Michal Valko
email: michal@cs.pitt.edu

Hamed Valizadegan

Branislav Kveton
email: branislav.kveton@technicolor.com

Gregory F Cooper

Conditional Anomaly Detection Using Soft Harmonic Functions: An Application to Clinical Alerting

Timely detection of concerning events is an important problem in clinical practice. In this paper, we consider the problem of conditional anomaly detection that aims to identify data instances with an unusual response, such as the omission of an important lab test. We develop a new non-parametric approach for conditional anomaly detection based on the soft harmonic solution, with which we estimate the confidence of the label to detect anomalous mislabeling. We further regularize the solution to avoid the detection of isolated examples and examples on the boundary of the distribution support. We demonstrate the efficacy of the proposed method in detecting unusual labels on a real-world electronic health record dataset and compare it to several baseline approaches.

Introduction

With the advances in health-care and with more data being handled and stored electronically, the opportunities increase for machine learning to improve the health care. Despite continuous improvement in medical practice, medical errors remain a very serious problem. According to recent data, medical errors are the 8-th leading cause of death in the US population [START_REF] Kohn | To Err Is Human: Building a Safer Health System[END_REF]. We aim to identify medical errors that correspond to unusual patient-management decisions, such as ordering a medication. Our hypothesis is that patient-management decisions that are unusual with respect to past patients may be due to errors and that it is worthwhile to raise an alert if such a condition is encountered. Typical systems for medical error detection rely on the clinical knowledge, such as expertderived rules. Extracting such knowledge is costly, time-consuming, and very difficult with medical practice constantly changing. Machine learning can offer a viable alternative: using past medical records to detect anomalies. Detecting anomalies in the patientmanagement decisions has the potential to help avoid medical errors, which could lead to improved quality of care and decreased costs.

Traditional anomaly detection methods used in data analysis are unconditional and look for outliers with respect to all data attributes [START_REF] Markou | Novelty detection: a review, part 1: statistical approaches[END_REF]. The conditional anomaly detection (CAD) problem [START_REF] Hauskrecht | Evidence-based anomaly detection[END_REF][START_REF] Song | Conditional anomaly detection[END_REF] seeks to detect unusual values for a subset of variables given the values of the remaining variables. In the special case when the target variable is a class label, the problem is often called mislabeling detection. While we focus on this special case of anomalies in the class label, the main objective is different: we are interested in constructing a system that can raise an alert when an anomalous label of a new example is observed. Formally, we want to solve the following problem:

Problem statement ( ): Given a set of n past observed examples (x i , y i ) n i=1 (with pos- sible label noise), check if any instance i in recent m examples (x i , y i ) n+m i=n+1 is unusual.
In general, we seek to reliably identify anomalies on the response (decision or class) variables for all possible values of the context (input or feature) variables.

Not knowing the underlying model, that generates the (attributes, label) pairs, may lead to two major complications. First, a given instance may be far from the past observed data points (e.g. patient cases). Because of the lack of the support for alternative responses, it is difficult to asses the anomalousness of these instances. We refer to these instances as isolated points. Second, the examples on the boundary of the class distribution support, also known as fringe points, may look anomalous due to their low likelihood.

Because the underlying conditional distribution of the data is unknown, a non-parametric approach that looks for the label consistency of the instances on their neighborhood (e.g. k-nearest neighbor or k-NN) can be very useful [START_REF] Papadimitriou | Crossoutlier detection[END_REF]. The problem with relying on models such as k-NN is that they fail to detect clusters of anomalous instances. Our approach differs from typical local neighborhood approaches in two important aspects. First, it respects the structure of the manifold and accounts for more complex interactions in the data. Second, it solves the problem of isolated and fringe points by decreasing the confidence in predicting an opposite label for such points through regularization.

Background

Label propagation on the graph is widely used for semi-supervised learning (SSL). The general idea is to assume the consistency of labels among the data which are 1) close to each other and 2) lie on the structure (manifold/cluster). The two examples are the Consistency Method of Zhou et al. [START_REF] Zhou | Learning with local and global consistency[END_REF] and the Harmonic Solution of Zhu et al. [START_REF] Zhu | Semi-supervised learning using gaussian fields and harmonic functions[END_REF], which are the instances of unconstrained regularization [START_REF] Cortes | Stability of transductive regression algorithms[END_REF]. Let G be the similarity graph with the nodes corresponding to {x i } n+m i=1 and with the weighted edges W encoding pairwise similarities between the nodes. We denote by L(W ) the (unnormalized) graph Laplacian defined as L(W ) = D -W where D is a diagonal matrix whose entries are given by d ii = j w ij . In the transductive setting, the unconstrained regularization searches for soft (continuous) label assignment such that it maximizes fit to the labeled data and penalizes for not following the manifold structure:

= min ∈R n ( -y) T C( -y) + T K , ( 1
)
where K is a symmetric regularization matrix and C is a symmetric matrix of empirical weights. C is usually diagonal and the diagonal entries often equal to some fixed constant c l for the labeled data and c u for the unlabeled. In a SSL setting, y is a vector of pseudotargets such that y i ∈ {±1} is the label of the i-th example when the example is labeled, and y i = 0 otherwise. The appealing property of ( 1) is that it can be computed by the following closed form solution:

= (C -1 K + I) -1 y (2)

Methodology

We now propose a way to compute the anomaly score from (2). The output of (1) for the example i can be rewritten (sgn refers to the sign function) as:

i = | i | × sgn( i )
(3) SSL methods use sgn( i ) in (3) as the predicted label for i. For an unlabeled example, the closer the value of i is to ±1, the more consistent labeling information was propagated to it. The key observation, which we exploit in this paper, is that we can interpret | i | as a confidence of the label. We define the anomaly score s i as the absolute difference between the actual label y i and the inferred soft label i :

s i = | i -y i |. (4)
We will now address the problems of isolated examples by setting K = L(W ) + γ g I, where we diagonally regularize the graph Laplacian. Intuitively, such a regularization lowers the confidence value | i | of all examples; however it reduces the confidence score of outlier points relatively more. In the fully labeled setting, the hard harmonic solution [START_REF] Zhu | Semi-supervised learning using gaussian fields and harmonic functions[END_REF] degenerates to the weighted k-NN. To alleviate this problem, we allow labels to spread on the graph by using soft constraints in the unconstrained regularization problem (1). In particular, instead of c l = ∞ we set c l to a finite constant and we set C = c l I. With such a setting we can solve (1) using ( 2):

= c -1 l L(W ) + 1 + γ g c l I -1
y.

(5)

To avoid computation of the inverse, we may calculate (5) by solving a system of linear equations. We then plug the output of ( 5) into (4) to get the anomaly score. We will refer to this score as SoftHAD score.

