Absence of superconducting dome at the charge-density-wave quantum phase transition in 2H-NbSe2
Owen Moulding, Israel Osmond, Felix Flicker, Takaki Muramatsu, Sven Friedemann
OOne-sided Competition of Charge-Density-Wave order and Superconductivity in2H-NbSe Owen Moulding, Israel Osmond, Felix Flicker, Takaki Muramatsu, and Sven Friedemann ∗ HH Wills Laboratory, University of Bristol, Bristol, BS8 1TL, UK Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Department of Physics,Clarendon Laboratory, Parks Road, Oxford, OX1 3PU, United Kingdom (Dated: Monday 8 th June, 2020)The interplay of incompatible ground states is a key concept in physics and central to manymaterials classes from monolayer graphene and transition metal dichalcogenides to cuparte high-temperature superconductors. Tuning the balance between these ground states reveals competition,coexisitence, and cooperation - sometimes even within the same system. Here, we present compre-hensive high-pressure Hall effect and magnetic susceptibility measurements of the charge-density-wave (CDW) and superconducting state in 2H-NbSe . We show that superconductivity is enhancedupon suppressing CDW order at high pressures whilst CDW order is insensitive to a suppressionof superconductivity in high magnetic fields. With a realistic model calculation we show that thesuppression of the CDW state at high pressures is described by the stiffening of the underlying barephonon modes, whilst the suppression of the superconducting state at low pressures is due to theloss of density of states inside the CDW phase. We conclude that the competition between the twostates is one-sided in 2H-NbSe . Our work clarifies the interplay of superconductivity and CDWorder in prototypical 2H-NbSe and highlights routes to optimise superconductivity in materialswith competing order. The interplay of competing orders is of fundamentaland practical interest [1–4]. Controlled switching be-tween phases promises new applications in data storageand sensing [1]. On a fundamental level, understandingthe interplay between ground states provides importantinsight into the mechanism underlying each ground stateand can reveal new phenomena at the border of orderedphases [3, 5]. For instance, a large body of work focuseson the interplay of superconductivity and charge order incuprate high-temperature superconductors [4, 6].With both superconductivity and charge-density-wave(CDW) order ensuing from the opening of a gap on (partsof) the Fermi surface, a mutual competition between thetwo states was originally expected [7–9]. As an alterna-tive, superconductivity in a dome around quantum crit-ical points was suggested to be promoted by quantumfluctuations of the ordered state with prominent exam-ples in heavy-fermion antiferromagnets [10], CDW sys-tems [11], and the CDW and pseudogap order in cupratesuperconductors [12]. In addition to competition andpromotion, superconductivity and charge order can co-exist for instance by opening a gap on different parts ofthe Fermi surface. 2H-NbSe is a prototypical materialhosting both CDW order and superconductivity.The interplay of CDW order and superconductivity in2H-NbSe remains disputed [13, 14]. CDW order setsin at T CDW ∼
33 K while superconductivity is presentbelow T c = 7 . at ambient pressure [15–18]. Super-conductivity opens gaps of different size on most of theFermi surface while the CDW opens a gap on small partsof the zone-corner niobium-derived Fermi surface sheetsonly [13, 14, 16, 19–22]. The separation of the CDWand superconducting gaps in k -space was interpreted as a hallmark for coexistence of the two ordered states but fur-ther studies suggested that superconductivity is boostedby CDW order [13], some studies suggested a promotionof superconductivity by the soft modes present at thequantum critical point of the CDW order [23, 24], andsome studies suggested a mutual competition for den-sity of states between the CDW and superconductivity[14, 25, 26]. Here, we use comprehensive high-pressureand magnetic field tuning of the CDW and superconduct-ing states to reveal a partial but one-sided competitionof the CDW with the superconductivity. We find thatthe suppression of CDW order with pressure is correlatedwith a rise in T c , but a suppression of the superconduc-tivity by magnetic field does not yield a rise in T CDW .Our high-pressure Hall effect measurements show thesuppression of T CDW under pressure in Fig. 1. Athigh temperatures, the Hall coefficient, R H , is weaklytemperature dependent and does not change with pres-sure. At T CDW , R H ( T ) shows a large drop and a signchange naturally associated with the Fermi surface re-construction due to the CDW [27, 28]. The contribu-tion to the Hall coefficient from the CDW, ∆ R H ( T, P ) = R H ( T, P ) − R H ( T, . , is calculated by subtractingthe non-CDW form well above the critical pressure. Inthe derivative d ∆ R H / d T , the CDW transition manifestsas a pronounced peak as shown in Fig. 1(b). T CDW ( P ) as-sociated with the maximum in d ∆ R H / d T shifts to lowertemperature as pressure is increased in good agreementwith T CDW ( P ) extracted from resistivity measurementsas well as with previous results of T CDW as highlighted inFig. 2 and in section SI of the supplemental information.The benefit of analysing the Hall coefficient is that thestrong signature can be traced to higher pressures where a r X i v : . [ c ond - m a t . s up r- c on ] J un FIG. 1.
Suppression of the CDW transition in 2H-NbSe under pressure. (a) The temperature dependenceof the Hall coefficient. Inset shows the pressure dependence R H ( P ) measured at µ H = 10 T at selected temperatures.Straight lines highlight linear fits to the data. (b) Deriva-tive of the Hall coefficient ( d ∆ R H / d T ) was calculated aftersubtracting the high-pressure background, i.e. ∆ R H ( T, P ) = R H ( P, T ) − R H ( T, P = 5 . . Arrows indicate T CDW ex-tracted as the maximum. the signature in resistivity is lost (cf. Ref [29]). We ob-serve the CDW transition in ∆ R H ( T ) up to a pressure of . .The suppression of T CDW is confirmed by the isother-mal pressure dependency of the Hall coefficient R H ( P ) shown in the inset of Fig. 1(a). R H ( P ) exhibits a pro-nounced kink associated with the critical pressure of theCDW phase, P CDW ( T ) (cf. intersecting linear fits in the FIG. 2.
