Walid Bechkit
email: wbechkit@hds.utc.fr

Yacine Challal
email: ychallal@hds.utc.fr

Abdelmadjid Bouabdallah
email: bouabdal@hds.utc.fr

Vahid Tarokh

A Highly Scalable Key Pre-Distribution Scheme for Wireless Sensor Networks

Keywords: Wireless sensor networks, security, key management, network scalability, secure connectivity coverage I

Given the sensitivity of the potential WSN applications and because of resource limitations, key management emerges as a challenging issue for WSNs. One of the main concerns when designing a key management scheme is the network scalability. Indeed, the protocol should support a large number of nodes to enable a large scale deployment of the network. In this paper, we propose a new highly scalable key management scheme for WSNs which provides a good secure connectivity coverage. For this purpose, we make use for the first time of the unital design theory. We show that the basic mapping from unitals to key pre-distribution allows to achieve an extremely high network scalability. Nonetheless, this naive mapping does not guarantee a high key sharing probability. Therefore, we propose an enhanced unital-based key pre-distribution scheme providing high network scalability and good key sharing probability lower bounded by 1e -1 ≈ 0.632. We conduct analytical analysis and simulations to compare our solution to main existing ones regarding different criteria including storage overhead, network scalability, network connectivity, average secure path length and network resiliency. The obtained results show that our approach enhances considerably the network scalability while providing high secure connectivity coverage and good overall performances. Moreover, the obtained results show that at equal network size, our solution reduces significantly the storage overhead compared to main existing solutions.

tions, sophisticated security services are required [START_REF] Zhou | Securing wireless sensor networks: A survey[END_REF]. Key management is a corner stone for many security services such as confidentiality and authentication which are required to secure communications in WSNs, the establishment of secure links between nodes is then a challenging problem in WSNs.

Because of resource limitations, symmetric key establishment is one of the most suitable paradigms for securing exchanges in WSNs. On the other hand, because of the lack of infrastructure in WSNs, we have usually no trusted third party which can attribute pairwise secret keys to neighboring nodes, that is why most existing solutions are based on key pre-distribution. Over the last decade, a host of research work dealt with symmetric key pre-distribution issue for WSNs and many solutions have been proposed in the literature [START_REF] Eschenauer | A key-management scheme for distributed sensor networks[END_REF][3][4] [START_REF] Castelluccia | A Robust Key Pre-distribution Protocol for Multi-Phase Wireless Sensor Networks[END_REF][6] [START_REF] Yu | A robust group-based key management scheme for wireless sensor networks[END_REF][8] [START_REF] Zhu | Leap: efficient security mechanisms for large-scale distributed sensor networks[END_REF][10] [START_REF] Perrig | Spins: security protocols for sensor netowrks[END_REF] [START_REF] Maala | Hero: Hierarchcal key management protocol for heterogeneous wsn[END_REF]. Nevertheless, in most existing solutions, the design of key rings (blocks of keys) is strongly related to the network size, these solutions either suffer from low scalability (number of supported nodes), or degrade other performance metrics including secure connectivity, storage overhead and resiliency in the case of large networks.

In this work, our aim is to tackle the scalability issue without degrading the other network performance metrics. For this purpose, we target the design of a scheme which ensures a good secure coverage of large scale networks with a low key storage overhead and a good network resiliency.

To this end, we make use, for the first time, of the unital design theory for efficient WSN key predistribution. Indeed, we propose a naive mapping from unital design to key pre-distribution and we show through analytical analysis that it allows to achieve an extremely high scalability. Nonetheless, this naive mapping does not guarantee a high key sharing probability. Therefore, we propose an enhanced unital-based key pre-distribution scheme that maintains a good key sharing probability while enhancing the network scalability. A preliminary work and few discussions were presented in [START_REF] Bechkit | A new scalable key pre-distribution scheme for wsn[END_REF].

The contributions of our work are many folds and can be summarized in the following points:

• We review the main state of the art of symmetric key management schemes for WSNs that we classify into two categories: probabilistic schemes and deterministic ones. We further refine the classification into sub-categories with respect to the underlying concepts and techniques used in key exchange and agreement.

• We introduce, for the first time, the use of unital design theory in key pre-distribution for WSNs.

We show that the basic mapping from unitals to key pre-distribution gives birth to an extremely highly scalable scheme while providing low probability of sharing common keys.

• We propose an enhanced unital-based key pre-distribution scheme in order to increase the network scalability while maintaining a good key sharing probability. We prove that adequate choice of our solution parameter should guarantee high key sharing probability lower bounded by 1e -1 while ensuring a high network scalability.

• We analyze and compare our new approach against main existing schemes, with respect to different criteria: storage overhead, energy consumption, network scalability, secure connectivity coverage, average secure path length and network resiliency. The obtained results show that our solution enhances the network scalability while providing good overall network performances.

Moreover, we show that at equal network size, our solution reduces significantly the storage overhead and thereby the energy consumption.

The remainder of this paper is organized as follows: section 2 presents related works on key management for WSNs. We give in section 3 a background on unital design and we propose a basic mapping from unitals to key pre-distribution for WSNs, we analyze the main performances of the resulting scheme. In section 4, we explain the enhanced scalable unital-based construction that we propose and we analyze its different performances. In section 5, we compare our approach to the existing ones regarding different criteria; we give and discuss theoretical and simulation results. In section 6, we end up this paper with some conclusions.

II. RELATED WORKS: KEY MANAGEMENT SCHEMES FOR WSNS

Key management problems in WSNs have been extensively studied in the literature and several solutions have been proposed [START_REF] Zhang | Wireless sensor network key management survey and taxonomy[END_REF] [START_REF] Yener | Key distribution mechanisms for wireless sensor networks: a survey[END_REF]. In this work, we mainly classify symmetric schemes into two categories: probabilistic schemes and deterministic ones (see figure 1). In deterministic schemes, each two neighboring nodes are able to establish a direct secure link which ensures a total secure connectivity coverage. In probabilistic schemes, the secure connectivity is not guaranteed because it is conditioned by the existence of shared keys between neighboring nodes. We give in table I the definition of the five considered evaluation metrics, while we summarize in table II the main used symbols.

A. Probabilistic schemes

In probabilistic key management schemes, each two neighboring nodes can establish a secure link with some probability. If two neighboring nodes cannot establish a secure link, they establish a secure path composed of successive secure links.

Eschenauer and Gligor proposed in [START_REF] Eschenauer | A key-management scheme for distributed sensor networks[END_REF] the basic Random Key Pre-distribution scheme denoted by RKP. In this scheme, each node is pre-loaded with a key ring of k keys randomly selected from a large pool S of keys. After the deployment step, each node i exchanges with each of its neighbor j the list of key identifiers that it maintains. This allows node j to identify the keys that it shares with node i. If two neighbors share at least one key, they establish a secure link and compute their session secret key which is one of the common keys. Otherwise, they should determine a secure path which is composed by successive secure links. The values of the key ring size k and the key pool size |S| are chosen in such a way that the intersection of two key rings is not empty with a high probability. This basic approach is CPU and energy efficient but it requires a large memory space to store the key ring. Moreover, if the network nodes are progressively corrupted, the attacker may discover a large part or the whole global key pool. Hence, a great number of links will be compromised. [START_REF] Chan | Random key predistribution schemes for sensor networks[END_REF] a protocol called Q-composite scheme that enhances the resilience of RKP. In this solution, two neighboring nodes can establish a secure link only if they share at least Q keys. The pairwise session key is calculated as the hash of all shared keys concatenated to each other: K i,j = Hash(K s 1 K s 2 ... K s q ′ ) where K s 1 , K s 2 , ...K s q ′ are the q ′ shared keys between the two nodes i and j (q ′ ≥ Q). This approach enhances the resilience against node capture attacks because the attacker needs more overlap keys to break a secure link. However, this approach degrades the network secure connectivity coverage because neighboring nodes must have at least Q common keys to establish a secure link. [START_REF] Chan | Random key predistribution schemes for sensor networks[END_REF] a perfect secure pairwise key pre-distribution scheme where they assign to each possible link between two nodes i and j a distinct key K i,j . Prior to deployment, each node is pre-loaded with P c × n keys, where n is the network size and P c is the desired secure coverage probability. Since we use distinct keys to secure each pairwise link, the resiliency against node capture is perfect and each captured node does not reveal any information about external links.

Chan et al. proposed in

Chan et al. proposed also in

The main drawback of this scheme is the non scalability because the number of the stored keys depends linearly on the network size. [START_REF] Du | A key management scheme for wireless sensor networks using deployment knowledge[END_REF] an enhanced random scheme assuming the node deployment knowledge.

Du et al. proposed in

Nodes are organized in regional groups to which are assigned different key pools, each node selects its keys from the corresponding key pool. The key pools are constructed in such a way that neighboring ones share more keys than distant pools. This approach allows to enhance the probability of sharing common keys as well as the resilience against node capture attacks. However, the application of this scheme is restrictive if the deployment knowledge is not possible.

In [START_REF] Liu | Establishing pairwise keys in distributed sensor networks[END_REF], Liu and Ning proposed a key management scheme in which nodes are pre-loaded with bivariate polynomials instead of keys. A global pool of symmetric bivariate polynomials (f (x, y) = f (y, x)) is generated off-line and each node i is pre-loaded with a subset of polynomials f (i, y). If two neighboring nodes share a common polynomial, the session key is derived by computing the polynomial value at the neighbor identifier. This approach allows to compute distinct secret keys which enhances the resilience against node capture. However, it requires more memory to store the polynomials and induces more computational overhead.

In [START_REF] Blom | An optimal class of symmetric key generation systems[END_REF], Blom proposed a λ-secure symmetric key generation system in which each node i stores a column i and a row i of size (λ+1) of two matrices G and (D.G) T respectively where :

D (λ+1)×(λ+1)
is a symmetric matrix, G (λ+1)×n is a public matrix and (D.G) T is a secret matrix. The matrix of pairwise keys of a group of n nodes is then K = (D.G) T .G. Yu and Guan [START_REF] Yu | A robust group-based key management scheme for wireless sensor networks[END_REF] used the Blom's scheme to key pre-distribution in group-based WSNs. Nodes are deployed into a grid and to each group is assigned a distinct secret matrix. Using deployment knowledge, the potential number of neighboring nodes decreases which requires less memory. The application of this solution gives good results in the case of node deployment knowledge which is not always possible.

