PRAMANA — journal of

c Indian Academy of Sciences °

physics

Vol. 66, No. 1 January 2006 pp. 219–225

Zeeman and orbital limiting magnetic fields in cuprates: The pseudogap connection LIA KRUSIN-ELBAUM1,∗ , TAKASADA SHIBAUCHI2 and GIANNI BLATTER3 1

IBM T. J. Watson Research Center, Yorktown Heights, New York 10598, USA Department of Electronic Science and Engineering, Kyoto University, Nishikyo-ku, Kyoto 615-8510, Japan 3 Theoretische Physik, ETH-H¨ onggerberg, CH-8093 Z¨ urich, Switzerland ∗ E-mail: [email protected] 2

Abstract. In cuprates, in a view where pairing correlations set in at the pseudogap energy scale T ? and acquire global coherence at a lower temperature Tc , the region Tc ≤ T ≤ T ? is a vast fluctuation regime. Tc and T ? vary differently with doping and the question remains about the doping trends of the relevant magnetic field scales: the field Hc2 bounding the superconducting response and the pseudogap closing field Hpg . In-plane thermal (Nernst) and our interlayer (tunneling) transport experiments in Bi2 Sr2 CaCu2 O8+y report hugely different limiting magnetic fields. Here, based on pairing (and the uncertainty principle) combined with the definitions of the Zeeman energy and the magnetic length, we show that both fields convert to the same pseudogap scale T ? upon transformation as orbital and Zeeman critical fields, respectively. The region of superconducting coherence is confined to the ‘dome’ that coincides with the usual unique upper critical field Hc2 on the strongly overdoped side. We argue that the distinctly different orbital and the Zeeman limiting fields can co-exist owing to charge and spin degrees of freedom separated to different parts of the strongly anisotropic Fermi surface. Keywords. Upper critical field; pseudogap; phase diagram; Fermi surface. PACS Nos 74.25.Dw; 74.25.Fy; 74.25.Op; 74.72.Hs

1. Relevant field/length scales and the coherence ‘dome’ Elucidating the fundamental length and energy scales associated with the superconductivity in the cuprates is essential for understanding the origin of the high transition temperature Tc . The values of upper critical field Hc2 are of particular importance, for they mark the onset of superconducting correlations and directly inform on microscopic parameters such as the coherence lengths in the superconducting state. The difficulty in mapping the Hc2 boundary is well-known: it lies in part in the hard-to-reach high field range, but more fundamentally in the large thermal fluctuation regime [1] and the loss of long-range phase coherence below Hc2 (T ), making this limiting field ‘fuzzy’ and hard to pinpoint using the usual

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Lia Krusin-Elbaum, Takasada Shibauchi and Gianni Blatter experimental tools, such as transport or magnetization. Complicating matters further is the normal state pseudogap [2] which dominates the phase diagram [3] and whose still unresolved connection to superconductivity is central to the issue of the onset of pairing and coherence. In this context, the stark difference in the doping dependencies of the pseudogap energy scale T ? and Tc has been wellestablished: T ? decreases (roughly linearly) with charge doping, while Tc follows the well-known superconducting ‘dome’ described by the phenomenological formula [4] Tc /Tcmax = 1 − 82.6(p − 0.16)2 . To track the coherence ‘dome’ and beyond in the H − p phase space, we begin by describing the origin of the three relevant magnetic field scales: (i) the orbital critical field Hc2 , (ii) the minimum intrinsic tunneling conductivity field Hsc , and (iii) Hpg – the Zeeman field corresponding to the onset of spin correlations and scaling linear with T ? ; this field is experimentally identified with the closing of the pseudogap [3]. In cuprates, the conventional derivation of the coherence length ξ through an evaluation of Hc2 = Φ0 /2πξ 2 from transport measurements has been unreliable due to the presence of a large vortex liquid regime [1] and the lack of sharp features in the resistivity. A feature that is accurately mapped from the field dependence of the interlayer c-axis resistivity, ρc (H) = σc−1 (H), is the peak at Hsc that corresponds to a crossover from the mostly Josephson (Cooper pair) tunnelling conductivity σJ (H) at low fields to that of quasi-particles, σq (H), at fields above Hsc . From the measurements of ρc (H) in Bi2 Sr2 CaCu2 O8+y (BSCCO) [3,5,6], we have found that throughout the entire doping range, Hsc (T ) is nearly T -exponential. This temperature dependence is easily derived following experimental observation that the c-axis conductivity is a two-channel tunneling process: σc ' σJ + σq ,

(1)

where σJ = σJ0 exp [U (H)/T ] is controlled by the thermally activated diffusive drift of pancake vortices hopping over the energy barriers U (H) in the CuO2 planes [7] ³ ´ H and σq = σq0 1 + H ? comes from tunneling of nodal quasi-particles [5]. σJ0 and c2

? ∼ Φ0 T ? 2 /¯h2 vF2 , where σq0 are the corresponding T → 0 extrapolations and Hc2 T ? coincides with the gap in the quasiparticle spectrum. Taking the derivative of eq. (1) with respect to H and recalling that U (H) ∼ U0 ln(a0 /ξ) ∝ ln(Hc2 /H) for a 2D vortex lattice [1], we obtain the expression for the field Hsc where the minimum in σc (maximum in ρc ) will occur:

· Hsc (T ) ≈ Hc2

σq0 T Hc2 ? σJ0 U0 Hc2

¸−T /U0 .

