Strain-induced long-range charge-density wave order in the optimally doped Bi₂Sr_2-xLa_xCuO₆ superconductor

Figure 2a, b show the temperature dependence of ac susceptibility under compressive and tensile strain measured at $H$ = 0 and 13 T, respectively. The crystal broke when the strain was more than $|$$\varepsilon$$|$ = 0.25 %. The strain dependence of $T_{\rm c}$ are summarized in Fig. 2c. Under the composite parameters of strain and high field $H^{\parallel c}$ = 13 T, $T_{\rm c}$ decreases symmetrically with compression and tension, as anticipated. This observation contrasts with the highly anisotropic response observed in the underdoped YBCO superconductor with orthorhombic structure [54, 55]. Because Bi2201 is a single-square CuO₂ plane cuprate with tetragonal structure [56], this suggests that the present result is the intrinsic strain response of superconductivity in cuprate. Notably, the strain response ($dT_{\rm c}$/$d\varepsilon$) becomes small above $|$$\varepsilon$$|$ = 0.1 %, and $T_{\rm c}$ does not decrease appreciably. The NMR measurements were performed at $H^{\parallel c}$ = 13 T below $|$$\varepsilon$$|$ = 0.25 %.

Figure 3a, b show the temperature dependence of ⁶³Cu-NMR $1/T_{1}T$ under various strains. First, near $T$ = 50 K, 1/$T_{1}T$ shows no clear strain response, suggesting that the pseudogap temperature $T^{*}$ is not strongly affected by strain. On the other hand, $1/T_{1}T$ decreases sharply below $T_{\rm c}$ as obtained by ac susceptibility in Fig. 2.

As shown in Fig. 3a, above $T_{\rm c}$, we found a peak at $T_{\rm CDW}$ for large strain beyond $\varepsilon=-0.152$ %, while such a peak is absent for $\varepsilon=-0.104$ %. The meaning of $T_{\rm CDW}$ will become clear later. $T_{\rm CDW}$ increases with increasing strain, reaching to $T_{\rm CDW}$ = 28 K at the maximum compression of $\varepsilon=-0.225$ %. The peak height also increases with increasing strain. The situation is similar under tensile strain (Fig. 3b). In general, 1/$T_{1}T$ shows a peak at a phase transition temperature. This occurs because the NMR relaxation rate diverges as the order parameter’s fluctuation slows down towards zero at the critical point. It is noteworthy that the peak in 1/$T_{1}T$ shows a qualitatively same behavior to the magnetic field-induced long-range CDW order found in the underdoped Bi2201 superconductors, signaling a phase transition [29, 44]. We summarized $T_{\rm c}$ and the observed phase transition temperature under strain in Fig. 4a.

Complementary to the insights from the temperature dependence of $T_{1}$, we investigated the phase transition at a fixed temperature by measuring the strain dependence of DOS and the full-width at half maximum (FWHM) of the ⁶³Cu-NMR center line. The spin part of the Knight shift, $K_{\rm s}$ (see Methods) is a direct measure of the spin susceptibility at the Cu-site and, consequently, the DOS, which correlates with the hole concentration $p$ [12]. The FWHM, on the other hand, provides information about the spatial distribution of the DOS and/or $p$.

Figure 5a displays the strain dependence of the ⁶³Cu-NMR center line at $T$ = 10 K. $K_{\rm s}$ (Fig. 5b) and the FWHM (Fig. 5c) were obtained from Gaussian fitting to the spectra (see Methods, Supplementary Fig. 4, and Note 4). $K_{\rm s}$ increases with strain up to $|$$\varepsilon$$|$ = 0.1 %, towards a constant, then decreases sharply beyond $|$$\varepsilon$$|$ = 0.15 % (see Fig. 5b). Note that $T$ = 10 K is below $T_{\rm c}(\varepsilon)$ up to $|$$\varepsilon$$|$ = 0.1 % (see Fig. 2c). Therefore, the increase of $K_{\rm s}$ under strain reflects the recovery of the Bogoliubov quasiparticle DOS with decreasing $T_{\rm c}$ up to $|$$\varepsilon$$|$ = 0.1 %. The estimated relative DOS (see Methods) from $K_{\rm s}$, which corresponds to the size of the Fermi arc [12, 13], is also indicated with secondary vertical axis of Fig. 5b. The reduction of the relative DOS from 0.5 to 0.3 also occurs at $|$$\varepsilon$$|$ = 0.15 %, indicating that a Fermi surface reconstruction to a smaller size occurred. Here, the observation that the NMR spectrum maintains its Gaussian shape while the $K_{\rm s}$ changes indicates that the strain-induced changes in the electronic state occur uniformly throughout the entire sample.