In [START_REF] Ruj | Fully secure pairwise and triple key distribution in wireless sensor networks using combinatorial designs[END_REF], Ruj et al. propose a trade-based key management scheme denoted Trade-KP. Given a finite set X of v elements, a Steiner trade t -(v, k) is defined to be two disjoint sets T 1 and T 2 of kelements blocks of X such that each set of t elements from X occurs in precisely the same number of blocks of T 1 as those of T 2 , and no set of t elements from X is repeated more than once in any of T 1 or T 2 . A steiner trade is said to be strong if any two blocks of T 1 and T 2 respectively intersects in at most two elements. Authors proposed a new trade construction: Having q a prime power and k (4 ≤ k < q), they construct T 1 and T 2 while the blocks of T 1 are represented by t 1 i,j = {(x, (xi + j) mod q) : 0 ≤ x < k}, where 0 ≤ i, j < q, and the blocks of T 2 are represented by t 2 i,j = {(x, (x 2 + xi + j) mod q) : 0 ≤ x < k}, where 0 ≤ i, j < q. Authors proved that the proposed construction results in a 2 -(qk, k) strong steiner trade. They proposed then a mapping to key pre-distribution where they associate to each element a distinct key and to each block of T 1 and T 2 a key ring. The key ring size is then equal to k and the scalability of the scheme is equal to 2q 2 .

After the deployment step, each two nodes can establish a direct secure link if they share exactly two common keys which are used to compute the pairwise session key. Based on the trade properties, authors prove that each pair of keys occurs either in exactly two nodes from T 1 and T 2 respectively or none of the nodes.

The main strength of the proposed scheme is the establishment of unique secret pairwise keys between connected nodes. However, this does not ensure a perfect network resilience as we prove later. Indeed, the attacker may construct a part of the global set of keys and then compute pairwise secret keys used to secure external links where the compromised nodes are not involved. Moreover, the proposed scheme provides a low session key sharing probability which does not exceed 0.25 as we show later.

B. Deterministic schemes

Deterministic schemes ensure that each node is able to establish a pair-wise key with all its neighbors. Many solutions were proposed to guarantee determinism.

A naive deterministic key pre-distribution scheme can be designed by assigning to each link (i,j) a distinct key K i,j and pre-loading each node with (n -1) pairwise keys in which it is involved where n is the network size. It is obvious that this solution is not scalable for large WSNs. Choi et al. proposed in [START_REF] Choi | The best keying protocol for sensor networks[END_REF] an enhanced approach allowing to store only (n + 1)/2 keys at each node.

For that purpose, they propose to establish an order relation between node identifiers and propose a hash function based key establishment in order to store only half of the node symmetric keys while computing the other half at each node. This approach allows to reduce the required stored keys to the half of network size, however, it is obvious that this scheme remains non scalable enough.

LEAP [START_REF] Zhu | Leap: efficient security mechanisms for large-scale distributed sensor networks[END_REF] make use of a common transitory key which is preloaded into all nodes prior to deployment of the WSN. The transitory key is used to generate pairwise session keys and is cleared from the memory of nodes by the end of a short time interval after their deployment. LEAP is based on the assumption that a sensor node, after its deployment, is secure during a time T min and cannot be compromised during this period of time. LEAP is then secure as far as this assumption is verified.

In [START_REF] Yener | Combinatorial design of key distribution mechanisms for wireless sensor networks[END_REF], C ¸amtepe and Yener proposed a new deterministic key pre-distribution scheme based on Symmetric Balanced Incomplete Block Design (SBIBD). The proposed mapping from SBIBD to key pre-distribution allows to construct m 2 + m + 1 key rings from a key pool S of m 2 + m + 1 keys such that each key ring contains k = m + 1 keys and each two key rings shares exactly one common key.

The main strength of the C ¸amtepe scheme is the total secure connectivity, indeed each two nodes share exactly one common key. However, the SBIBD scheme does not scale to very large networks. Indeed, using key rings of m + 1 keys we can generate only m 2 + m + 1 key rings. SBIBD based key pre-distribution was also used in [START_REF] Ruj | Key predistribution using combinatorial designs for grid-group deployment scheme in wireless sensor networks[END_REF] to guarantee intra-region secure communications in grid group WSNs.

In this work, we seek to design a scalable key management scheme which ensures a good secure coverage of large scale networks with a low key storage overhead. Basic schemes giving a perfect network resilience [START_REF] Chan | Random key predistribution schemes for sensor networks[END_REF] [17] achieve a network scalability of O(k) where k is the key ring size. The SBIBD [START_REF] Yener | Combinatorial design of key distribution mechanisms for wireless sensor networks[END_REF] and the trade [START_REF] Ruj | Fully secure pairwise and triple key distribution in wireless sensor networks using combinatorial designs[END_REF] based ones allow to achieve a network scalability of O(k 2 ). In this work, we propose new solutions achieving a network scalability up to O(k 4 ) when providing high secure connectivity coverage and good overall performances. For this purpose, we make use of the unital design theory in order to pre-distribute keys. We propose in what follows a basic mapping from unitals to key pre-distribution as well as an enhanced unital based scheme which achieves a good trade-off between scalability and connectivity.

III. UNITAL DESIGN FOR KEY PRE-DISTRIBUTION IN WSNS

WSNs are highly resource constrained. In particular, they suffer from reduced storage capacity.

Therefore, it is essential to design smart techniques to build blocks of keys that will be embedded on the nodes to secure the network links. Nonetheless, in most existing solutions, the design of key rings (blocks of keys) is strongly related to the network size, these solutions either suffer from low scalability, or degrade other performance metrics including secure connectivity and storage overhead. This motivates the use of unital design theory that allows a smart building of blocks with unique features that allow to cope with the scalability and connectivity issues.

In what follows, we start by providing the definition and the features of unital design theory.

We explain then the basic mapping from unital to key pre-distribution and evaluate its performance metrics. We propose finally an enhanced unital-based scheme which achieves a good trade-off between scalability and connectivity.

A. Background: Unital Design

In combinatorics, the design theory deals with the existence and construction of systems of finite sets whose intersections have specified numerical properties. Formally, A t-design (ν, b, r, k, λ) is defined as follows : Given a finite set X of ν points (elements), we construct a family of b subsets of X, called blocks, such that each block has a size k, each point is contained in r blocks and each t points are contained together in exactly λ blocks. For instance, the symmetric Balanced Incomplete Block Design (SBIBD) presented above is a (ν, b, r, k, λ) design, where [START_REF] Assmus | Cambridge tracts in mathematics[END_REF]. Each block contains m + 1 points and each point is contained in r = m 2 blocks. Each pair of points is contained in exactly one block together. We denote the Unital by 2design(

ν = b = m 2 + m + 1, r = k = m + 1 and λ = 1. A Unital design is a Steiner 2-design which consists of b = m 2 (m 3 +1)/(m+1) = m 2 (m 2 -m+1) blocks, of a set of v = m 3 + 1 points
m 3 + 1, m 2 (m 2 -m + 1), m 2 , m + 1, 1) or by (m 3 + 1, m + 1, 1)
design for simplicity sake.

Without loss of generality, we focus in this paper on Hermitian unitals which exist for all m a prime power. Other construction for m not necessarily a prime power exist in literature [START_REF] Assmus | Cambridge tracts in mathematics[END_REF]. Some Hermitian unital construction approaches were proposed in literature [START_REF] Betten | Unitals and codes[END_REF] [START_REF] Key | Some applications of magma in designs and codes: Oval designs, hermitian unitals and generalized reed-muller codes[END_REF].

A unital may be represented by its v × b incidence matrix that we call M. In this matrix rows represent the points P i and columns represent blocks B j . The matrix M is then defined as:

M ij =    1 if P i ∈ B j 0 otherwise
We give in figure 2 an incidence matrix of a 2-(9,3,1) hermitian unital. It consists of 12 blocks of a set of 9 points. Each block contains 3 points and each point occurs in 4 blocks. Each pair of points is contained together in exactly one block.

B. A basic mapping from unitals to key pre-distribution for WSNs

In this subsection, we start by developing a simple scalable key pre-distribution scheme based on unital design that we denote by NU-KP for the naive unital-based key pre-distribution scheme. We propose a basic mapping in which we associate to each point of the unital a distinct key, to the global set of points the key pool and to each block a node key ring (see table III). We can then generate from a global key pool of

|S| = m 3 + 1 keys, n key rings (n = b = m 2 (m 2 -m + 1)) of size k = m + 1 keys each one.
Before the deployment phase, we generate the unital blocks corresponding to key rings. Each node is then pre-loaded with a distinct key ring as well as the corresponding key identifiers. After the deployment step, each two neighboring nodes exchange the list of their key identifiers which allows to determine eventual common key. Using this basic approach, each two nodes share at most one common key. Indeed, referring to the unital properties, each pair of points is contained together in exactly one block which implies that two blocks cannot share more than one point. Hence, if two neighboring nodes share one common key, the latter is used as a pairwise key to secure the link; otherwise, nodes should determine secure paths which are composed of successive secure links.

C. Theoretical analysis 1) storage overhead: When using the proposed naive unital based version matching a unital of order m, each node is pre-loaded with one key ring corresponding to one block from the design, hence, each node is pre-loaded with (m + 1) disjoint keys. The memory required to store keys is then l × (m + 1) where l is the key size.

2) Network scalability: From construction, the total number of possible key rings when using the

naive unital based scheme is n = m 2 ×(m 3 +1) (m+1) = m 2 × (m 2 -m + 1)
, this is then the maximum number of supported nodes.

3) Direct secure connectivity coverage: When using the basic unital mapping, we know that each key is used in exactly m 2 key rings among the m 2 × (m 2m + 1) possible key rings. Let us consider two nodes u and v randomly selected. The node u is pre-loaded with a key ring KR u of m + 1 different keys. Each of them is contained in m 2 -1 other key rings among the possible m 2 × (m 2m + 1) -1 ones. Knowing that two pair of keys occurs together in exactly one block, we find that the blocks containing two different keys of KR u are completely disjoint. Hence, each node shares exactly one key with (m + 1) × (m 2 -1) nodes among the m 2 (m 2m + 1) -1 other possible nodes, Then, the probability P c of sharing a common key can be calculated as follows:

P c = (m + 1) × (m 2 -1) m 2 (m 2 -m + 1) -1 = (m + 1) 2 m 3 + m + 1 (1)
The evaluation of this naive solution shows clearly that the basic mapping from unitals to key pre-distribution gives a high network scalability which reaches O(k 4 ). Moreover, given a network size n, this naive scheme allows to reduce the key ring size up to 4 √ n. However, this naive solution results a low key sharing probability which tends to O( 1 k ). In order to improve the key sharing probability while maintaining a good scalability improvement, we propose in the next section an enhanced scalable and efficient unital-based key pre-distribution for WSNs.

IV. A NEW SCALABLE UNITAL-BASED KEY PRE-DISTRIBUTION SCHEME FOR WSNS

In this section, we present a new unital-based key pre-distribution scheme for WSNs. In order to enhance the key sharing probability while maintaining high network scalability, we propose to build the unital design blocks and pre-load each node with a number of blocks picked in a selective way.

