domain: sdv shs shs.eco

text:
Kassoum Ayouba

Jean-Philippe Boussemart

Henri-Bertrand Lefer

Hervé Leleu

Raluca Parvulescu
email: r.parvulescu@ieseg.fr

A measure of price advantage and its decomposition into output-and input-specific effects

Keywords: Data analysis envelopment, Price advantage, Technical efficiency, Value efficiency, allocative efficiency, European agricultural policy, CAP reforms

In this paper, we develop a price advantage measure obtained as the difference between efficiency scores computed with quantity-based and value-based data. This measure is interpreted as an increase in the firm's profit rate due to a favourable input and output price environment. Compared to traditional allocative efficiency measure, which uses only the evaluated firm's prices, price advantage measure considers prices of peers in the market. Our main contribution is to show that total-price advantage decomposes into the sum of output-and input-specific price advantages. The application of the measure to a French farm's dataset in the context of the successive common agricultural policy reforms -aiming at the liberalization of agricultural prices (1992-2013) -illustrates our theoretical model.

Introduction

Traditional technical efficiency gives indication on the ability of a decision-making unit (DMU) to increase its outputs (reduce its inputs) for a given level of inputs (outputs). This measure is based on physical quantities recorded at the DMU level. The introduction of the price dimension in this setting can be traced back to [START_REF] Farrell | The Measurement of productive Efficiency[END_REF] who introduced cost (overall) and allocative efficiencies as a measure of the producers' ability to optimally allocate their resources according to their own respective prices. [START_REF] Färe | Production Frontiers[END_REF] extended the initial approach, based on cost minimization, to revenue and/or profit maximizations. In this framework, the (input) allocative efficiency for example is usually computed indirectly as a residue of cost and technical input efficiencies [START_REF] Coelli | Technical, Allocative, Cost and Scale Efficiencies in Bangladesh Rice Cultivation: A Non-parametric Approach[END_REF]). An exception is [START_REF] Bogetoft | Allocative Efficiency of Technically Inefficient Production Units[END_REF] who aim at a direct measure of allocative efficiency.

To date, a few undesirable characteristics of the cost and allocative efficiencies have been pointed out in the literature. Recall that since the main objective of these efficiencies is the identification of the optimal allocation of output and/or input in terms of physical quantities for a given (known and fixed) system of relative prices,2 there is no prices comparison among producers. [START_REF] Tone | A Strange Case of the Cost and Allocative Efficiencies in DEA[END_REF] noted that in the case of price difference between DMUs, the cost and allocative efficiency measures present some limitations. For example, an equiproportional increase (or decrease) in the magnitude of inputs prices does not affect the estimated cost efficiency coefficient. To overcome this "strange" behaviour, he suggests the use of an enhanced radial efficiency measure that uses the levels of spending on each input instead of input quantities. In other words, [START_REF] Tone | A Strange Case of the Cost and Allocative Efficiencies in DEA[END_REF] introduced a new form of allocative and cost efficiencies based on a "value" production technology. [START_REF] Fukuyama | Economic inefficiency measurement of input spending when decision making units face different input prices[END_REF] and [START_REF] Färe | Resolving a Strange Case of Efficiency[END_REF] extended and generalized this approach to non-radial settings, using the directional distance function of [START_REF] Chambers | Benefit and distance functions[END_REF]. [START_REF] Sahoo | Cost, revenue and profit efficiency measurement in DEA: A directional distance function approach[END_REF] pursued in this direction. Specifically, they dropped the assumption that DMUs are price takers and proposed a "value" directional distance function on the basis of a technology set that includes all feasible (input) costs and (output) revenues. Note also that the cost efficiency model developed in [START_REF] Tone | A Strange Case of the Cost and Allocative Efficiencies in DEA[END_REF] served as a basis in [START_REF] Sahoo | Returns to growth in a non-parametric DEA approach[END_REF] for the estimation of returns to growth. This idea was also at the origin of [START_REF] Tone | Re-examining scale elasticity in DEA[END_REF] for the development of a new scheme for measuring scale elasticity of production. Finally, [START_REF] Sahoo | Non-parametric measurement of economies of scale and scope in noncompetitive environment with price uncertainty[END_REF] contrast the "value" based cost model to a "quantity" based one, which allows them to discuss scale and scope economies in a noncompetitive market characterized by heterogeneous firms and price uncertainty.

The question of the meaning of efficiency measures obtained with value-based data instead of quantity data has also been investigated. Starting from the observation that when DMUs face the same prices, quantity-based efficiency scores and value-based ones are identical and that when the prices faced by the DMUs are different, these scores differ as well, [START_REF] Cross | Farrell efficiency under value and quantity data[END_REF] showed that input-oriented radial value-based efficiency score can be multiplicatively decomposed as a purely technical efficiency score, a technology effect and a firm effect. On their side, [START_REF] Portela | Economic efficiency when prices are not fixed: disentangling quantity and price efficiency[END_REF] analysed cost efficiency by considering both prices and input quantities as decision variables. An input-price cost saving component was then integrated into the decomposition of this cost efficiency measure using a Bennet-type indicator. Although these measures clearly improve standard allocative efficiency, they still assume physical resource reallocation at the firm level for a given system of relative prices. [START_REF] Camanho | A generalisation of the Farrell cost efficiency measure applicable to non-fully competitive settings[END_REF] criticized the assumption of fixed prices in the cost efficiency assessment. They pointed out that the cost efficiency measure reflects only inputs efficiencies (technical and /or allocative efficiency) but not (market) price inefficiencies (deviation from fully competitive setting leading to price differences between firms), and then suggest a comprehensive framework that accounts for both inputs and market (price) inefficiencies and that circumvent the problems mentioned in [START_REF] Tone | A Strange Case of the Cost and Allocative Efficiencies in DEA[END_REF]. Specifically, they enhance the method of the measure of cost efficiencies to account for situation where the inputs prices can depend on negotiations or tend to qualitatively differentiate the resource available at each DMU. Some other approaches have dealt with the possibility that DMUs' prices are not identical, either from the perspective of imperfect markets (Kuosmanen et al., 2010, Fang and[START_REF] Fang | Cost efficiency in data envelopment analysis under the law of one price[END_REF] or from the perspective of price endogeneity, often complemented by price uncertainty [START_REF] Cherchye | Non-parametric production analysis in non-competitive environments[END_REF][START_REF] Ray | Input price variation across locations and a generalized measure of cost efficiency[END_REF][START_REF] Johnson | Allocative efficiency measurement with endogenous prices[END_REF].

We introduce in this paper the concept of "price advantage". Rather than decomposing the value technology further as in [START_REF] Cross | Farrell efficiency under value and quantity data[END_REF], our indicator is based on the difference between two inefficiency scores estimated with i) physical quantities of inputs and outputs, and ii) their respective values (costs and revenues). Our indicator is given an economic interpretation as the percentage increase in the profit rate due to a favourable price environment. In this sense, price advantage measure completes the picture provided by the allocative efficiency by emphasising producers' abilities to seize favourable market price environments for a given input and/or output mix. Our main contribution in this paper is to propose a decomposition of this difference into output-and input-specific price advantages. Thus, from a practitioner's point of view, we aim at providing a means for identifying those inputs and outputs, which are the main sources for price performance.

Our price impact measure is distinct from the "marketing efficiency" concept introduced in [START_REF] Singbo | Estimating farmers' productive and marketing inefficiency: an application to vegetable producers in Benin[END_REF] where a price index is introduced as an additional constraint in the technical efficiency model. In the same spirit, it is also different from the concept of "market efficiency" proposed by [START_REF] Camanho | A generalisation of the Farrell cost efficiency measure applicable to non-fully competitive settings[END_REF]. Indeed, in their work, each DMU is evaluated with a price vector chosen among other DMUs, be it the minimum price observed for each input (first measure for economic efficiency) or a DMU's price vector leading to the minimum cost (second measure for economic efficiency). One could qualify this approach as "normative" since optimal prices from other DMUs are applied to each evaluated DMU. Conversely, in our setting, a DMU is always evaluated with its own observed price vector. From a practitioner point of view, it is somewhat difficult to disentangle prices from the input physical quantities used. Thus, our measure adopts a more "positive" approach and is based on less restrictive assumptions. Moreover, our indicator is more general as it also accounts for output prices (profit setting) and proposes a decomposition of total price advantages into output-and input-specific price effects.

