Dependence of Upper Critical Field and Pairing Strength on Doping in Cuprates

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Science  03 Jan 2003:
Vol. 299, Issue 5603, pp. 86-89
DOI: 10.1126/science.1078422


We have determined the upper critical fieldH c2 as a function of hole concentration in bismuth-based cuprates by measuring the voltage induced by vortex flow in a driving temperature gradient (the Nernst effect), in magnetic fields up to 45 tesla. We found that H c2decreased steeply as doping increased, in both single and bilayer cuprates. This relationship implies that the Cooper pairing potential displays a trend opposite to that of the superfluid density versus doping. The coherence length of the pairs ξ0 closely tracks the gap measured by photoemission. We discuss implications for understanding the doping dependence of the critical temperatureT c0.

The superconducting state in a metal is completely suppressed if a sufficiently strong magnetic field is applied. In individual type-II superconductors, the field required—defined as the upper critical field H c2—is an important parameter because it determines the value of the coherence length ξ0(the size of the Cooper pair) as well as the strength of the pairing potential; the higher the field H c2, the stronger is the pairing potential and the smaller the pair size (1). In the phase diagram of the cuprates, superconductivity has been observed in the range of hole concentration 0.05 <x < 0.25. Many parameters of the superconducting state, notably the superfluid density and superconducting gap, have been measured as a function of x. The conspicuous exception isH c2, which is uncertain for reasons discussed below. Because even the basic trend of H c2versus x is unknown, the crucial question of whether the pairing strength, as distinct from the superfluid density, increases or decreases withx remains unanswered. We report measurements ofH c2 versus x in the Bi-based cuprates using the vortex-Nernst effect. In both single and bilayer systems, it was found that H c2 (and hence the pairing potential) steeply decreased as x increased. We show that ξ0 is intimately related to the gap measured by angle-resolved photoemission spectroscopy (ARPES) (2) and results from scanning tunneling microscopy (STM) (3,4).

In the Nernst effect (5–11), vortices in the vortex liquid state are driven down an applied temperature gradient –∇Tx. Their velocity v induces an electric field Ey =Bvx which may be detected with high sensitivity (the induction field Bz). The total Nernst signal e obs y =Ey /∣∇T∣ is the sum of the vortex signal ey and the carrier contributione N y:e obs y =ey +e N y.

Although our focus is on the Bi-based cuprates, we gain perspective and insight by first looking at the electron-doped cuprate Nd2–xCexCuO4(NCCO), in which the effects of fluctuations are weak and superconductivity is easily suppressed in fields of ∼10 T. Previous Nernst measurements have revealed a sizable carrier signale N y in NCCO (10, 11). We observed thatey has a distinctive “tent” profile that distinguishes it from e N y(fig. S1). The vortex signal ey at fixedT was initially zero until H exceeded the solid-to-liquid melting field H m(T) (Fig. 1A). In the liquid state, the vortex signal rises steeply to a maximum before falling monotonically to zero as H approaches H c2, with a similar profile occurring in low–critical T(low-T c) superconductors (7,8). The steep rise just above H m, which scales accurately with the in-plane resistivity ρ (dashed line), reflects the sharp increase in vortex mobility, whereas the fall at higher fields is caused by the field suppression of the condensate amplitude as H approaches H c2. The phase diagram in Fig. 1B, derived from ey , reveals an H c2 boundary similar to that observed in conventional type-II superconductors and well described by the Bardeen, Cooper, and Schrieffer (BCS) theory (1). It is weakly T-dependent at low T and decreases to zero linearly as T approaches T c0, the zero-field transition temperature [the linearH c2 near T c0 agrees with that in an earlier Nernst study (11)]. The vortex signaley rapidly vanishes aboveT c0 = 24.5 K, indicating that the amplitude of the order parameter Ψ vanishes at T c0. Hence, the phase diagram of NCCO is similar to the phase diagram of conventional superconductors (1), except that the vortex liquid region is greatly expanded. This example shows thatH c2 at finite T is reliably determined as the field at which the vortex-Nernst signal reaches zero: The tent profile of ey versus H (a sharp rise to a peak and a fall to zero at high fields) defines the entire vortex liquid region, in which ∣Ψ∣ remains finite. FromH c2(0) ∼ 10 T, we find ξ0 ∼ 58 Å in NCCO.

Figure 1

(A) The vortex-Nernst signaley versus H in Nd2–xCexCuO4(x = 0.15, T c0 = 24.5 K). For example, at 14 K, ey appears atH m = 1.1 T, rises to a peak atH* = 2.8 T, and decreases, withH c2 ∼ 5.8 T. The profile of ρ at 14 K (dashed line) matches the initial increase iney . Above T c0, the vortex signal rapidly vanishes (unlike in hole-doped cuprates). μ0 is the vacuum permeability. (B) TheT dependence of H m, H*, and H c2 derived from theey curves. The H c2 curve in NCCO is conventional and terminates close toT c0.

