PhD, LURPA Guillaume Merle

Ens Cachan

PhD, LURPA Jean-Marc Roussel

Jean-Jacques Lesage

Dynamic Fault Tree Analysis Based On The Structure Function

Keywords: Boolean function, fault trees, probability, reliability, symbol manipulation

This paper presents an algebraic approach allowing to perform the analysis of any Dynamic Fault Tree (DFT). This approach is based on the ability to formally express the structure function of DFTs. We first present the algebraic framework that we introduced to model dynamic gates and hence be able to determine the structure function of DFTs. Then, we show that this structure function can be rewritten under a canonical form from which the qualitative analysis of DFTs can be performed directly. We finally provide a probabilistic model of dynamic gates to be able to perform the quantitative analysis of DFTs from their structure function.

INTRODUCTION

Fault Tree Analysis (FTA) is one of the oldest, most diffused techniques in industrial applications, for the dependability analysis of critical systems [START_REF] Henley | Reliability Engineering and Risk Assessment[END_REF][START_REF] Leveson | Safeware: System Safety and Computers[END_REF][START_REF]Fault Tree Handbook With Aerospace Applications[END_REF]. When the interactions between events can be described by means of Boolean OR/AND gates only, so that only the combination of events is relevant, and not their sequence, Fault Trees are called Static Fault Trees (SFT). SFTs are commonly analyzed directly from their structure function, which is the logical expression between the top event and the basic events of the SFT. The qualitative analysis consists in determining the minimal cut sets -the minimal combinations of events which are sufficient to engender the top event -of the SFT. The quantitative analysis consists in computing the failure probability of the top event of the SFT. Dugan et al. [START_REF] Dugan | Fault Trees and Sequence Dependencies[END_REF][START_REF] Dugan | Developing a lowcost high-quality software tool for dynamic fault-tree analysis[END_REF] proposed a new model allowing to include various kinds of temporal and statistical dependencies in the SFT model, which is the Dynamic Fault Tree (DFT). The DFT is based on the definition of gates Priority-AND (PAND), Functional Dependency (FDEP), and Spare.

Even though the semantics of dynamic gates allows to model complex failure mechanisms that SFTs cannot take into account, DFTs cannot be analyzed by using regular approaches as their structure function cannot be determined. Other types of approaches are hence used to perform the analysis of DFTs. These approaches are mainly based on Zero-suppressed Binary Decision Diagrams (ZBDD) [START_REF] Tang | Minimal cut set/sequence generation for dynamic fault trees[END_REF], Continuous Time Markov Chains (CTMC) [START_REF] Coppit | Formal Semantics of Models for Computational Engineering: A Case Study on Dynamic Fault Trees[END_REF], Stochastic Petri Nets (SPN) [START_REF] Bobbio | Parametric Fault Trees with Dynamic Gates and Repair Boxes[END_REF], and dynamic Bayesian Networks (BN) [START_REF] Montani | DBNet, a tool to convert Dynamic Fault Trees into Dynamic Bayesian Networks[END_REF]. However, these approaches have limits in the analyses that they allow as well as in the distributions that can be taken into account, even though any distribution can, in most cases, be accommodated by numerical simulation.

In a previous article [START_REF] Merle | Probabilistic Algebraic Analysis of Fault Trees With Priority Dynamic Gates and Repeated Events[END_REF], the authors proposed to extend the approaches commonly used to analyze SFTs to DFTs. We hence proposed an algebraic framework allowing to determine the structure function of DFTs including dynamic gates PAND and FDEP, as well as an analytical approach allowing to perform the analyses from this structure function.

In this paper, we propose to extend the approach considered in [START_REF] Merle | Probabilistic Algebraic Analysis of Fault Trees With Priority Dynamic Gates and Repeated Events[END_REF] to the case of Spare gates. The main approaches allowing to analyze DFTs and their respective limits are presented in Section 2. The algebraic framework allowing to determine the structure function of DFTs is recalled in Section 3, and the behavioural and probabilistic models of dynamic gates are respectively presented in Sections 4 and 5. Finally, we illustrate our approach on a DFT example from the literature in Section 6.

STATE OF THE ART

Many approaches have been envisaged to analyze DFTs. In [START_REF] Tang | Minimal cut set/sequence generation for dynamic fault trees[END_REF], each dynamic gate of the considered DFT is replaced by the static gate corresponding to its logic constraints; the minimal cut sets of the resulting SFT are then generated by using ZBDDs, and these minimal cut sets are expanded to minimal cut sequences by considering the timing constraints. The authors of [START_REF] Coppit | Formal Semantics of Models for Computational Engineering: A Case Study on Dynamic Fault Trees[END_REF] propose to convert the DFT into a failure automaton which models the changing state of the system as failures occur. This failure automaton can then be converted into a CTMC, and the solution of the corresponding set of differential equations allows to determine the failure probability of the top event of the DFT. These two approaches have been implemented in Galileo [START_REF] Dugan | Developing a lowcost high-quality software tool for dynamic fault-tree analysis[END_REF]. Other model-based approaches can be used to perform the quantitative analysis of DFTs. For instance, in [START_REF] Montani | DBNet, a tool to convert Dynamic Fault Trees into Dynamic Bayesian Networks[END_REF], the whole DFT is converted into a dynamic BN and the failure probability of the top event of the DFT can be determined by using inference algorithms. In [START_REF] Bobbio | Parametric Fault Trees with Dynamic Gates and Repair Boxes[END_REF], the dynamic subtrees of DFTs are converted into a class of coloured SPNs called Stochastic Well-formed Net (SWN). This SWN can be converted into a CTMC to determine the failure probability of the top event of the dynamic subtree, and this failure probability can then be cast back into the original DFT. These prim

The exhausti e operators ca

BEHAVIO

The three tem determine the avioural mode and are illust Symbo

Table 1 -Beha

Regarding Sp Cold (CSP), W sider that CSP es. We hence ingle WSP ga nt in Sections must be ma ount the specif ally, we show amic gates al Ts directly. Behaviou The behavi an be simplifi vents:

Behavioural m

Let us consid mary event A

ܳ ൌ ܣ ή ܤ ൌ ܤ ή ሺܣ ܣ ் ൌ ሺܣ ٳ ܶሻ ܤ ் ൌ
In the case The structure behavioural m

ܧ ଶ ൌ ܷܫܯ ଵ ή ܯ ܷܫܯ ଵ ή ܯ ଵ ή ܯ ଶ ܯ ଵ ή ܯ ଷ ܯ ଶ ή ܯ ଷ ܯ ଷ ή ܯ ସ Finally,
ܧܶ ଷ ൌ ܷܤ ܧܶ ସ ൌ ܹܪ e function of s model of gate F ܷܫܯ ଶ ܷܫܯ ଵ ܯ ହ ܷܫܯ ଶ ή ଶ ή ܯ ଷ ܯ ଵ ή ܯ ଷ ή ܯ ସ ܯ ଵ ή ܯ ଷ ή ܯ ସ ܯ ଶ ή ܯ ସ ή ܯ ହ behavioural mo to determine t ൌ ܣ ή ʹܣ ή ሺͳܣ ܣ ή ͳܣ ή ሺܣ
e HECS will f ich models th e 5.

DFT of the HE cture function

be divided int he processing nt will be deno he memory sy will be denote the bus syst l be denoted by the application t will be denot he DFT of the

ܧܶ ଶ ܧܶ ଷ ܶ
, so their struc The canonical form of the structure function of the HECS can hence be determined as the conjunction of these 4 expressions.

ܷܵͳ ή ʹܷܵܤ ܹ ܱܲ ܹܵ subtree 2 can FDEP as ή ܯ ଷ ܷܫܯ ଵ ή ܯ ଵ ܷܫܯ ଶ ή ܯ ଶ ή ܯ ସ ܯ ଵ ܯ ଷ ή ܯ ହ ܯ ଵ ܯ ଷ ή ܯ ହ ܯ ଶ odel of

Qualitative analysis

The canonical form of the structure function of the HECS contains 23 (2 + 17 + 1 + 3) terms. On the one hand, 21 terms do not contain the temporal operator .ٱ They are static and can hence provide the minimal cut sets of the DFT:

ܷܫܯ ଵ ή ܷܫܯ ଶ ǡ ܷܫܯ ଵ ή ܯ ଷ ǡ ܷܫܯ ଵ ή ܯ ସ ǡ ܷܫܯ ଵ ή ܯ ହ ǡ ܷܫܯ ଶ ή ܯ ଵ ǡ ܷܫܯ ଶ ή ܯ ଶ ǡ ܷܫܯ ଶ ή ܯ ଷ ǡ ܯ ଵ ή ܯ ଶ ή ܯ ଷ ǡ ܯ ଵ ή ܯ ଶ ή ܯ ସ ǡ ܯ ଵ ή ܯ ଶ ή ܯ ହ ǡ ܯ ଵ ή ܯ ଷ ή ܯ ସ ǡ ܯ ଵ ή ܯ ଷ ή ܯ ହ ǡ ܯ ଵ ή ܯ ସ ή ܯ ହ ǡ ܯ ଶ ή ܯ ଷ ή ܯ ସ ǡ ܯ ଶ ή ܯ ଷ ή ܯ ହ ǡ ܯ ଶ ή ܯ ସ ή ܯ ହ ǡ ܯ ଷ ή ܯ ସ ή ܯ ହ ǡ ͳܷܵܤ ή ʹܷܵܤǡ ܹܪǡ ܱܲǡ ܹܵ (19
) On the other hand, 2 terms contain the temporal operator .ٱ They are dynamic and can hence provide the minimal cut sequences of the DFT.

