Quantum size effects, multiple Dirac cones and edge states in ultrathin Bi(110) films

The top and side views of Bi(110) surface with an A7 crystal structure are shown in figure 1(a) and (b), respectively. The pseudocubic surface unit cell is marked by a rectangle in figure 1(a), with the corresponding SBZ shown in figure 1(c). The cleaved Bi2212 surface shows a typical diffraction pattern with the long-range supermodulation along the $b_{S}$ direction, figure 1(d) [32]. The LEED pattern of a film with the nominal thickness of 3 BL, figure 1(e), displays both the substrate and overlayer-Bi related spots, an indication that films do not grow in a layer-by-layer mode, but follow a three-dimensional (3D) island growth. The similar growth mode was also observed for other substrates [31, 12, 18]. The observed pseudo-square LEED pattern indicates the formation of Bi(110) films. As the lattice parameters of Bi(110) surface ($a=4.54$ Å, $b=4.75$ Å) are smaller than the lattice parameter of Bi2212 ($a_{S}\approx b_{S}\approx 5.4$ Å), the Bi(110) reciprocal unit-cell is bigger than the Bi2212 one. The arc-like feature of the second-order diffraction spots is the signature of coexisting orthogonal domains originating from different lattice parameters, $a$ and $b$. Similar LEED pattern was reported in previous studies of Bi/Bi2212 system [18]. The LEED patterns indicate that the Bi-O-plane of the Bi2212 substrate and Bi(110) films are aligned so that $a\parallel b_{S}$ and $b\parallel b_{S}$.

At a nominal coverage of $\sim$ 5 BL, the substrate spots are no longer visible, figure 1(f). Considering the atomic structure of Bi(110) surface, the islands are expected to propagate along the zigzag chain as Bi atoms are only bonded along that direction, whereas the atoms in adjacent chains are not bonded directly in plane. Therefore, during the growth, migrating atoms tend to attach to the ends of the zigzag chains. STM studies on Bi(110) indicate that the longer side of islands is the zigzag direction (i.e., the shorter side of the unit-cell) [12]. If this is also true for the Bi/Bi2212 system, the obtained LEED pattern (figure 1(e)) implies that the Bi islands grow so that the $\bar{\Gamma}-\bar{X_{1}}$ and $\bar{\Gamma}-\bar{X_{2}}$ directions are along the $a_{S}$ and $b_{S}$ crystallographic axes of Bi2212. Indeed, in the previous STM study, it was apparent that islands nucleated along the two specific orthogonal directions [18]. However, these studies could not resolve whether the islands had BP or A7 structures, as both structures have very similar unit cells.

The experimental band dispersions along the $\bar{\Gamma}-\bar{M}$ direction of Bi(110) SBZ for the nominal film coverage of $\sim 2$ BL, $\sim 3$ BL, $\sim 3.5$ BL , $\sim 4$ BL and $\sim 5$ BL are presented in figure 2(a)-(e). With the increasing Bi coverage, a complex evolution of electronic bands indicates that multiple film thicknesses are present simultaneously, in accordance with our LEED results. To understand the origin of individual bands, the theoretical layer-dependent dispersions are reproduced from Ref. [24] (even-bilayer) and Ref. [8] (odd-bilayer), figure 2(f)-(j). The comparison enables the identification of spectroscopic features corresponding to the specific film thickness. For example, in the case of even-bilayer films, going from 4 BL to higher coverage, the electronic states marked by 1, 2, and 4 in figure 2(f)-(i) show very similar dispersions. On the other hand, the calculations predict quite a different behavior for the odd-bilayer films, with a linear dispersive band and Dirac cone at $\bar{M}$ (figure 2(g)-(j)) [8, 6]. Variation of spectral intensity of a particular band with the nominal Bi coverage is related to the change in the relative area covered by the corresponding thickness.

According to theory, in the 2 BL and 4 BL films (figure 2(f)) there should be essentially no electronic states crossing the $E_{F}$ and a spectral gap should be observed [24, 8]. Our results for the nominal film coverage of $\sim$ 2 BL, figure 2(a), show prominent signatures corresponding to 4 BL Bi(110) films, with some contribution from 3 BL and/or 6 BL thickness forming a hole-like band at $\bar{\Gamma}$. The states originating from 4 BL islands show a spectral gap along $\bar{\Gamma}-\bar{M}$, in agreement with the calculations.

