Electronic structure and signature of Tomonaga-Luttinger liquid state in epitaxial CoSb$_{1-x}$ nanoribbons

Recently, monolayer CoSb/SrTiO$_3$ has been proposed as a candidate harboring interfacial superconductivity in analogy with monolayer FeSe/SrTiO$_3$. Experimentally, while the CoSb-based compounds manifesting as nanowires and thin films have been realized on SrTiO$_3$ substrates, serving as a rich playground, their electronic structures are still unknown and yet to be resolved. Here, we have fabricated CoSb$_{1-x}$ nanoribbons with quasi-one-dimensional stripes on SrTiO$_3$(001) substrates using molecular beam epitaxy, and investigated the electronic structure by in situ angle-resolved photoemission spectroscopy. Straight Fermi surfaces without lateral dispersions are observed. CoSb$_{1-x}$/SrTiO$_3$ is slightly hole doped, where the interfacial charge transfer is opposite to that in monolayer FeSe/SrTiO$_3$. The spectral weight near Fermi level exhibits power-law-like suppression and obeys a universal temperature scaling, serving as the signature of Tomonaga-Luttinger liquid (TLL) state. The obtained TLL parameter of $\sim$0.21 shows the underlying strong correlations. Our results not only suggest CoSb$_{1-x}$ nanoribbon as a representative TLL system, but also provide clues for further investigations on the CoSb-related interface.


INTRODUCTION
Over the past decade, substantial works have been performed on monolayer FeSe/SrTiO 3 1-3 , where the cross-interface charge transfer and electron-phonon coupling are suggested to play important roles in enhancing the superconductivity 4-11 . Exploring similar systems could offer alternative research aspects in understanding the interfacial interactions and to facilitate establishing a general picture. Such efforts in cobalt-based compounds have led to a promising candidate, mono-layer CoSb/SrTiO 3 , which is predicted to adopt a tetragonal phase and similar band structures as monolayer FeSe/SrTiO 3 12 . Recently, a combined scanning tunneling microscopy/spectroscopy (STM/STS) and magnetization measurement reports possible superconductivity in two-dimensional (2D) CoSb with orthorhombic lattice 13 , where the triplet pairing symmetry is proposed 14 . Therefore, it is necessary to further explore the electronic properties of CoSb-based interface.
Depending on different growth conditions using molecular beam epitaxy (MBE), the CoSbbased interface is shown to be a versatile system not only manifesting as 2D thin film but nanowire with quasi-one-dimensional (Q1D) stripes, where a gap opening at Fermi level (E F ) is also observed 13 .
It thus serves as a rich playground for studying the electronic properties in different dimensions.
When the motion of electrons is confined in 1D, the Landau Fermi liquid theory tends to break down because the individual excitation is replaced by the collective mode, which is highly correlated. It leads to the Tomonaga-Luttinger liquid (TLL) model that has succeeded in understanding the distinctive characters of 1D systems, like the power-law-like behavior of various physical properties [15][16][17][18][19] . In order to understand the electronic properties under dimensional confinement in CoSb-related nanowires, it is critical to experimentally reveal their electronic structure by angleresolved photoemission spectroscopy (ARPES).
In this paper, we have fabricated CoSb 1−x nanoribbons on SrTiO 3 (001) substrates utilizing MBE 20 . The Q1D stripes on nanoribbons give rise to straight Fermi surfaces (FSs). CoSb 1−x /SrTiO 3 is slightly hole doped, where the interfacial charge transfer is opposite to that in monolayer FeSe/SrTiO 3 .
The near-E F spectral weight exhibits power-law-like suppression and obeys a universal tempera-3 ture scaling, demonstrating the underlying TLL nature. The determined TLL parameter of ∼0.21 reveals that CoSb 1−x is in the strongly correlated regime. These results establish a model TLL system and provide clues for further studies on the CoSb-related interface.

Surface and structural characterization of as-grown nanoribbons
The surface and structural characterization of CoSb 1−x are presented in Fig. 1.