A. Key Pre-distribution

Before the deployment step, we generate blocks of m order unital design, where each block corresponds to a key set. We pre-load then each node with t completely disjoint blocks where t is a protocol parameter that we will discuss later in this section. In lemma 1, we demonstrate the condition of existence of such t completely disjoint blocks among the unital blocks. In the basic approach each node is pre-loaded with only one unital block and we proved that each two nodes share at most one key. Contrary to this, pre-loading each two nodes with t disjoint unital blocks means that each two nodes share between zero and t 2 keys since each two unitals blocks share at most one element.

After the deployment step, each two neighbors exchange the identifiers of their keys in order to determine the common keys. If two neighboring nodes share one or more keys, we propose to compute the pairwise secret key as the hash of all their common keys concatenated to each other. The used hash function may be SHA-1 [START_REF]Secure hash standard[END_REF] for instance. This approach enhances the network resiliency since the attacker have to compromise more overlap keys to break a secure link. Otherwise, when neighbors do not share any key, they should find a secure path composed of successive secure links.

The major advantage of this approach is the improvement of the key sharing probability. As we will prove in next subsection, this approach allows to achieve a high secure connectivity coverage since each node is pre-loaded with t disjoint blocks. Moreover, this approach gives good network resiliency through the composite pairwise secret keys which reinforces secure links. In addition, we show that our solution maintains a high network scalability compared to existing solutions although it remains lower than that of the naive version.

B. Theoretical analysis

We denote in what follows by t-UKP the unital-based key pre-distribution scheme of parameter t (t is the number of pre-loaded blocks at each node). We note that the 1-UKP scheme matches the basic mapping presented in section 3.

1) storage overhead: When using the t-UKP scheme of order m, we pre-loaded each node with t(m + 1) distinct keys. Indeed, from the construction, we can see that t blocks pre-loaded in a given node are completely disjoint. So, each two blocks within a key ring do not intersect at any key. So, the memory required to store keys is then equal to l × t × (m + 1), where l is the key size.

2) Network scalability: Since each node is pre-loaded with t blocks from the m 2 × (m 2m + 1)

possible blocks of the unital design, it is obvious that the maximum number of key rings that we can reach is equal to n = m 2 t (m 2m + 1). This is the ideal case when all unital blocks are used. When using the random pre-distribution of unital blocks, we may generate a number of blocks slightly lower than this best value. We compute in what follows the minimum network size that can be supported by the random blocks distribution.

Lemma 1: Given t ≥ 2, each set of (t -1)(m + 1)(m 2 -1) + t blocks from an m order unital design contains at least one sub-set of t completely disjoint blocks.

Proof: As shown before, we know that each block of a unital design intersect with exactly (m + 1)(m 2 -1) other blocks at one key. We prove the proposition by induction: for t = 2, let T be a set of (m + 1)(m 2 -1) + 2 blocks of a unital design of order m and let us assume that each two blocks of T intersect at one point. So, each block of T intersects with the (m + 1)(m 2 -1) + 1 other blocks in T which contradicts the fact that each block intersects with exactly (m + 1)(m 2 -1) blocks of the global unital. Hence the proposition is true for t = 2.

Let us now assume that the proposition is true at the order t and check whether it is for t + 1. Let T be a set of t(m + 1)(m 2 -1) + t + 1 blocks of a unital of order m. Since the proposition is true at order t, it exists at least one subset T 0 of t disjoint blocks in T . Each of these blocks intersects with exactly (m + 1)(m 2 -1) other blocks. So the maximum possible number of blocks which intersect with T 0 is t(m + 1)(m 2 -1). Hence, among the remaining t(m + 1)(m 2 -1) + 1 blocks of T -T 0 , there exists at least one block which does not intersect with any block of T 0 . We deduce that T contains at least t + 1 completely disjoint blocks.

Proposition 1: Using the t-UKP scheme with a random pre-distribution of unital blocks, we generate at least m 2 (m 2 -m+1)-((t-1)(m 2 -1)(m+1)+t) t key rings.

Proof: Using a unital design of order m, the number of the generated blocks is equal to m 2 (m 2m + 1). From the t-UKP construction, we know that each key ring contains exactly t disjoint blocks. Following the lemma 1, we find that using the t-UKP scheme, we can generates at least m 2 (m 2 -m+1)-((t-1)(m 2 -1)(m+1)+t) t key rings.

3) Direct secure connectivity coverage: We discuss in what follows the direct secure connectivity coverage of the t-UKP scheme.

Proposition 2: Given t ≥ 2, using the t-UKP scheme, the secure connectivity coverage (the probability of key sharing between any two nodes) is given by:

P c = 1 -(1 -(m+1) 2 m 3 +m+1
) t 2 Proof: Let us consider two nodes u and v randomly selected. Each node is pre-loaded with a key ring containing t disjoint unital blocks:

KR u = {B u,1 ∪ B u,2 ∪ ... ∪ B u,t } and KR v = {B v,1 ∪ B v,2 ∪ ... ∪ B v,t }
Following equation (1) (Cf. Section 3.C), we find that the probability that two blocks B u,p and B v,q share one key is (m+1) 2 m 3 +m+1 while the probability that they don't share any key is (1

-(m+1) 2 m 3 +m+1
). Since all blocks of node u as well as those of node v are completely disjoint thanks to the proposed construction, the probability that the two nodes u and v don't share any key is then given by:

P (KR u ∩ KR v = ∅) = t p=1 t q=1 P (B u,p ∩ B v,q = ∅) = t p=1 t q=1 (1 - (m + 1) 2 m 3 + m + 1 ) = (1 - (m + 1) 2 m 3 + m + 1 ) t 2
The probability that two nodes share at least one key is then :

P c = 1 -(1 -(m+1) 2 m 3 +m+1 ) t 2
We plot and compare later the key sharing probability of the t-UKP scheme and show that we increase considerably the key sharing probability over the NU-KP scheme.

4) Network Resiliency:

We analyze in what follows the network resiliency of the t-UKP scheme.

Lemma 2: Using the t-UKP scheme of order m, the probability p(i) that two nodes share exactly i keys (0 ≤ i ≤ t 2 ) is:

p(i) =   t 2 i   (m + 1) 2 m 3 + m + 1 i × 1 - (m + 1) 2 m 3 + m + 1 t 2 -i
Proof: Let us consider two nodes u and v randomly selected. Each node is pre-loaded with a key ring containing t disjoint unital blocks:

KR u = {B u,1 ∪ B u,2 ∪ ... ∪ B u,t } and KR v = {B v,1 ∪ B v,2 ∪ ... ∪ B v,t }
Let us consider X p,q the variable giving the number of shared keys between two unital blocks B u,p and B v,q . Following equation (1) (Cf. Section 3.C), we find that X p,q takes only 1 and 0 values such that: P (X p,q = 1) = (m+1) 2 m 3 +m+1 while: P (X p,q = 0) = 1 -(m+1) 2 m 3 +m+1 X p,q follows then a Bernoulli distribution of parameter ( (m+1) 2 m 3 +m+1 ). The number of shared keys between KR u and KR v which represents ( t p=1 t q=1 X p,q ) follows then a binomial distribution (t 2 , (m+1) 2 m 3 +m+1 ). So, p(i) is given by :

p(i) =   t 2 i   (m + 1) 2 m 3 + m + 1 i × 1 - (m + 1) 2 m 3 + m + 1 t 2 -i
Proposition 3: When using the t-UKP scheme of order m, the network resiliency when x nodes are captured is:

R x = 1 - t 2 i=1
(1 -

m 3 (m -1) x × t m 2 (m 2 -m + 1) x × t ) i p(i) P c (2)
Proof: We recall that p(i) is the probability that two nodes share exactly i keys and that P c is the probability that two nodes share at least one key. So, P c = t 2 i=1 p(i) = 1p(0) which matches proposition 2.

Let us call LC, the event that a link is compromised, LC i the event that a link secured with i keys is compromised and NC x is the event that nodes are compromised. Let us compute first P (LC i |NC x ). When using the t-UKP scheme, the compromise of x nodes reveals exactly x × t unital blocks. We know that each key occurs in in m 2 unital blocks among the total number of m 2 (m 2 -m+1) blocks, the probability c that a key is uncompromised when x node are compromised is then the probability that the key does not occurs in any if the discovered blocks. So, we find that:

c =      m 2 (m 2 -m + 1) -m 2 x × t           m 2 (m 2 -m + 1) x × t      =      m 3 (m -1) x × t           m 2 (m 2 -m + 1) x × t     
The probability of compromising a given secret key composed of i keys is then equal to (1c) i . So, the probability that a given link is compromised when it is secured with i keys and when x nodes are compromised is given by: P (LC k |NC x ) = (1c) i . Let us now compute the resiliency of the global network. This can be computed as the average probability that a link be uncompromised when x nodes are captured and is given by :

R x = 1 -P (LC|NC x ) = 1 -t 2 i=1 P (LC i |NC x ) p(i) Pc = 1 -t 2 i=1 (1 -c) i p(i)

Pc

C. Choice of the t value

As we showed through performance analysis, the pre-distribution of t unital blocks in each node instead of one allows at the same time enhancing the key sharing probability and computing composite pairwise secret keys witch reinforce secure links. On the other hand, the use of the t-UKP scheme multiplies the storage overhead and decreases the network scalability over the naive version. The choice of the t value depends then on the application requirement in order to obtain the best tradeoff.

Indeed, when do not need to establish a secure link between each pair of nodes or when the length of secure paths is not a major concern, low values of t can be chosen. For instance, in many-to one WSNs where the key sharing requirement is reduced to the child-parent relationship, low values of t can be used in order to reach an extremely high scalable deployment. On the other hand, when

the key sharing probability and the length of secure paths are major concerns, t should be given a high value which allows to ensure a good key sharing probability.

In order to maintain a high key sharing probability and then low secure path length while maintaining a high scalability, we propose to choose t = √ m. Without loss of generality, we assume that m is a perfect square, if it is not the case, we can refer to the nearest integer to the square root of m. Indeed, this value allows to maintain a high scalability of O( √ mm 3 ) with a storage overhead of O( √ mm). As we will prove, this choice allows to reach a very good key sharing probability lower bounded by 1e -1 . We denote by UKP* the t-UKP scheme with t = √ m and we recall that a lower bound of the direct secure connectivity coverage of UKP* is a value L, such that the direct secure connectivity coverage of UKP* is always greater or equal to L for all m values.

Proposition 4: Let us consider the UKP* scheme (t-UKP with t = √ m), the limit of the direct secure connectivity coverage as m tends to infinity is equal to L = 1e -1 which is a lower bound the direct secure connectivity coverage of UKP*.

