Charge-density-wave order takes over antiferromagnetism in Bi$_2$Sr$_{2-x}$La$_x$CuO$_{6}$ superconductors

TL;DR

This study reveals that charge density wave order replaces antiferromagnetic order in Bi$_2$Sr$_{2-x}$La$_x$CuO$_6$ superconductors, highlighting the complex interplay between charge, spin, and pseudogap phenomena in high-temperature superconductivity.

Contribution

It demonstrates the emergence of charge density wave order induced by in-plane magnetic fields, surpassing antiferromagnetic order and scaling with the pseudogap temperature, providing new insights into cuprate phase competition.

Findings

CDW order appears above the superconducting dome in Bi$_2$Sr$_{2-x}$La$_x$CuO$_6$.

$T_{CDW}$ surpasses $T_N$ beyond a critical doping level.

$T_{CDW}$ scales with the pseudogap temperature $T^*$.

Abstract

Superconductivity appears in the cuprates when a spin order is destroyed, while the role of charge is less known. Recently, charge density wave (CDW) was found below the superconducting dome in YBa$_2$Cu$_3$O$_y$ when a high magnetic field is applied perpendicular to the CuO$_2$ plane, which was suggested to arise from incipient CDW in the vortex cores that becomes overlapped. Here, by $^{63}$Cu-nuclear magnetic resonance, we report the discovery of CDW induced by an in-plane field that does not create vortex cores in the plane, setting in above the dome in single-layered Bi$_2$Sr$_{2-x}$La$_x$CuO$_6$.The onset temperature $T_{\rm CDW}$ takes over the antiferromagnetic order temperature $T_{\rm N}$ beyond a critical doping level at which superconductivity starts to emerge, and scales with the psudogap temperature $T^{*}$. These results provide important insights into the relationship between spin order, CDW and the pseudogap, and their connections to high-temperature superconductivity.

Full text

I Introduction

High transition-temperature ($T_{\rm c}$) superconductivity is obtained by doping carriers to destroy an antiferromagnetic (AF) spin ordered Mott insulating phase. Although it is generally believed that the interaction responsible for the spin order is important for the superconductivity LeeNagaosaWen , the nature of the normal state is still unclear Uchidareview ; Fradkin . For example, in the region with low carrier concentration $p$ (0 $<p<$ 0.2), a pseudogap state emerges where partial density of states (DOS) is lost below a characteristic temperature $T^{*}$ well above $T_{\rm c}$ Timusk or even $T_{\rm N}$ Kawasaki . Although the nature of the strange metallic state is still under debate, it is likely connected to both spin and change fluctuations or even orders. In fact, experimental progress suggests that the spin and charge degrees of freedom are highly entangled.

For example, a striped spin/charge order was found around x$$\sim1/8 in La1.6-xNd0.4SrxCuO4 (LSCO) two decades ago Tranquada . More recently, various forms of charge order were reported in many other systems. Scanning tunneling microscopy (STM) in Bi2Sr2CaCu2O8+δ found a modulation of local DOS in the vortex cores where superconductivity is destroyed Hoffman , which was interpreted as due to halos of incipient CDW localized within the cores Sachdev ; KivelsonLee . Resonant elastic and inelastic x-ray spectroscopy (RXS) measurements found a short-range CDW with ordering vectors along the in-plane Cu-O bond directions, $\bf q$ =($\sim$0.3, 0) and (0, $\sim$0.3). The correlation length is $\xi_{a,b}\sim$ 50 Å  for YBa2Cu3O7-y (YBCO) Ghiringhelli ; ChangNatPhys and $\xi_{a,b}\sim$ 20 Å for the other systems CominBi2201 ; SilvaNetoBi2212 ; PengYY ; HashimotoBi2212 ; Tabis . Quite recently, it was suggested by 17O nuclear magnetic resonance (NMR) in YBCO that such CDW is of static origin WuONMR . In Bi2Sr2-xLaxCuO6+δ (Bi2201), most intriguingly, the onset temperature of the short-range CDW was found to coincide CominBi2201 ; PengYY with $T^{*}$ that is far above $T_{\rm c}$ or $T_{\rm N}$ Zheng ; Kawasaki .

