Mottness versus unit-cell doubling as the driver of the insulating state in 1T-TaS2

If a material with an odd number of electrons per unit-cell is insulating, Mott localisation may be invoked as an explanation. This is widely accepted for the layered compound 1T-TaS2, which has a low-temperature insulating phase comprising charge order clusters with 13 unpaired orbitals each. But if the stacking of layers doubles the unit-cell to include an even number of orbitals, the nature of the insulating state is ambiguous. Here, scanning tunnelling microscopy reveals two distinct terminations of the charge order in 1T-TaS2, the sign of such a double-layer stacking pattern. However, spectroscopy at both terminations allows us to disentangle unit-cell doubling effects and determine that Mott localisation alone can drive gap formation. We also observe the collapse of Mottness at an extrinsically re-stacked termination, demonstrating that the microscopic mechanism of insulator-metal transitions lies in degrees of freedom of inter-layer stacking.

I suggest briefly commenting these questions in the text if possible. 4. It is not clear from the present density plot of the dI/dV curves whether the gap is present very close to the step? I suggest showing the raw curves in the supplement.
Small/technical comments: 5. It would be better to use easily distinguishable colors for the atomic and cluster unit cells in Fig.  1b. 6. I suggest the T_C = a + b + c notation to be dubbed with T_C = 2a + c to be consistent with a number of the previous theoretical studies. 7. I would like to bring the authors' attention to the following three papers dealing with the three- This paper discusses STM/STS measurements on a new cleaving plane of bulk 1T-TaS2. Through STM imaging of a multiple-step structure on the TaS2 surface, the authors clearly identify three types of CDW stacking orders whose STS spectra yield very different electronic structure. The important observation here is that when the cleavage occurs between an on-top-stacked bilayer (BL) the STS still has a gap at EF. The authors view this split BL as an "unpaired" layer and claim that it should not have a gap without invoking Coulomb repulsion (i.e., Mott physics). The STM/STS data are of high quality and this is a very interesting observation; however, I find the interpretation not convincing enough to justify publication at this point, as described below.
The crux of the authors' argument is an unpaired surface layer hosting an odd number of electrons per unit cell which allows trivial gap opening mechanisms to be ruled out. However, this argument is not so convincing since the unpaired top layer has finite interlayer coupling to the layers underneath and can't be simply viewed as a decoupled layer. I believe it is important for the authors to show better evidence that for this stacking (and in the absence of correlation) there would indeed be a metallic surface state with an odd electron filling even though some interlayer coupling still exists. For a Mott-Hubbard gap to open this metallic surface state band has to have a small bandwidth and be half-filled, and the authors should provide more evidence that these conditions are fulfilled. Can the authors perform DFT simulations of this structure using a finite slab model to investigate the surface electronic properties? The problem here is that ARPES measurements and simulations have shown dispersion in the z-direction (PRB 96, 195147(2017), PRB 90, 045134 (2014)) and theoretical calculation of bulk "Tc-stacked" TaS2 have also exhibited metallic behavior with a ~400 meV bandwidth (PRL 122, 106404 (2019)). Both of these results suggest that there can be significant coupling and significant bandwidth for Tc-stacked layers, which goes against the authors' interpretation of a decoupled top layer with very narrow bandwidth (i.e., narrow enough to drive a Mott transition). I feel that this paper shows some very beautiful data, but that this issue must be addressed a little more thoroughly in order to support the "Mottness" claim mentioned in the paper's title.

Reply Letter
We sincerely thank both Referees for their insightful and helpful comments. Below we describe our revisions to the manuscript in response to these comments, including additional experimental results which were suggested. Throughout this Reply Letter we reproduce some text from the Referees' remarks and highlight these in red, and where we make additions or revisions to the text of the manuscript, those are highlighted in blue. This Reply Letter is followed by a comprehensive List of Changes to the manuscript, where the revisions and additions are listed (roughly) in order of their appearance in the manuscript.

Reply to Referee 1
First we thank the Referee for recognizing the novelty and significance of the current experimental results. The Referee's main concern, expressed in two parts, is that STM is strictly a surface measurement. Part 1: 1a) The authors can only guess the David star positions in the layers below the one they image.
The ambiguity originates from the finite correlation length (~3 to 10

