Superconducting correlations induced by charge ordering in cuprate superconductors and Fermi arc formation
aa r X i v : . [ c ond - m a t . s up r- c on ] M a y Superconducting correlations induced by charge ordering in cuprate superconductors and Fermiarc formation
E.V.L. de Mello ∗ and J.E. Sonier
2, 3 Instituto de F´ısica, Universidade Federal Fluminense, 24210-346 Niter´oi, RJ, Brazil Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada. Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada.
We have developed a generalized electronic phase separation model of high-temperature cuprate supercon-ductors that links the two distinct energy scales of the superconducting (SC) and pseudogap (PG) phases viaa charge-density-wave (CDW) state. We show that simulated electronic-density modulations resembling thecharge order (CO) modulations detected in cuprates intertwine the SC and charge orders by localizing chargeand providing the energy scale for a spatially periodic SC attractive potential. Bulk superconductivity is achievedwith the inclusion of Josephson coupling between nanoscale domains of intertwined fluctuating CDW and SCorders, and local SC phase fluctuations give rise to the Fermi arcs along the nodal directions of the SC gap. Wedemonstrate the validity of the model by reproducing the hole-doping dependence of the PG onset temperature T ∗ , and the SC transition temperature T c of YBa Cu O y and Bi − y Pb y Sr − z La z CuO δ . The results showthat the periodicity of the CDW order is controlled by the PG energy scale, and the hole-doping dependence ofthe SC energy gap is controlled by the charge ordering free energy. I. INTRODUCTION
Experiments using different methods have established theoccurrence of short-range, incommensurate static CDW corre-lations in a variety of high-temperature SC cuprates . Withthe exception of La-based cuprates in which CDW order isaccompanied by spin order, the charge order (CO) observedin different cuprate families appears to be similar. In zeromagnetic field, the CDW order is essentially two-dimensional.The wave vector of the CDW order is parallel to the Cu-O bond directions along the a and b axes, and decreases inmagnitude with increased charge doping. While much of theexperimental data cannot distinguish between checkerboard(bidirectional) or alternating stripe (unidirectional) CO, re-cent resonant X-ray scattering (RXS) experiments on under-doped
YBa Cu O (Y123) and an analysis of scanning tun-neling microscopy (STM) data for Bi Sr CaCu O δ in-dicate that the inter-unit-cell character is one of segregatedor overlapping unidirectional charge-ordered stripes. Further-more, it has been found that the CDW order possesses a d -wave intra-unit-cell symmetry with the modulated charge pri-marily on the O- p orbitals linking the Cu atoms . Sincethe SC order parameter also has d -wave symmetry, this localcharge or bond order symmetry supports theoretical propos-als that suggest the charge and SC order parameters are inti-mately intertwined. Some attribute the d - wave CO symmetryto quasiparticle scattering by antiferromagnetic (AF) fluctua-tions near a metallic quantum critical point, which also givesrise to the d-wave superconductivity . Alternatively, it hasbeen proposed that CDW order in cuprates is a consequence ofa pair-density wave (PDW) phase, in which the SC order pa-rameter is periodically modulated in space due to the Cooperpairs having finite momentum .The aim of our work is to establish a quantitative link be-tween the inter-unit-cell dependence of the CO resolved byRXS and imaged in real space by STM, and the energy gapsof the PG and SC phases. Our model is based on an intrin-sic propensity for mesoscale electronic phase separation be- low an onset temperature T PS that follows the hole-dopingdependence of the PG temperature T ∗ . This picture is sim-ilar to that previously advocated by Fradkin and Kivelson .We presume the onset of fluctuating CDW order domains at T ∗ , where STM measurements on Bi Sr CaCu O δ havedetected the emergence of charge stripes that extend into theoverdoped regime . The short-range static CO that has beenobserved by X-rays at a lower temperature T CO ≤ T ∗ is as-sumed to be confined to local regions where fluctuating CDWorder has become pinned by disorder. Contrary to this as-sumption, we note that in HgBa CuO δ (Hg1201) CDWorder observed by X-ray scattering vanishes already well be-low optimal doping . This seems to be due to the presenceof pairs of interstitial oxygens within the same unit cell spe-cific to Hg1201. Although not captured by our model, it isalso important to recognize that the pseudogap region marksthe onset of an intra-unit-cell magnetic order , a true phasetransition that modifies ultrasonic waves , an increase in an-tiferromagnetic correlations and global inversion-symmetrybreaking .Another important ingredient of our model is the experi-mental observation that the CDW periodicity is independentof temperature, leading us to surmise that the CDW periodic-ity is set by the onset of the PG at T ∗ . This infers that theCDW order is a consequence of the PG formation. At lowdoping ( p ≤ . ) where T CO decreases with decreasing dop-ing, CDW order is potentially suppressed by a slowing downof spin fluctuations and a tendency toward static SDW order.Compatible with experimental signatures of pairing or SC cor-relations persisting above T c , our model shows that CDWorder in the PG regime may induce SC domains that grow andconnect to establish bulk superconductivity at T c . II. SIMULATION OF THE CHARGE-ORDERED STATE
