Anomalous Dirac point transport due to extended defects in bilayer graphene

Charge transport at the Dirac point in bilayer graphene exhibits two dramatically different transport states, insulating and metallic, that occur in apparently otherwise indistinguishable experimental samples. We demonstrate that the existence of these two transport states has its origin in an interplay between evanescent modes, that dominate charge transport near the Dirac point, and disordered configurations of extended defects in the form of partial dislocations. In a large ensemble of bilayer systems with randomly positioned partial dislocations, the distribution of conductivities is found to be strongly peaked at both the insulating and metallic limits. We argue that this distribution form, that occurs only at the Dirac point, lies at the heart of the observation of both metallic and insulating states in bilayer graphene.


The model
Hamiltonian is briefly sketched in the Methods section. As a suggestion, either one fully explains the method --I recall there is no length constraint on this part --or fully reference other sources. The method section of this manuscript is just useless, if not frustrating for the reader. For example, in which space do the tau matrices operate (I assume sub-lattice space)? What is the relative value of the interlayer vs intralayer coupling? How do you exactly couple the leads to the scattering region in the Landauer approach (especially in case of "hard leads" how are oscillating wave-functions injected into the sample)?
3 (Minor point). It is not true that diffusive transport cannot be treated within Landauer approach. This is indeed a frequently misunderstood point. What cannot be treated (in the original Landauer approach) is incoherent transport, namely when the motion of the electrons is not phaseconserving because of lattice dynamics, for example. Impurity scattering, treatable within Landauer, can well result in diffusive transport (\sigma \propto 1/L).
Concerning the results: 1-Case of the individual defect: the conductivity approaches the minimal universal value if the dislocation is in high symmetry position, i.e. at the center of the sample. This behavior is expected if one assume closed boundary conditions ("hard leads"). It is no surprise that the authors find a similar behavior in the case of exponential behavior of the wave-function outside the leads. This is obviously not true if one assumes infinite leads of pristine bilayer graphene, for it would not make sense to speak of a high-symmetry position of the partial. Of course, this is an unreal limiting case, but makes clear how the leads can have a substantial effect on the model. The typical model situation for making minimum universal conductivity emerge is that one assumes doped bilayer graphene for the leads, and intrinsic graphene for the scattering region (ex. Ref 1. and 2.). Why did not the authors use the same approach here, which also gives the possibility of tuning the level of doping in the leads? 2. The manuscript suggest that the presence of parallel partials implies that electrons are confined in a series of quantum wells delimited by the position of the partials. I understand that imaginary value of kx is key to prevent localization due to destructive interference, but why in certain cases --the most symmetric ones --electrons can effectively tunnel from one well to the next, and in other case --the less symmetric --not? A necessary condition to re-evaluate my position is that this aspect is adequately discussed and convincingly clarified.
3. The author, speculate that the single model category that they treat, namely partials along armchair direction and perpendicular to the transport direction, does not hinder the generality of the result. I wonder how they would extend the disorder quantifier in the case of non parallel partials, which is indeed very common from experimental results.
Concerning the figures: 1. Insets of figure 2, given the units (Angstrom^-1) and the absence of reference to the 1D projected Brillouin zone are just not readable. I guess non-zero transmission only occurs in the vicinity of ky=2pi/3a, a being the periodicity along y axis. Still, what is the information that the reader should take from those plots?
To be honest, it has to be said that the idea is intriguing and the resemblance to the experimental results is satisfying. However, this is not per se a sufficient reason to accept it as the definitive explanation of the important problem addressed.
For these reasons, I cannot recommend publication in Nature Communication, but I am open to reconsider the manuscript if the points that I have raised will be fully met, if possible by means of the additional calculations and explanations suggested.