Intuitively, when the confidence is high but sgn( i ) = y i , we will consider the label y i of the case (x i , y i ) anomalous.

Backbone Graph

The computation of the system of linear equations (5) scales with cubic 1 time complexity. This is not feasible for a graph with more than several thousands of nodes. To address the problem, we use data quantization [START_REF] Gray | Quantization[END_REF] and sample a set of nodes from the training data to create G. We then substitute the nodes in the graph with a smaller set of k n + m distinct centroids which results in O(k 3 ).

Experiments

To evaluate our SoftHAD method, we compare it to the following baselines: (1) 1-class SVM approach in which we cover each class by a separate 1-class SVM [START_REF] Schölkopf | Estimating the support of a high-dimensional distribution[END_REF], (2) Quadratic discriminant analysis (QDA) model [START_REF] Hastie | The Elements of Statistical Learning[END_REF], (3) SVM classification model [START_REF] Vapnik | The nature of statistical learning theory[END_REF] with RBF kernel, and (4) Weighted k-NN approach [START_REF] Hastie | The Elements of Statistical Learning[END_REF] that uses the same weight metric W .

UCI ML Datasets

We first evaluated our method on the three UCI ML datasets [START_REF] Frank | UCI ML repository[END_REF] for which an ordinal response variable was available to calculate the true anomaly score. In particular, we selected 1) Wine Quality dataset with the response variable quality 2) Housing dataset with the response variable median value of owner-occupied homes and 3) Auto MPG dataset the response variable miles per gallon. In each of the dataset we scaled the response variable y r to the [-1, +1] interval and set the class label as y := y r ≥ 0. We randomly switched the class labels for three percent 2 of examples. The true anomaly score was computed as the absolute difference between the original response variable y r and the (possibly switched) label. Table 1 compares the agreement scores to the true score for all methods on (2/3, 1/3) train-test split. We see that SoftHAD either performed the best or was close to the best method.

Medical data

In this real-world experiment, we evaluated CAD on data extracted from electronic health records (EHR) of 4,486 patients. Our goal was to detect unusual lab test orders or medication administrations. We divided EHRs into two groups: a training set (2646 patients), and a test set (1840 patients). For each patient, we segmented the data according to the length of the patient 1 The complexity can be further improved to O(n 2.376 u ) with the Coppersmith-Winograd algorithm.

2 We also performed the experiments with 1% to 10% of switched labels with the same trends.

Wine Quality

Housing Auto MPG QDA 75.1% (1.3) 56.7% (1.5) 65.9% (2.9) SVM 75.0% (9.3) 58.5% (4.4) 37.1% (8.6) 1-class SVM 44.2% (1.9) 27.2% (0.5) 50.1% (3.5) wk-NN 67.6% (1.4) 44.4% (2.0) 61.4% (2.3) SoftHAD 74.5% (1.5) 71.3% (3.2) 72.6% (1.7)

Table 1. Mean anomaly agreement score and variance (over 100 runs) for CAD methods on the 3 UCI ML datasets.

stay where we considered all the patient data available at 8:00am each day. These patient instances were then converted into: (1) 9,282 features and (2) 749 labels/tasks -reflecting whether a particular lab was ordered or a particular medication was given within a 24-hour period. This segmentation led to 51,492 patient-state instances, such that 30,828 were used for training and 20,664 for testing. More details can be found in [START_REF] Hauskrecht | Conditional outlier detection for clinical alerting[END_REF].

Parameters for the graph-based algorithms

To construct G, we computed the similarity weights as:

w ij = exp -||x i -x j || 2 2,ψ /σ 2 ,
where ψ is a weighing of the features (we used Wilcoxon score [START_REF] Hanley | The meaning and use of the area under a receiver operating characteristic (roc) curve[END_REF]) and σ is a length scale parameter. We chose σ as 10% of the empirical variance of the Euclidean distances. For each label, we sampled an equal number of positive and negative instances to construct a k-NN graph. We set k = 75, c l = 1 and varied γ g and the graph size.

Scaling for multi-task anomaly detection In this dataset, we have 749 binary labels. We want to output an anomaly score which is comparable among the different tasks/labels so we can, for example, set a unified threshold when the system is deployed in practice. To achieve this score comparability, we propose a simple approach where we take the minimum and the maximum score obtained for the training set and scale all scores for the same task linearly so that the score after the scaling ranges from 0 to 1.

Results and Conclusion

For the dataset described above, we computed the SoftHAD anomaly scores according to (5). We asked the panel of 15 clinical experts to evaluate the 222 patient case-label pairs (selected from 749 × 20, 664 test case-label pairs), such that every case-label was evaluated by 3 experts who decided whether the alert was clinically relevant. We finally evaluated the performance of the CAD methods using the area under the ROC curve. In Figure 1, we compared SoftHAD vs. CAD using SVM with RBF kernel for different regularization settings. We sampled 200 examples to construct G (or train an SVM) and varied the γ g regularizer (or cost c for SVM). Scaling anomaly scores to the same range improved the performance of both methods and makes the methods less sensitive to the regularization settings. We outperformed SVM approach over the range of regularizers. In Figure 2, we fixed γ g = 1 and varied the number of examples we sampled from the training set to construct the similarity graph and compared it to the weighted k-NN. The error bars show the variances over 10 runs. Notice that the both of the methods are not too sensitive to the graph size. In future, we plan to extend the approach to the online anomaly detection. This can be beneficial for the deployment of our SoftHAD method in hospitals.

This research work was supported by grants R21LM009102, R01LM010019, and R01GM088224 from the NIH. Its content is solely the responsibility of the authors and does not necessarily represent the official views of the NIH.

Figure 2 .

 2 Figure 2. Medical Dataset: Varying graph size. Comparison of 1) SoftHAD and 2) weighted k-NN on the same graph.

Appearing in the ICML 2011 Workshop on Machine Learning for Global Challenges, Bellevue, WA, USA, 2011. Copyright 2011 by the author(s)/owner(s).


E Zio

R Bazzo

MULTIOBJECTIVE RELIABILITY ALLOCATION IN MULTI-STATE SYSTEMS: DECISION MAKING BY VISUALIZATION AND ANALYSIS OF PARETO FRONTS AND SETS

Keywords: Multi-State System Redundancy Allocation, Reliability, Availability, Multiobjective Decision Making, Pareto Front, Pareto Set, Level Diagrams

Reliability-based design, operation and maintenance of multi-state systems lead to multiobjective (multicriteria) optimization problems whose solutions are represented in terms of Pareto Fronts and Sets. Among these solutions, the decision maker must choose the ones which best satisfy his\her preferences on the objectives of the problem. Visualization and analysis of the Pareto Fronts and Sets can help decision makers in this task. In this view, a recently introduced graphical representation, called Level Diagrams, is here used in support of the analysis of Pareto Fronts and Sets aimed at reducing the number of non-dominated solutions to be considered by the decision maker. Each objective and design parameter is represented on separate "synchronized" diagrams which position the Pareto front points according to their proximity to ideal preference points and on the basis of this representation a two-step front reduction procedure is proposed. An application to a redundancy allocation problem of literature concerning a multi-state system is used to illustrate the analysis.