We take advantage from the context of prices liberalization and reinforcement of farmers' incentives to consider price signals in their decisions (from common agricultural policy, CAP hereafter, reforms) to show the relevance of our price advantage measure. Additionally, through the decomposition of total price advantages, we are able to characterize the origin of price advantages for each produce. This information is valuable for both practitioners and public authorities in markets ongoing a deregulation process. Specifically, we used a dataset of 11,967 observations from 1,186 farms located in the French department of Meuse for the period 1992-2013. After showing that price advantage measure is more informative about the state of producers' efficiency than the allocative efficiency measure, we show that subsequent to the adoption of the 2003 Mid-Term Reform (MTR hereafter), crops-specific price advantages decreased while milk-specific price advantages increased and bovine-meat specific-price advantages remained relatively constant.

The remaining of this paper is organized as follows. In Section 2, we present our methodology to estimate price advantage and introduce its decomposition into input-and output-specific price effects. Section 3 describes its empirical application in the French department of Meuse, and Section 4 presents a discussion of our methodology and results while Section 5 gives the conclusive remarks.

Methodology

Let N be the total number of producers, denoted as DMUs, using a common technology to transform M input quantities ( ) QI into S observed output quantities ( )

QO . If the technology T Q
satisfies properties such as: no positive outputs can be produced from zero inputs, free disposability, and convexity, we can define its data envelopment analysis (DEA) representation under constant return to scale according to [START_REF] Banker | Some models for estimating technical and scale inefficiencies in data envelopment analysis[END_REF] as:

1 1 {( , )| , 1,..., , , 1,..., , 0, 1,..., } N N Q n n n n n k k j j n n T QI QI k M QO QO j S n N QI QO λ λ λ = = = ≥ ∀ = ≤ ∀ = ≥ ∀ = ∑ ∑
(1).

Let PI and PO be the vector prices for the M inputs and the S outputs, respectively. For an observed DMU a ( )

1,..., a N = , a k
VI denotes the cost incurred for the input k ( )

1,..., k M =
. This cost is obtained as the product between physical quantity and its price:

a a a k k k VI QI PI = (2).
In the same way, DMU a's revenue from the output j ( )

1,..., j S = is defined as: a a a j j j
VO QO PO =

(3). Therefore, its observed total cost and total revenue are respectively given by:

1 M a a k k VI VI = = ∑ and 1 S a a j j VO VO = = ∑
DMU a's observed profit is given by the difference between total revenue and total cost:

a a a VO VI π = - (4).
Its observed profit rate is the ratio between observed profit and total revenue: Based on the idea introduced by [START_REF] Cross | Farrell efficiency under value and quantity data[END_REF] of multiplicatively decomposing the value efficiency score into a quantity-based efficiency score, a technology effect and a firm effect, we define in what follows the value technology T V based on the output values vector VO and the input costs vector VI.

1 1 {( , ) | , 1,..., , , 1,..., , 0, 1,..., } N N V n n n n n k k j j n n T VI VI k M VO VO j S n N VI VO λ λ λ = = = ≥ ∀ = ≤ ∀ = ≥ ∀ = ∑ ∑ (5).
The gap between a given production plan and the technology ( Q T for example) boundary is measured using the concept of the directional distance function. The usual directional distance function proposed by [START_REF] Chambers | Benefit and distance functions[END_REF] is defined as: . Properties of directional distance functions can be found in [START_REF] Chambers | Benefit and distance functions[END_REF]. In this traditional framework, β is a common scalar for all inputs and outputs defining a unique efficiency score for the DMU under evaluation.

( ) ( ) CCF M S M S DQ R R R R R + + + + + : × × × → ( ) ( ) { } sup : , CCF Q DQ R T β β β β + = ∈ - + ∈ QI QO
The standard measures for technical efficiency are either Shephard or directional distance functions. In the vein of Ritchie and Rowcroft (1996), [START_REF] Aparicio | Accounting for slacks to measure and decompose revenue efficiency in the Spanish Designation of Origin wines with DEA[END_REF]2015) who extended the analysis framework to account for slacks, our setting introduces a Färe-Lovell directional distance function where ( )

, M S R R + + = ∈ × QI QO β β β
is a vector with specific components for each input and output [START_REF] Färe | Measuring the technical efficiency of production[END_REF]. The main advantage of this approach is to exhaust all input and output slacks to obtain a full Pareto-efficiency measure. Since costs and revenues are known in our setting, we naturally use them in order to weight each component of the vector β. Clearly, whenever economic information on prices is not available, the distance function as defined by [START_REF] Chambers | Benefit and distance functions[END_REF] is useful by using a radial contraction and extension of the Q g vector. All input and output components are implicitly weighted equally. However, when prices are known, we use the individual value shares to weight distinctly inputs and outputs and a Färe-Lovell directional distance function seems more adequate.

M S M S DQ R R R R R + + + + + : × × × → is defined as: ( ) ( ) , ; , sup : , Q DQ T   + = - + ∈     QI QO QI QO QI QI QO QO O β β VI β VO QI QO g g QI β g QO β g e VO (7).
Likewise, let the directional distance function in the value space be defined as:

( ) ( ) , ; , sup : , V DV T   + = - + ∈     VI VO VI VO VI VI VO VO O β β VI β VO VI VO g g VI β g VO β g e VO (8).
This distance function is interpreted as the improvement in the profit rate as a result of the output revenue increase and/or the input cost reduction. While this is the most general case treating both inputs and outputs, one can restrict the analysis to either a revenue or a cost approach. For this, one should consider null components in either the input or the output direction respectively on the corresponding output-or input-value technology. We use the output-oriented model for the empirical application.

Total price advantage and its decomposition into output-and input-specific price advantages

We estimate the distance functions defined above in a linear programming framework. We choose the observed production plan of the evaluated DMU as the direction:

Q DQ T   + = - + ∈     QI QO QI a QO a a a a a I QI a O QO a O a β β β VI β VO QI QO QI QO e β QI e β QO e VO (9). with I M R ∈ e
is a row vector with all elements being equal to 1. This directional distance function can be computed from the following linear program (LP1):

( ) ( ) ( ) , , 1 1 , ; , 1 1,
..., 1 1,..., 0 1,..., 0 a QO a QI j j k k j k a N n n QO a j j j n N n n QI a k k k n n QO j VO VI DQ Max VO QO QO j S QI QI k M n N QO QI a a a a λ β β QI QO QI QO β β λ β λ β λ β = =   +     = ≥ + ∀ = ≤ - ∀ = ≥ ∀ = ≥ ∑ ∑ ∑ ∑ 1,...,S 0 1,..., QI k j k M β ∀ = ≥ ∀ = (LP1)
Recall that normalisation by revenue VO a is useful for providing an economic interpretation to the inefficiency as a profit rate growth, which completes the work of [START_REF] Cross | Farrell efficiency under value and quantity data[END_REF].

Besides, it allows to obtain a relative measure (the profit growth rate is in percentage) which allows for inter-DMU and inter-annual comparisons.

In the same way, directional distance function defined in the value space is obtained from the following linear program (LP2):

( ) ( ) ( ) , , 1 1 , ; , 1 1,..., 1 1,..., 0 1,..., 0 a VO a VI j j k k j k a N n n VO a j j j n N n n VI a k k k n n VO j VO VI DV Max VO VO VO j S VI VI k M n N VO VI a a a a λ β β VI VO VI VO β β λ β λ β λ β = =   +     = ≥ + ∀ = ≤ - ∀ = ≥ ∀ = ≥ ∑ ∑ ∑ ∑ 1,...,S 0 1,..., VI k j k M β ∀ = ≥ ∀ = (LP2). DMU a's technical efficient profit ( ) , a Q
π if it adopted optimal production decisions is defined as:

( ) ( ) , * * 1 VO 1 V a Q QO a QI a j j k k j k I π β β = + - - ∑ ∑ (10).
Value-based maximum profit ( )

, a V
π that a DMU a can get from an optimal choice of "pricequantity couples" for each production j and each input k is defined by:

( ) ( ) , * * 1 1 a V VO a VI a j j k k j k VO VI π β β = + - - ∑ ∑ .(11).
We define a technical inefficiency measure ( ) a TI which highlights the percentage increase in a DMU a profit rate by adopting optimal quantity decisions: According to the two above indicators (equations 12 and 13), DMU a is considered as technical and respectively value efficient when the corresponding measure is null. If the measure is strictly positive the DMU is said to be technical inefficient and respectively value inefficient.