Additional atomically resolved topographic imaging in Region 1 (see Supplementary Information) is used to determine the orientation of the SD clusters, …"
The above revision is listed as Item 8 in the List of Changes.
Additionally, it is important to distinguish between the in-plane domain walls, the 1D defects within a given layer, and the 'stacking faults' which the Referee mentions (described by Ritschel However, we now recognize it is worthwhile to give this disorder a thorough description. We add a short paragraph and the following figure (Fig. R1) in our Supplemental Information (in order to preserve the readability of the main text). This largely follows the comprehensive description given by Ritschel et al. [PRB 98, 195134 (2018)]. We also add a line in our main text to direct the reader to this information, and suitable citations of recent works discussing the dimerization and disorder. These are Items 4 & 16 in the List of Changes.
1b) The tunneling current can reach one layer beneath, as has been reported before and also stated by the authors when discussing the buried domain wall in Suppl. Fig. 1 It is probably not best to think that the STM can probe one layer beneath the uppermost one.
Rather, it is better to think that the STM may detect the influence, on the uppermost layer, of coupling to an underlying layer -i.e. the manifestation of the underlying layers in the measurement is indirect and the total signal is usually not well modelled as a linear combination or convolution of independent layer contributions. Why not? In STM, a good rule of thumb is that the tunneling probability falls off by about one order of magnitude for every Angstrom by which the tip-sample distance increases. This means that the second layer of a material with an interlayer spacing of ~6 Å yields a vanishingly small contribution to tunneling spectra. Therefore, we shouldn't expect the density of states of the underlying layer to appear in our dI/dV(V) curves in linear combination with that of the uppermost one.
Still, it is a good precaution to check the dependence of spectra on the tip-sample distance, as the Referee suggests. Figure R2 below shows that there is no significant dependence of the I(V) or dI/dV(V) curves on the tip-sample separation, for either of the two regularly observed surface terminations. The full detail can be found in the discussion newly added to the Supplemental Information. Please see Item 19 in the List of Changes. (Note: The sharp conductance peaks which appear in the Figure below, at the onset of the UHB at each surface, will be discussed in detail in an upcoming manuscript (in preparation).) Figure R2. Tip-height-dependence of dI/dV(V) and I(V) spectra. STM topography maps acquired at surfaces of paired (a) and unpaired (b) clusters (scale bar 1 nm). Spectra were collected in the center of SD clusters (at locations marked by the yellow dots) and sufficiently far from impurities that spectra were cluster-independent. (c)-(f) dI/dV(V) spectra, and also I(V) spectra, obtained at z set + z offset . (Here z set is the starting (set-point) tip height using V = 0.25 V, I set = 1.0 nA for (c) and V = 0.11 V, I set = 0.44 nA for (d), yielding a set-point tunnel resistance R gap = 0.25 GΩ in each case.) For the dI/dV curves in panels (a) and (b) V mod = 1 mV, and therefore the apparent increase in the gap size as compared to those shown in Figure 1(e) of the main manuscript is probably due to the narrower energy resolution function for the lock-in detection technique. The sharp peaks in conductance at the onset of the UHB will be discussed elsewhere.
The Referee points out that there is noticeable in-gap density of states for the apparent 'largegap' spectrum shown in Region III in Figure S1(d). This type of spectrum was seen in two domains found at two separate cleaved surfaces of only one 1T-TaS 2 platelet. Based on the relative rarity of this observation (found only locally and in only 2 out of 24 cleaved surfaces), and also on the tip-height dependence measurements shown above, we speculate that this ingap density of states is not due to tip-height effects, but instead may be due to the proximity below the surface of one of the inter-layer stacking faults described above, or some other type of stacking fault.
The tip-height-dependent data ( Figure R2  The Referee's point is well taken. The absolute distance between cluster sites is a necessary but not sufficient degree of freedom to model the situation. We understand that the orbital involved, located at the cluster center, is probably Ta 5d z 2 (PRB 96, 125138 (2017), for example), with lobes pointing out-of-plane, and the inter-layer orbital overlap should be very sensitive to the inter-layer registry, as well as the distance between sites. So a complete description must make the distinction between the intra-and inter-layer hoppings t ∥ and t ⊥ .
Probably the inter-layer hopping amplitude t ⊥ is very sensitive to the lateral displacement in stacking between layers, while t ∥ is not. (Domain walls might be assumed to strongly modify the local t ∥ , but we do not comment further on the electronic structure of domain walls in this work.) In light of this, we have re-written the relevant part of our discussion to frame the issue more carefully, and actually without explicitly referring to U and t parameters of the Hubbard model.
Revised version (repeated as Item 11 in the List of Changes): "The apparent role of Mottness in the unpaired SD layer also suggests that the metallic state also observed (Region 2 in Fig. 2)  In Figure S1(b) we show a map of in-plane displacement of the CDW for the field-of-view including both large-and small-gap domains. In this map, displacements with respect to an ideal reference lattice are expressed in polar coordinates as color (hue for direction and saturation for magnitude We thank the Referee for this observation. Below ( Figure R3), we show both the raw and normalized dI/dV curves associated with the dI/dV(x,V) plot in Figure 2(b). Each curve corresponds to one pixel along the long axis of the red-tinted rectangle in Figure 2(b), but is itself the average over the column of pixels spanning the short axis. This figure makes it clearer exactly where the Mott gap breaks down near the domain wall and steps. As is consistent with previous works, we see that the variation in spectra across the domain wall occurs over a length scale of ~3 nm (or ~3 CDW lattice spacings).
A short note has been added to the existing discussion in the Supplemental Information to draw attention to the above point. See items 7 and 21 in the List of Changes. 3.3 nm, and 3.4 nm, respectively, although these are overestimates due to the fact that the linecut is not perpendicular to the domain wall or steps.

It would be better to use easily distinguishable colors for the atomic and cluster unit cells in
The colors have been changed in the new version of Figure 1(b). The corresponding change has also been applied to the inset of Figure 1(d), and also to the similar diagrams in Figure S3.
Please see item 5 in the List of Changes. 6. I suggest the T_C = a + b + c notation to be dubbed with T_C = 2a + c to be consistent with a number of the previous theoretical studies.
Our introduction to the stacking vectors T A,B,C is now consistent with the established notation in previous literature. Specifically, we adopt the following expressions for the symmetrically distinct We have also added depictions of the atomic lattice vectors to Figure 1( Rep 7, 46048 (2017) We thank the Referee for pointing out these papers. Citations to the papers have been incorporated into the introduction section, and also in the discussion of Mottness at the 'unpaired' surface (see Reply to Referee 2 below). Please see both Items 1 and 4 in the List of  We thank the Referee for these insightful and important criticisms. 106404, show that the shifted-bilayer stacking (AL stacking in the notation of Lee et al., but 'ACAC' stacking in ours) substantially reduces the band width from ~400 meV down to 200~300 meV, and that a gap opens in absence of U. And Lee et al. specifically attribute the difference to the T C -stacking interfaces within the ACAC bulk structure. Therefore, it is reasonable that the inter-layer hopping between our T C -stacked ('unpaired') layer and the underlying bilayer may be weak enough that it may be considered as a somewhat decoupled layer amenable to analysis in isolation.
On metallicity of the unpaired layer in the absence of correlations: DFT calculations seem to be the only way to directly demonstrate that the system would be metallic without correlations, as we can't escape correlation effects in the real (experimental) world. However, we can argue that if the bulk bilayer stacking can be thought of as dimerized We have revised and added to the discussion of these points in the main text. Although it is not possible to draw a decisive conclusion, we have made an effort to make our reasoning as clear as possible to the reader. Please see item 10 in the List of Changes.
Can the authors perform DFT simulations of this structure using a finite slab model to investigate the surface electronic properties?
The large in-plane extent of the CDW unit cell poses a serious challenge: Each SD cluster contains 39 atoms (including sulfur), and a bulk unit-cell representing the 'ACAC' or any 'bilayer' stacking pattern contains 78 atoms, which is still manageable. Such bilayer bulk unit-cells were investigated using state-of-the-art techniques [Nature Physics 11, 328-331 (2015), PRB 87, been reported for a slab model including surfaces and a vacuum layer, beyond only a single isolated bilayer [PRB 90, 045134 (2014)]. This may be because a slab with a thickness of probably at least 5 bilayer unit-cells would be needed, at the bare minimum, which would already include 390 atoms. According to our correspondences with colleagues performing stateof-the-art DFT calculations, such calculations are "not impossible", but are very daunting, and probably represent a substantial project in their own right. Therefore, we leave this as a challenge to be tackled in the future by computational colleagues.
Replies to additional comments:  Fig. 2b?
The tip-induced band-bending (TIBB) effect occurs due to the electric field from the tip penetrating into an insulating sample. The outcome is that spectroscopic features are lifted to higher energies (or sunken to lower energies) while probing unoccupied (occupied) sample states. The energy shifts monotonically follow the bias (with the same sign), but of course are zero for zero bias, so are still small while detecting a band onset quite near to E F . We first point out that our dI/dV curves show that at both terminations of 1T-TaS 2 , the onset of occupied states (lower edge of the Mott gap) is only ~10 to 20 meV below the Fermi level, so that the TIBB is very weak at the point that the onset of these occupied states is detected. Therefore, the measured energy of this lower onset should be fairly accurate.
Furthermore, when measuring the onset of the UHB, the positive bias used would be expected to induce a rigid bending of the density of states to higher energies. This would soon bring the finite density of states from -10~20 meV upwards to the Fermi level, quickly resulting in effective screening of the tip's electric field by holes. For this combination of reasons, we don't anticipate a large TIBB effect which would impact any of the conclusion drawn in this manuscript.
It is better to demonstrate this experimentally, and we have performed the suitable measurements to do this, but these measurements yield somewhat anomalous results. In the well-established framework due to Feenstra, as well as being related to the bias, the size of the energy shift also depends on the tip-sample distance. Any shift is enhanced when the tip is close to the sample, so that for energies above E F , any particular spectroscopic feature with energy E 0 will appear at higher energy E 0 ' when the tip-sample distance decreases. i.e. dE 0 '/dz offset < 0. In principle, this effect should be observed in the additional tip-height-dependent conductance spectra we present in Figure R2 above. Here is it helpful to zoom in to the UHB edge, as shown below in Figure R4 below, where very sharp conductance peaks (to be discussed in detail in a forthcoming manuscript) allow us to track dE 0 '/dz offset . We find that in fact dE 0 '/dz offset > 0, which is not amenable to interpretation in the usual TIBB model established by Feenstra. Figure R4. Tip-height-dependence of conductance spectra. Conductance peaks at the UHB onset for paired (a) and unpaired (b) clusters, collected as described for Figure R1 above. In each case, the apex of the conductance peak, observed at the energy E 0 ' clearly shifts towards higher energy as the tip-sample distance increases, i.e. dE 0 '/dz offset > 0. This cannot be interpreted in terms of the usual TIBB model described by Feenstra [J. Vac. Sci. Technol. B 5, 923 (1987) and Nanotechnology 18, 044015 (2006)].
The above indicates that even if the usual band bending mechanism is in effect, another (as yet unknown) mechanism overrides it, and unfortunately the intrinsic gap size cannot be determined with greater precision at this time.
We have added a new section in our Supplementary Information to discuss these points, including the suggested citation (as well as a few more). For this, please see Item 19 in the List of Changes. Fig. 2b?

