Incoherent Cooper pairing and pseudogap behavior in single-layer FeSe/SrTiO$_3$

In many unconventional superconductors, the presence of a pseudogap - a suppression in the electronic density of states extending above the critical temperature - has been a long-standing mystery. Here, we employ combined \textit{in situ} electrical transport and angle-resolved photoemission spectroscopy (ARPES) measurements to reveal an unprecedentedly large pseudogap regime in single-layer FeSe/SrTiO$_3$, an interfacial superconductor where incoherent Cooper pairs are initially formed above $T_{\Delta}$ $\approx$ 60 K, but where a zero resistance state is only achieved below $T_{0}$ $<$ 30 K. We show that this behavior is accompanied by distinct transport signatures of two-dimensional phase fluctuating superconductivity, suggesting a mixed vortex state hosting incoherent Cooper pairs which persist well above the maximum clean limit $T_{c}$ of $\approx$ 40 K. Our work establishes the critical role of reduced dimensionality in driving the complex interplay between Cooper pairing and phase coherence in two-dimensional high-$T_c$ superconductors, providing a paradigm for understanding and engineering higher-$T_{c}$ interfacial superconductors.

achieved below T 0 < 30 K. We show that this behavior is accompanied by distinct transport signatures of two-dimensional phase fluctuating superconductivity, suggesting a mixed vortex state hosting incoherent Cooper pairs which persist well above the maximum clean limit T c of ≈ 40 K. Our work establishes the critical role of reduced dimensionality in driving the complex interplay between Cooper pairing and phase coherence in two-dimensional high-T c superconductors, providing a paradigm for understanding and engineering higher-T c interfacial superconductors.
Single-layer FeSe grown on SrTiO 3 (FeSe/SrTiO 3 ) has attracted interest due to its characteristics as an atomically thin, interfacially enhanced high-T c superconductor. 1 FeSe/SrTiO 3 exhibits a spectroscopic gap opening temperature (T ∆ ) between 60 to 70 K, 2-5 nearly one order of magnitude higher than that of bulk FeSe (8K), 6 and in excess of related electrondoped FeSe-based bulk compounds (≈ 40 K). 7,8 The combination of its high T c , relative simplicity, and inherently two-dimensional (2D) nature positions FeSe/SrTiO 3 as an ideal platform for exploring the importance of superconducting fluctuations and the possibility of interfacial enhancement in high-T c materials.
Nevertheless, significant challenges impede the systematic study of FeSe/SrTiO 3 , as its air sensitivity, variability in the post-growth annealing process, and potential impact of capping layers make meaningful comparisons across different techniques and studies, both in situ and ex situ , difficult. 9,10 Consequently, there remains a widely observed but heretofore unexplained discrepancy between the gap opening temperature T ∆ observed by ARPES (T ∆ ≈ 60 K) and the temperature at which a zero-resistance state has been measured by electrical transport, T 0 (T 0 < 30 K). 3,[11][12][13][14] A potential resolution to this puzzle is the existence of Cooper pair fluctuations above T c which are known to play an important role in twodimensional superconductors as well as underdoped cuprates, but have not been widely investigated for FeSe/SrTiO 3 .
To reveal the intrinsic nature of superconductivity and the pseudogap in FeSe/SrTiO 3 , we employ for the first time, a combination of angle-resolved photoemission spectroscopy (ARPES) and in situ resistivity measurements to simultaneously probe both the spectroscopic and electrical transport properties of pristine single-layer FeSe/SrTiO 3 samples in ultrahigh vacuum. Through a systematic investigation of a large number of such samples, we reveal the presence of intrinsic superconducting fluctuations over an unprecedentedly broad temperature range, as characterized by the window between the onset of spectroscopic gap T ∆ and the onset of zero resistance T 0 . This result establishes the essential role that reduced dimensionality plays in the superconductivity of FeSe/SrTiO 3 and resolves the long-standing confusion surrounding the critical temperature of FeSe/SrTiO 3 .
