Jean-Emile Dartois
email: jean-emile.dartois@b-com.com

Heverson B Ribeiro
email: heverson.ribeiro@b-com.com

Jalil Boukhobza
email: boukhobza@univ-brest.fr

Olivier Barais
email: barais@irisa.fr

Cuckoo: Opportunistic MapReduce on Ephemeral and Heterogeneous Cloud Resources

Keywords: cloud, scheduling, unused resources, ephemeral resources, spare resources, data placement, big data, mapreduce, hadoop, quantile regression

Cloud infrastructures are generally overprovisioned for handling load peaks and node failures. However, the drawback of this approach is that a large portion of data center resources remains unused. In this paper, we propose a framework that leverages unused resources of data centers, which are ephemeral by nature, to run MapReduce jobs. Our approach allows: i) to run efficiently Hadoop jobs on top of heterogeneous Cloud resources, thanks to our data placement strategy, ii) to predict accurately the volatility of ephemeral resources, thanks to the quantile regression method, and iii) for avoiding the interference between MapReduce jobs and co-resident workloads, thanks to our reactive QoS controller. We have extended Hadoop implementation with our framework and evaluated it with three different data center workloads. The experimental results show that our approach divides Hadoop job execution time by up to 7 when compared to the standard Hadoop implementation.

I. INTRODUCTION

Advances in technologies such as smart-phones and the Internet of things led us to a data deluge. According to recent estimations [?] by 2025 the amount of data generated will be about 160 zettabytes. MapReduce [START_REF] Dean | Mapreduce: simplified data processing on large clusters[END_REF] is a programming model proposed by Google for processing such large amounts of data while providing high performance and fault tolerance. Hadoop [START_REF] Shvachko | The hadoop distributed file system[END_REF] is an Open-source implementation of MapReduce that runs across clusters of a large number of computing nodes. Although processing massive data requires a significant amount of computing resources, maintaining such a large-enough dedicated infrastructures to process multiple types of jobs is undoubtedly expensive.

Cloud computing provides on demand access to scalable, elastic and reliable computing resources. Although these features make Cloud infrastructures good candidates for processing Hadoop workloads, a clear drawback is their operation cost. Furthermore, Cloud computing data centers are often over-provisioned in order to cope with workload variations [START_REF] Carvalho | Longterm SLOs for reclaimed cloud computing resources[END_REF] and nodes failure. This over-provisioning increases the Total Cost of Ownership (TCO) for Cloud providers and results in a low average resource utilization. In a previous study [START_REF] Dartois | Using quantile regression for reclaiming unused cloud resources while achieving sla[END_REF], authors shown that the average CPU usage lies between 20% to 50% on several data centers. Some studies proposed to reclaim these unused resources and offer them at a cheaper price [START_REF] Marshall | Improving utilization of infrastructure clouds[END_REF] to increase resource utilization. This led to a benefit increase of 60% for Cloud providers [START_REF] Carvalho | Longterm SLOs for reclaimed cloud computing resources[END_REF].

Therefore, a promising alternative for optimizing the cost of processing data-intensive applications on Cloud infrastructures is to opportunistically exploit their allocated but unused computing resources. In order to achieve that some challenges must be tackled.

• Cloud heterogeneity: Hadoop is designed to run on homogeneous and dedicated clusters of nodes, whereas Cloud infrastructures are built upon heterogeneous resources to avoid vendors lock-in effect and due to frequent hardware updates. Running Hadoop on environments with different computing capacities may significantly degrade its performance [START_REF] Zaharia | Improving mapreduce performance in heterogeneous environments[END_REF]. Hadoop distributes data chunks uniformly across nodes and expects them to run tasks with the same execution time. Thus, running Hadoop in heterogeneous systems requires data chunks to be reallocated to feed available computing slots. This creates network traffic overhead, and therefore degrades the system performance. Hence, Designing a resourceaware placement strategy is a necessary feature that can reduce subsequent chunk reallocation in order to avoid degrading the overall performance of Hadoop jobs. • Resources volatility: In Cloud systems, users are able to unilaterally reserve, consume and release computing resources on-the-fly. As a consequence, exploiting unused and ephemeral resources to run Hadoop jobs raises some issues. First, Hadoop relies on replication to provide reliable storage and fault tolerance. Relying on resources that may be preempted at any time increases the cost of maintaining replicas. Second, Hadoop does not replicate intermediate results. If resources holding these results are preempted, the resulting data vanishes and its associated reduce task may stall and require the re-execution of the initial map task. Clearly, the way with which users claim their allocated resources plays a fundamental role not only on the execution time of a Hadoop job, but also on its termination. Hence, it is necessary to have a precise estimation of the unused resources to exploit them efficiently in order to minimize the total cost of ownership and the cost of processing Hadoop applications on the Cloud. • Users SLA guarantee: When running Hadoop jobs on allocated but unused resources, one should hedge against violating SLA QoS guarantees for the users having reserved those resources. This may be caused either by bad interference between co-located users and deployed Hadoop jobs or by sudden changes in users behavior. Several studies have underlined that co-located jobs may interfere and result in unwanted performance glitches. This may be due to some hardware, system mechanisms (e.g. SSD, CPU, memory) or virtualization interference. On the other hand, in Cloud systems the nature of workloads is highly heterogeneous and their intensity may significantly vary (i.e., abrupt grow or shrink) according to users behavior [START_REF] Calzarossa | Workloads in the clouds[END_REF]. These unexpected changes makes prediction errors unavoidable. Thus, it is necessary to have the ability to quickly allocate, react and adapt the unused resource provisioning in a way to avoid degrading the QoS for the users that have reserved those resources.

In this paper we try to tackle the aforementioned challenges in order to make it possible to exploit unused resources on Cloud infrastructures in an opportunistic way to process Hadoop data-intensive workloads.

Our approach relies on three mechanisms. i) a Data placement planner to cope with cloud heterogeneity, ii) a Forecasting builder to predict resource volatility, and iii) a QoS controller to ensure users SLA guarantee by avoiding interference. The data placement planner relies on the Forecasting builder and decides about the distribution of Hadoop chunks to process according to resource availability. The Forecasting builder relies on quantile regression and machine learning algorithms to accurately predict the amount of unused resources and their availability (volatility) to feed the data placement planner. The QoS controller is used to avoid Hadoop interference on users workloads of the Cloud provider. It achieves that by increasing and decreasing the allocated resources to Hadoop containers on-the-fly.

We evaluated our approach using traces of three months of resources usage from three different data centers (i.e., two private companies, and one university). We compared native Hadoop job execution time with our solution. Our simulation results show that Cuckoo improves Hadoop native job execution time between 500% and 700% on ephemeral resources.

The remainder of this paper is organized as follows.

Section II presents some background information. Then, we describe our methodology in Section III. Section IV details the experimental evaluation we have performed. Section V presents some limitations of our approach. Section VI compares our approach with some related works. Section VII concludes the paper.

II. BACKGROUND

In this section, we introduce the key concepts of MapReduce paradigm through its Hadoop implementation.

A. MapReduce programming model

MapReduce is a programming model, inspired by Lisp programming language, and proposed for processing large data sets, potentially using hundreds or thousands of distributed machines [START_REF] Dean | Mapreduce: simplified data processing on large clusters[END_REF]. The MapReduce model hides complex tasks, such as partitioning large data sets, scheduling and executing programs across distributed computers, dealing with failures, and handling inter-machine communication from users. This is done with a simple abstraction based on two phases, namely map and reduce. For each phase, the user writes a specific function (i.e., one map and one reduce function). The map function takes an input data set and outputs a set of intermediate <key,value> pairs. After that, the intermediate pairs are grouped by the same key. Then, each set of values corresponding to a single key is forwarded to the reduce function. Finally, each reduce function merges the values trying to form a smaller set of values.

B. Hadoop framework architecture

Hadoop has a master/slave architecture organized into two main layers. The first layer consists of a single master node called Jobtracker and a set of slave nodes called Tasktrackers. The second layer consists of a master node called NameNode and slave nodes called DataNodes. Tasktrackers are in charge of executing map and reduce functions, while DataNodes store chunks of input data. Users interact with a Hadoop cluster by means of a ClientNode, which is used to send the input data, map and reduce functions to the cluster.

Specifically, a Hadoop user sends its map and reduce functions to ClientNode, which in turn, sends them the Jobtracker. Concurrently, ClientNode fetches the block allocation information (i.e., chunk-to-node mapping) from the NameNode. Then, the ClientNode splits the input file into even-sized data chunks and streams them to DataNodes, which are randomly replicated across the cluster for faulttolerance. Hadoop runs map and reduce tasks simultaneously. Each task occupies one single processing slot, which is released when the task is completed. When a Tasktracker has an empty slot, it sends a hearbeat message to the Jobtracker requesting a new task. The Jobtracker scheduler keeps assigning new tasks to available Tasktrackers until all the tasks are done. Hadoop scheduling algorithm favors data locality and does not consider other factors such as system load and fairness.

III. CUCKOO: A MECHANISM FOR EXPLOITING EPHEMERAL AND HETEROGENEOUS CLOUD RESOURCES

Our main goal is to provide a framework that leverages unused Cloud resources to run Hadoop jobs efficiently without interfering with the co-located workloads.

Our proposed architecture consists of the following actors.

• Farmers: data center owners. Organizations that seek to reduce their TCO by making unused computing resources available to other users. • Customers: two types: i) regular customers: organizations that buy regular Cloud resources from the operators, ii) ephemeral customers: organizations that want to process data-intensive applications on the Cloud at a lower cost. In this paper, we are more concerned with the latter. • Operators: organizations that are the interface between farmers and customers. Their objective is to minimize farmers TCO by offering unused resources to Customers while insuring SLA 1 .

