Modeling the Dynamics of Electric Field-Assisted Local Functionalization in Two-Dimensional Materials

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript reviews a comprehensive study on the evolution of local functionalization of 2D interface. This topic is closely relevant to the journal’s scope. The references are appropriately cited.

Below are some points the authors should address:

 

Fig.1 shows the finite element calculations for two different materials. The annotations require more consistent. For example, a 1nm graphene is used in Fig.1a, whereas TMDC flake (20 nm thick) is shown in Fig.1b. Please clarify why different gate voltages were applied for the two materials. In Fig. 1b, it is also unclear which region corresponds to the substrate, and which corresponds to the MoSe₂ flake. Additionally,

Please explain why a thickness of 1 nm for graphene was chosen for graphene, while 20 nm was used for MoSe₂ in the simulation. Why were the number of layers NOT treated as a simulation parameter?

 

What is the thickness of graphene presented experimentally in Fig.3?  Does the 2D flake thickness influence the oxidization or reduction region change?

Author Response

For research article
Response to Reviewer 1 Comments
1. Summary
Thank you very much for taking the time to review this manuscript. It is evident that the
collective feedback from the reviewers has contributed decisively to the improvement of the
manuscript. Please find detailed responses below and the corresponding revisions/corrections
highlighted/in track changes in the re-submitted files.
Point-by-point response to Comments and Suggestions for Authors
Comments 1: Fig.1 shows the finite element calculations for two different materials.
The annotations require more consistent. For example, a 1nm graphene is used in
Fig.1a, whereas TMDC flake (20 nm thick) is shown in Fig.1b. Please clarify why
different gate voltages were applied for the two materials.
Response to reviewer
Both cases of Figs. 1a and 1b will help us verify the validity of our model, so we set the
structure and operating conditions of the model (tip diameter, applied voltage, etc.) as
close as possible to those used experimentally in Refs [15] and [28]. We are aware that
this fact is not clarified in the text, so we have added the following paragraph:
Actions taken
On page 6, 1st paragraph, lines 215-219
In both cases we have employed multilayer structures and experimental conditions (tip radius, applied
voltage, etc) that align with those utilized in two experimental references: [28] for graphene, and [15]
for MoSe2. These cases are used to verify the validity of the model and are discussed below.
Comments 2: In Fig. 1b, it is also unclear which region corresponds to the substrate, and
which corresponds to the MoSe₂ flake.
Response to reviewer
In line with the above, the approximately 20 nm thick MoSe2 flakes used in the reference
experiment [15] were deposited on an electrically grounded gold layer.
Actions taken
On page 5, Fig. 1b.
To clarify this aspect, we have made thicker the line representing the substrate under the
 
2
MoSe2 layer and labeled it as a “gold substrate.” (see new Fig. 1b)
Comments 3: Additionally, please explain why a thickness of 1 nm for graphene was chosen
for graphene, while 20 nm was used for MoSe₂ in the simulation.
Response to reviewer
It is well known that the theoretical atomic thickness of a single, perfect graphene layer in
vacuum is approximately 0.345 nm. However, the thickness attributed to a graphene
monolayer transferred onto a certain substrate is typically higher, depending on the
conditions of the substrate surface and adsorbates. Even the most widely method for
thickness characterization at the nanoscale, atomic force microscopy (AFM), can yield a
variety of results in the range from 0.4 nm to 1.7 nm when measuring the monolayer
graphene thickness on a substrate, because of tip-surface interactions and influence of
imaging forces (see new Ref. [40]). In Ref. [38], the step measured between a Si substrate
and one monolayer graphene results about 1.5 nm, whereas a subsequent statistical
method derived from measurements on multilayer structures (up to 4 layers) allows to
propose an average thickness of 0.4 nm between graphene layers. In general, graphene
suppliers like Graphenea (https://eu.graphenea.com/products/monolayer-graphene-onquartz-
4-wafer ) and Nanographenex (https://nanographenex.com/single-layergraphene-
oxide-purity-99-9-size-1-nm/ ) establish a thickness close to 1 nm. In this work,
for simplicity, we have rounded the graphene thickness to 1 nm. Since graphene is a
highly conductive layer, variations of a few angstroms around this thickness do not
significantly alter the subsequent electrostatic results of the work.
In line with the previous point, the thickness of 20 nm for the MoSe2 flakes has been
replicated from the information provided in Ref [15]. As the MoSe2 layer is a wider
bandgap semiconductor and is located on a conductive gold substrate, we considered it
important to faithfully respect the thickness data due to possible electrostatic
implications.
Regarding the thickness of graphene oxide functionalized directly from graphene, our
own measurements using AFM indicate that it raises about 1 nanometer above the
graphene surface [28]. Other works report a 1 nm step over a mica substrate for GO flakes
(see Ref. [40]). In this work we have included our own determination, as they align
precisely with the outlined structure. So, in line with our measurements, we have
assigned a thickness of 2 nm to GO.
With regard to the thickness estimation for the oxidized MoSe2, which is considered a
non-stoichiometric compound (MoOx+SeO2), few data are found. Furthermore, Ref. [15]
reports that the thickness increases with time. Since the theoretical thickness of MoSe₂ is
approximately twice that of graphene [42], we considered an oxide thickness of 3 nm for
simplicity.
Actions taken
On page 6, 2nd paragraph, lines 221-242, we have added:
The thickness attributed to a graphene monolayer transferred onto a certain substrate is typically
 