Fernando Gargiulo
Reviewer #2 (Remarks to the Author) The Authors carry out numerical simulations with a model Hamiltonian to get insights into the electronic transport at the Dirac point in bilayer graphene with extended defects. They first analyze the transport through a single partial dislocation and then study that in the essentially macroscopically large system with a random network of partial dislocations. The Authors reveal the origin of the experimentally observed insulating and metallic transport states in bilayer graphene, which was not fully understood.
Taking into account the large amount of attention currently being paid to electronic properties of graphene and other 2D materials, the results reported in the manuscript should be of interest to the readership of Nature Communications, and their presentation is good. The manuscript is well written, so that I recommend it for publication provided that the Authors consider discussing the following issues: 1. As the readership of Nature Communications includes a broad range of materials scientists, chemists and physicists, perhaps the Authors could explain in more detail the physical origin of the dislocations they consider (strain, rippling, etc.) and emphasize that they are fundamentally different from the in-plane dislocations in graphene [see, e.g., O.Yazyev's works, Nat. Nanotech. 9 (2014) 755 and references therein], the picture that most readers will likely have in mind when dislocations in graphene are mentioned.
2. Following from the above, the Authors may also discuss the relationship between point defects, in plane (e.g., edge) dislocations and partial dislocations they consider. As the experiments [Nature Communications 4 (2013) 2098] indicate, in graphene splitting of point defects (e.g., divacancies) due to carbon bond rotations can be equivalent to the formation of several dislocations, which in tern give rise to buckling and in principle, may result in a change in the local stacking of the two sheets in the bilayer. Will the suggested model work if any disorder is present in the graphene sheets? If not, and if the two transport states would be completely smeared out, this could be another indirect confirmation for amazingly low concentration of defects in graphene. The manuscript submitted as a publication in Nature Communications "Anoma-2 lous Dirac point transport due to extended defects in bilayer graphene" ad-3 dresses a well-known and definitely important problem in the topic of the elec-4 tronic transport regime of ultra-clean samples of bilayer graphene, of which 5 approximately half exhibit metallic behaviur, otherwise insulating. Due to the 6 scarcity of other sources of scatterers such as phonons, other electrons, extrinsic 7 defects one is indeed led to ascribe the emergence of insulating behavior due 8 to the presence of stacking faults (also called partials). The authors suggest 9 that the spatial distribution of the defects determine the transport behavior. 10 However, the detailed mechanisms proposed by the authors is not convincing to 11 me (at least in the present form of the manuscript). Possibly, this is due to the 12 fact that the I do not fully understand the method and models employed. In 13 particular: 14 Concerning the model: 15 1. What are the boundary conditions used, besides the generic definition 16 "hard" (which, I assume, are U (0, L) = ∞) and "soft" leads?

17
This is a misunderstanding caused by our eccentric choice of notation (and 18 a typo -"exponential envelope") in the previous version: our lead/material for-19 malism is in fact exactly that used in previous calculations of minimal conduc-20 tivity, i.e., doped bilayer graphene for the leads and intrinsic bilayer graphene 21 for the material. The notation "soft" implied a finite lead doping, while the 22 notation "hard" the limit when this doping goes to infinity. 23 2. The model Hamiltonian is briefly sketched in the Methods section. As 24 a suggestion, either one fully explains the method -I recall there is no length 25 constraint on this part -or fully reference other sources. The method section of 26 this manuscript is just useless, if not frustrating for the reader. For example, in 27 which space do the τ matrices operate (I assume sub-lattice space)? What is the 28 relative value of the interlayer vs intralayer coupling? How do you exactly couple 29 the leads to the scattering region in the Landauer approach (especially in case 30 of "hard leads" how are oscillating wave-functions injected into the sample)? 31 We have both substantially extended the Methods section as well as incor-32 porated additional references into it. We hope it now offers a clear background 33 guide to the calculations performed in the manuscript. In particular, (i) the 34 construction of the Hamiltonian is now explained in detail and this should make 35 objects like the τ matrices much clearer to understand (which are indeed in sub-36 lattice space as the Referee correctly points out), (ii) the underlying tight-binding 37 method from which our continuum Hamiltonian is derived is described (includ-38 ing values of intra-vs. interlayer hopping constants), and (iii) the coupling of 39 the leads to the device is now both clearly elucidated as well as clearly sourced 40 (we follow closely the scheme of Refs. 1 and 2). Note that Ref. 15, which gathers 41 together the theoretical methods used for construction of the model Hamiltonian, 42 is still refereed to for theoretical background. As this is currently under review 43 at PRX we cannot give a more substantial reference, although the method we 44 employ has previously been used successfully to describe both the twist bilayer 45 1 [Phys. Rev. B 93, 035452 (2016)] and, perhaps more directly relevant, partial 46 dislocation networks in bilayer graphene [Nat Phys 11, 650653 (2015)].) 47 3 (Minor point). It is not true that diffusive transport cannot be treated 48 within Landauer approach. This is indeed a frequently misunderstood point.

49
What cannot be treated (in the original Landauer approach) is incoherent trans-50 port, namely when the motion of the electrons is not phase-conserving because of 51 lattice dynamics, for example. Impurity scattering, treatable within Landauer, 52 can well result in diffusive transport (σ ∝ 1/L).