INTRODUCTION

Solutions to multiobjective problems in reliability-based multi-state system design, operation and maintenance are not unique as they have to satisfy simultaneously various conflicting objectives.

The set of solutions for consideration of the decision maker (DM) forms the Pareto Set 1 in the space of the decision variables, and the Pareto Front 2 in the space of the objectives.

In practice, the solution to a multiobjective optimization problem is found by a search algorithm as a discrete approximation of the Pareto Front, from which the DM has to select one, or more, solutions of preference. For this task to be successful, the DM should be confronted with a relatively small number of solutions representative of feasible alternatives on the Pareto Front.

In this view, visualization techniques can represent valuable tools for analyzing the multidimensional Pareto Front and Set. For two-dimensional (and at times for three-dimensional) problems, it is usually possible to make an accurate graphical analysis of the Pareto Front and Set, but this becomes soon impractical for higher dimensions. Some of the most common techniques proposed for multidimensional visualization are [START_REF] Atkosoft | Survey of Visualization Methods and Software Tools[END_REF]:

 Scatter diagrams: The visualization consists of an array of scatter diagrams arranged in the form of an s x s matrix, where s is the number of objectives. Each objective function defines one row and one column of the matrix. The complexity of the representation increases notably with the number of dimensions.

 Parallel coordinates: A multidimensional point is plotted in a two-dimensional graph, one for the objective functions and one for the decision variables. Each dimension of the original data is transformed into a coordinate in the two-dimensional plot. This is a very compact way of presenting multidimensional information, but it soon loses clarity with large sets of data and the analysis becomes difficult.

 Interactive Decision Maps [START_REF] Lotov | Interactive Decision Maps, Approximation and Visualization of Pareto Frontier[END_REF]: Two-dimensional projections of the objective function space are used to display decision maps of contour lines parametrized by a third, colour-associated objective; the process of parametrizing the two-dimensional contours can become timeconsuming and cumbersome when many objectives are involved. 1 The so-called Pareto Set is the set containing all non-dominated solutions resulting from multiobjective optimization problems. Multiobjective optimization problems generally do not have a unique solution, but rather a set of solutions, which dominate all others but among them none is better. Dominance and non-dominance is determined by pair-wise vector comparisons of the multiobjective values corresponding to the pair of solutions under comparison. 2 The Pareto Front is the set of the values of the objective functions in correspondence of the solutions of the Pareto Set.

Recently, Level Diagrams3 have been introduced for visualizing multidimensional Pareto Fronts and Sets [START_REF] Blasco | A New Graphical Visualization of n-Dimensional Pareto Front for Decision-Making in Multiobjective Optimization[END_REF]; they can be employed a priori, interactively or a posteriori of the optimization process to help the DM defining his or her preferences during the solution selection phase. The visualization is based on a metric distance from an ideal solution which optimizes all objectives simultaneously; a solution colouring procedure can be adopted to visualize the DM preferences.

In the present work, Level Diagrams are drawn for the Pareto Front and Set of a multi-state system redundancy allocation problem of literature which involves three conflicting objectives: system availability to be maximized, system cost and weight to be minimized [START_REF] Messac | Physical programming for computational control[END_REF]. The decision variables define the system configurations, each variable indicating the number of components of a particular type allocated in the configuration to provide redundancy.

The Level Diagrams analysis is then exploited for establishing a two-step procedure of reduction of the number of possible solutions represented, which renders it easier for the DM to apply his or her preferences.

The remainder of the paper is organized as follows. Section 16.2 recalls the main ideas behind the Level Diagrams representation and the related procedure [START_REF] Blasco | A New Graphical Visualization of n-Dimensional Pareto Front for Decision-Making in Multiobjective Optimization[END_REF]. In Section 16.3, the multi-state system redundancy allocation problem of interest is formulated [START_REF] Messac | Physical programming for computational control[END_REF]. Section 16.4 reports the analysis of the Pareto Front and Set using Level Diagrams, with the proposed procedure of front reduction. Finally, some conclusions are drawn in Section 16.5.

LEVEL DIAGRAMS REPRESENTATION OF PARETO FRONTS AND SETS

Pareto Front and Set

Without loss of generality, let us consider a multiobjective minimization problem

    J D  min (16.1)
where

D p   ] ,...., [ 1   
is the vector of decision variables, D is the decision space and

           s J J J ,...., 1 
is the vector of objective functions.

Generally, there is a set of solutions to (1) dominating all others but among them none is superior.

Dominance is determined by comparing the values of the multiple objectives in correspondence of pairs of solutions: solution 1  dominates solution 2  , if

            2 1 2 1 : ,...., 1 , ,...., 1     k k i i J J s k J J s i        . ( 16

Level Diagrams

From the discrete set of Pareto solutions obtained by a multiobjective optimization algorithm, the decision maker is called to choose the best solution according to his or her preferences. This task can be quite difficult for high dimensional problems and must be aided by effectively visualizing the results of the multiobjective optimization.

Level Diagrams represent an interesting visualization tool which allows classifying the Pareto solutions according to their distance from the ideal solution, i.e., the one which optimizes all objectives simultaneously [START_REF] Blasco | A New Graphical Visualization of n-Dimensional Pareto Front for Decision-Making in Multiobjective Optimization[END_REF].

Assuming, in general, that l objectives are to be minimized and m maximized, with

m l s  
, each objective is normalized with respect to its minimum or maximum values on the Pareto Front: Front [START_REF] Blasco | A New Graphical Visualization of n-Dimensional Pareto Front for Decision-Making in Multiobjective Optimization[END_REF]. The norm here considered is the 1-norm :

    l i J J J J J i i i i i ,...., 1 min max min       (16.3) and     m i J J J J J i i i i i ,...., 1 min max max       (16.4) so that now,   s i J i ,...., 1 1 0     . ( 16
       s i i J J 1 1   , with   s J   1 0  (16.7)
The Level Diagrams are drawn as follows: each objective ( i J ) and each decision variable ( j  ) is plotted separately; the X axis corresponds to the objective or the decision variable in physical units of measurement, while the Y axis corresponds to the value   1  J for all the graphs; this means that all the plots are synchronized with respect to the Y axis, i.e., all the information for a single point of the Pareto Set will be plotted at the same level of the Y axis.

