Electron Pair Density Wave States
Detection of a Pair Density Wave State in UTe2
Essence: Three pair density wave (PDW) states coincident in wavevectors with the three charge density wave (CDW) states observed previously, are discovered and visualized in spin-triplet superconductor UTe2.
Schematic diagram of PDW visualization in UTe2: Left is the (0-11) termination layer, middle is the energy gap modulation and right the preexisting CDW state.
Spin-triplet topological superconductors should exhibit many unprecedented electronic properties and, although UTe2 may embody such bulk topological superconductivity, its superconductive order-parameter Δ(k) was unknown. Many diverse forms for Δ(k) are physically possible including intertwined density waves of spin (SDW), charge (CDW) and pairs (PDW) . The latter state exhibits spatially modulating superconductive order-parameter Δ(r), electron pair density and pairing energy-gap. To search for a PDW state in UTe2, we visualize the pairing energy-gap with μeV-scale energy-resolution using superconductive STM tips. We detect three PDWs, each with peak-peak gap modulations circa 10 μeV and at incommensurate wavevectors Pi=1,2,3 that are indistinguishable from the wavevectors Qi=1,2,3 of the prevenient CDW. Concurrent visualization of the UTe2 superconductive PDWs and the non-superconductive CDWs reveals that every Pi : Qi pair exhibits a relative phase δϕ≈π. From these observations, and given UTe2 as a spin-triplet superconductor, this PDW state should be a spin-triplet pair density wave.
Discovery of an Electron-Pair Density Wave in a Transition-Metal Dichalcogenide
Essence: Superconductive electron-pair density wave states are revealed to exist in a canonical TMD if it is superconducting and a CDW preexists. Phenomenology is consistent with Ginzburg—Landau theory for PDWs that are induced due to coexisting superconductivity and CDW.
Top left is the Se termination surface of NbSe2; top right is the scanned Josephson image of the electron-pair in the same FOV using Nb-tip; bottom is the SIS tunnel spectrum from Nb to NbSe2 showing how the Josephson signal appears V=0 as junction resistance is reduced into the MOhm range.
Pair density wave (PDW) states are defined by a spatially modulating superconductive order-parameter. To search for such states in transition metal dichalcogenides (TMD) we used high-speed atomic-resolution scanned Josephson-tunneling microscopy (SJTM). We detected a PDW state whose electron-pair density and energy-gap modulate spatially at the wavevectors of the preexisting charge density wave (CDW) state. The PDW couples linearly to both the s-wave superconductor and to the CDW, and exhibits commensurate domains with discommensuration phase-slips at the boundaries, conforming to those of the lattice-locked commensurate CDW. Nevertheless, we find a global δΦ∼±2π/3 phase difference between the PDW and CDW states, possibly owing to the Cooper-pair wavefunction orbital content. Our findings presage pervasive PDW physics in the many other TMDs that sustain both CDW and superconducting states.
Atomic-scale Structure of CuO2 PDW State Coexisting with Superconductivity
Essence: The so-called charge density wave (CDW) state of underdoped cuprates, which can be directly visualized at atomic scale by STM, is demonstrated - both theoretically and experimentally- to be more consistent with a d-symmetry pair density wave (PDW) state coexisting with d-wave superconductivity.
Left: Column shows the predicted Bogoliubov quasiparticle density of states of an the 8a0 periodic dFF pair density wave coexisting with d-wave superconductivity model for underdoped cuprates. Right: column shows the measured Bogoliubov quasiparticle density of states observed within the density waves of underdoped Bi2Sr2CaCu2O8.
Intense theoretical interest has focused on whether a pair density wave state (PDW) state could coexist with cuprate superconductivity. We used a strong-coupling mean-field theory of cuprates to model the atomic-scale electronic structure of an eight-unit-cell periodic, d-symmetry form factor, pair density wave (PDW) state coexisting with d-wave superconductivity (DSC). From this PDW+DSC model, the atomically-resolved density of Bogoliubov quasiparticle states N(r,E) is predicted at the terminal BiO surface of Bi2Sr2CaCu2O8 and compared with high-precision electronic visualization experiments using spectroscopic imaging STM. The PDW+DSC model predictions include the intra-unit-cell structure and periodic modulations of N(r,E), the modulations of the coherence peak energy Δp (r), and the characteristics of Bogoliubov quasiparticle interference in scattering-wavevector space (q-space). Consistency between all these predictions and the corresponding experiments indicates that lightly hole-doped Bi2Sr2CaCu2O8 does contain a PDW+DSC state.
