Observation of strong electron pairing on bands without Fermi surfaces in LiFe1-xCoxAs

In conventional BCS superconductors, the quantum condensation of superconducting electron pairs is understood as a Fermi surface (FS) instability, in which the low-energy electrons are paired by attractive interactions. Whether this explanation is still valid in high-Tc superconductors such as cuprates and iron-based superconductors remains an open question. In particular, a fundamentally different picture of the electron pairs, which are believed to be formed locally by repulsive interactions, may prevail. Here we report a high-resolution angle-resolved photoemission spectroscopy study on LiFe1-xCoxAs. We reveal a large and robust superconducting (SC) gap on a band sinking below the Fermi energy upon Co substitution. The observed FS-free SC order is also the largest over the momentum space, which rules out a proximity effect origin and indicates that the SC order parameter is not tied to the FS as a result of a FS instability.

order is also the largest over the momentum space, which rules out a proximity effect origin and indicates that the SC order parameter is not tied to the FS as a result of a FS instability.
Two main categories of theoretical descriptions arise when trying to describe the high-T c superconductivity of the iron-based superconductors (IBSCs): the weak coupling approach, which involves only the low-energy electronic structure near the Fermi energy (E F ) [11][12][13][14] , and the strong coupling approach, which emphasizes the local magnetic moments and strong Coulomb interactions [6][7][8][9][10] . In the former, superconductivity emerges as a Fermi surface (FS) instability and is in principle sensitive to FS changes. In particular, the superconducting (SC) gap is tied to the FS and its amplitude is strongly influenced by the nesting conditions. In the latter, the pairing is caused by local antiferromagnetic exchange couplings, well defined in the real space, which lead to a SC order parameter (OP) that is fixed in the momentum space and relatively insensitive to small changes of the electronic structure near the FS. In principle, one can distinguish between these two approaches and get critical information on the pairing mechanism of IBSCs by tracking precisely the evolution of the SC OP on bands for which the contributions to the FS vary drastically. In this respect, LiFe 1-x Co x As offers a perfect platform for this study because it undergoes a Lifshitz transition with one FS disappearing at small Co substitution 15,16 . We first look at the FS topologies of pristine LiFeAs and LiFe 0.97 Co 0.03 As, which are illustrated in Fig. 1. In agreement with previous studies, the substitution of Co introduces electron carriers and effectively moves the chemical potential upward 15-18 . Since the α FS shown in Fig. 1a only barely crosses E F in pristine LiFeAs 19,20 , a slight substitution of Fe by Co removes this tiny FS pocket at the Γ(0,0) point, and thus the system undergoes a Lifshitz transition 16 . The remnant intensity at E F around Γ in Fig. 1b is attributed to the limited energy resolution setting (~ 14 meV) for this normal state (NS) measurement (T = 30 K), which broadens the spectral width beyond E F .
To accurately determine the band top of the α band, we performed high-resolution (~ 3 meV) angle-resolved photoemission spectroscopy (ARPES) measurements in the vicinity of Γ for samples at three doping levels (x = 0, 1%, 3%, with onset Tc ≈ 18, 16, 15K, correspondingly). As seen in Figs. 2d-2f, the band top shifts to 4 meV and 8 meV below E F at Co contents of 1% and 3%, respectively. This shift is also clearly demonstrated by the energy distribution curves (EDCs) shown in Figs. 2j-2l. While the low-energy quasiparticle (QP) peaks of pristine LiFeAs are clearly cut off by the Fermi-Dirac (FD) function, those of the 1%Co and 3%Co samples shift below E F with small spectral weight at E F due the finite peak-width. We also measured the k z dispersion of the α band and confirmed its two dimensionality. As an example shown in the supplementary materials 21 , the α band of the 3%Co sample is sinking completely below E F at the Brillouin zone (BZ) centre for all k z planes. The disappearance of the α FS reduces the density-of-states (DOS) near E F and hence significantly suppresses the inter-band scattering between the α band and the electron FSs at the BZ corner, as seen from Figs. 2j-2l.