A. The Cuckoo Framework Architecture Overview

The Cuckoo framework relies on three modules, each of them addressing a specific challenge previously described in Section I (i.e., resource volatility, heterogeneity, SLA guarantees).

• Forecasting builder: This module predicts (as accurate as possible) the future resource utilization at the host level, and therefore the amount of unused resources and their availability. The Forecasting builder considers both CPU and memory as resources and its main goal is to estimate the volatility of • Data placement planner: This module uses the predictions of the Forecasting builder and applies a placement strategy in order to distribute data chunks across the cluster hosts. The Data placement planner solves the heterogeneity of available resources by tuning data chunks allocation according to CPU availability and volatility. • QoS controller: This module guarantees that running Hadoop jobs of ephemeral customers do not interfere with regular workloads of regular customers in order to ensure the SLA. The QoS controller continuously monitors the resource utilization to detect if regular customers could be impacted by ephemeral customers.

If it is the case some corrective actions are triggered.

It also has a preventive mechanism that consists in preserving a certain amount of unused resources to absorb workload variation. To this amount of preserved resources we refer to as safety margin. Figure 1 presents both actors and modules and shows how they interact among each other. The Customer starts by submitting (1) a Hadoop job using the ClientNode. Then, the JobTracker sends (2) a request to the Data placement planner to check if the Operator is able to provide enough resources to process the job within a time window of 24 hours. In order to verify that, Data placement planner retrieves 1 One may note that the farmer and the operator role can be played by the same organization (3) the latest resource predictions, which are continuously updated by the Forecasting builder module, and creates the block allocation information, which maps chunks to nodes for that specific job. After that, the JobTracker replies (4) to the ClientNode with either an acceptance or a rejection message depending on the amount of available resources. Following, the ClientNode fetches (5) the block allocation map and sends (6) the chunks to the DataNodes. Finally, the QoS Controller monitors (7) the real-time utilization of unused resources in order to adapt the amount of resources allocated to containers that run TaskTracker nodes, in the case of interference.

B. Forecasting Builder

The objective of the Forecasting Builder module is to estimate the future amount of used resources for each host. By doing so, it can to estimate the available resources for running Hadoop jobs.

State-of-the-art studies have discussed [START_REF] Amiri | Survey on prediction models of applications for resources provisioning in cloud[END_REF] how to select the appropriate learning algorithm(s) to forecast time series with learning algorithms such as Autoregressive (AR), Integrated Moving Average (ARIMA) and other more complex algorithms such as Recurrent Neural Network (RNN), Support Vector Machine (SVM), Long short-term memory (LSTM), Gradient Boosted Decision Trees (GBDT), and Random Forest (RF). In a previous work [START_REF] Dartois | Using quantile regression for reclaiming unused cloud resources while achieving sla[END_REF] we have shown that quantile regression is a relevant approach to reclaim unused resources with SLA requirements. This work has shown that quantile regression may increase the amount of savings by up to 20% compared to traditional approaches.

Quantile regression provides the accuracy of machine learning algorithms with the flexibility of quantiles. Moreover, quantiles make it possible to reason about the tradeoff between the amount of reclaimed unused resources and the potential SLA violations. In our work, we chose to use Gradient Boosting Decision Tree which gives the best tradeoff between prediction accuracy and training time in order to forecast a 24-hour time window. We use quantile regression to implement our forecasting builder module.

C. Data Placement Planner

The objective of the Data Placement Planner is to find the best data block mapping in order to minimize the overall execution time given the available resources by minimizing data transfers. To achieve that, we use a modified version of the Weighted-Round-Robin (WRR) algorithm. WRR is designed to handle hosts with different processing capabilities by assigning a different weight to each [START_REF] Katevenis | Weighted round-robin cell multiplexing in a general-purpose atm switch chip[END_REF].

In our approach, the weight was calculated by taking into account the predicted usage of resources and the processing capability for each host, estimated in GFLOPS 2 within a time interval of 24 hours. Then, data chunks are distributed proportionally to the weight assigned to each host. That is, hosts with higher weight receive more data chunks to process than hosts with lower values.

The safety margin value (described in Section III-D) denoted by sm is also taken into account. This value is used to remove the corresponding proportion of resources from the pool of unused resources that is the input of the Data Placement Planner. As mentioned before, the sm value is used to absorb unpredictable workload behavior and forecasting errors. For instance, if the forecast builder estimates that 12 cores are used in a 32 cores machine, then 20 cores would be available during the next 24-hour time window. If sm value is 10%, then 2 cores (from the 20 available) are removed from the the pool and only 18 cores are considered.

The following algorithms describe i) the process of calculating the host weight (Algorithm 1), and ii) the data chunk placement strategy (Algorithm 2). return weight 12: end function Calculating the host weight: : Algorithm 1 has four input parameters: host id , a safety margin (sm), time interval (ti) and sampling period (sp). The parameter ti is the time for which we calculate the weight. In our case, we used a ti 2 Giga Floating Point Operations per Second of 24 hours during which we measure hosts resources usage with a sampling period (sp) of 3 minutes, thus 480 measures every 24 hours.

Algorithm 1 Host Weight Calculation

First, we retrieve the maximum processing capacity (cap) for the selected host getFlops(host id ) at line 3. Then, we request the Forecasting Builder module to estimate the available amount of resources (predUsage) for this host ForecastingBuilder(host id , ti, sp) at line 4. The Forecasting Builder returns a set of ti/sp data prediction points. Then, we iterate over these predUsage data points to compute the weight of the selected host (lines 5-10).

For each predicted data point, we add the safety margin to the predicted load (line 6). Then, if the total used load is under 100 % (i.e., the host has some unused resources), the weight is calculated by subtracting from the total processing capacity of the host noted cap the predUsage and sm. We then multiply the result by the specified sampling period to integrate the duration (line 8). The higher the free CPU resource and capacity, the higher the weight.

Data placement strategy: Algorithm 2 starts by computing weights for each host using Algorithm 1. For each job, first we initilize a matrix of Booleans for the chunk mapping called blockAllocation at line 3 and we get the chunk replication factor used by Hadoop with function getReplication() at line 4. Second, for each chunk of the job (lines 5-12), we retrieve the estimated processing costs using getEstimatedTaskCost(chunk id ) function at (line 6). In this work, we have used fixed costs (see Section IV) but map or reduce tasks costs could be estimated as proposed by [START_REF] Khan | Hadoop performance modeling for job estimation and resource provisioning[END_REF]. Then, we select a host according to the assigned weights (based on WRR) at line 8. The hosts with higher weights are selected first. Next, we update the matrix blockAllocation to indicate that the chunk (chunk id ) has been placed on the chosen host (host i d) at line 9. Finally, we dynamically update the weight of the host by decreasing its value according to the used resources (cost) and to updated predictions (line 10). We repeat these three steps for the default number of replicas (i.e., nbReplicas) initialized for Hadoop (lines 7-12). If there is not enough resource for processing a chunk, a rejection message is sent. When all the chunks of all the jobs have been placed, we send the allocation matrix blockAllocation to the NameNode (line 14) and an acceptance message. Finally, the block allocation matrix is retrieved by the ClientNode.

D. QoS Controller

The QoS controller implements a mechanism that reacts to under estimation of the used resources from the Forecasting builder (e.g.,. As discussed before, prediction errors may exist due to an unexpected variation of the regular customers workloads. The reactive policy of the QoS controller checks if the regular customers workloads are using more than a predefined threshold of the safety margin (tuned to 50% in our experiments). In this case, the allocated resources for the send(job, blockAllocation) 15: end for ephemeral customers jobs must be reduced or completely released.

The QoS controller manages both the CPU and memory resources. In order to release resources the QoS controller proceeds as follows. The CPU control is done by adjusting dynamically the hard limits of the CPU cycles that a container is able to consume. In this way, Hadoop jobs cannot use more CPU than the amount of time set for the container. As a consequence, the map or reduce tasks will be slowed down without affecting regular customers workloads.

For the memory resource, Cuckoo has a more aggressive strategy and it acts as a system memory killer. Cuckoo kills proportionally the amount of map or reduce tasks necessary to free the safety margin related to memory occupation.

In case of CPU and/or memory starvation (i.e., only the safety margin resources are available) the container is killed.

IV. EXPERIMENTAL VALIDATION

This section describes the experiments conducted to validate the efficiency of the Cuckoo framework.

A. Experimental Methodology

We will try to answer three research questions (RQ) in order to tackle the 3 challenges mentionned in the section I (i.e., resource heterogeneity and volatility and QoS guarantee):

RQ1: What is the overall performance of Cuckoo compared to native Hadoop implementation ? RQ2: How does the forecasting builder accurately model the volatility of resources ?

RQ3: What is the effectiveness of Cuckoo with regards to the number of remote tasks?

We have conducted several experiments to evaluate our solution. We used a 3-months production data set from three different data centers and compared Cuckoo to standard Hadoop implementation. In our evaluations, we used two configurations related to the number of map tasks that is 514 and 640, corresponding to two different data sets of 40GB and 32GB respectively in a network of 50Mbps. In both cases we used 40 reduced-tasks. The processing costs for each Map and Reduce task is equal to 3100 FLOPS/Byte and 6300 FLOPS/Byte for the map and 1000 FLOPS/Byte for the reduce tasks.We set the chunk size to 64 MB and configured Hadoop to host three replicas for each chunk (default configuration) and each TaskTracker to run 20 slots of map and reduce tasks. According to [START_REF] Sammer | Hadoop Operations: A Guide for Developers and Administrators[END_REF], usually each task needs between 2 GB and 4 GB of memory which means that for a machine with 48GB of memory the Hadoop TaskTracker could run between 10 and 20 tasks in parallel.

