Y Bonhomme

P Girard

C Landrault

S Pravossoudovitch

Scan Cell Ordering for Low Power Scan Testing

Power consumption during scan testing is becoming a primary concern. In this paper, we present a novel approach for scan cell ordering which significantly reduces the power consumed during scan testing. The proposed approach is based on the use of a two steps heuristic procedure that can be exploited by any chip layout program before flip-flops placement and routing. The proposed approach works for any conventional scan design and offers numerous advantages compared with existing low power scan techniques. Reductions of average and peak power consumption during scan testing are up to 34% and 18% respectively for experimented ISCAS benchmark circuits.

Introduction

The System-on-Chip (SOC) revolution has brought some new challenges to both design and test engineers. Among these challenges, the power dissipation is one of the most important issues [START_REF] Crouch | Design-for-Test for Digital IC's and Embedded Core Systems[END_REF]. Generally, a circuit may consume more power in the test mode than in the normal mode due to the following reasons [START_REF] Zorian | A Distributed BIST Control Scheme for Complex VLSI Devices[END_REF]. First, the design-fortestability (DfT) circuitry embedded in a circuit to reduce the test complexity is often idle during normal operations but may be intensively used in the test mode. Second, the test efficiency has been shown to have a high correlation with the toggle rate; hence in the test mode, the switching activity of all nodes is often several times higher than the activity during normal operations. Third, in a SOC, parallel testing is frequently employed to reduce test application time, which may result in excessive energy and power dissipation. This elevated test power may be responsible for several kinds of problems: instant circuit damage, increased product costs, decreased system reliability, performance degradation, reduced autonomy of portable systems and decrease of overall yield. A survey of these problems is given in [START_REF] Girard | Survey of Low Power Testing of VLSI Circuits[END_REF]. The focus of this paper is on the problem of minimizing power dissipation during external scan testing, i.e. from an ATE (Automatic Test Equipment). Scan architectures are very popular and are commonly used to test digital circuitry in integrated circuits (ICs) or cores. However, scan-based architectures are expensive in power consumption as each test pattern requires a large number of shift operations with a high circuit activity [START_REF] Bushnell | Essentials of Electronic Testing[END_REF]. Of course, it is always possible to reduce average power during scan testing by simply scanning at a lower frequency. However, this increases test application time. Another solution is to add logic to hold the output of the scan cells at a constant value during scan shifting [START_REF] Hertwig | Low Power Serial Built-In Self-Test[END_REF]. The drawbacks of this approach are the area overhead and the performance degradation that it incurs. Some other solutions have been proposed recently to cope with the power problem during scan testing: low power ATPGs [START_REF] Wang | ATPG for Heat Dissipation Minimization for Scan Testing[END_REF][START_REF] Corno | A Test Pattern Generation Methodology for Low Power Consumption[END_REF], a scan path segmentation technique [START_REF] Whetsel | Adapting Scan Architectures for Low Power Operation[END_REF][START_REF] Saxena | A Scheme to Reduce Power Consumption During Scan Testing[END_REF], a static compaction technique [START_REF] Sankaralingam | Static Compaction Techniques to Control Scan Vector Power Dissipation[END_REF], two clock scheme modification techniques [START_REF] Sankaralingam | Reducing Power Dissipation During Test Using Scan Chain Disable[END_REF][START_REF] Girard | A modified Clock Scheme for a Low Power BIST Test Pattern Generator[END_REF], an interleaving scan architecture for multiple-scan circuits [START_REF] Lee | Peak-Power Reduction for Multiple-Scan Circuits during Test Application[END_REF], a test data compression technique for SOC [START_REF] Chandra | Combining Low-Power Scan Testing and Test Data Compression for System-on-a-Chip[END_REF], test scheduling techniques [START_REF] Chou | Power Constraint Scheduling of Tests[END_REF][START_REF] Iyengar | Precedence-Based, Preemptive, and Power-constrained Test Scheduling for System-on-a-Chip[END_REF], … Actually, a simple alternative solution for minimizing power consumption during scan testing is to use test vector ordering or scan cell ordering techniques. Test vector ordering has been investigated in [START_REF] Dabholkar | Techniques for Reducing Power Dissipation During Test Application in Full Scan Circuits[END_REF][START_REF] Girard | Reducing Power Consumption during Test Application by Test Vector Ordering[END_REF][START_REF] Girard | A Test Vector Ordering Technique for Switching Activity Reduction during Test Operation[END_REF] with the objective to define the order in which test vectors of a deterministic test set have to be applied to the circuit or core under test (CUT) to minimize the overall switching activity. Scan cell ordering has been investigated only in [START_REF] Dabholkar | Techniques for Reducing Power Dissipation During Test Application in Full Scan Circuits[END_REF], where two heuristics are proposed to determine the order in which the scan flip-flops of a given scan chain have to connected: a random ordering heuristic and a simulated annealing algorithm. Experimental results on small benchmark circuits show that scan cell ordering can reduce test power by 10-25% with no change in terms of fault coverage and test length. Larger benchmark circuits have not been experimented because they are intractable with the proposed heuristics, but an important comment reported in [START_REF] Dabholkar | Techniques for Reducing Power Dissipation During Test Application in Full Scan Circuits[END_REF] is that for large circuits with a huge number of scan elements, scan cell ordering combined with test vector ordering is much more efficient than test vector ordering alone. Hence, scan cell ordering must be selected as the first test power minimization strategy. In this paper, we propose a novel approach for scan cell ordering which significantly reduces the power consumed during scan testing. The proposed approach is based on the use of a simple procedure that can be exploited by any chip layout program before flip-flops placement and routing. The inputs to this procedure are i) a given set of scan flip-flops and ii) a sequence of deterministic test vectors with the corresponding output responses. The output is an ordered scan chain with minimum average and peak power consumption. To tackle this NP-hard problem efficiently, the heuristic procedure operates in two steps: the first one consists in determining the chaining of the scan cells so as to minimize the occurrence of transitions in the scan chain during shifting operations, the second one consists in identifying the input and output scan cells of the scan chain to limit the propagation of transitions during scan operations. An important feature of the proposed approach is that both scan-in and scan-out transitions (and not only scan-in transitions due to test vectors) are considered for solving the scan cell ordering problem. A comparison of the heuristic procedure with an exact method (on small benchmark circuits) demonstrates the efficiency of the proposed solution. Compared with existing low power scan techniques, our solution offers numerous advantages. The proposed approach works for any conventional scan design -no extra DfT logic is required. The fault coverage and the IC test time are let unchanged by the proposed approach. There is almost no penalty on the circuit performance. The area overhead, which is due to routing area, may be kept acceptably low when managed by an efficient chip layout program. The proposed approach is very easy to use in a classical DfT flow and has therefore a very low impact on the system design time. Reductions of average and peak power consumption during scan testing are up to 34% and 18% respectively for experimented ISCAS circuits. Note that the approach described in this paper can be used for full scan cores or ICs having one or multiple scan chains. It can also be combined with a scan vector ordering technique to provide additional reductions in test power. The remainder of the paper is organized as follows. Section 2 describes how power is estimated in the proposed approach. Section 3 is devoted to the description of the proposed scan cell ordering procedure. Results obtained on the benchmark circuits are reported in Section 4. A discussion on layout considerations during scan cell ordering are finally given in Section 5.

Power estimation for scan vectors

In a CMOS circuit, the predominant fraction of the power is dissipated when the circuit elements switch from logic 1 to 0 or vice versa. Static power consumption, which is caused by leakage current, is usually negligible and therefore ignored. During test, the elements in the circuit switch when the primary inputs change value or when the scan cells change value. In this paper, we assume that the primary inputs are controllable directly from the tester and hence are held constant during shifting operations. All the switching activity in the circuit is due to transitions in the scan cells. Consider a CUT with five flip-flops and the test vector 10001 being scanned in. Let the scan flip-flops be initially set to 00000. After the first clock cycle when the first bit has been scanned in, the scan flip-flops will be in state 10000. The state of the first flip-flop has changed from 0 to 1. This change will cause gates in the CUT to switch. The number of circuit elements that switch depends on the actual circuit. Next, the second bit is scanned in and hence both the first and second flip-flops switch. This process continues until the complete test vector has been scanned in. The test vector is then applied to the CUT and the output response is captured back in the scan chain. As the next test vector is scanned in, the output response from the previous vector is scanned out to the tester. Transitions in the output response being scanned out will also cause switching activity. Hence, we can divide the power dissipation during scan testing into two parts. The scan-in power, which is due to transitions in the test vectors during scan-in, and the scan-out power, which is due to transitions in the output responses during scan-out. To estimate the scan-in power dissipated by a test vector (or the scan-out power dissipated by the corresponding output response), a weighted transitions metric has been introduced in [START_REF] Sankaralingam | Static Compaction Techniques to Control Scan Vector Power Dissipation[END_REF]. This weighted transitions metric models the fact that the scan-in power for a given test vector (or the scan-out power for the output response) depends not only on the number of transitions in it but also on their relative positions. For example, consider the test vector b1b2b3b4b5 = 10001 where the bit b5 is first loaded into the scan chain. This vector has two transitions: transition t1 between b5 and b4, and transition t2 between b2 and b1. When this vector is scanned in, transition t1 passes through the entire scan chain. This transition dissipates power at every flip-flop in the scan chain. On the other hand, transition t2 dissipates power only at the first scan flip-flop during scan-in and hence causes less switching activity in the scan chain than transition t1. This weighted transitions metric has been shown to be strongly correlated to the switching activity at the internal nodes of the CUT during scan-in and scan-out operations. It was shown experimentally in [START_REF] Sankaralingam | Static Compaction Techniques to Control Scan Vector Power Dissipation[END_REF] that scan vectors (test vectors or output responses) with higher weighted transitions dissipate more power in the CUT. This metric is therefore a good way to accurately estimate the power consumed during scan testing and hence avoid timeconsuming and size limited simulations. In the proposed ordering procedure, when we need to estimate the scan-in power and the scan-out power dissipated by a complete test sequence, we use the number of weighted transitions as a measure. More formally, the number of weighted transitions in a test vector or in an output response is given by:

Weighted_Transitions = Σ (Size_of_Scan_Chain - Position_of_Transition)
Note that according to one considers a test vector or an output response, the intrinsic value of Position_of_ Transition changes. For example, consider the scan vector b1b2b3b4b5 = 00001. In the case where this vector is a test vector, the transition between b5 and b4 is in position 1 (b5 is the first bit being scanned in). In the case where this vector is an output response, the position of the transition is 4 (b5 is the first bit being scanned out). The scan-in power can be estimated by summing the weighted transitions of all test vectors in the test sequence. The scan-out power can be estimated by summing the weighted transitions of all output responses. The test power dissipated by the test sequence can be therefore estimated by adding the values of the scan-in power and the scan-out power.

