Electron-momentum dependence of electron-phonon coupling underlies dramatic phonon renormalization in YNi2B2C

Electron-phonon coupling, i.e., the scattering of lattice vibrations by electrons and vice versa, is ubiquitous in solids and can lead to emergent ground states such as superconductivity and charge-density wave order. A broad spectral phonon line shape is often interpreted as a marker of strong electron-phonon coupling associated with Fermi surface nesting, i.e., parallel sections of the Fermi surface connected by the phonon momentum. Alternatively broad phonons are known to arise from strong atomic lattice anharmonicity. Here, we show that strong phonon broadening can occur in the absence of both Fermi surface nesting and lattice anharmonicity, if electron-phonon coupling is strongly enhanced for specific values of electron-momentum, k. We use inelastic neutron scattering, soft x-ray angle-resolved photoemission spectroscopy measurements and ab-initio lattice dynamical and electronic band structure calculations to demonstrate this scenario in the highly anisotropic tetragonal electron-phonon superconductor YNi2B2C. This new scenario likely applies to a wide range of compounds.


Introduction
Interacting degrees of freedom in solids underlie new emergent ground states and competing phases with potential for new functionalities.Vibrations of the atomic lattice, i.e. phonons, can couple to electrons 1 , magnetic 2 , or orbital degrees of freedom 3 .In particular, electron-phonon coupling (EPC) received a lot of attention as a microscopically understood origin of superconductivity.Furthermore, EPC has recently been in the focus of investigations of materials with competing phases such as cuprates [4][5][6][7][8] and layered transition-metal dichalcogenides [9][10][11][12][13] .
Reduced phonon lifetimes, are typically related to nesting where large sections of the Fermi surface (FS) are connected by a single phonon wave vector [14][15][16] .The hallmark of nesting is a singularity or at least strong peak in the electronic susceptibility for a particular phonon wave vector q connecting parallel sections of the FS 17,18 .
Such phonon anomalies received a lot of attention recently in the context of charge density wave (CDW) formation, which is often driven by a soft phonon mode triggering a structural distortion at the CDW transition.In some CDW compounds, such as 1D conductors, this behavior is indeed related to the nesting of the FS 19,20 .Yet in others, such as 2H-NbSe2, nesting is absent [21][22][23][24][25] and cannot explain the soft-mode properties.While the CDW in 2H-NbSe2 originates from EPC 21,26 , it has been proposed that the phonon softening and broadening on cooling towards the CDW transition temperature   can be explained only by taking into account lattice fluctuations 24 , and anharmonic effects may play an important role 23 .Phonon anomalies in cuprates remain enigmatic though a strong response to the onset of superconductivity is evident 4,7 .In other cases, an expected phonon broadening related to nesting is absent 27 or incomplete 28 .In principle, the phonon momentum, q, dependence of the EPC itself, expressed in the EPC matrix elements   �⃗ + �⃗, �⃗  � �⃗ , can determine the wavevector of the phonon broadening in the absence of FS nesting 25,26 .Yet,   �⃗ + �⃗, �⃗  � �⃗ is expected to be temperature-independent for temperatures of the order of up to 10 3 K.
Here, we propose a scenario in which the electron momentum, k, dependence of the EPC matrix elements comes into play.In such a scenario, the   �⃗ + �⃗, �⃗  � �⃗ , is particularly large for electrons on certain parts of the FS.We show that such k-selective EPC can be strong enough to significantly broaden phonons even when k-integrated quantities like the electronic susceptibility   lack particular features at the phonon momentum q.In this case, the broadening can sensitively depend on the temperature-induced changes of electronic states at the Fermi energy.This scenario can explain large temperature-dependent phonon linewidths in the absence of both nesting and anharmonicity.
We demonstrate this scenario on the electron-phonon superconductor YNi2B2C (  = 15.2K).YNi2B2C is known for unusual low-temperature phonon line shapes reflecting the anisotropic superconducting energy gap [29][30][31] , EPC distributed over 160 meV in phonon energy 32 and phonons with unusual eigenvectors mediating superconductivity 33 .This work focuses on the microscopic origin of strongly increased linewidths of certain phonons observed over a large range of wavevectors in the normal state at low temperatures, i.e.  = 20 K.
