Exploration of the Hg-based cuprate superconductors by Raman spectroscopy under hydrostatic pressure
N. Auvray, B. Loret, S. Chibani, R. Grasset, Y. Guarnelli, P. Parisiades, A. Forget, D. Colson, M. Cazayous, Y. Gallais, Alain Sacuto
EExploration of the Hg-based cuprate superconductors by Raman spectroscopy underhydrostatic pressure
N. Auvray , B. Loret , S. Chibani , R. Grasset , Y. Guarnelli , P.Parisiades , A. Forget , D. Colson , M. Cazayous ,Y. Gallais , A. Sacuto ∗ Universit´e de Paris, Laboratoire Mat´eriaux et Ph´enom ` e nes Quantiques,CNRS (UMR 7162), 75013 Paris, France Laboratoire des Solides Irradi´es,Institut Rayonnement Mati ` e re de Saclay (IRAMIS),CEA, Ecole Polytechnique,91128 PALAISEAU Cedex, France Sorbonne Universit´e, Institut de min´eralogie,de physique des mat´eriaux et de cosmochimie,CNRS/MNHN/IRD (UMR 7590), 75005 Paris, France Universit´e Paris-Saclay, CEA, CNRS,SPEC, 91191, Gif-sur-Yvette, France (Dated: March 1, 2021)The superconducting phase of the HgBa CuO δ (Hg-1201) and HgBa Ca Cu O δ (Hg-1223)cuprates has been investigated by Raman spectroscopy under hydrostatic pressure. Our analysisreveals that the increase of T c with pressure is slower in Hg-1223 cuprate compared to the Hg-1201due to a charge carrier concentration imbalance accentuated by pressure in Hg-1223. We find thatthe energy variation under pressure of the apical oxygen mode from which the charge carriers aretransferred to the CuO layer, is the same for both the Hg-1223 and Hg-1223 cuprates and it iscontrolled by the inter-layer compressibility. At last, we show that the binding energy of the Cooperpairs related to the maximum amplitude of the d − wave superconducting gap at the anti-nodes, doesnot follow T c with pressure. It decreases while T c increases. In the particular case of Hg-1201, thebinding energy collapses from 10 to 2 K B T c as the pressure increases up to 10 GPa. These directspectroscopic observations joined to the fact that the binding energy of the Cooper pairs at theanti-nodes does not follow T c either with doping, raises the question of its link with the pseudogapenergy scale which follows the same trend with doping. I. INTRODUCTION
High- T c cuprate superconductors are one of the iconicquantum materials [1, 2]. Although discovered more than35 years ago, the complexity of their physics remains mis-understood. It calls for new concepts where the orders ofmatter are no longer independent of each other as in tra-ditional materials but they are intertwined [3]. In orderto understand their physics, many studies have alreadybeen carried out as a function of temperature T and car-riers concentration via the hole doping, p , leading to their( T − p ) phase diagram [2, 4]. It presents an insulatinganti-ferromagnetic phase at low doping. As the dopingincreases, an intermediate phase between the insulatorand the metal called the pseudogap phase emerges be-low T ∗ which harbors many orders of matter that appearto be interconnected. Some of them break translationalinvariance (charge density wave order), others time re-versal invariance (current loops order) or C rotationalinvariance (nematic order)[5–14]. At lower temperaturebelow a critical temperature T c , the superconductingphase arises. T c exhibits a dome like shape. The top ofthe dome called the optimal doping, separates the under-doped from the over-doped regime. The physics behindthis phase diagram remains widely debated and calls forthe development of new experiments and theoretical in-vestigations [3, 4, 15–21]. In order to get a better understanding of the cupratesphysics and in particular their superconducting phase, wecarried out Raman scattering measurements under hy-drostatic pressure at low temperature on the Hg-basedcuprates which have the most spectacular variation of T c with pressure. They exhibit an increase in T c of morethan 25 K for a pressure of 25 GPa, i.e. on average, anincrease of 1 K per GPa [22–25]. We will focus on theHgBa CuO δ (Hg-1201) and HgBa Ca Cu O δ (Hg-1223) compounds which have respectively a maximum T c ≈
95 K and ≈
135 K at ambient pressure ( ≈ planes, surroundedby blocks made up of HgO and BaO layers (cf. Fig. 1).These blocks are called charge reservoirs because the in-troduction of oxygen atoms within the HgO plane gen-erates a charge transfer via the BaO plane towards theCuO . This oxygen doping introduces hole charge carri-ers in the CuO plane [25, 28–30].In the cuprates Raman scattering has been used ex-tensively to track the energy scales of the phonons, themagnetic excitations, the superconducting gap, the pseu-dogap [31–39] or more recently the charge density wavegap [40–42]. Since Raman is a two photon scatteringprocess, by controlling the incoming and outgoing pho-ton polarizations, one can selectively probe both the lat-tice and the electronic excitations in different symme- a r X i v : . [ c ond - m a t . s up r- c on ] F e b CuOBaHg
CuOBaHgCa
O3 O2O1 O3 O2O1O4
Hg-1201 Hg-1223 O4 FIG. 1. Schemtic representation of the tetragonal crystalstructures (a) Hg-1201 and (b) Hg-1223. the oxygen atomsO1 and O4 are respectively related to the inner and outerCuO planes, O2 are the apical oxygen atoms, O3 are theoxygen atoms in excess inserted by the chemical doping inthe HgO plane. tries. In the superconducting phase, we will be focus-ing on the relationships that can be unveiled with pres-sure between lattice dynamics and T c and also betweenthe binding energy of the Cooper pairs associated withthe d -wave superconducting (SC) gap and T c . We willshow that the increase of T c with pressure is reduced inHg-1223 below 10 GPa in comparison to the Hg-1201compound due to an increase of the charge carriers con-centration imbalance between the inner and outer CuO planes of Hg-1223 with pressure. We find that the evolu-tion under pressure of frequency of the the apical oxygen,by which the charge transfer takes place, is mainly con-trolled by the inter-layer contraction. At last, we showthat the binding energy of the Cooper pairs related to themaximum amplitude of the d − wave SC gap along theprincipal axis of the Brillouin zone (BZ) called the anti-nodal region, does not follow T c with pressure. It decreasewhile T c increases with pressure. Our findings, togetherwith previous investigations that showed the binding en-ergy of the Cooper pairs at the anti-nodes decreases as T c increases with doping and follows the same trend asthe pseudogap energy scale [34, 41, 43–47], raise the ques-tion of its link to the pseudogap energy scale. II. EXPERIMENTAL METHODSA. Crystal growth and characterization
