Observation of gapped Dirac cones in a two-dimensional Su-Schrieffer-Heeger lattice

The Su-Schrieffer-Heeger (SSH) model in a two-dimensional rectangular lattice features gapless or gapped Dirac cones with topological edge states along specific peripheries. While such a simple model has been recently realized in photonic/acoustic lattices and electric circuits, its material realization in condensed matter systems is still lacking. Here, we study the atomic and electronic structure of a rectangular Si lattice on Ag(001) by angle-resolved photoemission spectroscopy and theoretical calculations. We demonstrate that the Si lattice hosts gapped Dirac cones at the Brillouin zone corners. Our tight-binding analysis reveals that the Dirac bands can be described by a 2D SSH model with anisotropic polarizations. The gap of the Dirac cone is driven by alternative hopping amplitudes in one direction and staggered potential energies in the other one and hosts topological edge states. Our results establish an ideal platform to explore the rich physical properties of the 2D SSH model.

Geng and coworkers have used a combination of experimental (ARPES measurements) and theoretical methods (first-principles and tight-binding calculations) to prove the existence of gapped Dirac cones at the Brillouin zone corners of Si/Ag(001). I have no objection on the longrange order of their system (LEED) and of their ARPES results, except for some minor details that will be mentioned later.
My main concern is that all proof for the existence of Dirac cones comes from the tight-binding model that yields -as each tight-binding model-periodic bands with relatively large gaps between them. In order to be convinced that these calculated bands could indeed correspond to Dirac cones, I would suggest: (i) that the authors give us the tight binding parameters for which the Dirac cones would be gapless and the corresponding band diagrams (ii) to explain how would the parameters of (i) "move" the Si adatoms in a way that would remain consistent with the structure proposed in Fig. 1a I also have some minor comments: (1) The authors wrote that the simulated STM image agrees with previous results. However, there is no reference to those previous results.
(2) The ARPES E-k maps are very clear. Nevertheless, the constant energy contours are less clear. I would suggest that the authors remove -or at least make fainter-the red lines in Fig. 3a that mask the experimental data. Moreover, I think that the author needs more successive contours from 0.6 eV to 1.1 eV that follow the evolution of the contours around M from dot-like to pockets that finally merge with each other. Finally, I suggest to make zoom panels around the M points.
(3) I suggest that the authors include the label of the corresponding high-symmetry point in the band diagrams shown in Fig. 3. The scaling of the horizontal axes is sometimes confusing as k=0.0 refers sometimes to the Gamma and sometimes to the X point.
(4) Looking at the first-principles calculations of Fig. 4a, I would expect that more features would be experimentally seen between the M_Si and the X point. Do the authors have any idea why the feature with the strongest intensity (according to calculations) is not observed by means of ARPES? I refer to the weakly dispersing feature at around 1eV.

Reviewer #3 (Remarks to the Author):
In this manuscript, the authors discuss the electronic properties of rectangular Si lattice on Ag(001) by using ARPES measurement and DFT calculations. The main claim of the manuscript is that the rectangular Si lattice on Ag(001) can be modeled as the anisotropic 2D Su-Schrieffer-Heeger (SSH) model, which might be the first realization of 2D SSH square lattice in Materials Science. The claim relies on the energy band gap observation by using ARPES measurement. The 2D SSH model is one of the recent hot topics in topological materials science. Since it shows the topological phase transition based on Zak phase. In this point, the manuscript treats the timely topic.
However, I think the analysis is relatively weak to scientifically support the evidence of realization of 2D SSH lattice to publish. The problem is following: with ARPES measurement. 2. The authors should discuss more carefully the effect of Ag substrate. At present, almost no data are presented in this aspect. 3. It is very unclear how the tight-binding model is derived from DFT result.
In conclusion, I cannot judge that the manuscript contains enough scientific materials. Thus, I cannot recommend the manuscript for publication.

C1
: Geng et al have reported on the electronic structure of Si on Ag(001). The system is synthesized epitaxially and measured using angle-resolved photoemission spectroscopy (ARPES). It is argued that this system is a realization of a theoretical construct known as the Su-Schrieffer-Heeger (SSH) model, which is a 2D topological system. This is substantiated by comparison to first-principle calculations and tightbinding calculations.