, a Q a a a TI VO π π - = (12).
We define a price advantage for the DMU a (PA a ) as: When the price advantage measure is strictly positive, we interpret this situation as a favourable price environment since the DMU's technical inefficiency (distance to its physical quantity benchmark) is greater than its value inefficiency (distance to the value benchmark). When the price advantage measure is negative we interpret this situation as an unfavourable price environment since the DMU's distance to the physical quantities benchmark is lower than its distance to the value benchmark. When the price advantage measure is null, then we consider that the DMU's price environment is neutral in the sense that its price system does not have an effect on its performances.

The total price advantage defined above (see equation 14) can also be written as the residual between technical and value inefficiencies, or as the gap between technically efficient profit ( )

, a T
π and value-efficient profit ( , a V π ) as a percentage of the observed revenue ( )

a VO . , , a Q a V a a a a PA TI VAI VO π π - = - = (15).
Indeed, we have that:

( ) ( ) , , * * * * 1 = , ; , , ; , a T a V a QO a QI a VO a VI a j j k k j j k k a a j k j k PA VO VI VO VI VO VO DQ DV a a a a a a a a QI QO QI QO VI VO VI VO π π β β β β       -   = - - -               = - ∑ ∑ ∑ ∑
The interpretation of the price advantage measured this way is equivalent to the one we gave above. 3 Given the use of input-and output-specific directions in our linear programs LP1 and LP2, we are able to propose specific price advantages for each output ( )

a j
PAO and respectively, for each input ( )

a k PAI . ( ) ( ) * * * * , 1, 2,..., QO VO a j j j a QO VO a j j j j a VO PAO j S VO β β β β α - = = - ∀ = (16),
where

a j
α is the share of the revenue obtained from output j in the total revenue:

a j a j a VO VO α = .
3 For example, when this measure is negative, it means that the profit rate obtained with the price-quantity couple of the benchmark is lower than the profit rate obtained with technical benchmark input and output quantities ( )

, , / / a V a a Q a VO VO π π ≥ . Consequently, this DMU is in an unfavourable price environment. ( ) ( ) * * * * , 1, 2,..., QI VI a k k k a QI VI a k k k k a VI PAI k M VO β β β β δ - = = - ∀ = (17),
where

a k
δ is the share of the cost spent on input k in the total revenue:

a a k k a VI VO δ = . 4
If the output-or input-specific price advantage is positive, then we infer that DMU a enjoys a favourable price environment for that output/input which can lead to an improvement in the DMU's profit rate. If the output/input-specific price advantage is negative then DMU a has suffered from an unfavourable price environment for that output/input and the measure gives the reduction in its profit rate due to the output price disadvantage. Obviously, this measure can also be null, meaning that the price environment has been neutral for that output/input and with no effect on the profit rate. For the same DMU a, its specific-output price advantages can be positive for some outputs/inputs, negative for some others and neutral for the rest.

Our main result is that the total price advantage can be algebraically decomposed into the sum of output-and input-specific price effects.

Proposition. Given our notations and the two linear programs LP1 and LP2 above we have that:

, 1, 2,..., and 1, 2,..., .

a a a j k j k PA PAO PAI j S k M = + ∀ = ∀ = ∑ ∑ (
β β β β β β β β - - + = + + + = - = - = ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑
Our measure of price advantage is comparable with the recent publications dealing with allocative efficiency [START_REF] Sahoo | Cost, revenue and profit efficiency measurement in DEA: A directional distance function approach[END_REF]) as a way to take into account the effect of prices on the DMU's performances. However, there are several distinctions between the two measures. The following paragraph seeks to discuss them.

Total price advantage and allocative efficiency: two complementary measures

Allocative efficiency allows identifying the optimal allocation of output and/or input in terms of physical quantities for a given set of relative prices. It is defined here as a residue between profit and technical inefficiencies. Consistent with this definition, we introduce below the profit inefficiency measure which represents the ratio of maximum profit ( ) ,* a π to observed revenue.

The maximum profit a DMU could get, given its own observed prices, is obtained through the following linear program (LP3):

,* , , 1 1 1 1 * * 1,...,S 1,...,M 0 1,..., . a a j k S M a a a a a a j j k k QO QI j k N n n a j j n N n n a k k n n Max PO QO PI QI QO QO j QI QI k n N ɶ ɶ ɶ ɶ ɶ ɶ λ π π λ λ λ = = = = = = - ≥ ∀ = ≤ ∀ = ≥ ∀ = ∑ ∑ ∑ ∑ (LP3).
This linear program (LP3) gives the optimal output and input quantities yielding maximum profit, for the DMU's system of relative output/input prices. Profit inefficiency ( )

π π π π π π - - - = - = - = (20).
Therefore, DMU a is allocative efficient if the optimal profit for its given output-and input-price system is identical to the optimal profit when the DMU is technical efficient. In this profit framework, the gap between ,* a π and , a T π highlights the additional profit the DMU a can get from an appropriate reaction to its own observed output and input prices.

However, as pointed out by [START_REF] Tone | A Strange Case of the Cost and Allocative Efficiencies in DEA[END_REF] allocative inefficiency measure is limited as it is based on the concept of relative prices. For a DMU producing several outputs with a given level of inputs, an equiproportionate increase in the prices of these outputs would not alter the DMU's allocative inefficiency. While a rich body of research seeks to improve this characteristic (Färe andGrosskopf 2006, Sahoo et al. 2014, etc.), their common feature is that they all assume resource reallocation at the firm level. In this sense, our work completes existing picture by emphasising producers' abilities to seize favourable market price environments for a given input and/or output mix. This justifies the wording chose for this measure as a price advantage as only price effects are measured while resource reallocation is not considered. While both allocative efficiency and price advantage measures can be obtained as residues, their definitions differ from one another. In this sense, no direct relationship can be established between them.

The next section is dedicated to an illustration of the price advantage measure and a comparison with the traditional allocative efficiency measure.

Empirical Application: Data and Results

This section presents an operational implementation of the price advantage concept. We start by describing the conditions under which France has applied CAP reforms in 2003 and 2009 and we show that our concept of performance is appropriate in the context of strong incentives to adapt to market signals. We further adapt our estimation strategy for this specific case. We then present our data, followed by the main results.

General context of the CAP reforms and estimation strategy

To comply with the World Trade Organization requirements, the European Union has progressively and structurally reformed its agricultural policy since the beginning of the 1990s.

The first major revision of the CAP -the MacSharry reform (1992) -replaced farmers' protection policies through guaranteed prices with a system of direct farm payments based on farming areas and livestock. The mid-term review (MTR; 2003) decoupled aid from production volumes to encourage farms to become market-oriented and reduce distortions in agricultural production and trade. Formally, European aid involved Single Payment (SP), the level of which was based on either the historical farm individual level, regional premium amounts, or a combination of both. Therefore, the European Commission expected farmers to respond better to market signals, orienting their practices towards market prices. The 2009 "health check" reinforced this move through gradual elimination of the remaining production-related payments by shifting to the SP scheme.

In France, the 2003 MTR of the CAP was applied from 2006 with a partial decoupling of aid for cereals, sheep, and cattle. To calculate this aid, French authorities used the individual-historical base (i.e. the average amount of subsidies received by the farm between 2000 and 2002). Regarding arable crops -for which the intervention price remained unchanged -the level of decoupling was 75% while the remaining aid was coupled to the surface. The 2009 health check reform specified that the remaining coupled aids for arable crops should disappear by 2010.

