Few-layer antimonene electrical properties

Antimonene -- a single layer of antimony atoms -- and its few layer forms are among the latest additions to the 2D mono-elemental materials family. Numerous predictions and experimental evidence of its remarkable properties including (opto)electronic, energetic or biomedical, among others, together with its robustness under ambient conditions, have attracted the attention of the scientific community. However, experimental evidence of its electrical properties is still lacking. Here, we characterized the electronic properties of mechanically exfoliated flakes of few-layer (FL) antimonene of different thicknesses (~ 2-40 nm) through photoemission electron microscopy, kelvin probe force microscopy and transport measurements, which allows us to estimate a sheet resistance of ~ 1200 $\Omega$sq$^{-1}$ and a mobility of ~ 150 cm$^2$V$^{-1}$s$^{-1}$ in ambient conditions, independent of the flake thickness. Alternatively, our theoretical calculations indicate that topologically protected surface states (TPSS) should play a key role in the electronic properties of FL antimonene, which supports our experimental findings. We anticipate our work will trigger further experimental studies on TPSS in FL antimonene thanks to its simple structure and significant stability in ambient environments.

Although antimony does not present a bulk gap, its nonzero topological invariant guarantees the presence of topologically protected surface states (TPSS), coexisting with bulk bands at the Fermi energy [24][25][26]. For a small number of atomic layers, antimonene could already behave as a 3D topological insulator because quantum confinement opens a gap in its bulk bands [27], but, when this occurs, the TPSS on opposite surfaces couple to each other and a gap opens at the Dirac point, partly degrading their topological properties. A minimum of approximately 7-8 layers (∼ 3 nm) is needed for a full decoupling of the TPSS on opposite surfaces [25,28], but bulk bands already cross the Fermi energy at this small thickness [26,27]. In this sense, FL antimonene is not so different from actual 3D topological insulators where the contribution of bulk bands at the Fermi energy is difficult to eliminate. Angle Resolved Photoemission Spectroscopy (ARPES) [9,25,26] measurements and, to a much lesser extent, transport experiments [27,28] have confirmed this issue in binary compounds such as Bi1-xSbx, Bi2Se3 or Bi2Te3 [26,29,30]. Ternary [31] and quaternary [32] compounds have also shown surface contribution in their conduction properties.
Since the surface/bulk conductivity ratio is typically a small fraction, it is difficult to isolate and benefit from the properties of the TPSS [33]. In fact, high-throughput numerical searches have shown that, in theory, stoichiometric topological semimetals outnumber topological insulators [34], thus is of much interest to show that topological semimetals still exhibit properties related with their non-trivial topology [35]. Therefore, finding topological semimetals with a large surface-to-bulk conductivity ratio where the exotic properties of TPSS can manifest in a more direct manner and, additionally, with simple structures and high stability, would increase their possibilities of practical use.
In this work, we present a local morphological and electronic study of mechanically exfoliated FL antimonene flakes of thicknesses between ∼ 2 and 40 nm (see Methods). Within this thickness range, the band structure fully reveals the decoupling of the top and bottom TPSS and the bulk bands present a finite contribution at the Fermi energy.  bends upwards to also cross the Fermi energy, giving rise to a peculiar Fermi surface with six hole pockets surrounding Γ. These pockets have a clear surface and helical character near Γ, but this character is partially lost as you move away from Γ and they progressively become bulk states [27]. Second, the appearance of three fully bulk electron pockets crossing the Fermi level close to the M points for over 5 layers. Moving away from Γ towards the M-point (K-point), ignoring the crossing of the bulk electron pockets, one crosses the Fermi level three times (once) as expected for a topological material. The localization properties of these surface states have been studied in Ref. [27].