The crystals used for Raman measurements under hy-drostatic pressure were prepared close to the optimaldoping where the SC transition temperature T c is max-imum. The Hg-1201 and Hg-1223 single crystals weresynthesized and annealed following the method describedin [48] and [49] respectively. We sieved the batch immedi-ately after annealing, in order to select crystals between 100 and 200 µ m in size. Samples with a naturally cleansurface were directly selected. Their critical temperature T c have been determined from DC magnetization suscep-tibility measurements under classical zero field cooling(ZFC) on a set of crystals of the same batch. A PPMSmagnetometer (Quantum Design) was used and a mag-netic field of 10 Gauss was applied. The DC magnetiza-tion curves of the Hg-1201 and Hg-1223 single crystalsat ambient pressure are displayed in Fig. 2 (a) and (b).The transition temperature T c and its width, ∆ T c , wereestimated by taking the maximum and the full width athalf maximum of the peak of the first derivative of eachDC magnetization curves, giving T c = 92 ± K and T c = 132 ± K for Hg-1201 and Hg-1223 respectively.The two sets of samples are slightly under-doped andwe name the ones selected for the Raman measurementsunder pressure (UD92K) for Hg-1201 and (UD132K) forHg-1223 respectively. - 4 . 0 x 1 0 -3 - 3 . 0 x 1 0 -3 - 2 . 0 x 1 0 -3 - 1 . 0 x 1 0 -3 - 2 . 0 x 1 0 -3 - 1 . 0 x 1 0 -3 T ( K ) M (emu)
H g - 1 2 2 3 ( b ) -4
02 x 1 0 -4 -4 -4 dM/dT T ( K ) T C = 1 3 2 K ( a ) H g - 1 2 0 1
M ( emu) -4 -4 -48 0 8 5 9 0 9 5 1 0 002 x 1 0 -4 -4 -4 T C = 9 2 K dM/dT T (K )
FIG. 2. DC magnetization curves as a function of temperatureof two sets of (a) Hg-1201 and (b) Hg-1223 single crystals.The first derivatives of the magnetization curves are reportedin the insets to underline the T c values and their full width athalf maximum. B. Polarized Raman experiments underhydrostatic pressure at low temperature
We have performed two different runs of Raman mea-surements under pressure for studying the Hg-1201 andHg-1223 compounds. Crystals were loaded inside a di-amond anvil cell with diamonds of diameter 800 µ m de-signed to withstand up to 10 GPa. The chamber betweenthe diamonds is a cylindrical hole, cut by laser througha stainless steel gasket. Rubies were added in the cellto act as in-situ manometers through their fluorescence.Using several rubies allowed us to control the uniformityof the hydrostatic pressure in the chamber. After load-ing the chamber with ultra-pure helium and sealing it,the chamber size is approximately 300 µ m of diameter by50 µ m of height (cf. Fig. 3). The diamond anvil cell was µ m300 µ m rubycrystal FIG. 3. Top view of the diamond anvil cell with diamondsof 800 µ m diameter before loading. The chamber between thediamonds is a cylindrical hole. After loading the chamber sizeis approximately 300 µ m (dashed circle). then installed in a cryostat including a helium inlet allow-ing us to tune in-situ the pressure applied on the lowerdiamond, which indirectly changes the pressure in thechamber. Raman measurements were performed throughthe Boehler-designed upper diamond of the anvil, with a532 nm laser wavelength and 4 mW of power (measuredbefore going through the cryostat windows and the dia-mond). Experiments were carried out using a JY-T64000spectrometer in triple grating (1800 grooves/mm) config-uration. The spectrometer is equipped with a nitrogenCCD detector. All the Raman spectra have been cor-rected for the Bose factor and the instrumental spectralresponse. They are thus proportional to the imaginarypart of the Raman response function χ (cid:48)(cid:48) ( ω, T ). The Ra-man responses in the different symmetries are obtainedfrom incoming and outgoing light polarizations. TheB g and B g symmetries were obtained respectively fromcrossed polarizations of the incoming and outgoing lightat 45 degrees and along the Cu-O bond direction of theCuO plane. The (A g + B g ) geometry was got fromparallel polarizations of the incoming and outgoing lightsalong the Cu-O bond direction. Raman scattering mea-surements at ambient pressure ( ≈ III. OVERVIEW OF THE SUPERCONDUCTINGAND NORMAL RAMAN RESPONSES OFHG-1201 AND HG-1223 CRYSTALS ATAMBIENT PRESSURE
Our first objective is to disentangle the electronic ex-citations from the Raman-active optical phonons in or-der to study the electronic signatures of the supercon-ducting phase of the Hg-1201 and Hg-1223 under hydro-static pressure. The superconducting Raman responsesat ambient pressure of a slightly under-doped (UD92K)Hg-1201 and an optimally doped (OP133K) Hg-1223 sin-gle crystals in three distinct geometries are reported in Fig. 4. They are made up of a broad electronic back-ground superimposed by narrow peaks due to opticalphonons. We focus first on the electronic background.
200 400 600 800
024 102030
500 1000 1500
UD 92 A + B Hg-1201
10K 100K (a) (c) '' A + B ( T ) ( a . u ) '' A + B ( T ) ( a . u ) '' B ( T ) ( a . u ) '' B ( T ) ( a . u ) B
10K 100K X2 (e) Raman shift (cm -1 ) Raman shift (cm -1 ) X2 B
10K 100K (d) B X2
11K 150K (f) '' B ( T ) ( a . u ) B X2 '' B ( T ) ( a . u )
11K 150K
OP 133 (b) A + B Hg-1223
11K 150K
130 260 400483 585
FIG. 4. Superconducting and normal Raman responses of theUD92K Hg-1201 and OP133K Hg-1223 in three distinct ge-ometries at ambient pressure. The straight lines and numbersindicate the locations of the vibrational modes. Dashes linesindicate the A g and B g contributions. A. Electronic part of the Raman response
Electronic Raman scattering is a particularly usefulprobe for studying the cuprates because we can selectdistinct parts of the BZ, the anti-nodal and nodal re-gions well known to have quite different electronic prop-erties [50]. In the B g geometry the Raman form fac-tor is (cos k x − cos k y ) and it predominantly probes theanti-nodal region where the superconducting gap and thepseudogap are maximal. Here k is the wave vector of theexcited electron. Likewise, in the B g geometry the Ra-man form factor is sin k x sin k y and it probes mostlythe nodal region where the superconducting gap and thepseudogap are minimal. In the A g geometry, the Ra-man form factor is more isotropic, with no symmetry-imposed nodes. Experimentally, we cannot access thepure A g component using linear polarizations, as it isalways associated with either a B g or B g component.The (A g + B g ) SC Raman spectra of (UD92K) Hg-1201and (OP133K) Hg-1223 single crystals (red curve at 10 Kin Fig. 4 (a) and (b)) show an extended hump in energymade up of two broad peaks, the A g and B g peaks. TheA g and B g peaks are respectively located around 400cm − and 600 cm − in the Hg-1201 spectrum and centeredaround 500 cm − and 800 cm − in the Hg-1223 spectrum.These features are indicated by gray dashed lines in thespectra and extensively studied in previous works [34, 51–53]. The B g SC peak alone are displayed in Fig. 4 (c) and(d). It corresponds to the pairs breaking peak related tothe maximum amplitude of the d − wave SC gap opening.No clear SC peak is detected in the B g Raman spectra ofHg-1201 and a relatively weak peak (close to 780 cm − )compared to the B g peak is detected in the B g Ramanspectra of Hg-1223 (cf. Fig. 4 (e) and (f)). This is ex-pected since the SC gap vanishes out in the nodal regionsprobed predominantly by the B g geometry. In particu-lar, the remarkably flat B g SC Raman response of the(UD92K) Hg-1201 will be exploited later for the elec-tronic Raman measurements under pressure. Notice that B g spectra of Hg-1223 contains an extra electronic con-tribution (around 580 cm − ) stemming from the chargedensity wave order as already discussed in our previousinvestigations [40, 41].