The experimentation has three phases: i) infrastructure initialization, ii) deployment, and iii) injection. The infrastructure initialization phase configures the physical machines (i.e., speed, number of cores, memory), the network (i.e., topology, available bandwidth, latency) according to the three data centers. Then, the deployment phase consists in launching the Hadoop and Cuckoo modules (e.g., Job-Tracker, DataNode, Data Placement Planner). Finally, the injection phase consists in increasing and decreasing the CPU and Memory load over time according to data centers traces. The injection is done by replaying the traces from three data centers. As we fixed the forecast window to 24 hours on three months, this gives 92 windows per host.

Our experiments were performed using Simgrid 3.20 simulation tool and a customized version of MRA++ MapReduce [START_REF] Anjos | Mra++: Scheduling and data placement on mapreduce for heterogeneous environments[END_REF] for handling the three phases.

B. Data sets

Each data set corresponds to a specific data center. The largest data center is PC-2 (i.e., private company 2) with 27 hosts providing a total of 3552 GFLOP/s and 3.8TB of RAM memory, followed by PC-1 and University. Table I shows the overall capacity of all data centers. Moreover, these data centers are heterogeneous. The PC-1 has six different configurations among its nine hosts, PC-2 has 13 different configurations among its 27 hosts, and finally the University data center has 6 different configurations. The average resource utilization of each data center is 13% for PC-1, 4% for PC-2 and 6% for University. These results motivated us toward reclaiming unused resources.

C. Experimental Results

We evaluated the overall execution time of a job according to the safety margin value, the number of remote tasks and the number of rescheduled tasks.

1) RQ1-Job Execution Time: To evaluate the benefits of Cuckoo compared to Hadoop we compared the job execution time according to different configuration of safety margin (i.e., 0%, 5%, 10%, 15%, 20%, 25%, 30%).

Figure 2 shows the resulting job execution time. We observe that the minimum median completion time of Hadoop is 417 minutes for PC-1, 336 minutes for PC-2 and 377 minutes for University. On the other hand, the best median job completion time for Cuckoo is 57 minutes for PC-1, 49 minutes for PC-2 and 71 minutes for University with a safety margin of 5%. The smaller dispersion and best completion time was observed in the case of a 5% safety margin for all data sets. Then, with a safety margin greater than 5%, we notice that the job execution time increases. This is due to the fact that by increasing the safety margin size it leads to a decrease on the amount of reusable resources and thus the slow down of the Hadoop jobs. In addition, we observed that the best safety margin is the same regardless the data set evaluated. This means that the prediction of the forecast builder is accurate. Cuckoo is 7 times faster than native Hadoop strategy for PC-1 and PC-2 and 5 times faster for the University.

2) RQ2-Effetiveness of the Forecast builder: As mentioned in Section I, one way to increase data center utilization is to reclaim unused resources to run ephemeral containers that are executing map or reduce tasks. However, such containers could be evicted by the QoS Controller in case of interference with the regular customers workloads (see Section III-D). It means that the task should be relaunched causing a resource waste. Figure 3 shows the number of relaunched tasks for the three data sets with a safety margin of 0%. The lower the number of relaunched tasks, the better the volatility taken into account by the Forecast builder.

We observe that with the standard implementation of Hadoop, more than 15% of the tasks were relaunched while only less than 5% are in case of Cuckoo for the three data sets (3 times less). This confirms that the our prediction module helps Cuckoo to handle the volatility of resources efficiently.

In Cuckoo, PC-2 has the higher percentage of relaunched tasks with about 5% while the University the slower, with less than 1%. This means that PC-2 is wasting more resources when compared to University and PC-1. Moreover, a high number of relaunched tasks may lead to remote tasks, as explained in Section IV-C3.

3) RQ3-Effectiveness of Cuckoo and Remote Tasks: In this experiment we evaluated the percentage of remote tasks generated by Cuckoo and we compare it to the ones generated by Hadoop standard implementation. We have measured data movement during the experiments (see Section IV-A). A task becomes remote when a TaskTracker has an empty slot and no more available local chunks. In this case, a chunk is downloaded to be processed. In heterogeneous environments this situation is more likely to happen. So, by estimating the number of remote tasks we evaluate the ability of Cuckoo to manage heterogeneity and volatility.

Figure 4 shows the percentage of remote tasks. One may observe that Cuckoo outperforms standard Hadoop for all data sets. In the case of PC-1, Cuckoo has reduced the percentage of remote tasks by about 7 times, while in PC-2 and University by 6 and 19 times respectively. This confirms that Cuckoo handles both heterogeneity and volatility of resources effectively. As discussed in Section I, remote tasks take longer to execute as compared to local tasks due to the required data transfer time.

We also observe that PC-2 has more remote tasks. This can be explained by the fact that PC-2 is the largest data center with 27 hosts and since data chunks are distributed across hosts, more data transfers are generated. So for a given number of chunks to process, the higher is the replication factor, the lower the number of data movements are. In addition, the lower the number of hosts, the lower data movements for a given replication factor.

We conclude that for all three tested datasets, Cuckoo outperforms Hadoop in all the cases. This is explained by the fact that our data placement and the predictive provisioning strategies together make Cuckoo volatility and heterogeneity aware, and therefore able to required less remote and relaunched tasks, when compared to Haddop standard implementation.

V. LIMITATIONS

There are some potential issues that may impact on the results of this study. We will consider these issues in a future work. In the following we highlight some of these issues.

• We did not consider memory, network and storage for the data placement decisions. These variables may have an impact on the overall performance. • The Map/Reduce tasks are single-threaded and cannot leverage the computing power of more than one core. • The sampling period of the data center traces is 3 minutes which means that we cannot measure violations with smaller time sampling. • We did not consider how co-located application workloads may interfere on Map or Reduce tasks. We have considered only a fixed capacity per host, while the capacity may depend on all running applications [START_REF] Dartois | Investigating machine learning algorithms for modeling ssd i/o performance for container-based virtualization[END_REF] VI. RELATED WORK

In this section, we discuss some studies that have already explored MapReduce in dynamic and heterogeneous environments.

Many studies have already shown that heterogeneity significantly impacts Hadoop performance. In [START_REF] Zaharia | Delay scheduling: a simple technique for achieving locality and fairness in cluster scheduling[END_REF] authors proposed to improve data locality by delaying the task scheduling by a small amount of time for short jobs, which significantly improved performance. In [START_REF] Zaharia | Improving mapreduce performance in heterogeneous environments[END_REF] authors proposed to prevent from incorrect execution of speculative tasks by defining a fixed threshold in which the scheduler is able to select tasks to speculate. Their approach have improved the response time of MapReduce jobs by a factor of 2 in large heterogeneous clusters. In both cases they have improved execution time of jobs in heterogeneous environment. In our case, we consider not only the heterogeneity but also the volatility of nodes. Some approaches considered the case of an open shared system and proposed a placement strategy that dispatches data chunks to hosts based on their availability rate [START_REF] Jin | Adapt: Availabilityaware mapreduce data placement for non-dedicated distributed computing[END_REF]. The drawback of such approaches is their assumption about the inter-arrival time of interruptions to be independent and identically distributed across the system. There are some studies that handled volatility in open systems by using a small set of dedicated nodes in order to ensure the minimum amount of resources required to execute MapReduce jobs [START_REF] Lin | Moon: Mapreduce on opportunistic environments[END_REF]. In [START_REF] Steffenel | Mapreduce challenges on pervasive grids[END_REF] authors considered the case of pervasive grids and monitored nodes capacity (i.e., processors and memory) to improve Hadoop scheduler decisions. Differently from those previous work, our contribution targets environments such as private Cloud data centers. In [START_REF] Chohan | See spot run: using spot instances for mapreduce workflows[END_REF], authors have investigated the use of volatile spot instances as accelerators in public Clouds without considering any SLA constraints, while in [START_REF] Anjos | Enabling strategies for big data analytics in hybrid infrastructures[END_REF] a hybrid infrastructure (i.e., mix of Cloud and volunteer computing) with up to 25% of unstable nodes is used to continuously execute Hadoop jobs without loosing performance.

Finally, some studies used a predictive approach to improve data locality and Hadoop performance. In [START_REF] Merabet | A predictive map task scheduler for optimizing data locality in mapreduce clusters[END_REF] authors used a linear regression model to predict the execution time of map tasks which is used to prefetch input data. Another approach [START_REF] Tang | Availability/network-aware mapreduce over the internet[END_REF] used the Naive-Bayes-classifier method to predict node availability and therefore improve Mapreduce scheduler performance.

Our work improves state-of-the-art solutions by simultaneously considering both heterogeneity and volatility. Besides that, our approach guarantees SLA without interfering with other workloads sharing the same physical resources.

VII. CONCLUSION AND PERSPECTIVES

Data center resources are underused. They are heterogeneous and their usage is not balanced among hosts. We argue in this paper that those resources could be used to process Big data Hadoop application at a low cost. In order to do so, several challenges need to be tackled: heterogeneity and volatility of resources and isolation with regards to regular customer workloads.

