Identification of a nematic pair density wave state in Bi₂Sr₂CaCu₂O_8+x

Our objective is exploration of intertwined PDW and DSC states in cuprates, by using SJTM. In principle, the total electron-pair density at location r, $n(\mathit{r})$, can be visualized by measuring electron-pair (Josephson) critical-current $I_{\text{J}}\left(\mathit{r}\right)$ from the sample to a superconducting scanning tunneling microscope (STM) tip (17). This is because $n(\mathit{r})\propto I_{\text{J}}^2\left(\mathit{r}\right)R_{\text{N}}^2(\mathit{r})$, where R_N is the normal-state junction resistance (18, 19). However, since typical thermal fluctuations far exceed the Josephson energy $E_{\text{J}}$ between SJTM tip and sample surface, a phase-diffusive steady-state at voltage $V_J$ drives an electron-pair current $I_{\text{P}}\left(V_J\right)=\frac{1}{2}{I}_{\text{J}}^{2}Z{V}_J/\left(V_J{}^2+V_{\text{c}}^2\right)$. Here $V_{\text{c}}=2eZ{k}_{\text{B}}T/\hslash$ with $Z$ the high-frequency impedance of the junction (20, 21). In the theory of such spectra, the maximum value of electron-pair current $I_{\text{m}}=(\hslash /8e{k}_{B}T)I_J^2$, providing the basis for atomically resolved SJTM visualization of the electron-pair condensate in superconductors (5–9) as

In that context, we study single crystals of Bi₂Sr₂CaCu₂O_8+x with CuO₂ plaquette hole-density $p\approx 0.17$, by using a dilution refrigerator based SJTM (5). The cryogenically cleaved samples terminate at the BiO crystal layer, and the d-wave superconducting scan-tip is prepared by exfoliating a nanometer-scale Bi₂Sr₂CaCu₂O_8+x flake from that sample surface (5, 9). A typical topographic image at T = 45 mK then consists of atomically resolved and registered surface Bi atoms (Fig. 1A) with the typical measured $I_{\text{P}}\left(V_J\right)$ shown as an inset. Using such tips and a virtually constant $R_{\text{N}}(\mathit{r})\approx 20 \text{MΩ}$ as determined from analysis of topography at the setup voltage, we image $I_{\text{P}}\left(V_J,\mathit{r}\right)$ and thus $I_{\text{m}}(\mathit{r})$, at $T = 45 mK$. The Fourier transform of the measured $I_{\text{m}}\left(\mathit{r}\right)$ image, $I_{\text{m}}\left(\mathit{q}\right)$, is shown in Fig. 1B. In such experiments on Bi₂Sr₂CaCu₂O_8+x, the $I_{\text{m}}\left(\mathit{r}\right)$ and $I_{\text{m}}\left(\mathit{q}\right)$ are both dominated by effects of the crystal supermodulation, a bulk 26-Å periodic modulation of unit-cell dimensions (22) whose general effects are discussed elsewhere (9). The focus of interest here is on the two PDWs observable in$I_{\text{m}}\left(\mathit{q}\right)$ as four relatively broad peaks surrounding the wavevectors (5) $\mathit{q}\approx \left(\frac{2\pi }{a}\right)\left(0,\pm 0.25\right);\left(\frac{2\pi }{a}\right)\left(\pm 0.25,0\right)$. Additionally, the somewhat heterogeneous electron-pair density of the DSC state is represented by the broad peak at $\mathit{q}=\left(\frac{2\pi }{a}\right)\left(0,0\right)$. The phenomena detected separately at these five wavevectors can reveal key information on the interplay of the five order parameter functions of Eqs. 1–3.