Figure 5c shows a FWHM increase of 0.2 MHz across $|$$\varepsilon_{\rm c}$$|$ = 0.15 %, suggesting the emergence of a distribution of the DOS, likely reflecting a spatial variation in the carrier density within the CuO₂ plane. Furthermore, the FWHM was found to increase above $|$$\varepsilon_{\rm c}$$|$ following the strain dependence of the mean-field model, $(\varepsilon-\varepsilon_{\rm c}$)^0.5 with $\varepsilon_{\rm c}=-0.152$ %. At $T$ = 10 K, the phase boundary is at $|$$\varepsilon_{\rm c}$$|$ = 0.15 %, as can be seen in Fig. 4a. Notably, the mean-field growth of the order parameter has also been observed in the magnetic field-induced long-range CDW order [26, 29].

Therefore, the strain dependence of $K_{\rm s}$ and the FWHM suggest a uniform growth of the order parameter [charge (hole) distribution amplitude] across all Cu sites above the critical strain $|$$\varepsilon_{\rm c}$$|$, indicative of a long-range order (second-order phase transition) at $\varepsilon_{\rm c}$.

Fig. 5d shows the data for 1/$T_{1}T$ at $T$ = 10 K as a function of strain, which were extracted from Fig. 3a, b. In contrast to the reduction of $T_{\rm c}$ upon applying strain, 1/$T_{1}T$ increases under strain. The observed behavior is qualitatively consistent with that of $K_{\rm s}(\varepsilon)$. However, 1/$T_{1}T$ continues to increase even though $T_{\rm c}$ changes little and $K_{\rm s}$ becomes constant above $|$$\varepsilon$$|$ = 0.1 %, reaching a maximum at the phase boundary $|$$\varepsilon_{\rm c}$$|$, and then decreases as $K_{\rm s}$ decreases (the FWHM increases) with further strain. The peak in 1/$T_{1}T$ at $|$$\varepsilon_{\rm c}$$|$ further supports the occurrence of a long-range order (second-order phase transition).

In addition to the $T_{1}$ for ⁶³Cu, we measured the $T_{1}$ for ⁶⁵Cu at $T$ = 10 K (Fig. 5d) to investigate the strain response of spin and charge fluctuations that equally exist at $\varepsilon=0$ [44]. The strain dependence of the ratio, ⁶⁵(1/$T_{1}$)/⁶³(1/$T_{1}$), is shown in Fig. 5e. When spin fluctuations are dominant, ⁶⁵(1/$T_{1}$)/⁶³(1/$T_{1}$) = 1.15, while when charge fluctuations are dominant, ⁶⁵(1/$T_{1}$)/⁶³(1/$T_{1}$) = 0.86. At $\varepsilon$ = 0, spin and charge fluctuations coexisted [⁶⁵(1/$T_{1}$)/⁶³(1/$T_{1}$) = 1] [44], however, we found that the fraction of the charge (spin) fluctuations increased (decreased) as the strain was increased and superconductivity was suppressed, and the charge fluctuations became dominant above $|$$\varepsilon$$|$ = 0.15 % [⁶⁵(1/$T_{1}$)/⁶³(1/$T_{1}$) = 0.86]. Therefore, the strain-induced phase transition is most likely due to a charge-originated one. To confirm that the order parameter of the phase transition is indeed of charge origin, we systematically measure NMR spectra under strain.

Figure 6a-f show the temperature dependence of the ⁶³Cu-NMR center line (Fig. 6a-c) and ^63,65Cu satellite lines (Fig. 6d-f), measured under the compressive strain $\varepsilon=-0.225$ %, zero strain $\varepsilon$ = 0, and the tensile strain $\varepsilon=+0.195$ %, respectively. The signal of Cu-metal near the center peak comes from the strain cell (see, Supplementary Fig. 4).