To tackle these issues, we have developed a heterogeneity and volatility aware data-placement strategy called Cuckoo. Volatility and heterogeneity are managed by our proposed forecasting and resource-aware data placement strategies. In addition, a QoS controller is used to avoid any interference with regular workloads by means of a safety margin.

Our results show that Cuckoo outperforms standard Hadoop implementation by a factor of 5 to 7, while avoiding any interference with regular customer workloads.

As future work we plan to evaluate check-pointing faulttolerance techniques to avoid using reserved resources for hosting intermediate data. Although we only used simulations in this work, we plan to evaluate our strategies using a testbed for experimental research, such as Grid5000. We are also planning to use GPU and other processing elements.

Figure 1 .

 1 Figure 1. Overview of the Cuckoo architecture

1 : 4 : 5 : 6 :+= sm 7 :

 14567 function CALW(host id , sm, ti, sp) cap = getFlops(host id ) predU sage = ForecastingBuilder(host id , ti, sp) for each load ∈ predU sage do load if load < 100 then 8: weight += (cap -((load/100) * cap)) * sp

Algorithm 2 5 : 8 :

 258 Data Placement Algorithm 1: weights = initW eights() 2: for each job ∈ JobT racker.all() do 3: blockAllocation[nbChunks, nbHosts] = f alse 4: nbReplicas = getReplication() for each chunk id ∈ job.chunks do 6: cost = getEstimatedT askCost(chunk id ) selectedH = selectHost id (hosts, weights) 9: blockAllocation[chunk id , selectedH] = true 10: weights[host id ] = updateW (selectedH, cost)

Figure 2 .

 2 Figure 2. Job Execution Time for standard Hadoop and Cuckoo

Figure 4 .

 4 Figure 4. Median Percentage Remote tasks comparison between Hadoop and Cuckoo

ACKNOWLEDGMENT

This work was supported by the Institute of Research and Technology b-com, dedicated to digital technologies, funded by the French government through the ANR Investment referenced ANR-A0-AIRT-07.


Dr Julen Munárriz
email: julenm@ucla.edu

Dr Mónica Calatayud

Dr Julia Contreras-García

Charles E Young

Drive East

Dr Julen Munarriz

Valence Shell Electron Pair Repulsion Theory Revisited: An Explanation for Core Polarization

Keywords: Chemical Bonding, VSEPR Model, Core Polarization, Quantum Crystallography, X-ray models

Valence Shell Electron Pair Repulsion (VSEPR) theory constitutes one of the pillars of theoretical predictive chemistry. It was proposed even before the advent of the "spin" concept, and it is still a very useful tool in chemistry. This article proposes an extension of VSEPR theory to understand the core structure and predict core polarization in main group elements. We show from first principles (Electron Localization Function analysis) how the inner and outer core shells are organized. In particular, electrons in those regions are structured following the shape of the dual polyhedron of the valence shell (3 rd period) or the equivalent one (4 th and 5 th periods). We interpret these results in terms of the "hard" and "soft" core character. All the studied systems follow this trend, providing a framework for predicting electron distribution in the core. We also show that lone pairs behave as "standard ligands" in terms of core polarization. The predictive character of the model was tested by proposing the core polarization of different systems not included in the original set (such as XeF 4 and Fe(CN) 6

3-) and checking the hypothesis by means of a posteriori calculations. From the experimental point of view, the extension of VSEPR to the core region has consequences for timely crystallography. In particular, it enables to explain the core polarization proposed in the resolution of accurate X-ray structures. This result is illustrated for the particular case of -silicon.

Introduction

The development of simple models that facilitate a reliable prediction and rationalization of molecular structures and properties is of great importance in current chemical research. [1,2] In this regard, a good example of the predictive character of a simple energy model is the widespread Valence Shell Electron Pair Repulsion (VSEPR) theory. Specifically, it allows for predicting molecular geometries in terms of repulsions between electron pairs. [3] This theory has proven to work well for main group-based systems, [4] with only some exceptions, especially for complexes bearing a d 0 central atom. [5,6] Hence, it constitutes, in conjunction with other models, such as Lewis structures and electronegativity, a basic pillar of "chemical intuition" and structural analysis in chemistry. [7,8] When developing the concepts behind VSEPR, the repulsion between electron pairs is used to explain the organization of the valence ligands and lone pairs. However, as its name states, this theory is generally applied to the valence shell, and less attention has been paid to the core electrons. [9,10] Herein we show that the principles that govern VSEPR theory can be applied to explain the spatial distribution of all the electronsvalence and core-in a given system. This generalization has obvious epistemological consequences, allowing to obtain a further understanding of the electronic structure of the core regions. Moreover, we also highlight its utility in the interpretation of new accurate crystallographic data. [11] Whereas VSEPR theory was postulated relying mainly on chemical intuition and the empirical observation of a great number of compositions and stoichiometries, we count nowadays with tools that are able to reveal electron localization from first principles. One of them is the Electron Localization Function (ELF). [12] The ELF is related with the probability of finding electron pairs with opposite spins, which makes it an ideal tool for the identification of Lewis entities (the ELF maxima typically appear at bonds and lone pairs). [13] It has demonstrated to be very useful in the understanding and prediction of chemical properties, [14] as well as in the study of a wide variety of reactions, ranging from organic to biochemical and organometallic, among many others. [15,16] Within the framework of this contribution, we should highlight the work performed by Gillespie et al., who were able to explain the distortions from the VSEPR geometry on different metallic compounds by applying the ELF to study the interactions between the valence shell ligands and the outer core regions. [9,17] This type of analysis holds an inherent interest in the interpretation of new and more accurate spectroscopies. High energy [18,19] and high pressure [20] processes have led to interesting novel chemical phenomena where core electrons become bare or even leave the atom. This introduces the need to understand the subtle core electronic structure. As an example, the inclusion of core deformations in the resolution of X-ray data is a current field of development in crystallography. [21] Crystallographic structures are usually resolved by resorting to the multipolar approximation. [22] In this approach, cores are kept frozen, whereas the valence deforms to fit the experimental structure factors by using multipoles. [23] However, it was recently proposed that for the appropriate resolution of accurate data, the core regions should also be allowed to deform. [11] Herein, we further confirm the existence of such deformations from a theoretical point of view, by reproducing and explaining the core deformations reported in -silicon. [24] The paper is organized as follows. Firstly, we introduce the concept of the dual and equivalent polyhedron, which will allow us to set the geometrical basis of the model. Then, we analyze the maxima of the ELF in the core region of a set of VSEPR-geometry molecules with central elements belonging to the representative groups of the periodic table, so as to unravel the connection between the core organization and the valence shell distribution. In particular, we interpret the localization patterns that arise in a number of VSPER molecular geometries, with central atoms belonging to the 3 rd , 4 th and 5 th periods. Within those, we firstly focus on the core structure of molecular systems with stoichiometries AX 6 , AX 4 , AX 5 and AX 3 (in decreasing order of symmetry). Then, we evaluate the effect that valence shell distortions have on the core structure. The consistency of the results allows us to propose some inductive general rules that enable to predict core organization, which are summarized at the end of the paper. Finally, the predictive power of the model is checked by i) explaining the experimentally determined core polarization of -silicon [24] and ii) predicting the core polarization of some test systems which are outside the group we used for elaborating the model.