53
Our intention here was just to contrast typical large area "dirty" graphene 54 (diffusive and incoherent transport) with the ultra-clean samples of the exper- As stated above, our modeling of the leads/device is very close to that of 75 Refs. 1 and 2, i.e. we use doped bilayer graphene for the leads and intrinsic (but 76 defected) bilayer graphene for the scattering region. Following the suggestion of 77 the Referee we have, in our extended Methods section, now included a figure   78 plotting the ratio of conductivities for two example insulating and conducting 79 partial configurations as a function of lead doping (Fig. 7). As may be seen, it 80 is only as the lead doping approaches the anti-bonding band edge, at which the 81 lead description breaks down, that a quantitative change in this ratio is seen.

82
Dirac point two state transport is thus robust to the value of the lead doping. This is an important question, and has prompted us to search for a more 92 intuitive explanation of our finding of two state transport at the Dirac point. 93 This is now explained in a new paragraph, lines 74-90 of the manuscript as well 94 as three additional panels to Fig. 2. In essence, each terrace is endowed with 95 a transmission resonance allowing charge transport only for a limited range of 96 evanescent momenta Im k x , with the resonance centre Im k (R) x = (ln L/l ⊥ )/L 97 and width (≈ 1/L) strongly dependent on the terrace length (see Fig. 1d). By 98 expressing the overall transport, via a standard formula for combining scattering 99 matrices, in terms of the transmission functions of the individual terraces, we 100 find an intuitive explanation in terms of single terrace resonances. For terraces 101 of similar length, that occur when the partial is at a high symmetry position, the 102 resonance centers have similar Im k x , and thus both terraces have common val-103 ues Im k x at which they are "open" to charge transport. Overall the system thus 104 conducts (see Fig. 1e). On the other hand for a low symmetry position of the 105 partial, the different terrace lengths lead to substantially different centers of the 106 evanescent resonances Im k (R) x , to no common values of evanescent momenta 107 for which both terraces are open, and so to a blocking of charge transport (see 108 Fig. 1f ).

109
This physics can also be understood in a rather intuitive way in terms of the 110 transport wavefunctions which indeed, as the Referee points out, give the ap-111 pearance of quantum well states (even though they are in fact the wavefunctions 112 for open and not quantum well boundary conditions). This a complementary 113 point of view to the multiple scattering analysis that underpins the transmission 114 resonance picture, and is based on the concept of consistent or incompatible 115 matching conditions between the terrace wavefunctions and partial dislocations 116 or leads. This is explained in detail in a second paragraph, lines 91-106. 117 We have also, using a similar multiple scattering analysis to that deployed 118 for the single partial case, addressed the question of why for the intermediate 119 disorder case either an insulating or conducting state can result from similarly 120 disordered terrace geometries. We find that multiple scattering at the bound-121 aries between terraces can lead to a resonant enhancement of transport, and 122 demonstrate this in Fig. 3 for two cases of intermediate disorder, along with 123 accompanying text in lines 131-145. 124 This more intuitive explanation of two state transport -basically hewing 125 closer to the language of transport -has, we believe, considerably improved the 126 manuscript and we are grateful to the Referee for raising this point. 3. The author, speculate that the single model category that they treat, 128 namely partials along armchair direction and perpendicular to the transport 129 direction, does not hinder the generality of the result. I wonder how they would 130 extend the disorder quantifier in the case of non parallel partials, which is indeed 131 very common from experimental results.

132
Non-straight and non-parallel partials, in addition to other structural com-133 plications such as possible AA segments and trilayer samples, will be treated in 134 a future work. For samples of non-straight partials the classification of disorder 135 obviously becomes more involved (one of the motivating reasons for our effective 136 1d model) however we envisage a likely classification strategy to involve the ad-137 dition two parameters to the descriptor of the partial: (i) the difference between 138 the left and right edge partial positions |x L − x R | and (ii) a "stiffness constant" 139 of the partial guiding a random walk between x L and x R . We suspect, however, 140 and preliminary results indicate, that the qualitative nature of our findings will 141 not be changed in such a situation. Our expectation would rather be that it will 142 lead to a smearing of the evanescent resonances, as for a terrace bounded by 143 non-straight partials an effective < L > average is performed. As may be seen The insets for these figures showed the corresponding transmission function, 159 intended primarily to convey whether the partial geometry resulted in an insu-160 lating or conducting state. As these inset panels were very unclear we have 161 now instead simply written the corresponding conductivity over each panel. The 162 region of non-zero transmission depends in a complex way on the partial con-163 figuration; for a single terrace the centre of the evanescent resonance peak is at 164 (ln L/l ⊥ )/L.

