P Bourdon
email: bourdon@sic.sp2mi.univ-poitiers.fr

O Alata
email: alata@sic.sp2mi.univ-poitiers.fr

G Damiand
email: damiand@sic.sp2mi.univ-poitiers.fr

C Olivier
email: olivier@sic.sp2mi.univ-poitiers.fr

Y Bertrand
email: bertrand@sic.sp2mi.univ-poitiers.fr

Geometrical and topological informations for MCMC based image segmentation

The image segmentation methods based on Markovian assumption consist in optimizing a Gibbs energy function which depends on the observation field and the segmented field. This energy function can be represented as a sum of potentials defined on cliques which are subsets of the grid of sites. The Potts model is the most commonly used to represent the segmented field. However, this model expressed just a potential on the classes for nearest neighbour pixels. In this paper, we propose the integration of global informations, like the size of a region, in the local potentials of the Gibbs energy. To extract these informations, we use a representation model well known in geometric modeling: the topological map. Results on synthetic and natural images are provided showing improvements in the obtained segmented fields.

Introduction

The main objective of image segmentation methods is to find areas of homogeneous pixels. In such context, there is two main researches axes: the boundary and the region based methods [START_REF] Gonzales | Digital image processing[END_REF]. Fusions of these two approaches have also been proposed. Nevertheless, whatever approach you are using, a partitioned image is the result of the segmentation operation. This partition is composed of subsets called "classes" for a set of homogeneous pixels, and "cluster" for a set of homogeneous and related pixels.

The interest of such treatment can be found in many fields like image compression [START_REF] Li | Image segmentation and compression using hudden Markov models[END_REF], biomedical image analysis [START_REF] Capelle | Unsupervised algorithm for the segmentation of 3-d magnetic resonance brain images[END_REF],. . . For two decades, Monte-Carlo Markov Chain (MCMC) methods have received increased interest. This is due to a rigorous mathematical background and the growing power of computers. From the original work of Geman & Geman on image restoration [START_REF] Geman | Stochastic relaxation, gibbs distribution, and the bayesian restoration of images[END_REF], these methods have been applied successfully to the problem of image segmentation using different implementations, i.e. using stochastic or deterministic algorithms [START_REF] Bouman | Multiple resolutions segmentation of textured images[END_REF] [START_REF] Kervrann | A markov random field model-based approach to unsupervised texture segmentation using local and global spatial statistics[END_REF].

In such approaches, the image is considered as a hierarchical field composed by an observation field and a segmented or label field. Different parametric models have been proposed to represent observation field's textures. [7] [8]. But in most cases, the commonly used Potts model [START_REF] Wu | The potts model[END_REF] is proposed for represent the label field. This model only takes into account the label values of the nearest neighbours in the Gibbs energy function. Geman & Geman [START_REF] Geman | Stochastic relaxation, gibbs distribution, and the bayesian restoration of images[END_REF] tried to add more informations to the Potts model by using the boundaries estimated during the optimization process. Some boundary configurations were penalized. In [START_REF] Geman | Boundary detection by constrained optimization[END_REF], this idea has been extensively used. A drawback can be seen to this approach in the sense that a priori weighted values were given to each boundary configurations.

In this paper, we propose to add informations to the energy function from the label field's geometry and topology. These informations, like the form or the size of a cluster, could be considered as "global" as opposed to "local" information provided by the Potts model. Of course, the Potts model will be still used cause of its regularization effect but potentials functions taking account "global" information will be added in the Gibbs energy function.

To get these additional informations, we propose the use of a combinatorial model: the topological maps. This structure encode all topological and geometrical informations contains in the image in an efficient way.

Our segmentation algorithm consists in two steps and it is unsupervised considering the number of classes and the model parameters. A first step allows to estimate the number of classes and the model parameters of a gaussian mixture. A Stochastic Expectation Maximization (SEM) algorithm is used [START_REF] Celeux | The sem algorithm : A probabilistic teacher algorithm derived from the em algorithm for the mixture problem[END_REF] as in [START_REF] Cariou | Unsupervised texture segmentation using 2-d ar modeling and a stochastic version of the em procedure[END_REF]. During a second step, a simulated annealing is done in order to obtain the final label field. This is the step in which we will introduce topological or geometrical informations obtained from the topological map.

In the second part, we start with a brief summary on the principles of markovian segmentation. Then, after having explained in the third part the operation of topological maps, we will propose in fourth part an integration model to include this map in a markovian segmentation algorithm. In the last, we propose some results on synthetic and natural images.

Markovian segmentation algorithm 2.1 Definitions

Let X = {X s , s ∈ S} a family of random variables on a regular grid S = {s 1 , s 2 , . . . , s N } which is a finite subset of Z and an image x = {x s , s ∈ S}, realization of X. Each X s has the same state space which can be Ω x = {0, 1, . . . , 255}, the set of gray pixel values, Ω x = R, . . . We suppose now that there is an another random field Z = {Z s , s ∈ S} called the label field with state space Ω z = {c 1 , c 2 , . . . , c K } for which a realization will be z = {z s , s ∈ S}. In our case, a label represents the class to which the site belongs.

From this definition, we suppose that X and Z are defined in a hierarchical way : for each given z s , we have a conditional model for X s which can be, for example an AR model or a Gaussian distribution [5] [8].

The likelihood method defines the a posteriori law P (Z = z/X = x) as the probability to get one specific realization of the labels field knowing the observations field:

P (Z = z/X = x) = P (X = x, Z = z) P (X = x) (1)
Using the Bayes formula, we can also write equation 1 in function of the a priori law:

P (Z = z/X = x) = P (X=x/Z=z)P (Z=z)

P (X=x) ∝ P (X = x/Z = z)P (Z = z) (2)
since we consider the observations field's probability P (X = x) as a constant value.

From now, we will simply write:

P (z/x) ∝ P (x/z)P (z) (3)
Maximizing this a posteriori law (MAP) is equivalent to getting the most probable labels field's realization, knowing the observations field.

Markov field and Gibbs distribution

We call V = {v s } a site neighborhood system for S. A neighborhood system has the two following properties: s / ∈ v s and s ∈ v r ⇔ r ∈ v s . A subset C ⊆ S is a clique if every pair of distinct sites in C are neighbors and C is the set of cliques. In our study, we will only use the neighboring system based on the four nearest neighbors (4-connecivity) and the eight nearest neighbors (8-connectivity). A random field Y = {Y s , s ∈ S} is a markovian field associated to the system v s if and only if:

P (Y = y) = P (y) > 0, ∀y ∈ Ω Y P (Y s /Y r , r ∈ S -{s}) = P (Y s /Y r , r ∈ v s ) (4)
From Hammersley-Clifford's theorem, the distribution function of a Markov random field follows a Gibbs' law:

P (y) = exp (-U (y)) Z y (5)
in which U is an "energy function". U can be written as:

U (y) = C∈C Φ C (y) (6)
where

Φ C : Ω |C| y → R is the potential corresponding at C with Φ C (y) = Φ C (y C ).
|A| is the cardinality of the A set and in the special case of a clique, it is called its order. y C denotes the set {y/y ∈ C}. Therefore, if the markovian assumption is done about the hierarchical field of the part 2.1, (X, Z), the equation 3 can be written in a general way as follows:

P (z/x) = exp(-U 1 (x, z) -U 2 (z)) Z(x) (7)
in which U 1 is an "energy function", depending on the conditional observation fields, and U 2 another energy function depending on the label field. Z(x) is a normalization constant depending on x. Let define now U (z) = U 1 (x, z) + U 2 (z) as x is the given observation field.

The MAP estimated of z (see part 2.1) can be written now:

ẑ = arg max z∈Ωz [P (z/x)] = arg min z∈Ωz [U 1 (x, z) + U 2 (z)] (8)
We have to optimize U : Ω |S| z → R. Simulated annealing (SA) based on Gibbs sampler has been shown to be adapted to such an optimization problem. In the next part, we present the stochastic model we have chosen and the form of the corresponding potentials.