We first argue that previously proposed explanations for the phonon properties in YNi2B2C, i.e., FS nesting 34 , can be ruled out.Then we demonstrate that results of our comprehensive inelastic neutron scattering (INS) and soft x-ray angle-resolved photoemission spectroscopy (SX-ARPES) measurements agree well with ab-initio lattice dynamical and electronic band structure calculations, which allows us to use these calculations to gain insights into the microscopic origin of EPC.The calculations highlight the importance of 2D electronic joint density of states (2D-eJDOS).It is defined as the usual electronic joint density of states (eJDOS) but evaluated for 2D slices of the reciprocal space at a specific component of k, kz, where the z-direction is defined to be parallel to the crystallographic c axis (Fig. 1a,b).The results highlight the decisive role of the interplay between the k-dependence of the EPC combined with a strongly k-dependent 2D-eJDOS, and explain strongly temperature-dependent phonon broadening in the absence of FS nesting and lattice anharmonicity.

Effect of electron-phonon coupling on phonons
Electron-phonon coupling renormalizes phonon energies and reduces the phonon lifetime through phonon emission or absorption by electron-hole pairs.Quantitatively, phonon energy is affected by the real part of the  dependent electronic susceptibility   , whereas the lifetime is determined by the imaginary part [see Feynman diagram in Fig. 1(a)].The former depends on electronic states near as well as far from the Fermi surface, whereas only electronic states near the FS contribute to the latter.
Thus knowledge of both electrons and phonons as well as the coupling between them is necessary to understand phonon self-energy in metals.It is easier to compare theory with experiments for the phonon lifetimes than for the peak positions, because electronic states near the FS can be accurately measured by Angle Resolved Photoemission (ARPES) and the phonon linewidths can be measured by neutron scattering.On the other hand, measurements of electronic states far from the FS that need to be included to obtain phonon peak shift due to interaction with electrons are a lot more challenging especially for the states that are unoccupied.
a In the following, we will use the term eJDOS.The equivalent term nesting function could cause confusion since conventional FS nesting plays absolutely no role in the reported effects.
Apart from their intrinsic  dependence, the EPC matrix elements   �⃗ + �⃗, �⃗  � �⃗ can amplify or suppress contributions to the phonon linewidth for electronic states at different k.This -selectivity has not yet been considered in the analysis of EPC.We show that it can have a profound impact on the lattice dynamical properties as exemplified by the behavior of the phonon broadening in YNi2B2C.

Lattice dynamics in YNi2B2C
YNi2B2C offers unique insights into the interplay between electronic and lattice degrees of freedom via superconductivity-induced phonon anomalies 29,31,36 .Phase competition of superconductivity with CDW order 34 , the role of FS nesting [37][38][39] as well as its lattice dynamics have been investigated extensively 29,40 including in our earlier work [31][32][33] .We first demonstrate that temperature and  dependence of phonon renormalization in YNi2B2C is inconsistent with either standard mechanism: FS nesting or anharmonicity.
Overall, ab-initio lattice dynamical calculations of YNi2B2C agree with phonon spectroscopy results at energies up to 160 meV with regard to both phonon energies and corresponding phonon linewidths due to the EPC [31][32][33] .The strongest phonon anomalies are observed in c-axis polarized transverse acoustic (TA) phonon modes 33 .One such mode at  = (0.5,0.5,0), the  point of the Brillouin zone (BZ), is measured at  = (0.5,0.5,7), i.e., in the BZ adjacent to  = (1,0,7) [Fig.2(a)].Another strong coupling TA mode is observed at  = (0.55,0,0), i.e., about halfway between the  and  = (1,0,0) points of the BZ.For simplicity, we call it here the /2 anomaly.We measured it at  = (0.45,0,7), i.e. close to the same BZ center as the  point phonon, i. [see Eq. ( 1)] can underlie a  dependence not reflected by the FS geometry.However, the matrix elements are expected to change only for temperatures on the order of an electron-volt corresponding to much higher values than those in Figure 2.
Anharmonic broadening of phonons is typically attributed to phonon-phonon scattering which is typically strong at elevated temperatures. 41,42At low temperatures anharmonic broadening can occur near or at structural phase transitions as it is argued for the case of the CDW transition in 2H-NbSe2 (TCDW = 33 K). 23In fact, it was argued that YNi2B2C is close to a structural phase transition related to the soft mode at  ≈ (0.5,0,0) only precluded by the onset of superconductivity. 34Hence, the large phonon linewidth just above   = 15.2K could still be anharmonic.However, previously reported superconductivity-induced changes of the line shape of the anomalous TA phonons rule out a non-electronic origin. 29This redistribution of phonon spectral weight 31 directly reflects the opening of the superconducting gap 2Δ (for more details see Fig. S1 and Supplementary Note 1).A model 36 which accurately describes these observations is purely EPC-based, which is possible only if phonon broadening in YNi2B2C occurs only related to electronic scattering, i.e., is EPC in nature.
