Pseudogap term in the magnetic response of cuprate superconductors

R. E. Walstedt, T. E. Mason, G. Aeppli, S. M. Hayden, and H. A. Mook

Phys. Rev. B 84, 024530 – Published 27 July 2011

Abstract

This paper presents a joint analysis of inelastic neutron scattering (INS) and nuclear magnetic resonance data on superconducting cuprates, with the aim of a detailed characterization of the dynamical susceptibility $χ^{''} (\vec{q}, ω)$ at low frequencies and at temperatures ranging from $T_{c}$ up to room temperature. Using the well-known relation between $1 / T_{1}$ and $χ^{''} (\vec{q}, ω)$ , the analysis shows that the temperature dependence of the nuclear relaxation rate $1 / T_{1}$ is controlled by a combination of dynamic spin-spin correlations between pairs of fluctuating moments in the CuO $_{2}$ plane and a time constant proportional to the integral of $χ^{''} (\vec{q}, ω)$ over the Brillouin zone. INS data on $χ^{''} (\vec{q}, ω)$ for ${La}_{1.86} {Sr}_{0.14} {CuO}_{4}$ (LSCO) and ${YBa}_{2} {Cu}_{3} O_{6.5}$ (YBCO6.5) are seen to obey $ω / T$ scaling above a transition temperature, then fall to very low values at low temperatures. Thus, there is no evidence in such data for magnetic pseudogap effects, which are known to be quite pronounced in YBCO6.5, but somewhat muted in LSCO. Our analysis of $T_{1}$ data shows, however, that above the transition temperature noted above there occurs the onset of another term in $χ^{''} (\vec{q}, ω)$ , which comes to dominate $1 / T_{1}$ at room temperature and above. We call this term “the pseudogap term” $χ_{P}^{''} (\vec{q}, ω)$ . The onset of $χ_{P}^{''} (\vec{q}, ω)$ coincides with the entry into a quantum-critical regime dominated by stripes, but could also be derived from low-energy fluctuations resulting from nearby phase transitions characterized by other types of order, such as ring currents. For YBCO6.5 this onset is at $T_{c} \sim 62 K$ , but for LSCO, it occurs 20–30 K higher than $T_{c}$ . We model the $q$ -space behavior of $χ_{P}^{''} (\vec{q}, ω)$ and discuss its prospects for observation via INS. The occurrence of the foregoing effects is suggested to be widespread among the superconducting cuprates.

Received 17 June 2010

DOI:https://doi.org/10.1103/PhysRevB.84.024530

Authors & Affiliations

R. E. Walstedt^1,*, T. E. Mason², G. Aeppli³, S. M. Hayden⁴, and H. A. Mook²

¹Department of Physics, University of Michigan, Ann Arbor, Michigan 48106, USA
²Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
³London Centre for Nanotechnology and Department of Physics and Astronomy, University College London (UCL), London WC1E 6BS, United Kingdom
⁴H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, United Kingdom

^*walstedt@umich.edu

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Vol. 84, Iss. 2 — 1 July 2011

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Figure 1
The spin-spin correlation factors $K_{n I a, b}$ defined in Eq. (5) [with $χ^{''} (\vec{q}, ω) = χ_{I}^{''} {(\vec{q}, ω)}_{a}$ ] are plotted as a function of temperature for LSCO. The coefficients $g_{1 a, b} (\vec{q})$ and $g_{3 a, b} (\vec{q})$ have twofold symmetry and therefore differ along the ( $a$ ) axis ( $K_{1 I a}$ and $K_{3 I a}$ ) and along the ( $b$ ) axis ( $K_{1 I b}$ and $K_{3 I b}$ ). Since the $K_{1 I a, b}$ ’s are negative, they are presented as $1 + K_{1 I a, b}$ , which is how they occur in Eq. (8).Reuse & Permissions
Figure 2
(a) Values of $τ_{e I}$ and $τ_{eff}$ determined as given in the text for LSCO are plotted vs $T$ . Values of $τ_{e I}$ (solid triangles) were calculated using $χ^{''} (\vec{q}, ω) = χ_{I}^{''} {(\vec{q}, ω)}_{a}$ in Eq. (6). The solid line is a linear regression showing approximate $ω / T$ scaling for $T > T_{str} \sim 50$ K. Values of $τ_{eff}$ obtained from $T_{1}$ data with Eqs. (7) and (8) are also plotted for $^{63} Cu$ (squares) and $^{17} O$ (circles), respectively. The Korringa-like behavior of the $τ$ ’s below $T_{str}$ is highlighted by the solid red line. Data for $^{17} O$ are scaled to that line for the analysis in (b). For a general comparison, values of $τ_{eff} (T)$ for YBCO7 obtained by Uldry and Meier (Ref. 28) for that system are shown as a dashed line. (b) Values of $τ_{e P}$ obtained with Eqs. (9) and (10) using the same $T_{1}$ data as above are plotted against temperature. For this purpose, values of the $K_{n P}$ ’s derived from a squared-Lorentzian model are used (see text). The $K_{n P}$ ’s with their fitted temperature dependence are shown in the inset. The solid line representing the behavior of $τ_{e I}$ from (a) is replotted here for comparison.Reuse & Permissions
Figure 3
Plots of $τ_{eff}$ , $τ_{e I}$ , and $τ_{e P}$ are presented for YBCO6.5 similar to the LSCO case in Fig. 2a. Values of $τ_{e I} (T)$ for YBCO6.5, calculated with Eq. (6) using INS data from Ref. 8, are shown as a solid line that obeys $ω / T$ scaling. Values of $τ_{eff}$ obtained with Eq. (7) for the $^{63} Cu$ and with Eq. (8) for the $^{17} O$ from $T_{1}$ data (Ref. 23) using values of $K_{n I a, b}$ are plotted as dashed-dotted lines. The disparity between the dashed-dotted lines for $^{63} Cu$ and $^{17} O$ is attributed to a disparity in incommensurate peak widths between NMR and INS samples and is corrected using adjusted peak widths (see text), leading to the curves showing filled squares ( $^{63} Cu$ ) and circles ( $^{17} O$ ). (b) Values of $τ_{e P}$ obtained with Eqs. (9) and (10) using $T_{1}$ data (Ref. 23) are plotted against temperature. Calculation of the $K_{n P}$ ’s is described in the text. A solid line representing the behavior of $τ_{e I}$ from INS data [see (a)] is replotted here for comparison. The $K_{n P}$ ’s with their fitted temperature dependences are shown in the inset.Reuse & Permissions
Figure 4
Normalized profile representing a scan of $χ^{''} (\vec{q}, ω)$ for LSCO, at low frequencies, through a center point $Q = (π, π)$ along the $q_{x}$ axis, with $q_{x}$ in units of $a^{- 1}$ . The solid line represents $χ_{I}^{''} {(\vec{q}, ω)}_{a} + χ_{P}^{''} (\vec{q}, ω)$ in terms of the fitting functions given in the text (Ref. 34). The dashed line shows the squared-Lorentzian profile of $χ_{P}^{''} (\vec{q}, ω)$ alone. The parameters used here correspond to peak widths, correlation factors, and relative amplitudes given in Fig. 2 for $T = 100$ K.Reuse & Permissions

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