Incoherent superconductivity well above T_c in high-T_c cuprates - harmonizing the spectroscopic and thermodynamic data
aa r X i v : . [ c ond - m a t . s up r- c on ] J u l Incoherent superconductivity well above T c inhigh- T c cuprates – harmonising the spectroscopicand thermodynamic data J G Storey , Robinson Research Institute, Victoria University of Wellington, P.O. Box 600,Wellington, New Zealand MacDiarmid Institute, Victoria University of Wellington, P.O. Box 600, Wellington,New ZealandE-mail: [email protected]
Abstract.
Cuprate superconductors have long been known to exhibit an energy gapthat persists high above the superconducting transition temperature ( T c ). Debatehas continued now for decades as to whether it is a precursor superconducting gapor a pseudogap arising from some competing correlation. Failure to resolve thishas arguably delayed explaining the origins of superconductivity in these highlycomplex materials. Here we effectively settle the question by calculating a varietyof thermodynamic and spectroscopic properties, exploring the effect of a temperature-dependent pair-breaking term in the self-energy in the presence of pairing interactionsthat persist well above T c . We start by fitting the detailed temperature-dependenceof the electronic specific heat and immediately can explain its hitherto puzzlingfield dependence. Taking this same combination of pairing temperature and pair-breaking scattering we are then able to simultaneously describe in detail the unusualtemperature and field dependence of the superfluid density, tunneling, Raman andoptical spectra, which otherwise defy explanation in terms a superconducting gap thatcloses conventionally at T c . These findings demonstrate that the gap above T c in theoverdoped regime likely originates from incoherent superconducting correlations, andis distinct from the competing-order “pseudogap” that appears at lower doping.PACS numbers: 74.72.Gh, 74.25.Bt, 74.25.nd, 74.55.+v Keywords : cuprates, scattering, specific heat, superfluid density, Raman spectroscopy,tunneling, optical conductivity
Submitted to:
New J. Phys. ncoherent superconductivity well above T c in high- T c cuprates (a) (b) (c) T * T * T c Hole Doping T e m pe r a t u r e Competing orderd ringgapPrecursor pairingpseudogap g orderudogap
Figure 1.
Candidate phase diagrams for the hole-doped cuprates. For simplicity onlythe superconducting and pseudogap phases are shown. (a) Precursor pairing scenario.(b) Competing or coexisting order parameter scenario. (c) Combined phase diagramproposed in this work.
1. Introduction
A prominent and highly debated feature of the high- T c cuprates is the presence of anenergy gap at or near the Fermi level which opens above the observed superconductingtransition temperature. It is generally known as the “pseudogap”. Achieving a completeunderstanding of the pseudogap is a critical step towards the ultimate goal of uncoveringthe origin of high-temperature superconductivity in these materials. For example,knowing where the onset of superconductivity occurs sets limits on the strength ofthe pairing interaction. The community has long been divided between two distinctviewpoints[1]. These can be distinguished by the doping dependence of the so-called T ∗ line[2], the temperature below which signs of a gap appear. The first viewpoint holdsthat the pseudogap represents precursor phase-incoherent superconductivity or “pre-pairing”. In this case of a single d -wave gap the pseudogap opens at T ∗ and evolvesinto the superconducting gap below T c . The underlying Fermi surface is a nodal-metal,appearing as an arc due to broadening processes[3, 4]. Here the T ∗ line merges smoothlywith the T c dome on the overdoped side (see figure 1(a)). The second viewpoint is thatthe pseudogap arises from some as yet unidentified competing and/or coexisting order.In this two-gap scenario the pseudogap is distinct from the superconducting gap with adifferent momentum dependence, likely resulting from Fermi surface reconstruction[5].The T ∗ line in this case bisects the T c dome and need not be a transition temperature inthe thermodynamic sense or “phase transition”, where it would instead mark a crossoverregion defined by the energy of a second order parameter given by E g ≈ k B T ∗ (seefigure 1(b)). Ironically, the multitude