As shown in Fig. 6g, the FWHM did not show a temperature dependence at $\varepsilon$ = 0. However, the FWHM of the satellite line increased significantly at $\varepsilon=-0.225$ % and $+0.195$ %, while the FWHM of the center line increased only slightly at low temperature. The FWHM of the satellite line increased by more than 2 MHz, compared to the center line FWHM of only 0.2 MHz, as observed in the strain dependence (Fig. 5c). In principle, the satellite line width is more affected by the distribution of the quadrupole interaction. This is because, in this experiment, the external magnetic field and the principal axis of the electric field gradient (EFG) at the Cu site are in parallel. In this situation, the center peak is originated from the Zeeman interaction, while the satellite peaks are originated from both the Zeeman and the quadrupole interactions (see Methods). Therefore, the fact that the center peak only slightly broadens while the FWHM of the satellite peaks increases significantly below $T_{\rm CDW}$ is evidence that a static distribution of the quadrupole interaction, rather than the Zeeman interaction, exists. Thus, Fig. 6g shows that a strain-induced long-range CDW order occurs in the CuO₂ plane below $T_{\rm CDW}(\varepsilon)$. The observed spectral changes, the satellite line being more prominently affected than the center line, are consistent with those found for magnetic field-induced long-range CDW order in underdoped cuprates [26, 29, 44]. Furthermore, the FWHM does not change or even increases below $T_{\rm c}(\varepsilon)$, which indicates that the strain-induced long-range CDW order and superconductivity are not mutually exclusive, and superconductivity can occur even in the long-range CDW ordered state (see Supplementary Fig. 5 and Note 5 for more detail).

As can be seen in Fig. 4a which shows the obtained phase diagram of the optimally doped Bi2201 superconductor under strains at ${H^{\parallel c}}$ = 13 T, unlike the underdoped YBCO superconductor [54, 55], we observed that both superconductivity and CDW exhibit symmetric strain responses. The present results also contrast with recent findings that there is essentially no strain effect on the charge order in underdoped La_1.875Ba_0.125CuO₄ [57]. The present study demonstrates the simultaneous observation of the two orders under strains in an optimally doped cuprate.

Bi₂Sr_2-xLa_xCuO_6+δ crystallizes tetragonal structure (space group $I4/mmm$) without a structural phase transition [56]. There is only one CuO₂ plane per unit cell, and the interplanar spacing is the largest in cuprates. The single crystal of Bi₂Sr_1.6La_0.4CuO_6+δ ($p$ = 0.162, $T_{\rm c}$ = 32 K) was grown by the traveling solvent floating zone method [56]. The hole concentration ($p$) was estimated previously [70]. The single crystal plates used for the measurement were cut from the same ingot as the crystal used in the previous NMR experiments [15, 12, 53, 29, 44]. Small and thin single-crystal platelet, sized up to $L_{0}\times W\times th$ = 2.00 $\times$ 0.62 $\times$ 0.12 ${\rm mm}^{3}$ where $L_{0}$ is the length of the strained part, $W$ is the width, and $th$ is the thickness along the $c$ axis was cleaved from the ingot. Strain was applied along the Cu-O bond direction, which was determined by Laue picture. The ^63,65Cu-NMR spectrum (Fig. 1b) and ⁶³(1/$T_{1}T$) below $T$ = 50 K (Fig. 1c) measured at $V_{\rm piezo}$ = 0 for reproducibility are consistent with the results measured on the bare sample. $T^{*}$ $\approx$ 160 K and $T_{\rm c}(H^{\parallel c}$ = 13 T) = 17.5 K are consistent with the previous results [15, 12, 53, 29, 44].