Most attempts to find H c2 in cuprates have relied on measuring the field profiles of the resistivity ρ in intense fields. We next show that this method is highly unreliable. The field profile of ρ at 14 K (Fig. 1A, dashed line) reveals that, aboveH m, it rises steeply toward the normal-state value ρN. Long before the field reachesH c2, ρ in the vortex-liquid state becomes indistinguishable from ρN. If we had used the “knee” in the profile of ρ to estimate H c2, we would have erroneously identified H c2 with the “ridge field” H*(T) defined (7) by the maximum in ey at each T. As shown in Fig. 1B, the ridge fieldH*(T) versus T has a positive curvature and remains strongly T-dependent as Tapproaches zero. We note that many H c2 curves derived from ρ in cuprates share these features ofH*(T) (12–14). The inset shows that at finite T, the trueH c2(T) is considerably higher thanH*(T). In the hole-doped cuprates, the difference between H* and H c2 is even greater (7).

With this caveat in mind, we turn to the single-layer cuprate Bi 2201 (Bi2Sr2–yLayCuO6) and the bilayer Bi 2212 (Bi2Sr2CaCu2O8), which are ideal for exploring the x dependence ofH c2 becausee N y is negligibly small, and the anomalously small H m(T) values allow the scaling studies described below to be extended over the broadest field range. In Bi 2201 (sample A1, x = 0.16, T c0 = 28 K),ey increases to a broad maximum and then decreases monotonically to zero at just above 45 T (fig. S2). The tent profile, similar to that in NCCO except for the higher field scales, again reveals how far the vortex liquid extends in the field.

A major difference between NCCO and the hole-doped cuprates arises from strong fluctuations in the phase of Ψ in the latter. Whereas the vortex signal in NCCO rapidly vanishes just aboveT c0, it remains large atT c0 in Bi 2201, La2–xSrxCuO4, and YBa2Cu3Oy (YBCO) and extends considerably above T c0. In all hole-doped cuprates examined to date (5–7, 9),T c0 has corresponded to the loss of long-range phase coherence (15–17), rather than the vanishing of ∣Ψ∣. Defining the field at which ey approaches zero as H c2(T) at eachT, we found that H c2(T) in Bi 2201 is nearly T-independent from 5 to 30 K (it goes to zero only at much higher T). Despite the non-BCS scenario in hole-doped cuprates, H c2(T) is still reliably obtained from the approach of ey to zero.

Comparison of ey versusH in several samples of Bi 2212 of different xrevealed a distinctive trend. Figure 2, A and B, compares theey -H curves in overdoped Bi 2212 (sample B1, x = 0.22, T c0 = 65 K) with those in underdoped Bi 2212 (sample B3, x = 0.087, T c0 = 50 K). The calibration ofx is discussed in (18). In sample B1 (Fig. 2A), the vortex Nernst signal closely resembled those in Bi 2201 and anticipated the scaling property to be described. In the underdoped sample (Fig. 2B), however, the curves were more stretched out along the field axis. In the temperature range of 40 to 50 K, ey was not appreciably diminished from its peak value at the maximum field 30 T, whereas in Fig. 2Aey had decreased from its peak by more than 60% by 30 T. The trend is summarized (Fig. 3A) for overdoped, optimally doped, and underdoped Bi 2212 (samples B1 to B3), using profiles ofey measured at their respectiveT c0's (65, 90, and 50 K in B1, B2, and B3, respectively). It is apparent that successively higher fields are needed to achieve comparable suppression of ey as we go from the overdoped to the optimum and to the underdoped sample. The same pattern is observed in Bi 2201 (with lower field data).

Figure 2

Comparison of the profiles ofey in (A) overdoped (OD) (T c0 = 65 K) and (B) underdoped (UD) (T c0 = 50 K) crystals of Bi2Sr2CaCu2O8. The curves in (A) peak at relatively low fields (5 to 10 T) and decrease by 50 to 60% when H reaches 30 T. The peaks in (B), however, lie much closer to 30 T.

Figure 3

(A) The curves ofey versus H in overdoped (sample B1), optimally doped (OPT) (sample B2), and underdoped (sample B3) Bi 2212 taken at their T c0 (65, 90, and 50 K, respectively). (B) The collapse of the three curves in (A) onto the template curve from Bi 2201 (sample A1 withT c 0 = 28 K) when plotted against h =H/H c2′. (C) Comparison of the scaled profiles ofey versus h in two other samples of Bi 2201 (measured up to 14 T) against the template curve in sample A1.

We found that, if we used the reduced field h =H/H c2′ as abscissa (the prime indicates the value at T c0), the three traces inFig. 3A accurately matched the template curve (18) of sample A1 (at Tc 0 = 28 K) with the right choice of H c2′ (Fig. 3B). Using the knownH c2′ value in A1 (50 T), we found thatH c2′ = 50, 67, and 144 T in B1, B2, and B3, respectively. A similar scaling was observed in three samples of Bi 2201 (Fig. 3C). The scaling behavior enabled us to accurately track thex dependence of H c2′ to field values considerably higher than 45 T.