of different techniques employed to study thepseudogap has lead to much confusion over the exact form of the T ∗ line.However, an alternative picture is beginning to emerge that encompasses bothviewpoints (see figure 1(c)). Small superconducting coherence lengths in the high- T c cuprates give rise to strong superconducting fluctuations that are clearly evidentin many techniques. Thermal expansivity[6], specific heat[7, 8], resistivity[9], Nernsteffect[10, 11, 12], THz conductivity[13], IR conductivity[14] and Josephson effect[15] ncoherent superconductivity well above T c in high- T c cuprates T c and does not track the T ∗ line[9, 11, 14, 8]which extends to much higher temperatures at low doping. An effective superconductinggap feature associated with these fluctuations which tails off above T c can be extractedfrom the specific heat [16]. And pairing gaps above T c have been detected by scanningtunneling microscopy in this temperature range[17]. Evidence for a second energy scale,which from here will be referred to specifically as the pseudogap, includes a downturn inthe normal-state spin susceptibility[18, 19] and specific heat[19], a departure from linearresistivity[20, 21, 22], and a large gapping of the Fermi surface at the antinodes byangle-resolved photoemission spectroscopy (ARPES)[23, 24, 25, 26]. The opening of thepseudogap at a critical doping within the T c dome can be inferred from an abrupt dropin the doping dependence of several properties. These include the specific heat jump at T c [19], condensation energy[19], zero-temperature superfluid density[19, 27], the criticalzinc concentration required for suppressing superconductivity[28], zero-temperature self-field critical current[29], and the Hall number[30], most of which represent ground-stateproperties. The last signals a drop in carrier density from 1 + p to p holes per Cu, andcan be explained in terms of a reconstruction from a large to small Fermi surface[5]. Ator above optimal doping the pseudogap becomes similar or smaller in magnitude thanthe superconducting gap and, since many techniques return data that is dominated bythe larger of the two gaps, it has been historically difficult to determine which gap isbeing observed. In this work it will be demonstrated explicitly that in this doping rangeit is in fact the superconducting gap persisting above T c that is being observed, therebyending the confusion over the shape of the T ∗ line.This work was inspired by two recent studies. The first by Reber et al .[31] fittedthe ARPES-derived tomographic density of states using the Dynes equation[32] I T DoS = Re ω − iΓ s p ( ω − iΓ s ) − ∆ (1)to extract the temperature dependence of the superconducting gap ∆ and the pair-breaking scattering rate Γ s . They found that ∆ extrapolates to zero above T c whileΓ s increases steeply near T c . They also found that T c occurs when ∆ = 3Γ s .Importantly, these parameters describe the filling-in behaviour of the gapped spectrawith temperature (originally found in tunneling experiments e.g. [33, 34, 35] and alsoinferred from specific heat and NMR[36]), as opposed to the closing behaviour expectedif ∆ was to close at T c in the presence of constant scattering.The second study, by Kondo et al .[37], measured the temperature dependence ofthe spectral function around the Fermi surface using high-resolution laser ARPES. Thiswas fitted using the phenomenological self-energy proposed by Norman et al .[38]Σ( k , ω ) = − i Γ single + ∆ ω + ξ ( k ) + i Γ pair (2)where ξ ( k ) is the energy-momentum dispersion, Γ single is a single-particle scattering rateand Γ pair is a pair-breaking scattering rate. The gap is well described by a d -wave BCS ncoherent superconductivity well above T c in high- T c cuprates T pair above the observed T c . Γ pair increases steeply near T c , with T c coinciding with the temperature where Γ pair = Γ single .The aim of the present work is to investigate whether other experimental properties areconsistent with this phenomenology. The approach is to fit the bulk specific heat using 2then, using the same parameters, calculate the superfluid density, tunneling and Ramanspectra, and optical conductivity. To reiterate, the focus here is the overdoped regimenear T c where the pseudogap and subsidiary charge-density-wave order are absent[39].