In the SC state, electrons are gapped toward higher binding energies and form a well-defined Bogoliubov quasiparticle (BQP) peak. Fig. 2j compares the representative EDCs of pristine LiFeAs across the Γ point above and below T c . The electronic states within the SC gap are significantly altered while the states at higher binding energies are only slightly modified by the Bogoliubov dispersion: where E k is the energy of the BQP, which will not show clear deviation from the normal state (NS) energy ε k , when ε k is much larger than the SC gap Δ k . Interestingly, we find that the electronic states of the 1%Co and 3%Co samples are clearly transferred to higher-binding energies as shown in Figs. 2k and 2l, indicating the opening of a SC gap on the α band, even though this α band is located below E F at these doping levels.
To see how the SC condensation affects the electronic states away from E F , we show the simulated EDCs without FD distribution in the NS and SC state in Fig. 3b This indicates that the Co substitution introduces impurity potentials, which is believed to play a destructive role in sign-reversal pairing, at least in the weak coupling regime 1 .
Following the procedure shown in Fig. 3, we extract the low-energy band dispersion below and above T c . Figs. 4a and 4b show the extracted data of the 1%Co sample in wide and narrow energy ranges, respectively. In agreement with Eq. 1, the band shift is the largest near the band top and quickly vanishes at higher binding energies. By using the NS data to fit the dispersion in the SC state, we extract the SC gaps of pristine LiFeAs, 1%Co and 3%Co and plot them as a function of the Co concentration in Fig. 4c. The SC gap on the α band remains almost constant while the associated FS topology undergoes a Lifshitz transition with the substitution of Co. Moreover, this SC gap is found to be the largest within the whole BZ space, which rules out the possibility of a proximity effect causing by the pairing on other FS sheets.
As discussed before, the inter-band scattering in the particle-particle channel is dramatically reduced due to the disappearance of the α FS. According to the weak coupling theories, the SC gap on the α band is expected to exponentially decrease when the DOS at E F goes to zero, as shown in Fig. 4c. This is clearly in contradiction with the experimental observations, which indicates that the SC pairing on the α band is strong and robust. It is also interesting to compare the observed SC gap with ε F , here defined as the energy difference from the band top to the chemical potential, as illustrated in Fig. 4a. Previous studies on the iron-chalcogenide superconductor FeTe  19,20,28,29 . Although this discrepancy can be removed by subtle modifications involving orbital fluctuations or small-q inter-band scattering 30,31 , our observation of strong SC pairing on the bands without FS here is beyond any reasonable mending within the weak coupling approach. Instead, it is naturally consistent with many strong coupling approaches [6][7][8][9][10]32 , in particular with the J 1 -J 2 model that predicts the strongest pairing at the zone centre [6][7][8][9] .
In summary, we have observed an unprecedented strong pairing on energy band without FS at the BZ centre of LiFe 1-x Co x As. The observed SC pairing strength is strong and robust against the reduction of the DOS near E F and the increasing impurity scatterings. The immunity of the SC pairing across the Lifshitz transition rules out the fundamental assumption of weak coupling theories that superconductivity is a FS instability. Our results clearly demonstrate that the pairing mechanism of the IBSCs resides in the strong coupling regime.

Method:
Single crystals of LiFe 1-x Co x As were synthesized by the self-flux method using Li 3 As, Fe 1-x Co x As and As powders as the starting materials. The Li 3 As, Fe 1-x Co x As and As    The small peak above E F in the blue EDC is due to the particle-hole mixing, which is a hallmark of SC condensation. By using the BCS spectral function to fit the EDCs in the NS and the SC state, we extract the SC gap. c, e and g, EDCs at the BZ centre in the NS and the SC state of pristine LiFeAs, 1%Co and 3%Co samples, respectively.
To extract the SC gap, the raw data shown in c, e and g are divided by FD function convoluted with the system resolution and shown in d, f, and h, respectively. The

Extraction of the SC gap without FS
To extract the SC gap from EDCs, we use functions: Here we assume a BCS spectral function in the occupied states, with the Γ k parameter not changing with binding energy. a i is a fitting constant, which is proportional to Γ k and u k 2 . The fitted results and extracted SC gap are shown in Fig. 3 of the main text.

Extracted band dispersions of LiFe 0.97 Co 0.03 As
In Fig. 4   As shown in the main text, the SC gap on the α band is large and robust.