In the following we present evidence that the strong TA phonon broadening in YNi2B2C originates from EPC that is strongly enhanced on parts of the Fermi surface, which are connected by the correct values of the phonon momentum.This scenario explains the results from phonon spectroscopy [see Figs.

Band structure from electron spectroscopy
There has been considerable discussion about FS nesting for  ≈ /2 in (Lu/Y)Ni2B2C including theoretical 34,37,43 and experimental work 38,39,44,45 .Yet, the clearest signature of electron phonon coupling, the   -induced phonon effect, appears over a large momentum region along the [100] direction 31 , which is inconsistent with nesting.In order to clarify the origin of this EPC, we performed SX-ARPES measurements on samples cut from the large YNi2B2C single crystal, which we used for our current and also previous INS studies [31][32][33] .We used both linear vertical, i.e.  , and linear horizontal, i.e.  polarization of the light in order to exploit photoemission matrix element effects. polarization is sensitive to the inner (outer) band for odd (even) values of   , whereas  polarization reveals inner and outer bands for both odd and even   .This is evident in Figures

Electronic joint density of states from density functional theory
The SX-ARPES study detailed above shows that the band structure of YNi2B2C is well described by our DFT calculations.Therefore, we rely on the analysis of the calculated FS (see Overall, our analysis demonstrates that the 2D-eJDOS is extremely sensitive to   and depends less on the phonon wave vector .

Momentum-dependent phonon broadening from inelastic neutron scattering
So far, we focused on the mechanism explaining the strong broadening of the two TA modes at  ≈ /2 and  = . While these effects are not sharply localized in  space, they do depend on the phonon wave vector 33 and only a rigorous experimental test enables an overall verification of the validity of the DFT calculations.
Therefore, we performed neutron scattering experiments with improved energy resolution at phonon wave vectors along as well as off the high-symmetry directions investigated previously 29,31,33 .We compare our results to abinitio calculations of    (see Eq. 1), which were calculated using the linear response technique or density functional perturbation theory (DFPT) 47 in combination with the mixed-basis pseudopotential method 48 .
For the present study, we measured phonons at wave vectors close to  = /2 and the  point at room temperature and  = 20 K and looked for peak broadening.We observed broad but clear momentum dependences, e.g., in measurements going away from the  point [see Fig. Finally, we note that the -averaged eJDOS is about 50% higher for the  point than for  ≈ /2 [Fig.-averaged eJDOS(q) associated with the nesting condition is responsible for a phonon anomaly.At increased temperatures, smearing of the Fermi function quickly suppresses this peak in the eJDOS(q) and reduces the phonon broadening. 18In our case, the peaks in the 2D − eJDOS ���������������  () [Figs.4(d)-(f)] are expected to also be strongly broadened and reduced in height with increasing temperatures.This is exactly what is observed in experiments.
Our results demonstrate that  selectivity of the EPC can boost the contribution of only a small part of the Fermi surface to the phonon lifetime.This new point of view has to be taken into account generally when assessing phonon anomalies in metallic systems.
This finding makes it necessary to reconsider our understanding of some well-known materials.NbSe2 is a model system for various fundamental aspects (phase competition as function of dimensionality 12 ; quantum phase transitions 54,55 ; 2D quantum metal state 56 ; Higgs mode in condensed matter physics 57,58 ).The correct view of EPC is indispensable ingredient for explaining many of the observed effects.In particular, NbSe2 is considered to be a showcase for -dependent EPC matrix elements featuring concomitant charge-density-wave (CDW) order (TCDW = 33 K) and phonon-mediated superconductivity (Tc = 7.2 K) 59 .Calculations 24 and experiments 25 have shown that FS nesting is absent and concluded that the periodicity of the CDW is determined by the momentum dependence of the EPC matrix elements.Thus, the phonon broadening and softening close to   is due to the EPC.However, lattice fluctuations overwhelm EPC away from   and explain the strongly reduced phonon renormalization at elevated temperatures 23,24 .We find that the calculated eJDOS across the CDW ordering wavevector   in 2H-NbSe2 is nearly featureless whereas the calculated linewidth due to EPC,    , shows a pronounced broad maximum around   in agreement with experiment 26 .The situation is, in fact, very similar to that observed for the /2 anomaly in YNi2B2C [see Figs.2(c) and 5(c)].In analogy to the results presented here, the strong temperature dependence of the phonon linewidth in 2H-NbSe2 could also originate from an interplay of -selective EPC and the electronic band structure.In fact, Flicker et al. 24 already considered an orbital-dependent EPC matrix element.Yet, it remains unclear whether the model calculations also support a strongly  dependent 2D-eJDOS.