165
To be honest, it has to be said that the idea is intriguing and the resem-  The Authors carry out numerical simulations with a model Hamiltonian to get 180 insights into the electronic transport at the Dirac point in bilayer graphene 181 with extended defects. They first analyze the transport through a single partial 182 dislocation and then study that in the essentially macroscopically large system 183 with a random network of partial dislocations. The Authors reveal the origin of 184 the experimentally observed insulating and metallic transport states in bilayer 185 graphene, which was not fully understood.

186
Taking into account the large amount of attention currently being paid to 187 electronic properties of graphene and other 2D materials, the results reported 188 in the manuscript should be of interest to the readership of Nature Communi-189 cations, and their presentation is good. The manuscript is well written, so that 190 I recommend it for publication provided that the Authors consider discussing 191 the following issues:  199 This is an important point and we have added such a discussion, and the 200 references suggested by the Referee, to the introductory part of the manuscript, 201 see lines 37-42. We are grateful to the Referee for the suggestion to place our 202 particular type of structural disorder in a broader context. 203 We would expect that most significant difference between stacking fault partial 204 dislocations and single layer dislocations resides in the fact that the former can 205 exist between structurally almost perfect graphene sheets, as only local strain 206 is required for their existence, and not the significant changes in local bonding 207 involved in single layer dislocations. As a consequence they are certainly to be 208 expected for growth situations in which a natural strain is imposed, for example 209 by a substrate, and indeed this is the case for bilayer graphene grown epitaxially 210 on SiC in which a dense network of dislocations is formed. 211 2. Following from the above, the Authors may also discuss the relationship 212 between point defects, in plane (e.g., edge) dislocations and partial dislocations 213 they consider. As the experiments [Nature Communications 4 (2013) 2098] 214 indicate, in graphene splitting of point defects (e.g., divacancies) due to carbon 215 bond rotations can be equivalent to the formation of several dislocations, which 216 in tern give rise to buckling and in principle, may result in a change in the local 217 stacking of the two sheets in the bilayer. Will the suggested model work if any 218 disorder is present in the graphene sheets? If not, and if the two transport states 219 would be completely smeared out, this could be another indirect confirmation 220 for amazingly low concentration of defects in graphene. 221 We believe that this is the case. In this respect it is worth noting the work 222 of Bao et al.  of the manuscript -in which two distinct types of bilayer 223 samples are reported: those having a very high mobility, for which two state 224 transport is observed, and those with a substantially lower mobility, found always 225 to be conducting. As the high mobility samples presumably exclude kinds of single 226 layer disorder likely to substantially disrupt the terrace wavefunction, we would 227 thus infer that terraces free from disorder are required for the mechanism we 228 describe to hold.

229
In other words, the scenario we advocate requires two graphene sheets that 230 have a very low concentration of in-plane defects, but nevertheless contain at 231 least one stacking fault in the form of a partial dislocation. We believe this is a 232 realistic situation, as in-plane defects are expected to have a higher energy cost  The only remark which they've left open is the one concerning generalization to their results to non-parallel partials. Their argument is that addressing this issue deserves a publication on its own, on which they are currently working. I take this statement as a gentlemen agreement that the authors will commit on publishing the result of this further piece of research, being they in agreement or in contradiction to the present claims.
I do recommend publication of this piece of research on Nature Communication.
Reviewer #2: Remarks to the Author: I am fully satisfied with the response of the Authors to my comments and questions. The manuscript has improved, and in my opinion it merits now the high Nature Communications standards of novelty and impact, so that I recommend it for publication in its present form.
I recognize that the authors took my remarks seriously and addressed my concerns almost completely.
The only remark which they've left open is the one concerning generalization to their results to non-parallel partials. Their argument is that addressing this issue deserves a publication on its own, on which they are currently working. I take this statement as a gentlemen agreement that the authors will commit on publishing the result of this further piece of research, being they in agreement or in contradiction to the present claims.
I do recommend publication of this piece of research on Nature Communication.
Response: The authors accept fully the terms of the proposed "gentlemens" agreement.
Reviewer #2 (Remarks to the Author): I am fully satisfied with the response of the Authors to my comments and questions. The manuscript has improved, and in my opinion it merits now the high Nature Communications standards of novelty and impact, so that I recommend it for publication in its present form.