High-pressure phase diagram of 2H-NbSe . (a)Experimental values of T CDW ( P ) are determined as the peakin d ∆ R H / d T as shown in Fig. 1, as the minimum in d ρ/ d T as shown in SI, and the kink in R H ( P ) as shown the inset ofFig.1(a). Theoretical values of T CDW ( P ) are calculated as de-scribed in methods section S V and section S VI of the Supple-mentary Material. Data from Refs. [23, 29] are included. Thesolid line marks a power-law fit to our experimental datasetsof T CDW ( P ) . (b) Experimental results for T c ( P ) are extractedfrom magnetisation measurements (Fig. 3). A detailed com-parison of T c ( P ) using different pressure media is given in SI. Theoretical values of T c ( P ) are calculated as described inS VI. The boundary of the CDW phase is reproduced from(a). Inset (c) shows the relation of the superconductivity andCDW transition temperatures. The solid line is a linear fit. inset of Fig. 1(a)). The position of P CDW ( T ) is in-cluded as black squares in the phase diagram in Fig. 2. P CDW ( T ) becomes virtually independent of temperaturefor T ≤
10 K , i.e. the kink in R H ( P ) is found at virtuallythe same pressure P CDW ( T ) = 4 . for and
10 K .In section SII of the supplementary information we showthat this result is also true if the Hall effect is probedin smaller magnetic fields. R H ( P ) indicates a sharp de-crease of T CDW ( P ) which could be interpreted as signa-ture of a first order transition at high pressures. This isfurther supported by the fact, that the amplitude of theCDW anomaly in ∆ R H ( T ) diminishes at . asshown in section S II of the supplemental information.However, no signs of first order have been observed inx-ray diffraction and models of the CDW including ourmodel below exclude a change from second to first orderat high pressures [23, 30].Our results are consistent with previous X-ray diffrac-tion measurements within experimental uncertainties[23, 30]. The critical pressure from the XRD studies is in-cluded as a green triangle in Fig. 2(a)). Our comprehen-sive Hall effect measurements provide for the first timea single dataset tracing T CDW ( P ) to the critical pres-sure. We fit the phase boundary extracted from our R H ( T ) and R H ( P ) datasets with a power-law yieldingan exponent . and an extrapolated critical pressure P CDW = 4 . (cf. solid line in Fig. 2(a)).Superconductivity is boosted under pressure in clearanticorrelation to the CDW. T c ( P ) is traced as the onsetof the diamagnetic signal in magnetic susceptibility mea-surements, χ , as presented in Fig. 3. The sharp onsetgives T c = 7 . at ambient pressure in good agreementwith our resistivity measurements (cf. S I of the supple-mentary information) and other published work [17, 18].With increasing pressure, T c ( P ) shifts to higher temper-ature whilst the transition remains very sharp. Above . , T c ( P ) saturates at . . The measurementspresented in Fig. 3 used argon as a pressure transmittingmedium (PTM) which remains hydrostatic up to
11 GPa [31]. We find very good agreement with T c ( P ) extractedfrom our resistivity measurements up to . – thelimiting pressure for hydrostaticity of the PTM glycerolused for the electrical transport measurements. The com-parison of different PTM and literature data in sectionS I show that the measurements with argon and glycerol(up to . ) reveal the intrinsic behaviour of super-conductivity in 2H-NbSe .The evolution of T c ( P ) is shown in Fig. 2(b) togetherwith the phase boundary of the CDW order establishedfrom our transport measurements. T c ( P ) is virtually con-stant for P ≥ . , i.e. outside the CDW phase. Atthe same time, T c is reduced by up to . inside theCDW phase in clear anticorrelation with T CDW ( P ) ashighlighted in Fig. 2(c). We do not observe a maximumin T c around the critical pressure of the CDW phase, andthus rule out a boost to superconductivity from the CDWcritical fluctuations.A detailed model using a structured electron-phononcoupling shows that the suppression of the CDW underpressure is driven by the stiffening of the underlying barephonon. In Fig. 2, the experimental transition tempera-tures are compared to the detailed model of the chargeorder and Fermi surface developed earlier by one of usas outlined in sections S V and S VI of the supplemen-tary material [32–34]. In our RPA calculations, the over-all magnitude of the electron-phonon coupling g is con-strained to reproduce T CDW ( P = 0) = 33 . (cf. S III ofthe supplementary information) and we keep g fixed forall pressures. To describe T CDW ( P ) , we assume a linear FIG. 3.
Enhanced superconductivity in 2H-NbSe un-der pressure. The volume susceptibility χ measured onwarming the sample in a magnetic field µ H = 0 . af-ter zero-field cooling. The transition into the superconduct-ing state inferred from the diamagnetic signal shifts to highertemperatures as the pressure is increased. Inset shows a pic-ture of the sample and ruby chips inside the pressure cell.Argon was used as a pressure medium. stiffening of the longitudinal acoustic phonons underly-ing the CDW formation consistent with high-pressure in-elastic x-ray studies [35] as detailed in section S V of thesupplemental information. From the good match withthe experimental phase boundary, we conclude that thesuppression of the CDW is indeed driven by the increaseof the bare phonon frequency whilst the electron-phononconstant remains unchanged.A partial competition for density of states (DOS) isthe main driver for the evolution of T c ( P ) . We use theexperimentally determined phase boundary (solid line inFig. 2(a)) to scale the evolution of the CDW phase toour pressure data as detailed in section S VI of the sup-plemental information. Inside the CDW phase, the DOSavailable for superconductivity is reduced due to the gap-ping of the inner K-pockets of the Fermi surface as illus-trated in Fig. 4 leading to a reduction of T c . As theCDW gap becomes smaller, the DOS available for super-conductivity becomes larger which in turn accounts foralmost the entire increase of T c and naturally explainswhy T c saturates above P CDW as can be seen in Fig. 2(b).Thus, we conclude that it is a partial competition forDOS which suppresses T c inside the CDW phase.A suppression of the CDW order inside the supercon-ducting phase is not observed. At zero pressure, no re-duction in the X-ray intensity of the CDW reflection isobserved at T c [15, 36]. At pressures up to . , the FIG. 4. Reduction of DOS as a function of CDW gap mag-nitude for the two Nb-derived bands at E F . Insets show theFermi surface in a wedge of the Brillouin zone for specificvalues of ∆ CDW and equivalent pressure. The plotted pointswere identified as the the points at which the RPA spectralfunction is within
15 % of its maximum value.