Our approach is to first simulate spatial modulations of theelectronic structure resembling experimentally resolved inter-
FIG. 1. Contour plots of the electronic density p ( r ) calculated on a square lattice of × unit cells, with average charge densities of p = 0 . in (a) and p = 0 . in (d). The charge order wavelengths are λ CO = 3 . a (a) and λ CO = 3 . a in (d), corresponding to thecharge order wave vectors determined by momentum-resolved X-ray probes . (b), (e) Corresponding spatial dependence of the free-energypotential V GL ( r ). The periodicity of the potential manifests in the periodic modulations of the charge density. (c), (f) Results of calculationsof the d-wave pairing potential ∆ d ( r ) displayed for a single domain over a × unit cell area (in meV unit). The spatial average value ofthe pair potential h ∆ d ( r ) i is 25.5 meV at p = 0 . in (c), and 43.8 meV at p = 0 . in (f). unit-cell CO modulations, using the time-dependent Cahn-Hilliard (CH) differential equation . Besides generating thedesired CDW order, the CH approach yields the associatedfree-energy modulations, which we assume scales with a pe-riodic attractive potential in the subsequent SC calculations.The starting point is the introduction of a time-dependent con-served order parameter associated with the local electronicdensity, u ( r , t ) = [ p ( r , t ) − p ] /p , where p is the average chargedensity and p ( r , t ) is the charge density at a position r in theplane. The Ginzburg-Landau (GL) free energy density of thesystem is of the form f ( u ) = 12 ε |∇ u | + V GL ( u, T ) , (1)where V GL ( u, T ) = − α [ T PS − T ] u / B u / ... is adouble-well potential that characterizes the electronic phaseseparation below T PS . The parameters α and B are constants,and ε controls the spatial separation of the charge-segregatedpatches. The CH equation obtained from the continuity equa-tion for the local free energy current density J = − M ∇ µ (where M is the charge mobility and µ = ∂f /∂µ is the chem-ical potential). Its solution is described in the Apendix I oncharge ordering simulations. For each time step the CH equa-tion is solved for u ( r , t ) , and p ( r , t ) is obtained. We adjust theparameters of the free energy such that when the periodicityof p ( r , t ) matches that of the experimentally observed CDWorder, the calculation is stopped and the solution is taken to bethe spatially-dependent static electronic density p ( r ) . Sincethe method described here does not generate an intra-unit-cell CO symmetry, it is applicable to systems that have SC pairingand CO symmetries other than d-wave.Figure 1(a) shows a simulation of alternating planar do-mains of -rotated charge stripes with an intra-domain pe-riodicity compatible with RXS data for detwinned Y123 at p = 0 . [8]. Within each domain the CO wavelength is λ CO = 3 . a , where the in-plane lattice constant is a =3 . ˚ A . The CO wavelength in Y123 measured by various X-ray methods increases with increased hole doping . Wehave used such data to generate similar CO striped patternsfor Y123 at p = 0 . (Fig. 1(d)) and p = 0 . .STM differential conductance maps for optimally-doped( T c = 35 K) and underdoped ( T c = 32 K and T c = 25 K) Bi − y Pb y Sr − z La z CuO δ [(Pb, La)-Bi2201] samples exhibit checkerboard patterns (indicative of the simultaneouspresence of both CDW domains) with . . a , . . a ,and . . a unit cell ( a = 3 . ˚ A ) periodicity, respectively.The increase in the CO wavelength with increased hole dop-ing agrees well with X-ray scattering and STM measurementson Bi2201 without cation substitutions . We have used thefollowing formula from Ref. 43 to calculate the average num-ber of holes per Cu for the (Pb, La)-Bi2201 samples in Ref.11: T c /T max c = 1 − p − . , where T max c = 35 K.This calculation yields p = 0 . and p = 0 . for the twounderdoped samples. Figs. 2(a) and 2(d) display CO checker-board patterns simulated by the CH equation that resemblethe STM differential conductance maps for (Pb, La)-Bi2201at p = 0 . and p = 0 . .The CO periodicity is manifest in the spatial dependenceof the free-energy potential V GL ( r ) , shown in Figs. 1(b) and FIG. 2. Contour plots of the electronic density p ( r ) calculated on a square lattice of × unit cells, assuming average charge densitiesof p = 0 . in (a) and p = 0 . in (d). The charge order wavelengths are λ CO = 4 . a in a and λ CO = 6 . a in (d), matching thecheckerboard wavelength of the STM conductance maps of underdoped ( T c = 25 K; p = 0 . ) and optimally-doped (Pb, La)-Bi2201 inRef. 