Evidence for competition between the superconducting and the pseudogap state in (BiPb){}_{2}(SrLa){}_{2}CuO{}_{6+\delta} from muon-spin rotation experiments

Evidence for competition between the superconducting and the pseudogap state in (BiPb)(SrLa)CuO from muon-spin rotation experiments


The in-plane magnetic penetration depth in optimally doped (BiPb)(SrLa)CuO (OP Bi2201) was studied by means of muon-spin rotation. The measurements of are inconsistent with a simple model of a wave order parameter and a uniform quasiparticle weight around the Fermi surface. The data are well described assuming the angular gap symmetry obtained in ARPES experiments [Phys. Rev. Lett 98, 267004 (2007)], where it was shown that the superconducting gap in OP Bi2201 exists only in segments of the Fermi surface near the nodes. We find that the remaining parts of the Fermi surface, which are strongly affected by the pseudogap state, do not contribute significantly to the superconducting condensate. Our data provide evidence that high temperature superconductivity and pseudogap behavior in cuprates are competing phenomena.

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The relevance of the pseudogap phenomenon for superconductivity is an important open issue in the physics of high-temperature cuprate superconductors (HTS’s). There are two main scenarios to be considered. In the first, the so-called ”precursor scenario“, the Cooper pairs are already formed at , the temperature at which the pseudogap opens first, but long-range phase coherence is not established until the sample is cooled below the superconducting transition temperature . In the second, the so-called ”two-gap“ scenario, the superconducting and the pseudogap state are not directly related with each other, and may even compete. Within this scenario the gaps in space, existing near the nodes and in the antinodal region of the Fermi surface, are due to the superconducting and the pseudogap states, respectively. This scenario gained support due to a number of recent experiments (1); (2); (3); (4); (5) which revealed that the antinodal gap remains unaffected as the temperature changes across , and generally its magnitude increases significantly in the underdoped region, where decreases. In contrast, the gap near the nodes scales with and obeys a well defined BCS temperature dependence (4). This interpretation also agrees with recent results from scanning-tunneling-microscopy experiments of Hanaguri et al. (6), suggesting that the incoherent antinodal states are not responsible for the formation of phase-coherent Cooper pairs. Consequently, superconductivity is caused by the coherent part of the Fermi surface near the nodes.

Measurements of the magnetic penetration depth can be used to distinguish between the above described scenarios. The temperature dependence of is uniquely determined by the absolute maximum value of the superconducting energy gap and its angular and temperature dependence. In addition, within the London model is proportional to the superfluid density via and, in case where the supercarrier mass is known, gives information on the supercarrier density .

Here we report on a study of the in-plane magnetic penetration depth in optimally doped (BiPb)(SrLa)CuO (OP Bi2201). The angular dependence of the energy gap in similar OP Bi2201 samples was recently studied by Kondo et al. (3) by means of angular-resolved photoemission (ARPES), where the observation of two spectral gaps that dominate different regions of the Fermi surface is reported. Our results reveal that is inconsistent with a model in which both of these spectral gaps are related to superconductivity, as well as with a superconducting gap of wave symmetry developed within the whole Fermi surface. Good agreement with the ARPES data was obtained within a model which assumes that the pseudogap affects the spectral density of the antinodal quasiparticles. Consequently, only carriers close to the nodes contribute to the superfluid density, while the weight of the coherent quasiparticle near the antinodes is negligible. This statement is also supported by comparing the zero-temperature value of for OP Bi2201 studied here with those of other OP HTS’s, such as CaNaCuOCl (OP Na-CCOC) (7) and LaSrCuO (OP La214) (8), having similar transition temperatures. It was observed that in superconductors where the superconducting gap is developed only close to the nodes (OP Bi2201 and OP Na-CCOC) the superfluid density is more than 50% smaller than in OP La214 where the wave superconducting gap is detected on the whole Fermi surface (9).

Details on the sample preparation for OP Bi2201 single crystals can be found elsewhere (10). The values of and the width of the superconducting transition, as determined from magnetization measurements, are 35 K and 3 K, respectively. The transverse-field muon-spin rotation (TF-SR) experiments were performed at the M3 beam line at the Paul Scherrer Institute (Villigen, Switzerland). Two OP Bi2201 single crystals with an approximate size of 420.1 mm were mounted on a holder specially designed to perform SR experiments on thin single crystalline samples. The sample was field cooled from above to 1.6 K in a series of fields ranging from 5 mT to 640 mT. The magnetic field was applied parallel to the crystallographic axis and transverse to the muon-spin polarization.