Also, how large is the lateral band bending effect in different domains in
It is well established that ordinary domain walls in 1T-TaS 2 can cause lateral band bending due to remaining charge density which we may think of as left over from partial SD clusters. It is reasonable to expect that step edges also host such remnant charge density. However, no obvious lateral band bend is seen near either of the step edges in Figure 2 In our experimental procedure, the samples were cooled to T = 77 K (far below the transition temperature into the C-CDW phase) for cleavage and, importantly, the head of the transfer rod which was subsequently used to insert samples into the STM was cooled along with them. This ensures that the cleaved surface retains information about the pre-formed bulk stacking order.
In almost all previous STM work which show dI/dV data which can be compared against our own, the temperature of sample cleavage was either reported to be room temperature [Nat.
Communs Moreover, even in our experiments, where cleavage through the pre-formed bulk charge order was achieved, the Type I spectrum was still seen for most cleaved surfaces. (Indeed, near the beginning of our own measurements, the first observation of the small gap surface (shown in Figure S1) was dismissed as an anomaly. Only later was it found to be robustly reproducible.) We have made an effort to further emphasize the importance of the cleavage and transfer temperatures in our main text, and in our Methods section, citing the works mentioned above.
Please see item 13 in the List of Changes.

List of Changes
Below, revisions are listed in order of their appearance in the manuscript. Please note that there are some revisions which are not prompted by the Referees' suggestions, but by correspondence with other colleagues, formatting considerations, the appearance of a new relevant manuscript, etc. The List of Changes has two parts: Main manuscript and Supplemental Information.

Main manuscript:
1) In accordance with Nature Communications format, the Introduction has been re-arranged.
The former introductory paragraph, minus citations and abbreviations, is now the Abstract. The first paragraph of the introduction is now as follows: "The origin of the spectral gap in many insulating materials is difficult to determine because, as well as band theoretic considerations such as the degree of band filling, electron-phonon interactions, strong electronic correlations [1,2] and other mechanisms generally can coexist and may all play some role. This is true in the decades-old chargedensity-wave compound 1T-TaS 2 , for which the debate over the nature of the lowtemperature insulating state has only intensified in recent years [3][4][5][6][7]. Though the proximate cause of this insulating state is under debate, its precursor is known to be an electron-phonon driven commensurate charge-density-wave (C-CDW) phase." The references used above are as follows: 2) An additional reference has been added in support of 1T-TaS 2 as a quantum spin liquid candidate. This is now reference Z: 3) The notation used to describe the possible stacking vectors has been changed:

Previous version:
"Neglecting the S atomic layers sandwiching the Ta layer, there are only three symmetrically inequivalent stacking vectors: T A = c, T B = a + c, and T C = a + b + c (with the latter two each having a group of symmetrical equivalents)."

Revised version:
"There are five symmetrically inequivalent stacking vectors, which may be collected into only three groups according to their length: T A = c, T B = ±a + c , T C = ±2a + c (or equivalently, ∓2a ∓b + c)." The corresponding change has also been applied to the later mention of the planar projection of T C when discussing the interpretation of Fig. 3c: "… we note that single-layer steps should result in an in-plane displacement, or phase jump ΔΦ, of the 2D projected CDW pattern from one terrace to the next, which should alternate between zero and non-zero (specifically ±2a, or equivalently, ∓2a ∓ b, the inplane projection of T C ), as is shown in Fig. 3a." 4) The introduction of the proposed 'dimerized' or 'bilayer' stacking structure has been expanded (new text is underlined) with additional references:

"… Ritschel et al and Lee et al recently challenged the rationale by which 1T-TaS 2 was thought to be a Mott insulator, showing that if the stacking alternates between vectors T A
and T C as previously suggested [18][19][20], such that the new supercell includes two SD clusters, ab initio calculations predict an insulator without the need to invoke strong e-e interactions [5,7]. (It has been established that the bulk stacking structure likely alternates between T A and a vector drawn randomly from three versions of T C related by rotations of 120°, in a partially disordered pattern -see Supplementary Information. The 'dimerization' of the stacking structure into bilayers, and the disorder, have also been discussed in the interpretation of recent experimental works [21,22].) Put simply, if the electronic unit cells contains two SDs …" New references:  "The fact that this surface was the most common outcome from cleavage (18 out of 24) indicates the energetic favourability of cleaving between BLs, rather than through them, suggesting non-negligible intra-BL bonding, or dimerisation."