THE MULTI-STATE SYSTEM REDUNDANCY ALLOCATION PROBLEM

The redundancy allocation problem (RAP) here considered regards the selection of the system redundancy configuration that maximizes availability and minimizes cost and weight. The system is decomposed in a number of subsystems, for each of which there are multiple component choices to be allocated in redundancy [START_REF] Kulturel-Konak | Efficiently Solving the Redundancy Allocation Problem Using Tabu Search[END_REF], [START_REF] Blasco | A New Graphical Visualization of n-Dimensional Pareto Front for Decision-Making in Multiobjective Optimization[END_REF].

The RAP case study here considered is taken from [START_REF] Messac | Physical programming for computational control[END_REF]. The system is made of . Each component can be in two states: functioning at nominal capacity or failed, at zero capacity. The collective performance of these binary components leads to a multi-state system behaviour. The types of components available are characterized by their availability, nominal capacity, cost and weight in arbitrary units (Table 16 The three objectives in mathematical terms are:

Availability:              u i i i A J 1 1 max max     , ( 16.8)
Cost : The problem has been solved in [START_REF] Taboada | MOMS-GA: a Multiobjective Multi-State Genetic Algorithm for System Reliability Optimization Design Problems[END_REF] using the MOMS-GA algorithm coupled with the Universal Moment Generating Function (UMGF) approach ( [START_REF] Ushakov | A Universal Generating Function[END_REF], [START_REF] Levitin | A New Approach to Solving Problems of Multi-State System Reliability Optimization[END_REF] and [START_REF] Levitin | Universal generating function in reliability analysis and optimization[END_REF]) for computing the system availability [START_REF] Messac | Physical programming for computational control[END_REF]. The resulting Pareto Front of 118 points is showed in Figure 16.1 in the objective functions space. In the Level Diagrams of all three objectives, one observes an initial decreasing trend of the norm value up to a minimum and then an increasing trend; this is reasonable, due to the conflict among the objectives. To understand this behaviour, let us consider for example Figure 16.2-a related to the availability objective: system design configurations of low availability (far from the optimal value) are characterized by small costs and weights (close to the optimal values) and as a result, the values of the 1-norm (7) are large; for values of the system availability increasing towards the optimal value, the norms decrease up to the minimum value, optimal with respect to all three objectives, after which the increase of cost and weight is such that the norms increase as the point in the Pareto Front moves away from the ideal one, optimal with respect to all three objectives. The Level Diagrams representation allows also a qualitative evaluation of the sensitivity of the results. Looking again at Figure 16.2 for the system availability objective, one can observe that the absolute value of the slope of the ascending part is larger than in Figure 16.2-b and Figure 16.2-c for the cost and the weight objectives, respectively: for example, system availabilities above 0.99 are achieved by rapidly moving away from the ideal point of minimum norm, due to significant cost and weight increases.

             u i m j ij ij i c J
An interesting feature is the presence of data clusters in the 1-norm Level Diagram of availability (Figure 16.2). This is due to the fact that whereas the cost and weight objectives grow "regularly" with increasing availability, the values that this latter objective can take are discontinuous, depending on the discrete redundant configurations which can be devised with different costs and weights but same availability values. This becomes clear if one considers Figure 16.3 in which the normalized values (3) or (4) of the three objectives ( 8), ( 9) and ( 10) are also plotted. Notice for example how at a given value of system availability (Figure 16. The observations thereby obtained suggest a Pareto Front reduction through the suppression of the configurations with higher values of 1-norm among those with similar values of one objective, availability. In Section 16.4.3, the results of this reduction will be shown to provide a less "crowded" Pareto Front, easier to be analysed by the DM for his or her preferences evaluation on the way to a final decision.

Decision Variables

In the case study considered, the decision variables define the system configurations in terms of the subsystem of redundant components. The configuration is contained in a vector of 29 discrete decision variables; each variable indicates the number of components of that particular type present in the configuration.

Figure 16.4 shows the Level Diagrams relative to the 7 decision variables characterizing the redundancy configuration of the first subsystem. Notice how some of the low capacity components (e.g., 6 and 7) are not selected in any of the Pareto Set configurations. On the contrary, the components of large capacity (the highest in the list of Table 16.1) are typically used, possibly in redundancy to ensure good values of availability. This is due to the definition of the multistate system availability, which considers not only the components availabilities but also their feeding capacities [START_REF] Levitin | A New Approach to Solving Problems of Multi-State System Reliability Optimization[END_REF]. In Figure 16.4, one also notices that to some decision variables values are associated a wide range of 1-norm values: for example, there is a large number of solutions in the Pareto Set with To systematically perform the Pareto Front reduction, a criterion must be established to define when two solutions on the Level Diagrams can be considered vertically aligned in a cluster of equal value of an objective (availability). To this purpose, let us consider a vector 1 J of length j n containing all the values of the availability objective of the Pareto Front sorted in ascending order, let min i be the position in the vector 1 J of the optimal, ideal point (optimal with respect to all the three objectives), i.e., the solution with the lowest values of the 1-norm [START_REF] Lotov | Interactive Decision Maps, Approximation and Visualization of Pareto Frontier[END_REF] and let 1 norm be the vector containing the values of the 1-norm [START_REF] Lotov | Interactive Decision Maps, Approximation and Visualization of Pareto Frontier[END_REF] for each solution on the Pareto Front. Since the differences of the availability values of two successively ranked solutions, i.e.,    

i J i J 1 1
1   are very different in the two branches of the Front (Figure 16.2 and Figure 16.3), two distinct criteria are needed to decide whether two solutions can be considered vertically aligned in a cluster on the left and right of the minimum value of the norm. Then, in the vertical cluster alignment, solutions with a higher value of the 1-norm, i.e., with a worse overall system performance, are discarded. In this work, the solution  

1  i  corresponding to   1 1  i J and to   1 1  i norm
, belongs to a vertical cluster alignment if the previous solution   i  is such that:

    3 1 1 10 1     i J i J for min 1 i i   (16.11)
i.e., on the right of the ideal point, while

    5 1 1 10 1     i J i J for j n i i   min (16.12)
i.e., on the left of the ideal point. Then, the solution  

i  is discarded if     i norm i norm 1 1 1   ;
(16.13) viceversa, the solution   shows, for example, the reduced 1-norm Level Diagram of the availability objective: notice, however, that although the shape is preserved, the clusters of solutions are not reduced to the best performing solutions, i.e., the minimum 1-norm solutions, as in the previous reduction procedure. for application in succession; a multi-state system redundancy allocation case study of literature has