For the estimation of the test power to be correct, an additional term has to be considered. This term corresponds to the power due to the transition that occurs when the first bit of a test vector differs from the last bit of the previous output response. In this case, the transition propagates through the entire scan chain, and the weight assigned to it is equal to the size of the scan chain. and output response R2, the numbers of weighted transitions are 6 and 3 respectively. For test vector V3 and output response R3, these numbers are 1 and 5 respectively. In addition, a transition will propagate through the entire scan chain when the first bit of test vector V2 will be scanned in. This comes from the difference between the value of the first bit of V2 and the value of the last bit of output response R1. The weight associated to this transition is 4, which corresponds to the size of the scan chain. Assuming an initial state 0000 for the scan flip-flops, the total number of weighted transitions produced by the test sequence is (4+5+6+3+1+5) + (4) = 28. For comparison purpose in the ordering procedure, this number gives an estimation of the test power dissipated by a test sequence.

V1 = 0 1 1 0 V2 = 0 1 0 1 V3 = 0 1 1 1 ff1 ff2 ff3 R1 = 0 1 0 0 R2 = 1 0 0 0 R3 = 1

Scan cell ordering procedure

In this section, we show how test power can be minimized by appropriately ordering the scan cells of a given scan chain. The inputs to the proposed procedure are i) a given set of scan flip-flops and ii) a sequence of deterministic test vectors with the corresponding output responses. The output is an ordered scan chain with minimum test power.

To tackle this NP-hard problem efficiently, the heuristic procedure operates in two steps: the first one consists in determining the chaining of scan cells, the second one consists in identifying the input and output scan cells of the scan chain. These two steps are now described in details.

Determining the scan cell chaining

The first step of the scan cell ordering procedure consists in determining the order in which the scan cells have to be connected to minimize the occurrence of transitions in the scan chain during scan-in and scan-out operations. To this end, we first consider the set of scan vectors (test vectors and output responses) used during scan testing, and we assume an initial order for the scan flip-flops. More formally, we assume that flip-flop 1 corresponds to bit 1 of each scan vector, flip-flop 2 corresponds to bit 2 of each scan vector , …, flip-flop n corresponds to bit n of each scan vector. For example, consider the test sequence shown in Figure 2, which is composed of four test vectors (V1 to V4) and four output responses (R1 to R4). The scan chain has four flip-flops, hence scan vectors are four-bit long. The initial order of the scan cells in the scan chain is depicted on the figure. According to the above description, flip-flop 1, denoted as ff1, corresponds to bit 1 in each scan vector, flip-flop 2, denoted as ff2, corresponds to bit 2, and so on (Figure 2.a). represents the total number of bit differences between two flip-flops for the complete test sequence, and reflects the number of transitions that may be generated in the corresponding portion of the scan chain by connecting these two flip-flops together. From this weighted graph, the problem then amounts to finding an Hamiltonian cycle of minimum cost in the graph. The cost of a cycle is obtained by summing the weights on edges belonging to this cycle. This problem is equivalent to the well known traveling salesman problem, which is well known to be NP-hard (the number of possible solutions is (n-1)!/2 -n being the scan chain length) and for which different polynomial-time approximation algorithms can be used [START_REF] Johnson | The Traveling Salesman Problem : A Case Study in Local Optimization[END_REF]. Among these solutions, greedy algorithms represent a good tradeoff between computation time and efficiency of the computed solution. We therefore implemented an heuristic solution based on a greedy algorithm (with a complexity equal to O(n 2 )) to find the scan cell chaining that minimizes the occurrence of transitions in the scan chain during scan-in and scan-out operations. The greedy algorithm starts from an initial state which is always scan cell ff1 in our case (it is reported in [START_REF] Gondran | Graphes et Algorithmes[END_REF] that the choice of the initial state is not so crucial in a greedy algorithm considering the number of vertices in the graph to be sufficiently high). Next, the algorithm operates in such a way that, at each stage of decision, a subset of the scan cells is dealt with and considered as definitively assigned.

ff1 ff3 ff2 V1 = 1 0 0 1 R1 = 0 1 0 0 V2 = 0 1 0 1 R2 = 0 0 1 0 V3 = 1 1 1 1 R3 = 1 0 1 1 V4 = 1 0 1 0 R4 = 1 0 0 1 ff1 ff2 ff3
For the example test sequence of Figure 2.a, the solution found by the greedy algorithm is the following chaining of scan cells: ff1-ff4-ff2-ff3-ff1. This solution represents the best solution to minimize the occurrence of transitions in the scan chain during shifting operations.

Identifying input and output scan cells

From a given chaining of the scan flip-flops, the second step of the ordering procedure consists in defining both the input scan cell and the output scan cell of the scan chain.

Appropriately defining the input and output scan cells allows to minimize the propagation of transitions in the scan chain during shifting operations. Let us consider again a scan chain composed of n scan cells. The chaining obtained from step 1 of the ordering procedure can now be represented by a simple oriented cyclic graph in which each vertex represents a scan cell and each edge represents the connection between two scan cells. For illustrative purpose, the oriented cyclic graph for the example test sequence considered in Figure 2.a is depicted in Figure 3. At this stage of the procedure, defining the input and output scan cells of the scan chain can be done by cutting the graph on a selected edge. The number of possible solutions to cut the graph is obviously equal to n, which also corresponds to the number of different scan chains that can be formed at this stage of the ordering procedure. Each cutting solution differs in the number of transitions that can be generated and propagated in the corresponding scan chain.

In order to find the scan chain that leads to the lowest test power, and hence identify both an input scan cell and an output scan cell, we evaluate the n possible solutions in terms of weighted transitions generated during scan-in and scan-out operations, and then select the solution having the lowest number of weighted transitions. This solution represents the optimal solution to the scan cell ordering problem, hence allowing to definitively identify the less power consuming scan chain.

To this end, we use the weighted transitions metric discussed in Section 3, and evaluate the n possible sets of scan vectors that each corresponds to a given cutting solution. For example, considering the test sequence given in Figure 2.a, the four possible sets of scan vectors which are evaluated to determine the less power consuming scan chain are shown in Figure 4. Note that in each set, the first, third, fifth and seventh vectors are the test vectors. The remaining scan vectors are the output responses.

1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 0 0 1 1 1 0 0 f f 1 f f 4 f f 2 f f 3 W T t o ta l = 1 9 + 4 = 2 3 0 1 1 0 0 0 0 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 0 1 1 0 0 0 1 1 0 f f 3 f f 1 f f 4 f f 2 W T t o t a l = 1 7 + 4 + 4 = 2 5 0 0 1 1 1 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 1 1 f f 2 f f 3 f f 1 f f 4 W T t o ta l = 2 3 + 4 = 2 7 1 0 0 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 1 1 0 1 1 0 0 1 1 1 0 0 1 f f 4 f f 2 f f 3 f f 1 W T t o t a l = 2 5 + 4 = 2 9 V 1 R 1 V 2 R 2 V 3 R 3 V 4 R 4 V 1 R 1 V 2 R 2 V 3 R 3 V 4 R 4

Figure 4: Identification of input and output scan cells

For each set of scan vectors in Figure 4, i.e. for each possible scan chain, the total number of weighted transitions has been evaluated and is reported at the bottom of the figure. This number represents the number of weighted transitions in test vectors and output responses plus the number of weighted transitions due to opposite values between the first bit of a test vector and the last bit of the previous output response (cf. Section 3). In this example, the lowest value of WT total is 23 and is obtained from the first possible scan chain. Hence, this scan chain (ff1-ff4-ff2-ff3) is the less power consuming scan chain, and ff1 is identified as the input scan cell and ff3 as the output scan cell.

Compared with existing low power scan techniques, our solution offers numerous advantages. The proposed approach works for any conventional scan design -no extra DfT logic is required. The fault coverage and the IC test time are let unchanged by the proposed approach.

There is almost no penalty on the circuit performance. The area overhead, which is due to routing area, may be kept acceptably low when managed by an efficient layout synthesis program. The proposed approach is very easy to use in a classical DfT flow and has therefore a very low impact on the system design time.

Experimental results

The benchmarking process described here was performed on circuits of the ISCAS'89 [START_REF] Brglez | Combinational Profiles of Sequential Benchmark Circuits[END_REF] 1. All experiments are based on deterministic testing from the ATPG tool "TestGen" of Synopsys [START_REF]Tg 3.0.2 User Guide[END_REF]. The missing faults in the fault coverage (FC) column are the redundant or aborted faults. The first part of Table 1 shows the number of scan cells and the number of gates for each benchmark circuit. The primary inputs and primary outputs were not included in the scan chain, but were assumed to be held constant during scan-in and scan-out operations. In the second part, we report the test length of each test sequence and the corresponding fault coverage. Note that these results are not influenced by the scan cell ordering procedure described in this paper.

The results in terms of power savings achieved by the proposed ordering technique are summarized in Table 2.

For each circuit, we have reported the peak power and the average power obtained first from a random ordering (first part of Table 2) and next with the proposed ordering technique. In all cases, the scan flip-flops have been initialized with random logic values. Results concerning random ordering are an average over ten runs. Peak power and average power are expressed in milliWatts. The last part in Table 2 shows the reduction in peak power and average power dissipation expressed in percentages. These results on benchmark circuits show that peak power reduction of up to 18% and average power reduction of up to 34% can be achieved with the proposed ordering technique. An important comment on these results is that the peak power reduction appears to be irregular and sometimes poor. This is due to the fact that in some cases, the highest current in the CUT appears during application of the test pattern and capture of the corresponding response. As this operation is unchanged with scan cell re-ordering, the reduction in peak power appears to be low. In fact, it is higher than it appears. The reason is that an elevated current consumed in a design can cause damage to the circuit if it occurs during more than one clock cycle. As the switching activity is reduced in the clock cycles preceding and following the capture clock cycle, the peak power is in fact much lower than the peak power without scan cell re-ordering.

Branch and Bound

Proposed

Table 4: Comparison between a Branch and Bound technique and the proposed ordering

For the reader's interest, we report in Table 4 comparative results between an exact method for scan cell ordering, e.g. the Branch and Bound technique [START_REF] Held | The Traveling Salesman Problem and Minimum Spanning Trees[END_REF], and the proposed ordering procedure. Results are given only for small circuits because applying the Branch and Bound technique to larger circuits is intractable from a CPU time point of view. Table 4 reports the average and peak power consumed in the CUT during scan testing first from a scan cell ordering produced by a Branch and Bound technique, and next from a scan cell ordering produced by the proposed technique. The two last columns in Table 4 show the ratio between the two techniques. As can be seen, the proposed heuristic procedure based on a greedy algorithm provides almost the same results than those achieved by using an exact method such as the Branch and Bound technique. Considering real-size circuits composed of thousands of gates and flip-flops, for which the scan cell ordering problem is intractable with an exact method, these results highlight the efficiency of the proposed ordering technique.