01 02 03 0
01 02 0 ( c )
R a m a n s h i f t ( c m - 1 ) H g - 1 2 2 3
U D 1 1 7 c ’’B1g( w , T) (a.u) R a m a n s h i f t ( c m - 1 ) O P T 1 3 3 B c ’’B1g( w , T) (a.u) ( a )( b ) H g - 1 2 2 3 9 3 58 1 0
U D 1 3 1 c ’’B1g( w , T) (a.u) ( f ) c ’’B1g( w , T) (a.u) c ’’B1g( w , T) (a.u) H g - 1 2 2 3
U D 1 1 7 c ’’B1g( w , T) (a.u) ( d ) B H g - 1 2 2 3
O P T 1 3 3 ( e )
H g - 1 2 2 3
U D 1 3 2
FIG. 5. Double structure of the B g superconducting gap ofthe Hg-1223 (a)-(c) as a function of doping (d)-(f) as functionof temperature for each doping. Dotted lines are guides forthe eyes. A more detailed analysis reveals that the B g superconducting peak in the Raman spectrumof the three-layer Hg-1223 cuprate presents a shoulder(around 700 cm − ) on its left side that we do notobserve in the Raman spectrum of the one-layer Hg-1201cuprate (see panels (c) and (d) Fig. 4). To have a betterunderstanding of its origin, we followed the evolutionof this shoulder as a function of doping. The shoulderthat we observe close to the optimal doping turns intoa double peak with under-doping. This is displayed inFig. 5 (a)-(c). The frequency difference between the twopeaks increases with under-doping. These two peakscould be related to two superconducting gaps inducedeither by an inter, or by an intra-unit cell dopinginhomogeneity of Hg-1223. Our experimental findingstend towards an intra-unit cell doping inhomogeneitydue to a charge carrier concentration imbalance between the inner and the outer CuO planes of one singleunit cell. Indeed, the Raman spectra measured on theHg-1223 crystal for a given doping level, are the samewhatever the location of the laser spot on the crystalsurface. This means there is no trace, at least at thescale of few ten microns, of inter cell inhomogeneity ofthe oxygen doping. Secondly, we always detect only twodistinct superconducting peaks (cf. Fig. 5 (a)-(c)) andnot more, which seems inconsistent with a distributionof spatial oxygen content over the area illuminatedby the laser spot. The observation of only two peakswhich deviate from each other in energy as the dopingdecreases is rather in favor of the existence of two SCgaps linked to the inner and outer CuO planes whichwould have different carriers concentrations. This is inagreement with previous works on Bi-2223 three layerscuprate [54, 55]. Thirdly, the two SC peaks detected inthe Raman spectra seem to disappear simultaneously(within our temperature accuracy) as the temperatureis raised (cf. Fig. 5(d)-(f)). This suggests that the twogaps are interconnected as it should be the case if theyare originate from the inner and outer plane of thesame unit cell and obviously not if the two gaps comefrom distinct regions of the crystal with different oxygencontents. The inner plane being more distant from thereservoir blocks by which the charge transfer takes place,earlier studies have concluded that the charge carriersconcentration of the CuO outer plane is higher thatthe one of the inner plane in the under-doped Hg-1223[29, 30, 56–59]. The B g superconducting gap energyis known to be larger upon reduced doping in Hg-1223[40, 41]. We can thus assign the high energy peak to theSC gap related to the inner plane and the lower one tothe SC gap of the outer plane. B. Phononic part of the Raman response
In Fig. 4 (a) and (b)) few narrow peaks (marked bystraight lines) are superimposed to the (A g + B g )electronic background. They correspond to Ramanactive phonons associated with the Hg-1201 and Hg-1223 structures as well as parasitic phases. Overall,we detect three types of phonons. The first type isassociated with the vibrational modes of the ideal stoi-chiometric Hg-1201 and Hg-1223 structure that belongto the D h space group. We then expect 2 A g + 2 E g and 5 A g + 1 B g + 6 E g pristine Raman active even(gerade) modes for the Hg-1201 and Hg-1223 structurerespectively. The second kind of vibrational modesis associated with defect stemming from symmetrybreaking induced by insertion of oxygen atoms inthe HgO layers which makes Raman active some odd(ungerade) modes. At last, the third kind of vibrationalmodes come from parasitic phases which are depositedon the crystal’s surface. They stem from residualoxides of synthesis precursor phases that subsistseven in the cleanest crystal surfaces we could select.A summary of these three kinds of phonons observedin Hg-1201 and Hg-1223 structures is reported in Table 1. IV. OVERVIEW OF THE SUPERCONDUCTINGAND NORMAL RAMAN RESPONSES OFHG-1201 AND HG-1223 CRYSTALS UNDERHYDROSTATIC PRESSURE
The A g + B g Raman superconducting responses of(UD92K) Hg-1201 and (UD132K) Hg-1223 single crys-tals measured inside the anvil cell at 0.4 GPa are reportedin Fig. 6. We have nearly the same spectra as those ofFig. 4 but with a poorer signal-to-noise ratio and few ad-ditionnal parasitic phonons (at 570 and 620 cm − ). Wesee an asymmetric hump with a maximum around 400cm − in the Hg-1201 which is made of two components(cf. Fig. 4). The Hg-1223 spectrum exhibits two broadpeaks, one centered around 480 cm − and the second onearound 760 cm − which are assigned the A g and B g components of the spectrum (cf. Fig. 4). On the topof these broad electronic peaks few narrow peaks associ-ated with vibrational modes are detected. The pristine A g + B g phonons related to the oxygen motions of theHg-1223 and Hg-1201 structures are indicated by blackstraight lines in panel (a) and (b) of Fig. 6.
01 02 0
01 02 03 0
R a m a n s h i f t ( c m - 1 )R a m a n s h i f t ( c m - 1 )R a m a n s h i f t ( c m - 1 ) ( a ) c ’’A1g+B1g( w , T) (a.u) A + B c ’’A1g+B1g( w , T) (a.u) H g - 1 2 0 1U D 9 2H g - 1 2 2 3U D 1 3 2 A + B T = 1 4 K ( b )
O 4 ’ O 2O 4T = 4 K c ’’A1g+B1g( w , T) (a.u) A + B ( c )
8 G P a7 G P a6 G P a5 G P a4 G P a3 G P a2 G P a1 G P a1 0 G P a0 . 4 G P a
H g - 1 2 2 3
9 G P a6 G P a5 G P a3 G P a2 G P a1 G P a c ’’A1g+B1g( w , T) (a.u) H g - 1 2 0 1 A + B ( d ) FIG. 6. Superconducting Raman responses in the A g + B g geometry of (a) the Hg-1201 and (b) Hg-1223 crystal at 0.4GPa. Zoom on the frequency range of the A g + B g Ramanspectra of (c) Hg-1201 and (d) Hg-1223 compounds under hy-drostatic pressures. The framed frequencies mark the pristineoxygen modes (see Table 1). The dotted lines are guides forthe eyes.
