Unconventional Coherence Peak in Cuprate Superconductors

Unconventional superconductors, such as $d$-wave and $s^{\pm}$-wave superconductors, are characterized by a varying sign of the superconducting gap function in momentum space, with their Cooper pairs widely believed to originate from electron-electron correlation rather than electron-phonon coupling[1]. Therefore, unconventional superconductors exhibit distinct physical properties, such as the absence of the Hebel-Slichter coherence peak [2, 3, 4, 5, 6, 7, 8, 9, 10]. In conventional superconductors with uniform superconducting gaps, the density of states (DOS) diverges at the gap energy, leading to a huge enhancement of spin-lattice relaxation rate $1/T_{1}$ just below the critical temperature $T_{\rm c}$ in nuclear magnetic resonance (NMR) experiment, called the Hebel-Slichter coherence peak[11]. Conversely, this Hebel-Slichter coherence peak is absent in unconventional superconductors[10], such as d-wave cuprate superconductors, where the DOS divergence persists. Consequently, detecting coherence peaks poses a significant challenge for unconventional superconductors. In this work, we reveal a novel unconventional coherence peak in high-temperature cuprate superconductors using nuclear quadrupole resonance (NQR), introducing a distinctive feature of unconventional superconductivity.

The nucleus with the spin $I>1/2$ is nonspherical and possesses an electric quadrupole moment $Q_{\alpha\beta}$, where $\alpha$ and $\beta$ are spatial directions x, y, z. This results in the nucleus having an electrostatic energy that varies depending on its orientation within an electric field gradient (EFG) $V_{\alpha\beta}=\tfrac{{\partial^{2}V}}{{\partial\alpha\partial\beta}}$ generated by the local potential $V$ from its surrounding environments[12]. Hence, the electrostatic energy splitting between different nucleus spin $|I|$ is utilized in NQR, analogous to how NMR employs the magnetic energy splitting of each spin $I$ under a magnetic field. Furthermore, the electric quadrupole moment is a valuable tool for investigating electric field dynamics, akin to how the magnetic moment is used to study spin dynamics. These tools complement each other in exploring electromagnetic fluctuations in condensed matter[13]. A key dynamic signal is the spin-lattice relaxation rate, which describes how the nuclei arrive at their thermal equilibrium via the process of spin-lattice relaxation and is proportional to the summation of the imaginary part of the dynamical susceptibility. In the conventional $s$-wave superconducting state, the rate can be expressed as[14]

where $T_{1{\rm N}}$ and $T_{1{\rm S}}$ are the relaxation times in the normal state and the superconducting state, respectively. $N_{0}$ is the DOS in the normal state and $N_{\rm S}(E)=N_{0}E/\sqrt{E^{2}-\Delta^{2}}$ for $E>\Delta$ is the DOS in the superconducting state. $f(E)$ is the Fermi distribution function and $\Delta$ is the superconducting gap. The sign of the coherence factor $\left({1\pm{{\Delta^{2}}\mathord{\left/{\vphantom{{\Delta^{2}}{E^{2}}}}\right.\kern-1.2pt}{E^{2}}}}\right)$ is contingent on the nature of the perturbation causing the transition. Magnetic relaxations, being not time-reversal invariant, have $\left({1+{{\Delta^{2}}\mathord{\left/{\vphantom{{\Delta^{2}}{E^{2}}}}\right.\kern-1.2pt}{E^{2}}}}\right)$ which enhance the divergency of $N_{\rm S}(E)$ and the magnetic relaxation rate $1/T_{1}^{\rm mag}$ exhibits a Hebel-Slichter coherence peak in conventional superconductors[14]. In contrast, quadrupole relaxations, being time-reversal invariant, have $\left({1-{{\Delta^{2}}\mathord{\left/{\vphantom{{\Delta^{2}}{E^{2}}}}\right.\kern-1.2pt}{E^{2}}}}\right)$ which compensate for the divergency of $N_{\rm S}(E)$ and the quadrupole relaxation rate $1/T_{1}^{\rm quad}$ drops rapidly below $T_{\rm c}$ in conventional superconductors[15, 16]. In unconventional superconductors, $1/T_{1}^{\rm mag}$ drops or decreases gradually below $T_{\rm c}$, without a coherence peak being observed[2]. These phenomena contradict the logarithmic divergence of $N_{\rm S}(E)$ in cuprate superconductors[17]. Additionally, the cuprate superconductor gap sign varies in momentum space, significantly impacting the coherence factor and potentially leading to a shift of the coherence peak to $1/T_{1}^{\rm quad}$.