Recently, phonon anomalies related to CDW order in cuprates have attracted large scientific interest.In particular, a strong phonon broadening was observed at and below the onset of CDW fluctuations competing with superconductivity 4,8,6 .The abrupt decrease of the phonon linewidth in YBa2Cu3O6.6 on entering the superconducting state 4 indicates a high-sensitivity to the opening of the superconducting energy gap on the FS mediated by the EPC.Yet, the mechanism determining the periodicity of the CDW remains unclear.The reported sharp  dependence of the phonon broadening is reminiscent of a FS nesting-type origin but the nesting could not be identified in the electronic band structure. selective EPC in concert with large 2D-eJDOS on a small part of the FS is a novel approach for an improved understanding of these observations.indicates that the phonon intensity was too weak to be observed -in agreement with DFPT structure factor calculations.Size of symbols (dots, circles) scale with the strength of the phonon renormalization within each panel.

Opening of the superconducting gap 2Δ observed in phonon spectroscopy
Below we explain in some more detail the superconductivity-induced changes of the phonon line shape observed for the TA phonon modes in YNi2B2C both at q = (0.5,0.5,0) [Fig.S1(a)] and q = (0.55,0,0) [Fig.S1(b)].Previously, some of us reported on this phenomenon in Ref. 1 .3][4] .experiments resulting in a sharp intensity increase at 2Δ.The predicted intensity increase corresponds to the one we observe at Es, i.e. 2Δ(T=2K) = 5.7 meV for the TA phonon at q = (0.5,0.5,0).In Ref. 1 , we were able to employ this signature of the superconducting gap in phonon spectroscopy in order to carefully determine the temperature dependence of 2Δ.
The phonon at q = (0.55,0,0) [Fig.S1 For the current subject regarding the phonon renormalization in the normal state of YNi2B2C, the superconductivityinduced changes of the phonon line shape are important.They demonstrate that the linewidth at low temperatures of T ≈ 20 K is sensitive to changes of the electronic states at the FS and, thus, EPC in nature and not related to anharmonic broadening as observed in 2H-NbSe2 6 .

Momentum dependent phonon renormalization
Phonon spectroscopy with INS is done at absolute wave vectors Q = τ + q, where τ denotes the center of a Brillouin zone (BZ) and q is the reduced wave vector within this BZ.Phonon energies and linewidths are defined within the first Brillouin zone (BZ) and, thus, do not depend on τ.In contrast, the phonon intensity varies strongly with τ.For example, the TA mode at the M point discussed in our work has a large intensity at Q = (0.5,0.5,7) [Fig.1(a)] but cannot be observed at (0.5,0.5,L) with L = 2,4,6. 7We note that the accurate determination of phonon linewidths requires sizeable intensities.Thus, we could only evaluate the temperature dependence of the phonon energy in some cases with weak scattering intensities.In other cases, even this was not possible anymore.
In Figure 4 of the main manuscript we show results only in reduced wave vectors q = (h,k,0) for the sake of clear assignments.The full momentum dependent investigation, however, can only be presented in absolute wave with ℎ = 0 − 0.4.The latter are the points given also in Fig. 4(a).We did not include more points in Fig. 4(a), since the assignment in units of the reduced wave vector q would have been unclear with regard to measurements done in the BZ adjacent to τ = (0,0,8) which are discussed below.