A. Study on the oxygen pristine modes of Hg-1223and Hg-1201 with pressure.
We displayed in Fig. 6 (c) and (d) a spectral rangezoom of the Raman spectra of panels (a) and (b). Thefrequencies of the oxygen pristine modes of the Hg-1223at 260, 483 and 585 cm − (see Table 1) increase with pres-sure (see red dotted lines in panel (c)). The feature cen-tered around 380 cm − is probably a parasitic mode (seeTable 1). In the Hg-1201 spectra (panel (d)), the apicaloxygen mode at 594 cm − (framed) increases with pres-sure (see red dotted line). It is located in between twoparasitic modes at 570 and 630 cm − (see Table 1) whosethe full width at half maximum is three time larger.These two peaks disappear at high pressure presumablybecause the structure of the parasitic phases evolves un-der pressure. We can also notice that two peaks at 470and 550 cm − appear with pressure (see arrows). Theyare likely defect modes (see Table 1) that come out due tomechanical stress which induce a redistribution of mobileoxygen in the HgO layer [68]. The normalized frequenciesof the oxygen pristine modes of the Hg-1223 and Hg-1201structure as a function of pressure, P , are displayed inFig. 7 TNC=Tc(P)/Tc(0) w N = w ( P ) / w ( ) P r e s s u r e P ( G P a )
O 2 ( H g - 1 2 0 1 ) O 2 ( H g - 1 2 2 3 ) O 4 ( H g - 1 2 2 3 ) O 4 ’ ( H g - 1 2 2 3 ) T
N C ( H g - 1 2 2 3 ) T
N C ( H g - 1 2 0 1 )
FIG. 7. The normalized frequencies of the oxygen vibrationalmodes of Hg-1223 and Hg-1201 as a function of pressure P .The normalized frequency is obtained by dividing the fre-quency at pressure P by this value at ambient pressure. Thenormalized critical temperature T c N = T c ( P ) /T c (0) is alsoreported. The T c ( P ) values come from ref. [25, 69]. A clear trend emerges: the normalized frequencies ofthe oxygen vibrational modes of the Hg-1223 and Hg-1201 compounds increase almost linearly with pressureup to 7 GPa. This behavior is usually expected with pres-sure [70–73]. The CuO layer of Hg-1201 being a sym-metry plane (cf. Fig. 1), the vibrational modes relatedto the (O4) and (O4’) oxygen motion in the CuO arenon Raman active in this structure. Therefore, we willfocus on the (O2) vibrational mode which is present bothin the Hg-1201 and Hg-1223 structure. Remarkably, theslopes ( dω N dP ) of the apical oxygen (O2) phonon over theentire pressure range are nearly the same in the Hg-1201 Hg-1201
Pristine mode 165 (Ba) 594 (O2)Defect mode 130 (Cu-Ba-Hg’) 260 (O1’) 461 (O1) 542 (O3)
Hg-1223
Pristine mode 120 (Ba) 260 (O4’) 483 (O4) 585 (O2)Defect mode 130 (Cu-Ba-Hg’) 400 (O1)Parasitic mode 205 310 380 570 625(Hg-Ba-O) (Ca-Cu-O) (Hg-O) (Ba-Cu-O) (Ba-Cu-O)TABLE I. Enumeration of the A g + B g vibrational modes (cm − unit) detected in the Raman spectra of Hg-1201 and Hg-1223single crystals. The pristine modes are all associated with vertical motions along the c -axis. The prime denotes counter phasedisplacement. The lattice dynamical calculation can be found in ref.[60] and previous attempts for assigning the Raman activemodes in Hg-1201 and Hg-1223 structures can be found in ref.[61–64]. Some of defect modes and parasitic phases can be foundin earlier works [61–67]. The (Ba-Cu-O), (Ca-Cu-O) and (Hg-O) compositions are generic terms which can involve differentparasitic phases namely: BaCuO , Ba CuO , Ba Cu O / , Ca CuO and HgBaO . and Hg-1223 structures (see red and green full squares).On the contrary, the slopes of the normalized T c , dT c N dP associated with the Hg-1201 and Hg-1223 compounds aredifferent (see red and green open stars). This means thatthe pressure evolution of T c in these structures cannot beattributed to the apical oxygen mode dynamics alone.The number of CuO planes and their respective chargecarrier concentration must also be taken into account. Inparticular, previous investigations [29, 30, 56, 58, 59, 74]revealed that the carriers concentration imbalance be-tween the CuO of multilayer superconductors such asHg-1223 was limiting the increase of T c . This can bevisualized in Fig. 7. The T c N slope of Hg-1201 is muchhigher than the Hg-1223 likely due to the difficulty toperform an efficient charge transfer by pressure to theinner plane of Hg-1223 as it will be shown in the nextsection.At this stage, it is interesting to make a comparison be-tween the effects of pressure (P) and doping (p) on thedynamic of the apical oxygen mode in the Hg-1201 andHg-1223 structure. The apical oxygen atom (O2) is thebridge between the HgO and the CuO planes (see Fig. 1)and its motion is along the c axis. The apical oxygenmode hardens with pressure (cf. Fig. 7) while it softenswith doping [48, 49, 62, 63]. Its softening is due to itscoupling with charge carriers which increases with dop-ing [75–77] and is a common feature of many cupratessuch as Hg-1201, Y-123 and Hg-1223 [48, 49, 62, 78–80]. On the other hand, its hardening with pressure islikely due to the inter-plane contraction along the c axis[58, 81] which is much stronger with pressure than withdoping. A way to compare the c axis contraction underpressure and doping is to evaluate it as a function of the T c change. According to the refs.[25, 28, 48, 49, 58, 81],the ∆ c ∆ T c ( P ) ratio is approximately one order of magnitudelarger than the ∆ c ∆ T c ( p ) . Our estimate is ∆ c ∆ T c ( P ) ≈ − ˚AK − and ∆ c ∆ T c ( p ) < − ˚AK − . This justifies why theapical oxygen mode hardens under pressure even if apply-ing pressure favor the charge transfer and an increase ofthe charge carriers which softens the apical oxygen mode in the case of chemical doping [58, 69, 82]. Despite thesecontrasting trends in the dynamics of the apical modewith pressure and doping, understanding the roles thatpressure and doping play on T c remains a major issue.We can cite two emblematic experimental facts that showtheir effects are different on T c . First, applying a pres-sure on an under-doped Hg-1212 or Hg-1223 compoundallows us to reach a much higher T c than that which canbe obtained by doping [69]. Secondly, starting from op-timally doped Hg-1201 and Hg-1223, T c decreases withdoping (over-doped regime) while it increases with pres-sure [22, 25]. Consequently, the effects of pressure anddoping on T c are complex and additional parameters tothe dynamics of the apical mode have to be consideredsuch as the shortening of the oxygen bond lengths in theCuO plane according to ref. [83]. B. Study on the bare electronic superconductingRaman response of Hg-1223 and Hg-1201 withpressure.
We focus now on the superconducting electronic Ra-man response under pressure. This is achieved by sub-tracting the most intense phonons after fitting them bylorentzian profiles [84]. The bare electronic Raman re-sponses in B g and B g geometries of Hg-1223 and Hg-1201 free of phonons are reported in Fig. 8. In AppendixA are shown the raw data with phonons.The B g electronic response of the UD132 Hg-1223compound (cf.Fig. 8 (a)) exhibits a hump which decreasesin energy with pressure. This hump is the SC pair break-ing peak and only appears below T c as shown in the Ra-man spectra at low and high pressure in Fig. 5 (b) andFig. 10 (a) respectively. The pair breaking peak centeredaround 2∆ ≈
800 cm − at 0 .
51 0 ( c )
H g - 1 2 0 1
U D 9 2 B
051 0
8 G P a 1 0 G P a 4 G P a 5 G P a 6 G P a 7 G P a c ’’B1g( w , T) (a.u) H g - 1 2 2 3
U D 1 3 1 B ( a )
51 0 B ( d ) R a m a n s h i f t ( c m - 1 )R a m a n s h i f t ( c m - 1 ) ( b ) c ’’B2g( w , T) (a.u) B FIG. 8. Sets of Hg-1223 and Hg-1201 Raman responses free ofphonons under hydrostatic pressures in B g and B g geome-tries. The sets of Hg-1223 and Hg-1201 data were obtainedat 4 K and 14 K respectively. (a)). This double component has already been identified(in section III) as the two gaps associated with the innerand outer CuO planes of the Hg-1223 structure whenthese two planes do not have the same charge carrierconcentration. As the pressure increases up, the chargecarrier concentration imbalance is accentuated betweenthe inner and outer planes. This makes it possible todetect again (but this time inside the cell) the splittingof the hump related to two superconducting gaps. Thiscan be seen by the two red trails in the color map whichcorrespond to the two SC peaks, see Fig. 10 (e). The B g Raman spectra of Hg-1223 under pressure and lowtemperature presents also a hump associated with a SCpeak (see Fig. 8 (b)). It corresponds to the weaker SCgap feature associated with the nodal region (see sec-tion III). This hump is more and more visible as pressureincreases and its frequency slightly increases as the pres-sure increases. This is highlighted in the colour map ofFig. 11 (e).We focus now on the B g SC Raman response functionof the Hg-1201 compound in Fig. 8 (c). The B g Ramanresponse at 0.4 GPa (red curve) exhibits an unexpectedbroad peak centered around 380 cm − in addition to theexpected B g pair breaking peak at 600 cm − (cf.Fig. 4 ( b ) - 1 ) c ’’B1g( w , T) (a.u) Pressure (GPa) R a m a n s h i f t ( c m - 1 ) ( e )