11. (b), (e), Corresponding spatial dependence of the free-energy potential V GL ( r ). (c), (f), Results of calculations of the d -wave pairingpotential ∆ d ( r ) displayed over a × unit cell area (in meV unit). The spatial average value of the pair potential h ∆ d ( r ) i is 9.2 meV at p = 0 . in (c), and 15.8 meV at p = 0 . in (f). V GL ( r ) mediates the attractivetwo-body SC interaction. In particular, we assume the fluctu-ating periodic potential has the same periodicity as the staticCO detected experimentally and has a time-averaged potentialwell depth that is proportional to the depth of the static peri-odic potential. In what follows, we make the approximation h V GL ( r ) i , where h V GL ( r ) i is the spatial average of V GL ( r ) over a × unit cell area. III. SUPERCONDUCTING CALCULATIONS IN THECHARGE-ORDERED STATE
Next we use the free-energy simulations and experimen-tally determined input parameters for the optimally-dopedcompounds to deduce the SC energy gap ∆ SC , the pseudo-gap ∆ PG , T c and T ∗ for the underdoped samples. To derivethe local SC gap we solved the Bogoliubov-deGennes (BdG)equations via self-consistent calculations based on a HubbardHamiltonian [Eq. B1]. The calculations were performed fora sub-lattice about the center of the simulated charge densitymaps, using periodic boundary conditions and governed byself-consistent conditions for a spatially-varying d-wave pair-ing potential ∆ d ( r ) and hole density p ( r ) [Eq. B4) and B5].We find that the spatial-average h ∆ d ( r ) i decreases with a re-duction of p (below p = 0 . ), but increases with decreas-ing λ CO . The latter behavior is because as the two holes areforced closer together by the narrower confining potential the binding energy of the two-body interaction increases. The re-sults on the CuO plane shown in Figs. 1(c), 1(f), 2(c) and2(f) indicate that in our approach the PDW is a consequenceof the CDW.The values of h V GL ( r ) i at optimal doping were multi-plied by a scaling factor, such that the calculations (Figs. 1(f)and 2(f)) generate an average value of the pairing potential h ∆ d ( r ) i for p = 0 . that is close to the experimentally es-timated value of the low-temperature SC gap ∆ SC . To cal-culate h ∆ d ( r ) i for the underdoped samples (Tables I and II),this same scaling factor was subsequently applied to the re-spective values of h V GL ( r ) i . For (Pb, La)-Bi2201, the valueof h V GL ( r ) i varies little with doping, and hence p and λ CO are responsible for the hole-doping dependence of h ∆ d ( r ) i .Experimental estimates of the SC gap for the p = 0 . and0.141 samples are not reported, but the calculated values of h ∆ d ( r ) i for the underdoped samples (Table II) roughly fol-low the trend expected if the ratio ∆ SC /k B T c is independentof p . In contrast to (Pb, La)-Bi2201, the doping dependenceof λ CO in Y123 is weaker, and the CH simulations of chargestripes are characterized by a significant change in h V GL ( r ) i with doping (Fig.4). Consequently, the depth of the periodicpotential plays an important role in the calculation of the dop-ing dependence of h ∆ d ( r ) i for Y123. The calculated valuesof h ∆ d ( r ) i at p = 0.09 and 0.16 agree well with an empir-ical relation for ∆ SC ( p ) that describes a number of high- T c cuprate superconductors . The calculated result at p = 0 . falls below this universal curve, which is consistent with thewell-known plateau of T c ( p ) for Y123 near 1/8 hole doping.An estimate of T c is obtained by self-consistently solving E J (p,T) /k B = T (a) p = 0.09p = 0.12p = 0.16 E J ( p , T ) / k B ( K ) T (K) p = 0.16p = 0.126 E J ( p , T ) / k B ( K ) T (K) p = 0.141E J (p,T) /k B = T (b) FIG. 3. (Color online) Calculated values of the superconducting tran-sition temperature. (a,b) The temperature dependence of the aver-age Josephson coupling energy h E J ( p, T ) i (divided by Boltzmannconstant k B ) for Y123 at p = 0 . , . , and 0.16 in (a), and(Pb, La)-Bi2201 at p = 0 . , . , and 0.16 in (b). The val-ues of T c correspond to the intersections of the dashed straight line h E J ( p, T ) i /k B = T with the h E J ( p, T ) i /k B curves and are markedby the arrows. the BdG equations with a temperature-dependent GL potential V ( T, p ) = V (0 , p )[1 − T /T PS ( p )] , (2)where we take V (0 , p ) ≈ h V GL ( r , p ) i and T PS ( p ) , the onsetof phase separation transition, is taken to be equan to T ∗ ( p ) in the calculations. Because of the BdG approach and theabove equation, the value of h ∆ d ( r , T ) i decreases with in-creasing temperature, but