For cattle, 100% of suckler-cow and calf-slaughter premiums and 40% of adult-slaughter premiums remained coupled. Regarding milk production in 2004-2007, intervention prices decreased by 25% for butter and 15% for milk powder. Milk production quotas were expected to increase by 2% in 2008 and by 1% every year until 2015. However, France froze this measure after the 2009 collapse of milk prices. To compensate for the price decrease, an entirely decoupled dairy premium per ton of quota was created in 2004. The 2009 health check maintained the individual-historical base to calculate payments but specified that all coupled aids should disappear by 2010 for arable crops and by 2012 for any remainders (except suckler cows, for which 75% of the aids remained coupled).

In this frame of gradual liberalization of agricultural prices, we believe that our measure of output-oriented price advantage can be useful to analyse farmers' decisions.

Meuse sample

Agricultural activity in the Meuse department predominantly involves both crop farming and animal husbandry (Ecoscopie de la Meuse 2000;[START_REF] Kuosmanen | Firm and industry level profit efficiency analysis using absolute and uniform shadow prices[END_REF][START_REF] Sahoo | Returns to growth in a non-parametric DEA approach[END_REF]. The main contributors to the total agricultural revenue by share are: crops (except fodder), milk and dairy, and cattle and calves. Moreover, over the last 40 years, the number of crop-specialized farms increased while those specializing in both crops and animal husbandry, and those specializing in bovines (milk, meat, or both) and sheep decreased. Further, the size of the farms increased steadily, consistent with the general trend in France.

Our dataset of farms observed in Meuse reflected these changes. Panel data from 1,186 farms, with 11,967 observations were available for the period 1992-2013. 5 The unbalanced panel comprises farms with an average age of 10 years. Table 1 shows the number of farms in our panel from 1992 -2013, indicating its representativeness in terms of the Meuse department's dataset.

According to the general agricultural census, there were 1,983 farms with these specializations in 2000 (31%) and 1,448 in 2010 (32%). For our empirical model we used an output-oriented model which is an adjustment of the global model. This choice is justified, firstly by the recent CAP reforms that were aiming at the liberalization of (agricultural) output prices; secondly, this model has a practical justification related to the unavailability of individual prices for our inputs, as it will be mentioned more in detail below.

In this specific setting, we derive an output price advantage which indicates the effect of the output price environment on the farm's revenue. For this, we defined an output-oriented directional distance function in the output-quantity space ( ) Table 2 presents the mean values and trends of the outputs (constant 2010 prices=100) and inputs. As one can notice, the main source for revenue for an average farm in our dataset is crops, followed by milk and bovine meat. While the trend for each of these productions is positive, it is strongest for bovine meat revenue. At the same time, one notices that the time trend has been positive for all inputs except labour. The fastest growing trend is for intermediate inputs. Note. Trends were estimated using a linear regression between the variable expressed in log and time.

, ; 0,
We estimate LP 4-6 for each year separately. Thus, we dispose of yearly estimations for their technical, value and revenue inefficiency. Based on these estimations, we compute a yearly price advantage and allocative inefficiency for each DMU. Consequently, our estimators are not affected by technical change since any potential shift in the frontier between two years is included in both quantity and value technologies. As a result, the price advantage defined as the difference between two yearly scores is free from technical change.

Allocative inefficiency and price advantages: a joint analysis

Figure 1 shows that allocative inefficiency for DMUs in our dataset ranges 3 -6%. 6 Based on this indicator, one is tempted to say that farms in our dataset are particularly efficient with regards to their relative price system. Furthermore, the proportion of allocative efficient farms varies from 38.9% (in 1994) to 59.3% (in 2013) and was 47.4% for the whole period (1992-2013). However, allocative inefficiency is computed given the observed set of prices for each DMU and does not include any inter-DMU comparison of prices. On the other hand, price advantage is calculated by comparing the set of prices and outputs between DMUs. Moreover, our findings on allocative inefficiency seem to contrast the graph for price advantage. Figure 1 shows the variability of price impact is much wider, with successive situations of favourable price environments (positive values) and of unfavourable price environments (negative values). Further, a net deterioration starting in 2003 can be observed.

Figure 1. Evolution of the per year mean allocative inefficiency and price advantage

Based on a very low, insignificant correlation coefficient between allocative inefficiency and price advantage measures, we can infer that they are independent. 7 Table 3 shows that all combinations between the two measures are possible across the study period. Moreover, a large share of farms (approximately 45%) had either a favourable price environment but were allocative inefficient or had an unfavourable price environment and were allocative efficient. At the same time, we notice that, on average, a relatively low proportion of farms (16.38%) had a favourable price environment and were allocative efficient. During the analysis period, two distinct evolutions may be noted: i) There was an improvement in the proportion of allocative-efficient farms. However, two opposing effects occurred. The proportion of allocative-efficient but suffering from an unfavourable price environment increased, while the proportion of allocative-efficient farms enjoying a favourable price environment decreased. The first evolution was stronger than the latter. ii) There was a decrease in the proportion of farms enjoying a positive price advantage, regardless of whether they were allocative efficient or inefficient. 8

If we consider that farms need to optimize their production decisions with respect to their own set of prices (allocative efficiency), the results suggest that the farms in our dataset performed well.

7

The correlation coefficient of 0.03 was not significant at <5%. 8 The proportion of farms that are neutral from their price impact perspective is constant over time.

However, if we compare prices across farms, results are more mitigated since some performed very well, while others performed very poorly. Moreover, allocative inefficiency assumes that farms modify their input/output mixes, which is an acceptable assumption for mid-to long-run analysis. However, one can expect that for short-run analysis, farms operate with a given mix and thus, our measure of price advantage (or price environment), makes sense. The rest of this article is dedicated to an analysis of the price advantages for farms in our dataset.

These findings are reinforced by the period trends presented in table 4 below. As one can notice, the share of allocative efficient farms has had a positive trend (0.66%, sig. at <5%) during the whole period of study. In contrast, the share of farms benefiting from a positive price environment has had a negative trend (-0.94%). Moreover, while the share of allocative efficient farms enjoying a price advantage has registered a declining trend (-1.19%) throughout the period of study the share of farms of allocative efficient farms but suffering from a negative price environment has had a positive trend (0.93%).

Output-oriented price advantage

Total output-oriented price advantage measure gives an indication of the price environment of farms which can be favourable, unfavourable, or neutral.9 Figure 2 shows the proportion of farms in each category during the analysis period. We notice that the share of farms with a neutral price impact is relatively constant. While the share of farms that have enjoyed a favourable price environment was comparable (with some exceptions) to those with an unfavourable price environment before the application of the reform (1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005), a larger share of farms perform in negative price environment after the reforms. This is confirmed in Table 5 which shows a drastic drop in the mean for the share of farms with a favourable price environment after the introduction of the reform and a simultaneous increase in the mean for the share of farms with a negative price environment. Moreover, farms obtaining negative price impacts were preponderant after 2006. The analysis of the evolution of total-price advantage may conceal some contrasting evolutions according to the type of output and its market structure. Indeed, the intended targets for the 2003 MTR differed according to the type of production. We pursued a decomposition of this effect into the three output-specific price impacts.

Decomposition of the total-price advantage into output-specific price advantages

Table 6 presents the percentage of farms producing a specific output and the valence of their price advantages as per the algebraic decomposition. Over the analysis period, two contrasted effects can be emphasized: the increase in the proportion of farms evolving in positive milkspecific price environments and the decrease in the proportion of the farms performing in cropspecific positive price environments. Figure 3 shows the effect of the 2003 reform on the relative distribution of farms, taking into account that this reform attempted to decouple aids received for the crop production. Our indicator shows that while farms in our dataset used to extract their price advantages from crop production before the reform, they used milk production to obtain their price advantages postreform. Whether it is from a practitioner point of view or a policy evaluation perspective, the decomposition proposed here has a certain appeal. A practitioner could use it in order to identify precisely the main sources for price advantages (or disadvantages). Also, within a larger strategy of policy evaluation, our proposed decomposition can be helpful for measuring whether, generally speaking, farmers respond adequately to the proposed incentives compared to the previous period.