(2)

Indeed, at low temperatures this high-field Hsc (T ) can hardly be distinguished in the experiments from a T -exponential behavior [8] shown in figure 1, giving the zero temperature limit Hsc |T →0 → Hc2 (0). Experimentally, the zero temperature values of Hsc as a function of hole doping follow a parabolic dependence Hsc (p) ∼ 1.4Tc (p), defining the ‘coherence dome’ in the figure (see also [9]).

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Figure 1. Doping dependence of the peak field Hsc (half-purple squares) in Bi2 Sr2 CaCu2 O8+y in the T → 0 limit. Hsc (0, p) (in T) ∼ 1.4Tc (in K) shown as half-orange squares, shapes the ‘superconducting dome’ defined through the presence of large interplane Josephson currents and hosting the ‘conventional’ superconducting phase with recombined quasi-particles. A similar dome-shape is derived from the magnetization measurements of systematically doped La2−x Srx CuO4 (LSCO), shown as red dots (from ref. [9]). Inset illustrates the nearly T -exponential temperature dependence of Hsc (T ) in Bi2 Sr2 CaCu2 O8+y with p = 0.225, pointing to the T = 0 value of Hsc .

2. How limiting magnetic field scales relate to the pseudogap Wang et al, in a series of recent papers [10–12], deduced values of an orbital limN – beyond this field the charge pairing amplitude is iting ‘upper critical field’ Hc2 N expected to vanish. The Hc2 obtained from a disappearance of a huge Nernst signal – attributed to vortex-like excitations – was found to decrease rapidly with increased doping. This in turn implies that the Cooper pairing potential and the superfluid density follow opposite trends versus charge doping. To understand this N observation, we need to know how the Nernst-derived Hc2 relates to the gap T ? observed by angle-resolved photoemission spectroscopy (ARPES) [13]. The idea is that pairing correlations are quenched through localization in a magnetic field H p once the magnetic length a0 = Φ0 /H becomes smaller than the pair correlation length ξ ? = ¯hvF /αT ? (here vF is the Fermi velocity, and α is a numerical of order unity [14]). The Nernst-derived magnetic field well matches this condition [10] and N hence can be ascribed as an orbital critical magnetic field Hc2 ; it scales quadratically N in T ∗ , µB Hc2 ∼ T ∗ 2 /mvF2 , where µB is the Bohr magneton. Pramana – J. Phys., Vol. 66, No. 1, January 2006

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N Figure 2. Comparison of Hpg , Hc2 in BSSCO, and the transformation from ? Hpg to Hc2 via eq. (3), where we used vF ≈ 1.8 ± 0.2 eV ˚ A and α ≈ 0.6; note the single ‘Hc2 ’ scale in the low doping regime. A sketch in the inset highlights N ? the splitting of the Hc2 (0) and Hc2 (0) lines when the ‘superconducting dome’ N Hsc (0, p) is reached. Hc2 data (half-green diamonds) is from scaling near Tc . N Light yellow band follows Hc2 (0) (half-green squares) obtained as described in the text.

N has been derived from our An even higher critical magnetic field Hpg > Hc2 c-axis interlayer tunneling transport [3]. In these experiments, the recovery of the normal ungapped state c-axis conductivity indicates that the pseudogap T ? closes at a much larger field scale Hpg (∼ factor of two on the low doping side, see figure 2). This limiting field also relates to the pseudogap energy scale T ? , but via the linear Zeeman relation kB T ? = gµB Hpg , where g ∼ 2 is the Land´e g-factor of the Cu2+ ions. Correspondingly, one deduces that spin-singlets are unpaired at the pseudogap closing field Hpg . Indeed, the field Hpg relates to T ? via the linear Zeeman scaling irrespective of whether the applied field is across CuO2 planes or in-plane [15]. The observed field anisotropy is only that of the g-factor [16], further supporting the view that the pseudogap is of spin-singlet origin.

3. Quadratic relationship between orbital and Zeeman limiting fields N By considering the equivalence of the limiting fields Hpg and Hc2 to the same ? N pseudogap energy scale T but via different routes, ‘orbital’ for Hc2 and ‘Zeeman’ for Hpg , we derive [17] how the two fields relate as a function of doping p,

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N as a function of doping for LSCO and BSCCO follow Figure 3. (a) Tc /Tonset the same curve; BSCCO is taken to follow this trend. The outcomes of both methods are in close correspondence. Values of the orbital limiting field from N Nernst measurements, Hc2 in ref. [10], were obtained near Tc . To evaluate N the zero-T values, we either (i) simply match Hc2 to Hsc (0) in the overdoped N regime, where Hsc (0) coincides with the usual Hc2 (see ref. [5]) or (ii) read it N N out from the ‘universal’ Hc2 (T )/Hc2 (0) vs. T /Tc shown in (b).