Proof: : Following proposition 2, when using the UKP* (t-UKP with t = √ m), the secure connectivity coverage is given by :

P c = 1 -(1 -(m+1) 2 m 3 +m+1
) m . Since P c (m) is a strictly decreasing function defined on the interval [0,1], the limit as m tends to infinity exists and is equal to L such as: (0 ≤ L ≤ 1). We have:

L = lim m→+∞ 1 -(1 - (m + 1) 2 m 3 + m + 1 ) m 1 -L = lim m→+∞ (1 - (m + 1) 2 m 3 + m + 1 ) m ln (1 -L) = lim m→+∞ m × ln (1 - (m + 1) 2 m 3 + m + 1 ) ln (1 -L) = lim m→+∞ m × (m + 1) 2 m 3 + m + 1 × ln (1 -(m+1) 2 m 3 +m+1 ) (m+1) 2 m 3 +m+1 ln (1 -L) = -1 L = 1 -e -1
So, the secure connectivity coverage function is a strictly decreasing function having a limit of L = 1e -1 ≈ 0.632 as m tends to infinity. This limit L is then a lower bound of the direct secure connectivity coverage when using the UKP* scheme.

V. PERFORMANCE COMPARISON

In this section, we compare the proposed unital-based schemes to existing schemes regarding different criteria (We recall that metric definitions are given in table I).

A. Network scalability at equal key ring size

We compare in figure 3 the scalability of the proposed unital based schemes against that of the SBIBD-KP and the Trade-KP ones. The network scalability of the t-UKP schemes is computed as the average value between the maximum and the minimum scalability. The network scalability of the SBIBD scheme is computed as m 2 + m + 1 where m is the SBIBD design order and m + 1

is the key ring size. We compute the salability of the Trade-KP scheme as 2q 2 where q is the first prime power greater than the key ring size k, this value allows a achieve the best session key sharing probability using the Trade-KP scheme as we proved in [START_REF] Bechkit | A new scalable key pre-distribution scheme for wsn[END_REF]. The figure shows that at equal key ring size, the NU-KP scheme allows to enhance greatly the scalability compared to the other schemes;

for instance the increase factor reaches 10000 compared to the SBIBD-KP scheme when the key ring size exceeds 100. Moreover, the figure shows that the t-UKP schemes achieve a high network scalability. We notice that the higher t is, the lower network scalability is. Nevertheless, 2-UKP and 3-UKP give better results than those of the SBIBD-KP and the Trade-KP solutions. Even we choose t = √ m as we propose (UKP*), the network scalability is enhanced. For instance, compared to SBIBD-KP scheme, the increase factor reaches five when the key ring size equal to 150. We plot in figure 4 the same results separately with linear scales which illustrate clearly the network scalability enhancement when using our solutions.

Authors in [START_REF] Chan | Random key predistribution schemes for sensor networks[END_REF], assess the network scalability of random schemes including the RKP and the Qcomposite ones regarding to the desired network connectivity and to the network capacity to maintain secure links while some nodes are compromised. They defined for that a threshold f m called the limited global payoff requirement. The later can be explained as the level of compromise past where the adversary gains an unacceptably information on the other pairwise secret keys. Depending on P c and f m they defined the maximum number supported network size. Authors present results for P c = 0.33 and f m = 0.1 and show that the network scalability with a key ring size of 100 is about 300 for RKP scheme and between 600 and 700 when using Q-composite schemes. The scalability of the same schemes with a key ring size of 400 is respectively of about 1200 and between 2700 and 2800. We can see clearly that our solutions allow to reach much better network scalability than the random schemes under the suggested parameters.

B. Key ring size at equal network size

In this subsection, we compare the required key ring size when using the unital-based, the SBIBD-KP and the Trade-KP schemes at equal network size. We compute for each network size the design order allowing to achieve the desired scalability and we deduce then the key ring size, the obtained results are reported in figure 5. The figure shows that at equal network size, the NU-KP scheme allows to reduce extremely the key ring size and then the storage overhead. Indeed the enhancement factor over the SBIBD-KP scheme reaches 20. When using the t-UKP schemes, the results show that the higher t is, the higher required key ring size is. However, this value remains significantly lower than the required key ring size of the SBIBD-KP and Trade-KP schemes. Moreover, we can see clearly in the figure, that at equal network size, the UKP* scheme provides very good key ring size compared the SBIBD-KP and the Trade-KP schemes. For instance, the key ring size may be reduced over a factor greater than two when using the UKP* compared to the SBIBD-KP scheme.

C. Energy consumption at equal network size

In this subsection, we compare the energy consumption induced by the direct secure link establishment phase. Since each node broadcasts its list of key identifiers, the energy consumption can be computed as :

E = E tx • k • log 2 (|S|) + η • E rx • k • log 2 (|S|)
where E tx (resp. E rx ) is the average energy consumed by the transmission (resp. reception) of one bit, k is the key ring size, η is the average number of neighbors and log 2 (|S|) represents the size of a key identifier in bits that we round up to the nearest byte size.

We compare the energy consumption of our solutions against SBIBD-KP and Trade-KP. The results plotted in figure 6 show that at equal network size, the NU-KP scheme consumes very small amount of energy to exchange the low number of key identifiers. We also note that the higher t is, the higher the consumed energy is. This is due to the increased number of stored keys and thereby the increased number of exchanged identifiers. Finally, the figure shows clearly that UKP* scheme consumes less energy than the SBIBD-KP and the Trade-KP schemes. This matches our expectation since the energy consumption is strongly correlated to the number of stored keys.

D. Network connectivity at equal key ring size

We compare in this subsection, the network secure connectivity coverage of the different schemes.

First, we plot in figure 7 (a) the key sharing probability when using the unital based schemes (NU-KP, t-UKP and UKP*). The figure shows that the NU-KP scheme provides a bad direct secure connectivity coverage which decreases significantly when the key ring size increases. Indeed, the key sharing probability is low and tends to O( 1 k ) as k tends to infinity. Otherwise, the obtained results show that the higher t is, the better the direct secure connectivity coverage is. Indeed, loading nodes with many blocks from unital design allows to increase significantly the key sharing probability. The figure shows moreover that the UKP* scheme gives very good connectivity results. For instance, the direct secure connectivity coverage remains between 0.82 and 0.66 when the key ring size is between 10 and 150. As the key ring size is high, the direct secure connectivity of UKP* approaches 1e -1 ≈ 0.632, which we proved to be a lower bound.

In a second time, we compared the direct secure connectivity coverage of the UKP* to those of the other existing schemes. The Trade-KP scheme allows to reach a direct secure connectivity lower than 0.25. Indeed, we proved in [START_REF] Bechkit | A new scalable key pre-distribution scheme for wsn[END_REF] that the key sharing probability of the Ruj et al. Trade-KP scheme [START_REF] Ruj | Fully secure pairwise and triple key distribution in wireless sensor networks using combinatorial designs[END_REF] is equal to k(k-1) 4q 2 where 4 ≤ k < q and q is a prime power. k is the key ring size and we chose q to be the first prime power greater than k which ensures the best key sharing probability. The figure 7 (b) shows that the secure connectivity of the RKP* scheme remains much better than that of Trade-KP one. Indeed the direct secure connectivity coverage of RKP* has a good lower bound of L = 1e -1 ≈ 0.632 while that of the Trade-KP have an upper bound of 0.25. We compared also the secure connectivity of the UKP* scheme to those of the RKP and the Q-composite schemes.

We assume for these random approaches that the global key pool size is equal to the square of the key ring size which allows to achieve a good network resiliency. The results show that UKP* scheme provides a better secure connectivity coverage than the RKP scheme and much better than the Q-composite schemes with Q = 2, Q = 3, etc. Indeed, using the Q-composite scheme, two nodes must share at least Q keys to be able to establish a secure link which degrades significantly the secure connectivity coverage (see figure 7 (b)).

Although the unital-based scheme UKP* increases significantly the network scalability and provides a good key sharing probability greater than 0.632, this metric remains lower compared to SBIBD-KP which ensures a perfect key sharing probability. However, our scheme allows to attend a total secure connectivity thanks to the secure path establishment.

We also studied the average secure path length when using different key pre-distribution schemes including our solutions. For this purpose, we conducted simulations while referring to the results given in [START_REF] Shakkottai | Unreliable sensor grids: coverage, connectivity and diameter[END_REF] in order to construct a grid deployment model which guarantees the network physical connectivity and coverage. The results showed that the UKP* scheme provides a good average secure path length between 1.18 and 1.36 when the key ring size is between 10 and 150. It does not exceed 1.37 even the key ring size is very high. In others terms, when using the UKP* scheme, two-thirds of possible links in the network will be secured directly while practically all the other third links can establish a 2-hop secure path. We give some numerical results about the average secure path length in the last subsection.

E. Network resiliency at equal key ring size

We compare in this subsection, the network resiliency of the unital-based schemes to those of the Trade-KP and the SBIBD-KP ones. We notice that the proposed trade based construction given in [START_REF] Ruj | Fully secure pairwise and triple key distribution in wireless sensor networks using combinatorial designs[END_REF] allows to have a unique pairwise key per secure link, this key is computed as the hash of a unique pair of initial keys. However the overall network resiliency is not perfect because the compromise of some key rings may reveal other pairwise secret keys used to secure external links in which the compromised nodes are not involved. We proved that the resiliency of the Trade-KP scheme is given by: (see proof in appendix A)

R x = 2q 2 -4q + 2 x + 4(q -1) 2q 2 -4q + 2 x -1 2q 2
x where x is the number of comprised nodes and q is the Ruj et al. trade construction parameter.

On the other hand, following the study presented in [START_REF] Yener | Combinatorial design of key distribution mechanisms for wireless sensor networks[END_REF], the network resiliency R x of the SBIBD-KP scheme is given by:

R x = m 2 x m 2 + m + 1 x
Where m is the SBIBD design order. Finnaly, the network resiliency formula of unital based schemes was given in proposition 3.

We compare in figure 8 the network resiliency at equal number of compromised nodes for |KR| = 68. The figure shows that the NU-KP scheme provides a good resiliency compared to other schemes.

Using the t-UKP, the higher t is, the lower network resiliency is at equal number of compromised nodes. This is due to the number of compromised unital blocks which is multiplied by t. On the other hand, the figure shows that the UKP* scheme improves the network resiliency over the SBIBD-KP scheme by 20%. It also gives a better network resiliency then the Trade-KP scheme when the number of compromised nodes exceeds 60.