Stochastic modeling

The conditional field X is now supposed to be composed of K textures. So, we have K probability laws describing the gray pixel value variations in x for the different clusters belonging to the K textures. Different choices can be done for this probability laws [START_REF] Barker | Image segmentation using Markov random field models[END_REF]. In this paper, we are making the assumption that our observation field is a mixture of K i. i. d. (independently and identically distributed) gaussian process. This assumption allows us to use Maximum Likelihood based algorithm like SEM [START_REF] Celeux | The sem algorithm : A probabilistic teacher algorithm derived from the em algorithm for the mixture problem[END_REF] for estimating K and the different model parameters, θ k = {µ k , σ k }, k = 1 . . . K, where µ k and σ k are respectively the average and the standard deviation of the texture k. Now, we suppose that K and θ = {θ k , k = 1 . . . K} are known. Then, we will have:

P (x/z) = s∈S f (x s /z s ) (9)
with f (x s /z s ) is a gaussian law N (µ zs , σ zs ). Therefore, we can write U 1 (x, z) as:

U 1 (x, z) = s∈S (x s -µ zs ) 2 σ 2 zs + log (σ zs ) (10)
This energy can be seen as a sum of local potential based on one order cliques:

Φ 1 s (x s , z s ) = (x s -µ zs ) 2σ 2 zs + log (σ zs ) (11)
It remains to propose a model for the label field. In many papers, the Potts model is used [START_REF] Barker | Image segmentation using Markov random field models[END_REF]. This assumption corresponds to a general hypothesis on the label field: the different classes can be permuted without any influence on the probability law. We propose to use a more general formalism mixing the Potts model, as it is used in [START_REF] Bouman | Multiple resolutions segmentation of textured images[END_REF], and the general auto-model from Besag [START_REF] Besag | Spatial interaction and the statistical analysis of lattice systems[END_REF] in order to include some additional informations to Potts model:

U 2 (z) = s∈S Φ s (z s ) + β i,j 1 (1 -δ (z i , z j ))+ β √ 2 i,j 2 (1 -δ (z i , z j )) (12)
where for p = 1, 2:

(i, j) ∈ S, i = j : i, j p ⇔ ∃C ∈ C/ {i, j} ∈ C, i -j 2 = √ p (13)
So, it is possible now to express the local probability law at site s ∈ S:

p s (z s /x s , z s ) = p s (z s /x s , v s ) = exp (-U s (x s , z s )) Z s (x s , v s ) (14)
with z s = {z s /s ∈ S -{s}} and

U s = Φ 1 s (x s , z s ) + Φ s (z s ) + Φ P s (z s ) Φ P s (z s ) = β r∈vs: s,r 1 (1 -δ (z s , z r ))+ β √ 2 r∈vs: s,r 2 (1 -δ (z s , z r )) Z s (x s , v s ) = zs∈Ωz exp (-U s (x s , z s )) (15)
Φ s must be chosen to respect the context of applicability of MCMC methods like SA [START_REF] Geman | Stochastic relaxation, gibbs distribution, and the bayesian restoration of images[END_REF]. Moreover, SA implies the introduction of a temperature T in the Gibbs distribution. The local probability can then be written as follows:

p s (z s /x s , z s ) = p s (z s /x s , v s ) = exp (-U s (x s , z s /T )) Z s (x s , v s , T ) (16)
Φ s will allow us to integer geometrical and/or topological informations, i.e. "global" informations, on the segmented field at each iteration. Let's present now the topological map.

Topological map

Topological maps allow to represent the nD regions segmented images. They encode at the same time topology and geometry of images. Topological maps are combinatorial maps with particular properties. So we are beginning by recalling the notion of combinatorial map before present the topological map structure.

Combinatorial maps

Combinatorial maps were introduced in the sixties by [START_REF] Cori | Un code pour les graphes planaires et ses applications[END_REF], at first as a planar graph representation model. They were extended by [START_REF] Lienhardt | Subdivision of n-dimensional spaces and n-dimensional generalized maps[END_REF] in dimension n to represent orientable or not-orientable quasi-manifold. They encode space subdivisions and all the incidency relations. They are made of abstract elements, called darts, on which are defined application, called β i . We are giving here the combinatorial map definition in n dimensions, that we can find for example in [START_REF] Lienhardt | Topological models for boundary representation: a comparison with n-dimensional generalized maps[END_REF]:

Definition 1 (combinatorial maps) Let n ≥ 0. A n combinatorial map, (or n-map) is an (n + 1)-tuple M = (B, β 1 , . . . , β n ) where :

1. B is a finite set of darts;

2. β 1 is a permutation on B;

3. ∀ 2 ≤ i ≤ n, β i is an involution on B; 4. ∀ 1 ≤ i ≤ n -2, ∀ i + 2 ≤ j ≤ n, β i • β j is an involution.
In this definition, there is an application β i for each space dimension which puts in relation two i dimension oriented cells. When two darts are linked with β i , they are said β isewed. Each space cell is implicitly represented by a set of darts. We can see figure 1

Topological map

A topological map's construction needs the cluster-based partition of the image, as opposed to its classes-bases partition we get from the markovian segmentation algorithm. So its construction will be preceded by a clusters growth algorithm, introduced to split all clusters belonging to the same class. The topological map's construction consists in taking the pixels' border elements, to coherently define the image's topology: the space is subdivided in elements and borders' elements, with which are provided incidence and adjacency relationships. Figure 2 represents a clusters-based segmented image and its interpixel borders. [START_REF] Bertrand | Topological encoding of 3d segmented images[END_REF][START_REF] Bertrand | 2 dimensionnal topological map and optimal extraction algorithm[END_REF]

pro- R 1 R 2 R 3 R 4 R 5 R 0 a. A segmented image R 0 R 1 R 2 R 3 R 4 R 5

b. Its clusters' interpixel borders

Figure 2: Interpixel Vision poses to redefine the topological map in the 2-dimensional space, introduced by Luc Brun in [START_REF] Bracquelaire | Image segmentation with topological maps and interpixel representation[END_REF], a simpler way, so it makes it easily expendable in n-dimensional spaces. This definition uses simplification levels we show below:

1. Level 0: Full map It is the starting point of the process. This map doesn't code the image's borders but simply all its interpixel elements. It is composed by squared faces, each of them represent a pixel of the image (figure 3).

Figure 3: Level 0: Full map 2. Level 1: Linel map Level 1 map is obtained from the level 0 map and encodes the image's interpixel borders. This is done by removing every edge of that level 0 that doesn't belong to a border line of the image (figure 4).

R 0 R 1 R 2 R 3 R 4 R 5
Figure 4: Level 1: Linel map 3. Level 2: Borders map It simplifies the linel map, by using edges to represent parts of a straight-line border instead of interpixels belonging to a border line of the image (figure 5).

Level 3: Topological map

From this last simplification level results the topological map, which now associates a single edge to each border (figure 6). This last level is the topological map which encodes all topological and geometrical information of the represented image. Moreover, it is minimal, stable for rigid transformations and unique. So a topological map is characteristic of an image. These properties ensure that this model is an efficient way to encode information contains in an image. So we are going to use this model to extract some global informations, for instance size or shape of clusters, number of adjacent clusters, clusters totally included into another. . . The following part proposes an exploitation of some of these informations.

R 0 R 1 R 2 R 3 R 4 R 5 Figure 5: Level 2: Borders map R 0 R 1 R 2 R 3 R 4 R 5

Integration of geometrical and topological informations 4.1 Unsupervised segmentation

The main purpose of this work is to make an algorithm able to treat any image. So we are looking for wanted or unwanted geometrical or topological features that could appear in the estimated label field. One of these unwanted features is the presence of isolated pixels or small clusters. So, the first potential Φ s (see equa-tions 14 and 15) we propose is a "size" potential: It imposes a minimal size, s min , under which the probability of pixel s to belong to a cluster will be disadvantaged. s min threshold's can be chosen proportional to the size of the image to segment. Disadvantaging a site's membership to a cluster also equals Size potential evolution in function of cluster size minimizing its a posteriori law, or maximizing the U energy of the Gibbs probability. The chosen "size" potential function (figure 7) is written:

Φ size s = γ (1 + 1 (|As(zs)>smin|) × (exp (-κ (|A s (z s ) -s min |)) -1)) (17)
with

1 (|As(zs)|>smin) = 1 if |A s (z s )| > s min 0 otherwise and A s (z s )
is the connected set of sites r such as z r = z s , i.e. the cluster at which the site s belongs. γ and κ are two constants which represents respectively the weight of the potential in the Gibbs energy function and the exponential decrease of Φ size s . To introduce this part, we have evoqued our intention to make an unsupervised algorithm. We are then now facing the difficulty to find wanted or unwanted features on the label field other than small clusters: we want the topological map to give us informations about the shape or the adjacency of clusters and use these informations to segment the image. But, advantaging or disadvantaging a cluster in function of its shape or adjacency is a choice that can only be made by the user, for a given image. So, this is the purpose of the next part.