Discussion

In an era of sub-micron technology, routing is becoming a dominant factor in area, timing, and power consumption. Therefore, considering routing impacts when we are looking for a solution to solve the scan cell ordering problem is needed. To incur minimum area overhead and power consumption due to routing, flip-flops should be selected so that chaining of them can lead to a minimum between distance. This is usually the criterion which is employed in any classical scan insertion technique (in addition to the delay criterion). In order to include the routing criterion in our proposed ordering procedure, we need to modify the two metrics used in each step of the procedure: the number of bit differences between each pair of scan flip-flops, and the number of weighted transitions of scan vectors. Modifying these metrics may consists for example in adding a weighting parameter that represents both (or either) area and power consumption overhead. Of course, this concerns only situations in which flip-flops placement is known a priori. In the case where flip-flops placement is not known a priori, the solution is to consider a chip layout program that can accept a fixed scan cells order (produced by our procedure) and in which the closest neighbor criterion is relaxed (having in mind in most situations, the main criterion for scan flip-flops placement relies on timing). Considering routing in our problem will be performed in a future work.

Figure 1 :

 1 Figure 1: An example set of test vectors and output responsesFor example, consider the test sequence shown in Figure1, which is composed of three test vectors and the corresponding three output responses. The scan chain has four flip-flops, hence scan vectors are four-bit long. The number of weighted transitions for test vector V1 is (4-3) + (4-1) = 4, and that of output response R1 is (4-1) + (4-2) = 5. Note that the intrinsic value of the position is different for test vectors and output responses. For test vector V2 and output response R2, the numbers of weighted transitions are 6 and 3 respectively. For test vector V3 and output response R3, these numbers are 1 and 5 respectively. In addition, a transition will propagate through the entire scan chain when the first bit of test vector V2 will be scanned in. This comes from the difference between the value of the first bit of V2 and the value of the last bit of output response R1. The weight associated to this transition is 4, which corresponds to the size of the scan chain. Assuming an initial state 0000 for the scan flip-flops, the total number of weighted transitions produced by the test sequence is (4+5+6+3+1+5) + (4) = 28. For comparison purpose in the ordering procedure, this number gives an estimation of the test power dissipated by a test sequence.

Figure 2 :

 2 Figure 2: The example test sequence and the weighted graph Next, we calculate the total number of bit differences between each pair of scan flip-flops, which represents the number of transitions that may be generated in the corresponding portion of the scan chain by connecting these two flip-flops together. For the example in Figure 2, calculating the total number of bit differences between each pair of flip-flops (on the complete sequence of test vectors and output responses) provides the following results: d(ff1,ff2) = 6, d(ff1,ff3) = 4, d(ff1,ff4) = 2, d(ff2,ff3) = 5, d(ff2,ff4) = 4, d(ff3,ff4) = 5. From these values of the bit differences between flip-flops, it is then possible to construct a complete undirected graph in which each vertex represents a flip-flop and each edge represents a possible connection between two flip-flops (Figure 2.b). The weight on each edge of the graph

Figure 3 :

 3 Figure 3: The oriented cyclic graph

Table 1 : Main characteristics of the experimented circuits First

 1 , structural characteristics and test parameters of the experimented circuits are reported in Table

				benchmark suites. Power
	consumption in each circuit was estimated by using
	PowerMill, a dynamic simulator provided by Synopsys
	[23], assuming a clock frequency equal to 200 MHz and a
	power supply voltage of 2.5 V. Experiments performed on
	each circuit have been done with technology parameters
	extracted from a 0.25µm digital CMOS standard cell
	library. The goal of the experiments we performed has
	been to measure the savings in test power obtained from
	the proposed scan cell ordering procedure.
	Circuit # scan cells	# gates Test length	FC
	s298	14	119	33	100 %
	s344	15	160	27	100 %
	s420	16	196	59	100 %
	s510	6	211	68	100 %
	s641	19	321	40	100 %
	s713	19	351	48	95.48 %
	s1423	74	749	44	98.99 %
	s5378	179	2225	145	99.05 %
	s9234	228	4678.5	249	93.99 %
	s13207	669	6395.5	354	98.99 %
	s15850	597	7987	279	97.84 %
	s35932	1728	16726.5	112	100 %
	s38417	1636	20446.5	325	99.70 %

  Concerning computing CPU time, ordering solutions are always obtained in less than one hour.

	Circuit	Ave. Random Ordering		Proposed Ordering
		peak	average	peak	average	peak	average
		[mW]	[mW]	[mW]	[mW]	reduction	reduction
	s641	30.31	1.01	26.55	0.71	12.4 %	29.6 %
	s713	33.90	1.09	27.53	0.72	18.8 %	34.1 %
	s1423	79.91	3.65	677.25	2.76	15.3 %	24.2 %
	s5378	250.5	12.69	250.4	8.97	0.5%	29.3%
	s9234	395.15	29.45 353.67	22.65	10.5 %	23.1 %
	s13207 651.39	48.97	609.9	39.55	6.4 %	19.2 %
	s15850 595.63	57.50	555.8	48.76	6.7 %	15.2 %
	s35932 1662.02 108.32 1610.5	96.03	3.1%	11.3%
	s38417 1899.56 135.01 1746.5 126.03	7.9%	16.5%

Table 2 : Power savings in the CUT

 2

Acknowledgment

The authors would like to thank Professor B. Rouzeyre from the LIRMM for his help on the way to implement the Branch and Bound technique used in our comparisons.


T Devillers
email: thibaut.devillers@cea.fr

M Jamet

A Barski

V Poydenot

P Bayle-Guillemaud

E Bellet-Amalric

S Cherifi

J Cibert

Structure and magnetism of self-organized Ge 1-x Mn x nano-columns

Keywords: numbers: 75.50.Pp, 75.75.+a, 61.46.-w

We report on the structural and magnetic properties of thin Ge1-xMnx films grown by molecular beam epitaxy (MBE) on Ge(001) substrates at temperatures (Tg) ranging from 80 • C to 200 • C, with average Mn contents between 1 % and 11 %. Their crystalline structure, morphology and composition have been investigated by transmission electron microscopy (TEM), electron energy loss spectroscopy and x-ray diffraction. In the whole range of growth temperatures and Mn concentrations, we observed the formation of manganese rich nanostructures embedded in a nearly pure germanium matrix. Growth temperature mostly determines the structural properties of Mn-rich nanostructures. For low growth temperatures (below 120 • C), we evidenced a two-dimensional spinodal decomposition resulting in the formation of vertical one-dimensional nanostructures (nanocolumns). Moreover we show in this paper the influence of growth parameters (Tg and Mn content) on this decomposition i.e. on nanocolumns size and density. For temperatures higher than 180 • C, we observed the formation of Ge3Mn5 clusters. For intermediate growth temperatures nanocolumns and nanoclusters coexist. Combining high resolution TEM and superconducting quantum interference device magnetometry, we could evidence at least four different magnetic phases in Ge1-xMnx films: (i) paramagnetic diluted Mn atoms in the germanium matrix, (ii) superparamagnetic and ferromagnetic low-TC nanocolumns (120 K ≤ TC ≤ 170 K), (iii) high-TC nanocolumns (TC ≥ 400 K) and (iv) Ge3Mn5 clusters.

I. INTRODUCTION

In the past few years, the synthesis of ferromagnetic semiconductors has become a major challenge for spintronics. Actually, growing a magnetic and semiconducting material could lead to promising advances like spin injection into non magnetic semiconductors, or electrical manipulation of carrier induced magnetism in magnetic semiconductors [1,2]. Up to now, major efforts have focused on diluted magnetic semiconductors (DMS) in which the host semiconducting matrix is randomly substituted by transition metal (TM) ions such as Mn, Cr, Ni, Fe or Co [3]. However Curie temperatures (T C ) in DMS remain rather low and TM concentrations must be drastically raised in order to increase T C up to room temperature. That usually leads to phase separation and the formation of secondary phases. It was recently shown that phase separation induced by spinodal decomposition could lead to a significant increase of T C [4,5]. For semiconductors showing T C higher than room temperature one can foresee the fabrication of nanodevices such as memory nanodots, or nanochannels for spin injection.

Therefore, the precise control of inhomogeneities appears as a new challenge which may open a way to industrial applications of ferromagnetism in semiconductors.

The increasing interest in group-IV magnetic semiconductors can also be explained by their potential compatibility with the existing silicon technology. In 2002, carrier mediated ferromagnetism was reported in MBE grown Ge 0.94 Mn 0.06 films by Park et al. [6]. The maximum critical temperature was 116 K. Recently many publications indicate a significant increase of T C in Ge 1-x Mn x material depending on growth conditions [7,8,9]. Cho et al. reported a Curie temperature as high as 285 K [10]. Taking into account the strong tendency of Mn ions to form intermetallic compounds in germanium, a detailed investigation of the nanoscale structure is required. Up to now, only a few studies have focused on the nanoscale composition in Ge 1-x Mn x films. Local chemical inhomogeneities have been recently reported by Kang et al. [11] who evidenced a micrometer scale segregation of manganese in large Mn rich stripes. Ge 3 Mn 5 as well as Ge 8 Mn 11 clusters embedded in a germanium matrix have been reported by many authors. However, Curie temperatures never exceed 300 K [12,13,14,15]. Ge 3 Mn 5 clusters exhibit a Curie temperature of 296 K [START_REF] Massalski | Binary Alloy Phase Diagrams[END_REF]. This phase frequently observed in Ge 1-x Mn x films is the most stable (Ge,Mn) alloy. The other stable compound Ge 8 Mn 11 has also been observed in nanocrystal-lites surrounded with pure germanium [START_REF] Park | [END_REF]. Ge 8 Mn 11 and Ge 3 Mn 5 phases are ferromagnetic but their metallic character considerably complicates their potential use as spin injectors. Recently, some new Mn-rich nanostructures have been evidenced in Ge 1-x Mn x layers. Sugahara et al. [18] reported the formation of high Mn content (between 10 % and 20 % of Mn) amorphous Ge 1-x Mn x precipitates in a Mn-free germanium matrix. Mn-rich coherent cubic clusters were observed by Ahlers et al. [15] which exhibit a Curie temperatures below 200 K. Finally, high-T C (> 400 K) Mn-rich nanocolumns have been evidenced [19] which could lead to silicon compatible room temperature operational devices. In the present paper, we investigate the structural and magnetic properties of Ge 1-x Mn x thin films for low growth temperatures (< 200 • C) and low Mn concentrations (between 1 % and 11 %). By combining TEM, x-Ray diffraction and SQUID magnetometry, we could identify different magnetic phases. We show that depending on growth conditions, we obtain either Mn-rich nanocolumns or Ge 3 Mn 5 clusters embedded in a germanium matrix. We discuss the structural and magnetic properties of these nanostructures as a function of manganese concentration and growth temperature. We also discuss the magnetic anisotropy of nanocolumns and Ge 3 Mn 5 clusters.