N ? Hc2 (p) = Hc2 (p) ≡ α2

µB Hpg (p) Hpg (p). mvF2

(3)

From ARPES experiments [13], the Fermi velocity vF is found to be insensitive 2 N (p) ∝ Hpg (p). to doping. Hence, eq. (3) predicts a simple quadratic relation Hc2 ? N ∼ A comparison of Hc2 (T = 0) (see figure 2) and Hc2 (0), by using most recently measured values for the Fermi velocity [18] vF ' 2 eV ˚ A and α ≈ 0.6, collapses N (0) was obtained in two ways, the data in the underdoped regime p < 0.16. Hc2 N to Hsc (0) at p > illustrated in figure 3. One, by a simple matching of Hc2 ∼ 0.2, where Hsc (0) coincides with the usual Hc2 (0) (see refs [3,6]). Another, from the N N N (0) vs. T /Tonset curve implicit in the data of refs [11] and (T )/Hc2 ‘universal’ Hc2 N [12]; here Tonset is the onset temperature of the Nernst response. Close to optimal ? enters the doping, the scaled and the measured orbital fields part their ways: Hc2 N superconducting ‘dome’ while the Hc2 follows its edge, pointing to a remarkable distinction between the low- and the high-doping sides [13]. 4. Resolving ‘double jeopardy’: Separated spin and charge degrees N Having the two critical fields, Hc2 and Hpg , related to a single energy scale T ? , the question arises how one could dispose of the same correlation energy twice: via the N N orbital route at Hc2 and then again via the Zeeman effect at Hpg À Hc2 . This ‘double jeopardy’ is naturally resolved by a strongly anisotropic (truncated) Fermi surface [13], hosting separated charge and spin degrees of freedom. A starting point is the quantum spin-singlet liquid forming at the energy scale T ? – this spin-liquid ground state does not have any long range order and competes with the antiferromagnetic arrangement of spins [19–21]. With doping, the spin-liquid becomes more favorable, charge and spin degrees separate, and holes are expected to condense on the spin-liquid background, becoming phase coherent at Tc ¿ T ? .

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Figure 4. The break-up of the Fermi surface into regions describing spinsinglet pairs and charged holes: the spin-pairing opens up gaps at the (0, π) points – where the pseudogap energy T ? establishes correlations on the scale ξ? ∼ ¯ hvF /T ? . Upon doping, a truncated Fermi surface appears around the (π, π) diagonals. When charges pair up, they draw correlations from the spinsinglet background, and hence spin-singlet pairing at the Fermi surface corners and hole-pairing at the diagonals both derive from T ? .

In these scenarios, there is a breakup of the Fermi surface into regions hosting spinsinglet pairs and charged holes. The spin-pairing opens up gaps near the (0, π) points (see figure 4); the corresponding pseudogap energy T ? forms correlations on the scale ξ ? ∼ ¯hvF /T ? . Upon doping, a truncated Fermi surface appears around the (π, π) diagonals. When charges pair up, they draw correlations from the spinsinglet background, and hence spin-singlet pairing at the Fermi surface corners and hole-pairing at the diagonals derive from the same energy scale T ? [21,22]. We note that in these views, far in the overdoped regime, the separation is not well-defined. N The above considerations naturally lead to two field scales Hpg and Hc2 : the N charge degrees are connected to the orbital field Hc2 obtained from the Nernst transport. The in-plane Nernst transport reflects the dissipation due to nodal quasi-particles [23] in the vortex cores, with momenta nearly parallel to (π, π). N inhibits hole-pairing at the Fermi surface diagonals, but does Consequently, Hc2 not destroy the spin-singlet pairs around the Fermi surface corners – these spinsinglets are unpaired at the much higher Zeeman field Hpg . The breakup of the spin-singlets leaves its trace in the c-axis tunneling experiments [24]. Hence, the N identification of two limiting magnetic fields, Hc2 and Hpg , deriving from the same pseudogap energy scale T ? via an orbital and a Zeeman relation, respectively, finds a natural interpretation in terms of a reconstructed Fermi surface with separated charge and spin degrees of freedom.

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Zeeman and orbital limiting magnetic fields in cuprates Acknowledgments Measurements were performed at NHMFL supported by the NSF Cooperative Agreement No. DMR-9527035. We wish to thank B Brandt, A Lacerda, and C H Mielke for their technical assistance, and V Geshkenbein, A Koshelev, M Rice, and M Sigrist for important insights. TS is supported by a grant-in-aid for scientific research from MEXT.

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