3
higher than that of the monolayer in vacuum, depending on the conditions of the substrate surface
and adsorbates. Atomic force microscopy (AFM) yields a variety of results in the range from 0.4
nm to 1.7 nm when measuring the monolayer graphene thickness on a substrate, because of tipsurface
interactions and influence of imaging forces [38, 39]. In this work, for simplicity, we have
rounded the graphene thickness to 1 nm. Since graphene is a highly conductive layer, variations of
a few angstroms around this thickness do not significantly alter the subsequent electrostatic results
of the work. In the case of MoSe2 flakes of Ref. [15], where a number of monolayers are involved,
we have considered, for simplicity, that the stack behaves as a homogeneous semiconductor with
the monolayer properties and the reported thickness of 20 nm. Regarding the thickness of graphene
oxide (GO) functionalized directly on graphene, our own measurements using AFM indicate that
it raises about 1 nanometer above the graphene surface [28]. Other studies report a 1 nm step over
a mica substrate for GO flakes [40], although it has been confirmed that the GO thickness depends
on the degree of oxidation [41] In this work we have included our own determination, as it aligns
precisely with the outlined structure, assigning a thickness of 2 nm to GO. Regarding the thickness
estimation for the oxidized MoSe2, which is considered a non-stoichiometric compound
(MoOx+SeO2), there is currently a lack of data. Given that the theoretical thickness of MoSe₂ is
approximately twice that of graphene [42, 43] and in view of the measured step at the initial stages
of the oxide growth [15], we considered an oxide thickness of 3 nm for simplicity.
Comments 4: Why were the number of layers NOT treated as a simulation parameter? Does
the 2D flake thickness influence the oxidization or reduction region change?
Response to reviewer
We appreciate this observation, as it pertains to a key element of our model, and it deserves
an additional explanation. In 2D materials, a multilayer implies non-covalent stacking of
individual layers. This description suggests that each layer roughly maintains the
conductive properties characteristic of each monolayer. In this context, the procedure of
local field-assisted oxidation, which supplies oxyanions mainly from the wetting layer,
should primarily affect the top layer. In the vertical direction of a multilayer, towards the
lower layers, oxygen must pass through already oxidized regions to carry out oxidation. In
the horizontal, lateral direction, it is incorporated by a different mechanism, according to
the principles described in this study, and progresses much more quickly [22, 23]. This
mechanism is approximately independent of the oxidation in the vertical direction.
Therefore, the number of monolayers should not affect the general formulation described in
section 3.3.
However, this formulation includes the reduction of the electrostatic potential barrier, ΔW0,
induced by the application of a local electric field. The structure of the sample, in particular
the location of the ground potential relative to the probe, does influence the electric field
and hence the ΔW0 factor.
In the case of MoSe2 flakes of Ref. [15], where a high number of monolayers are involved,
we have considered, for simplicity, that the stack behaves as a homogeneous semiconductor
with the monolayer properties, and we focus on the horizontal expansion of the oxidized
spot. Given that the flake is deposited on a metal substrate, and the oxidized spot is only a
few times larger than the flake thickness, one might guess that the number of layers should
 