"The surface of unpaired SD clusters (Regions 1 & 4) represents a new and perhaps
qualitatively distinct system, which may allow us to disentangle the role of strong e-e interactions from that of unit-cell doubling in the electronic structure. First, we note that if the BL-stacked bulk charge order can be considered as dimerised, then breaking the dimerisation by terminating the structure with a layer of unpaired clusters may ordinarily be expected to leave a metallic `dangling bond' surface state, as is the case at the wellknown Si(111)-7×7 surface.
As opposed to previously suggested structure of purely T A -stacked SD cluster, which yields a large out-of-plane bandwidth and a one-dimensional out-of-plane metal [3,4], the ACAC stacking pattern realised here has been predicted to result in suppression of the out-of-plane bandwidth which can be attributed specifically to the T C stacking interfaces [7]. In this case, the T C -stacked, unpaired surface layer here may be sufficiently "The apparent role of Mottness in the unpaired SD layer also suggests that the metallic state also observed (Region 2 in Fig. 2)

12)
We found through discussions with other colleagues that the schematic in Figure. 3 (e) showing the CDW stacking configurations for the three observed surfaces needs to be easier to understand. A schematic that offers 1:1 comparison with the data in Figs. 2 shown. This schematic should also compare well against the one shown for the expected pristine step-terrace formation shown in (a). As below:

13) Concerning how cleavage and sample transfer temperatures enable our new observations:
In the main text (the modified part is underlined): 14) While making Figure R3 above, a small error was found: The dI/dV(x,V) data shown in "The depiction of the three-dimensional pattern of charge order shown in Fig. 1c is not comprehensive. The stacking vector T C has three symmetrically equivalent instantiations related to each by rotations of 120° (for example, 2a + b, -a + b, and -a -2b), and it has been established that the extended stacking pattern features a partial disorder in which T C varies randomly between these three. The coherence length for ordered stacking within this partially disordered pattern is ~3 to 10 unit-cells [1][2][3][4]. A depiction of the extended stacking pattern is shown in Fig. S1 2046-2060 (1991).

17)
A short note about lateral strain (or the absence thereof) near the domain boundary has been added to the Supplemental Information, in the discussion for Figure S2: 18) Through correspondence with other colleagues since the time of submission, we found that it will be helpful to include the topographic line profile across the step-terrace morphology shown in Figs. 2 & 3. The different apparent step heights between the various surface terminations are shown, and a possible origin of the difference is described:   19) The tip-height-dependent I(V) and dI/dV(V) results shown above in Figure R2 (and caption) have been added to the Supplemental Information, with an explanatory paragraph. Here we also comment on the best estimate of the actual gap sizes for each surfaces:

"E. Tip-and DW-induced band bending effects.
For each of the two regular types of 1T-TaS 2 surface, the dependence of the conductance spectrum on the tip-sample separation was measured, as shown in Fig. S7.
Although the absolute tip-sample separation generally cannot be known in order to compare one measurement to another, a rough proxy for it, which can be used for The slightly larger apparent gap size in the conductance spectra here, as compared to those shown in Fig. 1(a) [7,8]. In Feenstra's framework, it is predicted that in a semiconductor or insulator, a given unoccupied state ordinarily residing at E 0 should be observed at an energy E 0 ' which shifts increasingly

20)
In a new section, the following paragraph has been added to discuss the lateral band bending near the domain walls and step edges: "It is well established that ordinary domain walls in 1T-TaS 2 can cause lateral band bending due to remaining charge density left over from partial SD clusters. It is reasonable to expect that step edges also host such remnant charge density. However, no obvious lateral band bend is seen near either of the step edges in Figure 2(b). The previously reported band bending occurred in insulating domains, whereas in Figure 2(b) shown here, the domain wall (and one of the step edges) is adjacent to the metallic domain (Region 2) into the remnant charge density may be allowed to diffuse away. This may explain the absence of any obvious lateral band bending around the domain wall between Regions 1 and 2, and is consistent with previous observation [9]. By comparison, the usual lateral band bending, comparable with previous observation [10], is seen in Fig. S1(c) Figure R3 above, showing the spatially resolved raw and normalized dI/dV curves, has been added to the Supplemental Information, and a suitable note has been added to the main manuscript text to direct the reader to it (In the caption of Fig. 2).

22) The manuscript now includes a Data Availability statement:
"The data that support the findings presented here are available from the corresponding author upon reasonable request."

Reviewers' comments:
Reviewer #1 (Remarks to the Author): Unfortunately, the self-consistent understanding of the origin of the gap and the spectral shape of the key object of this paper -the unpaired layer -is still missing. I believe the spectral shape and its origin have to be elucidated prior to accepting the paper. Otherwise, I appreciate the authors' efforts in improving the paper and stand to my previous assessment of the importance the present results.
1. Origin of the spectrum of the unpaired layer in the simple case (T_C stacking). I am puzzled by the presence of additional peaks in the spectrum of the unpaired layer, compared to the well-understood spectrum of the paired one.
My naïve guess was that there is some kind of convolution of the spectra of the underlying paired layer and that of the unpaired -this would explain the additional peaks. However, the distance dependence presented by the authors in their reply shows that such simple picture is inapplicable.
The present authors' argument -the suppressed out-of-plane bandwidth -in the first approximation seems to account for the gap size only, not the spectral shape. However, the spectral shape change is the qualitative effect. It is very nicely seen in the waterfall plot of the spectra in Fig. S5b if one extends the position of lower and upper Hubbard bands from Region 3 (paired layer) to the other regions (see Fig. RR1 attached).
Below are my requests/questions (here I will use Figure 1e for clarity): 1a. Please, assign the peaks in the red and blue curves to Hubbard bands and CDW bands respectively (as it was done in e.g. NatComm 7, 10956 (2016) or NatComm 7, 10453 (2016)). 1b. Please, discuss the qualitative change in the spectral shape mentioned above. 1c. The spectrum of the unpaired layer seems to have two split Hubbard bands (e.g. the two peaks around -200meV in the red curve are symmetrically shifted with respect to the original LHB in the blue curve). Could it be that the Hubbard bands are split due to the two inequivalent hopping integrals that appear due to the asymmetry of the David star position in the unpaired layer with respect to those in the underlying paired layer? 1d. What happens to the CDW bands in the unpaired layer? 2. Origin of the spectrum of the unpaired layer in the complex case (T_B stacking). In the supplemental Figure S2e  Question 2a: Once and if the shift is corrected, how the region II in Fig. S2 is different structurally and spectrally from the region 2 in Figs. 1, 2 in the main text? Question 2b: Given the above, is it possible to prove the following statement you make in the updated manuscript: "The orbital localized at the SD centre is most likely Ta 5dz2 [9,27], and due to its anisotropic projection perpendicular to the plane, it is reasonable to expect the inter-layer orbital overlap to be very sensitive to the lateral displacement between layers. In this case, upon changing the stacking from TC to TB, we may speculate that the inter-layer overlap increases beyond the threshold for breakdown of the Mott state."?