CDW order is detected with X-ray diffraction inside theSC state [23, 30]. At pressures above . , the CDWis absent in high fields, i.e. outside the superconductingphase and thus cannot be suppressed below the criticalfield of the superconducting phase. This finding is consis-tent with the above modelling showing that the suppres-sion of the CDW is driven by the stiffening of the barephonon whilst the suppression of the superconductivityis driven by the partial competition for DOS.Our results lead to several profound conclusions aboutthe interplay of CDW order and superconductivity in 2H-NbSe . (i) The CDW order is reducing the electronicdensity of states available for superconductivity leadingto a reduction of T c . (ii) Superconductivity is not reduc-ing T CDW . Hence, (iii) the competition between CDW or-der and superconductivity is not mutual but rather one-sided with only superconductivity suppressed by CDWorder but not the other way around. (iv) No signatureof a boost to superconductivity from CDW fluctuationshas been observed.In summary, we conclude that the competition betweenthe CDW and superconducting states is one-sided in 2H-NbSe , clarifying the prototypical nature of 2H-NbSe .Crucially, 2H-NbSe shows markedly different behaviourcompared to cuprate superconductors where T c increasesunder pressure despite no change to T CDW up to [6]. The interplay of coexisting states is a central focus ofcurrent physics, of interest both in its own right as an in-triguing theoretical puzzle, but also owing to potentiallywide-ranging applications such as clarifying the originsof high-termperature superconductors. The results pre-sented here reveal the precise form of the interplay be-tween the low-temperature states in a prototypical test- case, helping pave the way for understanding the idea inits broader context.
METHODSSamples samples were grown by J. A. Wilson [17]using the vapour transport method and showed a highresidual resistivity ratio, ρ ( T = 300 K) /ρ ( T = 9 K) = 64 ,confirming the good crystal quality. Samples were cutwith a scalpel. Lateral sample dimensions have been ob-tained with an optical microscope at ambient pressure(see inset in Fig. 3), sample thickness, t , was estimatedfrom the sample mass and the lateral dimensions usingthe known density of 2H-NbSe . The associated uncer-tainty of
10 % results in a systematic relative uncertaintyof the Hall coefficient of the same amount. The magneticfield was applied along the crystallographic c direction. High-Pressure Measurements
High-pressure measurements used moissanite anvilscells with a culet size of µ m for both the electricaland magnetic measurements. Both types of measurementused metallic gaskets which were prepared by indenting450 µ m thick BeCu to approximately 60 µ m followed bydrilling a 450 µ m hole.Pressure was determined at room temperature by rubyflorescence, with multiple ruby flakes placed within thesample chamber as a manometer. The uncertainty ofthe pressure is taken as the standard deviation betweenpressure estimates from rubies across the sample cham-ber, both before and after a measurement. A comparisonwith the pressure obtained from ruby at room temper-ature and the superconducting transition of a piece oflead revealed good agreement to within . for thepressure cells used for magnetisation measurements. Electrical Transport Measurements
For the electrical measurements, six bilayer electrodeswere deposited on one anvil in a three-step process with-out breaking vacuum. Firstly, the anvil was cleaned usingan RF argon plasma etch, followed by sputtering
20 nm ofnichrome, and finally evaporation of
150 nm gold. To en-sure no electrical short between electrodes, any nichromeoverspray was removed using TFN etchant.Gold contacts were evaporated on top of the sample.Epo-Tek H20E silver paint was used to connect the sam-ples to the electrodes on the anvil. A four-probe ACmethod was used to measure the resistance with a cur-rent I = 1 mA . The six electrodes were used to measure V l , the longitudinal and V t , the transverse voltages, re-spectively. The Hall coefficient was calculated from theantisymmetric part of V t ( H ) under reversal of the mag-netic field H as R H = V t ( H ) − V t ( − H )2 H tI .
The effect of different pressure media is discussed in SI of the supplementary information.For the electrical measurements, the gaskets were in-sulated using a mixture of Stycast epoxy 2850FT andBN powder; the mixture was pressed between the anvilsto above the maximum pressure required for the experi-ments and then cured whilst pressurised. A µ m holethrough the insulation was drilled for the sample space. Magnetic Measurements
A Quantum Design Magnetic Property MeasurementSystem (MPMS) was used to measure the DC magneticmoment of the sample inside the pressure cell as detailedin section S IV of the supplementary information. Thetransition temperature, T c , has been determined as thetemperature where χ ( T ) has dropped by
10 % of the nor-malised step, i.e. close to the onset of the transition. Thisprocedure results in uncertainty less than .
05 K of T c . ACKNOWLEDGEMENTS
The authors would like to thank Jasper van Wezel,Jans Henke, Nigel Hussey, Hermann Suderow, andAntony Carrington for valuable discussion. The au-thors acknowledge supported by the EPSRC undergrants EP/R011141/1, EP/L025736/1, EP/N026691/1as well as the ERC Horizon 2020 programme under grant715262-HPSuper.The research data supporting this publication can beaccessed through the University of Bristol data repository[37].