2. Results
The Green’s function with the above self-energy (2) is given by G ( ξ, ω ) = 1 ω − ξ + i Γ single − ∆ ω + ξ + i Γ pair (3)The superconducting gap is given by ∆ = ∆ δ ( T ) cos 2 θ , where ∆ = 2 . k B T p and δ ( T )is the d -wave BCS temperature dependence. θ represents the angle around the Fermisurface relative to the Brillouin zone boundary and ranges from 0 to π/
2. The densityof states g ( ω ) is obtained by integrating the spectral function A ( ξ, ω ) = π − ImG( ξ, ω ) g ( ω ) = Z A ( ξ, ω ) dξdθ (4)The electronic specific heat coefficient γ ( T ) = ∂S/∂T is calculated from the entropy S ( T ) = − k B Z [ f ln f + (1 − f ) ln(1 − f )] g ( ω ) dω (5)where f is the Fermi distribution function. The temperature dependence of Γ pair isextracted by using it as an adjustable parameter to fit specific heat data under thefollowing assumptions: i) the superconducting gap opens at T p = 120 K, at the onset ofsuperconducting fluctuations; and ii) A linear-in-temperature Γ single ranging from 5 meVat 65 K to 14 meV at 135 K, similar to values reported by Kondo et al .[37]. A difficultyin applying this approach over the whole temperature range is that the T -dependenceof the underlying normal-state specific heat γ n must be known. Therefore attention willbe focused close to T c on Bi Sr CaCu O δ data[19] with a doping of 0.182 holes/Cu,where γ n can be taken to be reasonably constant. In practice the quantity fitted is thedimensionless ratio of superconducting- to normal-state entropies S s ( T ) /S n ( T ).Fits and parameters are shown in figure 2 for data measured at zero and 13 Tapplied magnetic field. Γ pair increases steeply near T c in a very similar manner tothe scattering rates found from the ARPES studies mentioned above. No particularrelationship between Γ pair , Γ single and T c is observed, however the peak of the specificheat jump occurs when Γ pair = ∆. In other words, once the pair-breaking becomes ofthe order the superconducting gap the entropy changes less rapidly with temperature,which intuitively makes sense. This appears to differ significantly with the result∆( T c ) = 3Γ s ( T c ) from Reber et al .[31], but note that fitting with the Dynes equation ncoherent superconductivity well above T c in high- T c cuprates
60 70 80 90 100 110 120 130 1400.00.51.01.52.02.53.0 singlepairn (13T) ( m J / ga t K ) Temperature (K)0T13T(0T) , s i ng l e , pa i r ( m e V ) Figure 2.
Fits (blue and magenta lines) made to the electronic specific heat ofslightly overdoped Bi Sr CaCu O δ ( p = 0.182 holes/Cu) at zero and 13 T fields(black lines) using the self-energy given by 2. Γ pair is the adjustable parameter and∆( T ) and Γ single ( T ) are assumed. (1) returns a smaller scattering rate Γ s equal to the average of Γ single and Γ pair . Apuzzling feature of the cuprate specific heat jump is its non-mean-field-like evolutionwith magnetic field[40, 41]. Rather than shifting to lower temperatures, it broadensand reduces in amplitude with little or no change in onset temperature. The fitsexplain this in terms of an increase in Γ pair with field, without requiring a reductionin gap magnitude. Note that taking ∆( H ) = ∆ p − ( H/H c2 ) from Ginzburg-Landautheory[42], the estimated reduction in the gap at 13 T near T c is only 7 to 2 percent forupper critical fields in the range 50 to 100 T. Other properties will now be calculatedusing the parameters in figure 2. The two scattering rates, Γ pair and Γ single are inserted into the anomalous Green’sfunction F as follows F ( ξ, ω ) = ∆( ω + ξ + i Γ pair ) (cid:16) ω − ξ + i Γ single − ∆ ω + ξ + i Γ pair (cid:17) (6)The superfluid density ρ s is proportional to the inverse square of the penetration depth( λ ) calculated from[43]1 λ ( T ) = 16 πe c V X k v x Z dω ′ dω ′′ lim q → (cid:20) f ( ω ′′ ) − f ( ω ′ ) ω ′′ − ω ′ (cid:21) × B ( k+q , ω ′ ) B ( k , ω ′′ ) (7) ncoherent superconductivity well above T c in high- T c cuprates B is given by the imaginary part of F . Fora free-electron-like parabolic band ξ ( k ) = ~ ( k x + k y ) / m − µ , v x = ~ k x /m = p ξ + µ ) /m cos θ and changing variables to ξ and θ gives1 λ ( T ) ∝ Z ( ξ + µ ) cos θ Z (cid:20) f ( ω ′′ ) − f ( ω ′ ) ω ′′ − ω ′ (cid:21) × B ( ξ, θ, ω ′ ) B ( ξ, θ, ω ′′ ) dω ′ dω ′′ dξdθ (8)The T -dependence of Γ pair causes a clear steepening of ρ s away from the BCS T -dependence, with the main onset being pushed down from T p to T c , see