FIG. RR2
See the yellow and white arrows construction

FIG. RR3
See the yellow and red arrows construction Reviewer #2 (Remarks to the Author): In the revised version of their paper, Butler et al. have included a significant amount of supplementary data and made changes throughout the main text. The experimental data is excellent and convinces me that this work should be published in Nature Communications, but the lack of theoretical support is still the weak point of this paper. I would recommend publication so long as the authors can address the following questions.
1. I agree with the authors that DFT with a 5 double-layer system would be quite a bit of work, and is perhaps too much to ask for in this manuscript. On the other hand, I feel that the arguments provided by the authors are still somewhat unconvincing to establish the metallic surface state without correlation. (One can easily come up with counter-arguments to the "dangling bonds" surface layer, for example, that the Tc-Ta-Tc-Ta structure can also be viewed as a perfect stacking of Tc-stacked bilayers.) Perhaps the authors should consider performing tightbinding modelling regarding the surface state. My intuition is that this alternating Ta-Tc stacking of the cluster SD orbital is very similar to the Su-Schrieffer-Heeger model (along the out-of-plane direction), where the hopping is smaller for the Tc stacking and larger for the Ta stacking. When the sample is cleaved across the bilayer, one breaks the stronger bond, and in the spirit of SSH, should get an edge/surface state (even in the presence of a finite weak bond strength) that lies in the gap (although in this case I suppose it would be a 2D band in the gap) (this is the "topologically nontrivial" case). And the other cleaving plane should not yield a surface state (i.e., the "topologically trivial" case). This can be verified more rigorously by solving a 3D tight-binding model using only the cluster SD orbital with in-plane hoppings, out-of-plane Ta hopping, and outof-plane Tc hopping. I think this would be very straightforward and would greatly strengthen the authors' story. I have seen related arguments presented in talks by Sung-Hoon Lee (I'm not sure if it is published).
2. Regarding the band bending effects, seeing the gap size increase with increased current seems quite normal. Others have seen this in other transition metal dichalcogenides. The idea here is that the ratio of the voltage drop between the first layer and second layer compared to the drop between tip and first layer increases as the tip gets closer. (This gets a bit complicated when the tip and sample have a large work function mismatch, where the contact potential dominates and VB and CB can, for example, bend in the same direction. I believe Feenstra's simulations used a large contact potential.) The discussion in the supplement should be modified to reflect this.

Reply Letter (round 2)
We are very grateful to both the Referees for their careful reading of our revised manuscript, and their valuable comments, which have again helped us to substantially improve it. Below we describe our revisions in response, including additional experimental results. As in the previous reply, we reproduce some text from the Referees' remarks and highlight these in red, and where we make additions or revisions to the text of the manuscript, those are highlighted in blue. The figure numbers continue on from the previous Letter (i.e. starting from Fig. R7).
This Reply Letter is followed by a comprehensive List of Changes to the manuscript, where the revisions and additions are listed in order of their appearance in the manuscript and Supplementary Information.

Reviewer #1 (Remarks to the Author):
Unfortunately, the self-consistent understanding of the origin of the gap and the spectral shape of the key object of this paper 'the unpaired layer' is still missing. I believe the spectral shape and its origin have to be elucidated prior to accepting the paper. Otherwise, I appreciate the authors' efforts in improving the paper and stand to my previous assessment of the importance the present results.

Origin of the spectrum of the unpaired layer in the simple case (T_C stacking). I am puzzled
by the presence of additional peaks in the spectrum of the unpaired layer, compared to the wellunderstood spectrum of the paired one. […] The present authors' argument 'the suppressed out-of-plane bandwidth' in the first approximation seems to account for the gap size only, not the spectral shape. However, the spectral shape change is the qualitative effect. It is very nicely seen in the waterfall plot of the spectra in Fig. S5b  1b. Please, discuss the qualitative change in the spectral shape mentioned above.
Please allow us to answer these two points together in the discussion below. In answer to the Referee's comment regarding the origins of particular spectral features, we are reluctant to definitively assign a particular peak to the CDW, even for the case of the paired cluster surface.
Previous reports, which appear to measure the surface which we have demonstrated is the paired surface, have ventured to identify certain spectral peaks with the CDW, but these have We are prepared to provide a more complete phenomenological description of the spectral feature at each surface, using newly included data below. But please note that the curves shown in Figure 1e of the main manuscript are not very suitable for the purpose of identifying or even describing all the spectral features, as those spectra were acquired very near the centers of particular SD clusters. We hope the relevance of this will become clear below. Please note that a T C -stacked SD cluster actually has three underlying 'nearest neighbors' (for a very loose definition of 'nearest neighbors'). The three sites below all have slightly different absolute distances. However, the T C -stacked site is very close to the point which is equidistant to the three neighbors below (just slightly more than half of one Ta-Ta lattice constant away).
Therefore, all three of the interlayer hopping integrals across the T C interface are probably very similar, and we do not expect the difference between them to be the cause of the apparent additional peak in the conductance spectrum.
1d. What happens to the CDW bands in the unpaired layer?
As shown in Fig. R7 above, the spatial distributions of the CB and VB features (imaged at +400 meV and -450 meV, respectively) are very similar for both surfaces. This is also a good indication that the in-plane behavior of CDW-related bands is not affected by the inter-layers stacking.
To conclude, within the scope of this manuscript, we contend that the important observations are, in brief, (i) that we have experimentally established that the alternating 'ACAC' stacking   In answer to the Referee's question, we hope that it is now clear that Region II as depicted above and in Fig. S2e can be self-consistently interpreted as a T C -stacked 'unpaired' layer, similar to Regions 1 and 4 in Fig. 2 of the main manuscript, and with a similar 'small gap' conductance spectrum. It is both structurally and spectrally distinct from Region 2 of Fig. 2 of the main manuscript, which is a T B -stacked unpaired layer, and is metallic (possibly due to Mottness-collapse driven by enhanced orbital overlap, in our interpretation).
Technical comment: in the supplementary section B the authors refer to the non-existent figures S1a and S1e, but probably mean S2a and S2e.
We thank the Referee again for all the above comments, and especially for the very careful reading of the manuscript. This error has been corrected in the revised version.