Results
With this picture in mind, we proceed to the experimental characterization of our exfoliated FL antimonene flakes (see Supplementary Figure 3 for characterization of FL antimonene flakes with different techniques). We start by analyzing flakes with different number of layers and lateral sizes in the range of several microns, deposited on highly doped Si substrates, with low energy and photoemission electron microscopy (LEEM/PEEM), in combination with synchrotron based Xray photoelectron spectroscopy (XPEEM) (Figure 2). The samples were first cleaned in ultra-high vacuum (UHV) (see Methods), ensuring flakes were free of oxide and contaminants, as demonstrated by the Sb 4d core level spectrum (inset in Figure 2a), which shows the doublet with the 4d 5/2 at a binding energy of around 32 eV. Figure 2a   These results are consistent with the DFT calculations shown in Figure 1. For 1-2 layers the material is fully insulating, while for 3-4 layers, corresponding to the majority of the thinnest flakes in Figure 2, the Fermi surface (see Figure 1) is barely developed. For thicker flakes, the integrated contribution to the density of states at the Fermi energy comes mostly from the electron pocket at Γ and the hole pockets around it. Note also that, due to the low electron mean free path at the recorded kinetic energy, the PEEM signal comes from the few topmost 2 or 3 surface atomic layers, around 100 meV below the EF. Hence, as the TPSS have their weight centered there, it is not preposterous to think that this PEEM signal could be associated with the TPSS of the top layer and not from the bulk pockets that extend throughout the flake width.
Next, we carry out an electrical characterization of FL antimonene flakes of different thicknesses.
Kelvin probe force microscopy (KPFM) measurements on flakes with thicknesses between ∼ 2 to 9 nm (∼ 2-3 to 21 layers) [1] (see Supplementary Material section S4) provides information on the Contact Potential Difference (CPD) variation, which is related to the work function change. As in other 2D materials [36], the measured CPD varies with increasing thickness and above a given   Figure 6) shows that, for 16 layers (approximately 6 nm, which closely corresponds to our thinnest flake), this second band already crosses the Fermi level. Thus, a higher number of bulk bands is expected to be present in the thicker flakes. Even so, the increasing number of bulk bands does not seem to appreciably increase the conductivity, suggesting a dominant contribution of the TPSS.

Discussion
It is instructive to compare the mentioned results with the case of graphene, where a linear increase of the sheet conductance (the inverse of the sheet resistance) has been observed with the number of layers [38,39] in contrast with our case, where it remains constant. We compare the sheet resistance (ρ2D = 1200 ± 300 Ω sq -1 ) with that obtained following the same procedure for FL graphene [37], ρ2D-G = 670 ± 60 Ω sq -1 . Graphene has 4 Dirac cones, with intervalley scattering typically suppressed by long-range scattering potentials. The Dirac cone of FL antimonene is surrounded by hole pockets which, to a first approximation, altogether give a similar carrier concentration to that of graphene in ambient conditions (from the band structure we estimate n2D ~ 3.5×10 13 cm -2 ). The helicity of the states in the hole pockets near the Dirac cone are opposite to that of the nearby Dirac states which also prevents inter-pocket back-scattering 19 . Thus, the resistivity of FL antimonene is expected to be comparable but larger than that of graphene, as observed experimentally. Combining the estimation of the carrier concentration from the band structure with the sheet resistance obtained from the transport measurements we can also tentatively estimate the mobility, µ, by = , where σ2D is the bidimensional conductance (σ2D = 1/ρ2D), e the electron charge, and n2D the bidimensional carrier concentration. We obtain a high mobility of µ ~ 150 cm 2 V -1 s -1 in ambient conditions. To put this figure in perspective, one can compare it with the mobility of black phosphorus, which is reported to be µBP ~ 50 to 1000 cm 2 V -1 s -1 depending on the thickness [40]. layers. We determine a sheet resistance of ∼ 1200 Ω sq -1 above this threshold independent on the sheet thickness. We also estimate the mobility of FL antimonene in ambient conditions, resulting in 150 cm 2 V -1 s -1 . Such a high value, in combination with its stability and simple structure, turns antimonene into a promising candidate for nanoelectronics and optoelectronics applications.
Hence, we envision that the richness in properties and phenomena of this material (including optoelectronic and energy-related) will pave the way for further studies on its many opportunities both from fundamental and practical points of view, with a particular focus in topological surface state physics, as for example for fault-tolerant quantum computation or as conducting channels with reduced dissipation in spintronic devices.