There is only one Cu site in the single CuO₂ layer cuprate Bi2201. For ^63,65Cu-NMR, the Hamiltonian is the sum of the Zeeman and nuclear quadrupole interactions [71] as $\mathcal{H}$ = $\mathcal{H}_{\rm z}+\mathcal{H}_{\rm Q}$ = $-^{63,65}\gamma\hbar{\bf I}\cdot{\bf{H}_{0}}(1+^{63,65}K)+\frac{h^{63,65}\nu_{\rm Q}}{6}[3{I_{z}}^{2}-I(I+1)+\eta({I_{x}}^{2}+{I_{y}}^{2})]$, where gyromagnetic ratio ${}^{63}\gamma$ (${}^{65}\gamma$) = 11.285 (12.089) MHz/T, ${}^{63,65}K$ is the Knight shift, and the nuclear spin $I$ = 3/2. ${}^{63,65}\nu_{\rm Q}$ = $\frac{3eqV_{zz}}{2I(2I-1)h}\sqrt{1+\eta^{2}/3}$ with ${}^{63}{Q}$ (${}^{65}{Q}$) = $-0.220(-0.204)\times 10^{-24}$ cm² [72] and $V_{zz}$ being the nuclear quadrupole moment and the EFG tensor [71]. The principal axis of the EFG is along the $c$ axis, which is parallel to the external magnetic field ${\bf H_{0}}$. The asymmetry parameter $\eta$ are defined as $\eta$ $=(|V_{yy}|-|V_{xx}|)/|V_{zz}|$, which is zero [73] and the strain (up to 0.225 %) in this study was not large enough $\eta$ to be clearly appeared in the measurement results. The ^63,65Cu-NMR spectra were taken by sweeping the rf frequency by using a phase-coherent spectrometer. As shown in Fig. 1b, when ${\bf H_{0}}$ and the principal axis of the EFG are in parallel, the center and the two satellite lines between $|m\rangle$ and $|m-1\rangle$, ($m$ = 3/2, 1/2, -1/2), at the resonance frequency $f_{m\leftrightarrow m-1}=^{63,65}\gamma H_{0}(1+^{63,65}K)-(^{63,65}\nu_{\rm Q})(m-1/2)$ are obtained for ⁶³Cu and ⁶⁵Cu, respectively [71].

${}^{63}K$(%) is obtained from the peak frequency ($f_{\rm c}$) of the center line as $K(\%)$ = 100 $\times$ $\frac{f_{\rm c}-\gamma H_{0}}{\gamma H_{0}}$. The Knight shift is denoted by $K$ = $K_{\rm s}$ + $K_{\rm orb}$, where $K_{\rm s}$ and $K_{\rm orb}$ are the spin and orbital contributions, respectively. $K_{\rm s}$ is proportional to the spin susceptibility $\chi_{\rm s}$ [71] and thus reflect the DOS. $K_{\rm orb}$ is known to be 1.21 % [12]. Therefore, we can estimate the relative DOS at $T$ = 10 K as $\frac{N(10K)}{N_{\rm 0}}$ = $\frac{K_{\rm s}(10K)}{K_{\rm s}(T\geq T^{*})}$ = $\frac{[K(\varepsilon)-1.21\%]}{0.3\%}$, where $N_{\rm 0}$ is the DOS above $T^{*}$ and $K_{\rm s}(T\geq T^{*})$ = 0.3 % was previously determined [12].

In the long-range CDW ordered state, a spatial modulation $\delta\nu_{\rm Q}(\bf r)$ [$\gg\delta K(\bf r)$] appear to split or broaden the spectrum [29, 74, 75]. The ⁶³Cu-NMR high frequency satellite line ($-1/2\leftrightarrow-3/2$ transition, $f$ $\approx$ 177 MHz) is used to detect a long-range CDW order in the same way as in the previous report of the magnetic field-induced long-range CDW order [29].

The ^63,65Cu-NMR $T_{1}$ were measured at the center peak ($+1/2\leftrightarrow-1/2$ transition, $f$ $\approx$ 148 and 158 MHz, respectively). To obtain $T_{1}$, the time dependence of the spin-echo intensity after the saturation of the nuclear magnetization $M$ (recovery curve) was fitted by the theoretical function [76]. Typical recovery curves used to determine $T_{1}$ are shown in Supplementary Fig. 7 and Note 7. In general, in strongly correlated electron systems, $T_{1}$ probes the spin fluctuation through the hyperfine coupling constant $A_{\bf q}$ as ^63,65$(1/T_{1}^{\rm M}$) $\propto$ ${}^{63,65}\gamma$$k_{\rm B}T\sum_{\bf q}\left|A_{\bf q}\right|^{2}\chi_{\perp}^{\prime\prime}({\bf q},\omega_{0})/\omega_{0}$, where $\omega_{0}$ is the NMR frequency and ${\bf q}$ is a wave vector for a spin order [77]. On the other hand, in the case that $T_{1}$ probes the EFG (charge) fluctuation, then it is dominated by the quadrupole moment as ^63,65($1/T_{1}^{\rm Q})\propto 3(2I+3)(^{63,65}{Q^{2}})/[10(2I-1)I^{2}]$ [78]. Therefore, the origin of the nuclear spin lattice relaxation process for the cuprate can be identified from the ratio, ⁶⁵(1/$T_{1}$)/⁶³(1/$T_{1}$) [44]. When spin fluctuations dominate,⁶⁵(1/$T_{1}$)/⁶³(1/$T_{1}$) = (${}^{65}\gamma$/${}^{63}\gamma$)² = $\gamma_{\rm R}^{2}$ = 1.15, while charge fluctuations dominate, ⁶⁵(1/$T_{1}$)/⁶³(1/$T_{1}$) = (${}^{65}{Q}$/${}^{63}{Q}$)² = ${Q}_{\rm R}^{2}$ = 0.86 [44]. In the superconducting state below $T_{\rm c}$, 1/$T_{1}T$ reflects the temperature dependence of the quasiparticle density of states, $N(E{\rm{}_{F}})$. The $T_{1}$ measurements under strain in this study were performed in the pseudogap state below $T$ = 50 K. If a long-range CDW order occurs, $1/T_{1}T$ shows a peak at $T_{\rm CDW}$ [29].