To separately visualize the two PDWs we inverse Fourier transform $I_{\text{m}}\left(\mathit{q}\right)$ in Fig. 1B for wavevectors $\pm {\mathit{P}}_{\mathit{x}}\approx \left(\frac{2\pi }{a}\right)\left(0,\pm 0.25\right)$ and $\pm {\mathit{P}}_{\mathit{y}}\approx \left(\frac{2\pi }{a}\right)\left(\pm 0.25,0\right)$ respectively, using the $\mathit{q}$-space regions represented inside the colored circles. The resulting $I_{{\text{m}}_x}\left(\mathit{r}\right)$ and $I_{{\text{m}}_y}\left(\mathit{r}\right)$ images shown in Fig. 1 C and D, respectively, reveal the spatial arrangements of the two orthogonal PDW states (in the presence of a predominant DSC.) They are relatively coherent 4a₀-periodioc modulations with amplitudes differently disordered, and without obvious correlation between them. Clearly, there is a proliferation of edge-dislocation 2π topological defects (23) in $I_{{\text{m}}_{x,y}}\left(\mathit{r}\right)$, whose ongoing study will be reported elsewhere. Here we concentrate on the interplay of between the simultaneously observed PDW and DSC states.

To do so, we parameterize the two PDW state signatures in terms

Fourier component selection at $\pm {\mathit{P}}_{\mathit{x}}=\left(\frac{2\pi }{a}\right)\left(0,\pm 0.25\right)$ and $\pm {\mathit{P}}_{\mathit{y}}=\left(\frac{2\pi }{a}\right)\left(\pm 0.25,0\right)$ then yieldssuch that

(SI Appendix, section 1). The DSC state characteristic, ${\mathit{\text{A}}}_0\left(\mathit{r}\right),$ is defined similarly by inverse Fourier transform within the yellow circle surrounding wavevector $\mathit{q}=\left(0,0\right)$. Identical values of ${\text{σ}}_0={\text{σ}}_x={\text{σ}}_y=3 \hspace{0.17em}\text{nm}$ are applied throughout. Using this approach, Fig. 2 A–C respectively show ${\mathit{\text{A}}}_x\left(\mathit{r}\right)$, ${ \mathit{\text{A}}}_y\left(\mathit{r}\right)$ and ${\mathit{\text{A}}}_0\left(\mathit{r}\right)$ visualized simultaneously in the field of view (FOV) of Fig. 1A. Thus, it becomes possible to directly explore the phenomenology of intertwining the DSC and PDW states (SI Appendix, Section 2). Fig. 2D presents measured $\langle {\mathit{\text{A}}}_x+{\mathit{\text{A}}}_y\rangle$ versus ${\mathit{\text{A}}}_0$, where the average is carried out over all locations $\mathit{r}$ at which ${\mathit{\text{A}}}_0\left(\mathit{r}\right)$ has the value indicated on the abscissa. The positive slope $s=0.0389$ through zero indicates that, throughout the variations in ${\mathit{\text{A}}}_x\left(\mathit{r}\right)$, $A_y\left(\mathit{r}\right)$ and $A_0\left(\mathit{r}\right)$(Figs. 1 C and D and 2 A–C), the PDW and the DSC states are mutually enhancing on the average. This conclusion is independent of any inadvertent heterogeneity in the normal-state Josephson junction resistance ${\mathit{\text{R}}}_N\left(\mathbf{r}\right)$ in Eq. 6, because it gets divided out by the definition of $s$. The implication of Fig. 2 is that, within the GLW context (Eq. 4 and refs. 1–3), the DSC and PDW states are attractive ($({\beta }_{c1}-\left|{\beta }_{c2}\right|<0)$ electronic phases (SI Appendix, section 4) at zero magnetic field, although there is evidence of repulsion when strong gradients are present as in the vortex core.