In the TF geometry the local magnetic field distribution inside a superconductor in the mixed state, probed by means of SR, is determined by the coherence length , and the penetration depth . In extreme type-II superconductors () the distribution is almost independent of and the second moment of becomes proportional to (11). To describe the asymmetric (see Fig. 1), the SR time spectra were analyzed by using a two-component Gaussian expression (12). The second moment of was further obtained as (12):


Here , , and are the asymmetry, the relaxation rate, and the mean field of the th component, and  MHz/T is the muon gyromagnetic ratio. The analysis was simplified to a single Gaussian lineshape in the case when the two-Gaussian and the one-Gaussian fits result in comparable . The superconducting part of the second moment was obtained by subtracting the contribution of the nuclear moments measured at as (12). Since the magnetic field was applied along the crystallographic axis, our experiments provide direct information on .

Figure 1: (Color online) (a) Dependence of of OP Bi2201 on the applied magnetic field measured at  K. The black solid line corresponds to obtained by using the numerical calculations of Brandt (11) ( nm, ) for a superconductor with an isotropic energy gap. The solid red line represents expected in case of a wave superconductor. The blue dotted curves show the local magnetic field distribution calculated by means of the maximum entropy Fourier-transform technique at  K and  mT, 40 mT, and 640 mT. (b) Field dependence of the skewness parameter . The solid line is the average value .

Fig. 1a shows the dependence of on the applied magnetic field measured after field cooling the OP Bi2201 sample from down to 1.6 K. The distributions were calculated using the maximum entropy Fourier-transform technique for  mT, 40 mT, and 640 mT (see Fig. 1a). In the whole range of fields ( mT mT) is asymmetric. The asymmetric shape of is generally described in terms of the so-called skewness parameter [ is the th central moment of ]. is a dimensionless measure of the asymmetry of the lineshape, the variation of which reflects underlying changes in the vortex structure (14). In the limit and for realistic measuring conditions for an ideal triangular vortex lattice (VL). It is very sensitive to structural changes of the VL which can occur as a function of temperature and/or magnetic field (14); (15). Fig. 1b implies that in OP Bi2201 is almost constant [] and is smaller than the expected value of 1.2, which is probably caused by distortions of the VL due to pinning effects. It is known that Pb substitution in double-layer Bi2212 HTS’s enhances pinning quite substantially (13).

It should be noted here that addition of Pb does not change and the in-plane superfluid density (16), but makes OP Bi2201 more 3-dimensional. To estimate the anisotropy coefficient ( is the axis component of the penetration depth) we performed torque magnetization experiment on one of the crystals studied (17). A value of was found, which is more than 10 times smaller than obtained on OP Bi2201 without Pb by Kawamata et al. (18).

Figure 1 indicates that the magnetic field dependence of is not monotonic: with increasing field goes through the broad maximum at around 20 mT. The black solid line in Fig. 1a, calculated within the model of Brandt (11), corresponds to for an isotropic wave superconductor with  nm and ( nm was obtained from the value of the second critical field  T (19)). From Fig. 1a we conclude that the experimental depends much stronger on the magnetic field than expected for a fully gaped wave superconductor. As shown by Amin et al. (20) for a superconductor with nodes in the energy gap a field dependent correction to arises from its nonlocal and nonlinear response to an applied magnetic field. The solid red line represents the result of the fit by means of the relation:


which takes the nonlinear correction to for a superconductor with a wave energy gap into account (21). Here the parameter depends on the strength of the nonlinear effect. Since Eq. (2) is valid for intermediate fields ( is the first critical field) only the data points above 40 mT were considered in the analysis.

Figure 2: (Color online) (a) Dependence of of OP Bi2201 on measured at  mT. Lines represent the theoretical curves obtained by assuming different symmetries of the superconducting energy gap (see Fig. 3a). (b) Dependence of the skewness parameter on . The blue dotted curves represent distributions below ( K) and above ( K) the VL melting temperature.

We now discuss the dependence of . Figure 2 displays measured at  mT. Below 20 K is linear in as expected for a superconductor with nodes in the gap, consistent with the conclusion drawn from the analysis of the data (see discussion above and Fig. 1). To ensure that is determined primarily by the variance of the magnetic field within the VL we plot in Fig. 2b the corresponding . It is constant from 1.6 K to  K and drops to zero at  K, where becomes fully symmetric. A similarly sharp change of with temperature was observed in Bi2212 and was explained by VL melting (14); (15). Correspondingly, we conclude that for temperatures  K the variation of reflects the intrinsic behavior of the in-plane magnetic penetration depth .

