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A pseudogap term in the magnetic response of the cuprate superconductors

We combine neutron scattering (INS) and NMR/NQR nuclear spin lattice relaxation rate data to deduce the existence of a new contribution to the magnetic response (dynamic susceptibility) in cuprate superconductors. This contribution, which has yet to be observed with INS, is shown to embody the magnetic pseudogap effects. As such, it explains the long-standing puzzle of pseudogap effects missing from cuprate INS data, dominated by stripe fluctuations, for the dynamic susceptibility at low energies. For La_(1.86)Sr_(0.14)CuO_(4) and YBa_(2)Ba_(3)O_(6.5), the new term is the chief contributor to nuclear spin lattice relaxation at T>>T_(c).

Cuprate superconductors are well-known for their unusual normal metallic state properties. Prominent among these is an extended 'pseudogap' regime located above the dome of superconducting transition temperatures T c (see e.g. a typical phase diagram [1]). Theorists have attempted to connect features in the normal state phase diagram with the phenomenon of high temperature superconductivity itself. There are three main approaches. The first focuses on the disappearance of the Mott-Hubbard antiferromagnet, which is transformed into a valence bond state where mobile holes are naturally paired [2]. The second concentrates on 'stripe' correlations as providing the environment needed for superconductivity [3].
The third hypothesizes that the cross-over into the pseudogap regime is actually a phase transition to a state with concealed long range order [4]. Experimentally, we have been able to use inelastic neutron scattering (INS) as a function of temperature, composition and magnetic field to map stripe order and fluctuations [5][6][7]. In the joint NMR and INS analysis presented here we identify a novel low-frequency signal which is correlated with pseudogap formation.
In this Letter we present a new, joint analysis of INS and NMR (T 1 ) data for the systems mentioned above, in which we deduce the existence of a pseudogap fluctuation term χ ′′ P ( q, ω), which has not been identified by INS up to now. Thus, we write χ ′′ ( q, ω) = χ ′′ I ( q, ω) a + χ ′′ P ( q, ω), where χ ′′ I ( q, ω) a is the INS-measured term with incommensurate peaks along the a axis. The term χ ′′ P ( q, ω) introduced here, which is nonzero in the fluctuating stripe phase, will be modeled below to interpret the T 1 data. Thus, not only does the strongly evidenced occurrence of such a term clearly explain the hitherto baffling omission of a pseudogap effect from data for χ ′′ ( q, ω) [6,7], it also accounts for the disparate behavior of T 1 for the planar 63 Cu and 17 O nuclear spins in these systems [20,21]. We also show that the thermal and q-space behavior of χ ′′ P ( q, ω) is such that it could have easily been missed up to now by INS experiments on these systems. In sum, the present analysis addresses a major deficiency in our understanding of the anomalous normal-state physics of cuprates and will be of wide interest to theorists and experimentalists alike.
For the nuclear relaxation analysis we employ the formulation of the T 1 process pioneered by Uldry and Meier [22], in which the relaxation rates are written (1) for 63 Cu and 17 O, the two nuclear species of interest. In these equations A ab , B, C a and C b are hyperfine tensor components in units of Gauss per unit of spin [20]. The quantities K n represent the normalized dynamical spin-spin correlation functions [22][23][24]. Thus, K n = 4 S i · S j , where S i and S j are n th neighbor spins. The additional subscript a, b indicates, for n = 1 and 3, that the bond axis (i.e. r ij ) lies along the a or b crystalline axis. Using the fluctuation-dissipation formulation of T 1 [25,26], we express the K n 's in terms of χ ′′ ( q, ω) a , taking a as the discommensuration axis. The K n 's (n = 1,2,3) for the first three neighbor pairs in the CuO 2 plane may then be written where g 1a,b = cos(q a,b a); g 2 = cos(q a a)cos(q b a); and g 3a,b = cos(2q a,b a) in an obvious notation.
Note that K 2 is independent of the discommensuration axis. It is clear that |K n | ≤ 1 . Eq. (1) and (2) also employ the key parameter proportional to the 'local susceptibility', which acts as a correlation time that includes the particle statistics of the relevant carriers. τ ef f may also be estimated directly from INS data.