figure 3(a).The same result can be obtained using one scattering rate equal to the average of Γ single and Γ pair at each temperature. When plotted in terms of reduced temperature T /T c ,there is a very good match with experimental data from optimally doped cuprates(figure 3(b)). The data, taken by different techniques, includes a YBa Cu O − δ (YBCO) crystal[44] and film[45] with T c ’s near 90 K, as well as a (BiPb) (SrLa) CuO δ crystal[46] with a T c of 35 K. This raises the question as to whether the mootedBerezinskii-Kosterlitz-Thouless universal jump in superfluid density may not simply beattributable to the rapid increase in pair breaking scattering rate near T c arising fromfluctuations on a pairing scale that exceeds T c [47]. Although the tail above T c is notevident in the selected experimental data, it is observed elsewhere in the literature[48].There is a resemblance to an approximate strong-coupling T -dependence (dotted linein figure 3(a)), calculated from a rescaled BCS gap of magnitude ∆ = 2 . k B T c closing at T c = 94 K, in the absence of strong pair-breaking. However, as will beseen in the following sections, this interpretation of ρ s ( T ) is inconsistent with otherobservations. The suppression in superfluid density with field bears a qualitativesimilarity to field-dependent measurements on a YBCO thin film[49], but because of thatsample’s apparent low upper critical field the calculated suppression is much smaller inmagnitude. The current-voltage curve for a superconductor-insulator-superconductor (SIS) tunneljunction is calculated from[50] I ( V ) ∼ Z g ( E ) g ( E − eV )[ f ( E ) − f ( E − eV )] dE (9)where g ( E ) is the density of states given by 4. The tunneling conductance dI/dV isplotted in figure 4 for several temperatures around T c . The evolution of the spectrawith temperature is very consistent with experimental observations[51, 35, 52, 53, 54].These show a filling-in of the gap with temperature and a broadening and suppressionof the peaks at 2∆, with little or no shift in their positions. This is contrary to theexpected shift toward zero voltage that would occur for a strong coupling gap closingat T c in the absence of pair-breaking scattering. A depression persists above T c andvanishes as T p is approached, where the superconducting gap closes. Remember that apseudogap is not included in these calculations. The linearly sloping background seen ncoherent superconductivity well above T c in high- T c cuprates ( ) / ( T ) T/T c -2 ( single , pair ) H=0T YBCO crystal YBCO film Bi2201 crystal (b) d-wave BCS /k B T p =2.14 -2 ( single , pair ) H=0T -2 ( single , pair ) H=13T T c =94K /k B T c =2.9 ( ) / ( T ) Temperature (K)(a) T p Figure 3. (a) Normalized superfluid density calculated using the parameters from fitsto the specific heat in figure 2 (red and green lines). (b) Comparison of the calculatedzero-field superfluid density with experimental data from [44, 45, 46]. ncoherent superconductivity well above T c in high- T c cuprates -0.2 -0.1 0.0 0.1 0.20.00.20.40.60.81.01.21.4 d I/ d V ( a r b . un i t s ) bias (eV) T(K) 70 80 90 100 112
Figure 4.
SIS junction tunneling conductance at several temperatures around T c calculated using the parameters from fits to the specific heat in figure 2. Inset:Experimental data reproduced from [51]. in the experimental data can be reproduced by adding a linear-in-frequency term, asseen in ARPES[55], to Γ single . Another property that supports the persistence of ∆ above T c is the Raman B responsegiven by[56] χ ′′ ( ω ) = X k ( γ B k ) Z dω ′ π [ f ( ω ′ ) − f ( ω ′ + ω )] × [ A ( k , ω ′ + ω ) A ( k , ω ′ ) − B ( k , ω ′ + ω ) B ( k , ω ′ )] (10)The Raman B vertex γ B k ∝ cos k x − cos k y ∼ cos 2 θ probes the antinodal regions ofthe Fermi surface where ∆( k ) is largest. Changing variables from k to ξ and θ gives χ ′′ ( ω ) ∝ Z dξdθ cos (2 θ ) Z dω ′ [ f ( ω ′ ) − f ( ω ′ + ω )] × [ A ( ξ, θ, ω ′ + ω ) A ( ξ, θ, ω ′ ) − B ( ξ, θ, ω ′ + ω ) B ( ξ, θ, ω ′ )] (11)The superconducting Raman B response function with the normal-state responseat 122 K subtracted is shown in figure 5(a) for several temperatures around T c . The ncoherent superconductivity well above T c in high- T c cuprates ’’ s () - ’’ n () ( a r b . un i t s ) Raman shift (cm -1 ) T (K) 70 76 80 85 90 100 112 B Antinodal (a)