II. SAMPLE GROWTH

Growth was performed using solid sources molecular beam epitaxy (MBE) by co-depositing Ge and Mn evaporated from standard Knudsen effusion cells. Deposition rate was low (≈ 0.2 Å.s -1 ). Germanium substrates were epi-ready Ge(001) wafers with a residual n-type doping and resistivity of 10 15 cm -3 and 5 Ω.cm respectively. After thermal desorption of the surface oxide, a 40 nm thick Ge buffer layer was grown at 250 • C, resulting in a 2 × 1 surface reconstruction as observed by reflection high energy electron diffraction (RHEED) (see Fig. 1a). Next, 80 nm thick Ge 1-x Mn x films were subsequently grown at low substrate temperature (from 80 • C to 200 • C). Mn content has been determined by x-ray fluorescence measurements performed on thick samples (≈ 1 µm thick) and complementary Rutherford Back Scattering (RBS) on thin Ge 1-x Mn x films grown on silicon. Mn concentrations range from 1 % to 11% Mn.

For Ge 1-x Mn x films grown at substrate temperatures below 180 • C, after the first monolayer (ML) deposition, the 2 × 1 surface reconstruction almost totally disappears. After depositing few MLs, a slightly diffuse 1 × 1 streaky RHEED pattern and a very weak 2 × 1 reconstruction (Fig. 1b) indicate a predominantly twodimensional growth. For growth temperatures above 180 • C additional spots appear in the RHEED pattern during the Ge 1-x Mn x growth (Fig. 1c). These spots may correspond to the formation of very small secondary phase crystallites. The nature of these crystallites will be discussed below.

Transmission electron microscopy (TEM) observations were performed using a JEOL 4000EX microscope with an acceleration voltage of 400 kV. Energy filtered transmission electron microscopy (EFTEM) was done using a JEOL 3010 microscope equipped with a Gatan Image Filter . Sample preparation was carried out by standard mechanical polishing and argon ion milling for cross-section investigations and plane views were prepared by wet etching with H 3 PO 4 -H 2 O 2 solution [20].

III. STRUCTURAL PROPERTIES

In samples grown at 130 • C and containing 6 % Mn, we can observe vertical elongated nanostructures i.e. nanocolumns as shown in Fig. 2a. Nanocolumns extend through the whole Ge 1-x Mn x film thickness. From the high resolution TEM image shown in Fig. 2b, we deduce their average diameter around 3 nm. Moreover in Fig. 2b, the interface between the Ge buffer layer and the Ge 1-x Mn x film is flat and no defect propagates from the interface into the film. The Ge 1-x Mn x film is a perfect single crystal in epitaxial relationship with the substrate. In Fig. 2c is shown a plane view micrograph of the same sample confirming the presence of nanocolumns in the film. From this image, we can deduce the size and density of nanocolumns. The nanocolumns density is 13000 µm -2 with a mean diameter of 3 nm which is coherent with cross-section measurements. In order to estimate the chemical composition of these nanocolumns, we further performed chemical mapping using EFTEM. In Fig. 2d we show a cross sectional Mn chemical map of the Ge 1-x Mn x film. This map shows that the formation of nanocolumns is a consequence of Mn segregation. Nanocolumns are Mn rich and the surrounding matrix is Mn poor. However, it is impossible to deduce the Mn concentration in Ge 1-x Mn x nanocolumns from this cross section. Indeed, in cross section observations, the columns diameter is much smaller than the probed film thickness and the signal comes from the superposititon of the Ge matrix and Mn-rich nanocolumns. In order to quantify Mn concentration inside the nanocolumns and inside the Ge matrix, EELS measurements (not shown here) have been performed in a plane view geometry The background was carefully substracted from pre-edge images. Bright areas correspond to Mn-rich regions. [19]. These observations revealed that the matrix Mn content is below 1 % (detection limit of our instrument). Measuring the surface occupied by the matrix and the nanocolumns in plane view TEM images, and considering the average Mn concentration in the sample (6 %), we can estimate the Mn concentration in the nanocolumns. The Mn concentration measured by EELS being between 0% and 1%, we can conclude that the Mn content in the nanocolumns is between 30 % and 38 %. For samples grown between 80 • C and 150 • C cross section and plane view TEM observations reveal the presence of Mn rich nanocolumns surrounded with a Mn poor Ge matrix. In order to investigate the influence of Mn concentration on the structural properties of Ge 1-x Mn x films, ten samples have been grown at 100 • C and at 150 • C with Mn concentrations of 1.3 %, 2.3 %, 4 %, 7 % and 11.3 %. Their structural properties have been investigated by plane view TEM observations.

For samples grown at 100 • C with Mn concentrations below 5 % the nanocolumns mean diameter is 1.8±0.2 nm. The evolution of columns density as a fonction of Mn concentration is reported in figure 3b. By increasing the Mn concentration from 1.3 % to 4 % we observe a significant increase of the columns density from 13000 to 30000 µm -2 . For Mn concentrations higher than 5 % the density seems to reach a plateau corresponding to 35000 µm -2 and their diameter slightly increases from 1.8 nm at 4 % to 2.8 nm at 11.3 %. By plotting the volume fraction occupied by the columns in the film as a function of Mn concentration, we observe a linear dependence for Mn contents below 5 %. The non-linear behavior above 5 % may indicate that the mechanism of Mn incorporation is different in this concentration range, leading to an increase of Mn concentration in the columns or in the matrix. For samples grown at 100 • C, nanocolumns are always fully coherent with the surrounding matrix (Fig. 4a).

Increasing the Mn content in the samples grown at 150 • C from 1.3 % to 11.3 % leads to a decrease of the columns density (fig 3b). Moreover, their average diameter increases significantly and size distributions become very broad (see Fig. 3a). For the highest Mn concentration (11.3 %) we observe the coexistence of very small columns with a diameter of 2.5 nm and very large columns with a diameter of 9 nm. In samples grown at 150 • C containing 11.3 % of Mn, the crystalline structure of nanocolumns is also highly modified. In plane view TEM micrographs, one can see columns exhibiting several different crystalline structures. We still observe some columns which are fully coherent with the Ge matrix like in the samples grown at lower temperature. Nevertheless, observations performed on these samples grown at 150 • C and with 11.3% Mn reveal some uniaxially [19] or fully relaxed columns exhibiting a misfit of 4 % between the matrix and the columns and leading to misfit dislocations at the interface between the column and the matrix (see fig. 4b). Thus we can conclude that coherent columns are probably in strong compression and the surrounding matrix in tension. On the same samples (T g =150 • C, 11.3% Mn), we also observe a large number of highly disordered nanocolumns leading to an amorphous like TEM contrast(fig. 4c). In conclusion, we have evidenced a complex mechanism of Mn incorporation in Mn doped Ge films grown at low temperature. In particular Mn incorporation is highly inhomogeneous. For very low growth temperatures (below 120 • C) the diffusion of Mn atoms leads to the formation of Mn rich, vertical nanocolumns. Their density mostly depends on Mn concentration and their mean diameter is about 2 nm. These results can be compared with the theoretical predictions of Fukushima et al. [5]: they proposed a model of spinodal decomposition in (Ga,Mn)N and (Zn,Cr)Te based on layer by layer growth conditions and a strong pair attraction between Mn atoms which leads to the formation of nanocolumns. This model may also properly describe the formation of Mn rich nanocolumns in our samples. Layer by layer growth conditions can be deduced from RHEED pattern evolution during growth. For all the samples grown at low temperature, RHEED observations clearly indicate two-dimensional growth. Moreover, Ge/Ge 1-x Mn x /Ge heterostructures have been grown and observed by TEM (see Fig. 5). Ge 1-x Mn x /Ge (as well as Ge/Ge 1-x Mn x ) interfaces are very flat and sharp thus confirming a twodimensional, layer by layer growth mode. Therefore we can assume that the formation of Mn rich nanocolumns is a consequence of 2D-spinodal decomposition.

For growth temperatures higher than 160 • C, cross sec- tion TEM and EFTEM observations (not shown here) reveal the coexistence of two Mn-rich phases: nanocolumns and Ge 3 Mn 5 nanoclusters embedded in the germanium matrix. A typical high resolution TEM image is shown in figure 6. Ge 3 Mn 5 clusters are not visible in RHEED patterns for temperatures below 180 • C. To investigate the nature of these clusters, we performed x-ray diffraction in θ -2θ mode. Diffraction scans were acquired on a high resolution diffractometer using the copper K α radiation and on the GMT station of the BM32 beamline at the European Synchrotron Radiation Facility (ESRF). Three samples grown at different temperatures and/or annealed at high temperature were investigated. The two first samples are Ge 1-x Mn x films grown at 130 • C and 170 • C respectively. The third one has been grown at 130 • C and post-growth annealed at 650 • C. By analysing x-ray diffraction spectra, we can evidence two different crystalline structures. For the sample grown at 130 • C, the θ -2θ scan only reveals the (004) Bragg peak of the germanium crystal, confirming the good epitaxial relationship between the layer and the substrate, and the absence of secondary phases in the film in spite of a high dynamics of the order of 10 7 . For both samples grown at 170 • C and annealed at 650 • C, θ-2θ spectra are identical. In addition to the (004) peak of germanium, we observe three additional weak peaks. The first one corresponds to the (002) germanium forbidden peak which probably comes from a small distortion of the germanium crystal, and the two other peaks are respectively attributed to the (002) and (004) Bragg peaks of a secondary phase. The c lattice parameter of Ge 3 Mn 5 hexagonal crystal is 5.053 Å [21] which is in very good agreement with the values obtained from diffraction data for both (002) and (004) lines assuming that the c axis of Ge 3 Mn 5 is along the [001] direction of the Ge substrate. In summary, in a wide range of growth temperatures and Mn concentrations, we have evidenced a twodimensional spinodal decomposition leading to the formation of Mn-rich nanocolumns in Ge 1-x Mn x films. This decomposition is probably the consequence of: (i) a strong pair attraction between Mn atoms, (ii) a strong surface diffusion of Mn atoms in germanium even at low growth temperatures and (iii) layer by layer growth conditions. We have also investigated the influence of growth parameters on the spinodal decomposition: at low growth temperatures (100 • C), increasing the Mn content leads to higher columns densities while at higher growth temperatures (150 • C), the columns density remains nearly constant whereas their size increases drastically. By plotting the nanocolumns density as a function of Mn content, we have shown that the mechanism of Mn incorporation in Ge changes above 5 % of Mn. Finally, using TEM observations and x-ray diffraction, we have shown that Ge 3 Mn 5 nanoclusters start to form at growth temperatures higher than 160 • C.