Thus, we shall proceed by comparing the latter values of τ ef f with those obtained from T 1 data, using reasonable estimates of the other parameters in Eqs.
We note that T c ∼ 35K for this sample [6].
The latter results are to be compared with estimates of τ ef f (T) derived from T 1 data [19,20] using Eq.(1) and (2). To do this, we employ values of the HF constants derived from shift data [30] and values of the K na,b 's calculated with Eq.(3), using a two-peak form factor fitted to INS data [6]. The K nIa,b 's so determined vary only gradually with temperature.
As an example we mention values at T = 100K: K 1Ia = -0.75; K 1Ib = -0.97; K 2I = 0.73; Since only a single rate was observed [20], it is presumed either that a flip-flop mechanism is present to maintain a single 17 O nuclear spin temperature, or the stripe domain boundaries are fluctuating, so that each site automatically averages the two rates to yield a composite rate (given by Eq. (2)). Results so obtained are plotted in Fig. 1(a) as blue squares and red circles for the 63 Cu and 17 O nuclear spins, respectively. For comparison, the value of τ ef f deduced for YBCO7 by Uldry and Meier [22] is shown as a dashed line, reflecting the fact that the T 1 process in LSCO is substantially stronger than that for YBCO7.
Regarding the estimated values of τ ef f for LSCO in Fig. 1(a), note that there is approximate Korringa-like behavior for all three data sets below T = T Str , where the agreement is good considering that there are no adjustable parameters. The slight disparity in magnitudes is attributed to HF constant errors or differences in the widths of incommensurate peaks (i.e., of the K n 's) among samples (see the YBCO6.5 case below). Since the curves for τ ef f rise high above those for τ eI at T > T Str , the T 1 data give clear evidence for an additional term in χ ′′ ( q, ω) as stated above. We emphasize that there is no other realistic possibility among the well-documented T 1 mechanisms in solids. Also noteworthy is the drastic difference between the τ ef f curves derived from the 63 Cu and 17 O T 1 data. Values of τ ef f for these two measurements must actually be the same, so that the correlation properties of the new term χ ′′ P ( q, ω) are evidently rather different from those of χ ′′ I ( q, ω) a .
Our next step is to take explicit account of the two terms in χ ′′ ( q, ω) and rewrite Eqs. (1) and (2) as where K nP and τ eP are defined using χ ′′ P ( q, ω) in Eqs. (3) and (4), respectively. In Eqs. (5) and (6) the T 1Ic 's are calculated with Eqs.(1) and (2), respectively, using τ ef f = τ eI , and K n = K nI . We may now use Eqs. (5) and (6) to extract estimates of τ eP from data for both 1/ 17 T 1c and 1/ 63 T 1c . However, it is necessary to model χ ′′ P ( q, ω) in order to make systematic estimates of the K nP . For this purpose we follow Aeppli, et al. [6], taking a squared Lorentzian form of unit amplitude χ ′′ P ( q, ω)/χ ′′ P (peak) = q 4 w /(q 2 w + q 2 x + q 2 y ) 2 , from which the K nP 's follow via Eq.(3). Since 17 T 1c (Eq.(6)) varies rapidly with K 1P , while 63 T 1c (Eq. (5)) is more weakly dependent on the K nP 's, the width parameter q w may be varied with temperature to bring the τ eP 's into coincidence. Results of this procedure for LSCO are presented in Fig.1(b), with the corresponding values of K nP shown in the inset.