Reply to Referee 2
Reviewer #2 (Remarks to the Author): In the revised version of their The Referee suggests that it is easy to come up with counter-arguments to the expectation of a metallic state at the unpaired surface in the absence of correlations. However, the example the Referee gives, proposing that each set of T C -stacked layers could also be considered as the appropriate even-electron cell (the Peierls-like dimer), does not seem applicable, for reasons we would like to explain below. Additionally, on this point: even if we imagine that the dimer is actually formed across the T C interface, our other observed surface (T A -stacked, and with the larger gap) would then represent the 'unpaired' layer resulting from the broken dimer. As *both* surfaces are seen to be gapped, the alternative choice of dimer arrangement doesn't allow one to escape the conclusion that electronic correlations must open the gap at (at least) one of the two surfaces.
Still, we take the Referee's point that we need to affirmatively explain why, in the absence of correlations, we should normally expect a metallic state at the unpaired (T C ) surface, and after discussions with our theorist colleagues we have added such a through and comprehensive argument, described below. It is in line with the one described for the SSH model by the Referee, but we show its validity for a system extended in 3D (i.e. stacked 2D layers with a termination facing the vacuum). Please note that after discussions with our colleagues, we have determined that performing tight-binding calculations would not meaningfully add to the argument as presented below, as the important behavior of the 3D model in question can be understood easily enough without demonstrating it explicitly through calculations. Also, it is valid for the range of relevant in-plane and out-of-plane hopping strengths, beyond any specific choice we may make for a given calculation.
The argument which follows has been added to our Supplementary Information, and a line of text directing readers to it has been added to the discussion of the unpaired surface in our main text: If we consider a simple layered system with alternating interlayer hoppings t A and t C , but which terminates with a vacuum beyond the top surface (see Figure R9 below), we can first imagine the case in which t A = t C (left of Fig. R9), which clearly corresponds to a case of half-filling and a metallic state is realized. Note that this also corresponds to the purely T A and purely T C stacked 1T-TaS 2 cases, both of which are unsurprisingly predicted, using DFT calculations, to be metallic in the absence of electronic correlations [Ritschel et al., PRB 98, 195134 (2018)]. Now if the ratio t A /t C > 1 is varied, describing the case that the bulk of the structure becomes dimerized but the top layer is unpaired, we can proceed onwards to the extreme case in which t C ≪ t A (middle of Fig. R9), at which point it is clear that the top layer is completely decoupled, and again must be half-filled, and with an in-plane bandwidth determined by its own intra-layer hopping, which we can call t ∥ . As the ratio t A /t C > 1 is increased, we should expect the bulk of the material to pass through a metal-insulator transition. But importantly, there is no point between these two extremes at which the metallic state in the uppermost layer will disappear. Figure R9. Ubiquity of surface metallicity in absence of electronic correlations, for a dimerized, layered system with an unpaired top layer.
The detailed behavior of this model is that the bulk will have a gap determined by the difference in inter-layer hoppings, |t A | -|t C |, and the surface layer will retain a metallic band of bandwidth 6t ∥ (for a triangular lattice). Because the relevant orbitals are of d z 2 character, we can realistically expect t A to be larger than either t ∥ or t C , and so the bandwidth 6t ∥ may not necessarily span the dimerization gap 2∆ = 2(|t A | -|t C |), but a half-filled band should nevertheless persist at the surface for any set of relative strengths of t ∥ , t A and t C , as long as t A /t C > 1. Given that we should expect that t A /t C > 1 in the physical system under discussion (as thoroughly argued above), this provides a very strong argument that the best explanation for the observed gap is strong electronic correlations, and this fully justifies the use of the term 'Mottness' in our title.
In relation to the discussion here, we have added a note of Acknowledgment in our manuscript to thank Dr. Motoaki Hirayama and Dr. Ryotaro Arita for their invaluable advice.
The discussion of the origin of the spectral gap at the unpaired surface in the main text has been modified, and substantially streamlined, in order to mesh with the above argument, which Here the Referee's comments are correct by themselves: Seeing the gap size increase with increased current would be quite normal, and we agree. However, in our data ( Fig. R4 in our previous Reply Letter and reproduced below), the apparent gap size increases with *decreasing* current. The ratio of the voltage drop between the first and second layers to the drop between the tip and first layer should ordinarily increase as the tip gets closer, which, for positive bias, enhances the upward tip-induced shift for a given spectral feature. But what we observe is that the upward band-bending is apparently *suppressed* rather than enhanced as the tip gets closer, or even that there is a downward bending which is being enhanced. See below: Figure R10. Tip-height-dependence of conductance spectra. Conductance peaks at the UHB onset for paired (a) and unpaired (b) clusters. In each case, the apex of the conductance peak, observed at the energy E 0 ' clearly shifts towards higher energy as the tip-sample distance increases.
However, the Referee points out that the difference in tip and sample work functions plays an important role in the TIBB phenomenon, and this point is very helpful to us. We now realize that if the sample work function is significantly larger than that of the tip, the apparent tip-induced potential can be non-zero even for zero applied bias, and the condition of zero tip-induced bending can be found at a non-zero applied bias, corresponding to the energy W 0 = W sample - We again sincerely thank the Referee for these very helpful comments, which have helped to solve a long-standing problem for us.
In light of the above however, we still re-iterate that regardless of the sign and underlying mechanism of this tip-induced shift, its effect is quite small (on the order of only a few meV in the regime typical for STM measurements, as seen in Fig. R10 above). It not significant enough to impact on any of the conclusions drawn in this manuscript. We put forward our estimates of the band gap sizes for the respective surfaces as the best available so far, while making note of the above-mentioned tip-induced effects. 2) We have revised again the interpretation of the spectral gap at the unpaired surface, including a note to direct the reader to the comprehensive discussion which we have added to the Supplementary Information (see below). The revised text is as follows.

List of Changes
"The surface of unpaired SD clusters (Regions 1 & 4)  3) A note of thanks to both Dr. Motoaki Hirayama and Dr. Ryotaro Arita have been added to our Acknowledgments section. 4) Additional funding information (KAKENHI grant no. JP19H05602) has been added in the Acknowledgments section.