F. Numerical results

We provide in table IV numerical results comparing network scalability, direct secure connectivity coverage, and average secure path length of the three schemes (SBIBD-KP, Trade-KP and UKP*) at equal key ring size. We notice that we provide the average network scalability (number of nodes) when using UKP* scheme. On the other hand, we compute the average secure path length based on simulations. We refer in these simulations to the results given in [START_REF] Shakkottai | Unreliable sensor grids: coverage, connectivity and diameter[END_REF] in order to construct a grid deployment model which ensures the network physical connectivity and coverage. Numerical results show that the unital-based key pre-distribution scheme UKP* increases the network scalability over the SBIBD-KP and the Trade-KP scheme while maintaining high secure connectivity coverage. For instance, the network maximum size is increased by a factor of 3 and 4.8 when the key ring size is equal to 68 and 140 respectively compared to the SBIBD-KP scheme. In addition, we maintain a high connectivity over 0.63 which ensures a low average secure path length which does not exceed 1.37.

VI. CONCLUSION

We proposed, in this work, a scalable key management scheme for WSNs. We make use, for the first time, of the unital design theory. We showed that a basic mapping from unitals to key pre-distribution allows to achieve an extremely high network scalability while giving a low direct secure connectivity coverage. We proposed then an efficient scalable unital-based key pre-distribution scheme providing high network scalability and good secure connectivity coverage. We discuss the solution parameter and we propose adequate values giving a very good trade-off between network scalability and secure connectivity. We conducted analytical analysis and simulations to compare our new solution to existing ones, the results showed that our approach provides a good secure coverage of large scale networks with a low key storage overhead and a good network resiliency.

VII. ACKNOWLEDGEMENT

This work is made as part of the Picardie regional project under reference I159C. The authors wish to thank the Picardie regional council in France and the European Regional Development Fund (ERDF) for funding and supporting this project.

APPENDIX A NETWORK RESILIENCY OF THE TRADE-BASED KEY PRE-DISTRIBUTION SCHEME

Using the Ruj et al. trade construction [START_REF] Ruj | Fully secure pairwise and triple key distribution in wireless sensor networks using combinatorial designs[END_REF], the two sets T 1 and T 2 contain q 2 key rings each one.

Let us assume that x nodes are compromised and let us compute the probability that a given pair of keys K i and K j is known (We recall that two nodes can establish a secure session key if they share exactly two common keys).

From construction, we know that each key occurs in exactly q blocks in T 1 and q blocks of T 2 , and that each pair of keys occurs in one key ring from T 1 and one key ring from 2 . So, we find that among the 2q 2 possible key rings, two contains the pair K i and K j , 2q -2 contain only K i , 2q -2 contain only K j and then 2q 2 -4q + 2 not contain any key of K i and K j . Fig. 6. Energy consumption at equal network size: we consider a grid deployment model [START_REF] Shakkottai | Unreliable sensor grids: coverage, connectivity and diameter[END_REF] where η is set to 4 log(n) (n is the network size). The latter value ensures the physical network connectivity and coverage [START_REF] Shakkottai | Unreliable sensor grids: coverage, connectivity and diameter[END_REF]. Etx and Erx are set to the values of CC1000 radio configuration, i.e. 1625 nJ and 1156 nJ resp.

[24]

Fig. 1 .Fig. 2 .Fig. 3 .

 123 Fig. 1. Classification of symmetric key management schemes for WSNs

Fig. 4 .Fig. 5 .

 45 Fig. 4. Network scalability at equal key ring size (linear scale)

Fig. 7 .Fig. 8 .

 78 Fig. 7. Network connectivity at equal key ring size

Represents the maximum number of generated key rings which corresponds to the maximum number of supported nodes.

Storage overhead

Measures the memory required to store keys in each node. We exclude the memory required to store the key identifiers since it is negligible compared to the key ring storage overhead (2-logarithm of the maximum number of keys). Direct secure connectivity coverage Defines the fraction of secured direct links among possible links in the network; it is computed as the probability that a given pair of neighboring nodes are able to establish a direct secure link. Average secure path length When two neighboring nodes have no common keys, they should establish a secure path composed of successive secure links. This metric measures the average length in hop count of these secure paths. Network resiliency against node capture We define the network resiliency R x as the fraction of uncompromised external secure links when x sensor nodes are captured.


Sultan Demirdis

Yanina Fasano

Shigeru Kasahara

Takahito Terashima

Takasada Shibauchi

Yuji Matsuda

Marcin Konczykowski

Cornelis H Pastoriza

Jacominus Van Der Beek

S Demirdiş

C J Van Der Beek

Disorder, critical currents, and vortex pinning energies in isovalently substituted BaFe_2(As_1 -xP_x)_2

Keywords: numbers: 74.25.Op, 74.25.Sv, 74.25.Wx, 74.62.En, 74.70.Xa

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Recent vortex imaging studies performed on iron pnictide superconductors show evidence for nanoscale inhomogeneity 1,2 being at the origin of the low-field critical current density and the highly disordered vortex structures in these materials. [3][4][5][6][7][8][9][10][11][START_REF] Kawano-Furukawa | Abstract book of ICSM[END_REF] Notably, in Ba(Fe 1-x Co x ) 2 As 2 , the critical current density j c and vortex distributions imaged by Bitter decoration could be consistently analyzed, provided that spatial heterogeneity, on a scale of several dozen nm, both of the critical temperature T c and the vortex line energy ε 0 , is taken to be responsible for flux pinning. 2 At higher magnetic fields, of the order of several tenths of Tesla, nano-scale heterogeneities are inefficient in pinning flux lines. 1,[START_REF] Van Der Beek | [END_REF]14 The critical current density is then most likely determined [START_REF] Van Der Beek | [END_REF] by the scattering of quasiparticles in the vortex cores associated with the presence of atomic-size defects in the crystal, 15,16 leading to weak collective pinning. 16,17 A good candidate for these defects are the dopant atoms themselves. 1,2,[START_REF] Van Der Beek | [END_REF]14 The nature of the dopant atoms is essential for this mecha-nism; charged defects lead to different scattering than uncharged defects. [START_REF] Van Der Beek | [END_REF] This weak collective pinning contribution to the critical current density manifests itself as a plateau-like behavior in a j c (B) plot. It is present in all charge-doped iron-based superconductors, as well as in Ba(Fe 1-x Ru x ) 2 As 2 . 18 On the other hand, in isovalently substituted BaFe 2 (As 1-x P x ) 2 there is no indication of weak collective pinning, [START_REF] Van Der Beek | [END_REF] which qualifies the material as "clean" with respect to charge-doped iron-based superconductors. Given recent claims 11,[START_REF] Kawano-Furukawa | Abstract book of ICSM[END_REF] that in certain iron-based superconductors, the vortex configuration is more ordered than what was hitherto observed, [3][4][5][6][7][8][9][10] it is interesting to see whether the absence of weak collective pinning has any impact on the spatial configuration of vortices. From a magnetic force microscopy (MFM) study at magnetic fields up to 100 Oe, Yang et al. claim that vortex configurations in hole-doped single crystalline Ba 1-x K x Fe 2 As 2 (with x = 0.28 and x = 0.4) are more ordered than in, e.g. Ba(Fe 1-x Co x ) 2 As 2 . 2 Using an analysis method similar to that of Ref. 2, they report pinning forces that are one order of magnitude smaller on average. They also claimed the observation of local triangular vortex order in the optimally doped material, even though closer scrutiny (see section III E below) reveals the vortex ensembles to be no more ordered than those in Ba(Fe 0.925 Co 0.075 ) 2 As 2 . 2 Finally, the authors 11 reported on the observation of remarkable vortex chains, both in underdoped (x = 0.28) and optimally doped (x = 0.4) Ba 1-x K x Fe 2 As 2 . The presence of these was attributed to vortex pinning by the twin boundaries arising from the orthorhombic structure, at least in the underdoped material. Furthermore, neutron scattering experiments on the vortex lattice in isovalently substituted BaFe 2 (As 1-x P x ) 2 were performed at T = 2 K by Kawano-Furukawa et al.. [START_REF] Kawano-Furukawa | Abstract book of ICSM[END_REF] No vortex Bragg peaks were found for the optimally substituted compound. However, after annealing the samples at 500 • C, a distorted triangular vortex lattice was observed; this became more ordered as the applied magnetic field was increased from 0.7 to 7 T. [START_REF] Kawano-Furukawa | Abstract book of ICSM[END_REF] These results suggest that the disorder responsible for pinning in BaFe 2 (As 1-x P x ) 2 is extrinsic in nature.

In this work we present and analyze sustainable current density measurements, magneto-optical imaging, and Bitter decoration experiments performed on BaFe 2 (As 1-x P x ) 2 single crystals with different x. In contrast to some reports, 19 we find strong pinning, presumably by nanoscale heterogeneity, as the only observed pinning contribution in fields up to 5 T. As in Ba(Fe 1-x Co x ) 2 As 2 and other materials, 6 there is no evidence for any extended triangular order in the vortex ensemble; thus, the strong pinning contribution in itself suffices to generate the extreme disorder of the vortex ensemble. The spatial configuration of vortices in isovalently substituted BaFe 2 (As 1-x P x ) 2 does not present large vortex density fluctuations such as observed in charge-doped Ba(Fe 1-x Co x ) 2 As 2 single crystals, 2 a fact that is attributed to the different temperature depen-dences of the superfluid density in the two materials. The quantitative analysis of the vortex configurations in terms of the pinning energy confirms that pinning disorder is somewhat less effective in BaFe 2 (As 1-x P x ) 2 than in Ba(Fe 1-x Co x ) 2 As 2 , and that it depends on the P-content x. Analysis of the pinning energies, pinning forces, and the critical current density as function of Pcontent yields consistent estimates of the effective pin density. This clearly decreases upon increasing the Pcontent, a behavior that tracks the composition dependence of the scattering rate in the normal state. The main features of our results can be understood in terms of the composition dependence of the vortex line energy, implying that local variations of the superfluid density is a good candidate for the origin of the vortex pinning. The local variation of the dopant atom density as well as the variation of the sustainable current density with composition argues against any possible spatially phaseseparated superconducting and anti-ferromagnetic states of the material as being at the origin of pinning.

II. EXPERIMENTAL DETAILS

Experiments

have been performed on BaFe 2 (As 1-x P x ) 2 single crystals grown by the self-flux method, 20 and characterized using Energy Dispersive X-Ray Spectroscopy (EDX) and EDX mapping in a scanning electron microscope (SEM). Crystals with manifest chemical heterogeneity were discarded from further study. The crystals described below present no impurity phases, within the experimental limits of accuracy < ∼ 1%. Magnetic flux penetration in crystals with different substitution levels (x = 0.27 -0.49) was characterized using the magneto-optical imaging (MOI) method 1,14,21 before further experiments. The MOI technique notably allows one to discard samples with macroscopic defects, and also, to extract calibrated flux density profiles. The sustainable current density j s for magnetic fields up to µ 0 H a = 500 G (50 mT) was obtained from the gradient of the local magnetic flux density B perpendicular to the crystal surface, using the Bean model. 22,23 Given the thickness-to-width ratio of these crystals, d/w ∼ 0.25, one has µ 0 j s ∼ 3 dB/dx. 24 The crystal inhomogeneity, and notably the local distribution of T c was characterized using the differential magneto-optical (DMO) method. 14,25 Measurements in higher magnetic fields were performed using micron-sized Hall probe arrays, 26 tailored in a pseudomorphic GaAlAs/GaAs heterostructure, as well as using a Superconducting Quantum Interference Device (SQUID)-based magnetometer.