Results and Discussion

Historically, the organization of the outer core in metals has been studied by means of an analysis of the electron density maxima, within the Quantum Theory of Atoms in Molecules (QTAIM) approach. [25] A principle known as Ligand Opposed Charge Concentration (LOCC) was put forward. It states that the core localization maxima are located in the opposite direction of the ligands so as to minimize the repulsion between electron pairs, as exemplified in Figure 1a for a tetrahedral and an octahedral geometry. [26] Nonetheless, a mere opposition of electron pairs does not always explain the organization of the electron localization maxima predicted by the ELF. For example, for a tetrahedral system such as SiF 4 , the core ELF maxima have the shape of an inverted tetrahedron (in agreement with LOCC principle). However, for an octahedral system such as SF 6 , the ELF localization maxima does not lead to LOCCs (which would result in an octahedron with the same orientation as the ligands, as depicted in Figure 1a). On the contrary, a cubic organization is observed. This leads to a reformulation of the principle guiding the outer core organization. Along this contribution, we show that in general, core organizations do not respond to an electronic localization opposed to ligands, but to a general organization in terms of the overall geometrical disposition of the ligands and the period to which the central atom belongs. This leads to the appearance of the dual and the equivalent polyhedra. The dual polyhedron (DP) is the one that results from placing a vertex in the middle of the face of the original one (see Figure 1b). This principle allows to explain the shape of the localization maxima observed in the two previous examples, since the dual polyhedron of an octahedron is a cube (and vice versa); and that of a tetrahedron is another tetrahedron with an inverted orientation (see Figure 1b). These results anticipate the core structure being intimately related to the valence shell, and thus, the pertinent extension of VSEPR theory to the core regions. If this is the case, what happens as we go to the inner core? Whereas the previous seldom studies were carried out in the outer core, [9] the concept of the dual polyhedron has pushed us to analyze the whole core structure of central atoms belonging to the 3 rd , 4 th and 5 th periods. As previously stated, we have paid special attention to the relation between the ligand distribution in the valence shell and the core structure, and how distortions in the valence shell affect the core organization. AX6 stoichiometries. First, we considered simple octahedral molecules (O h symmetry). The details of the calculations are presented in the Experimental Section. In particular, we began by AX 6 (Z) geometries (A = S, P and Cl, in conjunction with X = H, F and Cl and the pertinent molecular charges, see Table S5). Notice that since the ELF respects the symmetry of the system, it only shows one maximum situated at the core position for the first core shell (2 nd period elements). Hence, the ELF can only be used to analyze the outer core polarization of central atoms having at least two core shells, that is, which belong to the 3 rd period onwards. In all the studied cases, the core region reflected the repulsion with respect to the valence shell, leading to the corresponding dual polyhedron: the localization maxima in the core for this set of molecules have the shape of a perfect cube (see Figure 2a for the case of PF 6 -). In order to unravel whether this is a common trend for the outer core, the central atoms were substituted by those belonging to the 4 th and 5 th periods. Specifically, we considered Se, As, Br (4 th row), Te, Sb and I (5 th row). To our surprise, for these systems a different kind of core polarization was observed. Namely, the outer core directly polarizes following the valence shell, that is, the ELF maxima in the core are ligand oriented, having the shape of the equivalent polyhedron (EP). As a consequence, octahedra are observed for the first core shell in 4 th and 5 th row elements, as exemplified in Figures 2b and2c for AsF 6 -and SbF 6 -, respectively (see Table S5 for more examples). Since relativistic effects may be important in these systems, [27] we applied the relativistic zeroth order regular approximation (ZORA) in 5 th row calculations, as explained in the Experimental Section. We believe this finding paves the way to the extension of the chemical hard and soft character to the core regions. [28] Atoms belonging to the 3 rd period would exhibit a hard core character, and thus counter-polarize with respect to the electron pairs in the valence shell (DP). On the contrary, the outer core of atoms of the 4 th row on would exhibit a soft core character. Hence, they undergo direct polarization (EP) with respect to the valence ligands. Notice that the inner core shells are also organized in terms of the dual and equivalent polyhedra (see the whole core structure depicted in Figure 2). AX4 stoichiometries. The previous results prompted us to extend the analysis to tetrahedral molecules (T d symmetry). In particular, general stoichiometries AX 4 (Z) with central atoms belonging to the 3 rd (A = Si, Al, P), 4 th (A = Ge, Ga, As) and 5 th (A = Sn, In, Sb) periods, in conjunction with X = H, F and Cl and the pertinent molecular charges were considered (see Table S6). Interestingly, the same general behavior as for AX 6 systems was observed. Namely, direct counter-polarization happened for 3 rd row atoms. The dual polyhedron of the ligand shell was reproduced in the core, obtaining an inverted tetrahedron, as depicted in Figure 3a for SiCl 4 . On the contrary, 4 th and 5 th row atoms undergo direct polarization. As a consequence, the equivalent polyhedron was obtained in the core region; that is, a tetrahedron oriented in the same direction as the ligand shell (see Figure 3b for GeCl 4 ). Again, the same kind of behaviororganization in terms of the EP and the DP-is found in the inner core. As presented in Figure 3c for Ge (4 th row), the outer core has the shape of the EP of the valence shell, while the inner core localization maxima form the DP of the outer core; see Figure S1 for additional details. AX5 stoichiometries. Less symmetrical valence shell geometries were also evaluated. In particular, we considered trigonal bi-pyramidal and triangular molecules, both with D 3h symmetry. For trigonal bi-pyramids, we analyzed AX 5 (Z) stoichiometries (A = Si, P and S for the 3 rd row, A = Ge, As and Se for the 4 th row, and A = Sn, Sb and Te for the 5 th row; in conjunction with X = H, F, Cl and the proper charges, as shown in Table S7). Noteworthy, the ELF maxima in the outer core of 3 rd row atoms have the shape of an inverted triangle (see Figure 4a for PH 5 ). That is, the core localization maxima adopts the shape of the DP of the equatorial triangle. As for the previous systems, this situation inverts when moving to 4 th row atoms, which directly polarize with respect to the valence shell (see Figure 4b). Thus, the electron pairs in the outer core adopts the shape of a trigonal bi-pyramid (EP). Also in the agreement with the previous results, the inner core of 4 th and 5 th row atoms is structured in terms of the EP and DP, as shown in Figure S2.

AX3 stoichiometries. The same principles were shown to hold for triangular molecules, AX 3 (Z) (A = Al, Si for the 3 rd row, A = Ga, Ge for the 4 th row and A = In, Sn for the 5 th row, with X = H, F, Cl, and the appropriate charges; see Table S8). In molecular geometries with a central 3 rd row atom, the outer core counter-polarizes, adopting the shape of a trigonal bi-pyramid with the central vertex opposed to the valence shell ligands (DP) and the two apical vertices perpendicular to the central molecular plane (see Figure 4c). As depicted in Figure 4c, the DP polyhedron in AlH 3 (and in general AX 3 systems) appears. However, it is accompanied by another two axial localization regions. It should be noted that within the D 3h group, the axial and equatorial maxima form two different reducible representations. The appearance of these "extra" maxima on top of the predicted ones requires further analysis. As expected, the outer core from the 4 th row on undergoes direct polarization (see Figure 4d). Thus, the electron localization maxima have the shape of a triangle oriented towards the ligands (EP). The inner core shells are also structured in terms of the DP and EP, as shown in Figure S3. Geometry distortions. The possibility of inducing core deformations was also analyzed. First, we checked the effect of varying the central atom-ligand distances. Nonetheless, attempts to invert the core polarization by expanding and compressing the ligand shell did not lead to core structural changes. This result prompted us to analyze the effect of introducing symmetry variations. For that, we substituted one of the ligands in the valence shell by a different one. This procedure is represented in Figure 5a for the transformation of SiF 4 into HSiF 3 . As a consequence, the molecular geometry evolves from tetrahedral (T d symmetry) to trigonal pyramid (C 3v symmetry). The geometry of the localization maxima in the core was modified according to the variations in the valence shell. In particular, we obtained the DP of a trigonal pyramid (a distorted tetrahedron), which is another trigonal pyramid pointing in the opposite direction. Moreover, we evaluated this effect in 4 th row atoms, by substituting Si by Ge. As expected, the outer core underwent direct polarization, and the electron localization maxima had the same geometry as the valence shell, i.e. a trigonal pyramid, see Figure 5a. It is worth noticing that this effect is transferred to the inner core, whose localization maxima have the geometry of the DP of the outer core: another trigonal pyramid. This way, we show that the modifications and distortions in the valence shell significantly affect all the core regions. Another way to induce core modifications by means of geometry distortions without changing the VSEPR group is substituting a ligand by a lone pair (and introducing two more electrons in the central atom). Figure 5b shows the evolution of SiF 4 to PF 3 . Notice that the lone pair has the same effect as the ligand variation. Namely, the original tetrahedron in the core of SiF 4 transforms into a trigonal pyramid as a consequence of the formal substitution of a fluoride ligand by a lone pair. The same effect is observed when taking into account 4 th and 5 th row atoms. For the particular case of the substitution of P by As, the outer core directly polarizes towards the valence ligands and the lone pair, forming the EP; that is, an elongated trigonal pyramid pointing in the same direction as the original one. Then, as for the previous approach (fluorine substitution by hydrogen), the lone pair effect in the outer core also influences the inner core, where a complementary elongated trigonal pyramid appears. Since the lone pair lacks a core, this shows that the effects hereby observed are directly related to electron-electron interactions, and not to a crystal field effect given by the external potential (i.e. by the cores of the ligands). -silicon. Pushed by the previous results, we tested the ability of the model proposed herein to explain the core polarization experimentally determined in -silicon. [24] The extension of the multipolar approximation to the core has shown that the Si outer core is polarized in the shape of a tetrahedron, which is in inverted position with respect to the tetrahedron formed by the first coordination sphere. This result is in agreement with our model, according to which the Si atom undergoes counter-polarization with respect to the bonded ligands (as previously shown for SiX 4 geometries, Figure 3a). This way, the Si core polarization has the shape of the DP of the valence ligands, that is, an inverted tetrahedron. In order to further confirm this result, we considered the model system reported in the original paper, Si(SiH 3 ) 4 , and analyzed the ELF maxima in the Si core region. [24] Notice that the expected behavior (an inverted tetrahedron in the Si core) was observed, as depicted in Figure 6. The agreement between the experimental results and the theoretical calculations further supports the validity of the theory.

The model. At this point, it may be convenient to summarize the general trends that can be derived from the previous results. We have found that, for the studied systems, all the central elements (which belong to the same periodic table row) behave equally in terms of core polarization, independently of the ligands that constitute the valence shell. More specifically, in all cases, the outer core of atoms belonging to the 3 rd period undergoes counter-polarization, and the core localization maxima exhibit the structure of the dual polyhedron of the ligand shell. We interpret this behavior to be a consequence of the system tendency to minimize repulsive interactions between electrons in the valence and the outer core shell. Notice that according to our results, the core electrons do not necessarily form electron pairs. As such, we can observe, for example, a cube as a localization pattern for the outer core of the 3 rd period elements, which does not lead to 2 electrons per vertex.

For the 4 th and 5 th periods, the outer core structure directly polarizes in the direction of the valence shell polyhedron. Thus, the localization maxima in the core have the shape of the direct polyhedron of the ligand shell. This highlights the softer nature of the core in 4 th and 5 th row atoms. This concept of "hard" and "soft" cores is in agreement with the relationship expected between volume, compressibility and conceptual DFT hardness. [29] Bigger atoms are expected to be easier to deform and polarize, [30] also leading to less compressible materials. [31,32] The coherence of the results support the predictive character of the polarization model we propose. Hence, it should be possible to predict the core structure of representative elements just by taking into account the position in the periodic table and the coordination number.