TLL origin of the spectral weight suppression
Notably, the ARPES spectral weight of CoSb 1−x gradually decreases as approaching E F (Fig. 2d-g and Supplementary Fig. 2). We examine the temperature evolution of energy distribution curves (EDCs) at k F of Sample I (cut#1 in Fig. 2b). As plotted in Fig. 3a, the spectral weight is continuously depleted towards E F , and there is no Fermi edge and well-defined quasiparticle peak. The suppression near E F is clearer by comparing the k-integrated EDCs (cut#1 in Fig. 2b) to a polycrystalline Au spectrum with a steep Fermi edge in Fig. 3b. Although the nanoribbons misaligned from [100] and [010] orientations may give rise to "polycrystallinity" and momentumaveraged spectra with reduced weight near E F , it would not deplete the density of states (DOS) at E F . As shown in Fig. 3c, similar spectral weight suppression is also observed in the STS spectra obtained on an individual nanoribbon. Therefore, the misaligned nanoribbons have minor effect 7 on the spectral weight suppression near E F . This spectral weight depletion is not likely due to the disorder effect either. As suggested by the Altshuler-Aronov theory 23 , the DOS near E F of disordered metals is the summation of a finite constant and a power-law-like function 24 , which is distinct from our observation that the DOS nearly decreases to zero at E F . The similar momentum broadening of α band in Sample I and III indicates similar level of disorder scattering and similar angular distribution of nanoribbons (see Supplementary Fig. 4), further implying the origin of spectral weight suppression is intrinsic. In a Peierls instability, the Q1D FS topology would induce a large charge density wave (CDW) susceptibility, which may lead to a CDW order or sizable CDW fluctuations 25 . The CDW order could be excluded in our case as the STS data in Fig. 3c exhibit continuous suppression towards E F without any signature of CDW gap formation, in contrast to the well-defined gap structure in the STS spectra of other typical Q1D CDW systems 26,27 . On the other hand, the CDW fluctuations could cause a pseudogap opening, which may explain the spectral weight suppression 28,29 . However, the suppressed spectral weight in a pseudogap would be filled up following a linear function of the temperature 30 , which is not observed in our case (Fig.   3a).
The spectral weight suppression can be well accounted for by potential TLL state in CoSb 1−x .
Serving as the spectroscopic fingerprints of TLL, the power-law-like suppression and scaling behavior for low-energy DOS have been observed in several Q1D materials, like (TaSe 4 ) 2 I 17 , Li 0.9 Mo 6 O 17 31 , Au/Ge(001) 32 , Bi/InSb(001) 33 , and K 2 Cr 3 As 3 34 . By a linear fit to the integrated spectra at 25 K in the double-logarithmic plot (Fig. 3d), the power-law-like manner of DOS can be straightforwardly visualized. Because the finite temperature effect has not been considered, the 8 curves at higher temperatures deviate from the simple power law below the binding energy of ∼20 meV, which have also been observed in K 2 Cr 3 As 3 34 .
To quantitatively model the DOS according to TLL theory with finite temperature effect included, we fit the integrated spectra based on the following formula for photoemission spectral function 35 , in calculations (Fig. 2j). It calls for future experiments on monolayer CoSb films to reveal the underlying correlation strength.

DISCUSSION
It is worth noting that the Coulomb blockade from the size effect has been observed on isolated CoSb-related nanowires 13 . Statistics of the topography in Fig. 1a gives the total length of 38±6 nm for connected nanoribbon domains, corresponding to an average gap size of ∼3-5 meV from the Coulomb blockade studied by STS 13 . From this perspective, it is only ∼10% of the leading edge position (binding energy of ∼40 meV) of spectral weight suppression in Sample I (Fig. 3a), indicating the Coulomb blockade would only have a minor contribution to the spectral weight suppression. Nevertheless, the Coulomb blockade has been observed by photoemission measurements in several interfacial materials, and been suggested as the dynamic final state effect during the photoemission process, which would induce statistical spectra broadening in the energy shifts 44,45 .
Therefore, the Coulomb blockade may have a decoration effect on the spectral shape of Sample I, making a contribution to the spectral weight suppression near E F , while the good agreement of our data with the power-law analysis indicates the dominant role of the TLL state in CoSb 1−x nanoribbons.
In summary, we have fabricated CoSb 1−x nanoribbons with Q1D stripes on SrTiO 3 (001) sub-strates and investigated the electronic structure. The Q1D FS topology is revealed. CoSb 1−x /SrTiO 3 is slightly hole doped, where the interfacial charge transfer is opposite to that in monolayer FeSe/SrTiO 3 .
The spectral weight near E F shows power-law-like suppression and obeys a universal temperature scaling, suggesting CoSb 1−x belongs to the TLL. The corresponding TLL parameter of ∼0.21 uncovers its strongly correlated nature. Our study not only demonstrates a canonical TLL system, but also sheds light on the understanding of the physics in CoSb-based interface.