Next, to search for the hypothetical (2, 3) nematic PDW state we define a nematic order parameter

Analyzing the data from Fig. 2 A and B in this way generates $\mathcal{N}(\mathbf{r})$ as shown in Fig. 3A. Again, we note that Eq. 10 provides an empirical definition for a nematic PDW state based directly on $I_{\text{m}}(\mathit{r})$ data, i.e., independent of any possible variations in ${\mathit{\text{R}}}_N\left(\mathit{r}\right)$. Fig. 3A then reveals the existence of strong PDW nematicity in Bi₂Sr₂CaCu₂O_8+x. Indeed, the histogram of all $\mathcal{N}\left(\mathit{r}\right)$ values from Fig. 3A, Inset demonstrates that $\left|\mathcal{N}\left(\mathit{r}\right)\right|>0.3$ for 45% of the sample area and thus that these nematic PDW states can predominate. To exemplify, Fig. 3B shows examples of measured ${\mathit{\text{I}}}_{\mathit{m}}\left(\mathbf{r}\right)$ along the x axis (y axis) in its left (right) panels, both within domains where $\mathcal{N}\left(\mathit{r}\right)\gg 0$. Fig. 3C shows equivalent exemplary data for domains where $\mathcal{N}\left(\mathit{\text{r}}\right)\ll 0$. Overall, we find that a robust nematic PDW state, consisting of nanoscale domains of opposite nematicity in electron-pair density, occurs in Bi₂Sr₂CaCu₂O_8+x at $p\approx 0.17$.

But this discovery begs the question of what it is that sets the size and location of the nematic PDW domains. One clue comes from recent reports (24) that, in La_2-xBa_xCuO₄ away from $p\approx 0.125,$ introducing Zn atom substitution at 1% of Cu sites leads to a cascade transition from two-dimensional (2D) superconductivity to three-dimensional superconductivity with falling temperature. The inference derived from these studies is that the Zn atoms pins PDW order locally. In cuprates, at each Zn impurity atom there is a superconductive impurity state with energy $E\approx -1\hspace{0.17em}\text{meV}$, and also a powerful suppression of electron-pair condensate $n(\mathit{r})$ as exemplified by SI Appendix, Fig. S2F. In Bi₂Sr₂CaCu₂O_8+x the Zn impurity states can be imaged directly in superconducting-tip differential conductance imaging, by finding the local maxima in $Z(\mathit{r})\equiv \text{I}(20\hspace{0.17em}\text{mV},\mathit{r})-\text{I}(-20\hspace{0.17em}\text{mV},\mathit{r})$, where $\text{I}\left(\text{V}\right)$ is the superconductor–insulator–superconductor single-electron tunnel current. These maxima occur because the coherence peaks of SJTM tip density-of-states near $E\approx \pm 20\hspace{0.17em}\text{meV}$ are convoluted with the Zn impurity state density-of-states peak $E\approx -1\hspace{0.17em}\text{meV}$ to produce a strong jump in tunnel currents $\text{I}\left(E\approx \pm 20\hspace{0.17em}\text{meV},\mathit{r}\right)$. Fig. 4A shows the resulting image $Z(\mathit{r})\equiv \text{I}\left(20\hspace{0.17em}\text{mV},\mathit{r}\right)-\text{I}\left(-20\hspace{0.17em}\text{mV},\mathit{r}\right)$ from which each Zn impurity-state maximum is identified by blue circles (SI Appendix, section 3). In Fig. 4B we show the amplitude of nematic order parameter $\left|\mathcal{N}\left(\mathit{r}\right)\right|$ with the sites of Zn impurity resonances overlaid as blue dots. Visually the Zn sites seem to occur near the domain boundaries in $\left|\mathcal{N}\left(\mathit{r}\right)\right|$. This can be quantified by plotting the histogram of the distribution of distances between each Zn impurity atom and its nearest PDW domain wall and comparing to the expected average distance of uncorrelated random points. The result shown in Fig. 4C reveals that the Zn sites occur highly preferentially near the boundaries of the PDW Ising domains. This direct visualization indicates that the PDW nematic domains are pinned by interactions with Zn impurity atoms at the Cu sites and is therefore highly consistent with deductions from the transport studies of Zn-doped La_2-xBa_xCuO₄ (24).