In this regard, we tested the predictive ability of the model by applying it to the structure of three different systems, which are not related with those considered for developing the previous discussion. In particular, we selected XeF 4 and [Fe(CN) 6 ] 3-, whose core structures are shown in Figure 7a and 7b (see Table S11 for the cartesian coordinates).  The choice of XeF 4 lies on the fact that it has two lone pairs. Thus, it allows to test whether we can extend the results obtained for one-lone-pair systems to those with more lone pairs. Since Xe belongs to the 5 th period, we would expect direct polarization of the outer core. The molecular geometry has the shape of a perfect square plus two lone pairs, which together form a compressed octahedron, as depicted in Figure 7a. Then, according to the previous results, the expected geometry for the outer core is a slightly compressed octahedron, which is the one that was found when performing the calculation (see Figure 7a). After that, we attempted to predict the core structure of a system bearing a transition metal. For that, we selected [Fe(CN) 6 ] 3-, as a well-known example of a metallic complex where VSEPR theory holds. [START_REF] Gillespie | The VSEPR Model of Molecular Geometry[END_REF] Since Fe belongs to the first transition metal row, we expect counter-polarization in the outer core. Thus, taking into account that the molecular geometry is octahedral, we expect that the localization maxima in the outer core have the shape of the DP of that octahedron, that is, a cube. This hypothesis was checked by performing the calculation, and, effectively, a cube shape was revealed for Fe outer core, as shown in Figure 7b. It is important to notice that the main analysis performed in this work was done with main group elements. Hence, we are cautious with the extension of these trends to systems bearing transition metals, and more systems should be studied to stablish general and consistent trends. Finally, we checked our ability to predict the core polarization in a crystal structure, as a test for crystallographic core polarization studies. We selected stishovite, a high pressure polymorph of silica. [START_REF] Kroll | [END_REF] This system has a tetragonal P4 2 /mnm symmetry, in which Si is in an elongated octahedral coordination environment. Then, as Si belongs to the 3 rd period, it should counter-polarize with respect to the valence shell and the core localization maxima should have the shape of an elongated cube (the DP of an elongated octahedron). In order to test this prediction, we constructed a cluster model, similar to the one considered for the analysis of -silicon (see Figure S7 and Table S11). Our prediction is in agreement with the calculation (see Figure 7c).

 Notice that the study of these kind of systems was proposed by one of the reviewers after the initial model development.

Conclusions

In conclusion, we have shown that the same physical principles that led to the development of VSEPR theory, that is, the molecular geometries adapting to minimize repulsion between electron pairs, can be used for predicting the core structure. Moreover, we have applied this model to understand in more detail the electrons organization in atoms with several core shells. Two different behavior patterns have been identified. On the one hand, for nuclei belonging to the 3 rd period, the electron localization maxima in the core orient toward the center of the polyhedral faces defined by the valence ligands, so as to minimize electron repulsions. On the contrary, for 4 th and 5 th periods, several core shells appear, and the outer one orient toward the ligands, undergoing direct polarization. This observation suggests a pertinent extension of the chemical concepts of hardness and softness to the core region: 3 rd row atoms would be harder than those belonging to the 4 th and 5 th periods. As a consequence, the spatial disposition of the outer core electrons of elements belonging to the 3 rd period is that that minimizes electron repulsions with the valence. On the contrary, the outer core of atoms in the 4 th and 5 th rows undergoes direct polarization toward the valence shell. Besides, we have demonstrated that there is an intimate relation between the ligand shell and the core, as valence distortions are transferred to all the inner core shells. Finally, we have shown that the lone pairs behave as "standard" ligands in terms of core organization. The principle hereby described not only confirms the existence of core polarization, but also explains the experimentally found core spatial distributions in -silicon. Hence, these basic principles can be used to help understand and predict core polarization in X-ray accurate data.

Experimental Section

Geometry optimizations and wavefunction calculations for ELF analyses were obtained through DFT calculations, by means of the Gaussian 09 program package. [START_REF] Frisch | Gaussian 09, Revision D.01[END_REF] The CAM-B3LYP exchange-correlation functional was applied, [START_REF] Yanai | [END_REF] in conjunction with the DGDZVP basis set, so as to be able to explicitly reproduce the electronic structure of the inner core of all the considered elements (up to the 5 th period). [37] The validity of the previous methodology was checked by using other functionals: B3LYP, [38] PBE [39] and PBE0, [40] in combination with bigger basis sets: 6-311G(d,p), [41] cc-pVTZ [42] and def2-TZVP. [43] We also applied Hartree-Fock (HF) to analyze whether the absence of electron correlation has a significant effect on the results. As explained in the Supporting Information (Tables S1-S4), the results obtained by varying the computational conditions led to consistent results. The calculation of the ELF and the analysis of its critical points were performed with the PROMOLDEN code. [44] Images were produced with VESTA. [45] As relativistic effects may have an influence on the core structure of atoms belonging to the 4 th and 5 th periods, relativistic calculations were performed on a set of molecules so as to confirm the validity of nonrelativistic calculations. We considered the zeroth-order regular approximation (ZORA) as implemented in the Orca 4.0 suite. [46] An all-electron ZORA-def2-TZVP basis set was used in all the calculations, [43] in conjunction with auxiliary SARC/J basis. [47] We considered representative molecules of all the geometries included in the main discussion, with central atoms belonging to 3 rd , 4 th and 5 th rows. In particular, the following set of molecules was chosen: AX3: AlF3, GaF3, InF3; AX4: SiCl4, SiF4, GeF4, SnF4; AX5: PF5, AsF5, SbF5; AX6: SH6, SeH6, TeH6. In all cases, the same results as for non-relativistic calculations were obtained for the whole core structure of the 3 rd and 4 th periods and the outer core of the 5 th period. Specifically, the inner core of InF3, SnF4 and SbF5 molecules presented inverted polyhedra as for non-relativistic calculations. As a result, in order to provide accurate and reliable results, molecules involving 5 th row central atoms were calculated by using relativistic calculations.

Figure 1 .

 1 Figure 1. Schematic representation of a) the LOCC, and b) the dual polyhedron of a tetrahedron and an octahedron.

Figure 2 .

 2 Figure 2. Outer and inner core polyhedra for a) PF6 -, b) AsF6 -and c) SbF6 -.Fluoride ligands are depicted in blue, and core ELF maxima in purple.

Figure 3 .

 3 Figure 3. Outer and inner core polyhedra for a) SiCl4, b) GeCl4 and c) Ge core of GeCl4. Chloride ligands are depicted in green and core ELF maxima in purple.

Figure 4 .

 4 Figure 4. Outer core polyhedra for a) PH5, b) AsH5, c) AlH3 and d) GaH3. Hydrogen atoms are depicted in light pink, and core ELF maxima in purple.

Figure 5 .

 5 Figure 5. Evolution of the localization maxima in the core for the transformation of SiF4 into a) HSiF3 and GeHF3, b) PF3 and AsF3. Fluoride ligands are depicted in blue, lone pairs in orange and core maxima in purple.

Figure 6 .

 6 Figure 6. Model for -silicon. The second coordination sphere has been replaced by hydrogen atoms. Si in blue, hydrogen in beige and ELF maxima in pink.

Figure 7 .

 7 Figure 7. Outer core polyhedra for a) XeF4, b) [Fe(CN)6] 3-and c) the model system for stishovite (the complete structure for this model is presented in FigureS4).Core ELF maxima are depicted in purple and lone pairs in orange.

Acknowledgements

J.M. thankfully acknowledges financial support from the Spanish Ministerio de Ciencia, Innovación y (FPU14/06003 and EST17/00161) and from the Universidad de Zaragoza, Fundación Bancaria Ibercaja and Fundación CAI (CB 6/17). J.C.-G. expresses her gratefulness to CALSIMLAB and the ANR within the Investissements d'Avenir program under reference ANR-11-IDEX-0004-02. All the authors thank the computational resources and technical expertise provided by the Laboratoire de Chimie Théorique (Sorbonne Université). We also thank Emilie Huynh, Hoai Dang and Daniel l'Anson who participated in the early stages of the work. J. M. is especially grateful to Alberto Pérez-Bitrián for helpful scientific discussions.

In this paper, the extension of VSEPR Theory to predict the electronic distribution of the core region is explored. The results show that there is an intimate and predictable relation between the valence and the core shells, broadening the scope of VSEPR to the core regions. This way, we shed light into the factors governing core polarization, which are important in current crystallography research.


F Boige
email: boige@laplace.univ-tlse.fr

F Richardeau

S Lefebvre

M Cousineau

SiC Power Mosfet in Short-Circuit operation : ELECTRO-THERMAL MACRO-MODELLING COMBINING PHYSICAL AND NUMERICAL APPROACHES WITH CIRCUIT-TYPE IMPLEMENTATION

Keywords: electrothermal simulation, model validation, MOSFET, short-circuit, silicon carbide

The purpose of this paper is to describe, for the first time, a global transient electrothermal model of (SiC) power MOSFETs during accidental short-circuit (SC) operations. The developed models allow to analysis an inverter-leg malfunctioning. A thermal model of the SiC MOSFET dies combined with extensive experimentations allow to develop models of the gate leakage current and of the drain saturation current during SC events. After verifying the robustness of the proposed elementary models, an original global circuit model with an easy implementation using a commercial circuit simulation tool is proposed and discussed. This circuit model can be used in order to develop protection schemes against SC faults.

