Identification of a nematic pair density wave state in Bi2Sr2CaCu2O8+x

Contributed by J. C. Séamus Davis; received April 15, 2022; accepted June 23, 2022; reviewed by Hai-Hu Wen and Mats Granath
July 27, 2022
119 (31) e2206481119

Significance

Although compelling evidence now exists that a pair density wave (PDW) state occurs in hole-doped CuO2 materials, its microscopic phenomenology and mechanism remain a mystery. A prevalent theoretical concept has been that the two-dimensional strongly correlated electrons in these materials can spontaneously generate a unidirectional, lattice-commensurate (striped) PDW state that is not induced by any preexistent charge density wave order. Here, in samples with virtually no charge density wave phenomenology, we show by direct scanned Josephson tunneling microscopy visualization of the modulating electron pair density that the PDW state in Bi2Sr2CaCu2O8+x is a vestigial nematic phase, with impurity-atom pinned Ising domains of strongly unidirectional and commensurate pair density modulations.

Abstract

Electron-pair density wave (PDW) states are now an intense focus of research in the field of cuprate correlated superconductivity. PDWs exhibit periodically modulating superconductive electron pairing that can be visualized directly using scanned Josephson tunneling microscopy (SJTM). Although from theory, intertwining the d-wave superconducting (DSC) and PDW order parameters allows a plethora of global electron-pair orders to appear, which one actually occurs in the various cuprates is unknown. Here, we use SJTM to visualize the interplay of PDW and DSC states in Bi2Sr2CaCu2O8+x at a carrier density where the charge density wave modulations are virtually nonexistent. Simultaneous visualization of their amplitudes reveals that the intertwined PDW and DSC are mutually attractive states. Then, by separately imaging the electron-pair density modulations of the two orthogonal PDWs, we discover a robust nematic PDW state. Its spatial arrangement entails Ising domains of opposite nematicity, each consisting primarily of unidirectional and lattice commensurate electron-pair density modulations. Further, we demonstrate by direct imaging that the scattering resonances identifying Zn impurity atom sites occur predominantly within boundaries between these domains. This implies that the nematic PDW state is pinned by Zn atoms, as was recently proposed [Lozano et al., Phys. Rev. B 103, L020502 (2021)]. Taken in combination, these data indicate that the PDW in Bi2Sr2CaCu2O8+x is a vestigial nematic pair density wave state [Agterberg et al. Phys. Rev. B 91, 054502 (2015); Wardh and Granath arXiv:2203.08250].
The existence and phenomenology of PDW states are fundamental issues within the challenge to understand cuprate correlated superconductivity (14). Their definitive characteristic is a periodically modulating electron-pair density which can now be visualized directly at the atomic scale, by using scanned Josephson tunneling microscopy (59) (SJTM). This new capability allows one to explore both the microscopic electronic structure of the cuprate pair density wave (PDW) state and the interactions between it and the other electronic orders. Understanding the latter is essential, because intertwining the d-wave superconducting (DSC) and PDW order parameters allows a plethora of distinct electron-pair orders to appear (13), but which one actually occurs in the various cuprates is unknown.
The simplest quantum condensates of electron pairs are defined by a homogeneous superconductive order-parameter
Δ0(r)=Δ0eiϕ(r) 
[1]
for which ϕ(r) is the electromagnetic gauge symmetry breaking macroscopic quantum phase. Heuristically, one may think of Δ0kγckck, where ck; ck are electron annihilation operators of opposite spin and momentum and γ  the pairing strength, as the amplitude of the electron-pair condensate wavefunction. A unidirectional PDW state is also a superconductor, but one that modulates the condensate order parameter spatially at wavevector Px such that
Δx(r)=ΔPx(r)eiPxx+ΔPx(r)eiPxx.
[2]
In a tetragonal crystal, an orthogonal PDW state can also exist modulating with at wavevector Py along the y-direction as
Δy(r)=ΔPy(r)eiPyy+ΔPy(r)eiPyy.
[3]
Because there are then five complex-valued scalar order parameter functions, when these three macroscopic quantum phases are intertwined, a plethora of global electron-pair order parameters becomes possible (13, 1012).
Theoretical analysis of how the DSC and two primary PDW states become intertwined requires an intricate Ginzburg–Landau–Wilson (GLW) free energy density functional (13, 10). Using such an approach, excellent success has been achieved in understanding the induced order parameters, the most prominent of which are the induced charge density wave (CDW) states with order parameters (2, 3) ρPx,Py(r)Δ0*ΔPx,Py+ΔPx,Py*Δ0 and ρ2Px,2Py(r)ΔPx,Py*ΔPx,Py. However, to fully understand the PDW of cuprates, one is not limited to studying such secondary or induced states, because SJTM imaging can give direct access (59) to the primary electron-pair orders. The structure and intertwining between PDW and DSC states are described by a subset of terms from the overall GLW free energy density
F=βc1|Δ0|2(|ΔPx|2+|ΔPy|2+|ΔPx|2+|ΔPy|2)+βc2[Δ02(ΔPxΔPx+ΔPyΔPy)*+(Δ02)*(ΔPxΔPx+ΔPyΔPy)],
[4]
where only the lowest order coupling terms are considered and the gradient terms are ignored. Among the possible global electron-pair orders are (2, 3) unidirectional or bidirectional modulated PDW phases eiPxx;eiPyy with fixed amplitude, as in the Fulde–Ferrell (13) state; and unidirectional or bidirectional modulated PDW amplitudes |ΔPx|cos(Pxx);|ΔPy|cos(Pyy)  with fixed phase, as in the Larkin–Ovchinnikov (14) state. An intriguing hypothetical state has been a nematic pair density wave (2, 3) with order parameter
N(|ΔPx|2+|ΔPx|2)(|ΔPy|2+|ΔPy|2)
[5]
and such states may also exhibit vestigial nematic phases (15, 16). But how the DSC and PDW orders are intertwined in cuprates, and their consequent global electron-pair order parameter, are all unknown.