The algebraic term ܣ ή ʹܣ ή ሺͳܣ ٱ ܣ ሻ ή ሺͳܣ ٱ ʹܣሻ indicates that A1 must fail before A a and A2 and hence corresponds to the two minimal cut sequences ሾͳܣǡ ʹܣǡ ܣ ሿ and ሾͳܣǡ ܣ ǡ ʹܣሿ. The algebraic term ܣ ή ͳܣ ή ሺʹܣ ٱ ܣ ሻ ή ሺʹܣ ٱ ͳܣሻ indicates that A2 must fail before A a and A1 and hence corresponds to the two minimal cut sequences ሾʹܣǡ ͳܣǡ ܣ ሿ and ሾʹܣǡ ܣ ǡ ͳܣሿ. The minimal cut sequences of the DFT hence are ሾͳܣǡ ʹܣǡ ܣ ሿǡ ሾͳܣǡ ܣ ǡ ʹܣሿǡ ሾʹܣǡ ͳܣǡ ܣ ሿǡ ሾʹܣǡ ܣ ǡ ͳܣሿ (20)

Quantitative analysis

The probabilistic model of dynamic gates presented in Section 5 allows to determine the failure probability of the top event of the 4 subtrees considered in Section 6.2.

For instance, in the case of subtree 1, the structure function of the subtree is given in (18). We have shown in Section 6.3 that the two algebraic terms of (18) correspond to the four minimal cut sequences given in (20). The structure function for TE 1 can hence be rewritten as ܧܶ ଵ ൌ ܣ ή ሺͳܣ ٱ ʹܣሻ ή ሺʹܣ ٱ ܣ ሻ ʹܣ ή ሺͳܣ ٱ ܣ ሻ ή ሺܣ ٱ ʹܣሻ ܣ ή ሺʹܣ ٱ ͳܣሻ ή ሺͳܣ ٱ ܣ ሻ ͳܣ ή ሺʹܣ ٱ ܣ ሻ ή ሺܣ ٱ ͳܣሻ (21) and the probability of these four disjoint algebraic terms can be determined from the probabilistic formulas recalled in Section 5 so as to determine the failure probability of TE1:

ሼܶܧ ଵ ሽሺሻ ൌ න ቆන ቆන ଶ ሺሻ ୳ ୵ ቇ ሺǡ ሻ ୲ ୵ ቇ ଵ ሺሻ ୲ න ቆන ቆන ሺǡ ሻ ୵ ୴ ቇ ଵ ሺሻ ୵ ቇ ଶ ሺሻ ୲ න ቆන ቆන ଵ ሺሻ ୳ ୵ ቇ ሺǡ ሻ ୲ ୵ ቇ ଶ ሺሻ ୲ න ቆන ቆන ሺǡ ሻ ୵ ୴ ቇ ଶ ሺሻ ୵ ቇ ଵ ሺሻ ୲ (22)
The failure probability of the top event of the DFT of the HECS can hence be determined by using the inclusionexclusion formula. It can be noted that this failure probability does not depend on the distribution considered for basic events as the probabilistic model of dynamic gates can accommodate any distribution for basic events.

In the particular case of exponential distributions with the failure rates given in Table 2, we obtain an unreliability of 95.92% after 100 hours. We have retained this mission time because it is the one retained in [START_REF]Fault Tree Handbook With Aerospace Applications[END_REF], even though the quantitative results obtained are different as basic components are considered as repairable in [START_REF]Fault Tree Handbook With Aerospace Applications[END_REF].

Basic component Failure rate (h -1 ) A1, A2, A 10 -4 M1, M2, M3, M4, M5 6 x 10 -5 MIU1, MIU2 5 x 10 -5 BUS1, BUS2 10 -6 HW 5 x 10 -5 SW 3 x 10 -2 OP 10 -3

Table 2 -Failure rates of the basic events of the DFT of the HECS, from [START_REF]Fault Tree Handbook With Aerospace Applications[END_REF] As the exponential distribution is not necessarily the most suitable to model the failure of components as it does not take into account their aging, the failure probability of the HECS could be computed by considering other more suitable distributions, such as the Weibull distribution, for instance.

CONCLUSION & PROSPECTS

In this paper, we presented the behavioural and probabilistic model of Spare gates. On the one hand, the behavioural model allows to take into account any type -Cold, Warm, or Hot -of Spare gate and to determine the structure function of any DFT under a canonical form thanks to the behavioural model of gates PAND and FDEP from [START_REF] Merle | Probabilistic Algebraic Analysis of Fault Trees With Priority Dynamic Gates and Repeated Events[END_REF]. The qualitative analysis of DFTs can then be performed directly from this canonical form. On the other hand, the probabilistic model allows to perform the quantitative analysis of any DFT from the canonical form of its structure function. It can be noted that, as this probabilistic model does not depend on the distribution considered for basic events, any distribution can be accommodated during the quantitative analysis.

Ongoing work is currently addressed to the elaboration of efficient algorithms allowing to determine the structure function of DFTs and to perform their analysis directly from this structure function. Besides, the set of minimal cut sequences obtained with this approach is not necessarily minimal as it may contain redundant minimal cut sequences.

Figure 1

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Figure 3

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We should hence define one or many minimization criterion and develop optimization algorithms allowing to reduce this set of minimal cut sequences.


Sandrine Da Col

Michael Darques

Olivier Fruchart

come

Reduction of magnetostatic interactions in self-organized arrays of nickel nanowires using atomic layer deposition

Reduction of magnetostatic interactions in self-organized arrays of nickel nanowires using atomic layer deposition Ordered arrays of magnetic nanowires are commonly synthesized by electrodeposition in nanoporous alumina templates. Due to their dense packing, strong magnetostatic interactions prevent the manipulation of wires individually. Using atomic layer deposition we reduce the diameter of the pores prior to electrodeposition. This reduces magnetostatic interactions, yielding fully remanent hysteresis loops. This is a first step towards the use of such arrays for magnetic racetrack memories.

The progress of the areal density of magnetic recording has been exponential over half a century. However it shall soon reach a halt because of thermal stability and considering as an ultimate bound that the grain size in current hard disk drives is not much larger than one order of magnitude the dimensions of atoms. Securing progress on the long run requires a rupture in concepts, such as the three-dimensional magnetic racetrack memory, where bits are encoded in the form of magnetic domain walls along vertical magnetic nanowires arranged in dense arrays 1 .

Competitive third dimension racetrack memories would require dense arrays of nanowires with diameter a few tens of nanometers and length several tens of micrometers. While these cannot be achieved by top-down techniques, self-organized nanoporous templates obtained by anodization 2 (possibly long-ranged ordered thanks to an initial step of lithography such as nanoimprint 3 ) and filled by electrodeposition meet these requirements, and have been used and optimized for many years 4,5 . A bottleneck of this approach is that although anodization processes may be varied to tune the pitch D of the array, the ratio of pore diameter d with D is essentially fixed and not much less than a fraction of unity 3 . This yields dense arrays with strong magnetostatic interactions between nanowires, preventing their use as independent entities as required in memories. At a macroscopic level, this manifests itself as a loss of remanence of the arrays, some wires reversing spontaneously under the demagnetizing influence of their neighbors.

It was confirmed experimentally that magnetostatic interactions are enhanced when the d/D ratio increases, based on chemical etching of the pores after anodization 6 , which is consistent with models of demagnetizing coefficients 7 . On the contrary, one would need to decrease the pore diameter after anodization, to decrease magnetostatic interactions to the required level. For this purpose we have used atomic layer deposition (ALD), able to yield conformal deposits even on large aspect ratio structures such as nanoporous materials 8-10 .

a) Electronic mail: sandrine.da-col@grenoble.cnrs.fr This allowed us to reduce pore diameters from 50 nm down to below 20 nm, keeping the initial array pitch D =105 nm. A dramatic decrease of magnetostatic interactions followed, which resulted in fully remanent hysteresis loops of the arrays.

Porous alumina membranes were prepared by a twostep anodization process of aluminum 11 . We used 0.5 M oxalic acid at 15˚C, potential 40 V during 17+7 h, and a 30 min pore opening step in phosphoric acid. This yields pores organized over domains of lateral size several micrometers with pore length t = 66 µm, pore diameter d = 46 nm and interpore distance D = 105 nm (figure 1.a).