Application of a high magnetic field is useful to diagnose the interplay between various orders in the cuprates. When a high magnetic field is applied perpendicular to the CuO2 plane, superconductivity can be suppressed substantially. In YBCO, 63Cu NMR at $H$ = 28.5 T revealed a long-range charge density modulation perpendicular to the CuO-chain in the sample with $p$ = 0.108 WuNature . RXS also indicated that a high field induces a correlation along the CuO-chain direction and modifies the coupling between CuO2 bilayers, thus causes a three-dimensional CDW Nojiri ; Changhighfield . These observations are consistent with early discovery of a Fermi-surface reconstruction by quantum oscillations Leyraud1 and a recent report of a thermodynamic phase transition LeBoeuf .

These findings have arisen much interests, but the origin of the CDW and its connection to superconductivity is yet unknown. As the long-range CDW onsets below $T_{\rm c}(H=0)$ and only emerges when the field is applied perpendicular to the CuO2 plane, a wide-spread speculation is that it is due to incipient CDW in the vortex cores Hoffman that becomes overlapped as the field gets stronger ChangNatPhys ; SilvaNetoBi2212 ; WuVortex . In fact, a field as large as 28.5 T applied in the CuO2 plane of YBCO did not bring about any long-range CDW WuNature . Also, the role of the CuO chain is unclear; in Bi2Sr2CaCu2O8+δ without a CuO chain, no long-range CDW was found CurroBi2212 .

In order to clarify the relationship between the intertwined AF spin order, CDW, pseudogap and superconductivity, we apply high magnetic fields up to 42.5 T parallel to the Cu-O bond direction ($H\parallel$ $a$ or $b$ axis) in Bi2Sr2-xLaxCuO6+δ where the pseudogap spans from the parent AF insulating phase to the overdoped superconducting regime Kawasaki ; Zheng . This material has no CuO chain and the application of an in-plane field does not create vortex cores in the CuO2 plane. Surprisingly, we discover a long-range CDW that emerges far above the superconducting dome for $H_{\parallel}$ $>$ 10 T. We find that such CDW order becomes the successor of the AF order beyond $p$ = 0.107 at which superconductivity starts to emerge. The $T_{\rm CDW}$ takes over $T_{\rm N}$, but disappears well before the pseudogap closes. Our results indicate that CDW can be well disentangled from other orders.

II Results

Evidence for a field-induced CDW in underdoped Bi2201. Figures 1a-1d shows the 63Cu-NMR satellite (3/2$\leftrightarrow$1/2 transition) lines for four compounds of Bi2Sr2-xLaxCuO6+δ at two representative temperatures at $H_{||}$ = 14.5 or 20.1 T. As seen in Fig. 1a, no change between $T$ = 100 K and 4.2 K is observed for the optimally-doped compound ($p$ = 0.162). However, the spectrum is broadened at $T$ = 4.2 K for $p$ = 0.135 (Fig. 1b), and a splitting $\pm\delta{f}$ of the spectrum is observed at $T$ = 4.2 K for $p$ = 0.128 and 0.114, as seen in Figs. 1c and 1d. The spectra at $T$ = 4.2 K for $p$ = 0.128 and 0.114 can be reproduced by a sum of two Gaussian functions. It is noted that at low fields below 10 T, the spectrum shows no appreciable temperature dependence in the whole temperature range. The NMR line splitting indicates a long-range order, as it measures an assemble of nuclear spins over the sample.

Figure 1e shows the field evolution of $\delta f$ for $p$ = 0.114. The $\delta f$ grows steeply at $H_{||}$ = 10.4 T and saturates above $H_{||}$ $\sim$ 14.5 T. Figure 1f shows the temperature dependence of $\delta f$ for $p$ = 0.114 under various fields. The $\delta{f}$ grows rapidly below $T$ $\sim$ 30, 55, and 60 K at $H_{||}$ = 11, 13, and above 14.5 T, respectively. These results indicate that a field-induced phase transition occurs in the underdoped Bi2201. The results are qualitatively similar to the results found in YBCO where the same transition line splits into two peaks due to the spatial modulation of the NQR frequency $\nu_{\rm Q}$ WuNature .