Reply:
We thank the reviewer for the in-depth review of our paper. The reviewer accurately summarized the main messages of our work.
C2: This is a nicely written paper, which walks the reader through the background of the SSH model and the intricacies of the Si/Ag(001) system. In principle it has the necessary novelty for Nature Communications, as the experimental realization of a model system is of fundamental significance. However, I have some concerns about the interpretations which should be clarified before I can recommend publication. In particular, the experiment itself is not decisive, since the topological signatures (edge states) are not observed. Therefore, the mapping onto the SSH model is based on the 2D band dispersions, which require further clarification. I elaborate below.

Reply:
We thank reviewer for the high evaluation of our work. We agree that "the mapping onto the SSH model is based on the 2D band dispersions, which require further clarification". In the following, we will address all the comments one by one. (Fig 3) the authors conclude that the measured bands are associated with the Si layer. However, when comparing Si/Ag(001) to pristine Ag(001) [e.g. panels (e)/(f) and (g)/(h)] the bands look qualitatively the same, with the only difference being a rigid shift of the band structure. Is it possible that only the Ag bands are relevant, with a rigid shift due to charge transfer from the Si? Reply:

C3. From the ARPES measurements
We thank the reviewer for the comment. The reviewer doubt that the Dirac-like bands might originate from the bulk bands of Ag, with a charge transfer induced energy shift. Here, we can exclude the bulk bands of Ag based on the following arguments: 1) Experimental identification of the Dirac-like bands as the Si-derived bands comes not only from the E-k dispersion in Fig. 3(e)-(j), but also from the constant energy contours (CECs). The Dirac-like bands behave as a closed pocket at certain binding energies, as shown in Fig. R1 we found that these bands have no apparent shift after the growth of Si. 3) As far as we know, it is difficult to shift the bulk bands by surface charge doping, especially for good conductors such as Ag. Surface doping typically has negligible effects compared to the large amounts of carriers in bulk materials, which can explain the absence of detectable energy shift of the Ag bulk bands. In addition, Si has a higher electron affinity than Ag, and therefore electrons will transfer from Ag to Si, leading to an upwards band bending of the Ag bands. However, panels (e)/(f) and (g)/(h) seem to suggest an opposite direction, although an opposite energy shift cannot explain the Dirac-like bands either. 4) The reviewer might notice the similarity of panels (e)/(f) and (g)/(h) with a rigid energy shift. First, we discuss (e) and (f). Apparently, the bulk bands of Ag do not move after the growth of Si, as indicated by the blue arrows. The variation of the spectral intensity of the Ag bulk bands originates from the photoemission matrix element effects and only E-k dispersions provide useful information here. Second, we discuss (g) and (h). To directly compare these bands, we superimposed the two bands together, with a rigid energy shift of the (h) towards higher binding energies, as shown in Fig. R2. There are three apparent differences: (1) the slopes of the Vshaped bands at the center; (2) the bandwidths; (3) the absence of downward dispersing bands beyond the M point in pristine Ag(001). Therefore, panels (e)/(f) and (g)/(h) cannot be explained by a rigid energy shift either.
Based on the above reasons, we can unambiguously exclude Ag bands as the origin of the Dirac-like bands.
Following the reviewer's comment, we added a sentence in the last paragraph of Section 2.2 of the main text: "Third, these bands behave as closed pockets at specific binding energies [see Fig. 3 Fig. 3   C4. Following up on this point: qualitatively, the strongest effect of Si deposition is to change the symmetry due to 3x3 construction. This is quite clear in the LEED data (Fig 2a). This should have a dramatic effect in the ARPES spectra by reducing the Brillouin zone size, with associated folding. However, the measured symmetries appear fully consistent with the Brillouin zone of pristine Ag(001). This again makes me skeptical about any role of the Si in the ARPES spectra.