the it remains finite in many regionsof the system for a significant range of temperature above Tc.This is consistent with the body of experimental results oncuprates mentioned earlier that are suggestive of persisting SCcorrelations above T c . Typical h ∆ d ( r , T ) i plots of threeY123 compounds are shown in Fig. A2. Next we assume thatbulk superconductivity is achieved via Josephson coupling be-tween different closely spaced patches of intertwined CO andSC pairing. We assume there is SC phase coherence within thepatches, and that there are many such closely spaced SC do-mains slightly above T c ( p ) forming junctions with an averagetunnel y proportional to the normal-state resistance immedi-ately above T c ( p ) . As explained previously , for a d -wavesuperconductor in single-crystal form it is sufficient to use the PGPG SC T ( K ) p T c :Y123 ref. (40,42)Y123: calc.Bi2201 ref. (11)Bi2201: calc. T* : Y123 ref. (42) Y123: calc. Bi2201 ref. (41) Bi2201: calc. SC V GL (r) 25 :Y123: calc.Bi2201: calc. CO :Y123 refs. (4,5,7)Bi2201 ref. (11) C O ( a ) , V G L ( r ) FIG. 4. Comparison of experimental and calculated values of T c and T ∗ versus hole doping. Also shown is the experimentally de-termined doping dependence of the charge order wavelength λ CO for both compounds, as well as the doping dependence of the calcu-lated absolute value of the spatial average of the free-energy potential |h V GL ( r ) i| . For display purposes |h V GL ( r ) i| is shown multiplied bya factor of 25. following relation for the average Josephson coupling energy h E J ( p, T ) i = π ~ h ∆ d ( r , T ) i e R n ( p ) tanh (cid:20) h ∆ d ( r , T ) i k B T (cid:21) (3)where h ∆ d ( r , T ) i = P Ni h ∆ d ( r i , p, T ) i /N is the pairingpotential. The quantity R n ( p ) is proportional to the normalstate in-plane resistivity ρ ab ( p, T ≥ T c ) . In what followswe assume R n ( p ) for the optimally-doped compounds, andin the case of (Pb, La)-Bi2201 use experimental values ofthe in-plane resistivity ratio ρ ab ( p ) /ρ ab ( p = 0 . to calcu-late R n ( p ) for the underdoped samples. Since the relation-ship between T c /T max c and the hole concentration p for (Pb,La)-Bi2201 is the same as for La-doped Bi2201 , we haveused available resistivity data for La-doped Bi2201 in ourcalculations shown in Table II. For orthorhombic Y123, weinstead used experimental values of the b -axis resistivity ratio ρ b ( p ) /ρ b ( p = 0 . from Ref. 47 to estimate R n ( p ) /R n ( p =0 . .As the temperature is lowered below T ∗ , thermal fluc-tuations diminish and long-range phase coherence betweenthe individual SC domains is established when k B T ≈h E J ( p, T ) i . The temperature T at which this occurs definesthe bulk critical temperature T c . Figs. 3(a) and 3(b) show thetemperature dependence of h E J ( p, T ) i for both compoundsat the different dopings. The intersection of the h E J ( p, T ) i curves with the k B T line yields values of T c in good agree-ment with the actual values for Y123 and (Pb, La)-Bi2201(Fig. 4 and Tables I and II). TABLE I. Experimental data for ∆ SC and ∆ PG determined fromuniversal curves that describe Y123 and a number of other high- T c cuprate superconductors (Ref. 44). An experimental value for ∆ SC at p = 0 . is omitted, since deviations from the universal curveare expected for Y123 near 1/8 hole doping, where T c plateaus. Thevalue of T ∗ ≈ K at p = 0 . is estimated from a linear extrap-olation of data in (Ref. 50). We tune the scaling factors explained inthe text to yield the blue and green values at optimum doping. Redvalues are calculated with the same parameters. p = 0 . p = 0 . p = 0 . λ CO ( a ) . ± .
16 3 . ± .
16 3 . ± . h V GL ( r ) i -0.156 -0.110 -0.105 ρ b (1 . T c )( µ Ω cm) ≈ ≈ ≈ ∆ SC (meV) ± - ± h ∆ d ( r ) i (meV 43.8 25.5 23.3 T c (K) ∆ PG
76 104 12476 93.4 ± ± T ∗
170 232 278170 209 ±
19 231 ± IV. THE PSEUDOGAP
Next, we use the free-energy simulations to make a simpleestimate of the PG, under the assumption that the PG appearsdue to the mesoscale phase separation that creates small do-mains of CO wavelength below T PS ≈ T ∗ . For Bi2201 weconsider a single-particle state bound to a two-dimensional(2-D) square box of depth h V GL ( r ) i and sides of length λ CO .For Y123 we consider a single-particle state bound to a stripe-like 2-D rectangular box of depth h V GL ( r ) i , width λ CO , andlength equivalent to the CDW correlation length, which ismuch longer than λ CO . We assume in both cases thatthe PG is proportional to the numerical solution of the corre-sponding 2-D Schr¨odinger equation for the ground state bind-ing energy. The proportionality factor is estimated using ex-perimental values of the pseudogap ∆ PG for Y123 and (Pb,La)-Bi2201 at p = 0 . (Tables 1 and 2), and the values of ∆ PG are calculated for the underdoped samples using the re-spective values of h V GL ( r ) i and λ CO . To further assess theaccuracy of the results for the underdoped samples, we con-vert ∆ PG to T ∗ using the experimental ratios T ∗ / ∆ PG = 170 K/76 meV and T ∗ / ∆ PG = 241 K/ 30 meV for optimally-doped Y123 and (Pb, La)-Bi2201, respectively. As shown inFig. 