Discussion

Our price advantage indicator is in the vein of the work initiated by [START_REF] Cross | Farrell efficiency under value and quantity data[END_REF]. However, the indicator proposed here is distinct from theirs in several regards. Firstly, we do not deal with the multiplicative decomposition of the value-efficiency score, but construct our indicator as a difference between two efficiency indicators. This allows us to give meaningful economic interpretation to our indicator as a percentage increase in the profit rate due to a favourable output and/or input price environment. Secondly, for our estimations, we use a Färe-Lovell directional distance function type which presents the advantage of eliminating slacks for each output and/or input used and which leads to a decomposition of the total price advantage as the sum of output-and/or input-specific price effects. At the same time, our price advantage measure is also distinct from the "marketing efficiency" concept introduced in [START_REF] Singbo | Estimating farmers' productive and marketing inefficiency: an application to vegetable producers in Benin[END_REF] or the "market efficiency" proposed in [START_REF] Camanho | A generalisation of the Farrell cost efficiency measure applicable to non-fully competitive settings[END_REF]. In the former, additional constraints on marketing outputs are added to a technical efficiency model. In the latter, each DMU is evaluated with a price vector chosen among other DMUs whereas in our approach a DMU is evaluated with its own values computed with their individual prices. Moreover, our indicator is more general as it includes both input and output dimensions and proposes a decomposition of total price advantages into output-and inputspecific price effects. This main methodological contribution of our paper is likely to rouse practitioners' interest who would be able to identify precisely those components of their mix, which contribute to their firm's price advantages (or disadvantages).

The theoretical model developed was then applied to a data set based on a collection of farms in the context of recent CAP reforms. The adaptation of the general model had to consider the specificities of both the political context and the availability of data. For this, an output-value technology had to be introduced. Consequently, this technology differs from the original value technology. While in the latter both outputs and inputs are in value terms and jointly determine the price advantage, in the output-value technology, only outputs are expressed in value terms. Since inputs are measured in quantity terms, the estimated price advantage can be attributed to the output dimension only.

Our results indicated that the share of farms enjoying positive price advantages drops considerably after the 2003 MTR. While some farms used the benefits of new price opportunities, most did not and suffered from price disadvantages. This could imply that due to the Reform, the latter lost the price rents they used to have on their produces. However, if we base our analysis on the concept of allocative efficiency only, we would be tempted to conclude that for the farms in our dataset, their efficiency slightly increased with time. Thus, one cannot conclude in favour of a possible association between this observed evolution and the 2003 MTR. Finally, given the decomposition of the total price advantage as the sum of output-specific price effects we observed a decrease in the crop price advantages after the adoption of the 2003 MTR. This phenomenon was accompanied by an increase in the milk price advantage and a stable level for the bovine-meat price effects.

This is, of course, a first empirical attempt to highlight the operationality of our concept of price advantages developed, from a theoretical point of view, in section 2. While we have paid attention to the selection of our sample and cleaned the data for potential outliers, the empirical analysis could be further improved by using robust estimators, such as the ones discussed in [START_REF] Daraio | Advanced robust and nonparametric methods in efficiency analysis: Methodology and applications[END_REF] or more recently in [START_REF] Ferreira | Economies of scope in the health sector: The case of Portuguese hospitals[END_REF].10

Conclusion

This paper is premised on the observations that allocative-efficiency measure, based on the DMU's own relative price system, assumes that reallocation of physical resources is possible at the DMU level. To complete the picture of the impact that prices have on the DMU efficiency, we develop the concept of "price advantage" which is computed at the observed input/output mix. Price advantage indicator is defined as the difference between technical efficiency scores measured with quantity-based data and value-based data.

This methodology was used to observe farms' economic efficiency evolution in the French department of Meuse over the period 1992-2013 during which successive CAP reforms were adopted. We observed that farms' price advantages decreased over time and mainly after the application of the 2003 MTR. According to the objectives of the Reform, farmers' price rents decreased on average. This conclusion could not have been drawn from the traditional analysis of allocative efficiency.

In subsequent studies we intend to identify exogenous explanatory factors for different price advantages. For example, the amount of aids received by a farm; the structure of aids, whether coupled or decoupled; and the degree of short/long-term indebtedness are all possible explanatory factors to the performance of farms in terms of the prices obtained, and deserve further investigation.

Appendix. Linear programs used for our estimations

Output-oriented distance function in the output-quantity space was estimated with LP 4 below. Note that in this program, we considered a control variable, Z corresponding to revenue obtained by the farm from other secondary produces. Finally, optimal revenue was estimated with LP 6:

  vector with all elements being equal to 1. This new directional distance function has an intuitive economic interpretation. The denominator of the objective function is the observed revenue ( O e VO ) while the numerator is the profit increase due to cost reductions and revenue increases measured by reaching the frontier in the direction defined by Q g . Therefore, DQ can be interpreted as an increase in the profit rate.

  ) in the value space. For a given DMU a, the directional distance function in the quantity space becomes:

  and a value inefficiency ( ) a VAI which gives the percentage increase in a DMU a profit rate by adopting optimal price and quantity decisions:

  a DMU's observed profit with regards to observed revenue ( ) as the residue between profit inefficiency (PROI a ) and technical inefficiency (TI a ) represents the gap between optimal profit ( a percentage of the actual (observed) revenue ( )

  VO was defined by assuming null input

Figure 2 .

 2 Figure 2. The evolution of the structure of farms according to the type of price advantage

Figure 3 .

 3 Figure 3. Sub-period means for the shares of farms obtaining either a favourable or an unfavourable output-specific price advantage.

  of the revenue of output j in the total revenue of DMU a.The following program was used to obtain output -oriented distance function in the output-value space.

QI QO QI,QO QI,QO;g ,g

	The technology ( ) Q T ∈ ⇔ DQ Q T can be characterized by the directional distance function, since ( ) 0 CCF ≥
		QI,QO;g , g	QI	QI g QO	g	QO	(6).
	where	Q g = (g ,g ) is a strictly positive vector defining the direction of projection on the QI QO
	frontier, and CCF in	CCF DQ stands for Chambers, Chung, Färe.

QI QO QI QO VI VO VI VO

  18).

	Proof.
	j	PAO	a j	k	PAI	a k	j	(			QO j	* VO	a	VO j	*	)	VO	a j		k	(	QI k	* VO	a	VI k	*	)	VI	a k
											QO j	* VO	a j				QI k	* VI	a k			VO j	* VO	a j	VI k	* VI	a k
								j												k							j	k
								DQ	(		a	,		a ; a VO	a	,		a	)	DV	(	a	,	VO	a a	;	a	,	a	)
								PA	a	.

Table 1 .

 1 Number of farms per year used to determine the benchmark frontier

	Before	1992 1993	1994	1995	1996	1997	1998	1999	2000	2001	2002	2003	2004	2005
	2003
	MTR	600	632	644	660	642	644	631	639	621	614	581	588	257	551
	After				2006	2007	2008	2009	2010	2011	2012	2013
	2003
	MTR				515	521	521	481	465	433	376	351

Table 2 .

 2 Period means and trend for the revenues of the three main aggregate outputs (in constant 2010 prices) and the four inputs

		Crop	Bovine meat	Milk	Utilised agricultural	Full-time	Intermediate	Fixed
		revenue	revenue	revenue	area (ha)	equivalent	inputs	capital
	Period mean	121 177	19 399	74 295	188	2.2	113 573	75 519
	Trend (%),	1.82	3.43	1.98	1.33	-0.4	1.57	0.61
	sig. < 5%

Table 3 .

 3 Period means for the share of farms according to their price environment and allocative inefficiency scores

	Allocative efficient	Allocative inefficient

Table 4 .

 4 Period trends (sig. at <5%) for the share of farms according to their price environment and allocative inefficiency scores

	Allocative efficient	Allocative inefficient
	(trend 0.66)	(trend -0.67)

Table 5 .

 5 Sub-period means of farms pre-and post-reform according to the valence of their total-

price advantage Positive price advantage Negative price advantage Neutral price advantage Pre-reform (1992-2005)

		42.31	46.69	11
	Post-reform (2006-2013)	27.87	60.82	11.3

Table 6 .