AUTHOR CONTRIBUTIONS
OM and SF conducted the electrical transport mea-surements, IO conducted the magnetic measurements.TM assisted with the high-pressure setups. FF per-formed the calculations. SF, OM, IO, and FF wrote themanuscript. The project was devised and led by SF.
ADDITIONAL INFORMATION
Data are available at the University of Bristol datarepository, data.bris, at https://doi.org/10.5523/ bris.xxxx [37].
COMPETING INTERESTS
The authors declare that no competing interests arepresent. ∗ [email protected][1] M. Nakano, K. Shibuya, D. Okuyama, T. Hatano,S. Ono, M. Kawasaki, Y. Iwasa, and Y. Tokura, Col-lective bulk carrier delocalization driven by electrostaticsurface charge accumulation, Nature , 459 (2012).[2] B. Baek, W. H. Rippard, S. P. Benz, S. E. Russek,and P. D. Dresselhaus, Hybrid superconducting-magneticmemory device using competing order parameters, Na-ture Communications , 3888 (2014).[3] S. Friedemann, W. J. Duncan, M. Hirschberger, T. W.Bauer, R. Küchler, A. Neubauer, M. Brando, C. Pflei-derer, and F. M. Grosche, Quantum tricritical points inNbFe , Nat. Phys. , 62–67 (2018), arXiv:1709.05099.[4] J. Chang, E. Blackburn, A. T. Holmes, N. B. Chris-tensen, J. Larsen, J. Mesot, R. Liang, D. A. Bonn, W. N.Hardy, A. Watenphul, M. v. Zimmermann, E. M. Forgan,and S. M. Hayden, Direct observation of competition be-tween superconductivity and charge density wave orderin YBa Cu O . , Nat. Phys. , 871 (2012).[5] C. Lester, S. Ramos, R. S. Perry, T. P. Croft, R. I. Be-wley, T. Guidi, P. Manuel, D. D. Khalyavin, E. M. For-gan, and S. M. Hayden, Field-tunable spin-density-wavephases in Sr Ru O ., Nat. Mater. , 373 (2015).[6] C. Putzke, J. Ayres, J. Buhot, S. Licciardello, N. E.Hussey, S. Friedemann, and A. Carrington, Charge or-der and superconductivity in underdoped yba cu o − δ under pressure, Phys. Rev. Lett. , 117002 (2018).[7] G. Bilbro and W. McMillan, Theoretical model of super-conductivity and the martensitic transformation in A15compounds, Phys. Rev. B , 1887 (1976).[8] M. Nunez-Regueiro, Extension of Bilbro-McMillancharge density wave-superconductivity coexistence rela-tion to quantum regimes: Application to superconduct-ing domes around quantum critical points, J. Magn.Magn. Mater. , 25 (2015).[9] G. Grissonnanche, O. Cyr-Choinière, F. Laliberté,S. René de Cotret, A. Juneau-Fecteau, S. Dufour-Beauséjour, M.-È. Delage, D. LeBoeuf, J. Chang, B. J.Ramshaw, D. A. Bonn, W. N. Hardy, R. Liang, S. Adachi,N. E. Hussey, B. Vignolle, C. Proust, M. L. Sutherland,S. Krämer, J.-H. Park, D. Graf, N. Doiron-Leyraud, andL. Taillefer, Direct measurement of the upper criticalfield in cuprate superconductors., Nat. Commun. , 3280(2014).[10] N. D. Mathur, F. M. Grosche, S. R. Julian, I. R.Walker, D. M. Freye, R. K. W. Haselwimmer, and G. G.Lonzarich, Magnetically mediated superconductivity inheavy fermion compounds, Nature , 39 (1998).[11] 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, Superconductivity in Cu x TiSe , Nat.Phys. , 544 (2006). [12] B. J. Ramshaw, S. E. Sebastian, R. D. McDonald, J. Day,B. S. Tan, Z. Zhu, J. B. Betts, R. Liang, D. A. Bonn,W. N. Hardy, and N. Harrison, Quasiparticle mass en-hancement approaching optimal doping in a high-T c su-perconductor, Science , 317 (2015), arXiv:1409.3990.[13] T. Kiss, T. Yokoya, A. Chainani, S. Shin, T. Hanaguri,M. Nohara, and H. Takagi, Charge-order-maximizedmomentum-dependent superconductivity, Nat. Phys. ,720 (2007).[14] S. V. Borisenko, A. A. Kordyuk, V. B. Zabolotnyy, D. S.Inosov, D. Evtushinsky, B. Büchner, A. N. Yaresko,A. Varykhalov, R. Follath, W. Eberhardt, L. Patthey,H. Berger, and B. Buchner, Two Energy Gaps and Fermi-Surface "Arcs" in NbSe , Phys. Rev. Lett. , 166402(2009).[15] D. E. Moncton, J. D. Axe, and F. J. DiSalvo, Study ofSuperlattice Formation in H -Nb Se and H -Ta Se byNeutron Scattering, Phys. Rev. Lett. , 734 (1975).[16] T. Yokoya, T. Kiss, A. Chainani, S. Shin, M. Nohara,and H. Takagi, Fermi surface sheet-dependent supercon-ductivity in 2H-NbSe ., Science , 2518 (2001).[17] J. A. Wilson and A. Yoffe, The transition metal dichalco-genides discussion and interpretation of the observed op-tical, electrical and structural properties, Adv. Phys. ,193 (1969).[18] A. Wieteska, B. Foutty, Z. Guguchia, F. Flicker,B. Mazel, L. Fu, S. Jia, C. Marianetti, J. van Wezel, andA. Pasupathy, Uniaxial Strain Tuning of Superconductiv-ity in 2 H -NbSe , arXiv:1903.05253 [cond-mat.supr-con](2019), http://arxiv.org/abs/1903.05253v1.[19] D. J. Rahn, S. Hellmann, M. Kalläne, C. Sohrt, T. K.Kim, L. Kipp, and K. Rossnagel, Gaps and kinks inthe electronic structure of the superconductor H -NbSe from angle-resolved photoemission at 1 K, Phys. Rev. B , 224532 (2012).[20] T. Valla, A. Fedorov, P. Johnson, P.