1  i  is discarded if     i norm i norm 1 1 1   . ( 16
shown their capabilities of leading to a significant reduction of the solutions to be considered by the decision maker.



  of non-dominated solutions is called the Pareto Set; the vector 

  solution  giving the worst value for the th i  objective. The distance to the ideal point with different norms, giving different views on the characteristics of the Pareto

  connected in series; redundancy can be provided to each unit by selecting components from i m types available in the market, 5 ,...., 1  i

  ,  = vector of the indexes of the system configuration, ij  = component of vector  , i.e., index of the th i  component in th j  subsystem, u = number of subsystems, i m = number of components in the th i  subsystem, i n max, = userdefined maximum number of redundant components that can be placed in subsystem i ,

Figure 16. 1 :

 1 Figure 16.1: Pareto Front in the objective functions space, obtained by the MOMS-GA algorithm

Figure 16 . 2 :

 162 Figure 16.2: Level Diagram representation for availability objective, J1 (a-left), cost objective, J2 (b-center), weight objective, J3 (c-right)

  3-a) correspond very different values of weight, creating the vertical alignment that can be seen in the Level Diagrams of Figure 16.2 and Figure 16.3. Something similar occurs in Figure 16.3-b, where at cost values around 20 correspond various values of weight, with consequent vertical alignment of the 1-norm values. These vertical alignments are caused by solutions which have approximately the same value of the objective represented in the Level Diagram, but different values of the other objectives.

Figure 16 . 3 :

 163 Figure 16.3 : Level Diagram for availability objective J1 and normalized objectives (a-left), cost objective J2 and normalized objectives (b-right)

Figure 16 . 4 :

 164 Figure 16.4 : Level Diagram representation of the 7 decisional variables values of subsystem 1

3

 3 to different values of the 1-norm on the Level Diagrams, i.e., different distances from the optimal point, ideal with respect to all the three objectives (top-left graph). ubiquitous in the space of the objective functions (Figure16.1) and the characteristics of the Pareto front may not be substantially affected if it were reduced to only Pareto Front and Set Reduction The Pareto Front and Set generated by a search algorithm often result in a crowded set of dominant solutions gradually different among them; on the contrary, less crowded Pareto Front and Set would be beneficial to the DM who must analyze the different solutions and identify the preferred ones. One way of proceeding to a reduction of the solutions contained in the Pareto Front and Set is that of focusing on the values of one objective function (in this case availability), and keep only few representative solutions of the clusters of solutions with approximately equal value of that objective function (availability) and different values of the other objective functions (cost and weight). The solutions retained as representative of a cluster at a given value of availability are those with lowest costs and weights. The clusters along the Pareto Front are then reduced to a few representative points different in terms of availability and optimal in terms of low costs and weights.

  .14) The results of the reduction performed on the Pareto Front under analysis are plotted in Figure 16.5: the Pareto Front is reduced from the original 118 points to 52, with the clusters indeed reduced to individual best points, i.e., those of lowest 1-norm at basically equal availability values.

Figure 16. 5 : 1 -

 51 Figure 16.5 : 1-norm Level Diagram representation of the reduced Pareto Front for the availability objective, J1

  center graph); looking at the Level Diagrams related to the other subsystems (here not shown)consideration for driving the reduction. In the final selection of the decision variables and values for the reduction of the Pareto Front and Set solutions, particular attention must be given to preserving as much as possible the representation of the region closest to the minimum 1-norm value in the Level Diagrams because it contains the best solutions, of minimum distance from the ideal point. Applying these considerations to the original Pareto Front and Set reduces the solutions to only 36. Figure 16.6

Figure 16 . 6 : 1 -

 1661 Figure 16.6: 1-norm Level Diagram representation of the reduced Pareto Front obtained by decision variables selection for the availability objective, J1

Figure 16. 7 : 1 -16. 5 CONCLUSIONS

 715 Figure 16.7 : 1-norm Level Diagram representation of the reduced Pareto Front obtained by the two-steps reduction, for the availability objective, J1 (a-left), the cost objective, J2 (b-center) and the weight objective, J3 (c-left)

Table 16 .1 : Characteristics of the components available on the market

 16 .1).Without loss of generality, component capacities are measured in terms of percentage of the maximum system demand. The different demand levels for a given period are given in Table16.2.

	Subsystem	Component	Availability Capacity (%)	Cost	Weight
	i	Type j
		1	0.980	120	0.590	35.4
		2	0.977	100	0.535	34.9
		3	0.982	85	0.470	34.1
	1	4	0.978	85	0.420	33.9
		5	0.983	48	0.400	34.2
		6	0.920	31	0.180	34.3
		7	0.984	26	0.220	32.6
		1	0.995	100	0.205	26.5
		2	0.996	92	0.189	22.4
	2	3	0.997	53	0.091	20.3
		4	0.997	28	0.056	21.7
		5	0.998	21	0.042	25.2
		1	0.971	100	7.525	42.1
	3	2	0.973	60	4.720	41.7
		3	0.971	40	3.590	40.8

Table 16 .2 : Demand levels

 16

Level Diagrams are a recently introduced technique for visualizing Pareto Front and Sets, which will be further explained in the chapter.

Acknowledgments:

The authors are thankful to Professor David Coit of Rutgers University for providing the Pareto Front and Set data.

This work has been funded by the Foundation pour une Culture de Securité Industrielle of

Toulouse, France, A0-2009-04.


M De

La Mare

S C Das

D Das

S Dhar

A Krier

M De La Mare

T D Das

N incorporation and photoluminescence in In-rich InGaAsN grown on InAs by liquid phase epitaxy

Dilute nitride InGaAsN layers with high In content have been grown on InAs substrates by liquid phase epitaxy using GaN as a precursor for N in the growth solution. Photoluminescence (PL) was obtained in the mid-infrared spectral range at temperatures between 4-300 K. Although Ga increases the InAs bandgap, the strong band anti-crossing effect from the N incorporation resulted in an overall bandgap reduction of 11 meV compared with InAs. The temperature-dependent PL exhibited a complicated behavior and showed an anomalous increase in intensity from 190 K to room temperature. This was due to the formation of a complex defect which behaves as a non-radiative recombination centre and prevents radiative band-band recombination at temperatures < 190 K. Above this temperature the PL increases as band-band transitions become allowed. The formation of this defect requires the presence of both Ga and N and becomes de-activated after a high temperature anneal. Raman spectroscopy confirmed the presence of phonon modes associated with In-N and Ga-N bonds confirming the incorporation of N using liquid phase growth.