The scaling behavior continued to hold at T belowT c0, if comparisons were made between curves at the same reduced temperature t =T/T c0. However, belowt ∼ 0.7, the vortex solid phase belowH m (where ey = 0) expanded considerably and precluded meaningful comparison for scaling behavior. This was not a serious drawback because the Tdependence of H c2 was very weak between 0 K andT c0 in the Bi-based cuprates (the prime onH c2′ is dropped hereafter).

We carried out the scaling comparison for five samples of Bi 2212 with 0.087 < x < 0.22, as well as in three samples of Bi 2201, and obtained the variation ofH c2 versus x displayed (Fig. 4A). As the doping xdecreases, H c2 in Bi 2212 undergoes a steep increase from 50 to 144 T, whereas in Bi 2201 it increases from 42 to 65 T. The gap amplitude Δ0 in Bi 2212, measured at lowT by ARPES (2), also increases as xdecreases (Fig. 4A, upper stripe). Most investigators regard this increasing Δ0 as the normal-state gap in the pseudogap state because it persists unchanged in size (2) to temperatures well above T c0.

Figure 4

(A). The x dependence of H c2 in Bi 2212 (solid squares) and in Bi 2201 (open squares) measured by scaling of Nernst profiles. As indicated, the upper gray stripe is the x dependence of the ARPES gap Δ0 (2) and the lower gray stripe isT c0. (B) Comparison of the coherence length ξ0 (solid squares) obtained fromH c2 and the Pippard length ξP(open circles) obtained from Δ0. The decay length of quasiparticle states imaged by STM (3) is shown as an open triangle.

We next show that our values for H c2 in Bi 2212 are in fact closely related to Δ0. The two quantities may be compared directly if we convert them to length scales. ExpressingH c2 as the coherence length via ξ0 = √(φ0/2πH c2), we found that ξ0 decreases from 26 to 15 Å as xdecreases from 0.22 to 0.087 (φ0 is the flux quantum). The gap Δ0 may be converted to the Pippard length by ξP =ℏ︀v F/aΔ0, wherev F = 1.78 × 105 m/s is the Fermi velocity (a = π for s-wave superconductors). The plots in Fig. 4B reveal that ξ0 and ξP are closely matched if we choose a = 3/2, consistent with extreme gap anisotropy. The agreement is strong evidence that the ARPES gap at low T and the Nernst experiments are probing the same length scale over a broad range ofx. Both experiments uncovered the same trend: The coherence length decreases by a factor of ∼2 as x decreases from 0.22 to 0.087. Moreover, the virtually T-independent behavior of H c2 in our experiment is consistent with the T-independent behavior in the ARPES gap. The comparison persuaded us that the ARPES Δ0 (2) represents the gap amplitude of the Cooper pairs.

High-resolution images of vortices in optimally doped Bi 2212 have been obtained by STM. The exponential decay of the quasiparticle state density yields a length scale of 22 Å (3). A longer length scale, ∼40 Å, is defined by the checkerboard pattern imaged outside the core (4). The shorter length of 22 Å is in good agreement with our ξ0 (Fig. 4B, open triangle), suggesting that the exponential fall-off is dictated by ξ0.

Recently, the field scale H pg for suppressing the pseudogap state in Bi 2212 has been estimated from thec-axis resistivity ρc versus H(19). Insofar as ρc probes changes in the single-particle density of states, rather than the pairing amplitude, the inferred H pg values should be distinguished from H c2 (indeed, the estimatedH pg values, 100 to 550 T, are much higher than our H c2 values).

To address our initial question, the present experiment establishes that H c2 increases as x decreases in both Bi 2201 and Bi 2212, which implies that the pairing potential is strongest in very underdoped samples. This finding may place strong constraints on theories of the pairing mechanism. The trend [opposite to that of the superfluid density n s(20, 21), which decreases as xdecreases] implies a scenario for underdoped cuprates that is radically different from that inferred from n sand T c0 alone. The physical picture is that, in hole-doped cuprates, the pairing potential Δ0 is maximal in the underdoped regime and falls rapidly with increased hole density, as anticipated in early resonating-valence-bond theories (15, 16). However, the smalln s at small x renders the condensate highly susceptible to phase fluctuations (17). AtT c0, spontaneous nucleation of highly mobile vortices destroys long-range phase coherence and the Meissner state (7, 22, 23), but vortex excitations remain observable to much higher T (5,6). As x is increased,n s and T c0 initially increase, but beyond x = 0.17,T c0 is suppressed by a steeply falling Δ0. The trade-off between n s and Δ0 informs the entire cuprate phase diagram and accounts naturally for the dome shape of the curve of T c0versus x.

An interesting implication may be inferred in the limit of smallx. If the trend in ξ0 in Fig. 4B persists, ξ0 becomes smaller than the separation d of Cooper pairs in this limit (d ∼ 25 Å at x = 0.05). For x < 0.05, we may treat the carriers as tightly bound pairs (bosons) that are well separated and too dilute to sustain long-range phase coherence. In uncovering the increased pairing strength at small x, our experiment provides clues that bosonic pairs may exist in this limit.

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Materials and Methods

Figs. S1 and S2


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