 6 Percentage (and trends) of farms producing a specific output and producing a specific output, according to valence of the output-specific price advantage

		Positive price advantage Neutral price advantage	Negative price advantage
	Crops	28.9 (trend -1,2)	25.7 (ns)	45.4 (trend 1,1)
	Bovine meat	25.9 (ns)	49.4 (ns)	24.7 (ns)
	Milk	38.1 (trend 1,2)	23.0 (ns)	38.9 (trend -1,3)

Note that[START_REF] Camanho | Cost efficiency measurement with price uncertainty: A DEA application to bank branch assessments[END_REF] extended the theoretical Farrell cost efficiency framework (that requires complete information on inputs prices) and addressed scenarios of incomplete price information.

Note that this cost-share gives a measure of the importance of one input compared to the others. As total revenue can be different from the total cost, the sum of these cost-shares is not always equal to 1.

Our observations are from the Centre d'Economie Rurale de La Meuse which audits farmers' accounts. Note that the dataset was financed as a part of an agreement with Institut national de la recherche agronomique (INRA).

Given allocative inefficiency definition in (20), a DMU is considered as efficient when the measure is equal to 0.

A neutral price environment is obtained in two situations; either the farm obtains the same inefficiency scores with the quantity-based and "value"-based technologies or, the farm is "efficient" according to the two measures. In our case, all instances of "neutral" price environments correspond to situations in which the farms were efficient.

We thank one of our anonymous reviewers for having pointed us into this direction for possible improvements in our empirical application.

components in the direction V g . Note that in this restricted framework corresponding to the output-value technology, only outputs are observed in value terms while inputs are (still) expressed in quantity or volume terms. This is a necessary condition in order to emphasize specific output-price advantages by keeping identical the input side. The interpretation of the output-oriented price advantage is similar to the one presented in the general case. When the obtained measure for DMU a is negative, the DMU's output pricequantity couple is suboptimal with regards to its price environment. In other words, the gap between observed revenue and "technically efficient revenue" obtained with optimal quantities is greater than the gap between revenue and "value-efficient revenue" obtained with optimal output price and quantity couples. In this context, this measure is interpreted as the potential decrease in the revenue measured in percentage due to an unfavourable output price environment. Conversely, when price advantage of a DMU is positive, the DMU would operate in a favourable price environment and the measure indicates the potential increase in the revenue in percentage due to the favourable output-price environment. Finally, when price impact is null, the evaluated DMU acts in a neutral price environment (value and technical efficiency are the same).

LP 1-3 were adapted in order to estimate output-oriented distance function in the output-quantity space and in the output-value space respectively (the resulting programs are presented in the Appendix LP 4-6).

The technology includes 3 outputs (S = 3 in LP 4-6) related to: crops (containing several produces, namely, wheat, barley, corn, peas, rapeseed, and sunflower), bovine meat (containing beef, young bulls, and cows), and bovine milk. The dataset provides necessary information regarding physical production and revenue obtained by the farm for each produce. First, to estimate technical efficiency, we used volumes of each output, which required that several produces needed to be aggregated. For example, a DMU crops volume is an aggregate of wheat, barley, corn, peas, rapeseed, and sunflower quantities. For that, we computed a price index common to all DMUs given by the average weighted individual prices. By using the same price index for all farms within the same year, all differences between farmers' output volumes reflected a quantity effect. Therefore, technical-efficiency measures based on volumes are equivalent to quantity estimations (LP 1) in our theoretical model. Second, to estimate value efficiency, we used observed farm's revenues for each produce based on individual prices. Consequently, our estimated price advantage is specific to each farm. Besides these three main outputs, farms in our dataset engaged in other activities (pig, sheep, and/or poultry production, market gardening, and arboriculture). We need to control for these side activities in order to reduce farm heterogeneity in our sample. Therefore, we introduced an additional constraint in LP 4-6 related to this control variable Z in LP 4-6 (cf. appendix), which is defined as farm revenue from these side activities. Since Z is a control variable, no efficiency score is associated and Z does not appear in the objective function. In terms of inputs (M = 4 in LP 4-6) we used total cultivated area, labour (full-time equivalent, be it family and/or non-family), intermediate inputs including operating costs (e.g. fertilizers, seeds, and pesticides) and other intermediate inputs (water, electricity, fuel, etc.) and capital cost including capital-expenditures aggregate depreciation (equipment, buildings), and agricultural contractors. The first two inputs are measured in physical quantities and the last two inputs are measured in volumes expressed in constant 2010 prices.


L Diallo

L Lechevallier

A Fnidiki

J Juraszek

M Viret

A Declémy

Characterization of nanostructure in low dose Fe-implanted p-type 6H-SiC Using Atom Probe Tomography

Keywords: Spin electronics, diluted magnetic semiconductor, atom probe tomography, nanoclusters

P-doped 6H-SiC substrates were implanted with Fe ions. During implantation the samples were maintained at 550°C with energies ranging from 30 to 160 keV in order to produce a diluted magnetic semiconductor (DMS) with a 2 % Fe homogeneous concentration to a depth of about 100 nm thickness. The effects of rapid thermal annealing on the microstructure were examined by atom probe tomography (APT). The study shows evidence of the formation of Fe rich nanoclusters after annealing which contain core magnetic phases that contribute to the magnetic properties.

Introduction

Doping a semiconductor with a 3d magnetic element (Cr, Mn, Fe, Co) confers ferromagnetic (FM) properties, while keeping the semiconductor nature of the material. The discovery of DMSs with relatively high Curie temperatures (T C ), such as (In, Mn) As [1], and especially (Ga, Mn) As [2], has initiated a major international research effort over the last two decades, in order to understand the origin and the mechanism of the magnetic properties of these types of materials. Many advances have been made in the growth and annealing of these samples to minimize the presence of secondary phases or compensate defects that are detrimental to the desired properties. The possibility to achieve the T C values higher than 300 K was forecasted for certain wide-gap DMSs with rather a high concentration of free holes [3]. This prognosis gave impetus to a great number of experiments aimed at the discovery of DMS materials with high T C . Many authors reported that they observed ferromagnetism in the corresponding objects above room temperature. However, most of these observations later turned out to be the consequence of the presence of precipitates in the studied specimens or the inclusion of other phases of transition metal (TM) compounds in solid solutions to which the DMS specimens belong [4]. This circumstance stimulated the appearance of theoretical works, showing that the ferromagnetic (FM) ordering in such specimens can be partly explained by the exchange interaction between charge carriers and doping magnetic ions [5]. Still, despite this intense research in the DMSs field, no T C higher than room temperature has ever been obtained. Moreover, the nature and origin of ferromagnetism in III-V compounds remain controversial [6]. Silicon Carbide (SiC) is a wide bandgap semiconductor that has been considered a possible candidate for spin electronics applications. SiC has a long history in materials research and device development and has already been commercialized for high-frequency and high power applications. A FM response has been experimentally observed in Ni, Mn and Fe doped 6H-SiC with the value of the T C varying from significantly below to close to room temperature. The authors assigned the magnetic signal to the true DMS behavior [7][8]. According to Dupeyrat et al. [9] and Diallo et al. [10], the formation of Fe 3 Si nanoscale particles is mainly responsible for the magnetic properties observed in 6H-SiC system implanted with a 6 at.% Fe ions concentration and annealed at a high temperature (> 900 °C). This is not the case according to ref 11 where authors demonstrate that the presence of Fe 3 Si is not the origin of FM ordering in Fe-doped SiC. It also indicated that traces of Fe-doping in SiC induce a high temperature FM arrangement. Typical FM order was established for low Mn-doping concentrations (10 -4 molar fraction) at around 250 K in 6H-SiC [12]. These authors found that defects-related effects other than the Mn content play the most important role in the magnetic ordering. Ferromagnetism was experimentally evidenced in 6H-SiC after neutron irradiation by Liu et al. [13]. Ab initio calculations using different computational techniques were used to theoretically study magnetic properties of SiCbased DMSs. Contradictions were also found in these results. Fe was found magnetic or nonmagnetic according to the structural configurations, while Cr and Mn were found to be magnetic at the substitutional Si site [14][15][16]. Recently Los et al. [16] reported that Fe in SiC matrix can exist in both magnetic and nonmagnetic states depending on the transition metal atom environment in the host matrix. For different impurity species, these authors found that the transition from the nonmagnetic to magnetic configuration and the corresponding changes of the atomic moments may take place either gradually, when the volume of the unit cell changes, or when the energy gap between the two states is overcome.