-A. Glans,C. McGuinness, K. Smith, E. Andrei, and H. Berger,Quasiparticle Spectra, Charge-Density Waves, Supercon-ductivity, and Electron-Phonon Coupling in 2H-NbSe ,Phys. Rev. Lett. , 086401 (2004).[21] J. D. Fletcher, A. Carrington, P. Diener, P. Rodière,J. Brison, R. Prozorov, T. Olheiser, and R. Giannetta,Penetration Depth Study of Superconducting Gap Struc-ture of 2H-NbSe , Phys. Rev. Lett. , 057003 (2007).[22] J. A. Galvis, E. Herrera, C. Berthod, S. Vieira, I. Guil-lamón, and H. Suderow, Tilted vortex cores and super-conducting gap anisotropy in 2H-NbSe , Communica-tions Physics , 30 (2018).[23] Y. Feng, J. Wang, R. Jaramillo, J. van Wezel, S. Haravi-fard, G. Srajer, Y. Liu, Z.-A. Z.-A. Xu, P. B. Littlewood,and T. F. Rosenbaum, Order parameter fluctuations ata buried quantum critical point., Proc. Natl. Acad. Sci.U. S. A. , 7224 (2012).[24] H. Suderow, V. Tissen, J. Brison, J. Martínez, andS. Vieira, Pressure Induced Effects on the Fermi Sur-face of Superconducting 2H-NbSe , Phys. Rev. Lett. ,117006 (2005).[25] B. J. Dalrymple and D. E. Prober, Upper critical fields ofthe superconducting layered compounds Nb − x Ta x Se ,Journal of Low Temperature Physics , 545 (1984).[26] K. Cho, M. Kończykowski, S. Teknowijoyo, M. A.Tanatar, J. Guss, P. B. Gartin, J. M. Wilde, A. Kreyssig,R. J. McQueeney, A. I. Goldman, V. Mishra, P. J.Hirschfeld, and R. Prozorov, Using controlled disorder to probe the interplay between charge order and super-conductivity in nbse2, Nature Communications , 2796(2018).[27] K. Yamaya and T. Sambongi, Low-temperature crystalmodification and the superconductive transition temper-ature of NbSe , Solid State Communications , 903(1972).[28] P. Knowles, B. Yang, T. Muramatsu, O. Mould-ing, J. Buhot, C. J. Sayers, E. Da Como, andS. Friedemann, Fermi surface reconstruction and elec-tron dynamics at the charge-density-wave transi-tion in ti se , Phys. Rev. Lett. , 167602 (2020),http://arxiv.org/abs/1911.01945v1.[29] C. Berthier, P. Molinié, and D. Jérome, Evidence for aconnection between charge density waves and the pres-sure enhancement of superconductivity in 2H-NbSe ,Solid State Commun. , 1393 (1976).[30] Y. Feng, J. van Wezel, J. Wang, F. Flicker, D. M. Sile-vitch, P. B. Littlewood, and T. F. Rosenbaum, Itinerantdensity wave instabilities at classical and quantum criti-cal points, Nature Physics , 865 (2015).[31] N. Tateiwa and Y. Haga, Evaluations of pressure-transmitting media for cryogenic experiments with di-amond anvil cell., Rev. Sci. Instrum. , 123901 (2009).[32] F. Flicker and J. van Wezel, Charge order from orbital-dependent coupling evidenced by NbSe , Nature Com-munications , 7034 (2015).[33] F. Flicker and J. van Wezel, Charge ordering geometriesin uniaxially strained NbSe , Phys. Rev. B , 201103(2015).[34] F. Flicker and J. van Wezel, Charge order in NbSe , Phys.Rev. B , 235135 (2016).[35] M. Leroux, I. Errea, M. Le Tacon, S.-M. Souliou, G. Gar-barino, L. Cario, A. Bosak, F. Mauri, M. Calandra, andP. Rodière, Strong anharmonicity induces quantum melt-ing of charge density wave in H − NbSe under pressure,Phys. Rev. B , 140303 (2015).[36] C.-H. Du, W. J. Lin, Y. Su, B. K. Tanner, P. D. Hat-ton, D. Casa, B. Keimer, J. P. Hill, C. S. Oglesby, andH. Hohl, X-ray scattering studies of 2H-NbSe , a super-conductor and charge density wave material, under highexternal magnetic fields, J. Phys. Condens. Matter ,5361 (2000).[37] O. Moulding, Doi: 10.5523/bris.xxx, Data repository for2H-NbSe article 2020 (2020). upplementary Information for“One-sided Competition of Charge-Density-Wave order and Superconductivity in2H-NbSe ” Owen Moulding, Israel Osmond, Felix Flicker, Takaki Muramatsu, and Sven Friedemann ∗ HH Wills Laboratory, University of Bristol, Bristol, BS8 1TL, UK Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Department of Physics,Clarendon Laboratory, Parks Road, Oxford, OX1 3PU, United Kingdom (Dated: Friday 5 th June, 2020)
INFLUENCE OF PRESSURE MEDIA AND SAMPLE PREPARATION ON SUPERCONDUCTIVITY
We have studied the influence of different pressure media and sample preparation on superconductivity underpressure in 2H-NbSe . In Fig. S1(a) we provide a comparison of two different pressure media used in this study:argon and glycerol. Argon was used for the measurements of the magnetic susceptibility presented in Fig. 3 of themain manuscript. Glycerol was used for all electrical transport measurements, e.g. Fig. 1 of the main manuscript.A second measurement of the magnetic susceptibility was done with glycerol as a pressure medium as presented inFig. S2.The comparison of T c ( P ) from these three measurements shows very good agreement up to P = 5 . . T c ( P ) is reduced for the samples in glycerol pressure medium. We attribute thisreduction of T c to a uniaxial compression along the crystallographic c axis of the sample as samples were mountedwith the crystallographic c -direction perpendicular to the anvil