Sample preparation.
We obtained FL antimonene flakes by mechanical exfoliation [1]. We placed a macroscopic freshly cleaved crystal of antimony (Smart Elements) in adhesive tape and after repetitive pealing, we directly transferred FL antimonene flakes to SiO2/Si and highly doped Si substrates. A primary optical microscopy inspection allowed us to locate the thinnest flakes that we later imaged by AFM in contact mode to measure their thickness [41]. We used OMCL-RC800PSA cantilevers from Olympus with a nominal spring constant of 0.39 N m -1 and low forces of the order of ∼ 1 nN to ensure that the flakes were not deformed by the tip. We used WSxM software (www.wsxm.es) both for the acquisition and processing of the AFM data [42,43].

PEEM/LEEM measurements.
The experiments have been carried out at the PEEM experimental station of the CIRCE beamline at the ALBA Synchrotron [44]. All measurements were done in a low energy and photoemission electron microscope from micrometer-sized FL antimonene crystals. Prior to the measurements, we carried out a careful cleaning process in ultrahigh vacuum (UHV), consisting of several annealing cycles at 400 ºC for 5 minutes in a hydrogen atmosphere (10 -6 mbar). This cleaning protocol ensured the flakes were free of oxide and contamination.

KPFM measurements.
We carried out simultaneous dynamic mode AFM for the topography and frequency modulation mode for the KPFM [45] in a single-pass scheme, using metallized AFM tips (Budget Sensors ElectriMulti75-G). We applied an AC bias voltage of amplitude 5 V and frequency 7 kHz to the tip. We performed the KPFM measurements in an inert Ar atmosphere to avoid CPD shielding by the presence of adsorbed water on the surface of the samples [36].

C-AFM measurements.
We contacted the FL antimonene flakes deposited on SiO2/Si substrates using gold nanowire electrodes through the SPANC technique [37]. Briefly, we deposited gold nanowires on the substrates with FL antimonene flakes and assembled them into nanoelectrodes by AFM manipulation. In this case, the so-fabricated gold nanoelectrodes were connected to a microscopic gold electrode, we then used a metallized AFM tip (Budget Sensors ElectriMulti75-G) as a second mobile electrode to acquire IV curves at different locations. We employed dynamic mode AFM to image the samples, with an amplitude set point of 15 nm (cantilever free amplitude 20 nm). Then, we stopped the tip over the points of interest, and we brought it down into contact. There we acquired several IV curves and after this we brought the tip back to dynamic mode AFM. Before and after each set of IV curves we checked that the tip had not changed by acquiring IV curves on the gold nanoelectrode, ensuring tip stability along the whole set of measurements. To avoid artifacts, we also always carefully check the dependence of the conductance with the applied load, selecting the optimal conditions [46]. This procedure ensures that any possible layer of contamination or oxide on the surface is pierced and the current measured would respond to the intrinsic properties of the material [47]. In addition, we obtain the sheet resistance of the few-layer antimonene flakes from the flakes geometry and the slope of the Resistance vs. Lentgh (RL) plots, which is independent of the contact resistance, thus obtaining reliable sheet resistance values within the uncertainty in the measured currents. RL plots acquired on the same flake but with different AFM tips can present different contact resistances, but they present the same slope (and therefore they lead to the same sheet resistance) (see Supplementary   Figure 10). Figure  1 were performed using DFT as implemented in the Vienna Ab initio Simulation Package (VASP) code [48,49]. We employed the local density approximation (LDA) [50] together with the projector-augmented-wave (PAW) [49] method which has been shown to correctly describe Sb films and predicts the bulk lattice constant [21] in excellent agreement with the experimental value of 4.30 Å [51]. Calculations are performed considering SOC and a plane wave basis set was used with a cutoff energy of 300 eV on a 21×21 Monkhorst-Pack [52] k-point mesh. A vacuum region of 16 Å along the z direction was used for each system in order to minimize the interaction between the periodic repetitions of the cells. Full structural relaxations were performed for each of the 2L-7L systems until atomic forces were smaller than 0.001eV/Å [53]. The relaxed structure of the 7L system was used for the 9L system employed for the conductivity calculations (see below). We have taken defects into consideration as a perturbation (to all orders), expanding the eigenstates of the system + defects in a basis of Bloch states near the Fermi energy of the defect-free system (see Ref. [27] for details). between two successive iteration steps converged to less than 10 -6 Hartree. A cutoff energy of 220