In recent communications [10,[17][18] it was shown that Fe atoms in the substitutional Si site lead to FM behavior in 6H-SiC. Here, we report about the characterization of the nanostructure in 2 at.% Fe implanted 6H-SiC by APT. This technique revealed a random distribution of Fe atoms in the SiC matrix in the as-implanted sample and the existence of Fe-rich nanoparticles in the annealed samples.

.

Material and Methods

Commercial (CREE) near (0001)-oriented n-type 6H-SiC substrates with a 200 nm thick uppermost ptype epitaxial layer (nA -nD ~10 +20 Al/cm 3 ) were co-implanted with 30 to 160 keV 56 Fe + ions. The dose reaches 10 +16 /cm 2 leading to a Fe concentration of about 2 at.% at depth between 20 nm and 100 nm from the sample surface, checked by Rutherford Backscattering Spectrometry (RBS) at the CSNSM-Orsay [19,20]. The samples were held at 550 °C during implantation in order to avoid amorphisation [21]. After the implantation, the samples were subjected to a rapid thermal annealing at different temperatures (900 and 1300°C) for 4 min. The principle of the APT technique is based on field evaporation of surface atoms from a sample prepared in the form of a sharp needle (radius < 50 nm). The chemical identification of the sample atoms is based on the time-of-flight mass spectrometry technique. The evaporated atoms are collected on a position-sensitive detector that provides both time of flight and position of the incoming ions. This combined information allows deducing the chemical nature of the evaporated ions and the initial position at the specimen surface (sharp tips) [22]. A 3D reconstruction of the analyzed volume is generated, giving access to the spatial distribution of atoms at the atomic scale. Local chemical composition of any arbitrary sub volumes can be obtained with a sub-nanometric resolution. In particular, concentration profiles with a depth resolution better than half an atomic plane can be generated in metals [23]. APT specimens were prepared through the lift-off method and standard milling [24] using a dual beam SEM/FIB Zeiss Nvision 40. In order to reduce Ga implantation and avoid damages in the region of interest, the Fe-implanted SiC samples were capped with 500 nm of Pt and a final polishing was performed at low acceleration voltage (2 kV). The samples were analyzed by a Laser Flexible Tomographic Atom Probe (LAWATAP) at 80 K in an ultrahigh vacuum chamber under a pressure of 10 -8 Pa. The used femtosecond laser pulse system was an amplified ytterbium-doped laser (AMPLITUDE SYSTEM s-pulse HR) with a pulse period of 350 fs and a 100 kHz repetition rate. The wavelength was fixed at 342 nm.

RESULTS and DISCUSSION

Structural and chemical composition

A typical mass spectrum of 6H-SiC implanted with a 2 at.% Fe concentration and annealed at 1300 °C is shown in Fig. 1. C atoms evaporate in the form of single or molecular ions in these conditions. The peak at 28 amu (atomic mass unit) is assigned to the 56 Fe 2+ isotope. Si 2+ ions are detected predominantly in the second charge state at 14, 14.5, and 15 amu. Al 3+ ions were also detected which correspond to the p-doping species in these samples. . The APT pattern evidences the presence of nano-sized Fe-rich nanoparticles, represented in this figure with different colors. The average concentrations of Si, C and Fe atoms over the entire analyzed volume are 50.1 ± 0.1%, 46.3 ± 0.1 and 1.9 ± 0.1%, respectively. The sum of these three elements being 98.3 at.%, the remainder (1.7 %) consists of Al and O atoms, corresponding to the initial electrical doping of SiC and the oxidation of the material, respectively. We also can observe that the Si and C concentrations are not exactly the same. This is due to the presence of molecular species such as C 2 and C 3 and to the difficulty to separate, for example, the two C + and C 2 2+ species obtained for the same M/n ratio (12 amu). Thus, it results that the uncertainty on the C concentration is perhaps weakly higher than ± 0.1 % but remains of the same order.

Nanoparticle compositions

The identification of clusters makes it possible to label each Si, Fe or C atom belonging to the different nanoparticles and thus to measure the average compositions of the Fe-rich nanoparticles. The average Fe concentration varies versus the average diameter of nanoparticles. It is 39, 51 and 64 at.% for nanoparticles having an average diameter of 6, 7 and 10 nm, respectively. For each identified nanoparticle, an associated mass spectrum is obtained and a radial profile is plotted. It can be noted that the largest nanoparticles contain several tens of thousands of Fe, Si and C atoms and the smallest ones contain a few tens of atoms, only. The core phases are identified by reporting the concentrations obtained in the central part of the nanoparticles in a sphere having a 1 nm diameter. The Fe atoms concentration varies greatly from one nanoparticle to another, indicating the presence of (FeSi) rich phases containing small C amounts. The values of the Si, C and Fe concentrations depend, as observed above, on the different atomic or molecular evaporated species as well as on the reconstruction artifacts of the APT which introduce position errors of atoms. Indeed, the difference of evaporation fields between the atoms located in the nanoparticles and in the matrix leads to local magnification effects in the reconstruction process, which decreases the spatial resolution. In our case, the nanoparticles are mainly composed of (Fe-Si) phases which are evaporated under a low evaporation field, close to the Fe evaporation field (i.e. E Fe ~ 33 V.nm -1 ), whereas the Si and C atoms of the SiC matrix are evaporated under a high evaporation field.

The evaporation fields of the matrix and the nanoparticles have been evaluated from the C, Si and Fe Kingham curves [25] [26]. In our samples, Fe and Si are mainly evaporated in the Fe 2+ and Si 2+ ions, as can be observed in the mass spectrum. We also obtain the two C + and C 2+ ions for the C atom. These considerations allow evaluating the Fe evaporation field close to 30 V/nm and the evaporation field of the matrix (only considering the C + and C 2+ ions amounts of the same order for the C atom) close to 40 V/nm. The Fe evaporation field determined from the Fe Kingham curve is consistent with the pure Fe evaporation field (33 V/nm). This is not surprising because the nanoparticles contain Ferich phases. The evaporation field of the SiC matrix atoms is higher than the one of the nanoparticles atoms. Thus, it can be assumed that the atoms of the SiC matrix are evaporated under a mean evaporation field E SiC ~ 40 V/nm, which is much higher than the evaporation field of nanoparticles [START_REF] Lardé | [END_REF].

The difference of evaporation fields between the matrix and the nanoparticles leads to an over-density of atoms inside the nanoparticles and a sub-density of atoms around the nanoparticles in the 3D reconstructions, a well-known artifact of the APT.

The ratio between the evaporation fields of the particles and the matrix is close to 0.75. This ratio is very important to better understand the local magnification effects and trajectory overlaps leading to the 3D reconstruction process of the samples. It has been shown that for ratios smaller than ~0.6, the trajectories of the evaporated atoms lead to a negative magnification and a mixing of the atoms contained in the particles [START_REF] Lefebvre | Atom probe tomography, Put theory into practice[END_REF]. This was not the case for our samples with a f ratio (0.75) higher than 0.6. It means that only the first atomic shells are impacted by this phenomenon and are intermixed with the neighborhood of the matrix. Only the smallest particles (with a diameter smaller than ~2 nm) can be considered by this artifact rather than the biggest particles (up to 3 or 4 nm diameter).

Taking into account the true concentrations in the nanoparticles core, we know the number of Fe, Si and C atoms (N Fe , N Si and N C , respectively) in the core and we can calculate the N Fe /N Si ratio. It appears that the values obtained for this ratio are very often close to 3.0, 2.0, 1.66 or 1, corresponding to the Fe 3 Si, Fe 2 Si, Fe 5 Si 3 and FeSi core phases. This confirms that the APT artifacts do not impact the core of the biggest particles. We do not obtain a N Fe /N Si ratio higher than 3.0. Of course, all of these phases are not pure Fe-Si phases; they all contain a small amount of C atoms. In all the cases the C concentration is always smaller than the Si concentration. For the biggest nanoparticles (diameter ~7 to 12 nm) and the two Fe-richest phases, the C core concentrations are smaller than 10 at.%. The N Fe /N Si ratios are 3.00 ± 0.05 and 2.00 ± 0.05 and thus obtained with good accuracy. We do not find the Fe 8 Si 2 C or FeSiC phases or other ternary phases with a C concentration higher than the Si concentration [28]. Moreover, the Fe 3 Si phase observed in the biggest nanoparticles of our samples has been characterized by Mössbauer spectrometry and exactly corresponds to the DO 3 structure [17]. Consequently, the C atoms enter the Fe 3 Si phase, randomly and likely in interstitial sites rather than substitutional sites, but maintain the DO 3 structure of this phase.