Reply:
We thank the reviewer for the valuable comment. We agree with the reviewer that "the measured symmetries appear fully consistent with the Brillouin zone of pristine Ag(001)". However, this does not contradict the symmetry of the Si lattice since each Dirac cone is centered at an M point of Si. The main concern of the reviewer might be the weak spectral weight in the first BZ of Si. In our opinion, there are several reasons: 1) Our DFT calculations including the substrate agree well with experimental results.
As shown in Fig 2) Stronger photoemission intensity in the higher-order BZ is very common because of the matrix element effect. The bands in the first BZ might be visible with different experimental setups, such as photon energies and incident angles. 3) Since the Si overlayer was grounded via the Ag(001) substrate, the emitted photoelectrons were compensated by the itinerant electrons from the substrate during the photoemission process. For Ag(001), the bulk sp bands serve as the electron source within ~3 eV of the Fermi level. The transition rates of electrons to nearby electronic states are higher because of the smaller change of k||, which might result in a larger photoemission cross-section of these bands. Therefore, the Si bands that are close to the bulk bands of Ag have stronger spectral weights. Similar effects have been observed in other surface systems, such as the 3×3 silicene/Ag(111) [PRL 122, 196801 (2019)].
In conclusion, despite of the absence of spectral weight at some M points of Si, we can still conclude that these bands are from Si layer, instead of the Ag (001) substrate.
To clarify this issue, we made the following changes in the main text: 1) We deleted the sentence "Notably, only bands at the second BZ are observable, which might originate from the matrix element effects in photoemission experiments." in second to the last paragraph of Section 2.2. 2) We added a new paragraph in the end of Section 2.2: "Notably, the Si-derived bands are only observable in the second BZ of Si in our ARPES measurements. This is a common phenomenon in photoemission experiments because of the matrix element effect. In addition, the bands in the second BZ of Si are closer to the bulk bands of Ag (001), and the transition rates of electrons from Ag (001) to these electronic states are much higher during the photoemission process, resulting in the stronger spectral weight. A similar phenomenon has been observed in (3×3)-silicene on Ag (111). " 3) We added a sentence "The bands near the ̅ point of Ag(001) have higher spectral weight, in agreement with our ARPES measurement results." in the first paragraph of Section 2.3.

C5.
The authors seem to argue that the relevant states are primarily Si-derived. But the Ag states may also be folded by the superstructure. Therefore, the calculated band structure (Fig 4a)  We thank the reviewer for the suggestion. Following the reviewer's request, we calculated the band structure of pristine Ag(001) and the unfolded band structure of reconstructed Ag(001) without Si, as shown in Fig. R4. Obviously, no "M"-shaped bands exist, in contrast with the DFT calculation and ARPES measurement results. Therefore, neither pristine Ag(001) nor reconstructed Ag (001) can realize the 2D SSH model. Based on the orbital projected band structures in Supplementary Fig. 5, we find that the Dirac bands are mainly contributed by the Si atoms. This further confirms that the 2D SSH model is realized by the Si atoms.
Following the reviewer's suggestion, we added Fig. R4 to the Supplementary  Information (Supplementary Fig. 4). We also added a sentence "In addition, neither pristine nor 3×3 reconstructed Ag(001) can reproduce the experimental results, as shown in Supplemental Fig. 4." in Page 6 of the main text.

Reply:
We thank the reviewer for the valuable suggestion. In the revised manuscript, we made the following changes: 1) We modified Fig. 4. The DFT and TB band structures along the same momentum cuts are shown in Fig. 4(a) and (c) for a side-by-side comparison. One can see a good agreement. The Fermi levels are different because our TB parameters are fitted to experimental results. 2) The TB band structures along all high-symmetry directions are shown in Fig. 4(d).

Reply:
We thank the reviewer for the comments. To make our arguments more convincing, we calculated the edge spectrum by both TB and DFT, and the results are shown in Fig. R5. From the TB calculations in Fig. R5(a) and (b), the nontrivial edge has two more edge states in the proximity of the Fermi level than the trivial edge. The DFT calculation results are much more complicated than the TB calculation results because more edge and bulk bands exist. However, the DFT calculated edge spectrum of the nontrivial edge has two more edge states than that of the trivial edge, as indicated by the white arrows in Fig. R5(a). The shapes of these two bands agree well with our TB calculation results. Therefore, our DFT calculations fully support the topological nature of the 2D SSH model.
In the revised manuscript, we added Fig. R5 to the Supplementary Information ( Supplementary Fig. 8). We also added a sentence "The existence of topological edge states is also confirmed by first-principles calculations, as shown in Supplementary  Fig. 8." during the discussion of the edge states.