Band structure calculations. Band structure and Fermi surface contour calculations in
Ry and an 11x11x1 k-grid have been used in all presented results. The unit cell structure was also geometrically optimized. The quasi-Newton eigenvector following method was executed for structural relaxation of all structures until the change in forces between two successive iteration steps was less than 10 -3 Hartree/Bohr. SOC was also employed and van der Waals interaction (vdW) correlation is considered by using the semiempirical dispersion-corrected density functional theory (DFT-D2) force-field approach. We have verified that the obtained band structure is essentially similar to that shown in Figure 1 (see Supplementary Figure 12).

Kubo conductivity calculations.
We have chosen a system of 9 antimonene layers for the calculation of the Kubo conductivity. This number guarantees that the bulk pockets are well developed and that the TPSS of opposite surfaces are completely decoupled. We first compute the DFT Kohn-Sham Hamiltonian with the CRYSTAL code [56,57] to which we have added SOC after self-consistency [58]. The lattice constant was considered (based on our previous work, Ref. [58]) to be a = 4.27 Å and the intra-and interlayer distances h = 1.52 Å and d = 3.68 Å, respectively.
For these calculations we utilized a small-core pseudopotential basis set [59] with 23 valence electrons. For better agreement with plane-wave calculations we considered 0.94 scaling factor for the last filled p-orbital. The SOC enhancement factor 65, was also defined based on our previous work where we verified the SOC implementation vs. fully relativistic non-colinear VASP calculation (Ref. [58]). LDA exchange [60] and VBH correlation [61] functionals were used on a 32 x 32 Monkhorst-pack k-grid. For convergence, we applied a Fock (Kohn-Sham) matrix mixing of 97% between subsequent SCF cycles and the convergence on total energy was set to 10 -9 Hartree.
We calculate the dc conductivity using finite-size Kubo formalism at room temperature. We have essentially followed the methodology developed in our previous work [27], but here we have fully implemented the exact evaluation of momentum matrix elements, as explained in Ref. [62]. The red bands shown in Figure 1 are the relevant ones in our calculations. The exfoliation procedure guarantees that the internal layers of our flakes are defect-free, but not the surfaces. We also expect the bottom layer to be more influenced by disorder than the top surface due to the substrate effects  Figure 4b was not changing anymore. Furthermore, we have smoothed the curves by averaging over the Fermi energy interval containing typically 11 levels.

Acknowledgments
We acknowledge financial support through the "María de Maeztu" Programme for Units of

S4. Kelvin probe force microscopy (KPFM)
We carried out KPFM measurements on flakes with thicknesses ranging from ∼ 2 to 9 nm (∼ 2-3 to 21 layers). 5 KPFM provides information on the Contact Potential Difference (CPD) variation, related to the work function change. 7 Figure S5 presents the results for two FL antimonene flakes with several terraces of different heights. The blue lines correspond to the real paths followed to plot the profiles of the variation of the CPD shown in Figure S5. The length for each section of the CPD profiles is proportional to the corresponding section in the topographical profiles. Thus, we could re-scale the CPD profiles to match the total length of the topographical profiles. Following this procedure, the CPD profiles shown better reflect the overall behavior of the flakes compared to having used direct profiles.

Supplementary
For terraces above ∼ 8 layers the CPD has appreciably decreased. The variation of the CPD contrast is inversely proportional to the variation of the work function in the sample (∆CPD = -∆φsample sample/e, where e is the magnitude of the electron charge), determined by the relative position of the Fermi levels, which are sensitive to the surface charge density or the presence of surface dipoles. 7 The measured CPD is higher for terraces below ∼ 8 layers and lower for thicker ones. Our DFT calculations in Figure S5g show this behavior, with the CPD actually decreasing with the number of layers, compatible with the KPFM measurements.