4
influence the electric field by bringing the ground plane closer or farther away from the
probe. Consequently, it will affect the speed of oxide expansion. In most structures the field
distribution is not readily obtained analytically, underscoring the importance of finite
element calculations on a particular structure such as that of Fig. 1b.
In the case of graphene, given its semi-metallic character and the fact that the graphene layer
is grounded at the sample’s periphery, adding a second layer below the one to be oxidized
must significantly impact the electric field. This bilayer structure is similar to locating a
ground plane just below the layer to be oxidized. Consequently, the progression of oxidation
should undergo significant changes respect those of Ref. [28]. We are currently obtaining
preliminary experimental results on bilayer graphene that confirm this hypothesis. The
application of the proposed model to these samples is left as future work.
So we decided to summarize the arguments in the “Discussion” section, including some
details as future work.
Actions taken
On page 16, 2nd paragraph, lines 558-573, we have summarized the previous discussion as
follows:
The number of single layers involved in the material is not a parameter directly included in the general
formulation described in section 3, but its influence is manifested through the modifications it can cause
in the electric field, and eventually in the reduction of the energy barrier, ΔW0, for the incorporation of
oxyanions. For instance, the MoSe2 flakes referenced in [15], to which the model is applied, are deposited
on a gold metal substrate that is grounded with respect to the probe voltage. The number of layers
contributes to the proximity of the ground plane to the probe, thereby influencing the electric field. In the
case of graphene, given its semi-metallic character and the fact that the graphene layer is grounded at the
sample’s periphery, adding a second layer below the one to be oxidized must significantly impact the
electric field. This structure is similar to locating a ground plane just below the layer to be oxidized.
Therefore, the progression of oxidation should undergo significant changes respect those of Ref. [28]. We
are currently obtaining preliminary experimental results on bilayer graphene that confirm this hypothesis.
The application of the proposed model to these samples is left as future work. In most structures, the field
distribution is not readily obtained analytically, underscoring the importance of finite element calculations
on a particular structure such as those of Fig. 1.
Comments 5: What is the thickness of graphene presented experimentally in Fig.3?
Response to reviewer
The graphene used for the experiment shown in Fig. 3 is a monolayer on insulating quartz
substrate, which aligns with those of Ref. [28] as well as the results obtained along the work.
Regarding its thickness, the reasoned response for Comment 3 applies.
Actions taken
On page 9, 1st paragraph, line 314, we have clarified:

"in a graphene monolayer on quartz substrate"

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The presented paper “Modelling the Dynamics of Electric Field-Assisted Local 2 Functionalization in Two-Dimensional Materials” by Fernando Borrás, Julio Ramiro-Bargueño, Óscar Casanova-Carvajal, Alicia de Andrés, Sergio J. Quesada and  Ángel Luis Álvarez is devoted to the first principal modelling of the expansion of the oxidized region over time in two-dimensional (2D) materials. 

The topic of this paper is interesting and important. The results seem to be correct and  physically meaningful.  However, the presentation of the paper should be modified. The authors claim that the paper is devoted to first principal mathematical calculations however almost no equations are provided in this manuscript. My first and most important suggestion would be to increase the presentation of mathematical model  and mathematical formalism.  Otherwise there is no point talking about first principal calculations.

 The abstract is written in a not clear way which doesn't show what the exact contribution why this is important.  While it looks unnecessarily long and misses a clear message,  it is really important that the novel results obtained by the authors are clearly mentioned in the abstract,  which must be later checked against the existing literature.

 The method is presented for “2D  materials” which is vague and has no reference to specific lattices.   

Since the field-assisted effects are addressed,  It would be also interesting and useful for the readers if you could also discuss some tunnelling effects, specifically, anomalous photon-assisted tunneling in graphene, as well as photon-dressed electronic states in topological insulators: tunneling and conductance.