Whether a coherence peak is present in $1/T_{1}^{\rm quad}$ motivated us to perform NQR on two isotopes ⁶³Cu and ⁶⁵Cu in oxygen-stoichiometric and underdoped YBa₂Cu₄O₈. This compound is an unconventional superconductor with a transition temperature $T_{\rm c}=81.5$ K[18], as confirmed by the temperature-dependent magnetization measurement in Fig. 6. As previously mentioned, NQR is particularly sensitive to the local chemical environment and can distinguish between Cu atoms situated in distinct positions. YBa₂Cu₄O₈ contains two Cu sites, namely, the chain Cu(1) and the planar Cu(2) (depicted in the upper inset in Fig. 1), offering a unique avenue to probe $1/T_{1}^{\rm mag}$ and $1/T_{1}^{\rm quad}$ and explore the coherence peak. The Cu(1) and Cu(2) sites have different EFG strengths, resulting in different resonance frequencies as shown in the spectra inserted in Fig. 1. Additionally, the distinct EFG tensors of Cu(1) and Cu(2) influence their electric dynamic behavior, which is discussed later. Two isotopes ⁶³Cu and ⁶⁵Cu lead to two NQR resonance peaks at each site, for a total of four peaks at both sites, as shown in the lower inset in Fig. 1 for the spectrum at $T_{\rm c}$.

The four NQR spin-lattice relaxation rates $R$ corresponding to the four peaks are measured and plotted in Fig. 1. For the planar Cu(2), the $R$ of ⁶³Cu(2) and ⁶⁵Cu(2) (green and purple lines, respectively) drop due to the reduction in DOS caused by the superconducting gap below $T_{\rm c}$, with noticeable kink behaviors observed at $T_{\rm c}$. The $R$ of Cu(2) above $T_{\rm c}$ keeps increasing and saturates around 200 K[19]. These data are consistent with previous reports in the whole temperature range[4, 6] and similar to YBa₂Cu₃O₇[5]. On the other hand, the $R$ of ⁶³Cu(1) and ⁶⁵Cu(1) (cyan and red lines, respectively) decrease with temperature decreasing above $T_{\rm c}$, resembling the behavior of a conventional metal. Notably, the $R$ of Cu(1) exhibits a slight increase just below $T_{\rm c}$, followed by a decrease at lower temperatures. These distinctive characteristics are highly unusual, suggesting a coherence peak akin to that observed in conventional superconductors.

Let us focus our attention on the superconducting transition and eliminate the linear temperature term in the metal by drawing $R$/T around $T_{\rm c}$ in Figs. 2(a) and 2(b). Both $R$/T of ⁶³Cu and ⁶⁵Cu at the Cu(1) site exhibit a coherence peak just below $T_{\rm c}$, whereas $R$/T of the Cu(2) site commences a quite steep descent below $T_{\rm c}$. The distinct temperature-dependence behaviors of Cu(1) and Cu(2) are attributed to their different EFG. $R$ can be expressed as a sum of magnetic relaxation rate $1/T_{1}^{\rm mag}$ and quadrupole relaxation rate $1/T_{1}^{\rm quad}$, $R=3/T_{1}^{{\text{mag}}}+1/T_{1}^{{\text{quad}}}$ for I = 3/2 NQR experiment[20]. Theoretically, the nuclear spin Hamiltonian of the interaction between quadrupole moment Q and EFG can be written as[12]

where $\nu_{\rm Q}$ is the quadrupole resonance frequency along the principal axis, $h\nu_{\rm Q}=3eQV_{zz}/2I(2I-1)$. $\eta=({V_{xx}-V_{yy}})/V_{zz}$ is an asymmetry parameter of the EFG, where $V_{xx},V_{yy}$, and $V_{zz}$ are the EFGs along the x, y, and z directions, respectively. Only the terms with $I_{+}$ and $I_{-}$ can flip spins and contribute to $1/T_{1}^{\rm quad}$, necessitating a sufficiently large $\eta$ to measure $1/T_{1}^{\rm quad}$. The planar Cu(2) is isotropic with a negligible $\eta$ [4], resulting in $1/T_{1}$ of Cu(2) being purely affected by magnetic relaxation and unable to detect EFG fluctuations[2]. Therefore, although there are charge fluctuations in the Cu-O plane detected by planar ¹⁷O with $\eta>0.2$ [21], they cannot be detected by Cu(2)[22]. On the other hand, the chain Cu(1) is anisotropic with $\eta$ = 0.85, causing $1/T_{1}$ of Cu(1) to be affected by both magnetic and quadrupole relaxation. If the coherence peak is due to magnetic relaxation, it should be observed at both Cu(1) and Cu(2) sites. The presence of the coherence peak exclusively at the Cu(1) site suggests its origin from quadrupole relaxation.

