Proximity-Induced Nodal Metal in an Extremely Underdoped CuO$_2$ Plane in Triple-Layer Cuprates

TL;DR

The study reveals a robust nodal-metal state in the extremely underdoped inner CuO$_2$ plane of Bi$_2$Sr$_2$Ca$_2$Cu$_3$O$_{10+\delta}$ triple-layer cuprates, with a large $d$-wave gap $\Delta_{0}$ ≈ 80–100 meV persisting well above $T_c$ and lacking a Fermi arc up to $T_{\text{pair}}$ ≈ 2$T_c$. Using ARPES across a wide doping range, the authors attribute this to strong proximity effects from the adjacent optimally doped outer planes, which enhance superconducting correlations in the inner plane. The inner plane exhibits electron-hole symmetric spectra near the node and a coherent IP peak even at very low IP doping, contrasting with AFM-pocket states seen in deeper underdoped multi-layer cuprates. Hubbard-model calculations partially reproduce the pseudogap but fail to capture the nodal-metal state, underscoring the crucial role of interlayer proximity and layer-number in high-$T_c$ superconductivity and helping explain why $T_c$ peaks at layer number three in multi-layer cuprates.

Abstract

ARPES studies have established that the high-$T_c$ cuprates with single and double CuO$_2$ layers evolve from the Mott insulator to the pseudogap state with a Fermi arc, on which the superconducting (SC) gap opens. In four- to six-layer cuprates, on the other hand, small hole Fermi pockets are formed in the innermost CuO$_2$ planes, indicating antiferromagnetism. Here, we performed ARPES studies on the triple-layer Bi$_2$Sr$_2$Ca$_2$Cu$_3$O$_{10+δ}$ over a wide doping range, and found that, although the doping level of the inner CuO$_2$ plane was extremely low in underdoped samples, the $d$-wave SC gap was enhanced to the unprecedentedly large value of $Δ_0\sim$100 meV at the antinode and persisted well above $T_{c}$ without the appearance of a Fermi arc, indicating a robust ``nodal metal''. We attribute the nodal metallic behavior to the unique local environment of the inner clean CuO$_2$ plane in the triple-layer cuprates, sandwiched by nearly optimally-doped two outer CuO$_2$ planes and hence subject to strong proximity effect from both sides. In the nodal metal, quasiparticle peaks showed electron-hole symmetry, suggesting $d$-wave pairing fluctuations. Thus the proximity effect on the innermost CuO${_2}$ plane is the strongest in the triple-layer cuprates, which explains why the $T_c$ reaches the maximum at the layer number of three in every multi-layer cuprate family.

Figures (13)

• Figure 1: $\bf{a}$ Electronic states on the Fermi surface. Upper panel: Pseudogap (PG) state above $T_{{c}}$, showing a Fermi arc in the nodal region. The Fermi arc is shown by a red curve. Lower panel: SC state below $T_c$, showing a point node of the $d$-wave SC gap. Here, the point node is shown by a red circle. $\bf{b}, \bf{c}$ The phase diagram of single- and double-layer cuprate superconductors Yoshida_PRL2009Kondo_Nature2009Kondo_PRL2007Gq_Zheng_PRL2005JC_Campuzano_PRL1999KTanaka_Science2006WSLee_Nature2007Vishik_pnas2012. The superconducting (SC) gap extrapolated from the node to the antinode $\Delta_0$, the antinodal pseudogap $\Delta^{*}$, and the pseudogap temperature $T^{*}$ are shown. Grey dashed lines are guide to the eye for $\Delta^\ast$ and $T^\ast$.
• Figure 2: ARPES spectra in the nodal region of the normal state ($T = 1.2T_c$) of Bi2223. a Intensity $E$-$k$ maps of UD80, OPT110, and OD110 measured for cuts with Fermi-surface angles ($\phi$) defined in panel d. Each intensity map has been divided by the Fermi-Dirac (FD) distribution function. b EDC divided by the FD function at several $k_{\rm F}$. Black and gray bars indicate peak positions on the occupied and unoccupied sides of $E_{\rm F}$, respectively. EDCs that show a peak at $E_F$ are shown in color. c The momentum dependence of the energy of the black peak for each sample. The Fermi arc for each sample is drawn using the corresponding color. Error bars in c represent an uncertainty of the spectral peak positions. d The momentum region where the point node and the Fermi arc exist shown by a circle and a thick line, respectively. See Sec. \ref{['S3']} and Fig. \ref{['fig:FigS3']} of SI for the Fermi arc lengths of all the samples.
• Figure 3: Energy gap evolution below and above $T_c$ in Bi2223. a, b momentum dependences of the energy gap of the OP and IP bands in the superconducting ($T\ll T_c$) and pseudogap states ($T=1.2T_c$). The nodal gap $\Delta_{0}$ is defined by the gap value extrapolated to the antinode $k=(\pi, 0)$ (blue arrows). In samples showing the "two-gap" momentum dependence, the larger energy gap extrapolated to the antinode $\Delta^\ast$ is indicated by red arrows. The IP of all the underdoped samples shows a $d$-wave pseudogap even at $T=1.2T_c$, indicating that $T_{\rm pair}>1.2T_c$. Error bars of energy gap represent standard deviations of the spectral peak positions.
• Figure 4: Temperature-dependent of the ARPES spectra of the UD75 sample. a Temperature dependence of EDCs at $k_{\rm F}$ from $\phi=0^\circ$ to 15.8${^\circ}$ divided by the FD function for the IP band of the UD75 sample. Black bars indicate the peak positions of the EDCs. Red EDCs show a peak at $E_{\rm F}$, indicating that the EDC's are on the Fermi arc. $\bf{b}$ Fermi arc length plotted as a function of $T/T_c$ in single-, double-, and triple-layered cuprates. The data of the single- and triple-layer cuprates are taken from Refs. Nakayama_PRL2009Kanigel_PRL2007. $\bf{c}$ Fermi-surface map of UD75 of Bi2223 at $T$ = 10 K. Red and black markers are replica Fermi surfaces due to the superstructure of the Bi-O layers and those of shadow bands, respectively.
• Figure 5: Variation of the energy gaps and characteristic temperatures in Bi2223. a$\Delta_{0}$, $\Delta^{*}$, $T_c$, and $T_{\rm pair}$ of Bi2223 plotted as functions of carrier concentration for each of the OP and IP. Energy gaps $\Delta$ and temperatures $T$ are scaled by $2\Delta=4.3k_{\rm B}T$.The changes in the Fermi surfaces of IP and OP are schematically illustrated for the temperature regions $T < T_c$, $T_c < T < T_{\rm{pair}}$, and $T_{\rm{pair}} < T$. Uncertainties in the energy gap, temperature, and hole concentration are shown by error bars. b Schematic illustration of the crystal structures of multi-layered cuprates. In the 5- and 6-layer cuprates, hole pockets have been observed in the innermost CuO$_2$ plane due to the AFM order Kurokawa_NatComu2022Kunisada_Science2020, whereas the $d$-wave-like pseudogap characteristic of a nodal metal was observed in the triple-layer cuprates.