cutlets [2, 3]. Pressure has been applied at roomtemperature and the pressure cells have been warmed to 300(1) K after application of pressure. We conclude thatglycerol provides high-qualtiy hydrostatic conditions for 2H-NbSe up to 5 . T c (P) differ significantly from previous studies extending to beyond the critical pressure ofthe CDW as highlighted in Fig. S1(b) [4, 5]. In order to identify the cause for these differences, we have studied therelevance of the pressure medium and sample preparation. As discussed above, we find a reduced T c in non-hydrostatic FIG. S1. Superconductivity in 2H-NbSe . (a) Comparison of measurements using argon and glycerol as pressure media. (b)Comparison of our measurements with previous studies beyond the CDW critical pressure [4, 5]. conditions for glycerol above 5 . T c observed by Smith et al [4] is due tothe usage of a solid pressure medium.Suderow et al. used methanol:ethanol which provides good hydrostatic conditions up to 10 GPa [6]. Yet, T c isreduced and a much smoother rise of T c ( P ) is observed by Suderow et al.. We could reproduce the behaviour seenby Suderow et al. in one measurement using pentane:isopentane as a pressure medium (sample 1 in Fig. S1(b)).Pentane:isopentane is very similar to methanol:ethanol and provides good hydrostatic conditions to 10 GPa [6]. Asecond sample measured in pentane:isopentane, however, followed the T c ( P ) of our argon measurements (sample 2 inFig. S2(b)). Thus, we conclude that the differences in T c ( P ) are not due to the pressure medium used as long as itprovides good hydrostatic conditions.We could identify a difference in the sharpness of the superconducting transition in χ ( T ) to correlate with thebehaviour of T c ( P ): For all samples showing the clear kink in T c ( P ) (as observed with argon PTM), transitions in χ ( T ) are very sharp (see Fig. 3 of the main text and Fig. S2(a) and (c)). For samples with a reduced T c and a smoothrise in T c ( P ), transitions in χ ( T ) are much broader (Fig. S2(b)). We note that the broad transitions in glycerol abovethe solidification pressure (Fig. S2(a) for P & . T c ( P ) and in χ ( T ) we identify the differences to arise from sample preparation. Indeed, sample 2 and our sample in argon havebeen screened for sharp transitions at ambient pressure before the study.With best hydrostatic conditions and sharp transitions in χ ( T ) correlated to a sharp kink in T c ( P ) at the criticalpressure of the CDW we conclude that the behaviour observed with argon as a pressure medium reveals the intrinsicbehaviour of 2H-NbSe under pressure. HIGH-PRESSURE HALL EFFECT MEASUREMENTS
The phase boundary T CDW ( P ) has been extracted from both d∆ R H / d T and R H ( P ) presented for measurementsin µ H = 10 T in the main manuscript. An important observation is the abrupt drop of T CDW at 4 . R H ( P ) in the inset of Fig. 1 of the main text.In Fig. 1 of the main manuscript we extract the CDW temperature from the position of the peak in d R H / d T . Nopeak is present for pressures P > . . . R H ( P ). In Fig. S4 we show R H ( P ) for further temperatures. At T ≤
10 K, a single kink is observed in R H ( P ). At 15 K and 20 K, two kinks arevisible in R H ( P ). The kink at 3 . . P CDW ( T ). At T ≤
10 K the steepest part of R H ( P ) has a similar steep slopecompared to 15 K and 20 K. Thus, we identify the low-pressure kink at 15 K and 20 K and the single kink at T ≤
10 Kwith the boundary of the CDW phase.At 15 K and 20 K, a second kink at 4 . R H ( P ) happens at the same pressure like the singlekink at 5 K and 10 K. The origin of this second kink remains elusive and we can only speculate that it is relatedto fluctuations of the CDW order for pressures below 4 . R H ( P ) extracted at µ H = 2 T, i.e. a field just above the critical field of the superconductingstate. We find the same behaviour: A kink in R H ( P ) at 4 . T CDW ( P ) at 4 . T CDW ( P ) is not due to theeffect of a finite magnetic field on the CDW transition. This argument is extended to lower fields by the observationof linear H c2 ( T ) at 4 . HIGH-PRESSURE RESISTIVITY MEASUREMENTS