Reply:
We thank the reviewer for the comment. In our opinion, the Tamm surface states of Ag(001) do not contribute to the Dirac-like bands in our ARPES measurements. There are three reasons. (1) The Tamm surface states of Ag(001) exist at the ̅ of Ag(001). These states are split from bulk d bands by the surface potential and are located at a binding energy of ~3.7 eV, which is much higher compared to the energy range of our experiment (within 3 eV of the Fermi level). (2) To the best of our knowledge, the Tamm states are very sensitive to surface potential distortion, e.g., adsorbates. Therefore, the Tamm states will completely disappear after growth of Si.
(3) Our DFT calculation results show that the Dirac-like bands are mainly contributed by the Si atoms, instead of the Ag atoms.
In conclusion, the Tamm states do not contribute to the Dirac bands of the Si/Ag(001) system. References: [Surf. Sci. 122, L629-L634 (1982)] [Phys. Rev. B 32, 4956 (1985)] [Surf. Sci. 178, 300-310 (1986)] [Solid State Commun. 67, 163-167 (1988)] [Phys. Rev. B 91, 125435 (2015)] [Phys. Rev. B 105, L241412 (2022)] C9. In general, the plotting of the first-principles calculations is unclear. This is partially due to the image resolution, but perhaps also due to the plotting style. Is it possible to plot in a way that makes the band dispersions more visible? Also, what do the marker sizes and colors represent? Reply: We thank the reviewer for the suggestion to help improve the data presentation. We replot our first-principles calculation results in green color and with 1.5 times of weight, as shown in Fig. 4(a) of the revised manuscript.
In the following, we explain the meaning of the marker size and color. The unfolding of the band structure gives a spectral function (not ( ) dispersion relation): ( , ) = ∑ ( ; , ) ( − ( , )) , where ( ; , ) = ∑ ⟨ , | , ⟩ ⟨ , | , ⟩ , , and , are the supercell (SC) and primitive cell (PC) wavefunction, respectively, N and n are the band index of SC and PC, respectively, and K and = + (G is the reciprocal vector of SC) are the momentum of SC and PC, respectively. Therefore, the marker size and color represent the weight that comes from the projection of SC wavefunction to the BZ of the PC. The marker size and color are related to enhancing the primary electronic bands. That is, the larger the size, the darker the color. When these bands are projected to the orbitals of specific atoms, an extra weight function W (N, K), i.e., the projected density of states of certain orbitals, should be included in the spectral function: ( , ) = ∑ ( ; , ) ( , ) ( − ( , )). The marker size and color represent the product of the weight that comes from the projection of the SC wavefunction to the BZ of the PC, and the weight comes from the projection of the SC wavefunction to the orbitals of specific atoms.
In the revised manuscript, we added a sentence "The marker size and color represent the spectral weight that comes from the projection of the supercell wavefunction to the BZ of the primitive cell." to the caption of Fig. 4. (first-principles and tight-binding calculations) to prove the existence of gapped Dirac cones at the Brillouin zone corners of Si/Ag(001). I have no objection on the long-range order of their system (LEED) and of their ARPES results, except for some minor details that will be mentioned later.

Reply:
We thank the reviewer for the in-depth review and high evaluation of our work. In the following, we will address all the comments one by one.

C2:
My main concern is that all proof for the existence of Dirac cones comes from the tight-binding model that yields -as each tight-binding model-periodic bands with relatively large gaps between them. In order to be convinced that these calculated bands could indeed correspond to Dirac cones, I would suggest: (i) that the authors give us the tight binding parameters for which the Dirac cones would be gapless and the corresponding band diagrams (ii) to explain how would the parameters of (i) "move" the Si adatoms in a way that would remain consistent with the structure proposed in Fig. 1a Reply: We thank the reviewer for the suggestions. (i) To obtain a gapless Dirac cone, we set the parameters: 1 = 3 = 0.25 eV, 1 = 2 = 0.45 eV, and 2 = 2 = 1.15 eV. With these parameters, the hopping integrals in either x and y directions are the same, and the on-site potentials of the two Si atoms in either x and y direction are also the same. These parameters will give rise to a gapless Dirac cone, as shown in Fig. R6. Since the hopping integrals in the x direction are different from those of the y direction, the Dirac cone is anisotropic, i.e., the Fermi velocities are different along the x and y directions.
(ii) Consider a 1D SSH model. A gapless Dirac cone emerges when both the hopping integrals and on-site energies are uniform. A 2D SSH model can be viewed as two 1D SSH models in the x and y directions, respectively. To obtain a gapless Dirac cone in the 2D SSH model, both 1D SSH models should have a gapless Dirac cone with the Dirac point at the same binding energy. That is to say, along each direction, the hopping integrals should be equal, as well as the on-site energies. A special case is that the hopping integrals and on-site energies along the x direction are the same as those along the y direction, respectively, which is the model proposed in Fig. 1(a). In that case, a gapless Dirac cone with isotropic Fermi velocity will emerge, as shown in Fig. 1(b) of the main text. When the hopping integrals are different along the x and y directions, the Dirac cone will become anisotropic, as shown in Fig. R6. In the Si/Ag(001) system, the alternate hopping integrals and staggered on-site energies result in an anisotropic and gapped Dirac cone.
Following the reviewer's suggestion, we added a new paragraph in Page 8 of the main text: "Consider a 1D SSH model. A gapless Dirac cone emerges when both the hopping integrals and on-site energies are uniform. A 2D SSH model can be viewed as two 1D SSH models in the x and y directions. When each 1D SSH model has a gapless Dirac cone with the Dirac points at the same binding energy, a gapless Dirac cone will emerge in the 2D SSH model. A special case is that the hopping integrals and on-site energies along the x direction are the same as those along the y direction, respectively, which is the model proposed in Fig. 1(a). In that case, a gapless Dirac cone with isotropic Fermi velocity will emerge, as shown in Fig. 1(b). When the hopping integrals are different along the x and y directions, the Dirac cone will become anisotropic, as shown in Supplementary Fig. 6. In the Si/Ag(001) system, the alternate hopping integrals and the staggered on-site energies results in an anisotropic and gapped Dirac cone." We also added Fig. R6 to the Supplementary  Information (Supplementary Fig. 6).