IV. MAGNETIC PROPERTIES

We have thoroughly investigated the magnetic properties of thin Ge 1-x Mn x films for different growth temperatures and Mn concentrations. In this section, we focus on Mn concentrations between 2 % and 11 %. We could clearly identify four different magnetic phases in Ge 1-x Mn x films : diluted Mn atoms in the germanium matrix, low T C nanocolumns (T C ≤ 170 K), high T C nanocolumns (T C ≥ 400 K) and Ge 3 Mn 5 clusters (T C ≈ 300 K). The relative weight of each phase clearly depends on the growth temperature and to a lesser extend on Mn concentration. For low growth temperature (< 120 • C), we show that nanocolumns are actually made of four uncorrelated superparamagnetic nanostructures. Increasing T g above 120 • C, we first obtain continuous columns exhibiting low T C (< 170 K) and high T C (> 400 K) for T g ≈130 • C. The larger columns become ferromagnetic i.e. T B > T C . Meanwhile Ge 3 Mn 5 clusters start to form. Finally for higher T g , the magnetic contribution from Ge 3 Mn 5 clusters keeps increasing while the nanocolumns signal progressively disappears. The ZFC peak at low temperature (≤150 K) can be attributed to the superparamagnetic nanocolumns. This peak widens and shifts towards high blocking temperatures when increasing growth temperature. The second peak above 150 K in the ZFC curve which increases with increasing growth temperature is attributed to superparamagnetic Ge3Mn5 clusters. The increasing ZFC-FC irreversibility at ≈ 300 K is due to the increasing contribution from large ferromagnetic Ge3Mn5 clusters. The nanocolumns signal completely vanishes after annealing at 650 • C for 15 minutes.

In Fig. 7a, the saturation magnetization at 2 Tesla in µ B /Mn of Ge 1-x Mn x films with 7 % of Mn is plotted as a function of temperature for different growth temperatures ranging from T g =90 • C up to 160 • C. The inset shows the temperature dependence of the magnetization at 2 Tesla after annealing at 650 • C during 15 minutes. Figure 7b displays the corresponding Zero Field Cooled -Field Cooled (ZFC-FC) curves recorded at 0.015 Tesla. In the ZFC-FC procedure, the sample is first cooled down to 5 K in zero magnetic field and the susceptibility is subsequently recorded at 0.015 Tesla while increasing the temperature up to 400 K (ZFC curve). Then, the susceptibility is recorded under the same magnetic field while decreasing the temperature down to 5 K (FC curve).

Three different regimes can be clearly distinguished. For T g ≤120 • C, the temperature dependence of the saturation magnetization remains nearly the same while increasing growth temperature. The overall magnetic signal vanishing above 200 K is attributed to the nanocolumns whereas the increasing signal below 50 K originates from diluted Mn atoms in the surrounding matrix. The Mn concentration dependence of the saturation magnetization is displayed in figure 8. For the lowest Mn concentration (4 %), the contribution from diluted Mn atoms is very high and drops sharply for higher Mn concentrations (7 %, 9 % and 11.3 %). Therefore the fraction of Mn atoms in the diluted matrix decreases with Mn concentration probably because Mn atoms are more and more incorporated in the nanocolumns. In parallel, the Curie temperature of nanocolumns increases with the Mn concentration reaching 170 K for 11.3 % of Mn. This behavior may be related to different Mn compositions and to the increasing diameter of nanocolumns (from 1.8 nm to 2.8 nm) as discussed in section III. ZFC-FC measurements show that the nanocolumns are superparamagnetic. The magnetic signal from the diluted Mn atoms in the matrix is too weak to be detected in susceptibility measurements at low temperature. In samples containing 4 % of Mn, ZFC and FC curves superimpose down to low temperatures. As we do not observe hysteresis loops at low temperature, we believe that at this Mn concentration nanocolumns are superparamagnetic in the whole temperature range and the blocking temperature cannot be measured. For higher Mn contents, the ZFC curve exhibits a very narrow peak with a maximum at the blocking temperature of 15 K whatever the Mn concentration and growth temperature (see Fig. 7b). Therefore the anisotropy barrier distribution is narrow and assuming that nanocolumns have the same magnetic anisotropy, this is a consequence of the very narrow size distribution of the nanocolumns as observed by TEM. To probe the anisotropy barrier distribution, we have performed ZFC-FC measurements but instead of warming the sample up to 400 K, we stopped at a lower temperature T 0 .

FIG. 9: Schematic drawing of the anisotropy barrier distribution n(EB) of superparamagnetic nanostructures. If magnetic anisotropy does not depend on the particle size, this distribution exactly reflects their magnetic size distribution. In this drawing the blocking temperature (TB) corresponds to the distribution maximum. At a given temperature T0 such that 25kB T0 falls into the anisotropy barrier distribution, the largest nanostructures with an anisotropy energy larger than 25kBT0 are blocked whereas the others are superparamagnetic.

If this temperature falls into the anisotropy barrier distribution as depicted in Fig. 9, the FC curve deviates from the ZFC curve. Indeed the smallest nanostructures have become superparamagnetic at T 0 and when decreasing again the temperature, their magnetization freezes along a direction close to the magnetic field and the FC susceptibility is higher than the ZFC susceptibility. Therefore any irreversibility in this procedure points at the presence of superparamagnetic nanostructures. The results are given in Fig. 10a. ZFC and FC curves clearly superimpose up to T 0 =250 K thus the nanocolumns are superparamagnetic up to their Curie temperature and no Ge 3 Mn 5 clusters could be detected. Moreover for low T 0 values, a peak appears at low temperature in FC curves which evidences strong antiferromagnetic interactions between the nanocolumns [22].

In order to derive the magnetic size and anisotropy of the Mn-rich nanocolumns embedded in the Ge matrix, we have fitted the inverse normalized in-plane (resp. out-of-plane) susceptibility: χ -1 (resp. χ -1 ⊥ ). The corresponding experimental ZFC-FC curves are reported in Fig. 10b. Since susceptibility measurements are performed at low field (0.015 T), the matrix magnetic signal remains negligible. In order to normalize susceptibility data, we need to divide the magnetic moment by the saturated magnetic moment recorded at 5 T. However the matrix magnetic signal becomes very strong at 5 T and low temperature so that we need to subtract it from the saturated magnetic moment using a simple Curie function. From Fig. 10b, we can conclude that nanocolumns are isotropic. Therefore to fit experimental data we use the following expression well suited for isotropic systems or cubic anisotropy:

χ -1 = χ -1 ⊥ ≈ 3k B T /M (T ) + µ 0 H ef f (T ). k B is the Boltz- mann constant, M = M s v
is the magnetic moment of a single-domain nanostructure (macrospin approximation) where M s is its magnetization and v its volume. The in-plane magnetic field is applied along [110] or [-110] crystal axes. Since the nanostructures Curie temperature does not exceed 170 K, the temperature dependence of the saturation magnetization is also accounted for by writting M (T ). Antiferromagnetic interactions between nanostructures are also considered by adding an effective field estimated in the mean field approximation [23]: µ 0 H ef f (T ). The only fitting parameters are the maximum magnetic moment (i.e. at low temperature) per nanostructure: M (in Bohr magnetons µ B ) and the maximum interaction field (i.e. at low temperature): µ 0 H ef f . In Fig. 11, the best fits lead to M ≈1250 µ B and µ 0 H ef f ≈102 mT for in-plane susceptibility and M ≈1600 µ B and µ 0 H ef f ≈98 mT for out-of-plane susceptibility. It gives an average magnetic moment of 1425 µ B per column and an effective interaction field of 100 mT. Using this magnetic moment and its temperature dependence, magnetization curves could be fitted using a Langevin function and M (H/T ) curves superimpose for T <100 K. However, from the saturated magnetic moment of the columns and their density (35000 µm -2 ), we find almost 6000 µ B per column. Therefore, for low growth temperatures, we need to assume that nanocolumns are actually made of almost four independent elongated magnetic nanostructures. The effective field for antiferromagnetic interactions between nanostructures estimated from the susceptibility fits is at least one order of magnitude larger than what is expected from pure magnetostatic coupling. This difference may be due to either an additional antiferromagnetic coupling through the matrix which origin remains unexplained or to the mean field approximation which is no more valid in this strong coupling regime. As for magnetic anisotropy, the nanostructures behave as isotropic magnetic systems or exhibit a cubic magnetic anisotropy. First we can confirm that nanostructures are not amorphous otherwise shape anisotropy would dominate leading to out-ofplane anisotropy. We can also rule out a random distribution of magnetic easy axes since the nanostructures are clearly crystallized in the diamond structure and would exhibit at least a cubic anisotropy (except if the random distribution of Mn atoms within the nanostructures can yield random easy axes). Since the nanostructures are in strong in-plane compression (their lattice parameter is larger than the matrix one), the cubic symmetry of the diamond structure is broken and magnetic cubic anisotropy is thus unlikely. We rather believe that out-ofplane shape anisotropy is nearly compensated by in-plane magnetoelastic anisotropy due to compression leading to a pseudo cubic anisotropy. From the blocking temperature (15 K) and the magnetic volume of the nanostructures , we can derive their magnetic anisotropy constant using Kv = 25k B T B : K≈10 kJ.m -3 which is of the same order of magnitude as shape anisotropy.

For growth temperatures T g ≥120 • C and Mn concentrations ≥ 7 %, samples exhibit a magnetic signal above 200 K corresponding to Ge 3 Mn 5 clusters (see Fig. 7a). As we can see, SQUID measurements are much more sensitive to the presence of Ge 3 Mn 5 clusters, even at low concentration, than TEM and x-ray diffraction used in section III. We also observe a sharp transition in the ZFC curve (see Fig. 7b, Fig. 12a and12b): the peak becomes very large and is shifted towards high block- ing temperatures (the signal is maximum at T =23 K). This can be easily understood as a magnetic percolation of the four independent nanostructures obtained at low growth temperatures into a single magnetic nanocolumn. Therefore the magnetic volume increases sharply as well as blocking temperatures. At the same time, the size distribution widens as observed in TEM. In Fig. 12a, we have performed ZFC-FC measurements at different T 0 temperatures. The ZFC-FC irreversibility is observed up to the Curie temperature of ≈120 K meaning that a fraction of nanocolumns is ferromagnetic (i.e. T B ≥ T C ). In Fig. 12b, in-plane and out-of-plane ZFC curves nearly superimpose for T ≤150 K due to the isotropic magnetic behavior of the nanocolumns: in-plane magnetoelastic anisotropy is still compensating out-of-plane shape anisotropy. Moreover the magnetic signal above 150 K corresponding to Ge 3 Mn 5 clusters that start to form in this growth temperature range is strongly anisotropic. This perpendicular anisotropy confirms the epitaxial relation: (0002) Ge 3 Mn 5 (002) Ge discussed in Ref. [12]. The magnetic easy axis of the clusters lies along the hexagonal c-axis which is perpendicular to the film plane.