Author Response

For review article
Response to Reviewer 2 Comments
1. Summary
Thank you very much for taking the time to review this manuscript. It is evident that the collective feedback from the reviewers has contributed decisively to the improvement of the manuscript. Please find detailed responses below and the corresponding revisions/corrections highlighted/in track changes in the re-submitted files.
Point-by-point response to Comments and Suggestions for Authors
Comments 1: The authors claim that the paper is devoted to first principal mathematical calculations however almost no equations are provided in this manuscript. My first and most important suggestion would be to increase the presentation of mathematical model and mathematical formalism. Otherwise there is no point talking about first principal calculations.
Response to reviewer
We have included a detailed description of how the formulation has been developed. Consequently we have renumbered equations throughout the manuscript.
Actions taken
On pages 9, 2nd paragraph, 327-368 (until page 10):
A preliminary formulation for the expansion of the oxide radius over time was previously proposed [28]. In that work, the probability that one oxyanion incorporates covalently to graphene, P1OH, is considered to obey the classical Boltzmann statistics, and depends essentially on the energetic barrier to break the 2D layer bonds. Below we present a detailed formulation derived from fundamental principles, which ultimately yields an expression for the increase in the radius of an oxide spot as a function of time.
After identifying the driving force behind the incorporation of oxyanions around oxidized areas by applying a local electric field, the formulation begins establishing the differential relationship between the attachment of oxyanions (OH-) and the resulting increase in the spot radius (rGO).
d(OH−)=2πrGOSOHdrGO (1)
where SOH represents the effective area of one oxyanion. SOH may be estimated from the degree of oxidation typically obtained by the different techniques. In case of field-assisted oxidation a typical O/C is 0.35 [28]. Chemical techniques can achieve O/C ratios as high as 0.8 by applying Hummers' methods [53], though this approach is not applicable in this case.
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The probability that one oxyanion close to the boundary covalently attaches to the material, P1OH, depends on the aforementioned energetic barrier (W0). This probability can be formulated according to the classical Boltzmann statistics. In this scenario, the dipolar field at the interface between the oxide and pristine graphene is expected to produce a reduction of the energetic barrier ΔW0 (as illustrated in Fig.2). This reduction is, in turn, dependent on the radius rGO. Then, under an external voltage, P1OH, is expressed as:
P1OH=K·e−(W0−ΔW0?rb?)kT=K
?·eΔW0(rGO)kT (2)
where K
?=K·e−W0kT , is a parameter in [s-1].
The total rate of incorporation of oxyanions, d(OH-)/dt, can then be described by the probability of attaching one oxyanion (P1OH) multiplied by the number of available sites (C atoms in the case of graphene) along the entire spot perimeter (NOH), as well as the probability of one site being occupied (PS) in terms of the density of present oxyanions around. It is at this factor that the role of relative humidity must be considered:
d(OH−)dt=P1OH·(NOH·PS)=P1OH·(2πrGO·f(RH)) (3)
By substituting (2) in (3),
d(OH)dt=2πr0f(RH)r?K
?eΔW0(r?)???? (4)
where r0 is the probe radius, and ??̌=????????0 is the dimensionless relative radius of the oxide in relation to that of the probe.
Taking advantage of the relation between (OH) and rGO in (1), we can finally write the expansion speed of the relative oxide radius, dř/dt, as:
dr?dt=1α0eΔW0(r?)???? (5)
where α0=r0AOH·f(RH)·K
? [s]. From which, by integration, a function t(ř) describing inversely the expansion of the relative radius along time is obtained.
t=α0∫ee(W0−ΔW0(r?GO))KTdr?GOr?GO1 (6)
Comments 2: The abstract is written in a not clear way which doesn't show what the exact contribution why this is important. While it looks unnecessarily long and misses a clear message, it is really important that the novel results obtained by the authors are clearly mentioned in the abstract, which must be later checked against the existing literature.
Response to reviewer
A new redaction of the abstract structured according to the sequence of points indicated in the abstract template has been proposed. In this new redaction, the exact contribution and its relevance are highlighted, together with the methodology proposed.
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Actions taken
On page 1, lines 18-35.
Electric-field-assisted local functionalization of materials is a resist-free technique generally applied at the nanoscale, which has been understood within the paradigm of the water meniscus. Using a home-made prototype the authors applied this technique at scales compatible with the biosensor industry (tens of microns). However, interpreting these results requires a different paradigm. The expansion of the oxidized region over time in two-dimensional materials under a localized electric field is modeled from first physical principles. Boltzmann statistics is applied to the oxyanion incorporation at the perimeter of the oxidized zone, and a new general relation between oxide radius and time is formulated. It includes the reduction of the energy barrier due to the field effect and its dependence on the oxide radius. To gain insight into this dependence whatever the layers structure, 2D material involved or electrical operating conditions, simple structures based on multilayer stacks representing the main constituents are proposed, where the Poisson equation is solved using finite element calculations. This enables to derive energy barriers for oxyanion incorporation at varying spot radii which are consistent with those resulting from fitting experimental data. The reasonable agreement obtained provides researchers with a new tool to predict the evolution of local functionalization of 2D layers as a function of the fabrication parameters: time, applied voltage and relative humidity, solely based on materials properties.
Comments 3: The method is presented for “2D materials” which is vague and has no reference to specific lattices.
Response to reviewer
The proposed mechanism is effective for the horizontal or lateral progress of local anodic oxidation and is particularly optimum for two-dimensional materials, because in this case, the possibility of oxidation towards lower layers is eliminated, and any interference with that process, although responding to an independent mechanism, is avoided. While the idea and formalism proposed in this work are straightforward, they represent a novel concept that has not been previously explored. It is important to note that, due to its simplicity, this formulation can be applied to any 2D material, regardless of its crystal lattice. However, we recognize the need for further clarification on this assertion, although some details are already indicated in the "4. Discussion" section. For instance, the covalent bond that attaches the oxyanion to the layer is represented by the W0 factor (the energy barrier for that bonding,as expressed in Eq. (2) of the manuscript). In section “4. Discussion,” it is noted that the anisotropic structure of the crystal lattice exhibited by MoSe2 results in a characteristic triangular shape at the onset of local anodic oxidation in a low relative humidity environment. In such conditions, the process is restricted by the supply of OH- (water meniscus) to an area just below the tip that is nanometer-sized. This favors the manifestation of lattice anisotropy. However, when the relative humidity increases and water meniscus as well as electric field lines spread, the triangular anisotropy is diluted, and the shape of the spot tends to become circular. In this regime, the oxidation becomes driven by the dipolar electric field at the perimeter of the oxidized spot, which is the main principle assumed in our formulation.