SJTM

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(r), can be visualized by measuring electron-pair (Josephson) critical-current IJ(r)  from the sample to a superconducting scanning tunneling microscope (STM) tip (17). This is because n(r)IJ2(r)RN2(r), where RN is the normal-state junction resistance (18, 19). However, since typical thermal fluctuations far exceed the Josephson energy EJ between SJTM tip and sample surface, a phase-diffusive steady-state at voltage VJ  drives an electron-pair current IP(VJ)=12IJ2ZVJ/(VJ2+Vc2). Here Vc=2eZkBT/ with Z the high-frequency impedance of the junction (20, 21). In the theory of such spectra, the maximum value of electron-pair current Im=(/8ekBT)IJ2, providing the basis for atomically resolved SJTM visualization of the electron-pair condensate in superconductors (59) as
n(r)Im(r) RN2(r).
[6]
In that context, we study single crystals of Bi2Sr2CaCu2O8+x with CuO2 plaquette hole-density p0.17, 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 Bi2Sr2CaCu2O8+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 IP(VJ) shown as an inset. Using such tips and a virtually constant RN(r)20  as determined from analysis of topography at the setup voltage, we image IP(VJ,r) and thus Im(r), at T = 45 mK. The Fourier transform of the measured Im(r) image,  Im(q), is shown in Fig. 1B. In such experiments on Bi2Sr2CaCu2O8+x, the Im(r) and Im(q) 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 Im(q) as four relatively broad peaks surrounding the wavevectors (5) q(2πa)(0,±0.25);(2πa)(±0.25,0). Additionally, the somewhat heterogeneous electron-pair density of the DSC state is represented by the broad peak at q=(2πa)(0,0). The phenomena detected separately at these five wavevectors can reveal key information on the interplay of the five order parameter functions of Eqs. 13.
Fig. 1.
Visualizing electron pair density n(r). (A) SJTM topographic image T(r) of BiO termination layer of Bi2Sr2CaCu2O8+x. (Inset) Average electron-pair current spectrum IP(VJ) measured in this FOV at T = 45 mK and RN20 MOhm, with maxima occurring at ±Im. (B) Power spectral density Fourier transform of Im(r), Im(q), as measured in FOV of A. Four broad PDW peaks surround the wavevectors P=(2π/a)(0,±0.25);(2π/a)(±0.25,0) as indicated by pairs of red and blue circles respectively. The DSC electron-pair density is represented by the broad peak at q=(2π/a)(0,0) as indicated by the yellow circle. (C) Fourier filtration of Im(q) P=(2π/a)(0,±0.25) as in B, to visualize the ±Px PDW modulating along the CuO2 x axis. (D) Fourier filtration of Im(q) P=(2π/a)(±0.25,0) in B, to visualize the ±Py PDW modulating along the CuO2 y axis.