For growth temperatures T g ≥145 • C the cluster magnetic signal dominates (Fig. 13b). Superparamagnetic nanostructures are investigated performing ZFC-FC measurements at different T 0 temperatures (Fig. 13a). The first ZFC peak at low temperature i.e. ≤ 150 K is attributed to low-T C nanocolumns (T C ≈130 K). This peak is wider than for lower growth temperatures and its maximum is further shifted up to 30 K. These results are in agreement with TEM observations: increasing T g leads to larger nanocolumns (i.e. higher blocking temperatures) and wider size distributions. ZFC-FC irreversibility is observed up to the Curie temperature due to the presence of ferromagnetic columns. The sec- ond peak above 180 K in the ZFC curve is attributed to Ge 3 Mn 5 clusters and the corresponding ZFC-FC irreversibility persisting up to 300 K means that some clusters are ferromagnetic. We clearly evidence the out-ofplane anisotropy of Ge 3 Mn 5 clusters and the isotropic magnetic behavior of nanocolumns (Fig. 13b). In this growth temperature range, we have also investigated the Mn concentration dependence of magnetic properties. In Fig. 14a, for low Mn concentrations (2.3 % and 4 %) the contribution from diluted Mn atoms in the germanium matrix to the saturation magnetization is very high and nearly vanishes for higher Mn concentrations (7 %, 9 % and 13 %) as observed for low growth temperatures. Above 7 %, the magnetic signal mainly comes from nanocolumns and Ge 3 Mn 5 clusters. We can derive more information from ZFC-FC measurements (Fig. 14b). Indeed, for 2.3 % of Mn, ZFC and FC curves nearly superimpose down to low temperature meaning that nanocolumns are superparamagnetic in the whole temperature range. Moreover the weak irreversibility arising at 300 K means that some Ge 3 Mn 5 clusters have already formed in the samples even at very low Mn concentrations. For 4 % of Mn, we can observe a peak with a maximum at the blocking temperature (12 K) in the ZFC curve. We can also derive the Curie temperature of nanocolumns: ≈45 K. The irresversibility arising at 300 K still comes from Ge 3 Mn 5 clusters. Increasing the Mn concentration above 7 % leads to: higher blocking temperatures (20 K and 30 K) due to larger nanocolumns and wider ZFC peaks due to wider size distributions in agreement with TEM observations (see Fig. 3a). Curie temperatures also increase (110 K and 130 K) as well as the contribution from Ge 3 Mn 5 clusters. Finally when increasing T g above 160 • C, the nanocolumns magnetic signal vanishes and only Ge 3 Mn 5 clusters and diluted Mn atoms coexist. The overall magnetic signal becomes comparable to the one measured on annealed samples in which only Ge 3 Mn 5 clusters are observed by TEM (see Fig. 7a). The magnetic properties of high-T C nanocolumns obtained for T g close to 130 • C are discussed in detail in Ref. [19].

In conclusion, at low growth temperatures (T g ≤120 • C), nanocolumns are made of almost 4 independent elongated magnetic nanostructures. For T g ≥120 • C, these independent nanostructures percolate into a single nanocolumn sharply leading to higher blocking temperatures. Increasing T g leads to larger columns with a wider size distribution as evidenced by ZFC-FC measurements and given by TEM observations. In parallel, some Ge 3 Mn 5 clusters start to form and their contribution increases when increasing T g . Results on magnetic anisotropy seems counter-intuitive. Indeed Ge 3 Mn 5 clusters exhibit strong out-of-plane anisotropy whereas nanocolumns which are highly elongated magnetic structures are almost isotropic. This effect is probably due to compensating in-plane magnetoelastic coupling (due to the columns compression) and out-of-plane shape anisotropy.

V. CONCLUSION

In this paper, we have investigated the structural and magnetic properties of thin Ge 1-x Mn x films grown by low temperature molecular beam epitaxy. A wide range of growth temperatures and Mn concentrations have been explored. All the samples contain Mn-rich nanocolumns as a consequence of 2D-spinodal decomposition. However their size, crystalline structure and magnetic properties depend on growth temperature and Mn concentration. For low growth temperatures, nanocolumns are very small (their diameter ranges between 1.8 nm for 1.3 % of Mn and 2.8 nm for 11.3 % of Mn), their Curie temperature is rather low (< 170 K) and they behave as almost four uncorrelated superparamagnetic nanostructures. Increasing Mn concentration leads to higher columns densities while diameters remain nearly unchanged. For higher growth temperatures, the nanocolumns mean diameter increases and their size distribution widens. Moreover the 4 independent magnetic nanostructures percolate into a single magnetic nanocolumn. Some columns are ferromagnetic even if Curie temperatures remain quite low. In this regime, increasing Mn concentration leads to larger columns while their density remains nearly the same. In parallel, Ge 3 Mn 5 nanoclusters start to form in the film with their c-axis perpendicular to the film plane. In both temperature regimes, the Mn incorporation mechanism in the nanocolumns and/or in the matrix changes above 5 % of Mn and nanocolumns exhibit an isotropic magnetic behaviour due to the competing effects of out-of-plane shape anisotropy and in-plane magnetoelastic coupling. Finally for a narrow range of growth temperatures around 130 • C, nanocolumns exhibit Curie temperatures higher than 400 K. Our goal is now to investigate the crystalline structure inside the nanocolumns, in particular the position of Mn atoms in the distorted diamond structure, which is essential to understand magnetic and future transport properties in Ge 1-x Mn x films.

FIG. 1 :

 1 FIG. 1: RHEED patterns recorded during the growth of Ge1-xMnx films: (a) 2 × 1 surface reconstruction of the germanium buffer layer. (b) 1 × 1 streaky RHEED pattern obtained at low growth temperatures (Tg <180 • C). (c) RHEED pattern of a sample grown at Tg =180 • C. The additional spots reveal the presence of Ge3Mn5 clusters at the surface of the film.

FIG. 2 :

 2 FIG. 2: Transmission electron micrographs of a Ge1-xMnx film grown at 130 • C and containing 6 % of manganese. (a) cross-section along the [110] axis : we clearly see the presence of nanocolumns elongated along the growth axis. (b) High resolution image of the interface between the Ge1-xMnx film and the Ge buffer layer. The Ge1-xMnx film exhibits the same diamond structure as pure germanium. No defect can be seen which could be caused by the presence of nanocolumns. (c) Plane view micrograph performed on the same sample confirms the columnar structure and gives the density and size distribution of nanocolumns. (d) Mn chemical map obtained by energy filtered transmission electron microcopy (EFTEM). The background was carefully substracted from pre-edge images. Bright areas correspond to Mn-rich regions.

FIG. 3 :

 3 FIG. 3: Nanocolumns size and density as a function of growth conditions. Samples considered have been grown at 100 • C and 150 • C respectively. (a) Mn concentration dependence of the size distribution. (b) columns density as a function of Mn concentration. (c) Volume fraction of the nanocolumns as a function of Mn concentration.

FIG. 4 :

 4 FIG. 4: Plane view high resolution transmission electron micrographs of different types of nanocolumns : (a) typical structure of a column grown at 100 • C. The crystal structure is exactly the same as germanium . (b) Partially relaxed nanocolumn. One can see dislocations at the interface between the columns and the matrix leading to stress relaxation. (c) Amorphous nanocolumn. These columns are typical in samples grown at 150 • C with high Mn contents.

FIG. 5 :

 5 FIG. 5: Cross section high resolution micrograph of a Ge/Ge1-xMnx/Ge/Ge1-xMnx/Ge heterostructure. This sample has been grown at 130 • C with 6% Mn. Ge1-xMnx layers are 15 nm thick and Ge spacers 5 nm thick. We clearly see the sharpness of both Ge1-xMnx/Ge and Ge/Ge1-xMnx interfaces. Mn segregation leading to the columns formation already takes place in very thin Ge1-xMnx films.

FIG. 6 :

 6 FIG. 6: Cross section high resolution transmission electron micrograph of a sample grown at 170 • C. We observe the coexistence of two different Mn-rich phases: Ge1-xMnx nanocolumns and Ge3Mn5 clusters.

FIG. 7 :

 7 FIG. 7: (a) Temperature dependence of the saturation magnetization (in µB/Mn) of Ge0.93Mn0.07 samples for different growth temperatures. The magnetic field is applied in the film plane. The inset shows the temperature dependence of a sample grown at 130 • C and annealed at 650 • C for 15 minutes. After annealing, the magnetic signal mostly arises from Ge3Mn5 clusters. (b) ZFC-FC measurements performed on Ge0.93Mn0.07 samples for different growth temperatures. The in-plane applied field is 0.015 T.The ZFC peak at low temperature (≤150 K) can be attributed to the superparamagnetic nanocolumns. This peak widens and shifts towards high blocking temperatures when increasing growth temperature. The second peak above 150 K in the ZFC curve which increases with increasing growth temperature is attributed to superparamagnetic Ge3Mn5 clusters. The increasing ZFC-FC irreversibility at ≈ 300 K is due to the increasing contribution from large ferromagnetic Ge3Mn5 clusters. The nanocolumns signal completely vanishes after annealing at 650 • C for 15 minutes.

FIG. 8 :

 8 FIG. 8: Temperature dependence of the saturation magnetization (in µB/Mn) of Ge1-xMnx films grown at 100 • C plotted for different Mn concentrations: 4.1 %; 7 %; 8.9 % and 11.3 %.

FIG. 10 :

 10 FIG. 10: (a) ZFC-FC measurements performed on a Ge0.887Mn0.113 sample grown at 115 • C. The in-plane applied field is 0.015 T. Magnetization was recorded up to different T0 temperatures: 30 K, 50 K, 100 K, 150 K, 200 K and 250 K. Curves are shifted up for more clarity. (b) ZFC-FC curves for in-plane and out-of-plane applied fields (0.015 T).

