Atomic-scale electronic structure of the cuprate pair density wave state coexisting with superconductivity

Abstract

<jats:p>The defining characteristic of hole-doped cuprates is<jats:italic>d</jats:italic>-wave high temperature superconductivity. However, intense theoretical interest is now focused on whether a pair density wave state (PDW) could coexist with cuprate superconductivity [D. F. Agterberg et al.,<jats:italic>Annu. Rev. Condens. Matter Phys.</jats:italic>11, 231 (2020)]. Here, we use a strong-coupling mean-field theory of cuprates, to model the atomic-scale electronic structure of an eight-unit-cell periodic,<jats:italic>d</jats:italic>-symmetry form factor, pair density wave (PDW) state coexisting with<jats:italic>d</jats:italic>-wave superconductivity (DSC). From this PDW + DSC model, the atomically resolved density of Bogoliubov quasiparticle states<jats:inline-formula><m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"><m:mi mathvariant="bold-italic">N</m:mi><m:mfenced><m:mrow><m:mi mathvariant="bold-italic">r</m:mi><m:mo>,</m:mo><m:mi>E</m:mi></m:mrow></m:mfenced></m:math></jats:inline-formula>is predicted at the terminal BiO surface of Bi<jats:sub>2</jats:sub>Sr<jats:sub>2</jats:sub>CaCu<jats:sub>2</jats:sub>O<jats:sub>8</jats:sub>and compared with high-precision electronic visualization experiments using spectroscopic imaging scanning tunneling microscopy (STM). The PDW + DSC model predictions include the intraunit-cell structure and periodic modulations of<jats:inline-formula><m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"><m:mi mathvariant="bold-italic">N</m:mi><m:mfenced><m:mrow><m:mi mathvariant="bold-italic">r</m:mi><m:mo>,</m:mo><m:mi>E</m:mi></m:mrow></m:mfenced></m:math></jats:inline-formula>, the modulations of the coherence peak energy<jats:inline-formula><m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"><m:msub><m:mi mathvariant="normal">Δ</m:mi><m:mi>p</m:mi></m:msub><m:mfenced><m:mi mathvariant="bold-italic">r</m:mi></m:mfenced><m:mo>,</m:mo></m:math></jats:inline-formula>and the characteristics of Bogoliubov quasiparticle interference in scattering-wavevector space<jats:inline-formula><m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"><m:mfenced><m:mrow><m:mi mathvariant="bold-italic">q</m:mi><m:mo mathvariant="bold">-</m:mo><m:mtext>space</m:mtext></m:mrow></m:mfenced></m:math></jats:inline-formula>. Consistency between all these predictions and the corresponding experiments indicates that lightly hole-doped Bi<jats:sub>2</jats:sub>Sr<jats:sub>2</jats:sub>CaCu<jats:sub>2</jats:sub>O<jats:sub>8</jats:sub>does contain a PDW + DSC state. Moreover, in the model the PDW + DSC state becomes unstable to a pure DSC state at a critical hole density<jats:italic>p</jats:italic>*, with empirically equivalent phenomena occurring in the experiments. All these results are consistent with a picture in which the cuprate translational symmetry-breaking state is a PDW, the observed charge modulations are its consequence, the antinodal pseudogap is that of the PDW state, and the cuprate critical point at<jats:italic>p</jats:italic>* ≈ 19% occurs due to disappearance of this PDW.</jats:p>

Scattering interference signature of a pair density wave state in the cuprate pseudogap phase

“…4. The initial Λ q; Δ 0 À Á features at T=4.2K are exactly as expected from theory and as observed by experiment at p = 0.17, for a PDW coexisting with a d-wave superconductor 35 . As temperature increases the characteristics remain strikingly unchanged except that the intensity become significantly weaker.…”

“…3a). These are the energy-gap modulations with period approximately 8a 0 , that have been previously reported 35,37,38 for samples with p % 0:17, and are the signature of a PDW state coexisting with dsymmetry superconductivity at low temperature. Fourier filtration of Fig.…”

“…Effi0 Zðq; EÞ is calculated at each temperature where Δ 0 ¼ 20 meV is the observed energy above which dispersive scattering interference is no longer detectable 10,35 . The measured temperature dependence of Λ q; Δ 0 À Á for 0:1T c ≤ T ≤ 1:5T c is shown in the left column in Fig.…”

“…12). The implication is that the PDW state which definitely exists at lowest temperatures [35][36][37][38] , continues to exist into pseudogap phase. But in that case, since that pseudogap is often (but not always) reported to support no supercurrents, it would have to be in a strongly phase fluctuating PDW phase 32,33,[49][50][51][52][53] .…”

Unconventional quantum vortex matter state hosts quantum oscillations in the underdoped high-temperature cuprate superconductors

“…The observed coexistence can potentially be explained by nonuniform models of superconductivity that are spatially modulated at a finite wave vector, such as the pair density wave (PDW) recently reported in experiments such as scanning-tunneling microscopy (31)(32)(33)(34)(35)(36)(37)(38)(39)(40)(41)(42)(43)(44). Unlike models of spatially uniform superconductivity that are characterized by nodal points (9, 19-21, 24, 29), models of finite wave vector superconductivity result in "nesting" over only a portion of the Fermi surface and consequently, yield a partially gapped Fermi surface and lines of gapless excitations (31).…”