Semi-supervised segmentation

Let's take two examples: an image representing geological cuts (figure 8) and the "House" image (figure 9). In the first case, the purpose of segmentation is to keep the identified seismic faults (see 8.b). Our algorithm, at its present state, is unable to get a satisfying segmentation of that image. But if we decide to use the fact that fails represent relatively long and narrow, mainly vertical, clusters, and ask it to advantage this type of clusters, the result could become convincing. In the case of the "House" image, the window to our left is a rather small cluster that could disappear on during the segmentation, although we would like to keep it. This cluster has an interesting propriety: it is adjacent to only one cluster (the wall). Here is another information we can easily get from the topological map. To reduce the risk to see this In MCMC methods, the final result of the Markov Chain process depends not on the initial law on X. So, we used the last iteration of the SEM algorithm as initialization of the SA. We start from an initial temperature T 0 = 1 and use the classical decrease scheme: T k = 0.99 k T 0 . We fix β = 1. notice on this first segmented image a few incoherences: badly-defined borders, a few "isolated" pixels, that don't look meaningless when showing at the original image, but that we don't want to show off on the final partition. To increase β is not a solution because it will produce a decrease in the accuracy of boundary estimation. However, additional informations about the image's clusters, such as their size, or adjacency, could be used to perform these kinds of enhancements. These informations, we call "global", as opposed to our previously used "local" informations, can be extracted from the topological map. Figure 10.d shows the estimated label field when using SA with the size potential with s min = 2, κ = 1.5 and γ = 10.

We can notice on figure 10.d the disappearing of isolated pixels and still exact boundaries. So, the size potential allows an enhancement of the segmentation process. Let's see now on a natural image.

Natural image

We used the "House" (128 × 128) image (see figure 11.a).

Conclusion

The algorithm we have proposed shows up as a markovian relaxation algorithm, to which are added global informations about the image, extracted from the topological map, as energy potentials, thanks to the Gibbs distribution. It should be able to deal with any image, as long as the user provides it an information, as simple as it can be, about the cluster he wants to advantage or disadvantage's characteristics. A possible way to handle this would be a user interface, on which choices such as "advantage long and narrow clusters", "advantage vertical clusters", "advantage clusters adjacent to only one cluster", etc. In the case of "difficult" images, we could also remove the learning phase (SEM) and directly provide the relaxation algorithm the image's stochastic model's parameters. Then it would be supervised segmentation. Moreover, there are many different possibilities to add some more topological or geometrical informations to the simulated annealing. We currently work at the definition of some criterion based on some of these informations. We think that these new criterion can strongly improve the result of our final segmentation.

  .a an example of an image and figure 1.b the corresponding combinatorial map. Each dart is represented by a segment, the β 1 relation by light grey arrows and the β 2 relation by dark grey arrows. β 1 put in relation a dart and the next dart of the same face. For example, the light grey face of the image is represented by four β 1 -sewed darts in the map. The adjacency between this face and the dark grey face is represented by two darts β 2 -sewed together. We are using the simplified representation (figure 1.c) that does not represent explicitly the applications, because it is more understandable. a. A 2D object. b. Correspond-

Figure 1 :

 1 Figure 1: An object and the corresponding combinatorial map represented by two different ways.

Figure 6 :

 6 Figure 6: Level 3: Topological map

Figure 7 :

 7 Figure 7: Φ size s

Figure 9 :

 9 Figure 8: Geological cut

Figure 10 .

 10 c shows the result of the SA after 200 iterations without taking account the potential on clusters. We can a. Original image. b. First step segmentation : SEM's last iteration. c. SA Without size potential. d. SA With size potential.

Figure 10 :

 10 Figure 10: Synthetic image

Figure 11 .

 11 b show the last iteration of the SEM algorithm we used as an initialization for SA algorithm. On figure11.c and 11.d, we can notice that the noise on the wall without the use of the size potential disappears when using it.(a completer)


Olivia Guest

Nicolas P Rougier

Diversity in Reproducibility

In our previous contribution, we proposed computational modelling-related definitions for replicable, i.e., experiments within a model can be recreated using its original codebase, and reproducible, i.e., a model can be recreated based on its specification. We stressed the importance of specifications and of access to codebases. Furthermore, we highlighted an issue in scholarly communication -many journals do not require nor facilitate the sharing of code. In contrast, many third-party services have filled the gaps left by traditional publishers (e.g., [START_REF] Binder | mybinder[END_REF]GitHub, 2007;Open Science Framework, 2011;[START_REF] Rescience | The ReScience Journal[END_REF]. Notwithstanding, journals and peers rarely request or expect use of such services. We ended by asking: are we ready to associate codebases with articles and are we prepared to ensure computational theories are well-specified and coherently implemented?

Scope of Evaluation

Dialogue contributions include proposals for: intermediate levels between replicability and reproducibility (Crook, Hinsen); going beyond reproducibility (Kidd); encompassing computational science at large (Gureckis & Rich, Varoquaux); and addressing communities as a function of expertise (French & Addyman). On the one hand, some replies discuss evaluation more broadly, empirical data collection, and software engineering. On the other hand, some delve into the details of evaluating modelling accounts. We will discuss the former first.

In Varoquaux's contribution, reproducibility includes replicability and code rot (e.g., in fMRI: [START_REF] Eklund | Cluster failure: Why fmri inferences for spatial extent have inflated false-positive rates[END_REF]. However, the titular computational reproducibility is orthogonal to maintaining a re-usable codebase. Software and hardware inevitably go out of fashion meaning codebases expire. Nevertheless, the overarching theory encapsulated by modelling software could withstand the effects of entropy if specified coherently, e.g., early artificial neural network codebases are not required to understand nor reproduce these models. Indubitably, there is a balance to be struck between reimplementation and re-use.

In contrast, Gureckis and Rich extend their scope to the empirical replication crisis in psychology. They mention that implicit knowledge often goes unpublished and thus only fully automated on-line experiments are computationally reproducible psychology.

Epistemically, empirical and software replication and reproduction are distinct from their modelling-related counterparts -they are six related endeavours. The difference between software for science (e.g., a statistical test) and science that is software (e.g., a cognitive model) is an important one to underline. In the former case the code is a tool, in the latter it constitutes an experiment. Notwithstanding, all such evaluations have scientific merit.

Levels of Evaluation

We mentioned two of the levels in which modelling work is evaluated. Unanimity is reached on replication as a minimum check, however some dialogue contributions go further. To wit, Hinsen separates this endeavor into three steps. Specifically we must check that a model is: bug-free; reproducible as presented; congruent with empirical data. These roughly map onto the levels of talking about modelling work more generally, as Kidd notes [START_REF] Marr | Vision: A Computational Investigation into the Human Representation and Processing of Visual Information[END_REF].

Implementation Level

With respect to the implementation level, as Crook explains, re-running code both within a lab and by others allows for checking for bugs and, importantly, if assumed-tobe-irrelevant variables, e.g., the random seed, are not driving the results. This also ensures documentation is appropriate. Success at this level indicates a model is replicable.

Model Level

To evaluate the quality of the specification, we may rewrite, i.e., reproduce, the model from scratch. This provides evidence for or against depending on the reimplementation's success. As Kidd mentions, and as we discovered [START_REF] Cooper | Implementations are not specifications: Specification, replication and experimentation in computational cognitive modeling[END_REF], this process allows us to: discern when implementation details must be elevated to the theory level and vice versa; evaluate the specification; and uncover bugs.

Theory Level

Many methods exist for testing theories. One such method involves computationally implementing a theory -another is to test predictions by gathering empirical data. As Crook points out, such data is also used to evaluate models and should be associated with the original article and codebase. In such cases, empirical data requires re-collecting. This is because if the phenomenon to-be-modelled, Hinsen warns, does not occur as described by the overarching theoretical account, then both theory and model are brought into question. "A* is a model of [...] A to the extent that [we] can use A* to answer questions [...] about A." (Minsky, 1965, p. 426)

Conclusions

Even though definitions for terms across the replies do not fully converge,1 all contributors agree that change is needed and imminent. A notable divergence of opinion can be found in the reply by French and Addyman, who believe specifications are less vital than we do. Importantly, we agree on some fundamentals: sharing codebases; linking articles with codebases; and reproducing models (e.g., [START_REF] Rescience | The ReScience Journal[END_REF].

In response to our question: Hinsen proposes modellers include a specific article section on evaluation; while Crook lists community-driven initiatives for sharing codebases and specifications. Crook hopes, as we do, for topdown publisher-enforced sharing of resources in partiallycentralised repositories. However, this does not preclude, and may in fact require, grassroots demands. If the scientific community rightfully yearns for change, we are required to act to make this happen.