FIG. 11 :

 11 FIG. 11: Temperature dependence of the inverse in-plane (open circles) and out-of-plane (open squares) normalized susceptibilities of a Ge0.887Mn0.113 sample grown at 115 • C. Fits were performed assuming isotropic nanostructures or cubic anisotropy. Dashed line is for in-plane susceptibility and solid line for out-of-plane susceptibility.

FIG. 12 :

 12 FIG. 12: (a) ZFC-FC measurements performed on a Ge0.93Mn0.07 sample grown at 122 • C. The in-plane applied field is 0.015 T. Magnetization was recorded up to different T0 temperatures: 50 K, 100 K, 150 K, 200 K and 250 K. Curves are shifted up for more clarity. (b) ZFC-FC curves for in-plane and out-of-plane applied fields (0.015 T).

FIG. 13 :

 13 FIG. 13: (a) ZFC-FC measurements performed on a Ge0.887Mn0.113 sample grown at 145 • C. The in-plane applied field is 0.015 T. Magnetization was recorded up to different T0 temperatures: 50 K, 100 K, 150 K, 200 K, 250 K and 300 K. Curves are shifted up for more clarity. (b) ZFC-FC curves for in-plane and out-of-plane applied fields (0.015 T).

FIG. 14 :

 14 FIG. 14: Temperature dependence of the saturation magnetization (in µB/Mn) of Ge1-xMnx films grown at 150 • C plotted for different Mn concentrations: 2.3 %; 4 %; 7 %; 9 %; 11.3 %. (b) ZFC-FC measurements performed on Ge1-xMnx films grown at 150 • C. The in-plane applied field is 0.025 T for 2.3 % and 4 % and 0.015 T for 8 % and 11.3 %.

VI. AKNOWLEDGEMENTS

The authors would like to thank Dr. F. Rieutord for grazing incidence x-ray diffraction measurements performed on the GMT station of BM32 beamline at the European Synchrotron Radiation Facility.


I Vogel

M.-L Flottes

C Landrault

Initialization of Partially LBISTed Sequential Circuits 1

Full scan is the most widely accepted and used DfT approach for large sequential machines. Nevertheless, in very dedicated cases it cannot be used mainly due to performance reasons as for example in high performance deeply pipelined CPU units. In this case full scan approach has to be replaced by partial scan. When trying to apply LogicBIST on partially scanned machines, the initialization problem of non-scan elements has to be solved. In this paper, we propose a nearly optimal algorithm to obtain a minimum set of memory elements to be initialized enabling to solve this initialization problem.

I.

Introduction

It is demonstrated that cyclic structures are responsible for initialization problems. Thus, our approach consists in selecting at least one memory element per cycle in the circuit and to transform this element into an equivalent one with state initialization facility. In order to lower the impact of the initialization hardware on BIST overhead, the technique must lead to break all cycles with a minimum number of transformations. This problem, called the Minimum Feedback Vertex Set (MFVS), is known as NP-hard. We propose a parameter-driven heuristic that can handle circuits with a large number of memory elements. First, the circuit is modeled by a digraph G(V,E). V is the set of vertices representing the set of FFs and E is the set of edges representing the combinational paths between them. Then, several pre-process reduction tasks preserving the MFVS are applied on the graph. Finally a heuristic is used to solve the MFVS problem on the reduced graph.

II.

Graph reduction

The first pre-processing task consists in partitioning the graph into Strongly Connected Components (SCCs). Each SCCs can be processed independently.

The MFVS of an initial graph G is the sum of the MFVSi computed on every SCCi in G. Then we define 3 rules able to reduce the graphs. Rule 1: In0Out0(G(V,E)) A vertex v i without incoming or outgoing edge cannot belong to a cycle. The vertex v i and its adjacent edges can be eliminated. Rule 2: In1(G(V,E)) A vertex v in V such that there is one and only one incoming edge (u,v) to v, then every cycle including v necessarily includes the vertex u. The vertex v can be eliminated. In order to maintain all the paths via v, the outgoing edges from v are replaced by outgoing edges from u. The same stands for a vertex v such that there is one and only one outgoing edge (v,w) from v (rule 2bis).

Rules 2(bis): Out1(G(V,E))

Rule 3: SelfLoop(G(V,E),MFVS)

This rule removes all self-loops vertices from the graph and put them automatically into the solution set (MFVS). Those three rules are applied iteratively on each SCC until there is no more possible reduction. Next figure shows an application of this pre-process task on a graph example.

Note that if a sub-graph Gi is empty at the end of the pre-process, the MFVS problem is solved for Conversely, if a sub-graph cannot be reduced to an empty graph, this means that cycles are still present in this partition and have to be detected and broken.

To do so, we use the heuristic defined below.

III. The GRASP Heuristic

The proposed method is issued from the metaheuristic: Greedy Randomized Adaptive Search Procedure (GRASP) proposed in [START_REF] Pardalos | A Greedy Randomized Adaptative Search Procedure for the Feedback Vertex Set[END_REF]. It is composed of two main steps. The first one, ConstructSol, builds up a solution -a set of nodes to break all the cycles-. This initial solution is not necessarily optimal. Then a second procedure, LocalSearch, explores the solution space from this initial solution in order to find a local minimum. These two procedures are run several times in order to explore different parts of the solution space. The CPU time increases with the user defined number of iteration (MaxIter). The variable BestSol stores the best solution computed so far.

GRASP(G,MaxIter)

BestSol= ; For k=1 to MaxIter InitSol=ConstructSol(G, ,clkseed); Sol=LocalSearch(InitSol); BestSol=UpdateSol(BestSol,Sol); end for;

a) Construction of an initial solution

The list of FFs to include in the initial solution is built up iteratively. The first task consists in selecting the best vertex candidates for breaking all the cycles (MakeCL). Then, the vertex v i to include in the initial solution is randomly selected among this list CL (RandomSelect). After each selection, the graph is updated by removing the selected node (UpdateGraph).

Finally, the procedure ReduceGraphSize containing all previously defined rules is applied before the next selection. The process iterates until the graph is empty. The candidate list CL is build up according to a gain representing the ability for a vertex to cut many cycles. The function used to evaluate this gain is:

F(v i ) = In(v i ) * Out(v i ). A vertex v i is included in the list CL if F(v i ) [Fmin + (Fmax- Fmin), Fmax]
where is a user tuning parameter. If =1, the algorithm is purely greedy. If =0, the selection is purely random.

When the graph includes few vertices, it is recommended to chose closed to zero and a high MaxIter value in order generate the optimal solution. Conversely, it is recommended to chose closed to one when the graph includes numerous vertices.

b) Local search in the initial solution proximity

The exploration of the solution space consists in checking if every vertex in the initial solution is redundant or not. The LocalSearch procedure returns a local minimum.

Solution LocalSearch(InitSol)

For every vertex v i in InitSol G'=New(G,v i ,InitSol); NewReduceGraphSize(G'); If G'= Remove(v i ,InitSol) end for return InitSol

IV. Experimental results

Experimental results have been performed on some ISCAS'89 benchmarks with =0.5 and MaxIter=1.

MFVS opti [2]

Our Columns 5 & 6 report our results. We obtain the optimal solution in shorter CPU time.

V.

Conclusion

We have described an algorithm enabling to solve the problem of the initialization of non-scan memory elements in partial scan approach. Based on a minimum edge cutting algorithm, this algorithm provides nearly optimum results in acceptable CPU time. In a very near future, it will be used to solve initialization problems on deeply pipelined CPU units provided by INTEL in the framework of this collaboration work.

  the solution is optimal.

Table 1 :

 1 MFVS experimental results

	s5378 179	30	13.4	30	0.27
	s9234 228	152	1.0	152	0.21
	s13207 669	310	3.7	310	0.5
	s15850 597	441	8.4	441	2.81
	s35932 1728	306	12.6	306	0.97
	s38417 1636 1080	44.6	1080	3.84
	s38584 1452 1115	37.1	1115	24.93

solution Circuit FFs (#FFs) CPU (sec) (#FFs) CPU (sec)


Jean-Marc Flesselles

Franck Pigeonneau

Kinematic regimes of convection at high Prandtl number in a shallow cavity. Régimes cinématiques de la convection à haut nombre de Prandtl dans une cavité allongée

Keywords: Fluid mechanics, natural convection, high Prandtl number Mécanique des fluides, convection naturelle, haut nombre de Prandtl

HAL is

Version française abrégée

Nous considérons une cavité allongée bidimensionnelle de hauteur H et de longueur L emplie d'un fluide incompressible de viscosité cinématique ν, de diffusivité thermique κ et de coefficient de dilatation volumique β avec P r = ν/κ ≫ 1. Le champ de gravité est noté g. Les deux parois verticales sont placées respectivement à des températures fixes T -et T + . On note ∆T = T + -T -et A = H/L.

Par une analyse phénoménologique réalisée dans le cadre de l'approximation de Boussinesq, nous montrons qu'il n'existe que deux régimes cinématiques selon la valeur de RaA 2 . Cette quantité correspond au rapport du temps caractéristique de diffusion de la chaleur suivant la verticale au temps d'advection longitudinale.

Dans le régime conductif obtenu à faible valeur de RaA 2 , la vitesse caractéristique est donnée par

u 0 ∼ Ra κ L avec Ra = gβ∆T H 3 νκ . (1)
Dans le régime convectif, observé à grand RaA 2 , la vitesse caractéristique a pour expression u 0 ∼ (RaA 2 ) -3/5 Ra κ L ∼ (gβ∆T /ν) 2/5 κ 3/5 L 1/5 .

(

) 2
Cette analyse confirme celle précédemment faite par Boehrer [START_REF] Boehrer | Convection in a long cavity with differentially heated end walls[END_REF] car nous retrouvons que le paramètre de contrôle est la quantité RaA 2 . Par contre, nous concluons qu'il n'y a que deux régimes cinématiques alors qu'une analyse basée sur la thermique montrait trois régimes. Dans le régime convectif, la vitesse caractéristique ne dépend plus de la hauteur, et augmente avec la longueur de la cavité.

Nous avons réalisé plusieurs simulations numériques du problème considéré dans cette note en faisant varier le nombre de Rayleigh de 1 à 10 9 et le facteur de forme de 1/2 à 1/20. La représentation de la composante longitudinale de la vitesse obtenue numériquement en fonction de RaA 2 confirme les prédictions théoriques. La transition entre les deux régimes est assez franche. Elle est observée pour RaA 2 ≃ 443.

Introduction

Despite the amount of work devoted to natural convection, the subject is not yet exhausted because of the variety of situations of interest, the number of possible control parameters and the subtlety of the effects at play. The purpose of this note is to determine the different kinematic regimes that occur in a two-dimensional cavity with end walls kept at fixed temperature and filled with a high Prandtl number fluid.