5)
In partial answer to the first Referee's queries, and to satisfy any curious reader, Fig. R7 above, along with a suitable description, has been added to the Supplementary   Information, showing the spatial distributions, indicated by conductance mapping, of the spectral features at each surface.
"In Fig. S2  This is consistent with the spatially resolved conductance data shown previously by Qiao et al. [7].
As can be seen in the lower panel, the spectrum for the unpaired surface shows several differences from that of the paired surface, aside from the smaller energy gap.
Most notable is the pair of peaks below the gap, at around -120 meV and -240 meV. The "We consider a simple layered system with alternating interlayer hoppings t A and t C , but which terminates with a vacuum beyond the top surface, as shown in Fig. S8 below. We can first imagine the case in which t A = t C (as on the left-hand side of Fig. S7), which clearly corresponds to a case of half-filling in which a metallic state is realized in the absence of electronic correlations. Note that this also corresponds to the purely T A and purely T C stacked 1T-TaS 2 cases, both of which are unsurprisingly predicted, using DFT calculations without on-site repulsion U, to be metallic [4]. Now if the ratio t A / t C > 1 is varied, describing the case that the bulk of the structure becomes dimerised but the top layer is unpaired, we can proceed onwards to the extreme case in which t C ≪ t A (middle of Fig. S7), at which point it is clear that the top layer is completely decoupled, and again must be half-filled, and with an in-plane bandwidth determined by its own intra-layer hopping, which we can call t ∥ . As the ratio t A = t C > 1 is increased, we should expect the bulk of the material to pass through a Peierls-type metal-insulator transition. But importantly, there is no point between these two extremes at which the metallic state in the uppermost layer will disappear.
The detailed behavior of this model is that the bulk will have a gap determined by the difference in inter-layer hoppings, |t A |-|t C |, and the surface layer will retain a metallic band of bandwidth 6t ∥ (for a triangular lattice). Depending on the relative strengths of t ∥ , t A and t C , the bandwidth 6t ∥ may not necessarily span the dimerisation gap 2Δ = 2(|t A |- stacking structure [8]. This suggests that the Peierls-like dimerisation should be thought of a straddling the T A -stacked interface, not the T C -stacked interface, and therefore that This also matches well with the basic intuitions that (i) the larger absolute clustercluster distance across the T C -stacked interface should lead to much smaller orbital overlap, and (ii) the lateral offset across the T C -stacked interface should also lead to much smaller overlap due to the d z 2 character of the relevant orbitals [7]. Given that the system under discussion, the unpaired surface, should correspond to the case in which t A = t C > 1 in the above model, this provides a strong argument that a metallic surface state is expected if electronic correlations are absent, and that such correlations are therefore best explanation for the observed spectral gap.
8) The discussion of the tip induced band bending effect in the Supplemental information has been modified, as follows: "  [10]. The authors have substantially improved the manuscript and addressed most of the issues raised. I appreciate the additional spatially-resolved spectra of split bilayer that elucidate the correlated origin of the spectral features. I still think that the qualitative splitting of the spectrum requires proper discussion prior to accepting the paper. Similarly, spectral feature assignment will improve the clarity of the message. I am happy to recommend the paper after adding the proper and selfconsistent discussion in the main text.
Below I reiterate two questions from my previous review as well as the part of the authors reply and then provide my further comments.
Previous question 1a: Please, assign the peaks in the red and blue curves to Hubbard bands and CDW bands respectively (as it was done in e.g. NatComm 7, 10956 (2016) or NatComm 7, 10453 (2016)).
Authors' reply to question 1a: Please allow us to answer these two points together in the discussion below. In answer to the Referee's comment regarding the origins of particular spectral features, we are reluctant to definitively assign a particular peak to the CDW, even for the case of the paired cluster surface. certainly the problem to be discussed, but cannot prevent the qualitative ascribing of the bands. With the above said, I cannot understand the authors' reluctance to ascribe the bands, as it reduces the uncertainty in understanding their results and supports their claim of correlated physics involved (CDW bands are intact as shown below). For the reasons described above and the references provided, I am adamant in my request.
Previous question 1c: The spectrum of the unpaired layer seems to have two split Hubbard bands (e.g. the two peaks around -200meV in the red curve are symmetrically shifted with respect to the original LHB in the blue curve). Could it be that the Hubbard bands are split due to the two inequivalent hopping integrals that appear due to the asymmetry of the David star position in the unpaired layer with respect to those in the underlying paired layer?
Authors' reply to question 1c: [...] Please note that a TC-stacked SD cluster actually has three underlying 'nearest neighbors' (for a very loose definition of 'nearest neighbors'). The three sites below all have slightly different absolute distances. However, the TC-stacked site is very close to the point which is equidistant to the three neighbors below (just slightly more than half of one Ta-Ta lattice constant away). Therefore, all three of the interlayer hopping integrals across the TC interface are probably very similar, and we do not expect the difference between them to be the cause of the apparent additional peak in the conductance spectrum.
New comment to 1c: I have to respectfully disagree with the last sentence. Simple construction shows (see figure attached) that the three nearest (in the geometric sense) neighbors are located at distances: B<sub>1</sub> = √(4a<sup>2</sup> + c<sup>2</sup>), B<sub>2</sub> = √(3a<sup>2</sup> + c<sup>2</sup>) and B<sub>3</sub> = √(7a<sup>2</sup> + c<sup>2</sup>). The CDW period is A<sub>CDW</sub> = a √(13) This allows us to write down: A<sub>CDW</sub> > B<sub>3</sub> >> B<sub>1</sub> > B<sub>2</sub>; and taking into account numerical values for a and c: A<sub>CDW</sub> ~ B<sub>3</sub>; B<sub>1</sub> ~ B<sub>2</sub>. The distances enter the hopping integrals exponentially (further approximation can take into account Wannier orbital shape). From here it is already clear that there is a set of hopping integrals that will determine fine features. Qualitatively, there are two limiting hopping integrals: one is related to A<sub>CDW</sub> ~ B<sub>3</sub> and another -to B<sub>1</sub> ~ B<sub>2</sub>. The role of this contribution to the splitting should be discussed in the manuscript. Next, the description should be self-consistent. The authors argue that the change of stacking from TC to TB causes Mottness collapse. TB stacking gives: A<sub>CDW</sub> > B<sub>3</sub> > B<sub>1</sub> >> B<sub>2</sub>, where B<sub>2</sub> = √(a<sup>2</sup> + c<sup>2</sup>) (see attached figure for construction). The B2 change is minor on going from TB to TC. The only other thing that changes is the asymmetry in the David star position in the split bilayer with respect to the underlying full bilayer (expressed in the distances hierarchy). Which of these two factors is responsible for the Mottness collapse then? On the qualitative level, it is highly likely that the smaller hopping in the TC case being not enough to collapse Mott state still modifies it by splitting the band and reducing the gap.
Authors' reply to question 1 in general: To conclude, within the scope of this manuscript, we contend that the important observations are, in brief, (i) that we have experimentally established that the alternating 'ACAC' stacking ('unit-cell doubling' in the title) is realized in 1T-TaS2, and that (ii) despite this unit cell doubling, we still see signs of a correlation-driven gap at the surface where the doubling is broken. (A more thorough argument for this has now been added. Please see our reply to the second Referee.) The detailed mechanisms determining the differing spectral shapes at each surface are certainly interesting and deserve investigation, but are beyond the intended scope of this manuscript, and beyond the suitability of the tools used here to elucidate confidently.
New comment: The qualitative features should be discussed and are within the scope of the paper for the reasons described in the two comments above.
Cut of the schematic in Figure 3a with the red and blue dashed ovals added on top to highlight the David star clusters in split bilayer on the normal bilayer respectively. B 1 and B 2 are the vectors connecting the David star centers where Hubbard bands are localized.  From the top view we can see that the David star cluster in the split bilayer (red) has three nearest neighbors in the normal bilayer (blue) underneath. The vectors between the centers (where Hubbard bands are localized) are: B 1 = 2a + b + c; |B1| = sqrt(7a 2 + c 2 ) B 2 = a + c; |B2| = sqrt(a 2 + c 2 ) B 3 = 3a + c; |B3| = sqrt(9a 2 + c 2 ) CDW period A = 3a + b = a*sqrt(13); a, b -in-plane atomic unit vectors; cout of plane atomic unit vector. One can clearly see that B2 ~ B3 ~ CDW period and A > B3 > B1 >> B2.

