Antiferromagnetic spin fluctuations in the metallic phase of quasi-two-dimensional organic superconductors

We give a quantitative analysis of the previously published nuclear magnetic resonance (NMR) experiments in the $\kappa \text{−}{\left(\mathrm{E}\mathrm{T}\right)}_{2}X$ family of organic charge-transfer salts. The temperature dependence of the nuclear-spin relaxation rate $1∕T_1$, the Knight shift $K_s$, and the Korringa ratio $\mathcal{K}$ is compared to the predictions of the phenomenological spin-fluctuation model of Moriya and Millis, Monien, and Pines (M-MMP), that has been used extensively to quantify antiferromagnetic spin fluctuations in the cuprates. For temperatures above $T_{\text{NMR}}\simeq 50\text{K}$, the model gives a good quantitative description of the data in the metallic phases of several $\kappa \text{−}{\left(\mathrm{E}\mathrm{T}\right)}_{2}X$ materials. These materials display antiferromagnetic correlation lengths which increase with decreasing temperature and grow to several lattice constants by $T_{\text{NMR}}$. It is shown that the fact that the dimensionless Korringa ratio is much larger than unity is inconsistent with a broad class of theoretical models (such as dynamical mean-field theory) which neglects spatial correlations and/or vertex corrections. For materials close to the Mott insulating phase the nuclear-spin relaxation rate, the Knight shift, and the Korringa ratio all decrease significantly with decreasing temperature below $T_{\text{NMR}}$. This cannot be described by the M-MMP model and the most natural explanation is that a pseudogap, similar to that observed in the underdoped cuprate superconductors, opens up in the density of states below $T_{\text{NMR}}$. Such a pseudogap has recently been predicted to occur in the dimerized organic charge-transfer salts materials by the resonating valence bond (RVB) theory. We propose specific experiments on organic superconductors to elucidate these issues. For example, measurements to see if high magnetic fields or high pressures can be used to close the pseudogap would be extremely valuable.

Figure 1

(Color online) Comparison of the measured nuclear-spin relaxation rate per unit temperature, $1∕T_{1}T$, with the predictions of the spin-fluctuation model for various organic charge-transfer salts. Panel (a) shows data for $\kappa \text{−}{\left(\mathrm{E}\mathrm{T}\right)}_2\text{Cu}\left[\text{N}{\left(\text{CN}\right)}_2\right]\text{Br}$ measured by Mayaffre et al. (Ref. 25). Panel (b) shows data for $\kappa \text{−}{\left(\mathrm{E}\mathrm{T}\right)}_2\text{Cu}\left[\text{N}{\left(\text{CN}\right)}_2\right]\text{Br}$ measured by De Soto et al. (Ref. 26). Panel (c) shows data for $\kappa \left(d8\right)\text{−}{\left(\mathrm{E}\mathrm{T}\right)}_2\text{Cu}\left[\text{N}{\left(\text{CN}\right)}_2\right]\text{Br}$ measured by Miyagawa et al. (Ref. 48). Panel (d) shows data for a $\kappa \text{−}{\left(\mathrm{E}\mathrm{T}\right)}_2\text{Cu}{\left(\text{NCS}\right)}_2$ powder sample measured by Kawamoto et al. (Ref. 49). The $1∕T_{1}T$ data are weakly temperature dependent at high temperatures, have a maximum at $T_{\text{NMR}}\sim 50\text{K}$, and drop abruptly below $T_{\text{NMR}}\sim 50\text{K}$, contrary to what one would expect for a Fermi liquid in which $1∕T_{1}T$ is constant. The remarkable similarities of these data result from the quantitative and qualitative similarity of the antiferromagnetic spin fluctuations in the metallic phases of these materials. The parameters that produce the best fits (solid lines) to Eq. (11c) are tabulated in Table . The spin-fluctuation model gives a good fit to the experimental data between $T_{\text{NMR}}\sim 50$ and room temperature which suggests strong spin fluctuations in the metallic states of $\kappa \text{−}{\left(\mathrm{E}\mathrm{T}\right)}_2\text{Cu}\left[\text{N}{\left(\text{CN}\right)}_2\right]\text{Br}$, $\kappa \left(d8\right)\text{−}{\left(\mathrm{E}\mathrm{T}\right)}_2\text{Cu}\left[\text{N}{\left(\text{CN}\right)}_2\right]\text{Br}$, and $\kappa \text{−}{\left(\mathrm{E}\mathrm{T}\right)}_2\text{Cu}{\left(\text{NCS}\right)}_2$. However, below $T_{\text{NMR}}$ the spin-fluctuation model does not describe the data well, indicating that some other physics dominates over the spin-fluctuation physics. In each figure the solid line is obtained from the approximate form for $1∕T_{1}T$ given by Eq. (13). To check that this approximation is reasonable, we also plot $1∕T_{1}T$ without any approximation, given by Eq. (11a), as a dashed line in panel (b). The full and dashed lines cannot be distinguished until well below $T_{\text{NMR}}$ and so we concluded that the approximation is excellent in the relevant regime. Note that the analysis on $1∕T_{1}T$ cannot differentiate between antiferromagnetic and ferromagnetic spin fluctuations (see Sec. 2A2), but the Korringa ratio strongly differentiates between these two cases and indicates that the fluctuations are antiferromagnetic (see Fig. 2). The nomenclature $\kappa \text{−}\text{Br}$, $d8\text{–Br}$, and $\kappa \text{−}\text{NCS}$ is used as shorthand for $\kappa \text{−}{\left(\mathrm{E}\mathrm{T}\right)}_2\text{Cu}\left[\text{N}{\left(\text{CN}\right)}_2\right]\text{Br}$, $\kappa \left(d8\right)\text{−}{\left(\mathrm{E}\mathrm{T}\right)}_2\text{Cu}\left[\text{N}{\left(\text{CN}\right)}_2\right]\text{Br}$, and $\kappa \text{−}{\left(\mathrm{E}\mathrm{T}\right)}_2\text{Cu}{\left(\text{NCS}\right)}_2$, respectively, in the figure keys.

Figure 2

(Color online) Comparison of the Korringa ratio $\mathcal{K}\propto 1∕T_{1}T{K}_s^2$ of $\kappa \text{−}{\left(\mathrm{E}\mathrm{T}\right)}_2\text{Cu}\left[\text{N}{\left(\text{CN}\right)}_2\right]\text{Br}$ measured by De Soto et al. (Ref. 26) with the prediction of the antiferromagnetic spin-fluctuation model. The best fit to Eq. (15) is indicated by the solid line. The Korringa ratio is larger than 1 which indicates that the spin fluctuations are antiferromagnetic ($\mathcal{K}<1$ for ferromagnetic fluctuations; see Sec. 2A2). The antiferromagnetic correlation length is found to be $2.8\pm 1.8$ lattice spacings at $T=50\text{K}$. Below $T=50\text{K}$ the Korringa ratio is suppressed, and the spin-fluctuation model does not explain this behavior. This is a clear indication that different physics is at play below 50 K.