4 and Table II, the calculated values of T ∗ for (Pb, La)-Bi2201 at p = 0.126 and 0.141 agree well with measurementsof the PG onset temperature for La-doped Bi2201 . Reason-able agreement is also obtained between the calculated andexperimental values of T ∗ for Y123 at p = 0 . and 0.12(Fig. 4 and Table I). As mentioned in the introduction we pre-sume the onset of CO domains at T ∗ above the long rangetemperature T CO ( p ) < T ∗ ( p ) , a behavior that has been de-tected by some different experiments . TABLE II. Experimental data for ∆ SC and ∆ PG from STM with theindicated T c values . The hole doping p was determined from the T c versus p relationship obtained by X-ray absorption experimentson (Pb, La)-Bi2201 and La-doped Bi2201 . The in-plane resistivity ρ ab data area for La-doped Bi2201 and T ∗ data are from intrinsictunneling measurements . We tune the scaling factor explained inthe text to yield the blue and green values at optimum doping. Redvalues are calculated without any extra parameters. p = 0 . p = 0 . p = 0 . λ CO ( a ) . ± . . ± . . ± . h V GL ( r ) i -0.1022 -0.1018 -0.1021 ρ b (1 . T c )( µ Ω cm) ∆ SC (meV) 15 - - h ∆ d ( r ) i (meV 15.8 13.2 9.2 T c (K)
35 32 2535.2 32.5 24.7 ∆ PG ±
12 45 ±
15 68 ± T ∗
241 355 446241 327.9 427.4
V. COMPARISON OF CALCULATED ANDEXPERIMENTAL PARAMETERS
Tables I and II contain values of experimental parameters(denoted by black text) used in the calculations for each com-pound, and the calculated parameters (denoted by red text).For each cuprate family the calculated values at p = 0 . (which are denoted by blue and green text) were multipliedby a scaling factor to match experimental values as follows:i) The proportionality constant between h V GL ( r ) i and theattractive pairing potential V of Eq. (B4) was adjusted toyield a calculated value of h ∆ d ( r ) i that approximately equalsthe experimental value of the SC gap ∆ SC at p = 0 . . Thisproportionality constant, once determined, was subsequentlyused for all other values of p .ii) The scaling factor between the normal resistance R n in Eq.(3) and the resistivity ρ b (1 . T c ) just above T c was adjusteduntil the calculated value of T c at p = 0 . approximatelyequaled the experimental value. This same scaling factor wasused for all other values of p .iii) The ground state binding energy of a single-particle in a2-D square (rectangular) box was multiplied by a proportion-ality factor so as to equal the PG of (Pb, La)-Bi2201 (Y123) at p = 0 . . Again, this same proportionality constant was usedfor the calculations at other dopings. VI. FERMI ARC FORMATION
Next we show that the phase separation approach consid-ered above is able to reproduce the ungapped portion of theFermi surface (Fermi arcs) that is known to occur near thenodal region just above T c . We start by recalling thatFigs. 1(c), 1(f), 2(c) and 2(f) show domains of SC orderparameter modulations. To each domain we assign a la-bel j and a complex SC order parameter ∆ d ( k, T ) exp( i Φ j ) ,where ∆ d ( k, T ) = ∆ ( T )[cos( k x a ) − cos( k y a )] =∆ ( T ) cos(2 φ ) , φ is the azimuthal angle measured from the x -direction in the CuO plane and ∆ ( T ) is the wave func-tion amplitude in the j th domain at temperature T . The d -wave symmetry implies larger supercurrents flowing in theCuO plane along the antinodal directions parallel to the Cu-O bonds, and vanishing values along the nodal directions φ = ± π/ and ± π/ . The local intrinsic SC phase Φ j andthe superfluid density n j ∝ ∆ d ( r j , T ) are canonically con-jugate variables , leading to large fluctuations of the phase Φ j along the nodal directions, where n j and ∆ n j vanish.This is due to the quantum uncertainty principle and we maywrite ∆Φ j ( φ ) ∝ / cos(2 φ ) to indicate the azimuthal depen-dence of the phase uncertainty, which has its maximum andminimum values along the nodal and antinodal directions, re-spectively. Furthermore, ∆Φ j has a clear dependence on theJosephson coupling. In particular, as shown in the previoussection, at T < T c all Φ j are locked together leading to longrange SC order, but at T > T c phase decoupling occurs be-cause h E J ( p, T ) i < k B T and concomitantly ∆Φ j increaseswith T up to the temperature at which h E J ( p, T ) i vanishes.In particular, ∆Φ j increases monotonically from near zeroat T c to very large values near T ∗ . Furthermore, ∆Φ j hasa large anisotropy when combined with the quantum effectsdiscussed above. The two distinct contributions are separa-ble, such that ∆Φ j ( p, T, φ ) = ∆Φ j ( p, T )∆Φ j ( φ ) . We dropthe index j because ∆Φ j ( φ ) is the same for all domains, andassume h E J ( p, T ) i is the same for all j . These considera-tions imply that just above T c the electrons ejected by ARPESfrom different domains come from regions where the SC or-der parameter has essentially the same Φ j along the antinodaldirections, and ∆Φ j ( φ ) ≈ . On the other hand, such phasecoherence is lost near the nodal directions where ∆Φ j ( φ ) ≈ π is maximum. Consequently, the average SC amplitude mea-sured by ARPES may be written as follows , h ∆ d ( p, T, φ ) i = | cos(2 φ ) | ∆Φ j ( p, T, φ ) Z ∆Φ j ( p,T,φ )0 h ∆( p, T ) i cos(Φ) d Φ= h ∆( p, T ) i| cos(2 φ ) | j ( p, T, φ ) sin[∆Φ j ( p, T, φ )] . (4)This expression contains the two distinct contributions thatweaken phase coherence, one from quantum oscillations thatdepends only on the azimuthal angle φ and one from ther-mal oscillations that competes with the average Josephsoncoupling, leading to ∆Φ j ( p, T, φ ) = ∆Φ j ( p, T )∆Φ j ( φ ) forall “j“. We may take ∆Φ( φ ) ∼ / cos (2 φ ) , which satisfiesthe expected inverse cosine dependence and the square makesit symmetric and always positive around the nodal directions( φ = ± π/ and ± π/ ). Now we can infer the functional formof ∆Φ( p, T ) noting that for T < T c , all Φ j are locked to-gether leading to long range order and ∆Φ j ∼ . On the other hand, for T ≤ T c all Φ decouple because h E J ( p, T ) i < k B T and ∆Φ > . Above T c , h E J ( p, T ) i decreases with increasing T and vanishes near T ∗ . Concomitantly ∆Φ increases. Thus,there are three distinct temperature dependent regimes:(i) T ≤ T c : Since ∆Φ ∼ , Eq. (4) is easy tosolve and we obtain the “bare“ expression h ∆ d ( r , T, φ ) i = h ∆ d ( p, T ) i| cos(2 φ ) .(ii) T > T ∗ : h ∆( p, T ) i ∼ and it is clear that there is no gap.(iii) T c < T : Taking into account the effect of h E J ( p, T ) i weassume ∆Φ( p, T ) = A [1 − h E J ( p, T ) i /k B T c ( p ) where A is aconstant. This expression vanishes at T c and increases mono-tonically with p , as expected from ARPES experiments .Thus, putting all together, ∆Φ( p, T, φ ) = [ A/ cos (2 φ )][1 −h E J ( p, T ) i /k B T c ( p ) and we obtain the value of A by com-paring with the onset of the measured gapless region fora given sample. To reproduce the measured gapless regionswe also assume in Eq. (4) that h ∆ d ( p, T, φ ) i ∼ whenever ∆Φ( p, T, φ ) ≥ π , due to destructive phase interference fromelectrons ejected from different domains.Specifically we use the ARPES measurements at T =102 K for the Bi2212 compound with T c = 92 K, whichshows a gapless region between ≤ φ ≤ to derivethe constant A . Equating ∆Φ( p ∼ . , T = 102K , φ =28 , ) = π , yields h ∆ d ( p, T, φ ) i ∼ and this is possi-ble if we take A = 2 . π . Note that φ is measured from the ( π, π ) to (0 , π ) direction of the Brillouin zone according toRefs. 51 and 52. With ∆Φ( p, T, φ ) determined, we may ap-ply the derived equation to any sample. In particular, we ap-ply this expression to the other two Bi2212 compounds mea-sured in Ref. 51. Some above T c values of h E J ( p, T ) i usedin the calculations are plotted in Fig.(5) for illustration pur-pose. Accordingly we obtain for the T c = 86 K compounda gapless region at . ≤ φ ≤ . , which compareswell with the experimental ≤ φ ≤ . For the under-doped T c = 75 K Bi2212 sample at T = 85 K, we obtain . ≤ φ ≤ . , which is also in good agreement with theexperimental result . We summarize the Fermi arc calcula-tions for the three samples in Fig. 6, where the results of the”envelope” phase factor of h ∆ d ( p, T, φ ) i R = | cos(2 φ ) | ∆Φ( p, T, φ ) × sin[∆Φ j ( p, T, φ )] (5)from Eq. (4) are in good agreement with the experiments .The arrows mark the experimentally determined onset of thegapless regions for each sample, as described above. VII. SUMMARY AND CONCLUSIONS
Our theory infers a fundamental link between the period-icity of the CDW and the PG and SC energy scales of high-temperature cuprate superconductors, and shows that withinthis framework it is possible to account for the onset temper-atures T c and T ∗ of two different cuprate families. We stressthat the only quantitative assumptions made in our calcula-tions for underdoped Y123 and (Pb, La)-Bi2201 pertain to anatural scaling factor, which we have determined by scalingcalculated free energy parameters to achieve values of the PG FIG. 5. Josephson coupling energy phase diagram and schematicFermi surface. The variation of the average Josephson coupling en-ergy with doping and temperature for Bi2212 (from Ref. 45). Abovethe onset of h E J i /k B = 0 K there is no SC gap. The correspondinggapless Fermi surface is depicted at the right of the phase diagram.In the region of the phase diagram where T ≥ T c , the average SCgap may be finite. However, combined thermal and quantum phasefluctuations may cause destructive interference in the ARPES dataalong the nodal directions, leading to the gapless Fermi arcs shownin the left of the figure.FIG. 6. The phase factor from the SC