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Actions taken
On page 16, 3rd paragraph, lines 574-591, following the reviewer's suggestion, we have rewritten the paragraph in section “4. Discussion”, where aspects about the crystal lattice of the 2D layer were considered.
With regard to the anisotropy exhibited by the lattice of some TMDs, the model proposed here is isotropic and does not consider dielectric properties in the form of an anisotropic tensor for the structures of Fig. 1. In particular, Ref. [15] presents an example in which, under conditions of low relative humidity (RH < 50%) and at the initial stages of the process, oxidation initiates in a triangular shape, inheriting the anisotropy of the crustal lattice. However, when RH increases (RH > 60%) and, therefore, both the water meniscus and the electric field lines spread, then the triangular anisotropy is diluted, and the shape of the spot tends to become circular. This experiment unveils the transition between two distinct oxidation regimes: when RH is low, the scale becomes smaller and the oxidation is limited by the oxyanion supply at the water meniscus. At this scale, oxidation may exhibit anisotropy according to the lattice structure. Alternatively, if RH is sufficiently high to guarantee oxyanion supply up to longer distances, the local anodic oxidation process tends to be dominated by the electrostatic dipolar field at the boundary between oxidized and non-oxidized regions, and the anisotropy is gradually lost with distance from the point where the potential is applied. This is an indication that reinforces the principles presented in this model. The calculations developed in Figs. 7 and 8 have been applied to these latter experimental conditions of Ref. [15]. This topic was previously discussed in section 3.4.
Comments 4: Since the field-assisted effects are addressed, It would be also interesting and useful for the readers if you could also discuss some tunnelling effects, specifically, anomalous photon-assisted tunneling in graphene, as well as photon-dressed electronic states in topological insulators: tunneling and conductance.
Response to reviewer
Let’s do some preliminary considerations on conventional tunnel effect in relation to the subject of this paper. During local anodic oxidation at the nanoscale, it has been recognized that the conventional tunneling effect may have an impact on oxide growth. In this case, the tunnel effect occurs between the tip and the layer to be oxidized, through the thin water bridge and the first oxide layers. In general, the diffusion of oxyanions is a slow process which contributes to space charge accumulation, which acts as a rate-limiting factor. In this case, electron tunneling is considered to help maintain the current flow, which is necessary to drive the ions towards the surface of the sample (see Ref. [25]).
This view does not correspond to our approach, since, as indicated in section “1. Introduction”, our work focuses on the electric field at the edge of the oxidized zone, and not below the probe tip, as is commonly done within the water meniscus paradigm. However, the dipolar structure formed under the application of voltage at the boundary between graphene oxide and pristine graphene could be susceptible to showing conventional tunneling effect for the free holes in graphene. In fact, the band alignment at the interface between GO and graphene is presumably of type I, because of the quasi-zero gap of graphene. However, the higher work function of graphene oxide tends to
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shift this alignment towards type II (Ref. 60). In any case, this results in a barrier for the free holes recombination (which precisely holds the bipolar action), which could be overcome by the tunnel effect, eventually including an additional current component. In our Ref. [28] different current components were studied during this process (Faradaic current, Mott-Gurney current and capacitive), although the tunnel component was not included as it was considered negligible. However, it is important to note that within our proposal, components that increase the parallel current, such as Mott-Gurney and tunnel current, should contribute to reducing the electric field (in contrast to what was expected for tunnel current within the meniscus paradigm). Consequently, the tunnel effect would eventually limit the oxidation rate.
After this introduction to the expected consequences of conventional tunnel effect applied to our model, let’s include some considerations about anomalous photon-assisted tunneling in graphene, as well as photon-dressed electronic states, in relation to the subject of this paper, local anodic oxidation. Firstly, we are grateful for this suggestion because photon-assisted local anodic functionalization is, to the best of our knowledge, an unexplored field that has the potential to be very exciting. In general, we think that any Floquet state contributing to tunneling will slow down the oxide progress. But we acknowledge the potential impact of light application on the electronic excitation of GO through electron-photon entangled states, which may contribute to enhancing the dipolar nature of the GO-graphene interface, and eventually the oxidation progress.
Going beyond the scope of this paper, the insulating GO spot, surrounded by a cloud of free holes concentrated in the near 30 nm, displays a specific architecture that may be considered the basis of a topological insulator, since it is insulator inside but highly conductive at the perimeter. From this point on, interaction with light of a specific frequency can create Floquet states in this interface, which change the conduction properties, opening up a wide range of possibilities.
Actions taken
On page 16, last paragraph, lines 599-620, we have summarized the previous discussion as follows:
Other effects that have not been taken into account in our model are the possibility of tunnel current from the free holes in graphene to GO. The band alignment at the interface between GO and graphene is presumably type I, given the quasi-zero gap of graphene. However, the higher work function of graphene oxide tends to shift this alignment towards type II [60]. In any case, this results in a barrier for the free holes recombination (which precisely holds the bipolar action). This barrier could be overcome by the tunnel effect, eventually including an additional current component. However, it is important to note that within our proposal, components that increase the parallel current, such as Mott-Gurney [28] and tunnel current, should contribute to reducing the electric field. Consequently, the tunnel effect would eventually limit the oxidation rate.
On the other hand, photon-assisted local anodic functionalization is an emerging field with significant potential. In general, it would be expected that any Floquet state contributing to tunneling will slow down oxide progress. However, the potential impact of light application on the electronic excitation of
GO through electron-photon entangled states should be considered on future works, as it may contribute to enhancing the dipolar nature of the GO-graphene interface and eventually the oxidation progress.
Finally, the insulating GO spot surrounded by a cloud of free holes concentrated in the near 30 nm, displays a specific architecture that may be considered the basis of a topological insulator, since it is insulator inside but highly conductive at the perimeter. From this point on, interaction with light of a specific frequency can create Floquet states in this interface. These states change the conduction properties, opening up a wide range of possibilities in Electronics.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