The film with a total nominal thickness of $\sim$ 3 BL, shows a few more states appearing (figure 2(b)). Comparison with the theoretical result (figure 2(g)) indicates that the additional states are mainly originating from the 6 BL islands. The presence of a linear dispersive band, marked by an arrow in figure 2(b), suggests an additional contribution from odd-bilayer islands, most likely from the 3 BL and/or 5 BL islands. The exact discrimination between them is difficult due to its extremely low intensity. Thus, the 3 BL nominal thickness film consists of 4 BL, 5 BL (and/or 3 BL), and 6 BL Bi(110) islands and small regions of uncovered Bi2212, as inferred from LEED (figure 1(e)). We note that there are some additional states which are visible near $E_{F}$ (0 to $-$0.3 eV) in between $\bar{\Gamma}$ and $\bar{M}$. These states are absent in the layer-dependent band calculations and their intensity is strongly suppressed with further Bi deposition.

Figure 2(c) shows the electronic states for the nominal coverage of 3.5 BL film. Now, the electronic states corresponding to 4 BL are suppressed, and the film is dominated by 6 BL Bi(110) islands. The electronic states 2 and 3 are well reproduced by theoretical calculations for 6 BL Bi(110) film (figure 2(h)). In addition, the presence of a less intense band 4 indicates a small contribution from 8 BL islands. Upon further Bi deposition with a nominal coverage $\sim$ 4 BL, the bands 4 and 5, originating from 8 BL Bi(110) islands get sharper and well-resolved as can be seen in figure 2(d). The linearly dispersing band near the $\bar{M}$ point also gained in intensity compared to the 3.5 BL film. This band shows a local minimum at $\sim$ k_∥=0.6 Å^-1 and its corresponding energy position of $\sim$ $-$0.57 eV matches well with the theoretical values of 5 BL Bi(110) film (figure 2(i)). Upon further Bi deposition, the local minimum further shifts to $\sim$ $-$0.47 eV (figure 2(e)) corresponding to the 7 BL Bi(110) film (figure 2(j)). The observed trend would suggest that the bands marked by 6 and 7 in figure 2(e) originate from the 10 BL islands.

The Fermi surfaces and constant-energy contours at $-50$ meV, along with the band dispersions for bare substrate and various film thicknesses are shown in figure 3: (a-d) bare Bi2212, (e-h) 2 BL, (i-l) 3 BL, (m-p) 3.5 BL and (q-t) 5 BL, respectively. In figure 3(a), the states originating from the Cu-O plane of the bare Bi2212 substrate are marked. The Fermi surface, dispersion and superconducting gap are consistent with our previous results on optimally doped Bi2212 [33, 34]. After the deposition of 2 BL film, the intensity of these substrate related states is strongly suppressed whereas an intense spot from the Bi film appears near the $\bar{X}$ point along the $\bar{\Gamma}$-$\bar{X}$ (figure 3(e) and (f)). The $k_{F}$ position of this spot agrees with the expected $k_{F}$ from the 4 BL Bi(110) islands [35, 24].

According to calculations, this band hosts a massive Dirac cone with a gap size of $\sim$ 70 meV, formed by the bonding and antibonding states of Bi-$p_{z}$ orbitals [24, 2]. In figure 3(g), the theoretical dispersions are shown on top of the experimental data. The bands for both $\bar{\Gamma}$-$\bar{X_{1}}$ (red dots) and $\bar{\Gamma}$-$\bar{X_{2}}$ (black dots) directions are shown, as they should coexist due to the formation of orthogonal domains, as discussed earlier. The red dots represent the Dirac cone. The more detailed picture of the experimental electronic structure of that Dirac state is given in figure 4. The Dirac point is located $\sim$ 150 meV below $E_{F}$. Such Dirac cone does not exist on the surface of bulk Bi(110) crystal [3].

In the case of 3 BL coverage, the FS (figure 3(i)) looks more complex and in addition to the 4 BL, the features from 3 BL and/or 5 BL and 6 BL are also visible. In figure 3(k) and (l), the hole-like band at $\bar{\Gamma}$, predominantly originating from 6 BL islands, creates a circular feature in the FS (figure 3(i)) and in the $-50$ meV constant-energy contour (figure 3(j)). The Dirac-cone at $\bar{M}$, marked by the red arrows in figure 3(i), (j) and (l) originates from the 3 BL and/or 5 BL islands. Overall, these features are in good agreement with the calculated FS of 3 BL/5 BL [21] and 6 BL [35] Bi(110) films. Interestingly, we also observe a set of new features that form extremely one-dimensional (1D) FS, running orthogonal to each other, as indicated by the white arrows in figure 3(i). These features are better visualized in the second derivative of spectral intensity, as shown in the supplementary information, figure S1. They were not observed on the Bi(110) surface of a bulk crystal [36] and do not appear in the layer-dependent FS calculations [21, 35]. We also note that they quickly fade away for thicker films (figure 3(q) and (r)). One feasible interpretation would be that these states are the edge-states of Bi(110) islands. Their 1D-like FSs are aligned with the edges of the Bi(110) islands, $\bar{\Gamma}$-$\bar{X}$. Their energy-momentum dispersions as indicated by arrows in figure 3(k) are also in good agreement with the calculated edge-states of Bi(110) ribbons [24, 2]. A direct side-by-side comparison of experimental and theoretical dispersion of these states is further presented in figure 5. The theoretical edge-state calculations for 6 nm wide Bi(110) nano-ribbon with BP-like (panel (a)) and the distorted BP-like structure (panel (c)) [19] are compared to the experimentally observed edge-state dispersion (panel (b)) of 3 BL films. The measured states resemble the edge-states from calculations. However, their relatively low-intensity and the presence of other bands make it difficult to discriminate between the BP-like and distorted BP-like structure of the film. We note that the Bi(110) edge-states were previously observed in the STM measurements [24, 25, 26], but their full energy-momentum dispersion has never been reported before.