The pore diameter was then reduced by depositing Al 2 O 3 by ALD. Contrary to chemical vapor deposition where all reactants necessary for growth are present simulaneously in the chamber, the ALD reaction is based on the saturating chemisorption of reactants sequentially injected in the deposition chamber along with a carrier gas (N 2 ). This cyclic process allows for a precise control of the deposited thickness, as one cycle is responsible for the formation of a stoechiometric monolayer of material. The saturated chemisorption allows one to perfectly match the substrate topography, however complex it may be.

The precursors used for alumina formation were trimethyl-aluminum and water, with a growth rate of 0.1 nm per cycle 12 . The exposure time was set to 60 s for each precursor for the homogeneous diffusion on the surfaces deep inside the pores. Several parts of the same porous template were exposed to 0, 35, 75, 100 or 150 ALD cycles. The empty membranes and wires released from the alumina were observed by scanning electron microscopy (SEM) and transmission electron microscopy (TEM), respectively (figure 1). The geometrical features of the array and nanowires after ALD are summarized in table I. All parameters are derived from SEM. The decrease of pore diameter with the number of ALD cycles is consistent with the growth rate observed on flat surfaces.

The electrodeposition was finally done for 1 h at -1 V in an acid electrolyte containing nickel ions after the backside was metallized by gold sputtering.

Single wires are expected to be essentially single- domain, and reverse their magnetization through nucleation of a domain wall at one end, followed by its quick propagation along the wire length 14 . In an array these single-domain wires interact through dipolar fields.

The intrinsic switching field distribution (SFD) of an assembly of wires if considered non-interacting is widened: starting from saturation the internal dipolar field is globally demagnetizing inside the array and adds up to the applied field, resulting in the early switching of the first wires with respect to their intrinsic coercive field. Similarly, when more than half of the wires have reversed the internal dipolar field in the array are stabilizing for the wires that have not yet reversed, delaying their reversal. Magnetization reversal processes of the arrays were studied by magnetization loops measured by vibrating sample magnetometer (VSM) at room temperature. We performed minor hysteresis loops, first to confirm the abrupt nucleation/propagation reversal mode, second to extract reliably the SFD. For each recoil point the backward slope is associated with reversible processes, while its difference with the forwards slope yields the density of switching events.

As shown in figure 2 (for d = 27 nm) the reversible contribution is nearly negligible. Wires with all other diameters exhibit the same behaviour. These results give a convincing basis to magnetization switching through nucleation and propagation. The major loops thus result most entirely from switching events, so that the SFDs can safely be extracted as the normalized derivative of these major loops. Only the major loop derivative is used hereafter.

The resulting SFDs coming from hysteresis loops measured for samples with wires of different diameters are shown in figure 3(a). The experimental values are fitted to a Gaussian curve centered around the coercive field.

The coercivity is only weakly depending on the wire diameter and follows a non-monotonic behavior, the reduction of diameter favoring higher reversal fields, counterbalanced at room temperature by increased thermal activation in low dimensions 15 . On the contrary the SFD is dramatically and monotonically reduced for smaller diameters, which we analyze quantitatively below.

Assuming wires homogeneously magnetized up to their very extremities, the stray field H d acting along a given wire may be calculated as arising from the so-called magnetic charges located at the ends of all other wires. We use the mean-field approximation, where surfaces charges are related to the average moment of the array at any step of the reversal. We consider a wire still not reversed, with magnetization up, and focus on the magnitude of H d in the vicinity of one end of the wire, where nucleation occurs (we consider the top end, without loss of generality). The maximum strength of the internal dipolar field H d felt by a given wire is given by 16 :

H d M s =- π 4 √ 3 d D 2 -η 3 8 + π 6 √ 3 d D 3 +η 3 9 16 + π 27 √ 3 d D 5 . (1)
where M s is the spontaneous magnetization and η the normalized depth along the wire length. For numerical application, we will use η = 1. As our model is a refinement of that of ref. 17 which was dealing with shorter wires, here we need to take into account the finite length of nucleation volumes, given the long length of the wires. The experimental strength of dipolar fields is estimated as the central width of the Gaussian of figure 3(a) encompassing 50 % of its area, which is 2 √ 2 erf -1 (0.5) σ, where σ 2 is the variance of the Gaussian.

The experimental and modeled strengths of dipolar fields are displayed in figure 3(b). They reveal a remarkable agreement, given the absence of adjustable parameter in the theory. This also hints at a very low intrinsic SFD of the wires if not in interaction.

As a result the strong reduction of the SFD allows us to recover fully remanent hysteresis loops for the arrays (insert of figure 3(a)), a principle that may be applied to any nanoporous template. This lifts a long-standing bottleneck of anodized nanoporous templates, where magnetostatic interactions usually dictate collective magnetization processes. This opens the way to the use of such nanowires still in their matrix for fundamental as well as applied purposes, where each nanowire may be addressed independently of the others. In racetrack memories however, the issue is more severe because domain wall propagation fields would be much smaller than nucleation field (the coercivity in the present study), so that the interactions should be much more reduced.

J. Debray is thanked for his help in samples preparation. .

In both cases the resulting field is vertical (z axis), and estimated at a reduced depth η = z/d, characterizing the center of the nucleation volume. 17 T. Wang, Y. Wang, Y. Fu, T. Hasegawa, T. Washiya, H. Saito, S. Ishio, F. S. Li, H. Oshima, K. Itoh, K. Nishio, and H. Masuda, App. Phys. Lett. 92, 192504 (2008)

S

  . Da Col, a) M. Darques, O. Fruchart, and L. Cagnon Institut Néel, CNRS et Université Joseph Fourier, BP 166, F-38042 Grenoble Cedex 9, France (Dated: 17 February 2011)

FIG. 1 .

 1 FIG. 1. Top view SEM image of empty alumina membranes (top) and TEM images of Ni nanowires (bottom) for (a,d) 0, (b,e) 100 and (c,f) 150 ALD cycles (same scale bar on all images).

FIG. 2 .

 2 FIG. 2. Major and minor hysteresis loops measured by VSM at room temperature on the sample with d =27 nm.

FIG. 3 .

 3 FIG. 3. (a) SFDs for different wire diameters determined from major hysteresis loops: Experimental values (symbols) are fitted with gaussian curves (solid lines) (b) Demagnetization field dependance with wire diameter: experiemental values (squares on dashed line) compared to the model (solid line).

2 √ 2 ;-

 22 3D 2 . 14 R. Hertel, J. Appl. Phys. 90, 5752-5758 (2001) 15 H. Zeng, R. Skomski, L. Menon, Y. Liu, S. Bandyopadhyay, and D. Sellmyer, Phys. Rev. B 65, 134426 (2002) 16 Equation 1 is a sum of 3 contributions.First, the opposite side is far from the wire end and is thus seen as carrying an average charge density -p M , giving rise to the first term in equation 1.Then, the nucleation side is treated in two parts, for the sake of accuracy, whose sum gives the second and third terms:-the ring of first neighbors is treated with point surface charges located on the emerging axis of each neighboring wire, giving rise to a termin the outer part starting at the center of second nearest neighbors, the charge density p M is modeled as homogeneous, giving rise to a term -

TABLE I

 I

	nALD	L (µm)	d (nm)	aspect ratio	p (%)
	0	10.0	46 ± 3	220 ± 10	17.3 ± 2.1
	35	13.0	39 ± 4	330 ± 30	12.7 ± 2.6
	75	20.0	32 ± 3	620 ± 70	8.5 ± 1.8
	100	19.0	27 ± 3	700 ± 90	6.0 ± 1.3
	150	28.5	18 ± 3	1900 ± 290	2.7 ± 1.0

. Pores and nanowires geometry depending on the number of ALD cycles nALD: length L, mean diameter d, aspect ratio and membrane's porosity p 13 .


Hervé Jégou

Romain Tavenard

Matthijs Douze

Laurent Amsaleg

SEARCHING IN ONE BILLION VECTORS: RE-RANK WITH SOURCE CODING

Keywords: nearest neighbor search, quantization, source coding, high dimensional indexing, large databases

Recent indexing techniques inspired by source coding have been shown successful to index billions of high-dimensional vectors in memory. In this paper, we propose an approach that re-ranks the neighbor hypotheses obtained by these compressed-domain indexing methods. In contrast to the usual post-verification scheme, which performs exact distance calculation on the short-list of hypotheses, the estimated distances are refined based on short quantization codes, to avoid reading the full vectors from disk.

We have released a new public dataset of one billion 128dimensional vectors and proposed an experimental setup to evaluate high dimensional indexing algorithms on a realistic scale. Experiments show that our method accurately and efficiently re-ranks the neighbor hypotheses using little memory compared to the full vectors representation.