C3: I also have some minor comments:
(1) The authors wrote that the simulated STM image agrees with previous results. However, there is no reference to those previous results.

Reply:
We thank the reviewer for pointing out our carelessness. The reference should be Ref. [30]. We have cited this paper here in the revised manuscript.

C4:
(1) The ARPES E-k maps are very clear. Nevertheless, the constant energy contours are less clear. I would suggest that the authors remove -or at least make fainter-the red lines in Fig. 3a that mask the experimental data. Moreover, I think that the author needs more successive contours from 0.6 eV to 1.1 eV that follow the evolution of the contours around M from dot-like to pockets that finally merge with each other. Finally, I suggest to make zoom panels around the M points.

Reply:
We thank the reviewer for the suggestions to improve the data presentation. We followed the reviewer's suggestion and made the following changes: 1) In Fig. 3(a), we changed the solid lines to dashed lines; 2) The thicknesses of both black and red lines in Fig. 3(a)-(d) are reduced. We did not remove the red lines because they provide guides for the eye for the E-k maps in Fig. 3(e)-(j). Supplementary Information (Supplementary Fig. 3) to show the evolution of zoomed-in constant energy contours (CECs) from 0.5 to 1.2 eV. The figure is duplicated as Fig. R7. All the red lines were removed. C5: I suggest that the authors include the label of the corresponding high-symmetry point in the band diagrams shown in Fig. 3. The scaling of the horizontal axes is sometimes confusing as k=0.0 refers sometimes to the Gamma and sometimes to the X point.

Reply:
We thank the reviewer for the suggestion. We added labels of high-symmetry points of Si in the revised Fig. 3(e,g,i,j). Figure 3(f) and (h) were not modified because pristine Ag(001) does not have these high symmetry points.

C6:
Looking at the first-principles calculations of Fig. 4a, I would expect that more features would be experimentally seen between the M_Si and the X point. Do the authors have any idea why the feature with the strongest intensity (according to calculations) is not observed by means of ARPES? I refer to the weakly dispersing feature at around 1 eV.

Reply:
We thank the reviewer for the valuable comment, which helped us to corrected a mistake.
After a careful check of the DFT calculation results, we found that the position of the blue dashed lines (guides for the eye) should be readjusted, as shown in Fig. R8(a) and (b). The feature with the strongest intensity corresponds to the Dirac-like bands that are observed by ARPES. The feature with weaker intensity, as indicated by the blue arrows in Fig. R8(b), might originate from the finite thickness of our slab calculations since this feature seems parallel with the strongest feature.
To further confirm the gapped Dirac cones in DFT calculations, we present the unfolded band structures along Cuts 1-4 that were measured by ARPES (Fig. 3 of the main text), as shown in Fig. R9. One can see that the calculated bands agree well with our ARPES measurements except that the calculated Fermi level should be slightly adjusted and the bands is a little deformed. The slight inconsistence may be caused by the following reasons. (1) The finite thickness in our DFT calculations because of the limited calculation resources; (2) A surface adsorption system is rather complex and the large number of atoms in our calculations may make the error larger. However, the qualitative agreement between our DFT calculation and ARPES measurement results can still provide compelling evidence for the existence of gapped Dirac cones in this system.
In the revised manuscript, we have corrected the blue dashed lines in Fig. 4 and Supplementary Fig. 5. In addition, we modified the first and second paragraphs in Section 2.3 to explain this issue (see the revised manuscript). We sincerely thank the reviewer for the valuable comment, which helped us to avoid mistakes.