Fig. S6 shows a zero-pressure resistivity trace and its temperature derivative for a sample from the same batch likethe high-pressure measurements. The CDW transition is observed at 33 . T CDW ( P ) as the minimum in d ρ/ d T as shown in the inset of Fig. S7.The upper critical field H c2 ( T ) has been extracted from temperature sweeps measuring T c at a fixed field. At eachfield T c was determined as the temperature where the resistivity reaches 10 % of the normal state value (see Fig. S8). FIG. S2. Superconducting transition of 2H-NbSe in a pressure cell using glycerol as the pressure transmitting medium (a).After subtraction of the background as described in the Methods section of the main manuscript, χ has been normalised χ norm ( T ) = ( χ ( T ) − χ (9 K)) / ( χ (9 K) − χ (4 K)). Measurements were done during heating up in µ H = 0 . We use the Ginzburg-Landau equation d µ H c2 d T (cid:12)(cid:12)(cid:12)(cid:12) T c = − . π k e ~ T c v (S1)to extract the Fermi velocity v F from the slope of the critical field. Our measurements show a linear slope over asimilar large temperature range as previous studies [9]. FIG. S3. Amplitude of the peak in d R H / d T . Main panel shows the height of the peak as a function of pressure with anempirical power-law fit to the data above 3 GPa giving a critical pressure P c = 4 . R H ( P ) at selected temperatures. Straight lines highlight linear fits to the data. BACKGROUND SUBTRACTION IN MAGNETIC MEASUREMENTS
A pressure cell mirror symmetric about the sample position was used. Thus, the MPMS software is able to reliablyfit a dipole function with the amplitude giving the total magnetic moment of the sample and pressure cell. We remove
FIG. S5. Pressure dependence of the Hall coefficient at µ H = 2 T for T = 5 K and T = 10 K. Straight lines show linear fits.The T=10 K data has been offset in the inset to match the high-pressure value of the T=5 K data. the background contribution arising from the pressure cell by subtracting a Curie-Weiss type contribution fitted tothe field-cooled measurement as illustrated in Fig. S9.The demagnetisation factor, D , of the sample was calculated using the rectangular prism approximation [10]. Themagnetic susceptibility, χ , is calculated from the sample magnetic moment m s , sample volume V s , and the appliedstatic magnetic field H as χ = m s V s H (1 − D ) . (S2)Small variations in χ at low temperatures are associated with uncertainty of the sample position relative to theSQUID pick-up coils. CALCULATION OF THE CHARGE-DENSITY-WAVE TRANSITION TEMPERATURE
We employed diagrammatic expansions based on the Random Phase Approximation (RPA), assuming the CDW todevelop from a structured electron-phonon coupling dependent on both the ingoing and outgoing electron momentaand the orbital content of the bands. This model, which has as its only free parameter the overall magnitude of theelectron-phonon coupling (fixed by T CDW ( P = 0)), has previously been shown to agree well with the full range ofexperimental observations on the charge ordered state in 2H-NbSe .The Random Phase Approximation provides the following expression for the softening of the bare (high-temperature)phonon mode Ω ( q ) as a function of momentum transfer q as the temperature decreases towards the CDW phasetransition at T CDW : Ω( q, T ) = Ω ( q )(Ω ( q ) − D ( q, T ))where D ( q, T ) is the generalized susceptibility to CDW formation [8, 11, 12]. D ( q, T ) is the Lindhard functionconvolved with the square of the electron-phonon coupling. As temperature decreases D ( q, T ) increases until, at T CDW and wavevector Q CDW ≈ . · Γ M , D ( Q CDW , T
CDW ) = Ω ( Q CDW ), and Ω softens to zero. A CDW withwavevector Q CDW results. The one free parameter in the model, the magnitude of the electron-phonon coupling, isset to give the measured value of T CDW = 33 . P = 0. (cid:2) (cid:1)(cid:1) (arb. units.) N b S e T C D W = 3 3 . 4 ( 2 ) K d r /d T (arb. units) T ( K ) FIG. S6. Zero-pressure resistivity measurements on a sample of 2H-NbSe from the same batch as used for the high-pressureresistivity measurements. Generally, the effect of increased pressure is to increase the frequency of phonons. Thus, we model high pressuresas an increase of Ω , the frequency of the longitudinal acoustic mode from which the CDW develops in 2H-NbSe . Wefind that the momentum transfer q at which D ( q, T ) peaks is largely independent of temperature down to T = 0 inagreement with temperature and pressure-dependent X-ray diffraction results [13]. Thus, we can obtain the pressuredependence of T CDW from D ( Q CDW , T = 0) using the ambient-pressure Q CDW .As a consequence of the increase of Ω ( Q CDW ) at higher pressures, a larger D ( Q CDW , T ) is required to reach Ω = 0necessary to achieve the CDW transition. As D increases with decreasing temperature, this corresponds to a decreasein T CDW with pressure. In our model calculations, we analyse the isothermal behaviour: For a fixed D ( Q CDW , T ),i.e. for a fixed temperature, we find the pressure P CDW ( T ) at which Ω( q, T ) = 0 corresponding to the boundaryof the CDW phase. The value of D ( Q CDW , T = 0) sets the maximum bare phonon energy (and therefore pressure)from which a CDW can develop. From this we extract a pressure scaling factor to fit the experimentally observedphase boundary. This scaling corresponds to a rate of stiffening of the bare phonon mode. Finally, by inverting therelation we obtain T CDW (P), the phase boundary plotted in Fig. 2 of the main manuscript.