Next, we consider the case of YBCO6.5. While LSCO has only a weak pseudogap, YBCO6.5 has stood from the earliest days as a classic pseudogap system [8][9][10]. There now exists for YBCO6.5 a fairly complete, quantitative INS data set, discussed by the authors in terms of dynamical stripes [7]. Low-frequency data for χ ′′ I ( q, ω) a exhibit clear-cut ω/T scaling, yielding the horizontal solid line in Fig. 2(a)  (1) and (2) with measured HF constants [30], to the widely disparate dash-dot curves for τ eP in Fig. 2(a). Such a discrepancy in the region below T c ∼ 62K suggests a sharp difference in the peak widths for χ ′′ I ( q, ω) a between the INS and NMR samples. Indeed, broadening the χ ′′ I ( q, ω) a peaks by a factor ∼ 2.5 leads to unification of the τ ef f curves at T < 62K, as shown by the blue square and red circle points in Fig. 2(a). This surprising broadening effect represents the difference between oriented powder samples used for the T 1 measurements [21] and single crystals used for the INS studies, most likely due to different degrees of chain oxygen ordering. Calculations of the coefficients K nIa,b made for Fig. 2(a)  Values of τ ef f deduced from Eqs.(1) and (2) agree very nicely with τ eI data (solid line) in Fig. 2(a) at T c ∼ 62K, again with no adjustable parameters. As with LSCO, the τ ef f curves show a sharp increase over τ eI and a strong divergence from one another at T > 62K. To find consistent values of τ eP for YBCO6.5, we again model χ ′′ P ( q, ω) using the squared Lorentzian form as for LSCO with the same exponential form for K 1P (T ). The result (Fig.2(b)) is quite successful. Curves for the K nP are shown in the inset. In this case K 1P (T ) = -0.87exp[−(T − 62)/725] decays a bit more slowly and begins with a somewhat narrower peak (q w ∼ 0.44 at T = 62K). The incommensurability ∼ 0.38, however, is less than q w , so that the progressively broadening peak of χ ′′ P ( q, ω) will form something of an elevated baseline for the incommensurate peaks. Such a background will be difficult to detect with unpolarized neutrons.
The τ eP curves in Fig. 1(b) and 2(b) are qualitatively similar, with τ eP vanishing nearly linearly as T declines toward T Str , while bending over towards room temperature. At the latter point, the new term contributes far more to 1/T 1 than the incommensurate 'stripe' fluctuations. Values of τ eI , which are considerably larger for LSCO than for YBCO6.5, obey ω/T scaling and also do not display a spin gap until the materials become bulk superconductors at T c . The experimental conclusion is therefore clear -the much-celebrated magnetic pseudogaps in these systems are gaps in the new term deduced from T 1 data, which accounts for more spectral weight than the incommensurate spin fluctuations at NMR frequencies.
Moreover, given the strong evidence that the stripe (incommensurate) fluctuations which dominate low-and medium energy neutron measurements compete with superconductivity, it is the pseudogap terms that are much more likely to form a pair binding texture.
Some time ago, in the first quantitative test of the magnetic fluctuation-dissipation the-orem a joint analysis of NMR-INS data on LSCO was presented [20]. That work was only a partial success because of its rather simple treatment of the T 1 process. At the time, a two-band model was called for, but there is no longer any clear motivation for such a model [31]. However, NMR shift analyses have been put forth recently giving evidence for a 'two-component' shift structure [32]. We suggest that the two-part structure for nuclear relaxation described in the present work could, via the Kramers-Kronig relation, form the basis in principle for a two-component NMR shift. In practice, there are no χ ′′ ( q, ω) data near q = 0 to provide a quantitative basis for a shift estimate. However, the proposed NMR shift structure is regarded as a natural extension of the present two-component model for χ ′′ ( q, ω). We emphasize that the latter model does not imply two independent bands of charge carriers.
In closing, we comment on the prospects for resolving the new fluctuation term χ ′′ P ( q, ω) with INS methods. The squared Lorentzian model form factor used here yields estimates for χ ′′ P ( q, ω) that are generally much broader and flatter than the incommensurate peaks reported to date. For LSCO, such a model yields an amplitude ∼10% of χ ′′ I ( q, ω) a for a scan through adjacent peaks. Such a result is compatible with experimental spectra for 50K ≤ T ≤ 100K [6]. Thus, resolving χ ′′ P ( q, ω) will require polarized neutrons with a far higher signal-to-noise ratio than what has been reported to date. Recent improvements in neutron scattering technique may render this feasible. However, we conclude that the χ ′′ P ( q, ω) term, whatever its precise shape and behavior may be, must exist, rendering χ ′′ ( q, ω) consistent with pseudogap studies using other probes. We suggest this inference to extend to other cuprates as well, offering a clear and broadly consistent picture of the pseudogap effect throughout this family of superconductors.
We thank B. S. Shastry and J. Zaanen for highly informative conversations. This