The overdoped end of the cuprate phase diagram
TThe overdoped end of the cuprate phase diagram
Thomas A. Maier,
1, 2
Seher Karakuzu, and Douglas J. Scalapino Computational Sciences and Engineering Division,Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6164, USA Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6164, USA Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA (Dated: April 29, 2020)Studying the disappearance of superconductivity at the end of the overdoped region of the cupratephase diagram offers a different approach for investigating the interaction which is responsible forpairing in these materials. In the underdoped region this question is complicated by the presenceof charge and stripe ordered phases as well as the pseudogap. In the overdoped region the situationappears simpler with only a normal phase, a superconducting phase and impurity scattering. Here,for the overdoped region, we report the results of a combined dynamic cluster approximation (DCA)and a weak Born impurity scattering calculation for a t − t − U Hubbard model. We find that adecrease in the d -wave pairing strength of the two-particle scattering vertex is closely coupled tochanges in the momentum and frequency structure of the magnetic spin fluctuations as the systemis overdoped. Treating the impurity scattering within a disordered BCS d -wave approximation, wesee how the combined effects of the decreasing d -wave pairing strength and weak impurity scatteringlead to the end of the T c dome. INTRODUCTION
In the overdoped region of the cuprate phase diagramthe normal phase exhibits properties similar to those ofa strongly correlated Fermi liquid [1–3]. The pseudogapas well as the charge and stripe ordered phases, whichcompete or coexist with superconductivity at lower dop-ing, are absent [4]. This is a region in which the results ofnumerical calculations are expected to be less sensitive toboundary conditions and lattice size effects. Here, usinga dynamic cluster approximation (DCA) [5] we study thedecrease in the strength of the d-wave pairing interactionfor a t − t − U Hubbard model in the overdoped regimeas the density x of doped holes per site increases [6, 7].Then, including impurity scattering within a disorderedBardeen-Cooper-Schrieffer (BCS) d -wave approximation[8], we examine the end of the T c dome.The Hubbard model we will study has a near-neighborhopping t , a next-near-neighbor hopping t and an onsiteCoulomb interaction U . H = − t X h i,j i σ c † iσ c jσ − t X hh i,j ii σ c † iσ c jσ + U X i n i ↑ n i ↓ − µ X iσ n iσ (1)The tight binding parameters give rise to a bandstructure ε k = − t (cos k x +cos k y ) − t cos k x cos k y and µ controlsthe filling, which we will measure in terms of the densityof holes x away from half-filling. In the following, theresults for t /t = − .
25 and
U/t = 7 . RESULTS
As discussed in the Supplemental Material section [9],previous DCA calculations [10, 11] have found that forthese parameters the pseudogap ends for x & .
15. Thisis the overdoped regime that we will study. In Fig. 1 wehave plotted the spin susceptibility χ ( q ) at T = 0 . t fordopings x = 0 .
15 and x = 0 . q = ( π, π ) is reduced and the ferro-magnetic (FM) spin susceptibility at small momentumtransfer is increased. However, as previously noted [6], asignificant response also remains at intermediate valuesof momentum transfer q = ( π/ , π/
2) and q = ( π, − π Im χ ( q , ω ) at dif-ferent momentum transfers for various dopings. Theseresults were obtained from a Maximum Entropy ana-lytic continuation of the DCA imaginary time data. Atlarge momentum transfer q = ( π, π ), one sees that thespin-fluctuation spectral weight is significantly reducedwith doping, while at intermediate values of momentumtransfers spin-fluctuations remain. For example, at theBrillouin zone boundary q = ( π, λ α φ α ( k ) = − TN X k Γ( k, k ) G ( k ) G ( − k ) φ α ( k ) (2) a r X i v : . [ c ond - m a t . s up r- c on ] A p r q x (cid:1) q y (cid:1) (cid:2) ( q ) (b) x = q x (cid:1) q y (cid:1) (cid:2) ( q ) (a) x = FIG. 1. The spin susceptibility χ ( q ) at T = 0 . t for (a) x = 0 .