Spectroscopic imaging STM and resonant X-ray scattering have produced a wealth of understanding of unidirectionality, commensuration and domain formation for CDW modulations in strongly underdoped cuprates. However, because photon scattering cannot (yet) detect PDW states, because the pseudogap in single particle tunneling masks the true electron-pairing energy gap and thus order parameter, and because SJTM visualization of PDW states is recent (5–9), equivalent issues for the cuprate PDW state are unresolved. Our SJTM visualization now demonstrates that the Bi₂Sr₂CaCu₂O_8+x PDW states tend strongly to be both unidirectional and commensurate, a situation widely predicted (25–34) as a consequence of strong-coupling physics within the CuO₂ Hubbard model. Further, we note that visualization of these characteristics in a robust PDW state occurs in Bi₂Sr₂CaCu₂O_8+x at a carrier density where the CDW modulations are virtually nonexistent (35, 36), as is the empirical case in the samples studied here (SI Appendix, Fig. S3). The implication, consistent with the eight unit-cell periodic energy gap modulations observed in single particle tunneling (37, 38), is that the Bi₂Sr₂CaCu₂O_8+x PDW states are not induced by the existence of a CDW and instead are the primary translation symmetry breaking state of hole-doped CuO₂. Furthermore, since the spatial configurations of the PDWs studied here appear uninfluenced by a preexistent CDW yet are obviously disordered, effects of chemical (and possibly dopant-ion) randomness on the PDW are adumbrated. Indeed, recent transport studies (24) provide experimental evidence that Zn impurity atoms pin the PDW order in La_2-xBa_xCuO₄. For comparison, our visualization of the Bi₂Sr₂CaCu₂O_8+x PDW nematic domains simultaneously with Zn scattering resonances demonstrates directly that the Zn impurity-atom sites occur predominantly within boundary regions between these domains. Overall, a vestigial state is one that only partially breaks the symmetry of the true ordered state. Here we find a nematic PDW state breaking the rotational (${\text{C}}_4$) symmetry of the lattice but not long-range translational symmetry, in what appears to be a disorder-pinned realization of a global nematic order. Hence, a plausible context in which to consider the PDW in Bi₂Sr₂CaCu₂O_8+x is as a new form of vestigial nematic state based on a disordered unidirectional density wave of electron pairs (39) rather than of single electrons (40).

Ergo, by using SJTM to simultaneously visualize the PDW and DSC states of Bi₂Sr₂CaCu₂O_8+x near $p\approx 0.17$ where no CDW phenomena obtrude, we demonstrate that they are intertwined in mutually attractive phases (Fig. 2). Conversely, separate imaging of the electron-pair density modulations of the two orthogonal PDWs reveals a robust nematic PDW state, with primarily unidirectional and lattice-commensurate electron-pair density modulations forming Ising domains of opposite nematicity (Fig. 3). Imaging the sites of Zn impurity-states finds them occurring preferentially in boundary regions of minimal nematic order (Fig. 4), implying that the PDW domains are pinned thereby. Generally, these data signify that the PDW state of Bi₂Sr₂CaCu₂O_8+x is a vestigial nematic phase of electron pairs, that it is locally unidirectional and lattice commensurate, and that it is not a subordinate but a primary electronic order of hole-doped CuO₂.

We acknowledge and thank Owen H. S. Davis and Shuqiu Wang for key discussions and suggestions. M.H.H. and J.C.S.D. acknowledge support from the Moore Foundation’s EPiQS Initiative through grant GBMF9457. W.R. and J.C.S.D. acknowledge support from the European Research Council under award DLV-788932. W.C. and J.C.S.D. acknowledge support from the Royal Society under award R64897. N.K., S.O., and J.C.S.D. acknowledge support from the Science Foundation of Ireland under award SFI 17/RP/5445. H.E. acknowledges support from Japan Society for the Promotion of Science KAKENHI (no. JP19H05823). P.D.J. acknowledges support by QuantEmX grant GBMF9616 from ICAM/Moore Foundation and by a Visiting Fellowship at Wadham College, Oxford.