Next we show that the field-induced phase transition is due to a charge order, but not spin order. Figures 2a, b, respectively, shows the temperature dependence of the satellite and the center lines for the sample with the lowest doping $p$ = 0.114. A spectrum broadening is also found in the center line (1/2$\leftrightarrow$-1/2 transition) at $T$ = 4.2 K, but it is much smaller than the satellite line. Figure 2c shows the temperature dependence of the NMR intensity obtained by integrating the spectrum at each temperature. The intensity has no anomaly above $T_{\rm c}$, indicating that there is no spin order. For antiferromagnetic insulator $p$ = 0.107 ($x$ = 0.08), the intensity decreases below $T_{\rm N}$ = 66 K because an internal field $H_{\rm int}$ = 7 T shifts the peak frequency far away Kawasaki . Furthermore, a possibility of striped phase formation leading to a wipe out effect found in LSCO Hunt can also be ruled out. It is noted that the possibility of a field-induced spin-density-wave order has already been ruled out previously for $p$ = 0.162 Mei .

Therefore, the splitting of the satellite line (Fig. 2a) and the broadening of the center line (Fig. 2b) is due to a distribution of the Knight shift $K_{\parallel}\pm\delta K_{\parallel}$ and the NQR frequency as $\nu_{\rm Q}\pm\delta\nu$ as observed in YBCO. Furthermore, the splitting $\delta f_{\rm satellite}$ = 1.22 MHz is much larger than $\delta f_{\rm center}$ = 0.271 MHz, indicating that the $\nu_{\rm Q}$ change is the main contributor to the observed line splitting. By a simple calculation (see Supplementary Note 1), we find that \delta$$K_{\parallel}\sim 0.05\pm 0.01\% and $\delta\nu$ = 2.5 $\pm$ 0.2 MHz can reproduce both the satellite and the center lines at the same time (shaded areas in Figs. 2a and 2b). The relation $\nu_{\rm Q}$ = 22.0 + 39.6 $p$ (see Supplementary Fig. 1) then yields a hole-concentration distribution $\delta p\sim 0.06\pm 0.01$ at the Cu site. Since there is no spin order here (Fig. 2c) as mentioned already, the splitting of the satellite line $\delta f$ ($\propto\delta p$) indicates a field-induced long-range charge distribution, i.e., a formation of CDW at low temperature in underdoped Bi2201.

Figure 2d, e shows the temperature dependence of the nuclear spin-lattice relaxation rate divided by $T$, $1/T_{1}T$ and spin-spin relaxation rate $1/T_{2}$ for $p$ = 0.114 obtained at two different fields. At $H$ = 9 T, both quantities decrease monotonically below $T^{*}$ = 230 K. At $H$ = 13 T, however, a pronounced peak was found in $1/T_{1}T$ at $T_{\rm CDW}$ =55 K. Such a peak in $1/T_{1}T$ is a characteristic of a CDW order SrPt2As2 . The $1/T_{2}$ also shows a sharp decrease at $T_{\rm CDW}$ =55 K. These results provide further evidence for a field-induced CDW phase transition.

$H$-$T$ phase diagram for underdoped Bi2201. To obtain the CDW onset temperature ($T_{\rm CDW}$) and the threshold field ($H_{\rm CDW}$) for $p$ = 0.114, we study the temperature dependence of the NMR spectra at various magnetic fields (Figs. 1e, f).

Figure 3 shows the $H$-$T$ phase diagram for $p$ = 0.114. Remarkably, the long-range CDW state in Bi2201 emerges at a temperature far above $T_{\rm c}$, in contrast to that in YBCO where CDW appears below $T_{\rm c}(H=0)$ WuNature ; WuVortex .

Relationship between CDW and superconductivity. Figure 4a, b shows the $H$-dependence and $T$-dependence of the satellite splitting $\delta{f}$ which allow us to obtain $H_{\rm CDW}$ and $T_{\rm CDW}$ for various doping levels. Figure 5 shows the hole concentration dependence of $H_{\rm c2}$, $H_{\rm CDW}$, and $T_{\rm CDW}$. The $H_{\rm c2}\sim$ 60 T for $p$ = 0.162 (see Supplementary Fig. 5) decreases with decreasing doping level but increasing again at $p$ = 0.114. Although the previous Nernst effect study on three Bi2201 samples ($p$ = 0.12, 0.16, and 0.19) did not take a closer look into the doping range as we did here ChangHc2 , our result is consistent with the results of YBCO WuVortex and La1.8-xEu0.2SrxCuO4 ChangHc2 .