The detailed momentum dependence of the experimentally observed phonon renormalization is in good agreement with the predicted momentum dependence of the electronic contribution to the phonon linewidth , where the symbol size in Phonon lifetimes are reduced when electrons are excited to energies larger than   by phonon absorption and relax by phonon emission.Since the energy and momentum are conserved, only occupied electronic states in a small energy window   −  ℎ  can contribute to this process, where  ℎ  is the energy of a phonon mode with wave vector  in the dispersion branch  [Fig.1(b)].Reduced lifetimes translate into large line widths of phonon scattering spectra.Two quantities are relevant for determining phonon line broadening due to the EPC: (1) The likelihood that a particular phonon with  ℎ  is scattered/absorbed by a particular electron in a state with wave vector  and energy    , which is expressed as the EPC matrix element   �⃗ + �⃗, �⃗  � �⃗ .(2) The imaginary part of the electronic susceptibility   ′′ reflects the number of electronic states at the FS which are connected by , i.e. the eJDOS, which is defined as ∑ �  �⃗ −   ��  �⃗ + �⃗ −   �  �⃗ and equivalent to the so-called nesting function () 35,a .Both   �⃗ + �⃗, �⃗  � �⃗ and   are in general  dependent and, therefore, their interplay in q-space determines the q-dependence of phonon linewidths [Fig.1(c)].Equation (1) describes the contribution    to the phonon linewidth  ∝ (phonon lifetime) −   ��  �⃗ + �⃗ −   �  �⃗ e.  =(1,0,7) [Fig.2(b)].The /2 anomaly is the only part of the phonon dispersion, which is not very well captured by ab-initio calculations in that the observed anomaly is stronger than predicted33  .Key enigmatic results are (1) that strong phonon broadening is observed over a wide range of phonon wave vectors  = (ℎ, 0,0), 0.4 ≤ ℎ ≤ 0.75, i.e.  ≈ /2, but quickly vanishes going away from  = (0.5,0.5,0) along the[110] direction and (2) that broadening of the TA mode at /2 is much stronger than at the  point.Both TA phonons display a pronounced broadening upon cooling from room temperature to  = 20 K easily identified in high energy resolution INS data (see methods) [Figs.2(a)(b)].The new data allows for a detailed study of the momentum dependence of phonon broadening (see Results section and Figs. 5, S7-S9).The observed broadening at  ≈ /2 is much stronger than at the  point [Figs.2(a)(b)].In contrast, the calculated eJDOS 33 along the [100] and[110] directions displays a peak only for  = (0.5,0.5,0) [Fig.2(c)].Yet, the calculations correctly predict the broad momentum range along the [100] direction over which phonon broadening is observed 33 .Therefore, nesting is unlikely to be the origin of the observed strong phonon renormalization.On the other hand, EPC matrix elements   �⃗ + �⃗, �⃗ 2(a)(b)] in the absence of FS nesting or lattice anharmonicity.
ℎ ≥ 650 eV have a resolution, which is sharp enough to reliably resolve the   -dispersion effects in YNi2B2C.bWe compared the experimentally observed band structure with density functional theory (DFT) calculations carried out in the framework of the mixed basis pseudopotential method 46 using the local density approximation (LDA) (for details see 33 ).Selected cuts of the FS in the   −   plane in Figs.3(a)-(c) and the   −   plane in Fig.3(d)(e) overlaid with corresponding DFT calculations (solid blue lines), show a good agreement (see alsoFigs.S2 and S3 in SI).We focus on the vicinity of the  point, i.e. at  = (0.5,0.5,   ), since the DFT predicts a square shaped FS around it for certain   values prone to nesting 33 .Our results demonstrate the significant evolution of the FS at : An elliptical FS stretches between the four visible  points at   = 23 r.l. u. [Fig.3(a),ℎ = 693 eV], a nearly squareshaped feature is found at   = 23.5 r. l. u. [Fig.3(b), ℎ = 725 eV], and a large nearly circular ellipse is observed at   = 24 r.l. u. [Fig.3(c), ℎ = 758 eV] with the long axis rotated by 90° relative to the orientation at   = 23 r.l. u..Note that due to the particular symmetry of the BZ, the positions of  and  points are reversed upon moving from   = 23 r.l. u. to   = 24 r.l. u. [see Fig. 3(f)].The experimentally observed spectral weight with a square-like shape for   = 23.5 r. l. u. is relatively diffuse, in contrast to the sharp FS expected from the calculated Fermi contour [Fig.3(b)].This originates from two features of the band structure in the vicinity of the M point:(1) The   dispersion of the inner band is very strong[Fig.3(d)(e)   and Fig.S4(c)] and (2) both the inner and outer bands forming the square have a shallow in-plane energy dispersion of less than 0.5 eV near   [see Fig.S3(c)].Hence, due to the uncertainty of the inner potential V0 used in rendering photon energy into   and to the limited   and energy resolution of the experiment, we pick up intensity inside the square.