Reply Letter (round 3)
We sincerely thank both Referees again for their continued careful reading and helpful comments, including Referee 2 for the endorsement of our manuscript for publication.
Below we respond to the two remaining comments from Referee 1, which have again helped us to improve the clarity of our message, and the manuscript as a whole. The Referee's comments are reproduced below in red, and as before, figure labels are continued on from our previous reply letter. This reply is again followed by a List of Changes, where revisions are listed in order of their appearance in the manuscript. We are now convinced that it is appropriate to label the peaks in spectral weight observed at -300 meV and below as originating from the CDW reconstruction, especially after reading in labelling has now been adopted in our Figure S2 in the Supplemental Information.
With regard to Fig. 1e, as stated in our earlier reply, it is not a suitable place to label the CDW features, because those spectra were collected at SD cluster centers, and therefore reflect predominantly the Mott-localised features. The images shown in Fig. S2 show that, in contrast, the local density of states associated with the CDW lies around the periphery of each cluster.
For this reason, we can fully label the CDW and Hubbard peaks only in Fig. S2, where curves collected at the cluster centers and peripheries are shown together. The labels in Fig. S2 have been altered accordingly, and the work by Rossnagel et al. has been cited.
Please note that although we don't label the UHB and LHB within Fig. 1e,   The Referee's suggestion is well taken. After considering the Referee's construction, and after further discussion with our theorist colleagues, we consider that a detailed calculation of the various energy scales associated with the hierarchy of overlaps is needed in order to draw even qualitative conclusions about the detailed spectral shape. An explicit calculation for such complex clusters is exceptionally challenging. However, we agree with the Referee's suggestion to discuss the possible origin of the spectral details. To that end, the following line has been As an aside, we also include a line of text which adds to the interpretation for the nature of the spectral gap at the paired surface, citing an additional reference which we see as very important for discussions which may follow on from ours, in light of the experimental observation of bilayer-type stacking (the modified or newly added text is underlined): "They also indicate that while the BL-stacked bulk structure of 1T-TaS 2 may satisfy the criteria for a simple band insulator [5,7], this does not preclude the presence of strong e-e correlations, and these have been evidenced in the recent observation of doublon excitations characteristic of a Mott state [6]. Indeed, it has previously been shown that a system described by a twolayered Hubbard model with inter-layer hopping t can exhibit a continuous crossover between the Mott and band insulating regimes [32]. Hence, for the paired surface and the BL-stacked bulk, with significant intra-BL orbital overlap, the two regimes may not be meaningfully distinct." The newly added reference is the following: The referee mentions two possible effects which could lead to Mottness-collapse upon going from T C to T B . The first is the reduction of the smallest nearest-neighbour (n-n) distance in the distance hierarchy, and the second is related to increased asymmetry (increasing imbalance between inter-cluster hoppings). The Referee seems to dismiss the first reason, suggesting that the change in the smallest n-n distance is minor. Below, by numerically evaluating a simple model we find that, on the contrary, the change in n-n distances could plausibly cause a bandwidth enhancement of the scale required to explain the Mottness-collapse.
As a first approximation, we may consider that the overlap integral for neighbouring orbitals is an exponentially decaying function of their absolute distance (in an s-wave approximation). The numerical result shown in Fig. R11, using |a| = |b| = 0.34 nm and |c| = 0.6 nm, verifies that if all three entries in the n-n distance hierarchy are included in a sum of exponentially decaying overlaps, then for any reasonable value of the decay constant , the total overlap for T B stacking is always larger than that for T C . (A 'reasonable' value of means that decay length-scale 1/ is realistically short -no more than 0.1 nm or so. In the limit of large 1/ the total overlap tends towards the coordination number, in this case three, regardless of stacking vector.) Please note that we identify and label the n-n distances in a different way from that used by the Referee, and that here B i (C i ) denote the n-n distances for T B (T C ) stacking.

Figure R11. Estimation of inter-layer orbital overlaps for three n-n, in an s-wave approximation, for T B and T C stacking configurations. (a) and (b) show the in-plane
projected n-n distance hierarchies in each case. (c) shows a comparison of the respective sums of the n-n overlaps for each case, given that each n-n distance enters into an exponential decay characterised by . The total overlap for T B stacking always exceeds that for T C stacking in the relevant range of , as shown by the ratio of the sums (blue curve).
This simple model yields greater overlap for T B than for T C , qualitatively consistent with the suggested mechanism of Mottness-collapse upon transition from T C to T B .
From the idea that overlap is larger for T B than for T C , it clearly follows that the overlap would be  Given the simplified framework described above, the argument for Mottness-collapse at the T Bstacked surface fits self-consistently: Due to change in n-n distances, the inter-layer overlap, and therefore bandwidth, is expected to be larger for T B stacking than for T C stacking, potentially explaining the observed Mottness-collapse.
An alternative mechanism --the second mechanism mentioned by the Referee --is that if the asymmetry of the stacking coordination and resulting hierarchy of n-n overlaps could modify the bandwidths and reduce (or close) the Mott gap. If this is the case for the T C -stacked layer, as mentioned above, then the increased asymmetry of the T B stacking coordination (see Fig. R11) might come with a relatively larger bandwidth enhancement, closing the gap. As mentioned above, a detailed calculation of the various energy scales associated with the hierarchy of overlaps is needed in order to draw conclusions about the peaks' splitting or bandwidths. An explicit calculation for such complex clusters is very challenging, and the contribution of this alternative mechanism is much harder to estimate, even qualitatively.
Ultimately, we are considering the hopping between orbitals centered in complex 39-atom clusters (if the top and bottom S layers are included, which have been neglected in the discussion so far), each with intricate internal bonding and orbital textures. The simplistic arguments we present above, or even explicit Slater-Koster-based tight-binding models which reduce the system only to the idealised 5d z 2 orbitals at the cluster centers, will clearly not be satisfying. A sufficiently detailed modelling of this complex system cannot be presented here, and we leave that as a target for future projects.
These arguments are now referred to in the main text, and largely reproduced in the Supplemental Information (see Item 7 in the List of Changes).
"The qualitative features should be discussed and are within the scope of the paper for the reasons described in the two comments above." Each of the Referee's concerns have been discussed in the revised version of the manuscript.
We hope the revised version removes any remaining obstacles to acceptance of the manuscript.

List of Changes
1) The usual identification of the two conductance peaks of the Type 1 spectrum have been added to the caption of Figure 1. 3) A brief discussion of the detailed spectral shape of the Type 2 spectrum has been added in the discussion section of the main text: "It is possible that the anisotropic coordination environment of the T C -stacked clusters results in a hierarchy of varying nearest-neighbour orbital overlaps which could explain the detailed spectroscopic features observed. The resulting modification of bandwidth may also be the cause of the apparent reduction of the Mott gap." 4) Additional discussion of the nature of the spectral gap at the paired (Type 1) surface has been included in the discussion section of the main text.
"They also indicate that while the BL-stacked bulk structure of 1T-TaS 2 may satisfy the criteria for a simple band insulator [5,7], this does not preclude the presence of strong ee correlations, and these have been evidenced in the recent observation of doublon excitations characteristic of a Mott state [6]. Indeed, it has previously been shown that a system described by a two-layered Hubbard model with inter-layer hopping t can exhibit a continuous crossover between the Mott and band insulating regimes [32]. Hence, for the paired surface and the BL-stacked bulk, with significant intra-BL orbital overlap, the two regimes may not be meaningfully distinct." In relation to the above, an additional reference is included:   Given the simplified framework described above, the argument for Mottness-collapse at the T B -stacked surface fits self-consistently: Inter-layer overlap, and therefore bandwidth, is expected to be larger for T B stacking than for T C stacking, potentially explaining the observed Mottness-collapse.
However, a satisfactory understanding of the mechanism of Mottness-collapse must consider the hopping between orbitals centered in complex 39-atom clusters (if the top and bottom S layers are included, which have been neglected in the discussion so far), each with intricate internal bonding and orbital textures. The simplistic arguments we present above, or even explicit Slater-Koster-based tight-binding models which reduce the system only to the idealised 5d z 2 orbitals at the cluster centers, will clearly not be satisfying. A sufficiently detailed modelling of this complex system cannot be presented here, and we leave that as a target for future projects." This has been fixed. Now θ is used for the polar coordinates.