15 and (b) 0.25. As the doping increases the antifer-romagnetic response weakens and there is an increase in theferromagnetic response.
Here G is the single particle propagator and Γ is the irre-ducible particle-particle scattering vertex and we haveused k = ( k , ω n ). At the superconducting transitiontemperature T c the leading eigenvalue of Eq. (2) goesto 1. For the doped Hubbard model the eigenfunction φ d ( k , ω n ) with the leading eigenvalue has d -wave sym-metry. At a doping x = 0 .
15 the DCA calculations give λ d ( T c ) = 1 with T c /t = 0 . t = 0 . T c ∼
65 K. Here we are interested in whathappens to the strength of the pairing interaction as x increases and the system is overdoped.Multiplying Eq. (2) by φ d ( k , ω n ) and summing over( k , ω n ) one obtains the following expression for λ d λ d = − T N P k,k φ d ( k )Γ( k, k ) G ( k ) G ( − k ) φ d ( k ) TN P k φ d ( k ) (3)Then inserting a complete set of states between the ver-tex Γ and the GG propagators and assuming that theleading d -wave eigenvalue is dominant, one obtains theseparable approximation λ d ( T ) ’ V d ( T ) P d ( T ) (4) ω χ ′′ ( q , ω ) x 0.250.20.15 (a) q = ( , ) ω χ ′′ ( q , ω ) x 0.250.20.15 (b) q = ( π , ) ω χ ′′ ( q , ω ) x 0.250.20.15 (c) q = ( π , π ) FIG. 2. The imaginary part of the DCA cluster spin suscepti-bility χ ( q , ω ) vs. ω for various cluster momenta and dopings x = 0 .
15 (blue), 0.2 (green), and 0.25 (red). The spin fluc-tuation spectral weight at large (AF) momenta is reduced asthe doping increases while the long wavelength (FM) spectralweight increases. For intermediate momenta q = ( π, with the strength of the pairing interaction V d ( T ) = − P k P k φ d ( k )Γ( k, k ) φ d ( k )( P k φ d ( k )) (5)and the non-interacting but dressed two-particle pairfieldsusceptibility P d ( T ) = TN X k φ d ( k ) G ( k ) G ( − k ) (6)In evaluating these expressions we will approximate the d -wave eigenfunction φ d ( k , ω n ) ∼ (cos k x − cos k y ) ( πT ) + ω c ω n + ω c (7)with ω c = t . This form provides a reasonable approxima-tion and is less noisy than using the DCA eigenfunction φ ( k , ω n ).To check the validity of the separable approximationfor λ d ( T ) given by Eq. (4) we have plotted λ d ( T ) andthe product V d ( T ) P d ( T ) versus T in Fig. 3 for differ-ent dopings. The close agreement between λ d ( T ) and V d ( T ) P d ( T ) seen in Fig. 3 arises from the fact that, while T λ d ( T ) ( ● ) , V d ( T ) P d0 ( T ) ( ▲ ) x FIG. 3. A comparison of the d -wave eigenvalue λ d ( T ) ( • ) ofthe Bethe-Salpeter equation (2) with the separable approxi-mation P d ( T ) V d ( T ) ( N ) given by Eqs. (5) and (6) for severalvalues of the doping x . The quality of the fit reflects thedominance of the leading d -wave eigenvalue. there are other singlet eigenstates of the Bethe-Salpeterequation such as extended s -wave and odd frequency p -waves, the singlet channel is dominated by the d -waveeigenfunction. Thus V d provides a measure of the d -wavepairing strength associated with the two particle scatter-ing vertex Γ( k , ω n , k , ω n ). Results for V d ( x ) at a lowtemperature T = 0 . t are shown in Fig. 4a. Here onesees the decrease of the d -wave coupling strength as thehole doping is increased.In spin-fluctuation theories of the pairing interaction, ameasure of the strength of the d -wave pairing interactionis given by V SF d = 3 ¯ U N X q Z ∞ dωπ Im χ ( q , ω ) ω cos q x (8) Using the DCA results for the cluster spin susceptibility χ ( q , ω ) and replacing ¯ U by U/