The $H_{\rm CDW}$ is slightly lower than that in YBCO, suggesting that CDW has a similar energy scale across different class of cuprates. However, the relationship between $H_{\rm CDW}$ and $H_{\rm c2}$ is completely different from that seen in YBCO where $H_{\rm CDW}$ scaled with $H_{\rm c2}$. Namely, $H_{\rm CDW}$ was the lowest at the doping concentration where $H_{\rm c2}$ was the smallest there WuVortex , which led to the suggestion that CDW can only be seen when the superconducting state is suppressed as the vortex cores become overlapped. In the present case, however, no vortex cores are created in the CuO2 plane. In fact, $H_{\rm CDW}$ and $T_{\rm CDW}$ are more related with doping concentration itself as can be seen in Fig. 5, rather than with $H_{\rm c2}$. Namely, the long-range CDW order is induced more easily closer to the AF phase boundary.

III Discussion

In this section, we discuss possible CDW form and the implication of the phase diagram we found. First, the result can be understood by an incommensurate 1-dimensional (1D) long-range CDW as follows, as the situation is similar to that observed at the in-plane Cu2F-site located below the oxygen-filled CuO chain in YBCO WuNature . For an unidirectional CDW state, the wave modulation causes a spatial distribution in the electric field gradient (EFG) and thus the NQR frequency so that $\nu$ = $\nu_{\rm Q}$+\delta\nu$$\cos$$(\phi(X)) Blinc ; SrPt2As2 , where $X$ (= $a$ or $b$ axis) is the modulation direction. The NMR spectral intensity ($I(\nu)$) depends on the spatial variation of $\phi(X)$ as $I(\nu)=1/(2\pi d\nu/d\phi)$. For an incommensurate order, $\phi(X)$ is proportional to $X$, so that the NMR spectrum shows an edge singularity at $\nu$ = $\nu_{\rm Q}$ $\pm$ $\delta\nu$, as $I(\nu)=1/(2\pi\delta\nu\sqrt{1-((\nu-\nu_{Q})/\delta\nu)^{2}})$ Blinc ; SrPt2As2 . By convoluting a broadening function, a two-peak structure can be reproduced. In such case, the quantity 2$\delta\nu$ corresponds to the CDW order parameter SrPt2As2 . We emphasize that the value of $\delta p$ is twice larger than that observed in YBCO WuNature , indicating that a larger CDW amplitude is realized in Bi2201. This difference may arise from the different crystal structure between the two systems. YBCO is a bi-layer system while Bi2201 is single-layered. When CDW has a different phase between two CuO2 planes, the ordering effect would be weakened or even canceled out.

It has been known that magnetic field works to induce a correlation along the CuO-chain direction and modifies the coupling between CuO2 bilayers in YBCO Nojiri ; Changhighfield . In the present case, the short-range CDW at zero magnetic field CominBi2201 ; PengYY could also be modified by $H_{\parallel}>$ 10 T to be a long ranged 1D CDW along the Cu-O direction.

Second, how about a 2D-CDW case? Recent resonant X-ray scattering measurement on Bi2201 $p$ $\sim$ 0.11 found a perfect 2D but local ($\xi\sim$ 20 $\AA$) CDW formation along the Cu-O bond direction with the wave vector ($Q^{*}$, 0) and (0, $Q^{*}$) ($Q^{*}$ $\sim$ 0.26) CominNatMat ; PengYY . On the other hand, STM measurement suggested a commensurate density-wave with the ordering vectors $\bf{q}_{\rm DW}$ = (0.25, 0) and (0, 0.25) DavisPNAS . In either case, if such a local CDW becomes long ranged with the same ordering vectors, a spatial distribution of the NQR frequency can be written as $\nu$ = $\nu_{\rm Q}$+\delta\nu_{X}$$\cos$$(\phi(X))+\delta\nu_{Y}$$\cos$$(\phi(Y)), where $X$ = $a$-axis and $Y$ = $b$-axis are the modulation directions Blinc . As suggested by RXS and STM measurements, when the CDW amplitudes are equivalent $\delta\nu_{X}$ = $\delta\nu_{Y}$, such CDWs yield $I(\nu)\sim-ln[(\nu-\nu_{\rm Q})/\delta\nu]/\delta\nu$ Blinc . In this case, the logarithmic singularity appears at $\delta\nu$ = 0 Blinc , which is different from the 1D case. It is obvious that we can not explain our experimental results with such 2D CDW. However, if the amplitude for each directions is different, $\delta\nu_{X(Y)}\gg\delta\nu_{Y(X)}$, two edge-singularities will appear Blinc . It is also interesting to note that if there are CDW domains with modulations $\nu_{X}$ = $\nu_{\rm Q}$+\delta\nu_{X}$$\cos$$(\phi(X)) and $\nu_{Y}$ = $\nu_{\rm Q}$+\delta\nu_{Y}$$\cos$$(\phi(Y)) in each domain, the NMR lineshape will be the same as in the 1D case.