The side-heated cavity is an academic model for many realistic situations of natural or technological flows. It has first been introduced by Batchelor [START_REF] Batchelor | Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures[END_REF] to study air circulation in buildings. In this case, the geometry -a tall and thin cavity-is characterized by a large aspect ratio A and the fluid, namely air, is characterized by a Prandtl number of order 1. The situation of a shallow cavity (with a small aspect ratio) also received much attention since it applies to a large number of geophysical flows, like estuaries or waste heat disposal. Studies concentrated on low Prandtl number fluids like air (P r = 0.7) or water (P r = 7) which are most commonly encountered, but a few authors have also studied high Prandtl number flows.

Boehrer published [START_REF] Boehrer | Convection in a long cavity with differentially heated end walls[END_REF] a comprehensive analysis of all available experimental data as well as possible theoretical scalings for the flow of a fluid in a two-dimensional shallow cavity with end heated walls, within the Boussinesq approximation. He deplored that no clear understanding of the various regimes that may occur in such an apparently simple flow would yet emerge and suggested a global picture of the convection regimes as well as the relevant control parameters.

According to him, his analysis applies to long cavities (0 < A < 0.25), provided the Rayleigh number is not excessively high (Ra < A -12 ), and filled with any fluid such that the Prandtl number is larger than unity. His main results are the following : the relevant control parameter is RaA 2 , where Ra is the Rayleigh number ; there exist three regimes, called conductive regime, transition regime and convective regime, the boundaries of which are at RaA 2 ≃ 10 2 and RaA 2 ≃ 10 4 . He also provides estimates for the temperature gradient in the core of the cavity.

Boehrer's analysis is based on a comparison of published data, together with a physical analysis of the different scaling, and a systematic re-plotting of all data according to the different hypothesis. However, Boehrer emphasis is essentially put on the temperature gradient within the core of the cavity.

Following Boehrer, we have reconsidered the case of high Prandtl number fluids. Our analysis is based on a physical analysis of the scaling laws and extensive computational fluid dynamics simulations. We deduce that, from a kinematic point of view, two regimes only exist. We provide the threshold between them and the laws for the velocity scale in each regime.

Formulation

We consider a two-dimensional rectangular shallow cavity of height H and length L filled with an incompressible fluid of constant kinematic viscosity ν and constant heat diffusivity κ. The heat expansion coefficient of the fluid is β. The cavity is in the gravity field g. One vertical side of the cavity is kept at a hot fixed temperature T + and the opposite side is kept at a cold temperature T -. The temperature difference is ∆T = T + -T -. Top and bottom sides are adiabatic. The fluid velocity vanishes on all sides. The Boussinesq approximation is assumed.

The continuity equation writes

u 0 L ∂u ∂x + v 0 H ∂v ∂y = 0, (3)
where x = x/L, y = y/H, u = u/u 0 , v = v/v 0 and overlines indicate dimensional variables. The scaling variables u 0 , v 0 for horizontal and vertical velocity scales are still unknown. Eq. ( 3) sets the relation between them:

v 0 ∼ (H/L)u 0 . (4)
Since the aspect ratio A = H/L is very small, v 0 is much smaller than u 0 : the flow is parallel (except near the lateral walls). With the use of eq. ( 4), the equations of motion are:

St ∂u ∂t + u ∂u ∂x + v ∂u ∂y = - δP ρ 0 u 2 0 ∂P ′ ∂x + ν u 0 LA 2 A 2 ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 (5)
A 2 St ∂v ∂t + u ∂v ∂x + v ∂v ∂y = - δP ρ 0 u 2 0 ∂P ′ ∂y + ν u 0 L A 2 ∂ 2 v ∂x 2 + ∂ 2 v ∂y 2 + β∆T gH u 2 0 θ (6)
St ∂θ ∂t + u ∂θ ∂x + v ∂θ ∂y = κ u 0 LA 2 A 2 ∂ 2 θ ∂x 2 + ∂ 2 θ ∂y 2 (7)
in which the Strouhal number St is defined by St = L/(u 0 t 0 ). In eq. (5-7), new unknown scaling variables for time, density and pressure have been introduced with the corresponding nondimensional variables:

t = t/t 0 , ρ = ρ/ρ 0 , P = P /δP . Nondimensional temperature is θ = (T -T -)/(T + -T -).
The scaling variables correspond to the maximum values attained within the domain where the scaling applies: it is the central region where the flow is parallel. In the next section, we determine these variables by a phenomenological analysis [START_REF] Bejan | Convection heat transfer[END_REF]. It relies on the least degeneracy principle, according to which the largest possible number of terms should be kept (see [START_REF] Germain | Méthodes asymptotiques en mécanique des fluides[END_REF][START_REF] Zeytounian | Modélisation asymptotique en mécanique des fluides newtoniens[END_REF]).

Phenomenological analysis

From eq. ( 7), one deduces that

κ/(u 0 LA 2 ) ∼ max(St, 1) (8)
We first consider the case where St ≫ 1, which means that the typical horizontal advection distance is much smaller than the length of the cavity. In this case, eq. ( 8) yields the time scale:

t 0 ∼ H 2 /κ. (9)
It is a characteristic thermal diffusion time over height H. When inserted into eq. ( 5), with the assumption of large Prandtl number (P r ≫ 1), this yields a relation between pressure and velocity scales:

δP/(ρ 0 u 2 0 ) ∼ ν/(u 0 LA 2 ). ( 10
)
This, with eq. ( 6), sets the u 0 velocity as proportional to the H based Rayleigh number:

u 0 ∼ Ra κ L with Ra = gβ∆T H 3 νκ . (11)
This equation may be written in a nondimensional way by introducing the Péclet number:

P e ∼ Ra with P e = u 0 L κ . (12)
It is easily checked that the St ≫ 1 condition also writes RaA 2 ≪ 1.

The Strouhal number may also be seen as the ratio of the horizontal advection time scale to the vertical thermal diffusion time scale. Hence we call this low Rayleigh number regime the conductive regime since the motion is controlled by the (long) thermal time scale. A temperature perturbation has time to equilibrate vertically because of thermal diffusion before being advected along the length of the cavity: isotherms are essentially vertical, which corresponds to a conductive solution.

On the other hand, if the Strouhal number becomes small enough, hence at large Rayleigh number, a thermal perturbation will travel along the cavity. This means that a horizontal boundary layer of thickness δ has appeared, within which the fluid moves to the vertical walls. This thickness is such that the Strouhal number associated to it is equal to unity:

κL/(δ 2 u 0 ) ∼ 1. (13)
Vertical length scales should be rescaled with δ instead of H. A similar analysis to the previous one may be performed. One finds that the Péclet number is no more equal to the Rayleigh number:

P e ∼ Ra(δ/H) 3 . ( 14
)
The size of the boundary layer is found by inserting eq. ( 14) in eq. ( 13). Finally one gets the final expression of the (nondimensional) velocity in the convective regime: The previous analysis sets on firm grounds the analysis and hypothesis of Boehrer: we recover the role of the quantity RaA 2 as the control parameter and we derive rigorously the relevance of the conductively controlled boundary layer, based on the advection time scale. In addition, we show that there exists only two kinematic convection regimes, whereas the current understanding focuses on the thermal field, and leads to a three regime classification.

P e ∼ Ra(RaA 2 ) -3/5 . (15)

Comparison to numerical data

In order to check the theoretical predictions (12) and (15), we have performed extensive numerical simulations of the problem expressed in the appropriate nondimensional variables. In a permanent state, the equations of motion write:

∂u ∂x + ∂v ∂y = 0, (16)
u ∂u ∂x + v ∂u ∂y = - ∂P ′ ∂x + P r RaA ∇ 2 u, (17)
u ∂v ∂x + v ∂v ∂y = - ∂P ′ ∂y + P r RaA ∇ 2 v + P r RaA 2 θ, (18)
u ∂θ ∂x + v ∂θ ∂y = 1 RaA ∇ 2 θ, (19)
where velocities are normalized by Raκ/L, and lengths by H. Eq. (16-19) are solved with a commercial CFD code in a rectangular box of height 1, and length 1/A, filled with a fluid of viscosity P r/RaA, of thermal conductivity 1/RaA, and of thermal expansion coefficient P r/RaA 2 . About 40 cases have been run, where the aspect ratio A was varied between 1/20 and 1/2 and the Rayleigh number Ra was varied in the range 1 to 10 9 . The Prandtl number was fixed, and equal to 1000. With the considered nondimensional variables, the maximum value of the x component of the velocity is equal to P e/Ra. Its value is reported a function of RaA 2 on Fig. 1. In accordance with the phenomenological analysis P e/Ra is constant at low RaA 2 , i.e. in the conductive regime. This constant value is easily calculated in the Ra → 0 limit. It is equal to 1/72 √ 3 ≃ 8.02 • 10 -3 [START_REF] Cormack | Natural convection in a shallow cavity differentially heated end walls. Part 1. Asymptotic theory[END_REF]. At high RaA 2 , i.e. in the convective regime, P e/Ra = 0.32(RaA 2 ) -3/5 , where the proportionality constant comes from a fit. The intersection between the two scalings provides the regimes boundary. It is rather sharp and occurs at RaA 2 ≃ 443.

Conclusion

The numerical data are in perfect agreement with our phenomenological analysis. These results both confirm and extend Boehrer's analysis: the relevant control parameter for natural convection of a high-Prandtl number fluid in a shallow box is indeed RaA 2 , which is the ratio of the heat diffusion time scale to the convection time scale. There are only two kinematic regimes, and the only condition lies on RaA 2 , provided P r > 1.

In the convective regime, that occurs when RaA 2 443, a boundary layer develops. As this new length scale sets in the problem, the velocity scale changes. When expressed in dimensional quantities, the velocity coming from eq. ( 15) writes:

u max = 0.32 gβ∆T ν 2/5 κ 3/5 L 1/5 . (20)
Hence, contrary to the intuitive understanding, the velocity does not depend on depth any more and increases with the length of the cavity, in full agreement with theoretical prediction and numerical results.

5 Figure 1 .

 51 Figure 1. Nondimensional maximum value of the x-velocity component in shallow cavities of various aspect ratios A vs. RaA 2 for a high-Prandtl number fluid.

Figure 1 .

 1 Figure 1. Valeur maximale de la composante x de la vitesse adimensionnée dans des cavités de différents facteurs de forme A en fonction de RaA 2 pour un fluide de haut nombre de Prandtl.

Email addresses: jean-marc.flesselles@saint-gobain.com (Jean-Marc FLESSELLES), franck.pigeonneau@saint-gobain.com (Franck PIGEONNEAU).

Acknowledgements

We thank X. Chavanne for numerous and fruitful discussions.