Introduction

The dilute III-V-nitride semiconductors are attractive for fundamental research as well as for a variety of practical applications [1][START_REF]Taylor and Francis 2004 Physics and Applications of Dilute Nitrides[END_REF][START_REF] Henini | Dilute Nitride Semiconductors[END_REF][START_REF]Dilute III-V Nitride Semiconductors and Material Systems[END_REF]. Because of the availability of cheaper substrates and mature GaAs technology, a large amount of work has been reported on GaAsN and quaternary InGaAsN layers grown on GaAs where the In content is less than 10%. Addition of In to GaAsN compensates for the strain produced by nitrogen and lattice matched layers have been grown on GaAs for In concentrations of 7% and nitrogen content of 2% for applications including high efficiency multilayer solar cells [5]. With 3 at% nitrogen, the material has been used for the fabrication of diode lasers emitting in the 1.3-1.55 µm optical communication range [6][7][8]. There are also a few reports on `high indium`InGaAsN with an In content > 50% and which is grown lattice-matched to InP substrates [9][10][11][12][13]. From theory, it has been established that nitrogen-induced band gap reduction in these materials occurs in a similar way to that in low indium content InGaAsN [14] and it exhibits some interesting phenomena, such as, weaker interaction between the nitrogen level and the conduction band as compared to that in low In content InGaAsN [10] and a change in nitrogen bonding configuration upon annealing [12]. Furthermore, for this alloy there are many applications within the technologically important mid-infrared (2-5 µm) spectral range [START_REF] Krier | The development of room temperature LEDs and lasers for the mid-infrared spectral range[END_REF][START_REF]Mid-Infrared Semiconductor Optoelectronics[END_REF] including; laser based remote sensing of pollutant and greenhouse gases, thermal imaging, biomedical analysis and infrared countermeasures for security, as well as in thermo-photovoltaic devices [START_REF] Cheetham | [END_REF]. Although midinfrared LEDs, lasers and photodetectors have been demonstrated [18][START_REF] Cook | Midinfrared electroluminescence from pentanary-quaternary heterojunction light-emitting diodes[END_REF][START_REF] Gainsb | AlGaInSb quantum well laser diodes operating above 200[END_REF][START_REF] Krier | Uncooled Photodetectors for the 3-5µm Spectral Range Based on III-V Heterojunctions[END_REF], the fabrication of efficient and cost-effective light sources and detectors is challenging because of the unfavorable imbalance in the density of states (DOS) due to the light electron effective mass in InAs [START_REF]Temperature dependence of mid-infrared electroluminescence in type II InAsSb/InAs multi-quantum well light-emitting diodes[END_REF][START_REF]InAs Basic Parameters[END_REF]. Recently, we have shown that it is possible to produce InAsN by molecular beam epitaxy (MBE) with good structural quality which exhibits photoluminescence (PL) emission in the mid-infrared [START_REF] Zhuang | 4.5 µm from InAsN[END_REF][START_REF] Zhuang | Photoluminescence in InAsN epilayers grown by molecular beam epitaxy[END_REF]. Provided that growth conditions are optimized, the electrical properties show that the mobility and residual carrier concentration are not so strongly affected by the introduction of N compared with GaAsN [START_REF] Gaasn Alloys | Strain enhancement during annealing of[END_REF][START_REF] Patanè | Electron coherence length and mobility in highly mismatched III-N-V alloys[END_REF][START_REF] Patane | Effect of low nitrogen concentrations on the electronic properties of InAs1-xNx[END_REF]. Good quality material has been demonstrated both on InAs and on GaAs substrates [START_REF] Zhuang | 4.5 µm from InAsN[END_REF][START_REF] De La Mare | Growth and characterization of InAsN/GaAs dilute nitride semiconductor alloys for the midinfrared spectral range[END_REF] where the incorporation of N and the material quantum efficiency can be improved by the addition of Sb during growth [START_REF] Krier | Molecular Beam Epitaxial Growth of InAsN:Sb for Mid-infrared Optoelectronics[END_REF]32]. More recently, dilute nitride quantum wells which exhibit mid-infrared PL emission have also been demonstrated [33].

The dilute nitrides offer excellent potential to readily access longer wavelengths and many possibilities to engineer novel device structures and to date MBE growth has been used to effectively obtain N incorporation. In this work we report on growth from solution using liquid phase epitaxy (LPE), which may offer an alternative route towards the production of high quality dilute nitrides provided that the N can be successfully incorporated. Previously, we have reported LPE growth of InAsN layers on InAs substrates where polycrystalline InN powder was added to the growth melt to supply N during growth [34]. In the present work we report on our attempts to improve on this by using GaN as a source of N instead of InN. One advantage of using GaN is that it is now available with 99.999% purity whereas the maximum purity available for commercial polycrystalline InN is 99.98%. We observed that it is possible to use GaN to successfully introduce N into the InAs crystal using LPE. We report on the growth of Inrich InGaAsN layers on InAs substrates with indium content > 97% which exhibit PL in the mid-infrared spectral range and which show some interesting temperature dependence. Raman spectroscopy and thermal annealing was used to probe the local arrangement of atoms around the incorporated N in the crystal.

Experimental Procedures

The InGaAsN epitaxial layers were grown using the conventional LPE technique in a horizontal sliding boat LPE reactor. High purity In (6Ns), obtained from JMC Ltd, UK, was first baked at 800 o C for 20 hrs under Pd-diffused hydrogen flow. The In solution was then saturated with As by placing it over an undoped single crystal InAs wafer at 600 o C for 1 hr followed by further baking at 700 o C for 15 hr. Next, precisely weighted quantities (upto 2 mol% in the melt) of 5N purity polycrystalline GaN grains (Kyma Technologies, USA) were added to the melt and baked at 700 o C for 1 hr. Finally, an n + (100) InAs substrate was mounted in the boat and growth was carried out for 5-10 min under a supercooling of 4-5 o C and a constant cooling rate of 0.3 o C/min. The resulting epitaxial layers exhibited smooth and shiny surfaces with a thickness in the range 3-5µm. The surface texture was found to degrade for GaN content above 4 mol% in the melt.