In Fig. 1, we show combined in situ resistivity and ARPES measurements conducted on the same sample of single-layer FeSe/SrTiO 3 . The Fermi surface (Fig. 1a) is comprised of electron pockets centered at the M point consistent with an electron doping of 0.11e − per unit cell, in good agreement with earlier reports, 2,4,15 and exhibits the expected spectroscopic signatures of superconductivity (a well-defined gap and band back-bending). In Fig. 1c, we show the sheet resistance, R s (T ) which exhibits a hump-like feature at 280 K, characteristic of heavily electron-doped bulk FeSe-derived compounds, 16 and a broad superconducting transition which onsets at T onset = 44±3 K, eventually falling below 0.1% of R 70K at T 0 = 29±0.2 K. When measured in situ, FeSe/SrTiO 3 samples exhibit residual resistivity ratios (RRRs, defined as R 300K /R 70K ) of ≈10, in contrast to RRRs of ≈ 1-2 for capped singlelayer films reported in the literature. 11 While samples remain robust for hundreds of hours and over numerous cooling/warming cycles when maintained under ultrahigh vacuum (red curve, Fig. 1d), pristine films deteriorate instantaneously upon exposure to atmosphere (black curve, Fig. 1d).
To explore this behavior more systematically, we perform detailed temperature-dependent measurements of the energy gap ∆(T ) using ARPES. In Fig. 2, we show a quantitative comparison between ∆(T ) and R s (T ) measurements on the same sample shown in Fig. 1. In   Fig. 2a, we plot over 100 energy distribution curves (EDCs) symmetrized about E F from 12 K to 94 K, measured at k F of the electron pocket, where false color represents the intensity of the EDCs. In Fig. 2b, we plot select EDCs extracted from the temperature series in Fig.   2a. Fig. 2c tracks ∆ as a function of temperature, defined as half the separation between quasiparticle peaks of the symmetrized EDCs from Figs. 2a and 2b, as well as the evolution of the spectral gap depth δ SW , defined as the difference between the coherence peak amplitude normalized to unity and the corresponding spectral weight at E F . In Fig. 2d we show R s (T ), as well as its derivative dR s /dT . As the superconducting transition is broad, we define three characteristic temperatures to describe the shape of the transition: T 0 , where the resistance reaches 0.1% of R s (70K); T onset , the intersection between the extrapolated normal-state sheet resistance and a linear fit to the superconducting transition region; and T * , where R s (T ) exhibits an inflection point in the paraconducting state (as determined by a local minimum in dR s /dT ). For the sample shown in Fig. 2, T 0 = 29±0.2 K, while T onset = 44±3 K, and T * = 72±4 K. Deep within the superconducting state (T < T 0 ), a clear superconducting gap (∆ = 12.8 ± 1 meV) and sharp Bogoliubov quasiparticle peaks are observed in the ARPES spectra. In the broad transition region where T 0 < T < T onset , the strength of the quasiparticle peak is gradually suppressed as temperature increases, accompanied by a rapid filling of spectral weight within the gap (Fig. 2, a and c), despite the energy separation between the peaks remaining largely constant. Upon increasing the temperature further (T onset < T < T * ), the energy gap continues to fill in at a more gradual rate, until eventually ∆ is no longer discernible above T ∆ = 73±5 K, a temperature that corresponds closely to This behavior is in stark contrast to what is observed in bulk conventional superconductors, where the resistivity drops abruptly to zero at the same temperature at which the superconducting gap opens (i.e, T 0 ≈ T onset ≡ T ∆ ). The most notable exception to this is underdoped cuprates, where the pseudogap at the d-wave antinode measured by numerous techniques including ARPES also opens at significantly higher temperatures than the bulk T c . 17 In contrast, in bulk Fe-based superconductors, it has been widely shown that Since T 0 in 2D superconductors can be strongly influenced by disorder, we have systematically investigated a large number of samples with varying degrees of disorder, using the extrapolated residual sheet resistivity R 0 as a metric, and controlled primarily through the post-growth annealing process. 26 A comparison with ARPES data shows close correspondence between R 0 and increased quasiparticle broadening, consistent with sample-to-sample variation in the disorder strength (Fig. S6). In Figure 4a, we show R s (T ) for a selection of single-layer FeSe/SrTiO 3 films, which clearly demonstrates the obvious dependence of T 0 and T onset on R 0 . As shown in Figure 4c, which summarizes all samples measured in this study, T 0 decreases linearly with increasing R 0 , approaching 40 K in the clean limit. The crossover from a superconducting to insulating regime occurs around R 0 ≈ 7.2 kΩ, close to the quantum of resistance for pairs, R Q = h/(2e) 2 , as would be expected for a 2D superconductor limited by phase fluctuations. 27 The importance of disorder on 2D phase fluctuations naturally explains the wide variation in T 0 and T onset values 3,11-14 previously reported in the literature from capped films (Fig 4b).