INTRODUCTION

Approximate nearest neighbors (ANN) search methods [START_REF] Datar | Localitysensitive hashing scheme based on p-stable distributions[END_REF][START_REF] Muja | Fast approximate nearest neighbors with automatic algorithm configuration[END_REF][START_REF] Torralba | Small codes and large databases for recognition[END_REF][START_REF] Weiss | Spectral hashing[END_REF] are required to handle large databases, especially for computer vision [START_REF] Shakhnarovich | Nearest-Neighbor Methods in Learning and Vision: Theory and Practice[END_REF] and music retrieval [START_REF] Casey | Content-based music information retrieval: Current directions and future challenges[END_REF] applications. One of the most popular techniques is Euclidean Locality-Sensitive Hashing [START_REF] Datar | Localitysensitive hashing scheme based on p-stable distributions[END_REF]. However, most of these approaches are memory consuming, since several hash tables or trees are required. The methods of [START_REF] Jégou | Improving bag-offeatures for large scale image search[END_REF][START_REF] Weiss | Spectral hashing[END_REF], which embeds the vector into a binary space, better satisfies the memory constraint. They are, however, significantly outperformed in terms of the trade-off between memory usage and accuracy by recent methods that cast high dimensional indexing to a source coding problem [START_REF] Sandhawalia | Searching with expectations[END_REF][START_REF] Jégou | Product quantization for nearest neighbor search[END_REF][START_REF] Jégou | Aggregating local descriptors into a compact image representation[END_REF], in particular the product quantization-based method of [START_REF] Jégou | Product quantization for nearest neighbor search[END_REF] exhibits impressive results for large scale image search [START_REF] Jégou | Aggregating local descriptors into a compact image representation[END_REF].

State-of-the-art approaches usually perform a re-ranking stage to produce a ranked list of nearest neighbors. This is always done in partitioning based method such as LSH [START_REF] Datar | Localitysensitive hashing scheme based on p-stable distributions[END_REF] or FLANN [START_REF] Muja | Fast approximate nearest neighbors with automatic algorithm configuration[END_REF], as the neighbor hypotheses are not ranked on output of the index. But as shown in [START_REF] Jégou | Product quantization for nearest neighbor search[END_REF], this post verification is also important for methods based on binary [START_REF] Jégou | Improving bag-offeatures for large scale image search[END_REF][START_REF] Weiss | Spectral hashing[END_REF] or quantized codes [START_REF] Sandhawalia | Searching with expectations[END_REF][START_REF] Jégou | Product quantization for nearest neighbor search[END_REF][START_REF] Jégou | Aggregating local descriptors into a compact image representation[END_REF], as the ranking provided on output of the large scale search is significantly improved when verifying the few first hypotheses. In all these approaches, reranking is performed by computing the exact Euclidean distance between each query and the returned hypotheses. For large datasets, this raises a memory issue: the vectors can not be stored in central memory and must be read from disk on-the-fly. Moreover, the vectors are to a large extent accessed randomly, which in practice limits the number of hypotheses that can be verified.

In this paper, we propose a new post-verification scheme in the context of compression-based indexing methods. We focus on the method presented in [START_REF] Jégou | Product quantization for nearest neighbor search[END_REF], which offers state-of-the-art performance, outperforming the FLANN which was previously shown to outperform LSH [START_REF] Muja | Fast approximate nearest neighbors with automatic algorithm configuration[END_REF]. It also provides an explicit approximation of the indexed vectors. Our algorithm exploits the approximation resulting from the first ranking, and refines it using codes stored in RAM. There is an analogy between this approach and the scalable compression techniques proposed in the last decade [START_REF] Taubman | High performance scalable image compression with EBCOT[END_REF], where the term "scalable" means that a reconstruction of the compressed signal is refined in an incremental manner by successive description layers.

In order to evaluate our approach, we introduce a dataset of one billion vectors extracted from millions of images using the standard SIFT descriptor [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF]. Testing on a large scale is important, as most ANN methods are usually evaluated on sets of unrealistic size, thereby ignoring memory issues that arise in real applications, where billions of vectors have to be handled [START_REF] Jégou | Improving bag-offeatures for large scale image search[END_REF][START_REF] Mikulik | Learning a fine vocabulary[END_REF]. The groundtruth nearest-neighbors have been computed for 10000 queries using exact distance computations. To our knowledge, this set is the largest ever released to evaluate ANN algorithms against an exact linear scan on real data: the largest other experiment we are aware of is the one reported in [START_REF] Lejsek | Nv-tree: An efficient disk-based index for approximate search in very large high-dimensional collections[END_REF], where a private set of 179 million vectors was considered. [START_REF] Camerra | iSAX 2.0: Indexing and mining one billion time series[END_REF] reports an experiment on 1 billion vectors, but on synthetic data with a known model exploited by the algorithm.

Experiments performed on this dataset show that the proposed approach offers an alternative to the standard post-verification scheme. The precision of the search is significantly improved by the re-ranking step, leading to state-of-the-art performance on this scale, without accessing the disk.

CONTEXT: COMPRESSION BASED INDEXING

In this section, we briefly review the indexing method of [START_REF] Jégou | Product quantization for nearest neighbor search[END_REF], which finds the approximate k nearest neighbors using a source coding approach. For the sake of presentation, we describe only the Asymmetric Distance Computation (ADC) method proposed in [START_REF] Jégou | Product quantization for nearest neighbor search[END_REF]. We also assume that we search for the nearest neighbor (i.e., k = 1), Let x ∈ R d be a query vector and Y = {y1, . . . , yn} a set of vectors in which we want to find the nearest neighbor NN(x) of x. The ADC approach consists in encoding each vector yi by a quantized version ci = qc(yi) ∈ R d . For a quantizer qc(.) with K centroids, the vector is encoded by bc = log 2 (K) bits, assuming K is a power of 2. An approximate distance dc(x, yi) between a query x and a database vector is computed as

dc(x, yi) 2 = x -qc(yi) 2 .
(

) 1
The approximate nearest neighbor NNa(x) of x is obtained by minimizing this distance estimator:

NNa(x) = arg min i dc(x, yi) 2 = arg min i x -qc(yi) 2 , (2)
which is an approximation of the exact distance calculation

NN(x) = arg min i x -yi 2 .
(

) 3
Note that, in contrast with the binary embedding method of [START_REF] Weiss | Spectral hashing[END_REF], the query x is not converted to a code: there is no approximation error on the query side.

To get a good vector approximation, K should be large (K = 2 64 for a 64 bit code). For such large values of K, learning a Kmeans codebook is not tractable, neither is the assignment of the vectors to their nearest centroids. To address this issue, [START_REF] Jégou | Product quantization for nearest neighbor search[END_REF] uses a product quantizer, for which there is no need to explicitly enumerate the centroids. A vector y ∈ R d is first split into m subvectors y 1 , ..., y m ∈ R d/m . A product quantizer is then defined as a function qc(y) = q 1 (y 1 ), ..., q m (y m ) ,

which maps the input vector y to a tuple of indices by separately quantizing the subvectors. Each individual quantizer q j (.) has Ks reproduction values, learned by K-means. To limit the assignment complexity, O(m × Ks), Ks is set to a small value (e.g. Ks=256).

However, the set of K centroids induced by the product quantizer qc(.) is large, as K = (Ks) m . The squared distances in Equation 2are computed using the decomposition

dc(x, y) 2 = x -qc(y) 2 = j=1,...,m x j -q j (y j ) 2 , (5)
where y j is the j th subvector of y. The squared distances in the sum are read from look-up tables. These tables are constructed on-the-fly for a given query, prior to the search in the set of quantization codes, from each subvector x j and the ks centroids associated with the corresponding quantizer q j . The complexity of the table generation is O(d × Ks). When Ks ≪ n, this complexity is negligible compared to the summation cost of O(d × n) in Equation 2.

This approximate nearest neighbor method implicitly sees multidimensional indexing as a vector approximation problem: a database vector y can be decomposed as

y = qc(y) + r(y), (6)
where qc(y) is the centroid associated with y and r(y) the error vector resulting from the quantization, called the residual vector. It is proved [START_REF] Jégou | Product quantization for nearest neighbor search[END_REF] that the square error between the distance and its estimation is bounded, on average, by the quantization error. This ensures, asymptotically, perfect search results when increasing the number of bits allocated for the quantization indexes.

The ADC indexing method is fully parametrized by the number of subvectors m and the total number of bits bc. In the following, we set bc = 8 (i.e., Ks = 256), as suggested in [START_REF] Jégou | Product quantization for nearest neighbor search[END_REF], which means that we use exactly m bytes per indexed vector.

RE-RANKING NEIGHBORS USING SOURCE CODING

Refinement: principle

The objective of the method proposed in this paper is to avoid the costly post-verification scheme adopted in most state-of-the-art approximate search techniques [START_REF] Datar | Localitysensitive hashing scheme based on p-stable distributions[END_REF][START_REF] Muja | Fast approximate nearest neighbors with automatic algorithm configuration[END_REF]. The idea is to take advantage of the information on the database point provided by the indexing. This is possible when using the ADC method [START_REF] Jégou | Product quantization for nearest neighbor search[END_REF]: this search algorithm provides an explicit approximation qc(y) of database vector y.

We first assume that the first retrieval stage returns a set of k ′ hypotheses. These vectors are the one for which a post-verification is required. For each database vectors yi, the error vector is equal to r(y) = y -qc(y).