INTRODUCTİON

Nowadays, due to mass production from major manufacturers, SiC power MOSFET is gradually used in several power applications such as inverter for PV panels and grid, UPS and railway traction drive. In that kind of applications, short-circuit (SC) faults can appear leading to large transient power dissipation and a dramatic risk of failure. Examples of these faults-types are described on Fig. 1. In order to develop a dedicated fast and robust protection against these faults, studies on the ruggedness of SiC MOSFET under short-circuit have been carried out [START_REF] Chen | Study of short-circuit robustness of SiC MOSFETs, analysis of the failure modes and comparison with BJTs[END_REF][START_REF] Wang | Temperature-Dependent Short-Circuit Capability of Silicon Carbide Power MOSFETs[END_REF][START_REF] Nguyen | Gate Oxide Reliability Issues of SiC MOSFETs Under Short-Circuit Operation[END_REF]. Nevertheless, until now, none electro-thermal models globally and accurately describe SiC power MOSFET behaviours under SC faults. Particularly, the gate-leakage current increases significantly during short-circuit, as depicted in Fig. 3, and the saturation current evolution is unusual compared to silicon (Si) devices. Indeed, SiC material can support much higher junction temperature leading to behaviour that cannot be seen in Silicon aat these temperature such as high temperature Fowler-Nordheim tunnelling [START_REF] Mbarek | Robustness Study of SiC MOSFET Under Harsh Electrical and Thermal Constraints To an in-depth physical failure analysis[END_REF] or high change in the channel mobility [START_REF] Potbhare | A Physical Model of High Temperature 4H-SiC MOSFETs[END_REF]. The significant gate-leakage current observed during the SC can directly affect the gate-voltage control of the device through the voltage drop across the gate driver resistor. Whatever the failure mode resulting of a SC event (failure of the gate with a short-circuit between gate and source or failure between drain and source after thermal runaway), the estimation of the critical energy capability and the corresponding junction temperature in short-circuit mode are necessary in order to determine the maximum delay time of the gate drive protection to be implemented.

The goal of the proposed study is to develop a new global and macro-model of SiC power MOSFETs. We focus in this paper on a particular device from ROHM™ (SCT2080KE) but the methodology and the model can be easily transposed to other devices from different manufacturers. Emphasis has been put on finding and fitting accurate models for transient gate leakage current (I G ) during the SC phase, drain saturation current (I Dsat ) and threshold voltage (V GS(th) ) according to complex and several parameters, particularly using estimated maximum junction temperature (T j ) at the top of the die. The original contribution of this paper to the classical MOSFET model is highlighted in Fig. 2. During SC events, large value of drain to source voltage and very large value of drain saturation current lead to a very large dissipated power density in the die resulting in a very fast increase of the junction temperature. A thermal model is developed and used in order to estimate the junction temperature during the SC event and then to develop models of the transient gate leakage current (I Gleak ) drain saturation current (I Dsat ) and threshold voltage (V GS(th) ).

Combining the gate-current and drain saturation current behaviour, the aim of this paper is to present how the electro-thermal modelling methodology of a SiC MOSFET under extreme stress has been built and evaluated for a practical use. In Section 2, after an introduction, a first dipole-type gateleakage current model function of four different parameters (dissipated energy, gate driver outputvoltage eq. to V buffer , gate resistance and drain to source voltage) is developed and presented. In Section 3, a 1D estimation model of the temperature distribution inside the chip is described. In Section 4, based on the results from this estimation, a physical model is fitted in order to precisely represent the drain saturation current evolution during the SC phase. Finally, in Section 5, simulations and robustness of the global behaviour, using PLECS software, are presented and compared with experimentations.

GATE-LEAKAGE CURRENT MODELLING

MODEL DETERMİNATİON

After few micro-seconds of short-circuit, a great transient current surge appears across the thin MOSFET gate oxide (typically a thickness close to 50nm). This current is most likely caused by a highly temperature dependent Fowler-Nordheim tunneling effect [START_REF] Nguyen | Gate Oxide Reliability Issues of SiC MOSFETs Under Short-Circuit Operation[END_REF] considering the important energy dissipated inside the chip (up to 10J/cm² in 12µs) combined with a great electric field (higher than 400V/m) applied between the polysilicon-gate and the SiC substrate of the device through the SiO 2 gate-oxide. However, the SC duration is much lower than the thermal time constant of the component die (

. In these conditions, only the die will be considered by the temperature increase during SC events and dissipated energy can be considered instead of the junction temperature in order to explain the time dependence of the gate leakage current during SC. Furthermore, as depicted in Fig. 3a, 3d, the gate-leakage current dynamic also depends on the gate voltage bias imposed by the gate-driver ( ), the drain to source voltage ( ) and the external gate resistor ( ) (not displayed). The aim of this section is to explain how the influence of each parameter has been identified and modelled. In order to reduce the complexity of the study, only the gate current increase, along the sequence n°3 in Fig. 3, is accurately modelled. As a result, the surge occurs from a fixed energy (0.8J or 5.9J/cm² of active chip area for all measurements as presented in Fig 3c) so the model will be valid from this value only. In practice, this threshold energy of the gate-stress represents a value close to the half of the failed critical energy of the device. For the other parameters, the model validity has to be large: , , . A "dipole-type" model has been chosen (Fig. 2) as an equivalent non-linear controlled current-load F IGleak between gate and source electrodes supplied by the driver voltage. This source is made of a voltage source (V buffer ) and a series gate resistor (R G ). On Fig. 4a are plotted current-load equations for four different buffer voltages, 18, 19, 20 and 21V. Different experimental points are also plotted on this figure (circles). For each buffer voltage, for points of identical dissipated energy (1, 1.05, 1.1 and 1.17 J) have been reported. It is remarkable that for a given energy and different buffer bias a load straight line, as a perfect ohmic behavior, is clearly identifiable and is significantly energy (temperature) dependent. In order to analyse the energy influence on the current-load equation, load straight lines are displayed in 3D representation (I G , V GS , E n ) in Fig. 4b. This representation is basically the same as depicted in Fig. 3. But the load characteristic of the gate driver can be visualised as a surface. Therefore, a polynomial surface (1) has been chosen and fitted with MATLAB™ to model the current-load variation function of V GS and the dissipated energy (E n ) on the validity area of the model.

With (

are the estimated factors.

This "response surface" approach has two main advantages. On one hand, MATLAB™ polynomial surface fitting algorithm from the fit function is powerful and robust. On another hand, the polynomial function and dipole-type model makes it easily computable and to be embedded into a commercial-type software. At this point, the model represents effects of V buffer, R G and E n on the gate leakage current value.

However, V DS variation, in Fig. 3d, is not yet added in the model. To do that, factors ( are, themselves, estimated for different V DS bias and pre-calculated in a table. It appears that these factors are linearly varying with V DS . Finally, the variation has been estimated and added to the model. The parameters value can be found in Table I.

I. Estimated factors function of V ds

RESULTS

The model has been estimated with experimental data from the component SCH2080KE from ROHM™. In order to determine the model robustness, two different tests were made. The proposed model was first compared with new experimental results which are different from those used for the model fitting (Fig. 5a). In a second step, the proposed model was compared with data obtained from another component SCH2080KE, i.e. for new experimental results (Fig. 5b). Moreover, the model can be easily adapted for other SiC MOSFET components from Rohm or other manufacturers. Indeed, in Fig. 5c, the model parameters has been estimated with experimental data from the component from Wolfspeed™. The simulation results are, in each case, very close to the experimental data. In conclusion, these results clearly validate the proposed numerical model and show its robustness in its full domain of validity.

TEMPERATURE ESTİMATİON

MODEL CONSTRUCTİON

During a short-circuit event, strong electrical field is distributed along the low doped epi-layer and a strong current density flows through the chip. This particular configuration induces a high and fast transient temperature gradient inside the chip within few micro seconds. With encapsulated components, the temperature gradient is not measurable in real time because the external case temperature do not represent the junction temperature of the chip [START_REF] Castellazzi | Short-circuit robustness of SiC Power MOSFETs: Experimental analysis[END_REF]. In order to fit physical models function of the temperature, such as dependant electron mobility or threshold voltage, an accurate temperature estimation is needed. All these physical models are required in order to model the drain saturation current during the SC. Moreover, the thermal model must be accurate and low simulation time consuming. To do that, the best compromise is a one dimensional transient thermal model across the chip depth [START_REF] Wang | Temperature-Dependent Short-Circuit Capability of Silicon Carbide Power MOSFETs[END_REF]. Two models were developed: a "simple model" with fast computation time (Fig. 6a) designed to be embedded into a circuit simulation tools named PLECS™ and a more complex model developed in order to simulate more finely the thermal physical effects (Fig. 6b).

Model construction common to the two models

During short-circuit transient, the dc bus voltage, V DC , is applied across the power device and leads to depletion layers with a width x p in the P-well (highly doped N a ) and x n in the N-drift region (low doped N d ). The resulting electric-field distribution in the space charge region is given by the integration of the Gauss 'law as described in [START_REF] Wang | Temperature-Dependent Short-Circuit Capability of Silicon Carbide Power MOSFETs[END_REF] as an example, neglecting the effect of the current, i.e. the charge displacement on the depleted charges.

Where is the total dielectric constant for 4H-SiC material and q is the electron charge. V DS is the electric field integral along the space dimension. Doping values and dimensions, presented in Fig 6 . are chosen according to the literature [START_REF] Baliga | Silicon Carbide Power Devices[END_REF]. It results that and at V DS =600V. The 1D approximation of the heat flux profile in the volume Q(x, t) [W/m 3 ] is depending on the electric field [V/m] function of the variable "x" and the drain-saturation current function of the variable "t" with the following formulation formulation based on the scalar product between the electrical field profile and the density current : Where is the experimental short-circuit current and the active area of the die.