The highest values of T onset ≈ 45 K reported here on pristine films are slightly higher than the maximum T onset observed in capped films from the literature (≈ 40 K), and are inconsistent with the singular report of T c > 100 K by Ge et al. 28 In contrast to T 0 , both T ∆ and T * show relatively little dependence on disorder (Fig. 4c), with the values of T ∆ reported here generally consistent with the values extracted from the literature using the same analysis method for our own data (Fig. 4b, grey symbols). 2,4,5,15,[29][30][31] The close correspondence of T ∆ and T * strongly suggests that the beginning of the resistive transition at T * is directly related to the appearance of Cooper pairs below T ∆ . This incoherent Cooper pairing persists within a high temperature pseudogap regime (T onset < T < T ∆ ) well above the temperature range where 2D BKT-like phase fluctuations are clearly observed (T < 40 K).
Taken together, these measurements present, for the first time, a self-consistent picture for the previously mysterious superconducting behavior of FeSe/SrTiO 3 . At low temperatures (T < T 0 ), the influence of phase fluctuations is minimal, resulting in sharp Bogoliubov quasiparticle peaks and a zero resistance state. As the temperature is increased, the zero resistance state is destroyed by a BKT-like vortex unbinding transition, at a temperature dependent on the level of disorder, while spectral weight begins to fill within the gap. Since T 0 should asymptote to T c in the clean limit for a 2D superconductor, 32  After growth films are progressively annealed until optimal superconducting properties are achieved, followed by deposition of 20 nm thick Au electrodes at the sample corners using a shadow mask to provide reliable 4-point electrical contact.
in situ electrical transport measurement In situ resistivity measurements are preformed using a custom-built UHV 4-point transport probe with a base temperature of 5.2 K and a base pressure of 7 × 10 −11 Torr. Contact is applied directly to the film using a set of Au-plated spring-loaded probes in a Van der Pauw geometry, with a nominal instrumental contact spacing of 7 mm. Resistance measurements were taken using a Keithley 6221/2182A current source/voltmeter combination in delta mode with a typical applied current of 1-10 uA. Sheet resistance values were calculated from the Van der Pauw equation such that: where π ln(2) is the Van der Pauw factor and 1.34 is an additional factor to account for the finite contact dimensions based on the known dimensions of the Au electrodes. 39 A subset of the films analyzed in Figure 4 were measured without gold electrodes present; in this case we instead use a correction factor of 1.1, based on finite-element analysis of the Van der Pauw correction factor for our known probe geometry.

ARPES Measurements
ARPES measurements were taken with a VG Scienta R4000 electron analyzer equipped with a VUV5000 Helium discharge lamp using He-I photons at 21.2 eV. The base pressure in the ARPES system is 5 × 10 −11 Torr. Energy resolution was nominally set at 12 meV for mapping and 9 meV for gap measurements. To avoid sample charging during ARPES measurement the film is grounded using a retractable contact pin built onto the sample manipulator. For gap measurements, the Fermi level is referenced to the measured Fermi edge of the Au electrodes.