(

) 7
The proposed method consists in reducing the energy of this residual vector to limit the impact of the approximation error on the dr dc Fig. 1. Illustration of the proposed refinement process. For each database vector y, the distance dc(x, y) = d(x, qc(y)) is computed to build the short-list of potential nearest neighbors. For selected y vectors, the distance is re-estimated by dr(x, y), which is obtained by computing the distance between y and its improved approximation dr = qc(y) + qr(y -qc(y)). estimated distances. This is done by encoding the residual vector r(y) using another product quantizer qr defined by its reproduction values Cr:

qr(r(y)) = arg min c∈Cr r(y) -c 2 , (8)
where the product quantizer qr(.) is learned on an independent set of residual vectors. Similar to qc(.), the set of reproduction values Cr is never exhaustively listed, as all operations are performed using the product space decomposition.

The coded residual vector can be seen as the "least significant bits", except that the term "bits" usually refers to scalar quantization. An improved estimation ŷ of y is the sum of the approximation vector and the decoded residual vector: ŷ = qc(y) + qr(r(y)).

(

) 9
As shown in Figure 1, this estimator will be used at search time to update the distance estimation between the query x and the database vectors y that are selected as potential neighbors:

d(x, y) 2 ≈ dr(x, y) 2 = qc(y) + qr(r(y)) -x 2 . ( 10
)
The refinement product quantizer qr is parametrized by its number of subquantizers and the total number of bits. Similar to qc, we use 8 bits per subquantizer. Therefore the only parameter for the refinement quantizer is the number m ′ of bytes for each code. The total memory usage per indexed vector is m + m ′ bytes.

Algorithm

This subsection details how the refinement codes are used to re-rank the hypotheses provided by the ADC. The resulting approach will be denoted by ADC+R in the experimental section. As for most indexing algorithms, we distinguish between the offline stage and the query stage, during which the system needs to be very efficient. The offline stage consists in learning the indexing parameters and building the indexing structure associated with a vector dataset. It is performed as follows.

1. The quantizers qc(.) and qr(.) are learned on a training set.

2. The vector dataset Y = {y1, . . . , yn} to be indexed is encoded using qc, producing codes qc(yi) for i = 1, . . . , n.

3.

The residual vectors are encoded, producing the codes qr(yi -qc(yi)) associated with all the indexed vectors.

Searching a query vector x proceeds as follows:

1. The ADC distance estimation is used to generate a list L of k ′ hypotheses. The selected vectors minimize the estimator of Equation 5, which is computed directly in the compressed domain [START_REF] Jégou | Product quantization for nearest neighbor search[END_REF].

2. For each vector yi ∈ L, the approximate vector ŷi is explicitly reconstructed using the first approximation qc(yi) and the coded residual vector qr(yi), see Equation 9. The squared distance estimator d(x, ŷi) 2 is subsequently computed.

3. The vectors of L associated with the k smallest refined distances are computed.

On output, we obtain a re-ranked list of k approximate nearest neighbors. The choice of the number k ′ of vectors in the short-list depends on parameters m, m ′ , k ′ and on the distribution of the vectors. In order for the post-verification scheme to have a negligible complexity, we typically set the ratio k ′ /k to 2.

Non exhaustive variant

Up to now, we have only considered the ADC method of [START_REF] Jégou | Product quantization for nearest neighbor search[END_REF], which requires an exhaustive scan of the dataset codes. Note however that the re-ranking method proposed in this paper can be applied to any method for which an approximate reconstruction of the indexed vectors is possible, e.g., [START_REF] Sandhawalia | Searching with expectations[END_REF]. In particular, in the experimental section we evaluate our approach with the IVFADC variant of [START_REF] Jégou | Product quantization for nearest neighbor search[END_REF], that avoids the aforementioned exhaustive scan by using an inverted file structure. This requires an additional coarse quantizer.

Adapting our re-ranking method to the IVFADC method is straightforward, as this method also provides an explicit approximation of the indexed vectors. In addition to the numbers m and m ′ of bytes used to encode the vector, this variant requires two additional parameters: the number c of reproduction values of the coarse quantizer and the number v of inverted lists that are visited for a given query. The main advantage of this variant is to compute the distance estimators only for a fraction (in the order of v/c) of the database, at the risk of missing some nearest neighbors if v/c is not large enough. Note finally that the memory usage is increased by log 2 (c) bits (typically 4 bytes), due to the inverted file structure. In the following, the IVFADC method used jointly with our re-ranking method will be denoted by IVFADC+R.

EXPERIMENTS

BIGANN: a billion-sized evaluation dataset

To evaluate ANN search methods, we propose a new evaluation dataset available online: http://corpus-texmex.irisa.fr. This benchmark, called BIGANN, consists of 128-dimensional SIFT descriptors (widely adopted image descriptors [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF]) extracted from approximately 1 million images. It comprises three distinct subsets:

• base vectors: one billion vectors to search in

• query vectors: 10000 vectors that are submitted to the system

• learning vectors: a set of 100 million vectors to compute the parameters involved in the indexing method.

The groundtruth has been pre-computed: for each query, we provide the k nearest neighbors that are obtained when computing exact Euclidean distance, as well as their square distance to the query Method m ′ recall@1 @10 @100 time/query ADC 0

Evaluation protocol

The search quality is measured by the recall@r measure, i.e., the proportion of queries whose nearest neighbor is ranked in the first r positions. The curve obtained by varying r corresponds to the distribution function of the ranks, and the point r=1 corresponds1 to the "precision" measure used in [START_REF] Muja | Fast approximate nearest neighbors with automatic algorithm configuration[END_REF] to evaluate ANN methods. Also, the recall@r is the fraction of queries for which the nearest neighbor would be retrieved correctly if a short-list of k = r vectors was verified using exact Euclidean distances. The efficiency is measured by actual timings.

Evaluation of the proposed approach

Unless explicitly specified, our approach is evaluated by querying in the whole BIGANN set (i.e., one billion vectors). The performance of the ADC and IVFADC algorithms are given for reference, and compared with the re-ranked versions ADC+R and IVFADC+R. In all experiments, we have set k=10000 and k ′ =20000. In addition, for the IVFADC+R we have fixed c=8192 and v=64, which means that the query is compared to approximately 1/128 th of the indexed vectors.

The re-ranking gain: We first consider the improvement brought by the re-ranking stage compared with the reference methods. Figure 2 shows the importance of this re-ranking stage on the recall@r measure: the performance of PQ+R (resp. IVFPQ+R) is significantly better than that of ADC (resp. IVFADC). These observations are confirmed by Table 1, which additionally shows that the impact of the re-ranking stage on the query time is limited. As already reported in [START_REF] Jégou | Product quantization for nearest neighbor search[END_REF], the IVFADC version is better than the ADC method, at the cost of 4 additional bytes per indexed vector. As expected, this observation remains valid when comparing IVFADC+R with ADC+R.

Performance for a fixed memory usage: Table 2 shows that, for a given memory usage (i.e., for m + m ′ constant), the re-ranking approach ADC+R achieves similar recall as ADC, at a lower computing cost. The search is approximately two times faster, as the bytes/vector m m ′ recall@1 @10 @100 ) and ADC+R for the same total amount of memory (bytes per indexed vector). In these experiments, the ADC+R approach with m = m ′ is approximately 2× more efficient that the ADC approach, as the cost of the the re-ranking stage is almost negligible, as shown Table 1. These experiments have been performed on the whole BIGANN set.

search time is dominated by the first retrieval stage, whose complexity is asymptotically linear in m and n for large values of n. One can also observe that near perfect neighbors are obtained by increasing the number m ′ of bytes used to re-rank the queries.

Impact of the dataset size: Figure 3 shows the impact of the database size on the recall@10 measure. The re-ranking stage becomes more important as the database grows in size, due to an increasing number of outliers. Interestingly, the quality of the search degrades more gracefully when using a precise post-verification scheme (with m ′ =16 bytes).

CONCLUSION

In this paper, following recent works on multi-dimensional indexing based on source coding, we propose a method to re-rank the vectors with a limited amount of memory, thereby avoiding costly disk accesses. Refining the neighbor hypotheses strongly improves recall at a rather low cost on response time. Our experimental validation is performed on a new public dataset of one billion vectors.

ACKNOWLEDGEMENTS

This work was partly realized as part of the Quaero Programme, funded by OSEO, French State agency for innovation.

Fig. 2 .

 2 Fig. 2.Searching in one billion vectors: impact of the re-ranking stage on the search accuracy (recall@r). m=8.

Fig. 3 .

 3 Fig.3. Impact of the database size on search performance measured by recall@10. m=8.7. REFERENCES

Table 1 .

 1 Performance and efficiency measured on 1 billion vectors, m=8. The query time is measured in seconds per query. The timings validate the limited impact of the re-ranking stage on efficiency.

			0.075	0.274 0.586	5.626
		8	0.258	0.683 0.951	5.686
	ADC+R	16	0.434	0.895 0.982	5.692
		32	0.656	0.970 0.985	5.689
	IVFADC	0	0.088	0.372 0.733	0.074
		8	0.262	0.701 0.962	0.116
	IVFADC+R	16	0.429	0.894 0.982	0.119
		32	0.630	0.977 0.983	0.120
	vector. The groundtruth for smaller sets (n=1M, 2M, 5M, 10M, ...,
	200M vectors) is also provided. As our own approach does not re-
	quire many training vectors, we only used the first million vectors
	from the learning set. All measurements (accuracy and timings)
	were averaged over the 1000 first queries.