2, as found in previousDCA studies [16], results for V SF d ( x ) versus x at T =0 . t are plotted in Fig. 4b. The change in χ ( q ) shownin Fig. 1 and the shift of the spin-fluctuation spectralweight with doping shown in Fig. 2 are reflected in thedecrease in V SF d as the doping increases. x V d ( x ) (a) x V d S F ( x ) (b) FIG. 4. a) The d -wave pairing strength V d ( x ), Eq. (5), at T = 0 . t obtained from the 2-particle scattering vertex Γplotted versus the doping x . (b) The d -wave spin-fluctuationpairing strength, V SF d ( x ) Eq. (8), at T = 0 . t versus x . The end behavior of the T c dome involves the effectsof impurities. Here we have in mind a situation in whichthe impurity dopants lie off of the CuO plane, adding xholes and giving rise to weak Born impurity scattering.Within the framework of a fluctuation exchange approxi-mation, Kudo and Yamada [17] found that the reductionof the Bethe-Saltpeter eigenvalue associated with the de-crease in strength of the pairing interaction caused by im-purity scattering is approximately off-set by the increaseof the spectral weight of the single particle propagator,leaving pair breaking as the dominant effect of the im-purity scattering. We will assume that this is also thecase here and use the Abrikosov-Gorkov [18] expressionfor the superconducting transition temperature given byln (cid:18) T c ( x ) T c ( x ) (cid:19) = ψ (cid:18)
12 + Γ( x )2 πT c ( x ) (cid:19) − ψ (1 / . (9)Here T c ( x ) is the putative superconducting transitiontemperature of the doped system without impurity scat-tering obtained by extrapolating the eigenvalue of theBethe-Salpeter equation λ d ( T c ( x )) to 1. Assuming thatthe impurity dopants lie out of plane, Γ( x ) is the nor-mal state Born impurity scattering rate, which we takeproportional to x Γ( x ) = Γ x (10)and ψ is the digamma function. Results for T c ( x ) versus x for various values of the scattering rate Γ per dopedhole are shown in Fig. 5. x T c Γ FIG. 5. The superconducting transition temperature T c ( x )(solid dots) for the pure system determined from an extrapo-lation of the Bethe-Salpeter eigenvalue λ d ( T ) to 1 for specificdopings. The red curve is a fit to these points. The additional T c ( x ) curves are solutions of the AG equation for different im-purity scattering strengths Γ . The transition temperaturesare normalized by the maximum T c value. Here T c ( x ) vanishes with an essential singularityexp( − t/V d ( x )) at an end point where V d ( x ) goes to zero.In the dirty d -wave theory, T c ( x ) approaches the endpoint x as ( x − x ) / with x determined by T c ( x ) / Γ( x ) = 2 γ/π (11)with γ ∼ . DISCUSSION AND CONCLUSION
We have used a combined DCA and weak Born im-purity scattering calculation for a 2D Hubbard modelto study the disappearance of superconductivity at theend of the overdoped region of the cuprate phase dia-gram. We have found that the decrease in the d -wavepairing strength with increasing doping is closely relatedto a similar decrease in the strength of the d -wave spin-fluctuation interaction. The additional effect of impurityscattering, taken into account within a disordered BCS d -wave approximation, is found to lead to a further re-duction of T c as the doping increases. Hence, in thiswork, the decrease of T c ( x ) in the overdoped regime re-flects both a decrease in the ”clean” transition tempera-ture T c ( x ) due to a reduction in the pairing strength andan increase in the impurity scattering rate with doping.Alternatively, ”dirty d -wave” models [8] in which T c ( x ) is constant and there is an increase in the Born impu-rity scattering, starting from a finite doping, to fit theobserved T c ( x ) have proved very useful. However, in thiscase where T c is a constant greater than the maximum T c , the required impurity scattering rate for the dirty d -wave model will be considerably larger than what wehave used. ACKNOWLEDGMENTS
The authors would like to thank P. J. Hirschfeld andS. A. Kivelson for their helpful comments. This work wassupported by the Scientific Discovery through AdvancedComputing (SciDAC) program funded by the U.S. De-partment of Energy, Office of Science, Advanced Sci-entific Computing Research and Basic Energy Sciences,and Division of Materials Sciences and Engineering. Anaward of computer time was provided by the INCITEprogram. This research used resources of the Oak RidgeLeadership Computing Facility, which is a DOE Officeof Science User Facility supported under Contract DE-AC05-00OR22725. [1] Cyril Proust, Etienne Boaknin, R.W. Hill, Louis Taille-fer, A.P. Mackenzie, “Heat Transport in a Strongly Over-doped Cuprate: Fermi Liquid and Pure d -wave BCS Su-perconductor”, Phys. Rev. Lett. , 147003 (2002).[2] A.F. Bangura et al. , “Fermi surface and electronichomogeneity of the overdoped cuprate superconductorTl Ba CuO δ as revealed by quantum oscillations”, Physical Review B , 140501.R (2010).[3] K.P. Kramer et al. , “Band structure of overdoped cupratesuperconductors: Density functional theory matching ex-periments”, Phys. Rev. B , 224509 (2019).[4] Doiron-Leyraud et al. , “Pseudogap phase of cuprate su-perconductors confined by Fermi surface topology”, Nat.Commun. , 2044 (2017)[5] T.A. Maier, M. Jarrell, T.C. Schulthess, P.R.C. Kent,J.B. White: “Systematic study of d -wave superconduc-tivity in the 2D repulsive Hubbard model”, Phys. Rev.Lett. , 237001 (2005).[6] Edwin W. Huang, Douglas J. Scalapino, ThomasA. Maier, Brian Moritz, Thomas P. Devereaux. “De-crease of d -wave pairing strength in spite of the persis-tence of magnetic excitations in the overdoped Hubbardmodel, Phys. Rev. B , 020503 (2017).[7] T.A.Maier and D.J.Scalapino: ”Disappearance of Super-conductivity in the Overdoped Cuprates” , Journal ofSuperconductivity and Novel Magnetism 33 (2020).[8] N. R. Lee-Hone, V. Mishra, D. M. Broun, and P. J.Hirschfeld, “Optical conductivity of overdoped cupratesuperconductors: Application to LSCO”, Phys. Rev. B , 054506 (2018).[9] see Supplemental Material.[10] K.-S. Chen, Z. Y. Meng, T. Pruschke, J. Moreno, and M.Jarrell: Lifshitz transition in the two-dimensional Hub- bard model, Phys. Rev. B , 165136 (2012).[11] Wei Wu, Mathias S. Scheurer, Shubhayu Chatterjee,Subir Sachdev, Antoine Georges, and Michel Ferrero:“Pseudogap and Fermi-Surface Topology in the Two-Dimensional Hubbard Model”, Phys. Rev. B 100, 214510(2019) Phys. Rev. X , 021048 (2018).[12] H. C. Robarts, M. Barthelemy, M. Garcia-Fernandez, J.Li, A. Nag, A. C. Walters, K. J. Zhou, S. M. Hayden:”Anisotropic damping of the spin fluctuations in dopedLa − x Sr x CuO studied by resonant inelastic x-ray scat-tering”, Phys. Rev. B , 214510 (2019).[13] L.J.P. Ament, M. van Veenendaal, T.P. Devereaux,J.P. Hill, and J. van den Brink, ”Resonant inelastic x-ray scattering studies of elementary excitations”, Rev.Mod. Phys. , 705 (2011).