Interplay of Superconductive and PDW Orders

To separately visualize the two PDWs we inverse Fourier transform Im(q) in Fig. 1B for wavevectors ±Px(2πa)(0,±0.25) and ±Py(2πa)(±0.25,0) respectively, using the q-space regions represented inside the colored circles. The resulting  Imx(r) and  Imy(r) 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  Imx,y(r), 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
Imx,y(r)=Ax,y(r) cos(Px,yr+δx,y(r)) Ax,y(r) cos(Φx,y(r)).
[7]
Fourier component selection at ±Px=(2πa)(0,±0.25) and ±Py=(2πa)(±0.25,0) then yields
Im(r) = =12πσ drIm(r) eiPx,y·re|rr|22σ02
[8]
such that
Ax,y(r)2(Re Im(r))2+(Im Im(r))2
[9]
(SI Appendix, section 1). The DSC state characteristic, A0(r), is defined similarly by inverse Fourier transform within the yellow circle surrounding wavevector q=(0,0). Identical values of σ0=σx=σy=3 nm are applied throughout. Using this approach, Fig. 2 AC respectively show Ax(r),  Ay(r) and A0(r) 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 Ax+Ay versus A0, where the average is carried out over all locations r at which A0(r) has the value indicated on the abscissa. The positive slope s=0.0389 through zero indicates that, throughout the variations in Ax(r), Ay(r) and A0(r)(Figs. 1 C and D and 2 AC), 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 RN(r) 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. 13), the DSC and PDW states are attractive ((βc1|βc2|<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.
Fig. 2.
Intertwined DSC and PDW order parameters. (A) Amplitude of Im(r) modulations Ax(r)  for PDW state with ±Px, from Fig. 1C. (B) Amplitude of Im(r) modulations Ay(r)  for PDW state with ±Py, from Fig. 1D. (C) Amplitude of DSC electron-pair density A0(r) derived from Eq. 6 with q = 0 as indicated by the yellow circle in Fig. 1B. (D) Ax(r)+ Ay(r) averaged over all locations r where A0(r) equals the abscissa value A0. The solid line is a linear fit to these data through (0,0). (Inset) The 2D histogram of the same data.

Discovery of Nematic PDW State in Bi2Sr2CaCu2O8+x

Next, to search for the hypothetical (2, 3) nematic PDW state we define a nematic order parameter
N(r)={Ax(r) Ay(r)}/{Ax(r)+ Ay(r)}.
[10]
Analyzing the data from Fig. 2 A and B in this way generates N(r) as shown in Fig. 3A. Again, we note that Eq. 10 provides an empirical definition for a nematic PDW state based directly on Im(r) data, i.e., independent of any possible variations in RN(r). Fig. 3A then reveals the existence of strong PDW nematicity in Bi2Sr2CaCu2O8+x. Indeed, the histogram of all N(r) values from Fig. 3A, Inset demonstrates that |N(r)|>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 Im(r) along the x axis (y axis) in its left (right) panels, both within domains where N(r)0. Fig. 3C shows equivalent exemplary data for domains where N(r)0. Overall, we find that a robust nematic PDW state, consisting of nanoscale domains of opposite nematicity in electron-pair density, occurs in Bi2Sr2CaCu2O8+x at p0.17.
Fig. 3.
Nematic PDW State of Bi2Sr2CaCu2O8+x. (A) Nematic order parameter N(r)={Ax(r) Ay(r)}/{Ax(r)+ Ay(r)} derived from Fig. 2 A and B. Domains of opposite nematicity occur with correlation length ξ15nm. (Inset) Histogram of all N(r) values in A, showing nonzero mean value. The magnitude N>0.3 for approximately 45% of the FOV, indicating a strong nematic interaction between the two PDW. (B, Left) Four examples of measured Im(r) along the x axis. (B, Right) Four examples of measured Im(r) along the y axis, both within domains where N(r)0. (C, Left) Four examples of measured Im(r) along the x axis. (C, Right) Four examples of measured Im(r) along the y axis, both within domains where N(r)0.
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 La2-xBaxCuO4 away from p0.125,  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 E1meV, and also a powerful suppression of electron-pair condensate n(r) as exemplified by SI Appendix, Fig. S2F. In Bi2Sr2CaCu2O8+x the Zn impurity states can be imaged directly in superconducting-tip differential conductance imaging, by finding the local maxima in Z(r)I(20mV,r)I(20mV,r), where I(V) is the superconductor–insulator–superconductor single-electron tunnel current. These maxima occur because the coherence peaks of SJTM tip density-of-states near E±20meV  are convoluted with the Zn impurity state density-of-states peak E1meV  to produce a strong jump in tunnel currents I(E±20meV,r). Fig. 4A shows the resulting image Z(r)I(20mV,r)I(20mV,r) 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 |N(r)| with the sites of Zn impurity resonances overlaid as blue dots. Visually the Zn sites seem to occur near the domain boundaries in |N(r)|. 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 La2-xBaxCuO4 (24).
Fig. 4.
Pinning of PDW nematic domains by Zn impurity atoms. (A) Locations of Zn impurity atoms Z(r) as detected in Z(r)I(r,20mV) I(r,20mV) are shown as blue circles. The identification and register of Zn impurity sites is discussed in SI Appendix, section 3. (B)  |N(r)|, the amplitude of the nematic order parameter from Fig. 3A, with the sites of Zn impurity resonances overlaid as blue dots. (C) The distribution of distances between each Zn impurity atoms and its nearest PDW domain walls (red). This is compared to the expected average distance if there is no correlation between Zn impurity atoms and the PDW domain walls (blue). Zn impurity atoms are concentrated near the PDW domain walls.