Fig. 6 depicts the boundary conditions. The top surface of the chip is assumed adiabatic due to the thick plastic coating around the die and the bottom is considered at a constant temperature (ambient temperature) due to the fact that the heat diffusion depth is much lower than the thickness of the die at the end of the short-circuit stress. This hypothesis has been confirmed by changing the boundary conditions (not presented here).

Model 1 distinctive characteristics

In the model 1, the thermal properties of the SiC: thermal conductivity (

) and specific heat (

) have been chosen as constant values. Moreover, the aluminium top layer is not simulated, to overcome this, a thicker P-well have been chosen. Finally, this model has been implemented on two software:

 with COMSOL™ software by solving the heat equation in 1D.  with PLECS™ by the mean of a Cauer network of RC-elements with a distributed power source. The Cauer model have been constructed by using a vectorised model [START_REF] Meynard | Analysis and Design of Multicell DCDC Converters Using Vectorized Models[END_REF] of 180 RC cells. The thermal resistors ( ) and capacitances ( ) are calculated as presented in [START_REF] Snead | Handbook of SiC properties for fuel performance modeling[END_REF]. With , the number of RC cells, the thermal conductivity of the material, the specific heat capacity, the material density, the active area of the die and the chip depth. .

The power source is sampled in the depletion layer, following the trapezoidal rule:

However, this power source represent the mean value of the power between 2 sampled points, as a consequence the Cauer network has been built as presented Fig. 7.

Model 2 distinctive characteristics

In the model 2, the aluminium top layer has been taken into account and included in the model. Moreover, the dependences of the thermal properties of SiC and aluminum with the temperature have been added into the model. The evolution of thermal conductivity and specific heat with the temperature can be found, respectively, in [START_REF] Snead | Handbook of SiC properties for fuel performance modeling[END_REF] and [START_REF] Hatch | Aluminum: Properties and Physical Metallurgy[END_REF], as well as the melting latent heat of aluminium (933K). The model has been implemented in COMSOL™ software. Fig. 7 : presented Cauer RC network

RESULTS

In Fig. 8, the junction temperature T j (x j ,t) is presented function of the short circuit duration for different drain biases. The presented results have been estimated from the two proposed models with two different solving process. In fig. 8a, the model 1 is solved with the two methods presented previously and the difference between the results is lower than 1%. The estimated temperature increases up to 1300K for 12µs of SC duration, these temperature values are close to results already observed in the literature [START_REF] Wang | Temperature-Dependent Short-Circuit Capability of Silicon Carbide Power MOSFETs[END_REF], [START_REF] Duong | Electrothermal simulation of 1200 V 4H-SiC MOSFET short-circuit SOA #x2020[END_REF] but obtained with other estimation methods. As the temperature is not measurable, the results between models can only be compared.

In the other hand, the model 2, presented Fig. 7b, is giving very different results: the temperature can be 500K higher from results obtained from the model 1, although model 2 is more realistic. This difference can be explained by the SiC thermal conductivity which tends to decrease with the temperature increase while the heat capacitor tends to increase. This behavior induces a great decrease of the thermal diffusivity of the heat and then a significant increase of the local temperature around the maximum power density region. Therefore, the more the temperature is high, the less the component can dissipate the heat flux. Moreover, this model hasn't been implemented in PLECS software with a vectorized RC network due to convergence issues and long computation time.

Finally, the Model 1 implemented in PLECS has been chosen as temperature reference for the saturation current fitting as well as for the electrothermal circuit-type model.

Fig. 1 .

 1 Fig. 1. (a) inverter-leg cross-conduction by faulty control signals (Type I SC). (b) Internal short-circuit by electrical breakdown of one of the two leg-switches (Type II SC).

Fig 2 .

 2 Fig 2. Enhanced electro-thermal SiC MOSFET circuit-type modelling. In black line, the classical MOSFET model. In red line, new features proposed in this article for short-circuit operation.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Experimental waveforms, from a lab. . set-up, during type-I SiC MOSFET short-circuit. (a) gate-leakage current and dissipated energy inside the chip. (b) drain current during SC (saturation current) and drain-source voltage. (c) & . (c) & .

Fig. 5 .

 5 Fig. 5. Model output (red lines) and experimental results (blue lines) of the gate leakage current evolution during SC. (a) Robustness results and validation obtained from test data which are not used for the model fitting but with the same device. (b) Robustness results and verification obtained from test data not used for the model fitting and with another device of the same reference. (c) New fitting results obtained with data used for another reference component C2M0080120D

Fig. 6 .

 6 Fig. 6. 1D thermal model of the SiC MOSFET (a) first thermal model with only SiC material and physical parameters not thermally dependent (b) 2 nd model with aluminum on top, and physical parameters thermally dependent.

Fig. 8 .

 8 Fig. 8. Estimated junction temperature T j (x j , t) in a short-circuit operation at V buffer =21V, T init (t=0)=298,15K. (a) Estimated temperature between Model 1 solved but COMSOL and solved by a Cauer network with PLECS. (b) Estimated temperature with model 1 and model 2 solved by COMSOL.

ACKNOWLEDGEMENTS

This research work received financial support from the French National Research Agency (ANR). The project name is "HIT-TEMS" and it is managed by CNAM Paris and SATIE Lab. at the ENS Paris-Saclay school.

DRAİN SATURATİON CURRENT MODEL

To accurately model the drain saturation current, as shown in Fig. 3, a representative physical model is needed. Available models depend of a lot of physical parameters that are not known over a wide temperature range. In order to estimate them, data from experiment are needed. Methodology is summarized in Fig. 9. Different models have been tested but finally, the following MOSFET model has been chosen due to its classical form [START_REF] Baliga | Modern power devices[END_REF]: With Where, is the oxyde capacitance per square area, Z and L are the width and length of the channel respectively. The carrier mobility in the channel, , is at least the parallel combinations of three different mobility functions with different sensitivity toward temperature [START_REF] Potbhare | A Physical Model of High Temperature 4H-SiC MOSFETs[END_REF]. The Matheissen's rule gives:

Where , , are mobility parameters at ambient temperature and a,b,c are sensitive thermal coefficients that have to be estimated in order to obtain a wide range temperature model. Otherwise, literature [START_REF] Baliga | Modern power devices[END_REF], gives a linear threshold voltage dependence with temperature:

Where x 1 and x 2 are unknown parameters. All the previous parameters are determined in two steps. At first, and are determined function of the temperature. To do that, an algorithm has been developed to find the best pair of parameters (x 1 , x 2 ) which minimize the difference between calculated with the equation ( 9) for different experimental data. Results are displayed in Fig. 10 (b), with the couple (x 1 =-0.025, x 2 =15.4) and K gm remain the same for different buffer bias (19V, 20V, 21V). In a second time, the mobility model ( 11) is fitted on profile previously determined using MATLAB's function fit. Results are evaluated in Fig. 10 (a).

Unfortunately at the current state, the model is not robust to V ds variation. Other models have been tested to take this parameter into account but without success. In these conditions, the proposed model will be used for a fixed voltage. The work will have to be continued in order to develop a model taking into account the voltage applied on the drain

IMPLEMENTATION IN PLECS

The models have been implemented in PLECS™, a software for simulations of electrical circuits. PLECS™ presents advantages of being embedded in MATLAB and of being able to quickly simulate complex electric topologies. The aim of this section is to present all the previous results implemented into a single circuit-type model as summarized in Fig. 2. The classical MOSFET model (in Fig. 2 black line) is completed (in red line) adding the previously presented gate leakage current model "F IGleak ", the drain saturation current model "F Idsat " and finally the gate threshold voltage model by "F VGth ". The different models are performed by PLECS function block or by code-C script. Moreover, the thermal model is constructed with a vectorized Cauer RC network as presented in the previous section. The PLECS schematic is displayed in Fig. 11a. This MOSFET model is itself connected to a voltage source between drain and source and connected to a MOSFET driver to turn it on/off (not displayed). The results of the whole simulation are shown in Fig. 11b. Clearly, the simulation results are much closed to the experimental waveforms during SC and the estimation error on the gate current surge is less than 1%. Our objective is fulfil: the simulation is well working and will be able to be used in more complex circuit, for example, short-circuit detection and more generally concerning the safety analysis of critical system. Moreover, the model clearly represent the effect of the buffer voltage bias variation. All the modelling procedure is automatized in MATLAB and this model could be easily obtained for other SiC MOSFET devices. However, strong coupling with PLECS circuit and COMSOL 1D model should be implemented and drain saturation current robustness to V DS voltage variation has to be also investigated in future works.

CONCLUSION

The aim of this paper was presenting a complete and new modelling methodology for SiC power MOSFET devices in high-current and high-temperature conditions. Firstly, the gate transient leakage current is modelled. In a second step the temperature is estimated inside the chip through two models and finally a physical model of the drain saturation current is given. After validating these different elementary models all the previous models are embedded into a circuit-type MOSFET model. This "block" could be used, as it is, to simulate fault operation into applications using SiC MOSFET components like a motor drive, for example. Moreover, the proposed model enhancement does not add lot of calculation time and is fully compatible with classical numerical or circuit-type MOSFET models. The whole modelling has been performed on the ROHM component SCT2080KE but the method is clearly practical for other SiC MOSFET such as Wolfspeed or STMicroletronics as demonstrated for the gate current modelling.