We do not wish to prescriptively enforce our terms and definitions -and we are open to suggestions, especially based on the use of such terms by computationally-based disciplines (e.g.,Mesnard & Barba

[START_REF] Binder | mybinder[END_REF][START_REF] Patil | A statistical definition for reproducibility and replicability[END_REF].


Claire A Berger

Dogukan A Deniz

Jamey A Gigliotti

James A Palmer

John A Hankinson

Yiran A Hu

Jean-Philippe A Turmaud

Renaud Puybaret

Abdallah A Ougazzaden

Anton A Sidorov

Zhigang Jiang

Walt A De Heer

Epitaxial graphene on SiC: 2D sheets, selective growth and nanoribbons

come   L'archive ouverte pluridisciplinaire

Introduction

Fifteen years of research on epitaxial graphene on SiC (hereafter called epigraphene) have largely demonstrated its potential not only as the best graphene nanoelectronics platform but also the best platform for a large variety of basic science studies of graphene. [START_REF] Berger | Epigraphene[END_REF] From its inception in 2001, it was realized that graphene for nanoelectronics needs an atomically well defined substrate. This requirement is satisfied by growing the material on single crystal substrates. Previous surface science studies of single layer graphite grown on hexagonal SiC substrate had demonstrated that it has characteristics of isolated graphene. In general, large scale integration calls for commercially available wafer scale substrates and best graphene science is demonstrated on flat, unstrained and un-rippled surfaces; all this makes SiC a substrate of choice for graphene. SiC is widely available at affordable price, as is clear from its widespread use as a substrate for LED lighting.

Epigraphene production methods have been developed allowing not only 2D sheets [START_REF] Berger | Ultrathin epitaxial graphite: 2D electron gas properties and a route toward graphene-based nanoelectronics[END_REF][START_REF] De Heer | Large area and structured epitaxial graphene produced by confinement controlled sublimation of silicon carbide[END_REF][START_REF] Emtsev | Towards wafer-size graphene layers by atmospheric pressure graphitization of silicon carbide[END_REF][START_REF] Camara | Anisotropic growth of long isolated graphene ribbons on the C face of graphite-capped 6H-SiC[END_REF][START_REF] Yakes | Conductance Anisotropy in Epitaxial Graphene Sheets Generated by Substrate Interactions[END_REF][START_REF] Robinson | Nucleation of Epitaxial Graphene on SiC[END_REF][START_REF] Michon | Effects of pressure, temperature, and hydrogen during graphene growth on SiC(0001) using propane-hydrogen chemical vapor deposition[END_REF][START_REF] Suemitsu | Epitaxial graphene formation on 3C-SiC/Si thin films[END_REF][START_REF] Yakimova | Morphological and electronic properties of epitaxial graphene on SiC[END_REF][START_REF] Robinson | Epitaxial Graphene Transistors: Enhancing Performance via Hydrogen Intercalation[END_REF] , but also intricately patterned structures to be grown on the substrate [START_REF] Sprinkle | Scalable templated growth of graphene nanoribbons on SiC[END_REF][START_REF] Hu | Structured epitaxial graphene: growth and properties[END_REF] . The ability to grow graphene nanostructures cannot be underestimated: if graphene is to be used in nanoelectronics, atomically defined graphene structures need to be patterned at precise locations. The methods presented here 12 14-16 (masking and more especially template growth on sidewall of SiC trenches) circumvent the detrimental effect of traditional plasma etching methods, and resulting nanoribbons show exceptional room temperature ballistic transport properties [START_REF] Baringhaus | Exceptional ballistic transport in epitaxial graphene nanoribbons[END_REF] . Furthermore, to achieve the best performance graphene needs to be annealed and encapsulated in order to mitigate the effects of adsorbates from exposure to the ambient atmosphere. An advantage of epigraphene is that it is resistant to high temperature and harsh radiation conditions. This review will focus on two main aspects of epigraphene material. First we will outline the quality of epigraphene 2D sheets when grown by the confinement controlled sublimation method; then we will present alternative methods to pattern graphene at the nanoscale with an emphasis on structured growth, and show paths toward large scale integration.

Near equilibrium Confinement Controlled Sublimation growth

The growth of graphene by SiC decomposition is non-conventional in the sense that it doesn't require an external source of carbon. Rather, at high temperature and in vacuum, the SiC surfaces decompose by silicon sublimation, resulting in a carbon rich surface that forms a graphene layer on the SiC [START_REF] Van Bommel | LEED and Auger electron observations of the SiC(0001) surface[END_REF] . The key to the growth of high crystalline quality graphene layer is to control the rate of silicon escape ΓSi from the surface, which is a balance between the rate of sublimation from and absorption onto the surface (Figure 1a) [START_REF] De Heer | Large area and structured epitaxial graphene produced by confinement controlled sublimation of silicon carbide[END_REF] . ΓSi is controlled by the temperature and the background pressure of silicon vapor. When the SiC surface is in equilibrium with the Si vapor (ΓSi=0), the formation of graphene is arrested.

The Confinement Controlled Sublimation (CCS) growth method relies on a near equilibrium growth condition by controlling ΓSi. Details can be found in Ref. 3. A SiC crystal is placed in a graphite enclosure (the crucible). During annealing the evolving Si vapor escapes through a small calibrated hole (see Figure 1b). A background of neutral gas pressure may be supplied that further slows down silicon diffusion. The graphite crucible is placed in a vacuum chamber. The system is pumped down to a base pressure of 10 -7 mbar with a turbomolecular pump, then uniformly heated using RF coils that heat the crucible, as shown figure 1c, to temperatures that can reach up to 2100˚C. The system is designed to have no heating elements inside the vacuum chamber, so that the SiC surface is exposed only to its own vapor. In our most recent furnace design [START_REF] Palmer | Pre-growth structures for nanoelectronics of EG on SiC[END_REF] , the induction susceptor is the graphite crucible itself. It is supported directly by the quartz tube which temperature remains well below its melting point even for the highest crucible temperature (2100ºC for more than 20 minutes), owing to the small actual contact area between the tube and the crucible. The compact design allows for fast pumping speed. The temperature is measured externally with a two-wavelength pyrometer and the assembly has very little thermal inertia, so that fast heating and cooling rate can be achieved up 150˚C/s in the whole temperature range. The temperature profile is completely computer automated with temperature ramps and plateaus fully configurable by the user. The RF-power is adjusted using a proportionalintegral-derivative controller that manages the feed-back loop between the measured and set temperatures. We have estimated [START_REF] Palmer | Pre-growth structures for nanoelectronics of EG on SiC[END_REF] , that gas pressure equilibration is reached in the crucible within 5µs, which is many orders of magnitude shorter than a typical growth time. This confirms that growth occurs very close to equilibrium. Further, since solid/ vapor pressure is an intensive property that does not scale with surface area when in equilibrium, the SiC chip area inside he crucible does not affect the growth rate.

To grow graphene on hexagonal 4H or 6H-SiC wafers a typical temperature cycle includes a degasing stage at 750˚C-800˚C for a few minutes, followed by a SiC surface structuring stage around 1100-1250ºC before the Si sublimation at high temperature (1400-1650˚C ). Optimum temperature and growth times are empirically determined, and stable recipes have been established for each type of structure, including graphene buffer layer, mono and bilayer on the Si-face, from monolayer to thick (more that 50 layers) on the C-face (see Figure 1d), and for confined growth on other than on-axis facets (for instance SiC sidewalls, see below). The exact temperature profile depends on the background pressure (vacuum or 1 atm Ar), the geometry (explicitly the hole size) and condition of the graphite crucible. We have noted a slow evolution with time of the growth parameters, which is due to a change in the adsorption of Si on graphite crucible walls after many growth cycles. The original conditions can be restored by baking the graphite crucible at high temperature.

Multilayer C-face

The near-equilibrium growth condition of the CCS method is favorable to grow very high quality epigraphene films. On the 4H/6H-SiC carbon face, typically 5-10 layers films are easily grown [START_REF] Berger | Electronic confinement and coherence in patterned epitaxial graphene[END_REF] and more than 40 layers can be produced [START_REF] Maysonnave | Terahertz Generation by Dynamical Photon Drag Effect in Graphene Excited by Femtosecond Optical Pulses[END_REF] , while single layer films are much harder to achieve [START_REF] Wu | Half integer quantum Hall effect in high mobility single layer epitaxial graphene[END_REF] . The number of layers is determined by both temperature (in the range 1450-1525˚C) and time (1 minute to several 1 hour cycles for very thick films).