6 G P a1 0 G P a ( a ) ( c )
4 G P a ( d )
FIG. 9. (Color online) B g Raman response in Hg-1223 un-der pressure at T = 4 K. (a)-(d) Electronic Raman responsefeaturing the superconducting peak at various pressures. Atmaximum 10 GPa pressures, the normal state response above T c is also displayed in order to underlined the superconduct-ing peak. (e) Contour plot highlighting the evolution of the B g spectral weight as a function of pressure. The dashedlines are guides for the eyes. ( a ) ( c )
2 G P a ( b )
6 G P a ( e ) c ’’B2g( w , T) (a.u) R a m a n s h i f t ( c m - 1 ) Pressure (GPa)
R a m a n s h i f t ( c m - 1 ) ( d )
0 G P a
FIG. 10. (Color online) B g Raman response in Hg-1223 un-der pressure at T = 4 K. (a)-(d) Electronic Raman responsefeaturing the superconducting peak. (e) Contour plot high-lighting the evolution of the B g spectral weight as a functionof pressure. The dashed line is a guide for the eyes. (c)). This extra peak is due to a leakage of an A g con-tribution to the B g Raman response. It likely comesfrom an increase in the birefringence of the pressurizeddiamonds. The B g and B g Raman spectra having beenmeasured one after the other for each pressure value,this leakage also exists in the B g Raman spectra. The B g Raman response of Hg-1201 (cf. Fig. 4 (e)), beingalmost flat, here, it is mostly dominated by the leak-age of the A g contribution and of an extrinsic back-ground common to the B g and B g geometry. Using the B g Raman response as a reference spectrum, we cansubtract these contributions from the B g one for eachpressure (for more details, see Appendix B). The pres-sure evolution of the B g electronic Raman response freeof leakage is displayed in Fig. 11. It is clear that the B g superconducting peak energy decreases by a factorof three with pressure from 600 cm − to 200 cm − (cf.Fig. 11 (a) and (b)). The evolution with pressure of the R a m a n s h i f t ( c m - 1 ) c ’’B1g - c ’’B2g (a.u.) Pressure GPa H g - 1 2 0 1 ( a )
R a m a n s h i f t ( c m - 1 ) - 0 . 7 41 . 01 . 5 ( b ) FIG. 11. (Color online) Following the B g SC gap in Hg-1201.(a) Difference between the B g and the B g Raman response(the latter one serving as a reference background), as a func-tion of pressure. (b) Contour plot highlighting the evolutionof the B g spectral weight with pressure. The dashed line isa guide for the eyes. characteristic SC peaks of the UD132 Hg-1223 and UD92Hg-1201 are summarized in Fig. 12. In panel (a), the B g SC peak associated with the Hg-1223 structure hastwo components detected by Raman measurements at ≈ B g inner plane component frequency decreasesmore slowly than the outer plane component. The fre-quency of the inner plane component does not decreasebelow 710 cm − while the one of the outer plane compo-nent continues to decrease down to 590 cm − at 10 GPa.We interpret these two distinct evolutions as a differentefficiency of the charge transfer with pressure betweenthe inner and the outer planes of Hg-1223. This energydecrease is much weaker than the one of the B g peak inthe Hg-1201 structure (cf. panel (b)). It is likely due to alarger inertia of the charge transfer induced by pressurein the triple layers than in the single one. Still, as thepressure increases, T c increases for both the Hg-1201 andHg-1223 structures. The increasing of T c with pressure isconfirmed by Raman measurements at high pressure (cf.Appendix D) that show the B g SC peak is still resolvedwell above the T c value measured at ambient pressure.Consequently, it is clear that the B g SC peaks of Hg-1223 and Hg-1201 do not follow T c with pressure. The B g SC peak frequency detected on the Hg-1223 spectraslightly increases with pressure following T c (cf. panel (a)of Fig. 12) but the A g SC peak observed on the Hg-1201spectra decreases with pressure (cf. panel (b) of Fig. 12).The A g Raman data are displayed in Appendix C.To summarize this part, the most important results arethat (i) the SC B g peak for both Hg-1201 and Hg-1223decreases drastically in frequency while T c increases withpressure, (ii) In the case of Hg-1201 single layer (whosecharge transfer is not altered by multi-layers), the B g SC H g - 1 2 2 3H g - 1 2 0 1 ( c )
P r e s s u r e ( G P a )
Raman shift (cm-1)
P r e s s u r e ( G P a ) A s c p e a k B s c p e a k ( b ) H g - 1 2 0 1
H g - 1 2 2 3 ( a )
P r e s s u r e ( G P a ) B i n n e r p l a n e p e a k B o u t e r p l a n e p e a k B B i n . p e a k B o u t . p e a k D /KBTC B p e a k FIG. 12. (Color online) Frequency evolution of the character-istic superconducting peaks in the Raman spectra of Hg-1223and Hg-1201 compounds. peak energy collapses from 10 to 2 K B T c (cf. panel (c)of Fig. 12). The two components of the B g SC peakof Hg-1223 also decreases in frequency but more slowlyfrom 10 and 6 K B T c . These frequency changes are signif-icantly larger than those obtained over the same pressurerange for the B g SC peak of YBa Cu O − δ (Y-123) [71].This is due to the fact that mercury-based compoundsare highly compressible along the c -axis compared to theother cuprates[81]. It is therefore apparent that the bind-ing energy of the Cooper pairs at its maximum value, cor-responding to the B g pair breaking peak, does not scalewith T c under pressure. It has already been shown thanthe B g peak energy scale detected in the Raman spec-tra decreases with doping like the pseudogap energy scalein several cuprate families [33, 41, 43, 46, 47], suggest-ing that the B g peak energy scale and the pseudo-gapenergy scale could be linked at least as a function of dop-ing. Can such a link be made as a function of pressure ?The hypothesis was already considered in an earlier Ra-man study under pressure on Y-123 [71] although at thetime it had not been established that the SC B g peakdid not follow T c with pressure as we report here. Theauthors had based themselves on the assumption thatthe pseudo-gap would be related to magnetic correlations[1, 2, 4, 15, 85–87] which are weakened as the hole concen-tration increase with pressure and thus, the pseudogapenergy scale should decrease. Based on this hypothesis,the B g peak softening with pressure was interpreted as asign of its connection to the pseudogap. This scenario de-serves to be explored. Unfortunately, there are very fewpressure dependent studies of the pseudogap in the liter-ature and the results are contradicting. Some data ad-vocate in favor of T ∗ as independent of pressure [88, 89],while others that T ∗ increases [90, 91], or decreases withpressure [92]. Additionally, to our knowledge, no directmeasurement of the pseudogap energy scale with pressurehas been yet carried out so far. So, it appears that no def-inite link between B g SC gap at the anti-nodes and thepseudogap energy scale can yet be made as a function ofpressure. Therefore, it appears crucial to investigate thepseudogap energy scale with pressure in order to clarifyits relationship with the binding energy of the SC gap atthe anti-nodes.
V. CONCLUSION
In summary, we have performed Raman measurementsunder hydrostatic pressure on the Hg-1223 and Hg-1201cuprate superconductors. Our analysis reveals that the T c increase with pressure is slowed down in the Hg-1223multi-layers compared to the Hg-1201 single layer dueto the inhomogeneous increase of the carrier concentra-tion inside the three CuO layers of the Hg-1223. We findthat the frequency dependence under pressure of the api-cal mode from which the charge transfer operates, is thesame for both the Hg-1223 and Hg-1201 cuprates andcontrolled by the inter-plane compressibility. Last butnot least, we show that the binding energy of the Cooperpairs related to the maximum amplitude of the d − waveSC gap at the anti-nodes (the B g SC peak) decreasesdrastically while T c increases with pressure. In particularfor Hg-1201, its energy collapses from 10 to 2 K B T c , in-triguingly reaching values below the weak-coupling BCSlimit[93]. These new experimental facts added to the for-mer one that the binding energy of the Cooper pairs atthe anti-nodes also decreases as T c increases with dop-ing, demonstrates that the binding energy of the Cooperpairs at the anti-nodes does not follow T c both with dop-ing and pressure. It is likely linked to the pseudogapenergy scale which follows the same trend with doping[41, 46]. However, a formal proof of this conjecture re-quires a measurement of the pseudogap energy scale as afunction of pressure. VI. ACKNOWLEDGMENTS
We thank the University of Paris, the Coll`ege deFrance and the Canadian Institute for Advanced Re-search (CIFAR) for their support. We acknowledge sup-port from the ANR grant NEPTUN n ◦ ANR-19-CE30-0019-01. Correspondence and request for materialsshould be addressed to A.S. ([email protected]).