1. Similar models are already being introduced in the literature. What is the new here and why it matters?
2. The introduction needs further revisions and references.
3. The line significant wetting layer but its need more quantification, that how much water is assumed and at what humidity the layers has formed.
4. The term “electric field rebound” what is this statement means and why this rebound occurs.
5. It would benefit if this work shows how well model matches the experimental work.
6. Include a comparative table comparing the performance with similar model.
7. Verify that all units and symbols are formatted consistently.
8. Add more recent references from the last few years.

Comments on the Quality of English Language

Need improvement 

Author Response

Response to Reviewer 3 Comments
1. Summary
Thank you very much for taking the time to review this manuscript. It is evident that the
collective feedback from the reviewers has contributed decisively to the improvement of the
manuscript. Please find detailed responses below and the corresponding revisions/corrections
highlighted/in track changes in the re-submitted files.
3. Point-by-point response to Comments and Suggestions for Authors
Comments 1: Similar models are already being introduced in the literature. What is the new
here and why it matters?.
Comments 2: The introduction needs further revisions and references.
Comments 8: Add more recent references from the last few years.
Response to reviewer
In this particular instance, we believe that it is appropriate to address the three points
simultaneously, given that most of the actions taken are related and encompassed within
the “1. Introduction” and “3. Results” sections.
As stated throughout the manuscript, our formulation focuses on the lateral or horizontal
expansion of oxidation, which is the main process that can occur in 2D layers. The progress
of local anodic oxidation in the vertical direction (height) at the nanoscale in bulk materials
has been extensively studied, even from first physical principles using the seminal
expressions by Cabrera-Mott and others. However, the lateral (horizontal/superficial)
expansion is a more complex phenomenon. On one hand, within the water bridge paradigm
it required understanding the electric field distribution and flux of oxyanions towards the
periphery of the water meniscus. And, on the other hand, it is challenging to describe the
lateral evolution of oxidation over time, since the water meniscus is a static concept. For this
reason, the key mathematical relationships that are often used are empirical fits to
experimental data, rather than universally applicable first-principles equations, as is the
novel approach described in our manuscript. This matters, not only because of the novelty
of the proposal (which focuses on the relevance of the dipole field at the periphery of the
oxidized zone, which had been overlooked in the past), but also because at this basic level,
the model includes the physical properties of the materials and the structures of the samples.
Thus, as stated in the abstract, “it provides researchers with a new tool to predict the evolution of
local functionalization of 2D layers as a function of the fabrication parameters: time, applied voltage
and relative humidity, solely based on materials properties.”. Eventually, as also mentioned in the
Introduction section, the value of this method for applications is evident as “it enables control
of extensive functional regions within the channel of field-effect transistor-based biosensors”.
Then, we have completed the section titled "1, Introduction," in which we discuss the
 