Figure 2 presents atomic force microscopy (AFM) and scanning tunneling microscopy (STM) images of few layer graphene on the C-face. At high resolution the graphene honeycomb lattice is revealed (Figure 2a), and large-area STM scans [START_REF] Haas | Structural characterization of epitaxial graphene on silicon carbide[END_REF] show that the top layer is extremely flat (Figure 2b) on terraces of tens of µm size (Figure 2c) in well formed C-face graphene. Note the vertical scale of less than 0.1 nm in the 400 by 400 nm STM scan of Figure 2b and the very low roughness (rms< 50pm) on the trace in Figure2d that extends over regions of different moiré [START_REF] Miller | Observing the Quantization of Zero Mass Carriers in Graphene[END_REF] . Extremely low roughness values are confirmed by x-ray diffraction (rms<5pm) [START_REF] Hass | The growth and morphology of epitaxial multilayer graphene[END_REF] . No grain boundaries have been seen in large-area STM scans (Figure 2d), indicating the top most graphene layer is continuous over the whole sample, which makes C-face epigraphene among the largest synthetic 2D crystal. The moiré patterns seen in Figure 2d provide clear evidence that adjacently stacked graphene layers are rotated creating a superstructure. Variations in the relative rotation of the layers change the moiré from place to place, with no change in the topographic height of the top layer (Figure 2d). On a larger scale, smooth graphene pleats are observed, as seen as bright lines in Figure 2c (see also SEM image of Figure 2e). The pleats (also referred to as puckers, ridges, wrinkles, folds, ripples, etc in the literature) result from the larger thermal contraction of SiC compared to graphene when the films are cooled down from the growth temperature (around 1500-1600˚C) to room temperature.

The rotational layer stacking in C-face graphene is new and explicitly is not turbostratic (random stacking of small grains). In CCS grown C-face, the stacking as determined by x-ray diffraction alternates 0±δθ degree/ 30± δθ degree oriented layers, with a variation δθ within typically a few degrees. [START_REF] Hass | The growth and morphology of epitaxial multilayer graphene[END_REF] Graphite (AB or rhombohedral) stacking in this case corresponds to stacking faults that have an occurrence of less than 15-19%. [START_REF] Hass | The growth and morphology of epitaxial multilayer graphene[END_REF] The properties of well grown CCS C-face multilayered epitaxial graphene (MEG) are exceptional and have been amply discussed in the literature (see for example review 1). It was soon realized that MEG is a model system to study graphene properties. This stems from a combination of factors: (i) Excellent structural quality, with very few defects and continuous sheets as shown by the absence of a D peak in Raman spectroscopy and by STM scans. (ii) Flatness of the layers, which excludes strain related gauge field effects and broadening effects in k-resolved surface spectroscopy measurements, such as Angle Resolved Photoemission Spectroscopy (ARPES). (iii) Extremely small interaction with the substrate and the environment, more specifically for the layers in the middle of the stack that are quasi-neutral (charge density n~5x10 9 cm -2 , that is at most 8 meV away from the Dirac point). (iv) Rotational stacking resulting in each layer in the stack having the electronic structure of monolayer graphene, explicitly not graphite.

Most prominently, the quasineutral layers show record high mobilities of 10 6 cm 2 /Vs at room temperature, as determined by infra-red magnetospectroscopy with no sign of dependence with temperature [START_REF] Orlita | Carrier Scattering from Dynamical Magnetoconductivity in Quasineutral Epitaxial Graphene[END_REF] . Transport measurements that include contribution from the negatively doped layer at the interface consistently show high (but reduced) mobilities of the order of 15,000 to 25,000 cm 2 /Vs, [START_REF] Berger | Electronic confinement and coherence in patterned epitaxial graphene[END_REF][START_REF] Dlubak | Highly efficient spin transport in epitaxial graphene on SiC[END_REF] and characteristics of graphene such as a Berry's phase of π and weak anti-localization. [START_REF] Berger | Epigraphene[END_REF] Spin transport is highly efficient in MEG with record spin diffusion lengths up to 285µm [START_REF] Dlubak | Highly efficient spin transport in epitaxial graphene on SiC[END_REF] . These layers show textbook graphene electronic structure that allow spectroscopic studies (STS, ARPES, magnetospectroscopy) down to instrumental resolution. [START_REF] Miller | Observing the Quantization of Zero Mass Carriers in Graphene[END_REF][START_REF] Orlita | Carrier Scattering from Dynamical Magnetoconductivity in Quasineutral Epitaxial Graphene[END_REF][START_REF] Sprinkle | First Direct Observation of a Nearly Ideal Graphene Band Structure[END_REF] Most notably, the fine structure of the Landau levels and their real space mapping was observed for the first time on a 2 dimensional gas system by STS measurement of the top C-face epigraphene layer. MEG is also very much sought for optical measurements due to the large area and homogeneity of the graphene layers residing on a transparent substrate. Moreover, since MEG is made of effectively decoupled graphene layers, the multiplicity enhances the signal without compromising the graphene characteristics, contrary to thin graphite. As an example of compelling result obtained from ultrafast optical spectroscopy studies in MEG the electron dynamic probed reveals the details of electronic relaxation in graphene, as an interplay between efficient carrier-carrier and carrier-optical-phonon scattering, and interlayer energy transfer effects. [START_REF] Mihnev | Microscopic Origins of the Terahertz Carrier Relaxation and Cooling Dynamics in Graphene[END_REF]

Monolayer C-face

Monolayer graphene is more difficult to produce due to the rapid growth rates on the C-face and the multiple nucleation sites. Usually isolated graphene patches up to tens of µm in size are produced by the CCS method [START_REF] Hu | Structured epitaxial graphene: growth and properties[END_REF][START_REF] Wu | Half integer quantum Hall effect in high mobility single layer epitaxial graphene[END_REF][START_REF] Zhang | Growth of large domain epitaxial graphene on the C-face of SiC[END_REF] at a graphitization temperature around 1500˚C. The graphene patches extend over multiple steps and show the characteristic pleat structure of C-face graphene (see Figure 3a). With optimized growth conditions (typically around 1500˚C for 10 min) larger patches and extended monolayer sheets can be produced as seen in AFM image of Figure 3a. An alternative method to CCS is to place a piece of graphite on the SiC chip, which provides some Si vapor confinement. This way narrow tapered stripes of monolayer graphene were produced [START_REF] Camara | Anisotropic growth of long isolated graphene ribbons on the C face of graphite-capped 6H-SiC[END_REF] .

The monolayer nature of the CCS produced graphene is assessed with combined AFM imaging and Raman spectroscopy (narrow 2D peak), as shown in Figure 3b. Raman spectroscopy is widely used to characterize graphene. The symmetrical and narrow graphene 2D peak observed in epigraphene, provides an unambiguous signature of graphene monolayers. The D and G peaks are in the same energy range as the second order SiC Raman peaks. A simple subtraction of the SiC contribution to the total spectra is generally performed but for monolayer graphene the similar intensity of the SiC and graphene peaks results in poorly defined graphene spectra. The Non-negative Matrix Factorization (NMF) decomposition technique [START_REF] Kunc | A method to extract pure Raman spectrum of epitaxial graphene on SiC[END_REF] allows the pure graphene and SiC spectra to be segregated from the raw data. This is done by decomposing a set of spectra, taken at different focal points of the exciting Raman laser light at, and just below the surface, into a final number of spectral components. Details of the method are provided in [START_REF] Kunc | A method to extract pure Raman spectrum of epitaxial graphene on SiC[END_REF] . A notable advantage of the technique is that the decomposition does not require prior knowledge of the components spectral profile, or even of the number of components needed (two in the case of graphene on SiC). The method applies to separate the contributions of stacked layers (such as graphene on SiC) or the contribution of patched films in inhomogeneous 2D maps. An example is given in Figure 3b, where the Raman spectrum of single layer graphene is extracted from the raw data. Significantly, the D peak is very small, attesting the good graphene quality.