Appendix A: Raw Raman data of Hg-1223 andHg-1201 under pressure
The Raman superconducting response of the slightlyunder-doped (UD92K) Hg-1201 and (UD132K) Hg-1223single crystals under hydrostatic pressure in B g and B g geometries are reported in Fig. 13. Similarly to Fig. 4we observe a broad electronic background superimposedby few weak narrow phonon peaks stemming from pris-tine, parasitic and defect modes. H g - 1 2 0 1U D 9 2 B c ’’B2g( w , T) (a.u) c ’’B1g( w , T) (a.u)
051 0
R a m a n s h i f t ( c m - 1 ) R a m a n s h i f t ( c m - 1 ) B
051 0
4 G P a 5 G P a 6 G P a 7 G P a 8 G P a 1 0 G P a 0 .4 G P a 1 G P a 2 G P a 3 G P a
H g - 1 2 2 3U D 1 3 2 B
051 0 B FIG. 13. Sets of the Hg-1201 and Hg-1223 Raman responsesunder hydrostatic pressures. The sets of Hg-1223 and Hg-1201spectra were obtained at 4 K and 14 K respectively.
Appendix B: Extraction of the B g aman signalunder pressure in Hg-1201 We display in Fig. 14, panel (a), both the B g and B g spectra of the (UD92K) Hg-1201 as a function ofpressure measured in the superconducting state at 14K. The subtraction of the B g spectra from the B g one(cf. panel (b)), allows us to eliminate the A g contribu-tion associated with the polarization leakage induced bypressure. We see that the B g SC peak decreases rapidlywith pressure. The 0 GPa spectra has been obtainedoutside the anvil cell and do not present any polariza-tion leakage. Note that the B g spectrum at 0 GPa doesnot exhibit any feature is almost flat as expected for aslightly under-doped Hg-1201 compound close to the op-timal doping level as mentioned previously. Appendix C: Evolution of the A g C peak withpressure in Hg-1201
We display in Fig. 15, the A g + B g Raman responseof the (UD92K) Hg-1201 as a function of pressure. Inthis geometry, the B g SC peak is weak in intensity, thisallows us to follow the A g SC peak which decreases inenergy with pressure.0 - 101- 101- 101- 1010246 c ’’B1g - c ’’B2g (a.u.) c ’’B1g - c ’’B2g (a.u.) c ’’B1g- c ’’B2g (a.u.) c ’’B1g - c ’’B2g (a.u.) c ’’B1g - c ’’B2g (a.u.) c ’’B1g - c ’’B2g (a.u.)
2 G P a
1 G P a
R a m a n s h i f t ( c m - 1 )
5 G P a c ’’B1g, B2g (a.u.) c ’’B1g, B2g (a.u.) c ’’B1g, B2g (a.u.) c ’’B1g, B2g (a.u.) c ’’B1g, B2g (a.u.) c ’’B1g, B2g (a.u.) B B H g - 1 2 0 1 ( a )
T = 1 4 K
3 G P a - 101 ( b )
H g - 1 2 0 10 G P a0 . 4 G P a
5 G P a
R a m a n s h i f t ( c m - 1 )
1 G P a
2 G P a
3 G P a
FIG. 14. (Color online) B g and B g Raman spectra of(UD92K) Hg-1201 compounds measured as a function of pres-sure.
Appendix D: Raman experimental evidence of T c ncrease with pressure in Hg-1201 and Hg-1223 In Fig. 16 is reported the SC B g peak detected at 9GPa and 10 GPa in Hg-1201 and Hg-1223 respectively.It is still observed above T c measured at ambient pressure(0 GPa) i.e: 92 K for Hg-1201 and 132 K for Hg-1223.Indeed, it is detected at 99 K and 144 K for Hg-1201 andHg-1223 respectively. This means that T c of Hg-1201 at9 GPa is at least greater than 99 K and that of Hg-1223is greater than 144 K. The change in T c as a function ofpressure for theses two compounds is then close to thatfound by transport measurements: namely 1 K/GPa [22–25].
051 0
1 G P a c ’’A1g+ B1g (a.u.)
051 0
6 G P a
H g - 1 2 0 1 ( a )
051 0
3 G P a ( b )
A 1 g
R a m a n s h i f t ( c m - 1 ) Pressure (GPa)
R a m a n s h i f t ( c m - 1 ) FIG. 15. (Color online) A g + B g Raman response in Hg-1201under pressure at T = 14 K. (a) Electronic Raman responsefeaturing the A g peak, whose frequency decreases under pres-sure; (b) Contour plot highlighting the evolution of the A g spectral weight as a function of pressure. The dashed line isa guide for the eyes. c ’’A1g + B1g ( w ) (a.u.) c ’’A1g + B1g ( w ) (a.u.) R a m a n s h i f t ( c m - 1 ) p e a k ( a ) R a m a n s h i f t ( c m - 1 ) ( b ) B p e a k B p e a k c ’’(90K) - c ’’(129 K) (a.u.) R a m a n s h if t ( c m -1 ) B p e a k c ’’(144K) - c ’’(154 K) (a.u.) R a m a n s h if t ( c m -1 ) FIG. 16. (Color online) A g + B g Raman response at highpressure for (a) (UD92K) Hg-1201 and (b) (UD132K) Hg-1223compound. The arrows indicate the location of the B g SCpeak. In the insets is displayed the subtraction between theRaman response above and below T c under pressure. The B g SC peak for the both compounds is still present well above T c at ambient pressure. [1] M. R. Norman, Science , 196 (2011),https://science.sciencemag.org/content/332/6026/196.full.pdf.[2] B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida,and J. Zaanen, Nature , 179 (2015).[3] E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Rev.Mod. Phys. , 457 (2015).[4] P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. , 17 (2006).[5] T. Hanaguri, C. Lupien, Y. Kohsaka, D. H. Lee,M. Azuma, M. Takano, H. Takagi, and J. C. Davis,Nature , 1001 (2004).[6] T. Wu, H. Mayaffre, S. Kr¨amer, M. Horvatic, C. Berthier,W. N. Hardy, R. Liang, D. A. Bonn, and M.-H. Julien,Nature , 191 (2011).[7] K. Fujita, C. K. Kim, I. Lee, J. Lee, M. Hamidian, I. A.Firmo, S. Mukhopadhyay, H. Eisaki, S. Uchida, M. J.Lawler, E. A. Kim, and J. C. Davis, Science , 612(2014).[8] R. Comin and A. Damascelli, Annual Reviewof Condensed Matter Physics , 369 (2016),https://doi.org/10.1146/annurev-conmatphys-031115-011401.[9] R. Arpaia, S. Caprara, R. Fumagalli, G. De Vec-chi, Y. Y. Peng, E. Andersson, D. Betto, G. M.De Luca, N. B. Brookes, F. Lombardi, M. Sal-luzzo, L. Braicovich, C. Di Castro, M. Grilli,and G. Ghiringhelli, Science , 906 (2019),https://science.sciencemag.org/content/365/6456/906.full.pdf.