fluctuation as function of theazimuthal angle. The envelope of the SC amplitude h ∆ d ( p, T, φ ) i according to Eq. (4). The arrows show the limits of the gaplessregion as determined from ARPES experiments on Bi2212 at T = T c ( p ) + 10 K for three dopings . and SC gap that agree with experimental values at one partic-ular doping. Our model is general in the sense that it can beapplied to other cuprate families, provided the doping depen-dence of the CDW order is known.Our approach generates a local free energy potential havinga spatial periodicity that matches that of the experimentallyobserved short-range static CDW order. Our calculations inthe framework of BdG theory yield different SC amplitudesin distinct charge-ordered domains that generally vanish onlyabove T c . Our approach is consistent with experiments that measure a finite SC amplitude above T c , and promotesthe scenario whereby the SC resistive transition marks the on-set of global phase coherence between SC domains. In ourmodel Fermi arcs appear above T c because there are large phase fluctuations along the nodal directions where the su-perfluid density vanishes. The increase of the arcs size with p is reproduced because the dependence of h E J ( p, T ) i on thetemperature changes rapidly with doping.Finally we address the experimental observations indicat-ing a competition between superconductivity and CO. WhileX-ray experiments show a decrease of the CDW diffraction in-tensity and correlation length below T c , these measure-ments seem to be detecting static charge correlations. StaticCDW order competes with superconductivity by reducing thenumber of charge carriers available for pairing. On the otherhand, our theory requires that dynamic CO is also present toinduce a fluctuating hole-pair potential that scales with V GL .While there is some evidence for CDW fluctuations fromoptical pump-probe and low-energy, momentum-resolvedelectron energy-loss spectroscopy experiments, there is cur-rently insufficient experimental information to assess the per-vasiveness or significance of fluctuating CO in cuprates. ACKNOWLEDGMENTS
We thank S. Ono and Y. Ando for providing us with theirresistivity data. We also thank Andrea Damascelli, Andr´e-Marie Tremblay and John Tranquada for informative discus-sions. Supported by the Brazilian agencies FAPERJ andCNPq (E.V.L.M.); the Canadian Institute for Advanced Re-search (CIFAR) and the Natural Sciences and Engineering Re-search Council of Canada (J.E.S.).
Appendix A: CHARGE ORDER SIMULATIONS
To describe the growth and development of spatial chargeinhomogeneity in the CuO planes we applied a theoryof phase-ordering dynamics, whereby the system evolvesthrough domain coarsening when quenched from a homoge-neous into a broken-symmetry phase. The time-dependent CHapproach provides a simple way to determine the time evolu-tion of the CO process . The CH equation can be written inthe form of the following continuity equation for the local freeenergy current density J = − M ∇ (( ∂f /∂u ) ∂u∂t = −∇ . J = − M ∇ (cid:20) ε ∇ u + ∂V GL ∂u (cid:21) , (A1)where M is the charge mobility that sets the phase separationtime scale. The order parameter varies between u ( r , t ) ∼ for the homogeneous system above the phase-separation onsettemperature T PS , and u ( r , t → ∞ ) = ± for the extreme caseof complete phase separation. We solved the CH equation bya stable and fast finite difference scheme with free boundaryconditions . The spatial dependence of the charge densityobtained by numerically solving the CH equation evolves withtime t = nδt , where n is the number of time steps δt . Whenthe order parameter is conserved, as in phase separation, thecharges can only exchange locally rather than over large dis-tances. This leads to diffusive transport of the order parame-ter. Consequently, at early times or small n, we obtain chargemodulations with periodicities of only a few lattice constants.Using different parameters and initial conditions we are ableto reproduce the experimentally determined CO patterns incuprates. Although these simulations are not the stable solu-tions of the CH equation (as is clear from the time evolutionof the simulations shown in Fig. 7), the aim here is to generateperiodic charge modulations with experimentally determinedwavelengths that can be subsequently used to calculate the SCgap and PG in our phase- separation model. For convergencethe time step δt and spatial step h ≈ /N for a square latticeof N sites must be such that δt ≤ h / (Ref. 55). For thecalculations here we used δt ≤ h / and h = 1 / .In the main paper we present detailed CO, T c and T ∗ cal-culations for six compounds. Three of the Bi2201 and threeof the Y123 families. In the next two paragraphs we give thevalues of some parameter used in the CH simulations. (Pb, La)-Bi2201 : Simulations with α = B = 1 , timesteps of n = 700 , , and ε = 0 . , . , . yield