Third, a 3D CDW will not show two edge singularities in $\nu_{\rm Q}$ distribution in any case Blinc . Therefore, we may exclude the possibility of a 3D CDW because, being different from the bilayer YBCO, the long distance between CuO2 planes produces no CDW correlation along the $c$-axis in Bi2201 PengYY and in our case the magnetic field is applied parallel to the CuO2 plane.

We now show in Fig. 6 how the long-range CDW emerges as the magnetic field is increased. As seen in the $H$-$p$ plane at $T$ = 4.2 K, the field-induced CDW emerges for $H_{\parallel}$ $>$ 10 T in the underdoped regime. At such high fields, upon increasing doping, the AF state with $T_{\rm N}$ = 66 K at $p$ = 0.107 changes to a CDW ordered state with $T_{\rm CDW}\sim$ 60 K at $p$ = 0.114. Upon further doping to $p$ = 0.162 where the pseudogap persists; however, the CDW order disappears. Although a detailed analysis is difficult, a similar field-induced CDW is also found when the magnetic field is applied perpendicular to the CuO2 plane (see Supplementary Fig. 6). Figure 7 compares the phase diagram with that for YBCO. In YBCO, the short-range CDW sets in far below $T^{*}$ and the field-induced CDW (FICDW) occurs inside the superconducting dome forming a dome-like shape WuNature , while in Bi2201 the short-range CDW sets in right at $T^{*}$ and the FICDW emerges far above the superconducting dome and coexists with superconductivity.

Figure 6 reveals several important things. First and most remarkably, down-shifting the $T^{*}$ curve in temperature coincides with the $T_{\rm CDW}$ curve. As can be seen more directly in Fig. 8, $T_{\rm CDW}$ scales with $T^{*}$. This may suggest that the psudogap is a fluctuating form of the long-range order found in this work, but more work is needed. Very recently, a polar Kerr effect ZXShen and an optical rotational anisotropy measurements Hsieh suggested that a possible phase transition takes place at $T^{*}$. However, we note that the probes used are ultrafast in time scale. In NMR measurements, the time scale is in the 10*-8* s range which is much slower than the optical measurements ZXShen ; Hsieh , and it is reasonable that $T^{*}$ is seen as a fluctuating crossover temperature. Second, $T_{\rm N}$ is succeeded by $T_{\rm CDW}$ beyond a critical doping level at which superconductivity emerges, pointing to the important role of charge degree of freedom in high-temperature superconductivity. However, the detailed evolution from AF to CDW order under high magnetic fields is unclear at the moment. It is a future task to clarify whether the evolution is a first-order phase transition or not. The result also calls for further scrutinies of the AF insulating phase. In fact, the entanglement of the spin and charge freedoms Fradkin ; TKLee was recently found to occur already in the insulating phase itself WangYY . In any case, our results show that CDW order is another outstanding quantum phenomenon that needs to be addressed on the same footing as the AF spin order. Finally, the first demonstration of the ability of using an in-plane field to tune the electronic state should stimulate more works that will eventually help to solve the problem of high-$T_{\rm c}$ superconductivity.

Methods

Samples. The single crystals of Bi2Sr2-xLaxCuO6+δ (Bi2201, $p$ = 0.114 ($x$ = 0.75), 0.128 (0.65), 0.135 (0.60), and 0.162 (0.40)) were grown by the traveling solvent floating zone method Liang ; Peng . The hole concentration ($p$) were estimated previously Ono . Small and thin single-crystal platelets, typically sized up to 2 mm-2 mm-0.1 mm, cleaved from an as-grown ingot, were used. The in-plane Cu-O bond direction ($a$ or $b$ axis) was determined by Laue reflection. $T_{\rm c}(H)$ is defined at the onset temperature of diamagnetism observed by ac-susceptibility measurement using NMR coil. $H_{\rm c2}$ is determined by fitting $T_{\rm c}(H)$ to the WHH formula WHH .