We observed that the number of nanoparticles in a volume depends on the size. The biggest nanoparticles are the least numerous but also the Fe-richest (Fe 3 Si and Fe 2 Si phases). When the nanoparticles become smaller, they are more numerous but also less Fe-rich (Fe 5 Si 3 and FeSi phases). When the nanoparticles become still smaller with a 2 or 3 nm order diameter, the core phase is hardly measurable due to the 1 nm order nanoparticle/matrix interface size and the APT artifacts described above.

Figure 3 represents Fe, Si and C atomic concentrations of the nanoparticles versus their average size. We can observe that, when the average size of the nanoparticles increases, the Si and C concentrations decreases and the Fe concentration increases. The C concentration decreases below 10 at.% for average size nanoparticles higher than 8 nm. For these nanoparticles, the Fe concentration is higher than 60 at.% allowing us to reach the Fe 3 Si phase composition. We also observe that the Fe-richest nanoparticles (with Fe 3 Si core phase) are obtained for the annealing performed at the highest temperature (here 1300°C). When the annealing temperature is lower than 1300°C (for example 1100 or 900°C but not shown here), we do not obtain the Fe 3 Si phase. When the annealing temperature decreases, the core phases becomes less Fe-rich. This is perfectly consistent with the values of the enthalpy of formation of the Fe-Si phases that increase with the Fe amount contained in the Fe-Si phases and consequently with the annealing temperature [29]. 4 shows a 3D reconstruction of a small volume containing only one nanoparticle (nanoparticle 3 taken as an example) and the corresponding concentration profile in the Z-direction of analysis. The cylinder used to determine the Si, C and Fe concentrations and plot the profile of Figure 4 (b) had a 0.5 nm diameter. Its size is sufficient to give an accurate profile of this nanoparticle. We can observe that the Fe, Si and C concentrations in the core of this nanoparticle are approximately 60, 30 and 10 at.% giving a N Fe /N Si ratio equal to 2. Thus, this nanoparticle contains a Fe 2 Si type core phase with a small C amount (the C concentration being much lower than the Si one). Taking into account the C concentration (10 at.%), we are here in the presence of a non-pure Fe 2 Si core phase as mentioned above, instead of a Fe 2(1-x) Si (1-x) C 3x type compound. According to these above observations, the present phase is the Fe 1.8 Si 0.9 C 0.3 phase which does not correspond to a Fe-Si-C stable ternary phase. Given that, all the nanoparticles are obtained in thin films samples and appear during 4 mn annealing and slow cooling processes. It is suitable to talk about Fe-Si phases with C amount rather than ternary Fe-Si-C phases because ternary particles predicted by ternary phase diagram were not observed. At the same time need to underline that the volume fraction of particles is not equilibrium one.

As-implanted sample

Figure 5 shows 3D reconstructions of the same volume of the Fe as-implanted 6H-SiC sample; the Si, C and Fe atoms are represented in red, green and blue, respectively. We note that this sample has not been annealed, but has been maintained at 550°C for the entire duration of the Fe implantation. This sample does not contain Fe, Si and C nanoparticles as can be observed in these 3 reconstructions. This sample is perfectly diluted, semiconductor, but it is non-magnetic. It is the application of an annealing that can lead to the appearance of the magnetism. This annealing must be sufficient to generate magnetism without being so high that it minimizes the appearance of nanoparticles.

Conclusion

This work, performed by APT, shows that 550°C Fe-implanted 6H-SiC samples with a 2 at.% Fe concentration are perfectly diluted after implantation but contain nanoparticles after annealing. These particles mainly contain Fe-Si phases rather than Fe-C phases because of the great chemical affinity between the Fe and Si atoms. Thus, for the great majority of nanoparticles, the Si concentration is always higher than the C concentration. It is notably the case for the most annealed sample (1300°C) in which we can observe the Fe-richest (Fe-Si) phases, such as the Fe 3 Si and Fe 2 Si phases. Other less Fe-rich phases, such as the Fe 5 Si 3 and FeSi phases, are also present in this sample, annealed only 4 minutes. All these phases are not pure (Fe-Si) phases but contain small C concentrations which are generally smaller than 10 at.% for the biggest nanoparticles. The annealing performed at lower temperatures (1100 and 900°C) also lead to the appearance of nanoparticles, but less Fe-rich. In the 1100 and 900°C annealed samples, we do not find the Fe 3 Si and Fe 2 Si phases. As mentioned above, the as-implanted sample is perfectly diluted but it is non-magnetic. It is the application of annealing (temperature, duration) that can lead to the formation of a magnetic sample but an annealing at too high a temperature also generates nanoparticles. To obtain a DMS with Fe-implanted 6H-SiC samples, it will be necessary to find an appropriate annealing (temperature and duration) and also perhaps an implantation temperature to generate magnetism in the samples to minimize the appearance of nanoparticles.

Fig. 1 .

 1 Fig. 1. Mass spectrum of the 1300°C annealed Fe-implanted 6H-SiC sample

Fig. 2 .

 2 Fig. 2. 3D reconstruction of the 1300°C annealed Fe-implanted 6H-SiC sample showing the spatial distribution of Fe atoms in a 70 x 70 x 73 nm 3 volume in a red color matrix. Nanoparticles are represented with different colors and their associated phases (Fe 3 Si, Fe 5 Si 3 , Fe 2 Si) are indexed with the numbers (1, 2, 3), respectively

Fig. 3 .

 3 Fig. 3. Fe, Si and C atomic concentrations of the nanoparticles versus the average diameter size

Figure

  Figure4shows a 3D reconstruction of a small volume containing only one nanoparticle (nanoparticle 3 taken as an example) and the corresponding concentration profile in the Z-direction of analysis. The cylinder used to determine the Si, C and Fe concentrations and plot the profile of Figure4(b) had a 0.5 nm diameter. Its size is sufficient to give an accurate profile of this nanoparticle. We can observe that the Fe, Si and C concentrations in the core of this nanoparticle are approximately 60, 30 and 10 at.% giving a N Fe /N Si ratio equal to 2. Thus, this nanoparticle contains a Fe 2 Si type core phase with a small C amount (the C concentration being much lower than the Si one). Taking into account the C concentration (10 at.%), we are here in the presence of a non-pure Fe 2 Si core phase as mentioned above, instead of a Fe 2(1-x) Si (1-x) C 3x type compound. According to these above observations, the present phase is the Fe 1.8 Si 0.9 C 0.3 phase which does not correspond to a Fe-Si-C stable ternary phase. Given that, all the nanoparticles are obtained in thin films samples and appear during 4 mn annealing and slow cooling processes. It is suitable to talk about Fe-Si phases with C amount rather than ternary Fe-Si-C phases because ternary particles predicted by ternary phase diagram were not observed. At the same time need to underline that the volume fraction of particles is not equilibrium one.

Figure 4 :

 4 Figure 4: (a) 3D reconstruction of a small volume (9 x 7 x12 nm 3 ) containing only one nanoparticle and: (b) Corresponding Si, C and Fe concentration profile in the analysis Z-direction

Figure 5 :

 5 Figure 5: 3D reconstructions of the Si (a), C (b) and Fe (c) atoms of the 550°C Fe as-implanted sample

Acknowledgments

This work was funded by Region of Normandy and the European Regional Development Fund of Normandy (ERDF) in the frame of the MAGMA project. The authors would like to thank F. Danoix, F. Vurpillot, H. Zapolsky and D. Blavette for useful and fruitful discussions about APT.

Corresponding authors: lindor.diallo@etu.uni-rouen.fr