checkerboard CO patterns with the desired wavelengths λ CO for (Pb, La)-Bi2201 (at p = 0 . , . and . ) near . a , . a , and . a , respectively. The fewer time steps re-quired to simulate the CO patterns of the underdoped samplesis indicative of a reduced charge mobility, and is consistentwith an increase of the normal-state resistivity. At later times(i.e., greater n) the periodic electronic structure evolves intoan irregular patch-like system of segregated low- and high-charge density regions. In addition, the length scale of the sys-tem increases with the phase separated regions forming largerdomains. This latter situation was considered previously .Fig. 7 shows CH simulations of u ( r , t ) at times beyond wherecheckerboard CO with λ CO = 4 . a is observed in (Pb, La)-Bi2201 at p = 0 . . Y123 : Simulations with α = 1, B = 5, ε = 0.0053, 0.0055,0.0058 and time steps n = 35 , , yield charge stripe pat-terns with the desired wavevectors Q = 0 . , . and0.287 r.l.u. ( λ CO = 1 /Q ) estimated from (Refs. 4, 5, and7) for Y123 at p = 0 . , . and 0.16, respectively. Notethat the values of n are much shorter than needed to simulatethe checkerboard CO patterns of (Pb, La)-Bi2201. Because ofthe fewer time steps, the simulations for Y123 are somewhatless sharp. Appendix B: Combined Bogoliubov-deGennes (BdG) andCahn-Hilliard (CH) Calculations
We performed self-consistent calculations with the BdGtheory (Refs.58 and 59) for each of the CH simulated chargedensity maps (Figs. 1(a) and 1(d), and Figs. 2(a) and2(d)). To calculate the SC pairing amplitude we assumedthe attractive interaction potential V scales with V minGL ∼h V GL ( r ) i . The SC calculations begin with the extended Hub-bard Hamiltonian . To describe the charge carriers dynam-ics in the CuO planes of the HTSC we consider this Hamil- tonian in a square lattice H = − X { ij } σ t ij c † iσ c jσ + X iσ µ i n iσ + U X i n i ↑ n i ↓ − V X h ij i σσ ′ n iσ n jσ ′ , (B1)where c † iσ ( c iσ ) is the usual fermionic creation (annihilation)operator at site i , the spin σ is up ↑ or down ↓ . n iσ = c † iσ c iσ isthe number operator, and t ij is the hopping between sites i and j . U is the magnitude of the on-site repulsion, and V is themagnitude of the nearest-neighbor attractive interaction. µ i isthe local chemical potential derived in the self-consistent pro-cess through which the local charge density is calculated bythe CH equation and is maintained fixed. For (Pb, La)-Bi2201we used nearest-neighbor hopping t = 0 . eV, next-nearest-neighbor hopping t = − . t , and third nearest-neighborhopping t = 0 . t derived from angle-resolved photoemis-sion spectroscopy (ARPES) dispersion relations . For Y123,we used the ARPES results t = 0 . meV, t = − . t and t = 0 . t (Ref. 61). The BdG mean-field equations are K ∆ d ( r )∆ ∗ d ( r ) − K ∗ u n ( r ) v n ( r ) = E n u n ( r ) v n ( r ) (B2)with Ku n ( r ) = − X R t r , r + R u n ( r + R ) + µ ( r ) u n ( r )∆ d u n ( r ) = X R ∆ d ( r ) u n ( r + R ) , (B3)and similar equations for v n ( r ) , where r+R is the positionof the nearest-neighbor sites, and µ ( r ) ≡ µ i is the localchemical potential. These equations are solved numericallyfor eignevalues E n ( ≥ self-consistently with the spatially-varying d -wave pairing potential ∆ d ( r ) = − V P n [ u n ( r ) v ∗ n ( r + R )+ v ∗ n ( r ) u n ( r + R )] tanh E n k B T , (B4)where V = V ( T, p ) was defined in Eq. 2. The results of h ∆ d ( r , T ) i are plotted in Fig. 8 for the three compounds ofthe Y123 system. Concomitantly, the spatially-varying holedensity of charge carriers is given by p ( r ) = 1 − X n [ | u n ( r ) | f n + | v n ( r ) | (1 − f n )] , (B5)where f n = [exp( E n /k B T + 1] − is the Fermi occupationfunction. It is important to emphasize that the spatially inho-mogeneous distribution of charge generated by the CH equa-tion for different dopings was kept fixed while the local chem-ical potential µ ( r ) was self-consistently determined in theconvergence process. This procedure incorporates the chargeinhomogeneity in the calculations in a natural way. FIG. 7. Two-dimensional CH simulations of u ( r, t ) for α = B = 1 , ε = 0 . and time steps (a) n = 900 , (b) n = 1500 , and (c) n = 9000 .These plots are continuation of the time evolution of Fig. 2(a) with n = 700 .FIG. 8. Example of calculated h ∆ d ( p, T ) i used to obtain T c : Theaverage SC amplitudes for Y123 from BdG Eq. B4 used in the cal-culations of h E J ( p, T ) i [see Eq. 3]. The low temperature limits of h ∆ d ( p, T = 0) i are also listed in Table I. ∗ Corresponding author: [email protected] J. Chang, E. Blackburn, T. Holmes, N. B. Christensen, J. Larsen,J. Mesot, Ruixing Liang, D. A. Bonn, W. N. Hardy, A. Wa-tenphul, M. V. Zimmermann, E. M. Forgan, and S. M.Hayden, “Direct observation of competition between supercon-ductivity and charge density wave orderin
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