The vortex ensembles in several crystals, of substitution levels x = 0.33, 0.36, and 0.49, were imaged using the Bitter decoration method, 2,27 at an applied field µ 0 H a = 20 G (2 mT) (see Section III E). In what follows, individual crystals will be identified as ( x = substitution level, sample number # ).

III. RESULTS

A. Spatial variation of the critical temperature Tc Figures 1 (a,b) presents DMO images of the exclusion of an applied field µ 0 H a = 1 G (0.1 mT) as one crosses the superconducting to normal transition of BaFe 2 (As 1-x P x ) 2 single crystals x = 0.27 #2 and x = 0.49 #1. These images reveal that T c is spatially heterogeneous. While inhomogeneity is especially pronounced in the underdoped samples, see crystal x = 0.27 #2, it is also observed in the overdoped crystals. A link between the heterogeneity observed in underdoped samples and a possible phase separation between of superconductingand Spin-Density Wave (SDW) antiferromagnetic phases, and/or that of orthorhombic and tetragonal structural domains is therefore not obvious.

The nous intensity I(r, T ) in the DMO images of Fig. 1.

T H (T ) is presented in Fig. 2a for crystal x = 0.27 #2.

The width of the superconducting transition obtained for the various crystals is presented in Fig. 2 b, where the error bar indicates the spread of T c in a given crystal, and the data points give the temperature where 50% of the crystal has become superconducting. BaFe 2 (As 1-x P x ) 2 single crystals x = 0.33 #1, and x = 0.36 #2, respectively. The former crystal is characterized by very weak bulk pinning and, as a result, a large influence of geometrical 28,29 and surface barriers. 30 The influence of the surface screening current leads to an in- The flux density profiles across the same crystals, depicted in Figure 4 (a,b) and(c,d) respectively, were extracted from the calibrated luminous intensities of the magneto-optical images in Fig. 3. One sees that, even for the same or comparable doping levels, very different flux density profiles can be obtained after zero-field cooling. Figure 4 (c,d) shows the Bean-like penetration of the magnetic flux inside crystal x = 0.36 #2, with no clear influence of a surface barrier, while the flux profiles for crystal x = 0.33 #1 in Figure 4 (a,b) show, apart from inhomogeneity, a large discontinuity in the magnetic induction at the sample edge, characteristic of a surface barrier. Given the very different behavior for nearly the same sample composition, the origin of the bulk critical current density in BaFe 2 (As 1-x P x ) 2 is most likely extrinsic. This is supported by the temperature dependence of the sustainable current density j s (T, B = 30 mT) of the studied samples, shown in Fig. 5. The absolute value of j s (T ) is widely disperse, even for crystals with the same doping level.

B. Sustainable current density js

In spite of the disparity, the flux pinning mechanism in all crystals is the same. Fig. 6 shows hysteresis loops of the local gradient of the magnetic induction dB/dx in fields of up to 2 T, obtained on crystals of different composition using the Hall probe-array magnetometry technique. 26 The hysteresis loops were measured at 6 K, at which flux creep has only a moderate influence. For all crystals, of all investigated substitution levels, one has the ubiquitous central peak at zero field, believed to be due to strong pinning by nm-scale disorder. 14 The mag- netic field dependence of the sustainable current density j s (B) was obtained from the value of dB/dx at given B. Fig. 7 shows j s (B) for the optimally substituted single crystal x = 0.36 # 2 at different temperatures. The j s (B) curve at the lowest T is representative of the fielddependent critical current density j c (B). The critical current density is characterized by a low-field plateau, followed by a j c ∝ B -1/2 decrease at higher fields, behavior that is typical of strong flux pinning by sparse pointlike defects. [START_REF] Van Der Beek | [END_REF]14,31 The contribution to j c (B) due to weak collective pinning of the vortex lines by atomic sized point pins, observed in all charge-doped iron based superconductors as well as in Ba(Fe 1-x Ru x ) 2 As 2 , is clearly absent in Fig. 7. The BaFe 2 (As 1-x P x ) 2 system can therefore be seen as a typical strongly pinning superconductor, in the sense that only large but sparse extrinsic point-like pins contribute to flux pinning. 31 At larger temperatures and fields, j s (B) decreases faster than B -1/2 , an effect attributed to flux creep.

Measurements carried out to larger fields using a SQUID magnetometer show that, contrary to the data presented in Ref. 19, the sustainable current density continues its monotonous decrease as function of magnetic field -there is no "fishtail" or "second-peak" effect 14 up to µ 0 H a = 5 T. The peak effect being usually associated with a weak collective pinning contribution to the critical current, we surmise that in the field range of interest the strong pinning mechanism is the only one at play.

C. Quantitative effect of flux creep

The influence of flux creep is assessed from relaxation measurements of the local flux density using the Hall probe magnetometry technique. Typical examples, shown in Fig. 9 (a), show that the creep rate S ≡ d ln(dB/dx)/d ln t typically amounts to a few percent. Nevertheless, j s is significantly affected by creep, such that it is determined as the solution of the relation U (j s ) = k B T ln [(t 0 + t) /τ ], rather than by the critical current density j c . 32,33 Here, t 0 is a time describing tran-sient effects at the onset of relaxation, and τ is a normalization time related to the sample inductance. 32,33 The dependence of the flux creep barrier U (j) on current density j can be extracted using various methods, including those of Maley et al. 34 and Abulafia et al.. 35,36 Applying the latter, we find [see Fig. 9(b)] that the creep barrier in optimally substituted BaFe 2 (As 1-x P x ) 2 follows

U (j) = U c j c j µ , (1)
with values of the exponent µ ∼ 1.5 -2. Therefore, the time and temperature-dependence of the sustainable screening current density is described by

j s = j c k B T U c ln t + t 0 τ -1/µ . ( 2
)
The impact of flux creep on the temperature dependence of the sustainable current is depicted in Fig. 9(c), which shows j s (T )-curves for crystal x = 0.33 #2, for different B. The curve in zero applied field is little affected by creep, and roughly follows the expected temperature dependence of the depairing current, j(0, T ) ∼ ε 0 (T )/Φ 0 ξ(T ) (Φ 0 = h/2e is the flux quantum). Here, the vortex line energy, ε 0 (T ) = Φ 2 0 /4πµ 0 λ 2 ab , proportional to the superfluid density n s ∼ λ -2 ab , is evaluated using the data for the in-plane penetration depth λ ab (T ) of Ref. 41, and the coherence length ξ(T ) ∼ ξ(0) (1 + T /T c )/(1 -T /T c ). The curves for varying applied field can then be well described by taking the creep barrier prefactor U c (T ) ∝ ε 0 (T ), j c (B, T ) ∝ j(0, T )B -1/2 , µ = 1.6, and ln [(t + t 0 ) /τ ] = 20. 36 Therefore, the temperature dependence of the screening current in fields larger than 0.1 T is essentially determined by flux creep.

D. Extraction of pinning parameters

We now analyze the j s (B)-curves measured at low temperature, representative of the critical current density j c (B), and which bear the hallmarks of strong pinning. These are the plateau at low magnetic field, 14,31

j c (0) = f p Φ 0 L = π 1/2 f p Φ 0 ε λ U p n i ε 0 1/2 , (B B * ) (3)
followed by a power-law decrease as a function of the flux density B, 14,31 which can be described as

j c (B) = f p Φ 0 L 2 ε λ a 0 π = f p Φ 0 ε λ U p n i ε 0 Φ 0 B 1/2 . (B B * ) (4)
The main parameter, L = (πU p n i /ε λ ε 0 ) -1/2 , is the average distance between effective defects pinning a single vortex in the low-field limit. 31 The crossover field B * is that above which the intervortex repulsion limits the number of effective pins per vortex, and f p is the maximum pinning force exerted by a single strong pin. U p /[J] is the pinning energy of a single strong pin, n i is the pin density, a 0 = (Φ 0 /B) 1/2 is the intervortex distance, and ε λ = λ ab /λ c is the penetration depth anisotropy.

Equations ( 3) and ( 4) show that the pinning force of a single strong pin f p = (Φ 3/2 0 ε λ /π) j 2 (0)/[∂j(B)/∂B -1/2 ] can be obtained from the experimentally measured low temperature, lowfield current density j c (0), and the slope ∂j c (B)/∂B -1/2 at intermediate fields. Reserving our attention to the crystals used in the Bitter decoration experiments presented below, we obtain, for an estimated ε λ = 0.15, 37 f p ≈ 8 × 10 -13 N for both crystals (x = 0.36#2) and (x = 0.49#1). This value is twice larger than that measured in Ba(Fe 1-x Co x ) 2 As 2 . 2 An evaluation at the highest measurement temperature of 0.8T c yields f p = 2 × 10 -14 N; however, this value is likely to be overestimated due to creep. Similarly, one can extract the length L = f p /Φ 0 j c (0). Figure 10 shows that L is of the order of several dozen to hundreds of nm, in accordance with the strong pinning hypothesis. Moreover, the distance between effective pins clearly increases as function of P-content x.

E. Vortex imaging by Bitter decoration

The Bitter decoration technique 2,27 was used to image the vortex ensemble in BaFe 2 (As 1-x P x ) 2 single crystals with three different substitution levels, x = 0.33, x = 0.36, and x = 0.49. The experiments were realized under field cooled (FC) conditions, with a field µ 0 H a = 20 G (2 mT) applied parallel to the c-axis of the crystals. The decoration experiment for the crystal with x = 0.33 was not successful, presumably due to the large value of the penetration depth. 38 The vortex configurations shown in Fig. 11, for crystals x = 0.36 #2, and x = 0.49 #1, are representative of what is observed over the entire specimens. From the decoration images, we obtain, for both crystals, the average value of the magnetic induction as B int = n v Φ 0 ≈ 19 G (with n v the vortex density). This is 1 G smaller than the applied field during the experiment. Notwithstanding Ref. 39, there is therefore evidence for Meissner exclusion of the magnetic flux. Moreover, the Meissner current manifests itself as a vortex-free "Meissner belt" along the edges of decorated crystal x = 0.49 #1, as well as near surface steps that appear during preliminary cleavage of the samples, as indicated in Figure 11. Long vortex chains reminiscent of those observed in Ref. 11 are also observed in the decoration images, for both investigated P-contents, however the chains are more pronounced in crystal x = 0.36 #2.