CALCULATION OF THE SUPERCONDUCTIVITY TRANSITION TEMPERATURE
Our calculation of T c as a function of pressure is based on the change of the density of states (DoS) at the Fermilevel, g E F (∆ CDW ), as a function of the CDW gap magnitude ∆
CDW .The total DoS g E F (∆ CDW ) is calculated from the contributions of the two Nb d z − r orbitals g Nb E F (∆ CDW ), capturedby our model of the Fermi surface developed in Refs. [8, 11, 12]. We add a contribution g Se E F (∆ CDW ) from the selenium
FIG. S7. Electrical resistivity at selected pressures. Inset shows the derivative d ρ/ d T . Data in the inset have been offset forclarity. Arrows indicated the minimum in d ρ/ d T on both panels.FIG. S8. Resistivity of 2H-NbSe normalised to the normal state resistivity at T = 9 K. Dotted line indicates 10 % which wasused to determine the critical field H c2 ( T ) presented in Fig. 4 of the main text. band such that the total matches the Sommerfeld coefficient [14].The DoS of the Nb orbitals was calculated as the sum over the Brillouin zone of the spectral function A ( E, k ): g Nb = 1 N X k ∈ BZ A ( E, k ) = − πN Im X k ∈ BZ G ( E, k ) FIG. S9. Magnetic moment of the pressure cell with 2H-NbSe sample on warming after zero-field cooling and on field coolingin µ H = 0 . where G ( E, k ) is the retarded electronic Green’s function at energy E and wavevector k . We calculated the Green’sfunction, including the CDW gap, using the Nambu-Gor’kov method [8]. For the wavevector dependence of the CDWgap we solved for the gap self-consistently at six high-symmetry points across the Brillouin zone, and used the resultsto create a six-parameter tight-binding fit. This calculation was previously shown to give a good match to scanningtunneling spectroscopy measurements of g ( E ) over a range of energies around E F [15]. Fig. S10 shows this result.Note that the CDW gap is centred 16 meV above E F ; nevertheless, it is g E F which is relevant for the formation ofsuperconductivity.We simulate the pressure dependence of T c assuming that ∆ CDW varies from the zero-pressure value ∆ = 12 meVdown to zero. We obtain the DoS of the Nb orbitals g Nb E F (∆ CDW ) as shown in Fig. S11. We assume a BCS temperaturedependence of ∆
CDW ( T ) at a given pressure to obtain g Nb E F (∆ CDW ) at the superconducting transition temperatureself consistently.We relate the CDW gap to pressure by scaling to the fitted T CDW ( P ) in Fig. 2(a): P = P CDW ( T = 0) " − (cid:18) ∆ CDW ∆ (cid:19) /n (S3)where n and P CDW ( T = 0) are the exponent and critical pressure from the fit to the phase boundary (solid line inFig. 2).In order for the total density of states g E F to be consistent with the Sommerfeld coefficient, we add a constant valueof 0 . − which is associated with the DoS from the Se-orbitals.We use the BCS expression to calculate the transition temperature of the superconducting state T c = 1 .
14Θ exp (cid:18) − g E F V (cid:19) where the coupling constant V = 0 .
035 was obtained to fit the zero-pressure T c using the zero-pressure g E F (which isconsistent with the experimentally determined DoS from the Sommerfeld coefficient). ∗ [email protected]
FIG. S10. Density of states g Nb ( E ) of the Nb orbitals as a function of energy for different values of the gap, normalised to thegap value at zero pressure. The energy is given relative to E F .FIG. S11. Density of states g E F (∆ CDW ) at the Fermi level as a function of the CDW gap size ∆
CDW .[1] A. Drozd-Rzoska, S. J. Rzoska, M. Paluch, A. R. Imre, and C. M. Roland, On the glass temperature under extremepressures, The Journal of Chemical Physics , 164504 (2007).[2] A. Wieteska, B. Foutty, Z. Guguchia, F. Flicker, B. Mazel, L. Fu, S. Jia, C. Marianetti, J. van Wezel, and A. Pa-supathy, Uniaxial Strain Tuning of Superconductivity in 2 H -NbSe , arXiv:1903.05253 [cond-mat.supr-con] (2019),http://arxiv.org/abs/1903.05253v1.[3] T. Sambongi, Effect of uniaxial stress on the superconducting transition temperature of NbSe , Journal of Low TemperaturePhysics , 139 (1975).[4] T. F. Smith, L. E. Delong, A. R. Moodenbough, T. H. Geballe, and R. E. Schwall, Superconductivity of NbSe to 140kbar, Journal of Physics C: Solid State Physics , L230 (1972).[5] H. Suderow, V. Tissen, J. Brison, J. Mart´ınez, and S. Vieira, Pressure Induced Effects on the Fermi Surface of Supercon- ducting 2H-NbSe , Phys. Rev. Lett. , 117006 (2005).[6] N. Tateiwa and Y. Haga, Evaluations of pressure-transmitting media for cryogenic experiments with diamond anvil cell.,Rev. Sci. Instrum. , 123901 (2009).[7] S. V. Borisenko, A. A. Kordyuk, V. B. Zabolotnyy, D. S. Inosov, D. Evtushinsky, B. B¨uchner, A. N. Yaresko, A. Varykhalov,R. Follath, W. Eberhardt, L. Patthey, H. Berger, and B. Buchner, Two Energy Gaps and Fermi-Surface ”Arcs” in NbSe ,Phys. Rev. Lett. , 166402 (2009).[8] F. Flicker and J. van Wezel, Charge order in NbSe , Phys. Rev. B , 235135 (2016).[9] B. J. Dalrymple and D. E. Prober, Upper critical fields of the superconducting layered compounds Nb − x Ta x Se , Journalof Low Temperature Physics , 545 (1984).[10] A. Aharoni, Demagnetizing factors for rectangular ferromagnetic prisms, Journal of Applied Physics , 3432 (1998),https://doi.org/10.1063/1.367113.[11] F. Flicker and J. van Wezel, Charge order from orbital-dependent coupling evidenced by NbSe , Nature Communications , 7034 (2015).[12] F. Flicker and J. van Wezel, Charge ordering geometries in uniaxially strained NbSe , Phys. Rev. B , 201103 (2015).[13] Y. Feng, J. van Wezel, J. Wang, F. Flicker, D. M. Silevitch, P. B. Littlewood, and T. F. Rosenbaum, Itinerant densitywave instabilities at classical and quantum critical points, Nature Physics , 865 (2015).[14] C. L. Huang, J.-Y. Lin, Y. T. Chang, C. P. Sun, H. Y. Shen, C. C. Chou, H. Berger, T. K. Lee, and H. D. Yang,Experimental evidence for a two-gap structure of superconducting NbSe : A specific-heat study in external magneticfields, Phys. Rev. B , 212504 (2007).[15] A. Soumyanarayanan, M. M. Yee, Y. He, J. van Wezel, D. J. Rahn, K. Rossnagel, E. W. Hudson, M. R. Norman, andJ. E. Hoffman, Quantum phase transition from triangular to stripe charge order in NbSe ., Proc. Natl. Acad. Sci. U. S. A.110