3(a)(c) where the spectral weight is located at the center and at the edges of the ellipse at the  point, respectively, with , but uniformly distributed with  [bottom right in Fig.3(a)].This is also visible on the   dispersion for   = 0.5 r.l.u.[Figs.3(d)(e)].With  polarization [Fig.3(d)], the experimental periodicity of the band structure looks like it is doubled in comparison to the calculations due to these matrix elements effects.In fact, we recover the correct periodicity using  polarization of the light [Fig.3(e)].Note that the data of Figure3(d) cut the  point in the   −   plane between the first and second BZ, while the data of Figure 3(e) cut the  point between the second and third BZ.However, Figures 3(a)-(c) indicate that variations of the photoemission matrix elements across different BZs are smaller than variations due to light polarization.
2(a)]: Already at close-by wavevectors, the broadening is clearly reduced [Fig.5(a)] and completely absent further away in momentum space [Fig.5(b)].More INS data taken along different directions for both anomalous modes are shown in Figure S7 of SI.The experimentally observed  dependence of phonon broadening [symbols in Fig. 5(c)] is in good agreement with the calculated one of    [color code in Fig. 5(c)].Similar agreement is found for phonon wavevectors with a nonzero l component (Figs.S8 and S9 and Supplementary Note 2).
2(c) and 4(a)], whereas the 2D − eJDOS ���������������  (  = 0.5 r. l. u. )[Figs.4(c),(e)(f) and Fig.S6] and   [Fig.5(c)] have similar values at these two wave vectors.This is further evidence that the electronic states at the Fermi level with   = 0.5 r. l. u. are selected by EPC matrix elements and mediate the coupling responsible for the broadening of the TA phonons in YNi2B2C.DiscussionEPC has been studied for a long time because it can stabilize emergent ground states as well as lead to phase competition.The particular role/impact of EPC matrix elements was discussed already in the 1970s,49,50  though experimental evidence for the decisive role of the momentum dependence of the EPC matrix elements was only reported in the last decade23,25,26,[51][52][53] .Still, only the  dependence of the EPC matrix element   �⃗ + �⃗, �⃗  � �⃗ was scrutinized.Our work takes the full  and  momentum dependence of   �⃗ + �⃗, �⃗  � �⃗ into account.We can explain not only phonon broadening due to EPC as a function of  but also its pronounced weakening at increased temperatures in the absence of FS nesting and anharmonicity.The peak of the 2D − eJDOS ���������������  as function of   [Figs.4(d)-(f)] accounts for a high sensitivity to temperature via the electronic band structure if the contribution of the electronic states with   = 0.5 r. l. u. to EPC is boosted by  -selective matrix elements.The peak of the 2D − eJDOS ���������������  () [Figs.4(d)-(f)] is superficially reminiscent of classic FS nesting scenario where the peak in the

Figure
Figure S1(a) shows the evolution through Tc=15.2K of the low temperature line shape of the TA phonon mode at the M point.On cooling from T = 300 K to 20 K, this phonon softens and broadens substantially, indicative of a strong EPC [see temperature dependences given in Figs.1(c) and S1(c)].However, the line shape remainsLorentzian to a very good approximation.On further cooling through Tc, the line shape starts to deviate strongly from a Lorentzian.In particular, a step-like increase at a certain energy Es ≈ 5.7 meV appears in the spectrum taken at T = 2 K [Fig.S1(a)].According to the theory developed by Allen et al.5  , a part of the low energy tail which lies below the value of the superconducting gap 2Δ is pushed up in energy to form a narrow spike at 2Δ.The theory is based on the full quantum mechanical treatment of EPC where vibrational and electronic excitations mix into hybrid modes.The finite spectrometer resolution should wash out the theoretically predicted spike in (b)] has a significantly lower energy than the M point phonon discussed above [see Fig.1(c)] and, moreover, has a much larger linewidth [Fig.S1(c)].As a consequence, the superconductivity-induced redistribution of spectral weight is even stronger, but the superconducting gap cannot be inferred from the data as easily, except for temperatures close to Tc where the gap is still small.For lower temperatures, most of the spectral weight condenses into a fairly sharp peak [Fig.S1(b)] whose energy isaccording to theory -somewhat below 2Δ.According to Ref.5  , this resonance can be regarded as a mixed vibrational or superelectronic collective excitation.Its position with respect to 2Δ has to be determined from calculations for the parameter values of this particular phonon.Again, the theory reproduces the observed line shapes sufficiently well to precisely determine the gap value.