The Ga content in the layers was measured by energy dispersive X-ray analysis (EDX) using a Sirion 200 electron microscope. The change in lattice constant of the layer, as a result of both Ga and N incorporation, was observed using high resolution double crystal x-ray diffraction (XRD) measurements. Photoluminescence (PL) measurements were done in the range 4-300K, where the test sample was held in a continuous flow He cryostat. An Ar + ion laser was used for excitation and the luminescence from the sample was analyzed through a 0.3m Bentham M300 monochromator and detected with a liquid nitrogen cooled InSb detector and a lock-in amplifier. The N-related local vibrational modes (LVM) were characterized using a Renishaw InVia micro-Raman spectrometer with a 576 x 400 pixel CCD detector cooled to -70 °C. Excitation was provided by a diode laser (100 mW at 785 nm, which is 1.58 eV, higher than the band-gap of all the InGaAsN samples). A spot size of ~1.6 µm 2 was obtained using the 50X objective lens of a Leica DM2500M compound microscope. The spectral resolution was ~1 cm -1 using a 1200 line/mm diffraction grating and a 50 µm entrance slit. A Rayleigh edge holographic notch filter was used to block out unwanted reflections. An exposure length of 10 seconds was used to minimize potential heating effects at room temperature, with scans from 100 to 1000 cm -1 taken at several locations on each sample. Each scan had 10 accumulations to reduce the effects of noise.

Results and discussion

From EDX measurements, we found that the material contained 2.3 at% Ga for growth from a melt containing 4 mol% GaN, which appears to be the upper limit for successful epitaxy of smooth layers.

High resolution x ray diffraction spectra for three samples grown with successively increasing amounts of up to 2 mol% GaN in the melt are shown in Fig. 1. In all three cases, we observed two clearly distinguishable peaks, one due to the InAs substrate and the other due to the epitaxial layer. Furthermore, with increasing GaN content in the melt, the separation between the peaks increased indicating a decrease of the lattice constant of the layer. Unfortunately, both N and Ga effect a reduction in the lattice constant and hence it cannot be said specifically which species is primarily responsible for the observed decrease in the lattice constant since N is not readily detected using EDX.

Figure 2 shows the 4 K PL spectrum of InGaAsN grown with 0.7 mole% GaN added to the melt together with that from an un-doped InAs homoepitaxial layer also grown by LPE for comparison. The InGaAsN spectrum consists of three peaks located at 2.97, 3.22 and 3.51 µm which have been labeled A, B and C respectively. The un-doped InAs exhibits two characteristic peaks located at 3.01 and 3.12 µm, which correspond to band-band and donor-acceptor transitions respectively, while the smaller peak located at 3.37 µm is attributed to defects or impurities within the band gap [35]. From the comparison, peak A can be attributed to band-band transitions in the InAs substrate, whereas the main peak B may be associated with emission from localized impurity or native lattice defect states close to the InGaAsN band edge.

Peak C originates from recombination involving a defect level lying deeper within the InGaAsN band gap.

Figure 3 shows the temperature dependence of the PL from the InGaAsN epitaxial layer. As the temperature increases peak A becomes more prominent while peak B quenches rapidly, consistent with thermal redistribution of carriers from localized states into the conduction band. However, we note that the InGaAsN band-band photoluminescence is not observed until higher temperature (> 160 K) because of strong non-radiative recombination as will be discussed below. Similarly, peak C becomes fully quenched at temperatures above 60 K. Both the peaks B and C exhibit almost no shift in peak position with increasing temperature which is typical of emission from a defect level.

Between 80 K and 190 K, peak A quenches. As the temperature increases above 190 K a new feature appears, beginning at approximately 3.25 µm and which evolves into a new emission band (labeled as D) which then dominates the spectrum. This band can be identified with band-band transitions in the InGaAsN since its behavior follows the Varshni empirical formula quite closely at elevated temperatures, as shown in Fig. 4a. The corresponding band gap of the InGaAsN alloy at each temperature was determined by extrapolation from a linear fit to the square of the near-band gap portion of the photoluminescence intensity and is shown in Fig. 4b. This is consistent with electron-hole recombination across a direct gap semiconductor with a parabolic density of states (DOS) where, . The extrapolation of the Varshni fit in Fig. 4a results in a value for the InGaAsN band gap as 0.405 eV at 0 K, which is smaller than the generally accepted value for InAs (0.416 eV) and confirms the incorporation of N which reduces the alloy band gap due to the band anti-crossing effect [36]. This is sufficient to counteract any band gap increase arising from the incorporation of Ga. EDX analysis on this sample was unable to detect N directly, but revealed that the Ga content in the InGaAsN was 2%. Consequently, using this value together with the band gap obtained from PL we may estimate the N content to be 0.3 % since the band gap reduction is entirely due to the incorporation of N.

A rather unusual feature of the emission band D is that it grows steadily in intensity with increasing temperature (> 160 K) and continues to increase up to room temperature where the experiment was stopped. This behavior is shown in figure 5 and was observed in all the InGaAsN samples characterized.

In general, PL intensity decreases with increasing temperature because of an enhancement of competing non-radiative recombination which is of the form:

………………….. (1)
In our samples we observe a clear deviation from this relationship as is evident from Fig. 5. We attribute the anomalous increase in band D to the existence of a non-radiative deep level defect within the InGaAsN band gap which acts as an efficient centre for non-radiative recombination at temperatures below 160 K. Regarding the nature of such a defect, it must be noted that the increased PL at higher temperatures is only observed in materials containing both Ga and N in the epitaxial layer. Figure 6 shows the high temperature PL of an InGaAs epilayer grown with 4 at% Ga in the melt under similar conditions. This material does not demonstrate any increase in emission with increasing temperature and follows closely the relationship described in equation (1). Growth of InAsN by LPE (or MBE) [34,[START_REF] Zhuang | Photoluminescence in InAsN epilayers grown by molecular beam epitaxy[END_REF] also results in no increase in PL emission. Consequently, the introduction of both Ga and N into the epilayer are likely to result in the formation of a complex deep level defect which changes its ionization state with increasing temperature, switching from an efficient non-radiative recombination centre at low temperature to an inactive neutral centre at higher temperatures which then enables band-band recombination to occur more efficiently. A similar increase in band-band PL with temperature has been observed in GaAsN, although this was interpreted on the basis of a spin blockade involving Ga self-interstitials and relies on preventing the capture of spin polarized free electrons by non-radiative recombination centers with the same spin orientation as the localized electrons [37]. The formation of such features have also been found to be thermodynamically favorable in the presence of N, while no such behavior was found in N-free GaAs epilayers. In our case, since anti-site defects (such as As vacancies and In As ) are known to form in InAs, the addition of Ga and N may easily result in the formation of complex defects with multiple charge states [38].