This work p = 0.21 p = 0.19 (b) N o r m a li z ed P ea k A r ea T/T c Figure 5. (a) Difference between the superconducting and normal-state (i.e. justabove T p ) antinodal ( B ) Raman response functions at temperatures around T c ,calculated using the parameters from fits to the specific heat in figure 2. (b) Normalizedarea under the curves in (a) compared with experimental values from [57] for dopings p =0.19 and 0.21. resemblance to experimental data, reported in [58, 59, 57], is striking. Like the tunnelingresults above, the peak at 2∆ broadens and reduces in amplitude and barely shifts withtemperature indicating that the gap magnitude is still large at T c [58]. Figure 5(b) showsthe normalized area under the curves in (a) versus reduced temperature ( T /T c =94K),together with data from [57]. The calculations show that data plotted in this way giveslittle indication of a gap above T c . The final property considered in this work is the ab -plane optical conductivity calculatedfrom[60] σ ( ω ) = e ω X k v ( k ) Z dω ′ π [ f ( ω ′ ) − f ( ω ′ + ω )] × [ A ( k , ω ′ ) A ( k , ω ′ + ω ) + B ( k , ω ′ ) B ( k , ω ′ + ω )] (12)where v ab ( k ) = p v x + v y . Again a change of variables is made from momentum toenergy and Fermi surface angle as follows σ ( ω ) ∝ ω Z dξdθ ( ξ + µ ) Z dω ′ [ f ( ω ′ ) − f ( ω ′ + ω )] × [ A ( ξ, θ, ω ′ ) A ( ξ, θ, ω ′ + ω ) + B ( ξ, θ, ω ′ ) B ( ξ, θ, ω ′ + ω )] (13)Spectra at several temperatures around T c are plotted in figure 6. A suppression is ncoherent superconductivity well above T c in high- T c cuprates () ( a r b . un i t s ) (cm -1 ) T(K) 122 112 100 90 85 80 76 70 Figure 6. ab -plane optical conductivity at several temperatures around T c calculatedusing the parameters from fits to the specific heat in figure 2. Inset: Experimentaldata reproduced from [61] visible below 2∆ at low temperature that fills in as temperature is increased. A gapclosing at T c would result in the onset of this suppression shifting to lower frequency.The calculations bear a strong qualitative resemblance to the overdoped data reportedby Santander-Syro et al .[61]
3. Discussion
As summarized in table 1 only the superfluid density, and more approximately thezero-field specific heat, can be interpreted by a strong-coupling gap closing at T c inthe absence of scattering. The non-mean-field T -dependence of all properties examinedin this work is instead well described in terms of a superconducting gap that persistsabove T c , in the presence of a steep increase in scattering. This result is insensitive tothe addition of linear-in-frequency terms or a cos 2 θ momentum dependence to Γ pair andΓ single . The scattering is further enhanced by magnetic field. What is the origin of thescattering and can it be suppressed to bring T c up to T p ? A rapid collapse in quasiparticlescattering below T c , also found in microwave surface impedance measurements[62],is expected when inelastic scattering arises from interactions that become gapped orsuppressed below T c [63]. The spin fluctuation spectrum is a plausible candidate andhas been investigated extensively[64, 65], although those calculations assumed that thesuperconducting gap closes at T c .The work presented here illustrates that the merging of the T ∗ line on the lightly ncoherent superconductivity well above T c in high- T c cuprates Table 1.
Interpretations of the experimental data.
Technique ∆ closing at T c ∆ above T c (strong coupling) plus scatteringSpecific heat X X
Penetration depth
X X
ARPES X Tunneling X Raman spectroscopy X Optical conductivity X overdoped side of the T c dome is not a product of the pseudogap per se, but ratherthe persistence of the superconducting gap into the fluctuation region between T c and T p . As doping increases, this region becomes narrower and experimental propertiesbecome more mean-field-like. Switching direction, as doping decreases the pseudogapopens, grows, and eventually exceeds the magnitude of the superconducting gap at theantinodes. When this occurs, the gap associated with T ∗ changes to the pseudogap.In other words, T ∗ is given by the larger of T p and E g / k B (see figure 1(c)). Such aninterpretation makes immediate sense of the phase diagram presented by Chatterjee etal .[66] Acknowledgments
Supported by the Marsden Fund Council from Government funding, administered bythe Royal Society of New Zealand. The author acknowledges helpful discussion withJ.L. Tallon.
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