The epitaxial graphene monolayer is further unambiguously characterized from the observation of the quantum Hall effect (QHE), [START_REF] Hu | Structured epitaxial graphene: growth and properties[END_REF][START_REF] Wu | Half integer quantum Hall effect in high mobility single layer epitaxial graphene[END_REF] as shown in Figure 3c -d for two samples of high mobility μ = 39,800 cm 2 /Vs (20,000 cm 2 /Vs) and charge density n = 0.19×10 12 cm -2 (0.87 ×10 12 cm -2 ), respectively. Note that unlike the graphene layers in the middle of the stack in MEG that are quasi neutral, the graphene layer at the interface with SiC is naturally n-doped with a few 10 12 cm-2 . Positive counter-doping, either by exposure to the ambient or with a top electrostatic gate is necessary to reduce the charge density close to the Dirac point. [START_REF] Wu | Half integer quantum Hall effect in high mobility single layer epitaxial graphene[END_REF] The samples in Figure 3 show two attractive characteristics of graphene for resistance metrology standard based on the quantum Hall effect, namely a robust Hall resistance quantization starting at low magnetic field and still observed at high temperature. The quantized ρxy=12.8kΩ=(2e 2 /h) -1 Hall plateau in Figure 3c has an extension of more than 4 Tesla with an onset below 3 Tesla, that is within the range of commercial room temperature electromagnets. A very well defined plateau is observed at 100K that is above the widely available liquid N2 temperature (Figure 3d). [START_REF] De Heer | Epitaxial Graphene Electronic Structure And Transport[END_REF] From a metrology perspective, note that the most accurate QHE plateau quantization that rivals the best GaAs 2DEGs QHE standards was measured in epigraphene [START_REF] Tzalenchuk | Towards a quantum resistance standard based on epitaxial graphene[END_REF][START_REF] Lafont | Quantum Hall resistance standards from graphene grown by chemical vapour deposition on silicon carbide[END_REF] .

Having established a reliable production method of high mobility monolayer graphene on the C-face, the samples can be used for field effect transistors (FETs). [START_REF] Guo | Record Maximum Oscillation Frequency in C-Face Epitaxial Graphene Transistors[END_REF][START_REF] Lin | 100-GHz Transistors from Wafer-Scale Epitaxial Graphene[END_REF] For this the samples are patterned in a dual source/gate and common drain structure (Figure 3e inset) that is designed for ultra-high frequency measurements. By having contacts and T-shape gates optimized to minimize access resistances and parasitic capacitances, C-face graphene FETs show record maximum oscillation frequency fmax=70GHz. [START_REF] Guo | Record Maximum Oscillation Frequency in C-Face Epitaxial Graphene Transistors[END_REF] The frequency fmax quantifies the practical upper bound for useful circuit operation. It was the first time that a power amplification was achieved at a frequency comparable to the current amplification (characterized by fT=110GHz).

Monolayer Si-face

Epigraphene on the Si-face is the most studied epigraphene form in the literature. Growth on the Silicon face proceeds from the SiC steps. This adds an inherent difficulty to produce extended monolayer sheets, because bilayers (or multiple layers) tend to form at step edges when full terrace graphene coverage is reached. As a matter of fact, the step edges act as electronic scattering centers, which has been related to a reduction of the carrier concentration on the sidewall of step, to a n/p junction or the presence of a bandgap at the step edge, or to the presence of bilayers/ multilayers (see Ref. [START_REF] Berger | Epigraphene[END_REF] and refs therein). The later have a detrimental effect to the homogeneity of the quantum Hall effect. Better control of epigraphene growth on the Si-face has been obtained by providing carefully balanced methane and H2. [START_REF] Michon | Effects of pressure, temperature, and hydrogen during graphene growth on SiC(0001) using propane-hydrogen chemical vapor deposition[END_REF] The fact that growth starts at a sidewall, can be tuned to an advantage by arresting the growth on the sidewall of steps (either natural steps or etched steps in SiC -see below in section 3.2). Another consequence of graphene growth initiated at steps is that the quality of the SiC surface prior to growth is essential. Rough surfaces result in inhomogeneous and defective graphene layers, which can be rationalized by the presence of multiple graphene nucleation centers for surface decomposition. Surface flattening (removal of polishing scratches) can be obtained by SiC etching at high temperature (typically 1600ºC) in a hydrogen environment (we use 3% H2 in Argon). Commercial SiC wafers are also available with surfaces treated by a chemical mechanical process, which provide good starting surfaces. For natural steps, another consideration is the width of the terraces that depends on the step bunching for a given miscut. We have used pre-growth annealing in various conditions to help organize the step-terrace structure for a particular desired surface state. These include annealing at moderate temperature (1100˚C-1200˚C) in vacuum, high temperature in argon, face-to-face SiC surface re-structuring and step pinning under an evaporated refractory cap. In some cases it can be beneficial to grow first the buffer layer (that is the semiconducting graphene layer bound to SiC on the Si-face) to help reduce the step displacement. An example of pre-growth SiC surface structuring is given in Figure 4a, featuring large atomically flat terraces with straight steps. The terraces are 20-40µm in width and extend over hundreds of µm in length (see also Figure 7e).

The capping technique consists in evaporating an amorphous carbon grid onto bare SiC. Upon thermal treatment the SiC steps are pinned under the amorphous carbon and steps displacement is confined to within each amorphous carbon enclosure. [START_REF] Palmer | Controlled epitaxial graphene growth within removable amorphous carbon corrals[END_REF] This results in step bunching at one end of the enclosure and step alignment along the amorphous carbon corral, as demonstrated in Figure 4b-c. This technique demonstrates a significant improvement in SiC surface structuring by providing not only flat terraces of large size (Figure 4c -bottom) but most importantly at a predefined location. Note also that amorphous C is easily removable by plasma etching after annealing.

Selective graphene growth

Masking techniques

Any graphene electronic application requires that graphene patterning. The principle of structured growth stems from the realization that patterning a 2D graphene sheet by the usual oxygen plasma removal of carbon atoms is very destructive and leaves roughed edges and uncontrolled orientation and termination. It is therefore advantageous to grow graphene directly at preset locations. Figure 5a-b presents two examples where masking methods were used to grow graphene into shape with no need for further etching. Graphene growth can be either stopped under an AlN mask [START_REF] Rubio-Roy | Structured epitaxial graphene growth on SiC by selective graphitization using a patterned AlN cap[END_REF] , slowed down under a Si-poor Si3-xN4 mask or promoted under a rich Si3+xN4 mask [START_REF] Puybaret | Scalable control of graphene growth on 4H-SiC C-face using decomposing silicon nitride masks[END_REF] . In Figure 5a, an 80nm thick AlN film was evaporated then patterned revealing bare SiC in the shape of a multiprobe Hall bar. After annealing (20 minutes at 1420˚C) multilayer graphene was grown on the 4H-SiC C-face only on the exposed SiC, while no graphene grows under AlN, as shown by the Raman 2D map of Figure 5a (bottom). In the case of the SiN mask, contrary to AlN, the mask vanishes during the heat treatment, so that there is no need for further mask removal; a differential in graphene growth is revealed where the mask was present. The four panels of Figure 5b show how the Si -rich Si3+xN4 patterned mask (a) was transferred into enhanced selective graphene growth as shown in the optical image (b), the Raman 2D peak (c) and 2D peak/G peak maps (d) with sub-micron resolution. Conversely graphene was used as a mask to prevent GaN growth in Figure 5c. Here holes 75 nm diameters are patterned in MEG (bright dots in the inset) and GaN crystals (triangles in the main panel) grow only on SiC in the holes. This structure is interesting to integrate GaN LED with graphene.

Sidewall facets

The concept of selective growth was applied to grow predefined nanoscale structures at desired locations without masking. [START_REF] Sprinkle | Scalable templated growth of graphene nanoribbons on SiC[END_REF] The technique relies on the fact that graphene grows faster on facets with orientation other that the (0001) basal plane of the Si-face, as was readily observed by the growth of multilayer graphene at step edges (see above). Facets can be the natural steps arising from the miscut angle relative to the (0001) plane; step bunching by annealing in controlled condition produces arrays of steps and terraces with rather uniform width and height. More interestingly, trenches of various shapes can be etched in SiC, [START_REF] Hu | Structured epitaxial graphene: growth and properties[END_REF] the sidewalls of which recrystallize into the crystallographic equilibrium facets of SiC (like (2-207)-facets) upon annealing. Trenches can be etched for instance along the 4H-SiC-(-1-120) and 4H-SiC-(1-100) directions in SiC providing graphene ribbons along the zigzag or armchair orientation, respectively owing to the epitaxial orientation of graphene on SiC (see figure 6a).