$1/T_{1}^{\rm mag}$ and $1/T_{1}^{\rm quad}$ can be decomposed by utilizing two isotopes ^63,65Cu, which possess the same spin I=3/2 but different gyromagnetic ratios and quadrupole moments[2]. Using isotopes can avoid the influence from form factors and principle axes directions[23], since they occupy the same atom site. The decomposition equations are $3/^{{\text{63}}}T_{1}^{{\text{mag}}}=\left({{}^{65}R-b\times^{63}R}\right)/\left({a-b}\right)$ and $1/^{63}T_{1}^{{\text{quad}}}=\left({a\times{}^{63}R-^{65}R}\right)/\left({a-b}\right)$, where the quotients $a=(^{65}\gamma/^{63}\gamma)^{2}=1.1477$ and $b=(^{65}Q/^{63}Q)^{2}=0.8562$ are known[24]. Before decomposing, the ratio of ${}^{65}R/^{63}R$ is calculated to assess the weight of $1/T_{1}^{\rm mag}$ and $1/T_{1}^{\rm quad}$, shown in Figs. 2(c) and 2(d). ${}^{65}R/^{63}R$ at the Cu(2) site remains a constant value around the magnetic quotient $a=1.1477$ for various temperatures, indicating negligible $1/T_{1}^{\rm quad}$ and the presence of only $1/T_{1}^{\rm mag}$. Conversely, ${}^{65}R/^{63}R$ at the Cu(1) site deviates from $1.1477$ around $T_{\rm c}$, suggesting a detectable $1/T_{1}^{\rm quad}$ component.

The magnetic and quadrupole relaxations of Cu(1) are decomposed using the relations mentioned above, and the results are illustrated in Figs. 3(a) and 3(b), respectively. The $1/T_{1}^{\rm mag}$ depicted in Fig. 3(a) drops below $T_{\rm c}$ due to the loss of DOS, aligning with the behavior of the $R/T$ at the Cu(2) site where solely magnetic relaxation occurs and $R/T=3/(T_{1}T)^{{\text{mag}}}$. In contrast, the $1/(T_{1}T)^{\rm quad}$ displays a prominent coherence peak just below $T_{\rm c}$, in sharp contrast to conventional superconductors where the coherence peak is evident in the $1/(T_{1}T)^{\rm mag}$ and absent in $1/(T_{1}T)^{\rm quad}$. This discrepancy elucidates the absence of a coherence peak at the Cu(2) site, as it is unaffected by quadrupole relaxation. It is noteworthy that the $1/(T_{1}T)^{\rm quad}$ is smaller than $1/(T_{1}T)^{\rm mag}$, underscoring the higher sensitivity of most nuclei, such as Cu, to magnetic fluctuations in dipolar interaction than to EFG fluctuations in quadrupole interaction. Moreover, in the superconducting state, on lowering the temperature, the quadrupolar relaxation diminishes faster than the magnetic one[22]. Consequently, the coherence peaks in Fig. 2 (a) have historically been inconspicuous and challenging to detect.

There is one thing we need to emphasize: The superfluid density of the system around $T_{\rm c}$ is dominated by the CuO₂ plane, since the bare transition temperature of the chain is much smaller than the plane owing to the strong fluctuations in one dimension [25, 26, 27, 28]. Therefore, although $1/T_{1}^{\rm quad}$ is measured at the chains, the peak from $1/T_{1}^{\rm quad}$ directly probes the charge dynamics of the CuO₂ plane. To simplify our discussion and arrive at a qualitative understanding, we focus on only the CuO₂ plane and leave the general discussion to the appendix. The newly observed $1/T_{1}^{\rm quad}$ peak is proportional to a $q$-summed charge susceptibility $\chi_{\rho}\left({q,E}\right)$ of cuprate superconductivity, underscoring the significance of both long and short wavelengths to $1/T_{1}^{\rm quad}$. Theoretically, $1/T_{1}^{\rm quad}$ can be written as