To further elucidate the nature of the defects responsible for the anomalous PL behavior, the sample was subjected to a thermal annealing treatment. The InGaAsN epitaxial layer was placed face down on a clean undoped InAs wafer and annealed at 750 °C for 15 mins in an ultra-pure hydrogen atmosphere. The resulting PL emission is shown in Fig. 7. Note that the overall intensity of the PL increased by approximately one order of magnitude (Fig. 5) and the peak at 2.97 µm is greatly increased while that of the peak near 3.22 µm has been drastically reduced The changes in the PL emission spectrum are most likely due to a re-arrangement of the local site symmetry around the incorporated N and Ga atoms, or the out diffusion of N as a direct result of the post growth anneal [39]. It should also be noted that the higher temperature increase in PL intensity is no longer observed after the annealing, this can also be seen clearly in Fig. 5 (solid points). From the Arrhenius plot of integrated PL intensity in Fig. 8 we found that the increase of PL intensity from 190 to 300 K in the as-grown material is associated with an activation energy of 26 meV. The activation energies for thermal quenching for the as-grown sample was 9 meV, which changed to 15 meV after annealing.

Fig. 9 shows the Raman spectra from two InGaAsN epitaxial layers grown from a melt containing 0.7 and 2 mol% GaN along with the spectrum from undoped InAs for comparison. Strong InAs transverse optical (TO) phonon and longitudinal optical (LO) phonon modes can be readily identified at 217 and 238 cm -1 , respectively [40] in all the three samples. A weak band between 250 and 300 cm -1 is found in InGaAsN only and can be attributed to GaAs TO (268 cm -1 ) and LO (293 cm -1 ) phonon modes [41,42]. A broad feature extending between 420 and 500 cm -1 is again found only in the InGaAsN samples. These peaks arise from the presence of N complexes and N-related local vibrational modes (LVM) in the material [41,42]. In the InGaAsN quaternary system, the Ga-N bond length (1.95 ) is less than the In-N bond length (2.14 ), resulting in a larger bond strength for Ga-N bonds, making their formation more favourable compared to that of In-N bonds. On the other hand, for the same reason, a Ga-N bond in an InAs lattice, with In-As bond length of 2.61 , is more strained compared to an In-N bond in the same lattice.

According to Klar et al [43], in as-grown InGaAsN, a nitrogen atom surrounded by four nearest neighbor Ga atoms (Ga 4 N), is favoured in the surface region, whereas in the bulk, the In 4 N configuration is preferred. They further suggested that after a high temperature anneal, the entire material is converted to the latter configuration. From the spectra in figure 8 two prominent LVMs can be identified at ~450 and 470 cm -1 associated with configurations such as In 4 N [41,44] and Ga 4 N [45], both of these are likely to exist in as-grown InGaAsN. However, it should be noted that it is difficult to unambiguously identify these N-related peaks due to the presence of 2 nd order InAs transitions present within the same region [46]. Furthermore, it has been shown experimentally [12] as well as theoretically [47] that even in high Incontent InGaAsN containing a small amount of Ga, there is a preferential formation of Ga-N bonds. This explains the almost equal Raman signals due to InN-like and GaN-like LVMs obtained in our samples containing about 3 at% Ga.

The Raman difference spectra between the 0.7 mol% and 0 mol% samples is shown in figure 10 and depicts the region of the broad feature centered around 450 cm -1 . From this spectrum 6 peaks are evident located at 380, 413, 439, 461, 487 and 496 cm -1 respectively. These peaks are related to the presence of N since the InAs related peaks have been subtracted, (the InAs peaks would normally appear in the minima of this spectrum). The InN-like mode at 439 cm -1 can be identified as the 14 N As local vibrational mode consistent with previously published data [12,46]. The two peaks located at 461 and 496 cm -1 we attribute to N-In 4 and N-Ga 2 In 2 clusters respectively [48]. The peaks at 413 and 487 cm -1 are related to higher order complexes and have been determined to be due to high order di-nitrogen In-N complexes [34] and doubly degenerate N-Ga 3 In cluster modes [48,49]. Finally, the peak located at 380 cm -1 is tentatively attributed to a gap mode caused by In vacancies [50].

Figures 5 and8 show that after a high temperature anneal, the PL dependence on temperature reverts to the more usual thermally activated quenching of equation ( 1). This is consistent with the Raman spectrum of Fig. 11, which shows that the N-related modes observed in the InGaAsN layer grown with 2 mol% GaN in the melt become broadened and shift to higher frequencies near 500 cm -1 . This is in agreement with previous findings where such behavior was related to a change in the atomic arrangement around the nitrogen atoms [51,52]. Although further work is required to fully identify the nature of the clusters and defects present in our InGaAsN it is apparent that using GaN as a source of N results in the formation of complex defects or clusters involving N which are responsible for the temperature dependent behavior of the PL.

Conclusion

In summary, InGaAsN layers with high In content have been grown on InAs substrates using liquid phase epitaxy by adding GaN to the InAs growth melt. Using this technique a maximum incorporation of 2.3 at% of Ga was obtained in the as-grown epitaxial layers. PL was obtained in the mid-infrared spectral range at temperatures between 4-300 K. Interpretation of the temperature dependent PL enabled us to extract the 0 K bandgap as 0.405 eV. Together with the x-ray diffraction and electron microprobe analysis results we were able to obtain the alloy composition as InGaAsN. Although Ga increases the InAs bandgap, the strong band anti-crossing effect from the N incorporation resulted in an overall bandgap reduction of 11 meV. The temperature-dependent PL exhibited a complicated behavior with increasing temperature and showed an anomalous increase in intensity from 190 K to room temperature. From our results, it seems likely that this is due to the formation of a complex defect which behaves as a nonradiative recombination centre. This prevents radiative band-band recombination at temperatures < 190 K. Above this temperature the PL increases presumably as the charge state of the defect changes and band-band transitions become allowed. The defect also becomes de-activated after a high temperature anneal. It further appears that the formation of this defect requires the presence of both Ga and N, as we have not found any such behavior in the PL of ternary InGaAs or InAsN epilayers. Raman spectroscopy confirmed the presence of phonon modes associated with In-N and Ga-N bonds. However, further work is required to elucidate the actual composition, site symmetry and charge states of the N related structure defects present in this material.

Fig 1 .Fig 2 .Fig 3 .Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .

 1236789 Fig 1. HRXRD rocking curves for three InGaAsN layers grown with increasing amounts of GaN added to the growth melt

Figure 10 :

 10 Figure 10: Raman difference spectra of 0.7 mol% InGaAsN . The position of N related configurations is indicated by the arrows above the spectrum while the InAs 2 nd order locations are indicated by the arrows below the spectrum.

Fig. 11 .-

 11 Fig.11. Raman spectrum for an InGaAsN layer grown from a melt containing 2 mol% GaN, compared with that from the same sample after a high temperature anneal.

Acknowledgements

The work was supported by the Department of Electronics, Government of India and the Engineering and Physical Sciences Research Council, UK. We would also like to thank Dr Frank Martin and Mr. Imran Patel of the Center for Biophotonics, Lancaster Environment Center (LEC) for Raman spectroscopy measurements.