As for natural steps, graphene grows preferentially on these facets; under controlled growth conditions at temperature around 1600˚C with the CCS method, graphene growth can be arrested on the sidewalls producing monolayers on them. The ribbon width is then determined by the step height and the facet angle from the basal plane (about 27 degrees). Because the steps in SiC are etched and annealed prior to graphene growth, the facets onto which graphene grows are smooth and atomically defined. This is best demonstrated in cross-sectional transmission electron microscopy (TEM). Figure 6b shows that a single graphene layer drapes over the (here armchair) step and merges into the buffer layer that is tightly bound on the (0001) surface. Significantly, the graphene layer on the main facet is at a significantly larger distance from SiC than the buffer layer on the (0001) face.

Ribbons of any nominal orientation can in principle be produced. However, we have observed that straight steps may in some cases become rounded after annealing [START_REF] Baringhaus | Exceptional ballistic transport in epitaxial graphene nanoribbons[END_REF] and conversely etched pillars show faceting 3 along the armchair direction, as seen in Figure 6cd. Rounding and faceting are sensitive to a number of factors such as step direction, growth condition (both temperature, time, and possibly other factors such as the type of heating) and the pinning of steps (for instance under amorphous carbon pads such as in figure 4b or by defects). The SiC trench height is a relevant parameter since shallow trenches (≤10nm) get washed out upon annealing and deep trenches tend to break into multiple facets revealing multiple parallel ribbons. Uniform and well formed graphene ribbons of various shapes are consistently realized on trenches in the range 15-35 nm deep.

An important point of consideration is the ribbon integrity for transport measurements. Growth of secondary ribbons may occur if there are transverse substrate steps that cross the trenches. Graphene growth on these substrate steps produces "side arms" on the main ribbon that have been found to affect the transport [START_REF] Baringhaus | Growth and characterization of sidewall graphene nanoribbons[END_REF] . An example is given on the AFM topographic and friction images of Figure 7a-d. The substep features are clearly visible on the terraces of Figure 7a (bright color) and give rise to ribbons seen as dark lines (low friction) in Figure 7b. A single ribbon can be measured by selecting a segment with no side arms and etching away the surrounding, as was done in the low temperature measurement in Ref. [START_REF] Baringhaus | Exceptional ballistic transport in epitaxial graphene nanoribbons[END_REF] . An alternative to get very long pristine ribbon is to choose a step/trench orientation parallel to the local miscut. That way, the terraces are flat (see Figure 7d and AFM trace in Figure 7e and very straight ribbons with no side arms can be formed over hundreds of microns length as shown in Figure 7d-f. Ribbons ~30-50nm wide show ballistic transport at room temperature [START_REF] Baringhaus | Exceptional ballistic transport in epitaxial graphene nanoribbons[END_REF] . This was unambiguously demonstrated in local four-point resistance measurements in UHV, showing a length independent resistance from 1 to 17 microns at room temperature. Importantly, ballistic transport was observed not only for specific graphene orientations but also for chiral and curved ribbons. In an early experiment 39 , a conductive AFM tip was brought into contact with a curved ribbon 35nm wide and 1 µm long provided with a contact at both ends (Figure 8a). The tip is scanned over the sample and the tip to contact resistance is recorded. The resistance is minimum when the tip is in contact with the ribbons, as shown in the color plot of Figure 8b. The minimum resistance R on each scan line is plotted as a function of the distance L to the contact in Figure 8c. After subtracting the gold contact and line resistance, we find R=Rcontact +(ΔR/ΔL) L, with Rcontact =0.9 (h/ e 2 ) and ΔR/ΔL =0.56 (h/e 2 )/µm. The contact resistance indicates there is only one conducting channel; in these conditions [START_REF] Datta | electronic transport in mesoscopic systems[END_REF] the slope ΔR/ΔL=(h/ e 2 )(L/λ) gives a mean free path λ=1.8µm. This value is longer that the ribbon length indicating ballistic transport. This result is remarkable because the ribbon is exposed to ambient conditions, which in general increases significantly the resistance in contrast to protected ribbons in UHV or encapsulated under a dielectric (Al2O3 like in Ref.17). We believe that scanning the tip over the ribbon cleanses the ribbon by sweeping away the contaminants. We have often observed accumulation of particles on the rougher buffer layer next to ribbons.

Large scale integration

By design, multiple ribbons can be produced all at once. Arrays of tens of thousands of ribbons have been grown on the sidewall of trenches patterned by photolithography [START_REF] Sprinkle | Scalable templated growth of graphene nanoribbons on SiC[END_REF] . An example of integration of 10,000 field effect transistors/cm 2 is presented in the optical image of Figure 9, where an aluminum gate (G) lies on a ribbon in between Pd/Au source and drain contacts (S -D).

Mainstream electronics and very large scale integration is based on silicon; it is therefore important to devise schemes to integrate graphene with Si wafers. The route we proposed [START_REF] Dong | Wafer bonding solution to epitaxial graphene-silicon integration[END_REF] is based on the standard industry methods using wafer bonding and the smart-cut technique. The idea here is to transfer a Si film from a Si wafer onto a graphitized SiC wafer. Figure 9c shows the principle of the design. The top-bounded Si wafer is ready for device patterning, and the graphene layer beneath is accessed through metallic vias managed through the sub-micron thick Si wafer. This 3D integration realizes the interconnection of the SiC supported graphene platform and the Si-based electronic wafer while preserving the integrity of graphene. Details have been published elsewhere; the main steps consist in the fabrication of graphene structures (either from patterned epitaxial graphene or from template growth on SiC sidewalls), evaporation of an alumina film that serves as a bonding layer between the SiC and the oxidized Si wafer, wafer bonding, and finally splitting of the Si wafer (smartcut) that leaves a thin crystalline Si layer on top. The main advantage of the process is that the graphene growth temperature (above 1500˚C) is not limited by the presence of silicon (melting point 1414˚C), or the limited SiC quality like in SiC epilayers on Si; the Si wafer is fully accessible and the stacking of wafers increases the areal density inspired by the 3D stacked layers Very Large Scale Integration Technology.

Figure 9d-e shows the top Si film bonded on a SiC chip The purple color corresponds to the strongly bonded regions. The zoom out optical image of Figure 9e shows an array of ribbons grown on parallel sidewall trenches as seen from the SiC (transparent) side of the bonded wafers. Note that the ribbons structure is intact after bonding.

Conclusion

We have reviewed the main characteristics of epitaxial graphene grown on SiC by the confinement controlled sublimation method and selective graphene growth techniques. Most recent results on structured growth demonstrate that graphene nanoribbons can be grown on the sidewall of either natural SiC steps or trenches etched in SiC. Ribbons of 20-40nm wide and more than 100µm long can be very well confined onto the sidewall and well isolated (no sidearms) from other graphene structures. Schemes for large scale integration of epitaxial graphene ribbons with Si wafer were also presented. (a) STM image: 4x4nm (From Ref. [START_REF] Haas | Structural characterization of epitaxial graphene on silicon carbide[END_REF] ). (b) STM image: 400x400nm. The blue square represents the size of the STM scan in (a) (From Ref. [START_REF] Haas | Structural characterization of epitaxial graphene on silicon carbide[END_REF] ). (c) AFM scan: 40µmx40µm. The white lines are graphene pleats. (d) 400nm long STEM image across areas of different moiré pattern, showing that the top layer is flat (rms<50pm) and continuous (From Ref. [START_REF] Miller | Observing the Quantization of Zero Mass Carriers in Graphene[END_REF] ) (e) Scanning Electron Microscopy image of a graphene pleat (From ref. [START_REF] Cambaz | Noncatalytic synthesis of carbon nanotubes, graphene and graphite on SiC[END_REF] ). (c) low field onset (3Tesla) for a sample of μ = 39,800 cm 2 /Vs and charge density n = 0.19×10 12 cm -2 . (From Ref. [START_REF] Hu | Structured epitaxial graphene: growth and properties[END_REF] )(d) high temperature quantization (n=0.87 ×10 12 cm -2 , μ = 20,000 cm 2 /Vs -From Ref. [START_REF] De Heer | Epitaxial Graphene Electronic Structure And Transport[END_REF] ). (e) High frequency transistor with a monolayer C-face graphene channel. Inset design of the dual gate transistor; main panel: the intercepts of U 1/2 = 1 (slope 20 dB/dec) gives a maximum oscillation frequency (fmax =70GHz) for a 100nm gate length. U is the Mason's unilateral gain (power gain). The current amplification H21 versus frequency gives a cutoff frequency fT =110 GHz. (From Ref. [START_REF] Guo | Record Maximum Oscillation Frequency in C-Face Epitaxial Graphene Transistors[END_REF] ) [START_REF] Puybaret | Scalable control of graphene growth on 4H-SiC C-face using decomposing silicon nitride masks[END_REF] ). -a-Optical image of the SiN pattern and -b-of subsequent MEG growth on SiC (graphene is light color) -c-Raman 2D map and -d-Raman 2D/G map, showing that graphene growth is promoted where the mask resided. Scale bar is 10 μm. (c) SEM images of 30nm thick GaN grown on 4H-SiC using punctured graphene as a mask (From Ref. [START_REF] Puybaret | Nanoselective area growth of GaN by metalorganic vapor phase epitaxy on 4H-SiC using epitaxial graphene as a mask[END_REF] ). No GaN grows on graphene, as exemplified on the right side of the image. Inset: SEM image of the epigraphene mask with 75 nm holes in it showing SiC (light dots). Scale bar in the inset is 200 nm. Figure 6. Side wall growth (a) Relative epigraphene to 4H/6H-SiC orientation and direction of trenches to grow ribbons of in principle defined zigzag or armchair direction. (b) cross-sectional TEM image of a 40nm high step covered with one layer graphene (From Ref. [START_REF] Palacio | Atomic Structure of Epitaxial Graphene Sidewall Nanoribbons: Flat Graphene, Miniribbons, and the Confinement Gap[END_REF] ). (c) 3D rendition of a facetted pillar after annealing (d) AFM topographic (top) and EFM (bottom) image of annealed pillars, showing that graphene is confined to the side walls (bright ring in the EFM images) (adapted From Ref. [START_REF] De Heer | Large area and structured epitaxial graphene produced by confinement controlled sublimation of silicon carbide[END_REF] ).