Discussion and Conclusion

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 (59), equivalent issues for the cuprate PDW state are unresolved. Our SJTM visualization now demonstrates that the Bi2Sr2CaCu2O8+x PDW states tend strongly to be both unidirectional and commensurate, a situation widely predicted (2534) 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 Bi2Sr2CaCu2O8+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 Bi2Sr2CaCu2O8+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 La2-xBaxCuO4. For comparison, our visualization of the Bi2Sr2CaCu2O8+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 (C4) 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 Bi2Sr2CaCu2O8+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 Bi2Sr2CaCu2O8+x near p0.17  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 Bi2Sr2CaCu2O8+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.

Data Availability

All study data are included in the article and/or SI Appendix.

Acknowledgments

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.

Supporting Information

Appendix 01 (PDF)

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Information & Authors

Information

Published in

Go to Proceedings of the National Academy of Sciences
Proceedings of the National Academy of Sciences
Vol. 119 | No. 31
August 2, 2022
PubMed: 35895680

Classifications

Data Availability

All study data are included in the article and/or SI Appendix.

Submission history

Received: April 15, 2022
Accepted: June 23, 2022
Published online: July 27, 2022
Published in issue: August 2, 2022

Keywords

  1. cuprate
  2. nematic
  3. pair density wave
  4. vestigial
  5. zinc impurity atom

Acknowledgments

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.

Notes

Reviewers: H.-H.W., Nanjing University; and M.G., University of Gotenburg.

Authors

Affiliations

Clarendon Laboratory, University of Oxford, Oxford OX1 3PU, United Kingdom
Wangping Ren1
Clarendon Laboratory, University of Oxford, Oxford OX1 3PU, United Kingdom
Niall Kennedy
Clarendon Laboratory, University of Oxford, Oxford OX1 3PU, United Kingdom
Department of Physics, University College Cork, Cork T12 R5C, Ireland
M. H. Hamidian
Department of Physics, Cornell University, Ithaca, NY 14850
S. Uchida
Institute of Advanced Industrial Science and Technology, Ibaraki 305-8568, Japan
Institute of Advanced Industrial Science and Technology, Ibaraki 305-8568, Japan
Peter D. Johnson
Clarendon Laboratory, University of Oxford, Oxford OX1 3PU, United Kingdom
Condensed Matter Physics & Materials Science Department, Brookhaven National Laboratory, Upton, NY 11973
Shane M. O’Mahony
Department of Physics, University College Cork, Cork T12 R5C, Ireland
J. C. Séamus Davis2 [email protected]
Clarendon Laboratory, University of Oxford, Oxford OX1 3PU, United Kingdom
Department of Physics, University College Cork, Cork T12 R5C, Ireland
Department of Physics, Cornell University, Ithaca, NY 14850
Max Planck Institute for Chemical Physics of Solids, D-01187 Dresden, Germany

Notes

2
To whom correspondence may be addressed. Email: [email protected].
Author contributions: W.C. and J.C.S.D. designed research; M.H.H., S.U., H.E., and J.C.S.D. performed research; W.C., W.R., N.K., P.D.J., and S.O. M. analyzed data; and W.C. and J.C.S.D. wrote the paper.
1
W.C. and W.R. contributed equally to this work.

Competing Interests

The authors declare no competing interest.

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