[10] B. Fauqu´e, Y. Sidis, V. Hinkov, S. Pailh`es, C. T. Lin,X. Chaud, and P. Bourges, Phys. Rev. Lett. , 197001(2006).[11] R. Daou, J. Chang, D. LeBoeuf, O. Cyr-Choini`ere,F. Lalibert´e, N. Doiron-Leyraud, B. J. Ramshaw,R. Liang, D. A. Bonn, W. N. Hardy, and L. Taillefer,Nature , 519 (2010).[12] C. Proust and L. Taillefer, Annual Reviewof Condensed Matter Physics , 409 (2019),https://doi.org/10.1146/annurev-conmatphys-031218-013210.[13] Y. Sato, S. Kasahara, H. Murayama, Y. Kasahara, E.-G. Moon, T. Nishizaki, T. Loew, J. Porras, B. Keimer,T. Shibauchi, and Y. Matsuda, Nature Physics , 1074(2017).[14] N. Auvray, B. Loret, S. Benhabib, M. Cazayous, R. D.Zhong, J. Schneeloch, G. D. Gu, A. Forget, D. Colson,I. Paul, A. Sacuto, and Y. Gallais, Nature Communica-tions , 5209 (2019).[15] D. J. Scalapino, Rev. Mod. Phys. , 1383 (2012).[16] S. Sachdev and R. La Placa, Phys. Rev. Lett. , 027202(2013).[17] Y. Wang, D. F. Agterberg, and A. Chubukov, Phys. Rev.Lett. , 197001 (2015).[18] S. Caprara, C. Di Castro, G. Seibold, and M. Grilli,Phys. Rev. B , 224511 (2017).[19] W. Wu, M. S. Scheurer, S. Chatterjee, S. Sachdev,A. Georges, and M. Ferrero, Phys. Rev. X , 021048(2018).[20] D. Chakraborty, M. Grandadam, M. H. Hamidian,J. C. S. Davis, Y. Sidis, and C. P´epin, Phys. Rev. B , 224511 (2019). [21] P. Choubey, S. H. Joo, K. Fujita, Z. Du, S. D. Edkins,M. H. Hamidian, H. Eisaki, S. Uchida, A. P. Mackenzie,J. Lee, J. C. S. Davis, and P. J. Hirschfeld, Proceedingsof the National Academy of Sciences , 323 (1993).[23] M. Nu˜nez-Regueiro, J. L. Tholence, E. V. Antipov, J. J.Capponi, and M. Marezio, Science , 97 (1993).[24] L. Gao, Y. Y. Xue, F. Chen, Q. Xiong, R. L. Meng,D. Ramirez, C. W. Chu, J. H. Eggert, and H. K. Mao,Physical Review B , 4260 (1994).[25] E. V. Antipov, A. M. Abakumov, and S. N. Putilin,Superconductor Science and Technology , R31 (2002).[26] S. N. Putilin, E. V. Antipov, O. Chmaissem, andM. Marezio, Nature , 226 (1993).[27] A. Schilling, M. Cantoni, J. D. Guo, and H. R. Ott,Nature , 56 (1993).[28] A. Fukuoka, A. Tokiwa-Yamamoto, M. Itoh, R. Usami,S. Adachi, and K. Tanabe, Phys. Rev. B , 6612 (1997).[29] H. Kotegawa, Y. Tokunaga, K. Ishida, G.-q. Zheng,Y. Kitaoka, A. Iyo, Y. Tanaka, and H. Ihara, Phys. Rev.B , 184504 (2002).[30] H. Mukuda, S. Shimizu, A. Iyo, and Y. Kitaoka, Jour-nal of the Physical Society of Japan , 011008 (2012),https://doi.org/10.1143/JPSJ.81.011008.[31] T. P. Devereaux and R. Hackl, Rev. Mod. Phys. , 175(2007).[32] G. Blumberg, M. Kang, M. V. Klein, K. Kadowaki, andC. Kendziora, Science , 1427 (1997).[33] M. Opel, R. Nemetschek, C. Hoffmann, R. Philipp, P. F.Muller, R. Hackl, I. Tutto, A. Erb, B. Revaz, E. Walker,H. Berger, and L. Forro, Phys. Rev. B , 9752 (2000).[34] M. Le Tacon, A. Sacuto, A. Georges, G. Kotliar, Y. Gal-lais, D. Colson, and A. Forget, Nat Phys , 537 (2006).[35] S. Blanc, Y. Gallais, M. Cazayous, M. A. M´easson, A. Sa-cuto, A. Georges, J. S. Wen, Z. J. Xu, G. D. Gu, andD. Colson, Phys. Rev. B , 144516 (2010).[36] Y. Li, M. Le Tacon, M. Bakr, D. Terrade, D. Manske,R. Hackl, L. Ji, M. K. Chan, N. Bariˇsi´c, X. Zhao,M. Greven, and B. Keimer, Phys. Rev. Lett. , 227003(2012).[37] S. Benhabib, A. Sacuto, M. Civelli, I. Paul, M. Cazayous,Y. Gallais, M. A. M´easson, R. D. Zhong, J. Schneeloch,G. D. Gu, D. Colson, and A. Forget, Phys. Rev. Lett. , 147001 (2015).[38] B. Loret, S. Sakai, S. Benhabib, Y. Gallais, M. Cazayous,M. A. M´easson, R. D. Zhong, J. Schneeloch, G. D. Gu,A. Forget, D. Colson, I. Paul, M. Civelli, and A. Sacuto,Phys. Rev. B , 094525 (2017).[39] B. Loret, Y. Gallais, M. Cazayous, R. D. Zhong,J. Schneeloch, G. D. Gu, A. Fedorov, T. K. Kim, S. V.Borisenko, and A. Sacuto, Phys. Rev. B , 174521(2018).[40] B. Loret, N. Auvray, Y. Gallais, M. Cazayous, A. For-get, D. Colson, M.-H. Julien, I. Paul, M. Civelli, andA. Sacuto, Nature Physics , 771 (2019).[41] B. Loret, N. Auvray, G. D. Gu, A. Forget, D. Colson,M. Cazayous, Y. Gallais, I. Paul, M. Civelli, and A. Sa-cuto, Phys. Rev. B , 214520 (2020). [42] L. Wang, B. Yu, R. Jing, X. Luo, J. Zeng, J. Li, I. Bialo,M. Bluschke, Y. Tang, J. Freyermuth, G. Yu, R. Sutarto,F. He, E. Weschke, W. Tabis, M. Greven, and Y. Li,Phys. Rev. B , 220509 (2020).[43] J. Tallon and J. Loram, Physica C , 53 (2001).[44] A. Kanigel, M. R. Norman, M. Randeria, U. Chatterjee,S. Souma, A. Kaminski, H. M. Fretwell, S. Rosenkranz,M. Shi, T. Sato, T. Takahashi, Z. Z. Li, H. Raffy, K. Kad-owaki, D. Hinks, L. Ozyuzer, and J. C. Campuzano,Nature Physics , 447 (2006).[45] O. Fischer, M. Kugler, I. Maggio-Aprile, C. Berthod, andC. Renner, Rev. Mod. Phys. , 353 (2007).[46] C. Bernhard, L. Yu, A. Dubroka, K. Kim, M. R¨ossle,D. Munzar, J. Chaloupka, C. Lin, and T. Wolf, Jour-nal of Physics and Chemistry of Solids , 3064 (2008),sNS2007.[47] N. Munnikes, B. Muschler, F. Venturini, L. Tassini,W. Prestel, S. Ono, Y. Ando, D. C. Peets, W. N.Hardy, R. Liang, D. A. Bonn, A. Damascelli, H. Eisaki,M. Greven, A. Erb, and R. Hackl, Phys. Rev. B ,144523 (2011).[48] A. Legros, B. Loret, A. Forget, P. Bonnaillie, G. Collin,P. Thu´ery, A. Sacuto, and D. Colson, Materials ResearchBulletin (2019), 10.1016/j.materresbull.2019.05.004.[49] B. Loret, A. Forget, J.-B. Moussy, S. Poissonnet, P. Bon-naillie, G. Collin, P. Thu´ery, A. Sacuto, and D. Colson, Inorganic Chemistry , Inorg. Chem. , 9396 (2017).[50] M. R. Norman and C. P´epin, Rep. Prog. Phys. , 1547(2003).[51] Y. Gallais, A. Sacuto, and D. Colson, Physica C