NMR. The 63Cu-NMR spectra were taken by sweeping the rf frequency at a fixed field below $H$ = 15 T but they were taken by sweeping the field at a fixed frequency above $H$ = 15 T. Measurements at $H$ = 14.5 T were conducted at Institute of Physics, CAS, Beijing, and those below $H$ = 14.5 T were conducted at Okayama University. High magnetic fields above $H$ = 15 T are generated by the Hybrid magnet in the National High Magnetic Field Laboratory, Tallahassee, Florida. For 63,65Cu, the nuclear spin Hamiltonian is expressed as the sum of the Zeeman and nuclear quadrupole interaction terms, $\mathcal{H}$ = $\mathcal{H}_{\rm z}+\mathcal{H}_{\rm Q}$ = $-^{63,65}\gamma\hbar{\bf I}\cdot{\bf{H}_{0}}(1+K)+(h\nu_{\rm Q}/6)[3{I_{z}}^{2}-I(I+1)+\eta({I_{x}}^{2}-{I_{y}}^{2})]$, where ${}^{63}\gamma$ = 11.285 MHz/T and ${}^{65}\gamma$ = 12.089 MHz/T, $K$ is the Knight shift, and $I$ = 3/2 is the 63,65Cu nuclear spin. The NQR frequency $\nu_{\rm Q}$ and the asymmetry parameter $\eta$ are defined as $\nu_{\rm Q}$ = $\frac{3eQV_{zz}}{2I(2I-1)h}$, $\eta$ $=$ $\frac{V_{xx}-V_{yy}}{V_{zz}}$, with $Q$ and $V_{\alpha\beta}$ being the nuclear quadrupole moment and the electric field gradient (EFG) tensor abragam . The principal axis $z$ of the EFG is along the $c$ axis and $\eta$ = 0 ZhengNuQ . Due to $\mathcal{H}_{\rm Q}$, one obtains the NMR center line and the two satellite transition lines between $|m\rangle$ and $|m-1\rangle$, ($m$ =3/2, 1/2, -1/2), at $\nu_{m\leftrightarrow m-1}=^{63,65}\gamma H_{0}(1+K)+(\nu_{\rm Q}/2)(3\cos^{2}\theta-1)(m-1/2)$ + second-order correction. The second term of the right side is the first order term in the presence of quadrupole interaction. Here, $\theta$ is the angle between $\bf{H}$ and EFG. The $T_{1}$ and $T_{2}$ were measured at the frequencies in the center peak ($m=1/2\leftrightarrow$ - 1/2 transition). The $T_{1}$ values were measured by using a single saturating pulse and were determined by standard fits to the recovery curve of the nuclear magnetization to the theoretical function for the nuclear spin $I$ = 3/2 Zheng . The $T_{2}$ values were obtained by fits to the spin-echo decay curve of the nuclear magnetization $I(t)$ to $I(t)=I(0)\exp{(-2t/T_{2})}$ Kawasaki .

Data availability

The data that support the findings of this study are available on reasonable request. Correspondence and requests for materials should be addressed to G.-q.Z.

Acknowledgments

We thank D.-H. Lee, S. Uchida, L. Taillefer, T. K. Lee, M.-H. Julien and S. Onari for useful discussion, and S. Maeda and D. Kamijima for experimental assistance. A portion of this work was performed at National High Magnetic Field Laboratory, which is supported by NSF Cooperative Agreement No. DMR-1157490 and the State of Florida. Support by research grants from Japan MEXT (No. 25400374 and 16H04016), China NSF (No. 11634015), and MOST of China (No. 2016YFA0300502 and No. 2015CB921304) is acknowledged.

Authors contributions

G.-q.Z planned the project. C.T.L synthesized the single crystals. S.K, Z.L, M.K, P.L.K, A.P.R and G.-q.Z performed NMR measurements. G.-q.Z wrote the manuscript with inputs from S.K. All authors discussed the results and interpretation.