Figures 12 (a coordination number. Even if there are some regions with several adjacent sixfold coordinated vortices, the insets to Fig. 12 (c) and(d) reveal that the latter represent less than half of the total (46% for crystal x = 0.49 #1, and 43% for crystal x = 0.36 #2). In fact, for the magnetic field of 20 G under study, the percentage of sixfold coordinated vortices is the same as in Ba(Fe 0925 Co 0.075 ) 2 As 2 (43%) 40 and in Ba 0.6 K 0.4 Fe 2 As 2 (44%). 11 Moreover, the coordination number histograms have the same width for the three materials, revealing similar disorder of the vortex ensemble. The absence of vortex lattice order is further brought out by the Fourier transforms of the vor- tex positions shown in the insert to Fig. 12 (a) and(b). Nevertheless, the spatial distribution of vortices in the two BaFe 2 (As 1-x P x ) 2 crystals presents smaller density fluctuations than previously observed in the Co-doped material. 2 Panels (c,d) of Fig. 12 show the distributions of nearest-neighbor intervortex distances. These have a mean value |r ij | = 1 µm, while the lattice parameter for a triangular perfect lattice of the same density is a = 1.075 Φ 0 /B = 1.12 µm. This shift is due to the existence of more densely packed vortices, notably in the chain-like structures. 6). Lower panels: pinning force distributions for BaFe2(As1-xPx)2 single crystals (x = 0.36 #2), (e,g), and (x = 0.49 #1), (f,h). Again, vortices located within a distance 10λ ab from the map edge are excluded from the histograms.

IV. DISCUSSION

In what follows, we adopt the procedure of Ref. 2 to determine the vortex interaction energy. For this, one needs to know the value of λ ab at the temperature T f at which the vortex ensemble is frozen. As in Ref. 2, we use the information that can be obtained from vortex lines situated near surface steps. Such steps may act as barriers, but, due to the circulation of the Meissner current, they also prevent vortex lines from being situated right at their edge. Inserting the height of the surface step in Fig. 11 (a), h = 1.3 µm, and the width of the vortex-free region close to the step, u = 1.2 µm, in Eq. (1) of Ref. 2, the value of the penetration depth at the freezing temperature is graphically estimated as λ ab (T f ) ≈ 700 nm. Using the temperature dependence of λ ab (T ) from Ref. 41, one obtains the freezing temperature of the vortex ensemble as T f ≈ 0.87T c . Thus, even though vortices are frozen at a relatively high reduced temperature, its value is lower than T f /T c = 0.95 found in Ba(Fe 1-x Co x ) 2 As 2 .

Using λ ab (T f ) and the vortex positions extracted from Fig. 11, the interaction energies of the individual vortices can be calculated as

E i int = j 2ε 0 K 0 |r ij | λ ab . (5)
Here K 0 (x) is the lowest-order modified Bessel function, and |r ij | is the distance from vortex i to vortex j. For each vortex i, only neighbors j contained within a circle of radius of 10λ ab (T ) are taken into account. This radius was chosen after verification that vortices situated at a larger distance do not significantly contribute to E i int . Figures 13(a-d) present the vortex interaction energies as color-coded maps, with the energy scale normalized by ε 0 (T f ), as extracted from the images of Fig. 11 (a) to (d) respectively. The maps show a globally homogeneous distribution; however, a number of denser regions exist. Histograms of the interaction energies for the maps (a) to (d) are presented in Fig. 13 (e) to (h). Note that the presence of the chain-like features with a denser vortex arrangement broadens the histograms for BaFe 2 (As 1-x P x ) 2 single crystal (x = 0.36 #2). Still, the energy distributions are considerably narrower than those found in Ba(Fe 1-x Co x ) 2 As 2 . 2 Furthermore, all distributions are centered about the average E int ≈ 3.5ε 0 , which corresponds to the interaction energy value (δpeak) of the perfectly traingular Abrikosov lattice for this particular vortex density. Therefore, in contrast to Ba(Fe 1-x Co x ) 2 As 2 , 2 no pinning-induced shift of the average value of the energy distribution histogram with respect to the δ-peak value is observed.

The large shift found in Ref. 2 was interpreted in terms of a large average pinning energy in the vicinity of T c , which can only be the result of T c heterogeneity. The absence of such a shift in isovalently substituted BaFe 2 (As 1-x P x ) 2 suggests that spatial inhomogeneity of T c is irrelevant for vortex pinning in this material. A probable reason for this is a smoother temperature dependence of the pinning potential. The pinning energy due to spatial variations of the vortex line energy, on a scale δz and with variance ∆ε 0 , writes U p ∼ ∆ε 0 (T )δz ∼ [∆n s (0)/n s (0)][∂ε 0 (T )/∂n s (0)]n s (T ) + [ε 0 /n s (0)][∂n s (T )/∂T c ]∆T c . At temperatures close to T c , the second contribution, due to spatial fluctuations of T c , dominates the pinning energy. 2 However, whereas in materials such as Ba(Fe 1-x Co x ) 2 As 2 -in which n s is nearly linear in 1 -T /T c as one approaches T c (see Fig. 15)this contribution is large, in BaFe 2 (As 1-x P x ) 2 with a smoother temperature dependence of n s this contribution vanishes. In fact, the T c heterogeneity within the decorated areas of the P-substituted material, such as observed by the DMO technique, does not result in qualitatively different vortex arrangements in different parts of the crystal [see Figure 1

f i = j 2ε 0 λ ab r ij |r ij | K 1 |r ij | λ ab (6)
following a similar procedure as used for the determination of E i int . Here K 1 (x) is the first order modified Bessel function. Since the rendered vortex configurations in Fig. 11 are in a stationary state at the freezing temperature T f , the calculated intervortex repulsive force must be balanced by the pinning force. The maps of Fig. 14 The maps of the local pinning force moduli and the interaction energy shown in Figs. 13 and 14 respectively are correlated. However, the respective probability distributions are clearly broader for the lower substitution level x = 0.36. As in Ref. 2, the ratio of the elementary pinning force per pin f p , extracted from the j(B) curves in section III A, and the value of |f i | obtained from Bitter decoration, allows one to evaluate an upper bound on L in an independent manner. Using the low-temperature value f p ∼ 8 × 10 -13 N yields L = 180 nm for crystal (x = 0.36 #2) and L = 230 nm for crystal (x = 0.49 #1). Figure 10 shows that these numbers are consistent with those directly extracted from the sustainable current density measurements.

Several reasons can be invoked to explain the observed enhancement of L with increasing P-content x. First is the increase of the superfluid density n s (x) as function of x. 38 Other possibilities are a decrease of the density n i of pinning centers for larger x, and the decrease of the penetration depth anisotropy ε -1 λ for higher substitution levels. We first investigate the effect of the increase of the superfluid density with x. Assuming that nm-scale fluctuations of the dopant atom density (with variance ∆x) are spatially isotropic, their effect on the pinning force f p ∼ δz ∇ε 0 (r)dz ∼ ∆ε 0 (7) and the pinning energy U p ∼ ∆ε 0 δz can be estimated by exploiting the dependence λ ab (x), 38

f p ∼ ∆ε 0 ∼ ∂ε 0 ∂λ ab ∂λ ab ∂x ∆x. (8)
The low-field critical current density (3) becomes

j c ∼ π 1/2 n i Φ 0 ε λ ∆ε 3/2 0 ε 1/2 0 = n i Φ 0 ε λ π ε 0 (x) 1/2 ∂ε 0 ∂λ ab ∂λ ab ∂x ∆x 3/2 ; (9)
the length between effective pinning centers

L ∼ ε λ π 1/2 ε 0 n i ∆ε 0 δz 1/2 = ε λ 2π 1/2 1 (n i δz) 1/2 1 ε 0 ∂ε 0 ∂λ ab ∂λ ab ∂x ∆x -1/2 . (10)
Assuming that for density fluctuations of the dopant atoms the pin density n i ∼ (δz) -3 scales as the inverse cube of the defect size, these expressions can be directly compared to the dependence of the critical current density on P-content (see Fig. 2c), as well as that of L (see Fig. 10). For δz = 100 nm and ∆x = 0.3 %, the qualitative trend with x of both quantities can be reproduced. Thus, the dependence of the penetration depth on Pcontent accounts, at least in the overdoped regime, for an enhanced probability of encountering larger critical current densities around optimal substitution. The similar dispersion of critical current density and pinning length data for different x around the model estimations ( 9) and (10) in Figs. 2c and 10 would then be due to similar disorder of the P-distribution in the different growth batches, eliminating the need to invoke a decreasing disorder with increasing x. Still, the evolution L(x) is reminiscent of that of the normal-state mean free-path, extracted by Shishido et. al. from de Haas-van Alphen oscillations of the Landau magnetization. 42 They reported that the mean free-path for the β orbits in BaFe 2 (As 1-x P x ) 2 single crystals increases from l ∼ 170 Å to 800 Å when P content varies from x = 0.41 to 1. This would imply the presence of structural defects that act as strong pinning centers that are unrelated to spatial composition fluctuations. Last but not least, the increase of ε λ (x), as inferred from the doping-dependent evolution of the Fermi surface, 42 and corresponding to a smaller Fe-pnictogen distance for large x, would lead to stiffer vortex lines and to less pinning for higher P-content. Finally, we address the clear presence of chain-like structures in the Bitter decoration images. The following hypotheses can be suggested to explain the appearance of these chains. The first is the possibility of native heterogeneity of the crystals under study, introduced during growth. The presence of line-like defects, or of linear agglomerates of point-like defects giving stronger local pinning, will lead to the appearance of vortex alignments or chainlike structures, much as this was found in e.g. Bi 2 Sr 2 CaCu 2 O 8+δ . [43][44][45][46][47] A second possibility is that, the vortex images having been obtained after field-cooling, and since the BaFe 2 (As 1-x P x ) 2 material under consideration show Meissner expulsion, a certain fraction of vortices must exit the material before the vortex ensemble is frozen at T f . The chains possibly correspond to flow channels for these exiting vortices, the flux in intermediate areas remaining pinned. 48 Finally, the multi-band character of BaFe 2 (As 1-x P x ) 2 may be responsible for the occurrence of vortex chains, with the possible existence of an attractive part in the inter-vortex potential. 49 We leave these questions open for further work.