[14] S. Wakimoto, K. Ishii, H. Kimura, M. Fujita, G. Dellea,K. Kummer, L. Braicovich, G. Ghiringhelli, L.M. Debeer-Schmitt, and G.E. Granroth, ”High-energy magnetic ex-citations in overdoped La − x Sr x CuO studied by neutron and resonant inelastic x-ray scattering”, Phys. Rev. B ,184513 (2015).[15] D. Meyers, H. Miao, A.C. Walters, V. Bisogni,R.S. Springell, M. d’Astuto, M. Dantz, J. Pelliciari,H.Y. Huang, J. Okamoto, D.J. Huang, J.P. Hill, X. He,I. Boˇzovi´c, T. Schmitt, and M.P.M. Dean, ”Doping de-pendence of the magnetic excitations in La − x Sr x CuO ”, Phys. Rev. B , 075139 (2017).[16] T. A. Maier, A. Macridin, M. Jarrell, and D. J. Scalapino:“Systematic analysis of a spin-susceptibility representa-tion of the airing interaction in the two-dimensional Hub-bard model”, Phys. Rev. B , 144516 (2007).[17] K. Kudo, and K. Yamada, ”Reduction of T c due to Im-purities in Cuprate Superconductors”, J. Phys. Soc. Jpn. , 2219 (2004).[18] A. A. Abrikosov and L. P. Gorkov, ”On the theory of su-perconducting alloys”, Sov. Phys. JETP , 1090 (1959). he overdoped end of the cuprate phase diagram SupplementalMaterial Thomas Maier and Seher Karakuzu
Computational Sciences and Engineering Division and Center for Nanophase Materials Sciences,Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6164, USA
D.J. Scalapino
Department of Physics, University of California,Santa Barbara, CA 93106-9530, USA (Dated: April 29, 2020) a r X i v : . [ c ond - m a t . s up r- c on ] A p r In Fig. S1 we show a section of a t /t − x phase diagram based upon DCA calculations by -0.3-0.2-0.10.0 0.1 0.2 x t ′ t x FS x PG FIG. S1: A part of the t /t − x phase diagram based on DCA calculations [1]. The solidcurve denotes a Lifshitz transition which separates a lower doped region which has ahole-like FS around the ( π, π ) point of the Brillouin zone from a region which has anelectron-like FS about the origin (0,0). A dashed curve, which separates from the Lifshitzcurve at larger doping marks the end of the pseudogap region. Here we study the decreaseof the d -wave pairing strength and the end of the T c dome for t’/t=-0.25 as the holedoping increases along the dash-dot line.Wu et al. [1]. Here a solid curve marks the Lifshitz transition at which the topology of theFermi surface changes from hole-like around ( π, π ) to electron-like around (0,0) as the holedoping increases. The dashed curve in Fig. S1 marks the end of the pseudogap (PG) regime.Similar to the cuprates, as discussed by Doiron-Leyraud et al. [2], the simulation finds thata PG does not open on an electron-like FS, confining the PG to a region of the t /t − x phasediagram in which x is less than the curve marking the Lifshitz doping in Fig. S1. However,for larger values of | t /t | there is a range of dopings below the Lifshitz doping in which thePG is also absent [1, 2]. In the main text, for t /t = − .
25, we examine the strength of the d -wave pairing and the end of the T c dome as the doping is increased along the dash-dottedline shown in Fig. S1. [1] Wei Wu, Mathias S. Scheurer, Shubhayu Chatterjee, Subir Sachdev, Antoine Georges, andMichel Ferrero: “Pseudogap and Fermi-Surface Topology in the Two-Dimensional HubbardModel”, Phys. Rev. B 100, 214510 (2019) Phys. Rev. X , 021048 (2018).[2] Doiron-Leyraud et al. , “Pseudogap phase of cuprate superconductors confined by Fermi surfacetopology”, Nat. Commun.8