Table 2 .

 2 Comparative accuracy of ADC (m ′ =0

	8	8 4	0 4	0.075 0.056	0.274 0.586 0.223 0.504
	16	16 8	0 8	0.245 0.258	0.671 0.952 0.683 0.951
	32	32 16 16 0	0.487 0.571	0.956 0.999 0.977 1.000
	64	64 32 32 0	0.791 0.832	1.000 1.000 0.999 1.000

In practice, we are often interested in retrieving the k nearest neighbors (k > 1) and not only the nearest neighbor. We do not include these measures in the paper, as qualitative conclusions for k=1 remain valid for k > 1.


C Diaz--Guerra

J Piqueras

J A Garcia

A Remon

C Opagiste

Cathodoluminescence and photoluminescence studies of sintered BaCuO2

Keywords: Cathodoluminescence, Scanning electron microscopy, Time--resolved photoluminescence

The capability of luminescence techniques to characterize BaCuO2 and in particular to detect the presence of this compound as impurity phase in high--T, superconductors (HTSC) has been investigated. Cathodoluminescence (CL) in the scanning electron microscope and photoluminescence (PL) have shown that the luminescence emission of BaCuO2 can be separated from those of Y--, Bi--and Tl--based HTSC. The main luminescence bands and their decay times in BaCuO2 are described.

introduction

In the study of physical and structural properties of high temperature superconductors (HTSC) ceramics, the presence of impurity phases is an important factor to be considered in the analysis of experimental results. Impurity phases are often related to precursors, and information is available on their nature and content in the matrix. In other cases however, unknown phases appear during treatments received by the samples which cause partial decomposition of the superconductors.

One of the phases which can appear as a result of decomposition of different HTSC systems is BaCuO2 whose physical properties have not been often reported. In particular, BaCuO2 is known to show luminescence properties [l] which makes luminescence techniques potentially useful for the detection of this compound in HTSC ceramics.

On the other hand, several luminescence bands are related to structural features of , and the knowledge of the luminescent behaviour of impurity phases appears necessary to separate the luminescence response of matrix and impurities. In the present work photoluminescence (PL) and cathodoluminescence (CL) in the scanning electron microscope (SEM) are used to investigate the luminescence of sintered BaCuO2.

Experimental method

Samples were prepared by a solid state reaction from a mixture of BaO2 and CuO precursor powders. After pressing the stoichiometric mixture in order to obtain the BaCuO2 phase, the pellets were heated in air at 980°C during 16 h. After regrinding, the same thermal treatment was applied. X-ray diffraction (XRD) investigations showed that the sample was nearly pure. Only small amounts of Ba2Cu3O6 impurity phase could be detected.

CL investigations were performed in a Hitachi S--2500 SEM equipped with a cooling stage system (Oxford Instruments) providing a continuous temperature control from 77--300 K. Accelerating voltage was 20 kV. For panchromatic CL images acquisition the light arising from the sample was concentrated by an optical lens on a H957--08 photomultiplier (PM), directly attached to a window of the microscope. In order to record CL spectra, an optical guide was used to feed the light to the entrance slit of an Oriel 78215 computer controlled monochromator, with the PM placed at the exit slit. Monochromatic CL spatial distribution was studied in a similar way fixing the desired wavelength in the monochromator.

Steady--state PL, time--resolved spectra (TRS) and lifetime measurements were performed using a CD900 spectrometer system (Edinburgh Instruments), with a R955 PM in a Peltier cooled housing. For steady--state spectral measurements, a 450 W Xe lamp was used as an excitation source and a δ Xe flash lamp with a 2 µs pulse of 285 nm (4.35 eV) light for TRS. Lifetime measurements were performed using the multichannel scaling technique. The samples were cooled, between 10 and 270 K, in a closed cycle helium cryostat.

A liquid nitrogen cooled Ge detector was used to study the PL emission in the 800--1800 nm spectral range in which no measurable signal was observed.

Results and discussion

At 10 K, the maximum PL intensity is obtained by using the excitation energy of 4.35 eV (285 nm). In the corresponding emission spectrum a wide band with peaks at 2.70 eV (459 nm) and 2.54 eV (487 nm) is observed (Fig. 1). The 2.54 eV band fits to a gaussian curve whose peak position and FWHM (0.73 eV) do not change by increasing the temperature up to about 200 K. At higher temperature the band broadens and the intensity decreases. Fig. 2 shows a series of spectra at different temperatures between 10 and 270 K. Above this Fig. 1. PL excitation and emission spectra of BaCuO2 recorded at 10 K temperature the intensity is too low to record spectra with the experimental system used. The evolution of total PL intensity with temperature is shown in Fig. 3 and above 140 K an exponential decrease is observed. CL and emissive images of the sample show an inhomogeneous distribution of luminescence not related to specific topographic features (Fig. 5). Fig. 6 shows a series of CL spectra, recorded at temperatures between 80 and 290 K, of an extended area of the sample including bright and dark regions. These spectra are, as the PL spectra, representative of an average emission in the sample. A broad CL band peaked at about 2.5 eV, and extending to higher energies, dominates the spectra. Another wide band centered near 3.0 eV appears as a shoulder of the main emission. The reported PL bands at about 2.55 and 2.70 eV seem to be included in the 2.5 eV peaked CL emission, while the 3.0 eV CL band is not observed in PL spectra. Such variations are often observed when imaged in the CL micrographs) typically show the spectrum that can be seen in Fig. 7 recorded at several temperatures. Here the low--energy band centered now at about 2.6eV dominates the emission.

In some areas of the sample, low-and high energy CL emission bands appear more clearly resolved (Fig. 8). CL monochromatic images recorded at 3.08 and 2.38 eV (Fig. 9) show that the spatial distribution of the luminescent centers is not the same, which indicates that at least two different radiative centers are involved in the emission.

The existence of luminescence emission from HTSC in the blue--green spectral range has been previously reported. In particular, CL and PL results obtained under similar experimental conditions as in this work, show that YBa2Cu307-x (YBCO), Bi2Sr2CaCu3O8+x, (Bi--2212) and Tl2Ba2CuO6+x (Tl--2201) exhibit luminescence bands at about 2.85 and 2.3 eV [2--8]. The latter band is related to oxygen deficiency or disorder phenomena in the oxygen sublattice, while the 2.85 eV band has been found to be also of intrinsic character and is attributed to an oxygen quasimolecule [9] or a complex involving F centers [10].

The present results show the difference of luminescence spectra of BaCuO2 and the above mentioned HTSC materials. CL and PL enable to detect possible to separate the luminescence of BaCuO2 as impurity phase, from the HTSC luminescence. On the other hand, the comparison of the present CL results with the previously mentioned work on CL microscopy and spectroscopy of HTSC provides some comments on the BaCuO2 luminescence. As CL images show, the distribution of CL intensity in BaCuO2 is highly inhomogeneous. Spectral analysis has shown that in bright and dark areas of the sample two luminescence bands, whose relative weights depend on the area considered, are present. In HTSC a qualitatively similar CL intensity distribution has been observed, and a number of experiments enabled the interpretation of the CL intensity fluctuations as due to local variations in oxygen content or degree of order. Since XRD of the samples used in this work show only the presence of small amounts of Ba2Cu3O6 impurity phase, the distribution of CL intensity would be related to structural defects and/or local changes in composition. Previous works on the crystal structure and composition of BaCuO2 have shown that large deviations from the ideal composition are usual. In particular, oxygen content can vary among different samples between 1.8 and 2.5 oxygen in the formula unit [11,12]; and local variations in a sample can take place as in the case of HTSC. Oxygen content changes and ordering effects in BaCuO2 involving oxygen atoms and Cu 2+ ions, have been reported in the past years [13,14].

Conclusions

Two luminescence bands in the visible range with fast and slow decay components, centered at about 2.70 and 2.55 eV, are detected in BaCuO2 by PL spectroscopy. CL spectra show two broad bands peaked near 2.5 and 3.0 eV. The first seems to include both PL emission bands, whereas the latter is not observed in PL spectra. CL microscopy has revealed that the 2.50 and 3.05 eV bands are associated with centers that show a different spatial distribution. The luminescence spectra of BaCuO2 differ from spectra of several investigated HTSC. This fact enables us to use luminescence techniques to detect the presence of BaCuO2 as an impurity phase in Y--, Bi--and Tl--based superconductors. Luminescence in the BaCuO2 sample investigated has an inhomogeneous distribution which could be related to local compositional and structural changes.

Fig. 2 .

 2 Fig. 2. BaCu02 PL spectra evolution with temperature.

Fig. 3 .

 3 Fig. 3. Total PL intensity evolution with temperature.

Fig. 4 .

 4 Fig. 4. Time resolved PL spectra recorded at 10 K for different delay times.

Fig. 5 .