A homemade piezoelectric-driven strain cell is used to compress and/or tensile single crystal plate. Following C. W. Hicks $et$ $al$ [79], as shown in Supplementary Fig. 1, we have arranged three commercially available piezoelectric actuators (PI, P-885.51) in a configuration of two outer and one inner actuators, and connected them with a titanium cell to allow compression and tension of the crystal plate in the center of the cell. A sample plate was fixed to the cell using epoxy (STYCAST 2850FTJ). A power supply (Razorbill Instruments, RP100) was used to drive the actuators. The actual displacement is determined by reading the capacitance of a homemade parallel-plate capacitor placed next to the sample with an LCR meter (Keysight, E4980AL). For NMR measurement, to avoid temperature-dependent changes in strain, we applied voltage to the actuators and monitored the strain value at each temperature.

In this paper, in line with previous strain experiments [80, 81, 57], we define strain ($\varepsilon$) using the displacement $x$ as $\varepsilon(\%)$ = 100 $\times$ $(x-x_{0})$ / $L_{0}$, where $L_{0}$ is the unstrained length of the crystal. We have verified that $x_{0}$ ($V_{\rm piezo}$ = 0) corresponds to the zero-strain (bare sample) within the error of NMR experiments and $T_{\rm c}$ measurements, based on the results of our investigation using Bi2201 (see Fig. 1c and Supplementary Fig. 1-4 and Note 1-4). The results measured with the sample fixed with the cell at zero volts ($V_{\rm piezo}$ = 0) all agree with the results measured with the bare sample (zero strain). As shown in Supplementary Fig. 2, Hooke’s law was confirmed within the strain range of the experiment, indicating that the strain was uniformly distributed in the sample.

Our cell does not have a force sensor to directly measure pressure alongside strain. Therefore, for reference, we estimated pressure values using the elastic constants of Bi₂Sr₂CuO₆ from prior first-principles calculations [82]. These estimated values are shown on the upper axis of Fig. 4a. Of note, our value is in good agreement with that reported in strain experiments on the related cuprate La_1.875Ba_0.125CuO₄ superconductor [57].

We measured ac susceptibility, ^63,65Cu-NMR spectrum, and nuclear spin-lattice relaxation time $T_{1}$. A high-resolution superconducting magnet (Oxford Instruments, AS600) is used to apply a constant magnetic field of $H$ = 12.951 T ($H^{\parallel c}$ = 13 T in this paper) along the $c$ direction to partially suppress superconductivity and to compare the results with those at $\varepsilon$ = 0 (Fig. 1a). The high field is also effective for NMR experiments on a tiny single crystal, as it can improve the signal-to-noise ratio. $T_{\rm c}(\varepsilon)$ was measured by recording the resonance frequency of the NMR coil during both cooling and warming the sample on the strain cell. To detect a long-range charge order, ⁶³Cu-NMR $T_{1}$, the center, and the satellite spectra were measured systematically. The strain dependence of $T_{\rm c}$ was obtained by using six single-crystal plates and three homemade strain cells (#3, #4, and #5). This was done to confirm the reproducibility of the results and to determine the strain amount at which Hooke’s law holds. The results for all six sample plates were the same until the plate broke, so the results are shown without distinction in Fig. 2. NMR experiments were performed on the sixth plate. The photographs in Supplementary Fig. 1 show cell #5.