ), that is the largest slope in∂ρ(T )/∂T . The bars indicate TCDW and its errors.

0.2

0.4

0

0.30.20.1

0

x= 10.7

0.540.4

X-ra

yin

tens

ity(re

lativ

eun

its)

T =300KLu(Pt1-xPdx)2In

20 40 60 80 100 1202 (deg)

Supplementary Figure 2 | Evolution of powder x-ray diffraction patterns as a function of Pd-content. In LuPd2In (top) allpeaks can be indexed with the cubic Heusler L21-type structure (Fm3m). The weak peaks confirm ordering within the Lu, Inand Pt sites. Upon decreasing Pd-content additional weak peaks appear. As deduced from T dependent measurements onLuPt2In (see Supplementary Fig. 3), some peaks are T independent and can thus be attributed to foreign phases seen in EDX.Other peaks (long gray bars) appear only below TCDW and therefore indicate a structural modulation below TCDW. The shortbars at the bottom mark Bragg peaks expected for the Fm3m structure.

5






80 90 100 110

0

350

400

450

500

550

a

2 Neutron (deg)

Neutron

intensity(arb.units)

T (K)

LuPt2Inλi = 4.1 Å

66 67 68 69 70

0

350

400

450

500

550

b

T (K)

2 X-ray (deg)

X-ray

intensity(arb.units)

LuPt2Inλi = 1.54 Å

Supplementary Figure 3 | Appearance of superstructure peaks at the CDW transition. In LuPt2In additional peaks appearbelow 500K in neutron diffraction patterns a at 2θ = 106.5◦ (Q = 2.42A-1) and in x-ray diffraction patterns b at 2θ = 66.7◦

(4.48A-1) and 70.0◦ (4.68A-1). A splitting of the main peaks could not be resolved, indicating that the CDW superstructurekeeps cubic symmetry. The integrated intensity of the CDW peak in a was used to analyse the T dependence of the orderparameter (see Fig. 1d).

b

I23

Pd-rich Lu(Pt1-xPdx)2In, high temperaturesPt-rich Lu(Pt1-xPdx)2In, low temperatures

_Fm3m

cubic Heusler L21-type structure

Pt

Lu

In

preliminary modelcubic CDW superstructureperiodic lattice distortion

a

Supplementary Figure 4 | Tentative model for the structure in the CDW phase. a, Tentative model for the low T structure inthe CDW phase as deduced from x-ray single crystal data. Prominent change is the rotation of Pt-cubes. b, Heusler structureobserved above TCDW.

6






0

10

20C

/T(m

Jm

ol -1K-2)

C 0Tn ∝T - 2

Cph +Ce

B = 0 T0.1 T

b

x=0.54

0.4 0.8 1.200

10

γ0

CSC /T

(mJ

mol -1K

-2)

Temperature T (K)

Tc

c

S = 1.8 mJ mol-1 K-1

S = 2.0 mJ mol-1 K-1

∝ exp(-∆E/ kbT )

-60 0 60

-5

0

5 Lu(Pt1-xPdx)2Inµ 0M

(mT)

a

T = 0.5K

x=0.54

Field µ0H (mT)

Supplementary Figure 5 | Superconducting properties of the sample with the highest Tc. a, The magnetization loopmeasured at T = 0.5K shows a hysteresis typical of a type-II superconductor with a large Meissner effect up to Hc1 ≈ 2.4mTand complete suppression of SC at Hc2 ≈ 74mT. b, Specific heat C(T ) in the SC (B = 0) and in the normal state (B = 0.1T).Below 0.4K C(T ) is dominated by the nuclear specific heat Cn(T ). A reliable estimation of Cn(T ) in the SC state can beobtained by extrapolating the high field results to B = 0 (see Supplementary Note 4). Subtracting this Cn(T ) and thephononic part Cph from the measured data we obtain an estimation of the electronic part CSC in the SC state (green dots inc). These data suggest that CSC(T ) follows either a high power law or more likely an exponential function below 0.2K.Extrapolating these data to T = 0 using an exponential function yields the entropy quenched in the SC state up to Tc. This isequal to that obtained in the normal state within accuracy of the experimental techniques (green areas), supporting thevalidity of the used approach.

-1 -0.5 0 0.5 10

2 3 4

7

8

9

10

11

aC / T = γ0 + β T 2

B = 0T

Lu(Pt1-xPdx)2In0.540.56

0.60.64

0.70.8

1

x= 00.10.20.30.4

0.460.5

0.52

C/T

(mJ

mol

-1K

-2)

Squared temperature T 2(K2)

430 K490 K550 K

Neu

tron

inte

nsity

(arb

.uni

ts)

Q = (2.42 0.05)Å-1

λi= 4.1Å

bLuPt2In

Energy (meV)

Supplementary Figure 6 | Softening of low energy phonon modes at the T = 0 quantum critical and at the T = TCDW

classical critical point. a, The slope of the C(T )/T versus T 2 linear fits corresponds to the coefficient β of the phoniccontribution to the specific heat. This plot clearly shows that β is significantly larger at the critical concentration x=0.54than at smaller or larger x-values (see Fig. 4b). A larger β implies a higher density of low energy phonons at x = xc. b, AtT = TCDW the intensity of inelastic neutron spectra at small energy transfers (±0.4meV) is significantly larger than attemperatures above or below the transition temperature TCDW = 490K, providing direct evidence for an increase in thelow-energy phononic DOS at TCDW.

7






0 10 20

1.02

1.04

1

x= 00.10.20.30.4

0.460.5

0.52

Lu(Pt1-xPdx)2In

Temperature T (K)

/5K

0 10 20

0.540.56

0.60.64

0.70.8

x = 1

Supplementary Figure 7 | Determination of the exponent n in the T dependence of ρ(T ). The exponent n is obtainedthrough a fit of ρ(T ) to the function ρ0 +ATn in the range 2K< T < 20K. Results for successive Pd-contents are shiftedupwards to improve clarity.

Γ X K Γ L0

5

10

15

c

Pho

non

ener

gyhν

(meV

)

Wave vector q

0.10

d

LuPt2In

LuPd2In

Phononic DOS (meV-1)

X Γ L W K Γ-6

-4

-2

0

2

a

Ele

troni

cen

ergy

E−ε

F(e

V)

Wave vector q

0 5 10

LuPt2In

b

totalPtLuIn

Electronic DOS (eV-1)

Supplementary Figure 8 | Results of DFT based calculations. Electronic band structure a and electronic DOS b of LuPt2Inobtained within local density approximation. The results for LuPd2In are very similar. Strongest changes near the Fermi levelbetween the Pt- and the Pd-compound are observed along the Γ-X-direction, where the flat band located just below εF inLuPt2In is pushed above εF in LuPd2In. Phonon dispersion c and phonon DOS d of LuPt2In (purple) and LuPd2In (orange)obtained with DFT based calculations. The strongest difference between the Pt- and the Pd-compounds is the strong shift ofthe lowest set of optical branches to higher energies in the latter one. These branches mainly corresponds to Pt/Pd vibrations.Therefore, this shift can be attributed to the much smaller atomic mass of Pd. The most interesting feature is the pronouncedsoftening of the lowest branch at the X-point in LuPt2In, which indicates an incipient lattice instability.

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