 5 Fig. 5. Secondary electron (a) and panchromatic CL (b) mode images of an extended area of the sample. luminescence spectra corresponding to different excitation techniques are compared. In particular PL under monochromatic excitation is known to be more selective, and in our sample excitation with 4.35 eV (285 nm) light favors the intense emission at about 2.70 eV. Excitation with electrons reveals, more clearly, a strong emission in the range 2.4--2.6 eV, as well as a different band at about 3.0 eV. CL spectra of selected areas show the inhomogeneity of the distribution of the different luminescent centers in the sample. In particular, strong emitting regions (as

Fig. 6 .

 6 Fig. 6. Temperature dependence of the BaCuO2 CL spectra.

Fig. 7 .

 7 Fig. 7. CL spectra recorded at several temperatures in a bright zone of the BaCuO, sample.

Fig. 8 .

 8 Fig. 8. CL spectrum of a small area of the BaCuO2 ceramic. BaCuO2 in a HTSC matrix. CL microscopy in particular, could be used to obtain spatial distribution in BaCuO2 phases. In addition to the different spectral features shown by BaCuO2 and the mentioned HTSC materials, the decay times of emission are also different. Previous time--resolved spectroscopic studies have shown two components of τ with values of 34 and 87 µs for YBCO (5), and 12 and 70 µs for Bi--2212 [4]. CL or PL mechanisms in BaCuO2 as well as the mechanisms causing the thermally stimulated luminescence [1] have not been described previously to our knowledge. The present data do not enable us to propose a luminescence mechanism, but makes it

Fig. 9 .

 9 Fig. 9. Monochromatic CL micrographs of the area showed in Fig. 5. (a) 2.38 eV (521 nm) image. (b) 3.08 eV (403 nm) image.

Acknowledgements

This work has been supported by DGICYT (Project PB93--1256) and by CICYT (Project MAT95--1184--E). C.D.G. acknowledges MEC for an FPI research grant.


Yu Ya Tomashpol'skii

N V Sadovskaya

C Díaz-Guerra

J Piqueras

Yu Ya Tomashpol'ski

L Ya

C Diaz-Guerra

Structural and composition changes in

come

__________________________________________________________________________________

Introduction

The influence of irradiation by fast electrons on the surface morphology, electronic structure, and chemical and phase composition is of utmost importance in superconductor research. [1][2][3][4][5][6] The investigation of irradiationinduced electronic processes in a superconductor broadens our concepts of its electronic structure and real defects. The intentional alteration of the nature of the properties in individual parts of the superconducting matrix can provide a basis for local modification of the surface. The isolation of zones subjected to the influence of electron bombardment has important bearing on electron-beam lithography in potential electronics applications.

In previous studies the effect of electron irradiation of high-temperature superconductors has been investigated by transmission and scanning electron microscopy, cathodoluminescence, Raman scattering, x-ray spectral microanalysis, and Auger electron spectroscopy. The irradiation of the surface of YBa 2 Cu 3 O 7-x ; by electrons with energies of 150-300 keV can stimulate the formation of oxygen vacancies and disordering of the material at O(4) and O(5) sites in the Cu-O planes. 1 Structural and composition changes induced by transformation of the phase composition and by oxygen losses in electron irradiation have been detected visually with the aid of a scanning electron microscope (SEM) (Ref. 2) and by spectral methods Based on the development of model representations of secondary electron emission processes, several secondary emission techniques have been developed in recent years with capabilities for performing phase analysis of individual grains of superconducting ceramics, for determining superconducting transitions in small local zones, and for measuring the oxygen content of yttriumcontaining crystals and bismuth-containing ceramics. [7][8][9][10][11][12][13][14][15][16] The model of Ref. 11 predicts an exponential dependence of the emission yield on the density of oxygen vacancies, which should make the secondary emission yield highly sensitive to the oxygen content, and model representations of the electronic subsystems in a superconducting matrix and the interaction of secondary electrons with them help to interpret the complex nature of experimental data on secondary emission from superconductors.

In this paper we demonstrate the efficacy of using secondary-electron measurements in conjunction with cathodoluminescence, which yields information on processes in the oxygen sublattice, and also together with x-ray spectral microanalysis and scanning electron microscopy to study the effects of local electron irradiation of superconductors, in the example of YBa 2 Cu 3 O 7-x and Tl 2 Ba 2 CuO 6+x ceramics.

Experimental

The YBa 2 Cu 3 O 7-x ceramic was prepared by standard ceramic technology from a stoichiometric oxide mixture, which was annealed first in air and then in an oxygen atmosphere. [START_REF] Mamontov | Tomashpol'skié, Sverkhprovodimost[END_REF] Minute quantities of the phases CuO x , BaCuO 2 , and Y 2 BaCuO 5 were detected together with the main phase. The superconducting transition temperature was 90 K. The procedure used to synthesize the Tl 2 Ba 2 CuO 6+x ceramic is described in detail in Refs. 18 and 19. The starting components were Tl 2 Ba 2 O 5 and CuO powders. Two different preparation techniques were used to obtain superconducting tetragonal or nonsuperconducting orthorhombic samples. Annealing of the samples at a low oxygen pressure showed that either type can be converted to a superconductor with an orthorhombic structure. Impurities of CuO x , Ba 2 Cu 3 O x (<0.1%), and Tl 2 Ba 2 O 5 (<5%) are detected, depending on the processing conditions. In our work we investigated three ceramic samples with Tc = 0, 52.4 K (orthorhombic), and 91.1 K (tetragonal).

A higher Tc corresponds to a lower oxygen content. We investigated the ceramic surfaces in the natural state and after polishing with diamond paste.

The secondary emission yield was measured by means of a high-vacuum (10 -7 Pa) modified BS-350 SEM with oil less pumping and a cold field-emission cathode. The output of a Thornley-Everhart secondary-electron detector was connected to a digital voltmeter, which recorded a voltage proportional to the secondary-electron current, and to a computer for processing of the signals. The measurement scheme is described in greater detail in Ref. 20. The error of determining the secondary emission yield did not exceed 1% and the energy and current of the electron probe were 16 keV and 10 -10 -10 -11 A, respectively. The primary-electron beam had scanning areas of 4.4 X 3 µm 2 , 23 X 16 µm 2 , 48 X 34 µm 2 , and 230 X 160 µm 2 .

The ceramic surfaces were locally irradiated with fast electrons for several seconds using a 25 kV accelerating voltage and a probe current >10 -7 A. The irradiated areas had dimensions of 100 X 80 µm 2 and 180 X 50 µm 2 .

The cathodoluminescence spectra were recorded by means of a Hitachi S-2500 electron microscope with a 25 kV accelerating voltage at temperatures from 80 K to 300 K. An optical lens was used in the measurements to focus light onto a photomultiplier mounted in the window of the sample chamber. The spectra were recorded through an optical fiber, which directed the light signal into a computerized monochromator.

The general plan of the investigations was as follows. After irradiation directly at the epicenter of the sample and along the direction from the epicenter toward the periphery, the secondary emission yield was measured, and the cathodoluminescence spectra were recorded. Elemental analysis of the composition and morphology observations were performed at the same locations by means of a JEM-35 CF SEM with a LINK attachment or a JSM-6400 SEM. The secondary emission in the intermediate zone was measured in 16 µm to 100 µm steps; a 1 µm step was used for the elemental composition measurements.

Results and discussion

After the surface had been irradiated, we distinguished its epicenter as determined by the scanning area of the electron-beam probe, an intermediate zone in which changes took place under the influence of the epicenter, and the periphery, i.e., the remaining zone unaffected by irradiation. The morphology of the surface was completely altered at the irradiation epicenter in both the yttrium-containing and the thalliumcontaining ceramics. The coarser relief and other morphological features indicate that irradiation is accompanied by melting with subsequent hardening and elements of recrystallization. The secondary emission yield measured over a large area in both ceramics is always higher than in the intermediate and peripheral zones. More localized measurements taking topographical irregularities into account indicate a higher emission yield in isolated parts of the epicenter. Here the cation elemental composition is always non uniform and in most cases differs from the almost stoichiometric composition in the periphery. In the case of the yttrium ceramic, for example, a Y 2 O 3 phase and copper depleted phases are detected in the epicenter; for the thallium ceramic significant thallium losses and barium enrichment are observed. In this light it is natural to associate the increase in the secondary emission yield with changes in the type of electronic structure in the epicenter toward semiconducting or insulator phases according to data in Refs. 8 and 10. The phase transformations associated with modification of the cation composition appear to be corroborated by the cathodoluminescence spectra. It is conceivable that the superposition of Y 2 O 3 and Y a Ba b O x phases is responsible for modifying the cathodoluminescence spectrum from the epicenter in the yttrium ceramic (in contrast with the periphery), and the appearance of the 590 nm band in the cathodoluminescence spectrum reflects the changes in the cation composition in the epicenter of the Tl 2 Ba 2 CuO 6+x ceramic. The contents of these phases increase as the irradiation time is prolonged. The changes in the intermediate zone are of considerable interest. Figure 1 shows the distribution of the secondary emission yield and the cation composition along the direction from the epicenter to the periphery in the YBa 2 Cu 3 O 7-x ceramic; the results of analogous measurements for the Tl 2 Ba 2 CuO 6+x ceramic are shown in Fig. 2. Since the metals composition remains stable in the periphery and in the intermediate zone up to the edge of the irradiated zone, and since the surface morphology is left unaltered, we identify the shape of the secondary emission distribution with non uniform oxygen losses. The boundary between the intermediate zone and the periphery, where the oxygen content is stabilized, is determined by the point on the yield curve separating its variable and constant parts. It is evident from the figures that the linear dimensions of the intermediate zone are much greater than those of the epicenter, exceeding 1000 µm. Of all the causes leading to surface oxygen losses, the most likely is the brief temperature rise due radiation heating of the epicenter with a temperature gradient toward the periphery. Rapid cool down prevents the losses of oxygen from being compensated by its diffusion from the interior depth. Effects of this nature, which lead to the thermally stimulated surface segregation of oxygen, have been observed in BaTiO 3 and SrTiO 3 (Ref. 21).