V. SUMMARY AND CONCLUSION

We have presented an overview of vortex pinning in single crystals of the isovalently substituted iron-based superconductor BaFe 2 (As 1-x P x ) 2 , in which we have attempted to correlate the sustainable screening current density as a function of temperature, field, and doping x, with the structural properties of the vortex ensemble. The critical current density in BaFe 2 (As 1-x P x ) 2 is, overall, very well described by the strong pinning scenario of Ref. 31, which allows one to extract elementary pinning forces (of the order of 10 -13 N) and the distance between effective pins. The latter is of the order of 100 nm, and increases as a function of doping level x. These values are consistent with those independently obtained by means of the magnetic decoration technique. Contrary to Ref. 19, we find no contribution of weak collective pinning to the sustainable current density, suggesting that P-atoms are not responsible for quasi-particle scattering. [START_REF] Van Der Beek | [END_REF] The sus-tainable current data are affected by flux creep, which prohibits one from drawing definite conclusions concerning the temperature dependence of pinning.

Bitter decoration reveals slightly more ordered vortex ensembles (the number of sixfold coordinated vortices is slightly higher) than those observed in chargedoped Ba(Fe 1-x Co x ) 2 As 2 . Also, the interaction energy and pinning force distributions in BaFe 2 (As 1-x P x ) 2 are much narrower than those in Ba(Fe 1-x Co x ) 2 As 2 , and are not shifted with respect to the interaction energy of a perfectly triangular vortex lattice with the same density. These observations exclude a role of spatial variations of the critical temperature T c in determining the frozen vortex state obtained upon field-cooling. The absence of the weak-collective pinning contribution to the critical current density in BaFe 2 (As 1-x P x ) 2 means that the strong pinning contribution is what generates the disordered vortex configurations.

The main features of strong vortex pinning in BaFe 2 (As 1-x P x ) 2 , such as the energy-and force histograms, the density of effective pins, and the evolution of the critical current density with P-content, can be understood using a model for heterogeneity of the superfluid density on the scale of several dozen to several hundred nanometers, due to an inhomogeneous distribution of the dopant atoms. A small spatial variance ∆x ( of the order of 0.3 % ) suffices to explain the magnitude of j c . This model explains why pinning is of extrinsic origin, and why the disorder can be readily annealed. [START_REF] Kawano-Furukawa | Abstract book of ICSM[END_REF] Even if the x-dependence of the critical current density can be well described without any assumption of less disorder in crystals with higher P-content, the reminiscence of the evolution with P-content of the mean distance between effective pinning sites and the normal-state meanfree path 42 suggests that a second type of pinning centers may be at play. This could be areas of enhanced local strain, more prominent at optimal doping, such as these arise from the very different Fe-As and Fe-P distances. 50

FIG. 1 :

 1 FIG. 1: (Color online) Differential magneto-optical images of the screening of a magnetic field µ0Ha = 1 G (0.1 mT) by BaFe2(As1-xPx)2 single crystals (a) x = 0.27# 2, and (b) x = 0.49#1. The intensity is proportional to the local magnetic flux density B(r). Thus, black areas are regions of excluded flux, while the light areas surrounding the crystal are traversed by the applied magnetic field. Rectangular frames in (a) indicate the regions where the transmittivity data of Fig. 2 (b) are determined. The frames in (b) denote the areas where the decoration images of Fig. 11 (b,d) were obtained.

FIG. 2 :

 2 Figures1 (a,b) presents DMO images of the exclusion of an applied field µ 0 H a = 1 G (0.1 mT) as one crosses the superconducting to normal transition of BaFe 2 (As 1-x P x ) 2 single crystals x = 0.27 #2 and x = 0.49 #1. These images reveal that T c is spatially heterogeneous. While inhomogeneity is especially pronounced in the underdoped samples, see crystal x = 0.27 #2, it is also observed in the overdoped crystals. A link between the heterogeneity observed in underdoped samples and a possible phase separation between of superconductingand Spin-Density Wave (SDW) antiferromagnetic phases, and/or that of orthorhombic and tetragonal structural domains is therefore not obvious.The T c -heterogeneity is quantified by the local transmittivity T H = [I(r, T ) -I(r, T T c )] / [I(r, T T c ) -I(r, T T c )], extracted from the lumi-

Figure 3 presents

 3 Figure 3 presents MOI of the magnetic flux penetration (after zero-field cooling) into superconducting

FIG. 3 :

 3 FIG. 3: Magnetic flux density distribution in BaFe2(As1-xPx)2 single crystals x = 0.33 #1 (a), and x = 0.36 #2 (b), after zero-field cooling to the indicated temperatures and the application of different magnetic fields. The top left panel of each subfigure shows the respective crystal, the white lines are those along which the profiles in Fig. 4 are extracted. The intensity in the other panels reflects the local flux density B(r). The bottom right-hand panel of (a) shows the trapped flux distribution in crystal x = 0.33 #1, after application and removal of µ0Ha = 22.1 mT.

FIG. 4 :

 4 FIG. 4: (Color online) Magnetic flux density profiles in BaFe2(As1-xPx)2 crystal x = 0.33 #1, at T = 19 and 26 K (a,b), and crystal x = 0.36 #2, at T = 19 and 22.2 K (c,d), after zero-field cooling and application of the magnetic field in successive steps ∆Ha. The Bean-like profiles in (c,d) are obtained from the MOI images of Fig. 3(b). The profiles in (a,b) are influenced by a surface barrier and correspond to the crystal of Fig. 3(a).

2 FIG. 5 :

 25 FIG. 5: (Color online) Temperature dependence of the sustainable current density js(T, B = 30 mT), determined from flux density profiles obtained from MOI images, for a variety of crystals with substitution levels 0.23 ≤ x ≤ 0.49. Samples denoted "S" concern a limited area of the sample with the same number. Dashed lines are guides to the eye.

FIG. 6 :

 6 FIG. 6: (Color online) Hysteresis loops of the spatial gradient dB/dx of the local magnetic induction on the surface of BaFe2(As1-xPx)2 single crystals of different substitution levels 0.27 ≤ x ≤ 0.58, measured using the Hall-probe magnetometry technique, 26 at T = 6 K.

FIG. 8 :

 8 FIG. 8: (Color online) (a) Magnetic hysteresis loops measured on BaFe2(As1-xPx)2 crystal x = 0.33 #2 using a SQUID magnetometer. (b) Sustainable screening current js as function of the applied field µ0Ha(in Tesla).

FIG. 9 :

 9 FIG. 9: (Color online) Magnetic relaxation in BaFe2(As1-xPx)2 crystal x = 0.33 #2. (a) Relaxation of the magnetic flux density at three positions on the crystal surface -the center, and positions 20 and 40 µm from the center. Data were taken at 10 K, after field cooling in 0.2 T and subsequently removing the field. (b) Experimental fluxcreep activation barrier U versus js, as obtained using the method outlined in Refs. 35,36. (c) Temperature dependence of the sustainable current density in zero applied field, and applied fields of 0.14, 0.2, 1, and 2 T. Measurements using the Hall probe-array technique (open symbols) were obtained from hysteresis loops such as shown in Fig.6, while data from MOI are obtained from the flux-profile gradient. The drawn line through the js(Ha = 0)-data corresponds to the expected T -dependence of the depairing current; the drawn lines for higher fields are obtained from this by taking the field-dependence of the critical current(3,4) into account, and by correcting for flux creep using Eq. (2).

FIG. 10 :

 10 FIG.10:(Color online) Average distance L between effective pins versus P content. L is obtained from js(B)-data at 5 K (red bullets •), and from Bitter decoration at T f = 0.87Tc (blue squares ). The drawn lines show the comparison with Eq. (10), for variations of the dopant atom density ∆x = 0.3 %, on a characteristic length scale δz = 100 nm.

FIG. 11 :

 11 FIG. 11: Bitter decoration images of BaFe2(As1-xPx)2 single crystals for an applied field µ0Ha= 20 G (a,c) x = 0.36 #2, and (b,d) x = 0.49 #1. The white arrows indicate the vortexfree Meissner belt observed (a) near a surface step and (d) at the edge of the crystal x = 0.49 #1. The regions where images (b) and (d) were obtained with respect to the entire crystal are depicted in Fig. 1.

  ) and (b) present the respective Delaunay triangulations of the vortex ensembles of Fig.11 (a) and (b). Here, blue dots represent vortices with sixfold coordination, while red dots represent vortices with different

FIG. 12 :

 12 FIG. 12: (Color online) Delaunay triangulation of vortex ensembles observed in BaFe2(As1-xPx)2 single crystals (a) x = 0.36 #2 and (b) x = 0.49 #1, and presented in Figs. 11 (a,c) respectively. Blue dots represent vortices with sixfold coordination, while red dots represent differently coordinated vortices. The insets show the Fourier transform of the vortex positions. (c,d) Nearest neighbor distance distributions for the respective triangulations. The insets present the respective coordination number distributions.

FIG. 13 :

 13 FIG.13:(Color online) (a-d) Color-coded maps of the normalized individual vortex interaction energies calculated from the images of Fig.11(a-d) using Eq. (5), and represented in the same configuration. Lower panels: normalized interaction energy distributions for BaFe2(As1-xPx)2 crystals (x = 0.36 #2) (e,g), and (x = 0.49 #1) (f,h). Vortices located within a distance 10λ ab from the map edge are excluded from the histograms. The interaction energy per vortex of the triangular lattice (δ-function) is represented by the central orange line in each histogram.

FIG. 14 :

 14 FIG.14:(Color online) Normalized color-coded maps of the modulus of the pinning force (per unit length), calculated from the images of Fig.11using Eq. (6). Lower panels: pinning force distributions for BaFe2(As1-xPx)2 single crystals (x = 0.36 #2), (e,g), and (x = 0.49 #1), (f,h). Again, vortices located within a distance 10λ ab from the map edge are excluded from the histograms.

FIG. 15 :

 15 FIG. 15: Temperature dependence of the superfluid density, ns(T )/ns(0) ∝ λ 2 ab (0)/λ 2 ab (T ) for various doping levels x of Ba(Fe1-xCox)2As2 (upper curves, small data points), of Ba0.55K0.45Fe2As2 (•) and BaFe2(As0.67P0.33)2(•) (both from Ref. 41).

Figure 14

 14 Figure 14 shows maps of the modulus of the pinning force for each individual vortex, calculated from

  (a) to (d) therefore represent the minimum local pinning force for each vortex, min(|f i |). The distributions of min(|f i |) shown in Fig. 14(e) to (h) allow one to estimate the average pinning force per vortex and per unit length. We obtain |f i | ∼ 3.5 × 10 -6 Nm -1 for crystal x = 0.49 #1 and |f i | ∼ 4.5 × 10 -6 Nm -1 for crystal x = 0.36 #2. These (high temperature) values are comparable to those found in Ba(Fe 1-x Co x ) 2 As 2 . 2

Acknowledgements

This work was made possible thanks to the support of the ECOS-Sud-MINCyT France-Argentina bilateral program, Grant No. A09E03. Work done in Bariloche was partially funded by PICT 2007-00890 and PICT 2008-294.