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existence of models that study the growth of oxidation in the vertical direction, although we
will be providing more detailed commentary on the empirical models that describe the
lateral expansion of oxidation.
In addition, new references have been added in two relevant aspects: i) those related to the
models that attempt to describe the process (section “1. Introduction”); ii) the justification
of some approaches or hypotheses of the work, for example, the choice of layers thickness
(section “3. Results”).
Actions taken
On page 2, 3rd paragraph, lines 73-90, we have added:
A number of studies have focused on the water meniscus shape [16-18] as well as on mathematical
formulations of the oxide growth in the vertical direction (height) over time within the volume provided
by this water bridge. [19-21] For a comprehensive overview of the existing models, please refer to [8].
However, for moderate to high RH values (e.g. RH > 70%), when a significant wetting layer is created
over the material, it has been observed that the oxidation progresses in the surface plane up to distances
orders of magnitude greater than those in the vertical direction towards the interior of the sample [22,23].
Experimental studies on Si and MoSe2 flakes have confirmed that the oxidized region in bulk materials
exhibits a distinctive "UFO" shape over time, slowly growing in height just below the probe, and much
faster in the lateral dimension. [10, 14, 15, 24] Therefore, the horizontal or lateral progress of oxidation
appears to follow a different and independent mechanism. Within the water meniscus paradigm, the lateral
expansion of oxidation is a complex phenomenon influenced by the electric field distribution within the
water bridge and wetting layer, together with the initial charge density formation and the subsequent lateral
diffusion of ionic species for a set of different reactions. [21, 25] Mathematical formulations have so far
employed empirical fits to experimental data [25-27], rather than universally applicable first-principles
equations. Table 1 provides a summary of various empirical models that have been utilized.
On page 6, 2nd paragraph, lines 221-242, we have included additional information regarding
the election of the various layer thicknesses within the modeled structures.
The thickness attributed to a graphene monolayer transferred onto a certain substrate is typically
higher than that of the monolayer in vacuum, depending on the conditions of the substrate surface
and adsorbates. Atomic force microscopy (AFM) yields a variety of results in the range from 0.4 nm
to 1.7 nm when measuring the monolayer graphene thickness on a substrate, because of tip-surface
interactions and influence of imaging forces [38, 39]. In this work, for simplicity, we have rounded
the graphene thickness to 1 nm. Since graphene is a highly conductive layer, variations of a few
angstroms around this thickness do not significantly alter the subsequent electrostatic results of the
work. In the case of MoSe2 flakes of Ref. [15], where a number of monolayers are involved, we have
considered, for simplicity, that the stack behaves as a homogeneous semiconductor with the
monolayer properties and the reported thickness of 20 nm. Regarding the thickness of graphene oxide
(GO) functionalized directly on graphene, our own measurements using AFM indicate that it raises
about 1 nanometer above the graphene surface [28]. Other studies report a 1 nm step over a mica
substrate for GO flakes [40], although it has been confirmed that the GO thickness depends on the
degree of oxidation [41] In this work we have included our own determination, as it aligns precisely
with the outlined structure, assigning a thickness of 2 nm to GO. Regarding the thickness estimation
for the oxidized MoSe2, which is considered a non-stoichiometric compound (MoOx+SeO2), there is
 
13
currently a lack of data. Given that the theoretical thickness of MoSe₂ is approximately twice that of
graphene [42, 43] and in view of the measured step at the initial stages of the oxide growth [15], we
considered an oxide thickness of 3 nm for simplicity
A total of 18 new references have been cited, of which 7 were reported in the last 5 years.
Comments 3: The line significant wetting layer but its need more quantification, that how
much water is assumed and at what humidity the layers has formed.
Response to reviewer
We acknowledge that the expression "significant wetting layer" used in section "5.
Conclusions" is imprecise. The context of the manuscript in which this expression is used
explains that the wetting layer is directly linked to the concept of relative humidity (RH).
The underlying idea is that the process needs to be carried out at a certain level of RH in
order to create a sufficient wetting layer. This layer is necessary for verifying the expansion
of oxidation according to the principles outlined in our model: a regime driven by dipolar
electric field, where the oxyanions supply comes from de close environment. The outcome
of this process depends on the specific characteristics of the material surface and their degree
of hydrophobicity. There is no specific "onset" for this RH, as the generation of a wetting
layer occurs gradually. Despite this, we can find references in literature on the RH level for
which the expansion of oxide is an evident phenomenon. With reference to Si, Kuramochi
et al. [22 and 23] observe that for RH > 60%, the oxide spot already expands to distances
orders of magnitude greater than the oxide thickness in the vertical direction. According to
the findings of Avouris et al., [10], the aspect ratio (height/width) of the oxide dots
demonstrates a substantial decrease as soon as RH reaches 61%. Regarding the work on
MoSe₂ flakes, Borodin et al. [15] note that the anisotropy in oxidation is lost in favor of
circular and larger spots for relative RH between 60-65%. With reference to graphene, in Fig.
4 of the manuscript we observe that the size of the oxide spot grows abruptly from 80%, but
from 60-70% we already observe a significant oxide progression. In this regard, note that in
section “1. Introduction”, we have already mentioned that a significant wetting layer is
generated “for moderate to high RH values (e.g., RH > 70%)”.
Actions taken
On page 17, 4th paragraph, lines 622-632, in light of the reviewer's suggestion, in section “5.
Conclusions” we have been more specific with the expression “significant wetting layer”:
In summary, a model grounded in physical principles has been developed to describe the phenomenon of
electric field-assisted local functionalization in 2D materials. The model illustrates how the oxidized spot’s
radius expands horizontally over time, driven by the dipolar electric field at the boundary between oxidized
and pristine material, which results from the applied voltage. To that end, Boltzmann statistics and relevant
physical quantities are taken into account, including the energy barrier for the incorporation of oxyanions
and its dependence on radius. This regime is particularly effective in conditions that ensure a substantial
wetting layer, facilitating the supply of oxyanions from the immediate environment. According to the
materials under consideration (silicon, graphene, and MoSe2 flakes), this typically takes place when the
relative humidity (RH) is greater than 60–70%.
 