where the $F(q)$ is the structure factor of the quadrupole interaction. A key aspect of this equation is the coherence factor $\left({1-{{\Delta_{k}\Delta_{k+q}}\mathord{\left/{\vphantom{{\Delta_{k}\Delta_{k+q}}{E^{2}}}}\right.\kern-1.2pt}{E^{2}}}}\right)$. For a d-wave superconductor, a prominent momentum q is around $q_{0}=(\pi,\pi)$. This $q_{0}$ excitation, along with the corresponding coherence factor, has previously resulted in a distinct spin resonance peak in neutron scattering of cuprate superconductors[29]. Similarly, this coherence factor around $q_{0}$ gives rise to a sign reversal $\Delta_{k}\Delta_{k+q_{\text{0}}}<0$, which cannot counterbalance the divergence from DOS in the superconducting state. Hence, this $q_{0}$ may lead to this unconventional coherence peak in $1/T_{1}^{\rm quad}$ upon entering the superconducting transition.

The unconventional coherence peak identified in the nuclear quadrupole relaxation rate of YBa₂Cu₄O₈ complements the relaxation rate pattern depicted in Fig. 4. In conventional superconductors, a Hebel-Slichter coherence peak is typically observed in the magnetic relaxation rate, as illustrated in Fig. 4 (a), whereas the quadrupole relaxation rate drops below $T_{\rm c}$ due to coherence factors, as illustrated in Fig. 4 (b). It reflects that the gap is isotropic without a sign reversal in conventional superconductors, as shown in Fig. 4 (c). Conversely, unconventional superconducting gaps exhibit a sign reversal attributed to electron-electron correlation. In such cases, the magnetic relaxation rate does not display a coherence peak around $T_{\rm c}$ experimentally, and the quadrupole relaxation rate acquires a coherence peak, as illustrated in Figs. 4(d) and 4(e). Historically, the absence of the Hebel-Slichter coherence peak in $1/T_{1}^{\rm mag}$ was widely discussed [8, 9, 7, 30, 31, 32, 33, 34]. One of the prevailing views is that the sign-changing gap is the reason for this absence, which eliminates the coherence factor contribution [31, 32, 30, 17]. Meanwhile, it will lead to a phenomenon that the $1/T_{1}^{\rm quad}$ be enhanced by the coherence factors just below $T_{\rm c}$ due to the gap sign changes at various positions around Fermi surfaces, with scattering between different signs becoming dominant, as shown in Fig. 4 (f).

The sharp contrasts observed in $1/T_{1}^{\rm mag}$ and $1/T_{1}^{\rm quad}$ from conventional superconductors provide a novel method to explore the unconventional nature of the pairing mechanism in unconventional superconductors. The spin-singlet Cooper pairs are widely believed to be formed above $T_{\rm c}$ owing to the small superfluid density in unconventional superconductors[35]. The unconventional peak observed in $1/T_{1}^{\rm quad}$ leads to a hallmark of forming phase coherent unconventional superconducting condensate below $T_{\rm c}$. This peak can be used to diagnose unconventional (sign-changing gap) superconductivity with significant quadrupole relaxation.

On the other hand, measurement conditions for $1/T_{1}^{\rm quad}$ are quite stringent, necessitating substantial EFG fluctuations, significant $\eta$ and the presence of two or more isotopes. Cuprates have charge-density wave (CDW) fluctuations in the phase diagram[36, 37], which provide opportunity to detect $1/T_{1}^{\rm quad}$. Although CDW fluctuations contribute to $1/T_{1}^{\rm quad}$, the peak found here cannot be due to CDW which competes with superconductivity and CDW order appears only when superconductivity is killed. Some cuprates contain chain Cu with large $\eta$ and two isotopes ^63,65Cu. Underdoped YBa₂Cu₃O_7-δ sharing similar electronic band structure with YBa₂Cu₄O₈ can also be utilized to identify the unconventional coherence peak[38]. However, measuring $1/T_{1}^{\rm quad}$ in other unconventional superconductors poses challenges, such as in iron arsenide where only one isotope of ⁷⁵As is present[39]. Theoretically, $1/T_{1}^{\rm quad}$ can be estimated by comparing the relaxation rate measured by NMR and NQR. However, the different principle axes and the external field make it hard to compare NMR and NQR directly[23]. Another method is to compare the relaxation rate of satellite peaks with the central peak. Nevertheless, achieving this requires exceedingly precise data that surpass current experimental precision levels[20]. As measurement precision improves, there may be opportunities to explore a wider range of systems. We hope our study will inspire extensive future experimental and theoretical investigations to elucidate whether the unconventional coherence peak is a universal characteristic of unconventional superconductivity and delve into its underlying mechanisms.