Figure 1 .

 1 Figure1. Growth of epigraphene by the confinement control sublimation method (adapted from Ref.[START_REF] De Heer | Large area and structured epitaxial graphene produced by confinement controlled sublimation of silicon carbide[END_REF] ) (a) Principle of graphene growth by thermal decomposition of SiC at high temperature, Si escapes and is partially re-absorbed on the SiC surface. (b) CCS growth furnace with induction heating elements and graphite crucible with calibrated hole to contain the Si vapor. (c) Schematics of the graphene layer growth on the top 6H/4H-SiC(000-1) and bottom (000-1) surfaces.

Figure 2 .

 2 Figure 2. Multilayer epigraphene on the C-face topography images.(a) STM image: 4x4nm (From Ref.[START_REF] Haas | Structural characterization of epitaxial graphene on silicon carbide[END_REF] ). (b) STM image: 400x400nm. The blue square represents the size of the STM scan in (a) (From Ref.[START_REF] Haas | Structural characterization of epitaxial graphene on silicon carbide[END_REF] ). (c) AFM scan: 40µmx40µm. The white lines are graphene pleats. (d) 400nm long STEM image across areas of different moiré pattern, showing that the top layer is flat (rms<50pm) and continuous (From Ref.[START_REF] Miller | Observing the Quantization of Zero Mass Carriers in Graphene[END_REF] ) (e) Scanning Electron Microscopy image of a graphene pleat (From ref.[START_REF] Cambaz | Noncatalytic synthesis of carbon nanotubes, graphene and graphite on SiC[END_REF] ).

Figure 3 .

 3 Figure 3. Monolayer graphene on the C-face. (a) AFM images of large monolayer patches draping over steps. (b) Raman spectroscopy of the monolayer C-face graphene in (a); the black trace is the raw data, the red trace is the graphene spectrum once the SiC Raman peaks have been subtracted by the NFM method. (c)-(d) quantum Hall effect in monolayer graphene showing properties of the ν=2 plateau:(c) low field onset (3Tesla) for a sample of μ = 39,800 cm 2 /Vs and charge density n = 0.19×10 12 cm -2 . (From Ref.[START_REF] Hu | Structured epitaxial graphene: growth and properties[END_REF] )(d) high temperature quantization (n=0.87 ×10 12 cm -2 , μ = 20,000 cm 2 /Vs -From Ref.[START_REF] De Heer | Epitaxial Graphene Electronic Structure And Transport[END_REF] ). (e) High frequency transistor with a monolayer C-face graphene channel. Inset design of the dual gate transistor; main panel: the intercepts of U 1/2 = 1 (slope 20 dB/dec) gives a maximum oscillation frequency (fmax =70GHz) for a 100nm gate length. U is the Mason's unilateral gain (power gain). The current amplification H21 versus frequency gives a cutoff frequency fT =110 GHz. (From Ref.[START_REF] Guo | Record Maximum Oscillation Frequency in C-Face Epitaxial Graphene Transistors[END_REF]

Figure 4 .

 4 Figure 4. Structuring SiC. (a) AFM topographic image of 4H-SiC large terraces structured before growth, and step profile, showing straight steps and flat terraces 20 to 40µm wide. (b)SiCStep pinning under an evaporated amorphous carbon grid. After annealing at 1350˚C, the steps accumulate at one side of the grid in each enclosure providing large terraces at a locations defined by the grid. (From Ref.[START_REF] Palmer | Controlled epitaxial graphene growth within removable amorphous carbon corrals[END_REF] )(c) AFM topographic profile within the grid (top) before and (bottom) after step bunching annealing, showing that a large terrace has developed.

Figure 5 .

 5 Figure 5. Selected growth. (a) Hall bar-shaped graphene multilayer (light grey) grown directly into shape using a AlN mask (dark) evaporated onto bare SiC. (Bottom) Raman map of the 2D graphene peak for the central area in (a), showing that graphene grows only in the unmask areas. From Ref. (14)(b) Graphene growth under a Si-rich SiN mask (From Ref.[START_REF] Puybaret | Scalable control of graphene growth on 4H-SiC C-face using decomposing silicon nitride masks[END_REF] ). -a-Optical image of the SiN pattern and -b-of subsequent MEG growth on SiC (graphene is light color) -c-Raman 2D map and -d-Raman 2D/G map, showing that graphene growth is promoted where the mask resided. Scale bar is 10 μm. (c) SEM images of 30nm thick GaN grown on 4H-SiC using punctured graphene as a mask (From Ref.[START_REF] Puybaret | Nanoselective area growth of GaN by metalorganic vapor phase epitaxy on 4H-SiC using epitaxial graphene as a mask[END_REF] ). No GaN grows on graphene, as exemplified on the right side of the image. Inset: SEM image of the epigraphene mask with 75 nm holes in it showing SiC (light dots). Scale bar in the inset is 200 nm.

Figure 7 .

 7 Figure 7. Perfecting sidewall ribbons. (a)-(b) Etched trenches with non flat bottom and top terraces. (a) AFM topographic and (b) Friction Force Microscopy (FFM) image (brown is low friction, yellow is the rippled, rougher buffer layer). Graphene grows on the main steps (vertical lines) as well as on the substeps on the terraces. (c)-(d) Straight isolated sidewall ribbons grown on bunched natural steps. (c) AFM and (d) FFM images for etched trenches with non flat bottom and top terraces. (e) Topographic profile along the red line in (c). (f) AFM 3D image of a long isolated ribbon grown on a step.

Figure 8 .

 8 Figure 8. Electrical transport in sidewall ribbons. (a) EFM image of an isolated epigraphene ribbon connected to two large graphene pads. (b) Tip to sample resistance color map showing that conduction is along the ribbon only (Resistance scale is on the right). (c) Plot of the resistance as a function of length along the ribbon in ambient condition.

Figure 9 .

 9 Figure 9. Large scale integration (a) Schematics of a field effect transistor (FET) with a sidewall ribbon as the channel connected to source S and drain D with a top gate G. (b) Integration of 10,000 FET/cm 2 (optical image) with parallel ribbons connected between Pd/Au contacts and provided with an aluminum gate (G) on an Al2O3 oxide (From Ref.[START_REF] Sprinkle | Scalable templated growth of graphene nanoribbons on SiC[END_REF] ). (c) Principle of epigraphene on SiC to Si wafer bonding, with a top Si wafer ready for CMOS technology and an underneath epigraphene circuit. Both are interconnected by metal vias through the thin Si layer. (d-e) Realization of the epigraphene on SiC to Si wafer bonding. (d) optical image of the bi-wafer chip from the Si side. The purple color corresponds to the strongly bonded regions. (e) optical image from the transparent SiC side, showing an intact trench array supporting the sidewall ribbons. (From Ref.[START_REF] Dong | Wafer bonding solution to epitaxial graphene-silicon integration[END_REF] ).

Acknowledgements

Financial support is acknowledged from AFOSR and NSF under grants No FA9550-13-1-0217 and 1506006, respectively. Additional support is provided by the Partner University Fund from the French Embassy.