408 -410 , 785 (2004).[52] W. Guyard, A. Sacuto, M. Cazayous, Y. Gallais,M. Le Tacon, D. Colson, and A. Forget, Phys. Rev.Lett. , 097003 (2008).[53] S. Benhabib, Y. Gallais, M. Cazayous, M.-A. M´easson,R. D. Zhong, J. Schneeloch, A. Forget, G. D. Gu, D. Col-son, and A. Sacuto, Phys. Rev. B , 134502 (2015).[54] S. Ideta, K. Takashima, M. Hashimoto, T. Yoshida,A. Fujimori, H. Anzai, T. Fujita, Y. Nakashima, A. Ino,M. Arita, H. Namatame, M. Taniguchi, K. Ono, M. Kub-ota, D. H. Lu, Z.-X. Shen, K. M. Kojima, and S. Uchida,Phys. Rev. Lett. , 227001 (2010).[55] G. Vincini, K. Tanaka, T. Adachi, L. Sobirey,S. Miyasaka, S. Tajima, S. Adachi, N. Sasaki, andT. Watanabe, Physical Review B , 144503 (2018).[56] M.-H. Julien, P. Carretta, M. Horvati´c, C. Berthier,Y. Berthier, P. S´egransan, A. Carrington, and D. Colson,Phys. Rev. Lett. , 4238 (1996).[57] H. Mukuda, N. Shiki, N. Kimoto, M. Yashima, Y. Ki-taoka, K. Tokiwa, and A. Iyo, Journal of the PhysicalSociety of Japan , J. Phys. Soc. Jpn. , 083701 (2016).[58] M. Mito, K. Ogata, H. Goto, K. Tsuruta, K. Nakamura,H. Deguchi, T. Horide, K. Matsumoto, T. Tajiri, H. Hara,T. Ozaki, H. Takeya, and Y. Takano, Phys. Rev. B ,064503 (2017).[59] S. Iwai, H. Mukuda, S. Shimizu, Y. Kitaoka, S. Ishida,A. Iyo, H. Eisaki, and S.-i. Uchida, in JPS ConferenceProceedings , Vol. 1 (Journal of the Physical Society ofJapan, 2014) pp. –.[60] V. Popov and V. Hadjiev, in
Lattice Dynamics ofHgBa2CuO4+d, Fabrication, Properties and Applica-tions of Low-Dimensional Semiconductors , NATO ASISeries, edited by M. Balkanski and I. Yanchev (SpringerNetherlands, Dordrecht, 1995) p. 239. [61] M. C. Krantz, C. Thomsen, H. Mattausch, and M. Car-dona, Phys. Rev. B , 1165 (1994).[62] A. Sacuto, A. Lebon, D. Colson, A. Bertinotti, J.-F.Marucco, and V. Viallet, Physica C: Superconductivity , 209 (1996).[63] X. Zhou, M. Cardona, C. W. Chu, Q. M. Lin, S. M.Loureiro, and M. Marezio, Physical Review B , 6137(1996).[64] L. Wang, X. Luo, J. Li, J. Zeng, M. Cheng, J. Freyer-muth, Y. Tang, B. Yu, G. Yu, M. Greven, and Y. Li,Physical Review Materials , 123401 (2018).[65] Y. T. Ren, H. Chang, Q. Xiong, Y. Q. Wang, Y. Y. Sun,R. L. Meng, Y. Y. Xue, and C. W. Chu, Physica C:Superconductivity , 273 (1993).[66] N. H. Hur, H.-G. Lee, J.-H. Park, H.-S. Shin, and I.-S.Yang, Physica C: Superconductivity , 365 (1993).[67] A. Sacuto, D. Colson, A. Forget, and J. Cayssol, Pro-ceedings of the International Conference on Materialsand Mechanisms of Superconductivity High TemperatureSuperconductors VI , Physica C: Superconductivity , 2253 (2000).[68] B. Lorenz and C. Chu, in
Frontiers in Superconduct-ing Materials (Springer Berlin Heidelberg, Berlin, Hei-delberg, 2005) pp. 459–497.[69] A. Yamamoto, N. Takeshita, C. Terakura, andY. Tokura, Nature Communications , 8990 (2015).[70] A. Goncharov, M. Muinov, T. Uvarova, and S. Stishov,High Pressure Research - HIGH PRESSURE RES ,500 (1992).[71] A. F. Goncharov and V. V. Struzhkin, Journal of RamanSpectroscopy , J. Raman Spectrosc. , 532 (2003).[72] T. Cuk, V. V. Struzhkin, T. P. Devereaux, A. F. Gon-charov, C. A. Kendziora, H. Eisaki, H.-K. Mao, andZ.-X. Shen, Phys. Rev. Lett. , 217003 (2008).[73] I. Aupiais, M. Mochizuki, H. Sakata, R. Grasset, Y. Gal-lais, A. Sacuto, and M. Cazayous, npj Quantum Mate-rials , 60 (2018).[74] A. Trokiner, L. Le Noc, J. Schneck, A. M. Pougnet,R. Mellet, J. Primot, H. Savary, Y. M. Gao, andS. Aubry, Phys. Rev. B , 2426 (1991).[75] M. Balkanski, K. P. Jain, R. Beserman, and M. Jouanne,Phys. Rev. B , 4328 (1975).[76] F. Slakey, S. L. Cooper, M. V. Klein, J. P. Rice, andD. M. Ginsberg, Phys. Rev. B , 2781 (1989).[77] B. Friedl, C. Thomsen, and M. Cardona, Phys. Rev.Lett. , 915 (1990).[78] C. Thomsen, R. Liu, M. Bauer, A. Wittlin, L. Genzel,M. Cardona, E. Sch¨onherr, W. Bauhofer, and W. K¨onig,Solid State Communications , 55 (1988).[79] A. Sacuto, M. Balkanski, O. Gorochov, and R. Surya-narayanan, Journal of Alloys and Compounds , 359(1993).[80] M. Bakr, S. M. Souliou, S. Blanco-Canosa,I. Zegkinoglou, H. Gretarsson, J. Strempfer, T. Loew,C. T. Lin, R. Liang, D. A. Bonn, W. N. Hardy,B. Keimer, and M. Le Tacon, Phys. Rev. B , 214517(2013).[81] J. H. Eggert, J. Z. Hu, H. K. Mao, L. Beauvais, R. L.Meng, and C. W. Chu, Physical Review B , 15299(1994).[82] Y. Ohta, T. Tohyama, and S. Maekawa, Phys. Rev. B , 2968 (1991).[83] E. Pavarini, I. Dasgupta, T. Saha-Dasgupta, O. Jepsen,and O. K. Andersen, Phys. Rev. Lett. , 047003 (2001). [84] M. Le Tacon, A. Sacuto, Y. Gallais, D. Colson, andA. Forget, Phys. Rev. B , 144505 (2007).[85] D. Scalapino, Physics Reports , 329 (1995).[86] P. W. Anderson, Science , 1705 (2007),https://science.sciencemag.org/content/316/5832/1705.full.pdf.[87] A. V. Chubukov and M. R. Norman, Phys. Rev. B ,214529 (2008).[88] N. Doiron-Leyraud, O. Cyr-Choini`ere, S. Badoux,A. Ataei, C. Collignon, A. Gourgout, S. Dufour-Beaus´ejour, F. F. Tafti, F. Lalibert´e, M.-E. Boulanger,M. Matusiak, D. Graf, M. Kim, J.-S. Zhou, N. Momono,T. Kurosawa, H. Takagi, and L. Taillefer, Nature Com-munications , 2044 (2017). [89] O. Cyr-Choini`ere, D. LeBoeuf, S. Badoux, S. Dufour-Beaus´ejour, D. A. Bonn, W. N. Hardy, R. Liang, D. Graf,N. Doiron-Leyraud, and L. Taillefer, Phys. Rev. B ,064513 (2018).[90] E. V. L. de Mello, M. T. D. Orlando, J. L. Gonz´alez,E. S. Caixeiro, and E. Baggio-Saitovich, Phys. Rev. B , 092504 (2002).[91] R. V. Vovk and A. L. Solovjov, Low Temperature Physics ,Low Temperature Physics , 81 (2018).[92] P. S. H¨afliger, A. Podlesnyak, K. Conder, and A. Fur-rer, Europhysics Letters (EPL) , Eur. Phys.Lett. , 260(2006).[93] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys.Rev.108