14
Comments 4: The term “electric field rebound” what is this statement means and why this
rebound occurs.
Response to reviewer
In this context, the term "electric field rebound" referred to an upturn or sharp increase in
the value of the electric field. This phenomenon occurs due to the dipole distribution of
charges at the boundary between oxidized and non-oxidized regions in response to an
applied voltage.
Actions taken
To avoid any confusion that this term may cause, the new wording is as follows:, on page 8,
1st paragraph, line 283, where it said “results in a significant rebound of the electric field”
now it says, “results in a significant upturn or sharp increase of the electric field”.
And on page 17, 5th paragraph, lines 636-637, where it said, “attributed to a substantial
rebound in the electric field”, now it says “attributed to a substantial increase in the electric
field”
Comments 5: It would benefit if this work shows how well model matches the experimental
work.
Response to reviewer
In view of the reviewer’s suggestion, we consider very appropriate to include a new Fig. 6
in subsection 3.3. Formulation of the expansion of the oxidized region over time in graphene,
which allows to verify how well the model matches the experimental work.
Actions taken
On page 12, a new Figure 6 with suitable captions (lines 430-433) have been included.
On pages 12-13, lines 425-438 the following comments have been added to describe results of
this new Fig. 6:
The result is shown in Fig. 6 as the blue solid line, where Vb = −30V and relative humidity RH = 95%
have been considered. The observed deviation in the spot behavior beyond two seconds as compared to
the model’s prediction, is associated with the deformation of the circular shape of the spots. At this limit,
for large spot diameters (rGO/r0 > 6), it is interpreted that the dipolar electric field driving the expansion is
comparable to the fluctuations caused by lattice defects (grain boundaries, dust particles, etc). This results
in irregular progress of oxidation.
Comments 6: Include a comparative table comparing the performance with similar models.
Response to reviewer
A new Table 1 has been added, compiling the most commonly used models and empirical
expressions for the lateral expansion of oxidation over time.
As illustrated in Fig.6 of Ref. [25], Fig. 1b of Ref. [26] and Fig. 4 of Ref. [15], the performance
of some of those models in fitting the experimental data is highly satisfactory. However, one
 
15
must be careful with this assessment, as empirical expressions with a sufficient number of
parameters offer great flexibility when adjusting to a variety of different trends, even if
different causes are mixed together. However, in the case of our model, as it is derived from
specific physical quantities, in case of not being able to adequately adjust a trend, it enables
us to clearly identify and discuss the causes for this deviation. This is precisely what has
been discussed in the paragraphs referencing Fig. 7 last paragraph on page 13, where we
have derived noteworthy conclusions regarding various regimes that may operate within
the scope of these measurements. We believe that this further reinforces the relevance of our
model.
Actions taken
Please refer to Section “1. Introduction” on page 3, line 94, where Table 1 has been included. This
table presents a compilation of the most commonly used empirical models to describe the lateral
expansion of oxidation over time, including specific expressions, parameters, and their
respective sources.
Comments 7: Include a comparative table comparing the performance with similar models.
Response to reviewer
We appreciate this suggestion. We have conducted a thorough review of the variables and
symbols, with their respective subscripts, and corrected some errors detected.

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

Accept