Visualizing the Energy-Bap Modulations of the Cuprate Pair Density Wave State
Essence: A PDW state was earlier discovered in underdoped cuprates as a modulated electron-pair density (Nature 532, 343 (2016)) or as an induced CDW inside the vortex halo (Science 364, 976 (2019)). Here, the concomitant energy gap modulations are observed with 8a0 periodicity.
Top left : Energy gap Δ(r) modulation image in Bi2Sr2CaCu2O8. Top right : Δ(q) the FT of top left. Bottom left: example of the energy gap modulation in real apace. Bottom right: Linecuts through Δ(q) showing sharp maxima at Q= 2p/8a0.
Although the signature of a cuprate PDW had been detected in SJTM Cooper-pair tunnelling, the signature in single-electron tunneling of a periodic energy gap Δ(r) modulation had not been observed. We discovered strong Δ(r) modulations in Bi2Sr2CaCu2O8+δ that having eight-unit-cell periodicity or wavevectors Q=2π/a0(1/8,0); 2π/a0(0,1/8). An analysis of spatial arrangements of Δ(r) modulations then reveals that this PDW is predominantly unidirectional, but with an arrangement of nanoscale domains. Simultaneous imaging of the local-density-of-states N(r,E) reveals electronic modulations with wavevectors Q and 2Q, as anticipated when the PDW coexists with superconductivity.
Magnetic-field Induced Pair Density Wave State in the Cuprate Vortex Halo.
Essence: We demonstrateBi2Sr2CaCu2O8+δ that the fundamental state generated by increasing magnetic fields in the underdoped cuprate superconductors is not a CDW as often reported, but is a PDW with approximately eight CuO2 unit-cell periodicity (λ=8a0) and predominantly d-symmetry form factor.
Left : Magneti-fied induced density of states N(r,V) modulation image in Bi2Sr2CaCu2O8 Right : N(q,V) the FT of left showing the predicted N(q,V) modulation at both Q= 2p/8ao and Q= 2p/4ao if the field induced state is a PDW
When very high magnetic fields suppress the superconductivity in underdoped cuprates, an exceptional new electronic phase appears. It supports remarkable and unexplained quantum oscillations and exhibits an unidentified density wave (DW) state. Although generally referred to as a "charge" density wave (CDW) because of the observed charge density modulations, theory indicates that this could actually be an electron-pair density wave state (PDW). To search for evidence of a field-induced PDW in cuprates, we visualized the modulations in the density of electronic states N(r) within the halo surrounding Bi2Sr2CaCu2O8 vortex cores. This reveals multiple signatures of a field-induced PDW, including two sets of N(r) modulations occurring at wavevectors QP and 2QP, both having predominantly s-symmetry form factors, the amplitude of the latter decaying twice as rapidly as the former, along with induced energy-gap modulations at QP . Such a microscopic phenomenology is in detailed agreement with theory for a field-induced primary PDW that generates secondary CDWs within the vortex halo. These data indicate that the fundamental state generated by increasing magnetic fields from the underdoped cuprate superconducting phase is actually a PDW with approximately eight CuO2 unit-cell periodicity (λ=8a0) and predominantly d-symmetry form factor.
Detection of a Cooper-Pair Density Wave in Bi2Sr2CaCu2O8+x
Essence: By developing scanned Josephson tunnel microscopy (SJTM) operating at 50mK, we discovered the existence of and visualized, a pair density wave (PDW) state in underdoped cuprates. This is the first PDW ever detected directly, and the first ever visualized.
Left: First ever scanned Josephson image of electron-pair density nP(r) in any superconductor, in this case underdoped in Bi2Sr2CaCu2O8. Right: nP(q) the FT of left showing the pair density modulations as local maxima.
In theory, however, electron pairs of a superconductor can exist with finite momentum Q and thereby generate states with spatially modulating electron-pair density. While never observed directly in any superconductor its was widely hypothesized that the cuprate pseudogap phase contains such a 'pair density wave' (PDW) state. We developed and used nanometer resolution scanned Josephson tunneling microscopy (SJTM) to image Cooper-pair tunneling from a d-wave superconducting STM tip to the condensate of Bi2Sr2CaCu2O8+x. Then, by using Fourier analysis of these SJTM images, we discover the direct signature of a electron-pair density modulation in Bi2Sr2CaCu2O8+x. The amplitude of these modulations is ~5% of the homogenous condensate density and their form factor exhibits primarily s/s′-symmetry. This is the first direct detection and visualization of a PDW state.