The 3.5 BL film (figure 3(m)-(p)), does not show any 4 BL-related features, while the 1D-like features are now suppressed. The spectra show mainly 6 BL (indicated in figure 3(m)) and 5 BL-related features (Dirac cone at $\bar{M}$).

In the case of 5 BL films, the elongated FS segments, marked by “8 BL” in figure 3(q) are not originating from the edge-states, but from the electronic states of 8 BL films, as shown in figure 3(s). The main differences between 3.5 BL and 5 BL films are that the circular feature (figure 3(q) and (r)) and the linearly dispersing bands (figure 3(s) and (t)) around $\bar{\Gamma}$ originate predominantly from the 7 BL islands, not from 6 BL. Furthermore, the presence of intense spots at $\bar{M}$ in figure 3(m) and (q), clearly suggest that the Dirac point in these films are very close to $E_{F}$, in good agreement with the theoretical results (figure 2(j)). The Dirac cone at $\bar{M}$ was also observed at the surface of bulk Bi(110) due to the presence of surface dangling bonds [3]. We also note the 3 BL, 3.5 BL and 5 BL films all show an arc-like feature along the $\bar{X_{1}}-\bar{M}-\bar{X_{2}}$ in the FS (figure 3(i), (m) and (q)) and in the constant-energy contour (figure 3(j), (n) and (r)). This feature is due to the presence of a shallow electron pocket that is also present in the theoretical calculations for 3 BL and 5 BL Bi(110) films [21].

Further, we want to point out that, although the ARPES is generally not sensitive to the edge-state, as the probed volume fraction of the edges is usually extremely small, in the case of Bi/Bi2212, the edge density might just be large enough to generate a detectable photoemission signal. In thinner films, the islands are relatively small, with widths much smaller than the island lengths [18], while in thicker films the islands generally grow bigger and the effective edge density within the ARPES probing region decreases.

As already noted, the theoretical results [24, 2] show that the 2 BL and 4 BL Bi(110) films should host gapped Dirac states along $\bar{\Gamma}-\bar{X}$ $-$ the prerequisite conditions to support QSH states [2]. Our results on 4 BL islands (figure 3(g) and figure 4) suggest that the films are metallic and without a significant gap at the Dirac point. The absence of a gap would suggest that the observed 1D-like states in our experiment might be the edge resonances, rather than in-gap edge-localized states. This would further suggest that it is unlikely that the Bi/Bi2212 system hosts QSH states. However, as the magnitude, the position relative to the Fermi level and the character of the gap are determined by the atomic buckling, lattice parameter, and charge transfer at the film-substrate interface, it is expected that by growing Bi films on appropriate substrates, the Dirac state could be tuned so that it supports a QSH effect [24, 2]. We also expect that an application of hydrostatic or uniaxial pressure could be used for fine tuning of a given system into a QHS state.

It is important that, unlike in Bi films on HOPG, where only even bilayer islands are formed, both even and odd bilayer films can be stabilized on Bi2212. This suggests that the substrate plays a crucial role in stabilizing the nucleation of islands that might be otherwise energetically unfavorable. The electronic fingerprint of the odd-bilayer islands is a Dirac-like band at $\bar{M}$ [6]. Theoretical study shows that this cone is a consequence of the dangling $p_{z}$ orbitals on the surface, the strong spin-orbit coupling and mirror symmetry of the lattice [6]. Thus, experimental observation of this feature confirms the existence of surface dangling bonds which in turn would imply the A7-like (bulk-like) surface structure of the odd-bilayer films.

A direct comparison of experimental data with the BP-like and A7-like structure can be seen in figure S2, suggesting that the surface structure of even-bilayer films is most likely very close to the BP-like. Further, considering the theoretical prediction that the Bi(110) films ($>$ 4 BL) with BP-like structure are topologically trivial and metallic [24, 2] would suggest that the edge-like states detected in our study are likely topologically trivial. The spin-polarised ARPES and STM studies would be helpful in addressing this issue.