Figure 3: (Color online) (a) Angular dependence of the energy gap of OP Bi2201 obtained in ARPES experiments (3). Lines represent the various models of the gap symmetries used to analyze the experimental data (see Fig. 2a). The inset shows schematically a part of the Fermi surface. The points ”A“ and ”B“ are close to the nodal () and the antinodal () region, respectively. (b) Temperature dependence of the energy gap in the nodal (curve A) and the antinodal (curve B) regions.

The dependence of was analyzed by assuming that the angular dependence of the energy gap in OP Bi2201 is similar to the one from recent ARPES experiments (3) (see Fig. 3a). In analogy with Refs. (3) and (4) it was also assumed that the energy gap in the nodal region changes with temperature in accordance with the weak-coupling BCS prediction (22), while the one near the antinodes is independent (see the corresponding lines ”A“ and ”B“ in Fig. 3b). The following cases were considered: (I) a monotonic wave gap (green dashed line); (II) a monotonic wave gap with suppressed quasiparticle weight in the antinodal region (solid orange line); (III) an analytical function, which follows the monotonic wave in the nodal region and changes to a behavior close to the antinodes (solid blue line). The dependence of the magnetic penetration depth was calculated within the local (London) approximation () using the following equation (23):


denotes the Fermi function. Here we also replace the prefactor of the integral with to account for the case when the superconducting energy gap is developed only on a part of the Fermi surface (in our case from to ). The results of this analysis are presented in Fig. 2a. The monotonic wave gap as well as the combined gap represented by the solid blue line in Fig. 3a can not describe the experimental . Full consistency between ARPES and SR data is obtained if one assumes a superconducting wave gap with only carriers in the region contributing to the superfluid. It should be noted here that the theoretical curves in Fig. 2 were not fitted, but obtained directly by introducing the angular dependence of the gap measured in ARPES experiments into Eq. (3), describing the dependence of the penetration depth within the London approach.

Compound SC gap region
(K) m)
OP Bi2201 35 7.8  
OP Na-CCOC 28 10.0    
OP La214 36 15.0    

Ref. (3), Ref. (7), Ref. (6), Ref. (8), Ref. (9)

Table 1: Transition temperature , zero-temperature in-plane magnetic penetration depth , and angular region where the superconducting wave gap is observed for OP Bi2201 (studied here), OP Na-CCOC, and OP La214.

Next we compare the zero-temperature values of for various OP HTS’s having comparable values and for which the angular dependence of the superconducting gap was measured (3); (6); (9). OP La214, which exhibits a fully developed superconducting gap, has an approximately 50% higher value of the superfluid density as compared to both OP Na-CCOC and OP Bi2201, having the superconducting gap opened only on a limited part of the Fermi surface (see Table 1). Assuming that the supercarrier masses are the same for all OP compounds listed in Table 1 (in analogy with reported for La214 and YBaCuO families of HTS’s (24)), the difference in the values of can be naturally explained by the different number of carriers condensed into the superfluid. In the case of OP Bi2201 and OP Na-CCOC, is strongly reduced because of the fraction of the states is no more available for the superconducting condensate due to the pseudogap.

To conclude, the in-plane magnetic penetration depth in optimally doped Bi2201 was studied by means of muon-spin rotation. By comparing the measured with the one calculated theoretically using a model consistent with ARPES measurement (3) we found that the superconducting gap in OP Bi2201 has wave symmetry, but only carriers from parts of the Fermi surface close to the node () contribute to the superfluid. This implies that the pseudogap affects the spectral density of the quasiparticles and, consequently, not all the states at the Fermi surface are available to participate in the superconducting condensate. Our results supports the scenario where the superconducting and pseudogap state are two distinct and competing phenomena. This statement is also consistent with the fact that the superfluid density in OP Bi2201 is strongly reduced in comparison with that in OP La214, where the superconducting gap and coherent quasiparticles are observed along the whole Fermi surface () (9).

This work was performed at the Swiss Muon Source (SS), Paul Scherrer Institute (PSI, Switzerland). The authors are grateful to Y.J. Uemura and R. Prozorov for stimulating discussions, and S. Weyeneth for performing torque experiments. This work was supported by the K. Alex Müller Foundation and in part by the Swiss National Science Foundation. Work at the Ames Laboratory was supported by the Department of Energy - Basic Energy Sciences under Contract No. DE-AC02-07CH11358.


  1. preprint: PREPRINT (May 8, 2018)


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