The oxygen losses in the intermediate zone are confirmed by cathodoluminescence data (Fig. 3). The 540 nm band, whose formation also involves a contribution from oxygen vacancies, 3,22 is very pronounced in the spectrum of the intermediate zone and is weaker in the spectrum of the peripheral zone. A reduction in oxygen content near the irradiated zone has also been observed by Raman scattering. The difference in the nature of the distribution of the secondary emission yield for the yttrium and thallium ceramics is attributable to specific characteristics of their electronic structures. Figure 4 For the yttrium ceramic the oxygen deficit is calculated from x-ray structural data, 9 and for the thallium ceramic the oxygen losses are estimated qualitatively from data 24 relating the transition temperature to the oxygen content; the oxygen loss data for the bismuth ceramic are taken from Ref. 23. In the tetragonal phase, as x is increased, the hole-type conductivity not only leads to the compensation of donor-level electrons, but also causes the density of holes to decrease in advance, so that the secondary emission yield rises considerably as the oxygen losses increase.

The nature of the dependence of the secondary emission yield on the oxygen losses is altogether different in Tl 2 Ba 2 CuO and Bi 2 Sr 2 CaCu 2 O 8+x . To explain the form of this function in Bi-containing superconductors, we consider a model incorporating two electronic subsystems (I and II), which contain (respectively) electrons generated by oxygen vacancies and weakly bound electrons equivalent to the number of holes. 23 This model is consistent with existing notions concerning electronic processes in high-Tc superconductors with ''excess'' oxygen, [24][25][26] where not only the existence of the electronic subsystems is postulated, but also those subsystems are localized in different parts of the crystal structure (e.g., holes in Cu-O planes, electrons at O(4) sites, etc.). It is also possible for the subsystems to interact when changes take place in the density of vacancies. 26 To account for the onset of the maximum of the curve of the secondary emission yield versus the oxygen content, we consider the basic expression for the secondaryelectron emission yield 11 :

where the factor βnλ describes the generation of a secondary electron (β is the secondaryelectron cross section of interaction with the primary electron, n is the carrier density, and λ is the escape depth of the secondary electron, the factor exp(-λ/L) refers to scattering of the secondary electron by electrons and phonons (L is the mean free path of the secondary electron), and B is the probability of the secondary electron overcoming the surface barrier. If we regard x as a parameter, it suffices to keep only the dominant exponential factor. The influence of this factor is essentially to differentiate between interaction of the secondary electron with electrons generated by oxygen vacancies (L e = 1 / α e n e ) and with weakly bound electrons equivalent to the number of holes (L h = 1 / α h n h ), where α e and α h are the cross sections of interaction of the secondary electron with electrons of subsystems I and II, respectively. Analyzing the exponential factor exp(-λ(α e n e+ α h n h )), as a function of the oxygen ''excess'' x, we see that as x is decreased (along with the oxygen content), the density of vacancy-generated electrons increases, while the density of holes (and of weakly bound electrons) decreases. The final result depends on the mechanism of interaction of subsystems I and II, the initial hole density n h , the difference between α e and α h , and the maximum possible oxygen losses. In Bi 2 Sr 2 CaCu 2 O 8+x , for example, the hole density drops appreciably for very small oxygen losses (hundredths to thousandths of a mole), 27 and the emission yield increases. With a further decrease in x the density of ''vacancion'' electrons increases, and the secondary emission yield decreases. As a result, the curve representing the dependence of the yield on the oxygen losses acquires a maximum. We are therefore looking at a qualitative analogy between Tl-and Bi-containing ceramics, 23 in which the secondary emission yield varies non monotonically with the oxygen content. Against the qualitative similarity of the mechanism underlying the interaction of the electronic subsystems we find quantitative differences associated, in particular, with the different initial hole densities.

In any case the secondary emission yield in either ceramic with x > 0 is higher than for x ≈ 0. When the oxygen content falls below the limiting value, the secondary emission yield depends on the emission properties of the new phases created by the disintegration of the matrix.

In discussing the analytical aspect, it is important to emphasize that secondary-electron emission spectroscopy is highly sensitive to small changes in the oxygen concentration (as little as 10 -5 mol; Ref. 23), and this consideration includes samples of superconductors with a surface oxygen concentration gradient.

Using a calibration curve previously obtained for YBa 2 Cu 3 O 7-x (Ref. 14), we have calculated the oxygen distribution in the intermediate zone (Fig. 1), from which it follows that semiconducting zones are formed as a result of oxygen losses in the superconducting matrix. Lacking an accurate calibration curve for the Tl 2 Ba 2 CuO 6+x ceramic, we have plotted a qualitative oxygen distribution curve on the surface (Fig. 2). These distributions nonetheless reveal a monotonic decrease in the oxygen concentration from the periphery toward the epicenter, and in fact this process creates a variation of the superconducting transition temperatures in the intermediate zone.

Conclusion

To we have used secondaryelectron emission measurements in conjunction with cathodoluminescence and x-ray spectral microanalysis to show that local electron irradiation produces zones around the epicenter with dimensions in the hundreds of micrometers, wherein oxygen losses are induced by a temperature gradient associated with brief transient heating and rapid cooldown. Quantitative measurements of the oxygen distribution have shown that regions with variable transition temperatures Tc and with semiconducting properties can form on the surface of a superconductor. The distribution of the secondary-electron emission yield in the intermediate zones is similar for Ti and Bicontaining ceramics and differs for a Ycontaining ceramic. The cause is rooted in the parameters of the electronic subsystems and their interaction as the oxygen content varies. Secondary-electron emission characterization of the zones formed by electron irradiation can provide a means for the preparation of standard samples suitable for comparison in quantitative measurements of oxygen content in oxid compounds. At the same time, there is an obvious need for spatial thermal effects to be taken into account in the electron-probe analysis of superconductors.

FIG. 1 .

 1 FIG. 1. Secondary-electron emission yield, cation composition, and oxygen content vs distance from the irradiation epicenter to the periphery for YBa 2 Cu 3 O 7-x ceramic (Tc = 90 K). Cation composition distribution curves: 1) Y; 2) Ba; 3) Cu.

FIG. 2 .

 2 FIG.2. The same as Fig.1, for Tl 2 Ba 2 CuO 6+x ceramic without final annealing (Tc = 0K). Cation composition distribution curves: 1) Cu; 2) Ba 2 ; 3) Tl.

3

 3

FIG. 3 .

 3 FIG. 3. Cathodoluminescence spectra from for Tl 2 Ba 2 CuO 6+x ceramic, recorded at 80 K from different zones: 1) intermediate zone; 2) periphery.

  shows the dependence of the secondary emission yield on the oxygen content in YBa 2 Cu 3 O 7-x , Tl 2 Ba 2 CuO 6+x and for comparison in Bi 2 Sr 2 CaCu 2 O 8+x . As mentioned in Ref. 23, in the orthorhombic phase of YBa 2 Cu 3 O 7-x the decrease in the hole density and the increase in the electron density with increasing value of x almost cancel each other, so that the secondary emission yield in the superconducting region changes insignificantly.

FIG. 4 .

 4 FIG. 4. Secondary emission yield vs oxygen losses for various ceramics. a) YBa 2 Cu 3 O 7-x ; b) Tl 2 Ba 2 CuO 6+x ; c) Bi 2 Sr 2 CaCu 2 O 8+x .For the yttrium ceramic the oxygen deficit is calculated from x-ray structural data,9 and for the thallium ceramic the oxygen losses are estimated qualitatively from data24 relating the transition temperature to the oxygen content; the oxygen loss data for the bismuth ceramic are taken from Ref.23.

Acknowledgements

The authors are grateful to S. G. Prutchenko for assistance with the elemental analysis. This work has been carried out under the auspices of RNTK AP SFSK (subprogram ''High Temperature Superconductivity,'' Project 93079) and has received support from the Russian Fund for Fundamental Research (Project 96-03-32068a), Grant HTECH.LG941034, DGIGYT Project PB 93-1256, and CICYT Project MAT 95-1184E.


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