István A Kovács

Jean-Christian Anglès D'auriac

Ferenc Iglói

Excess entropy and central charge of the two-dimensional random-bond Potts model in the large-Q limit

We consider the random-bond Potts model in the large-Q limit and calculate the excess entropy, SΓ, of a contour, Γ, which is given by the mean number of Fortuin-Kasteleyn clusters which are crossed by Γ. In two dimensions SΓ is proportional to the length of Γ, to which -at the critical point -there are universal logarithmic corrections due to corners. These are calculated by applying techniques of conformal field theory and compared with the results of large scale numerical calculations. The central charge of the model is obtained from the corner contributions to the excess entropy and independently from the finite-size correction of the free-energy as: limQ→∞ c(Q)/ ln Q = 0.74(2), close to previous estimates calculated at finite values of Q.

I. INTRODUCTION

Entropy represents a fundamental concept in different domains of science, such as in information theory 1 , in quantum systems 2 and in classical statistical mechanics. In quantum systems the entanglement entropy turned out a very important indicator of new and exotic phases and quantum phase transitions [2][3][4][5] . Its analogue in information theory is the mutual information and in classical statistical mechanics the excess entropy. Also in classical systems the excess entropy can be defined as mutual information, where the probabilities are given as Boltzmann weights [6][7][8] . In these calculations the system is divided in two (or more) parts, and the excess entropy is basically associated with the interface separating the subsystems. Due to this, the excess entropy is proportional to the surface of the interface, which is called the area law. In critical systems, however, usually there are universal corrections to the area law, which are logarithmic in the linear extent of the interface. In conformally invariant systems, such as in one-dimensional quantum and in two-dimensional classical models the central charge of the conformal field theory can be deduced from the sizedependence of the critical entanglement entropy 2-5 and mutual information 8 , respectively.

Concerning classical statistical mechanics most of the studies described in the previous paragraph are performed on non-random systems. It is known, however, that systems with quenched disorder are also conformally invariant, provided the properties of averaged quantities (magnetization, correlation function, etc.) are concerned. Therefore it is of interest to study the scaling properties of the excess entropy in random systems, too. For this purpose we consider the two-dimensional Q-state Potts model 9 in the presence of bond disorder 10 . In two dimensions (2D) the phase transition in the random bond Potts model (RBPM) is of second order [11][12][13] for any value of Q, even in the limit Q → ∞. The critical behavior of this model has been studied by different methods [14][15][16][17][18] , in particular the critical exponents and the central charge has been calculated, mainly for Q > 4, in which case the phase transition in the non-random model is of first order 19 . Special attention has been paid to the model in the large-Q limit, in which case the critical parameters are found to be smooth function of 1/ ln Q 20 . For example from the numerical data calculated at large, but finite values of Q the central charge is conjectured to be 20

lim Q→∞ c(Q)/ ln Q ≡ c ′ = 1/(2 ln 2) = 0.7213.
Later, it has been shown 21 that the model can be studied directly at the limiting value Q → ∞, when in the random cluster representation 22 the partition function of the model is dominated by one term, the so called optimal graph. This means that thermal fluctuations are negligible compared to disorder fluctuations, thus the critical behavior of the system is controlled by a so called infinite disorder fixed point 26 . The optimal graph of the RBPM has been calculated by a combinatorial optimization method 23 , which provides the exact value of the partition function for a given sample, i.e. for a given realization of the disorder. From the numerical data exact values of the critical exponents are conjectured [START_REF] Anglès | [END_REF]25 through an expected relation with the exactly known infinite disorder fixed point of the random transverse-field Ising chain 27 : x m = (5 -√ 5)/4 (bulk magnetization), x s m = 0.5 (surface magnetization) and ν = 1 (correlation length).

In this paper we study the scaling properties of the excess entropy in the RBPM in the large-Q limit. For this we consider a subset of bonds, Γ, and calculate the corresponding excess entropy, S Γ . If Γ is a closed loop, separating a subsystem, A, from the rest of the system, B, then S Γ is the mutual entropy S Γ = S A + S B -S A∪B . In the following section we show, that in the random cluster representation S Γ is simply given by the mean number of clusters in the optimal sets which are crossed by Γ. This type of problem has already been considered by two of us in the case of the non-random Potts model both for Q = 1, representing percolation 28,29 and for general values of Q ≤ 4 30 . Repeating the reasoning applied in these papers we show that the dominant term of S Γ represents the area law to which there are logarithmic corrections at the critical point due to corners and these are calculated by conformal techniques. The analytical conformal conjectures are then confronted with the results of large scale numerical calculations for different forms of the contour. These results involve the central charge of the RBPM, for which we calculate a precise estimate.

The rest of the paper is organized as follows. The model, its solution in the random cluster representation and the calculation of the excess entropy is presented in Sec. II. Numerical results for the corner contribution to the excess entropy are presented in Sec. III. Independent estimates for the central charge of the model through analyzing the finite-size correction of the free-energy is given in Sec. IV. Finally, Sec. V contains our conclusions.

II. RANDOM BOND POTTS MODEL IN THE LARGE-Q LIMIT

We consider the Q-state Potts model defined by the Hamiltonian 9

H = - i,j J ij δ(σ i , σ j ), (1)
in terms of the Potts spin variables

σ i = 0, 1, . . . ,Q -1.
The J ij > 0 couplings between nearest neighbor sites are i.i.d. random variables and in the following we restrict ourselves to the square lattice. In the random cluster representation 22 the partition function of the model at T = 1/β temperature is given by:

Z = G Q Ntot(G) ij∈G e βJij -1 (2)
where the sum runs over all bond configurations G, and in G the total number of connected components (clusters) are denoted by N tot (G). The mean number of clusters is given by:

N tot = ∂ln Z(Q) ∂ ln Q . ( 3
)
where • • • denotes thermal averaging and • • • stands for the average over quenched disorder.

Let us now introduce a subset of bonds, Γ, and fix all spins on Γ (in state 0, say) but leave the couplings unchanged. Then the partition function becomes:

Z fix = G Q Ntot(G)-NΓ(G) ij∈G e βJij -1 (4)
where N Γ (G) denotes the number of clusters which intersect Γ. Consequently:

N tot -N Γ = ∂ln Z fix (Q) ∂ ln Q . ( 5
)
Now let us take the large-Q limit, in which case the entropy scales as S ∼ ln Q, thus it is convenient to use the reduced entropy: S ′ = S/ ln Q and the reduced temperature:

T ′ = T ln Q (β ′ = β/ ln Q).
These reduced quantities are of O(1) at the phase transition region.

In terms of β ′ the partition function in Eq.(2) reads as

Z = G Q Ntot(G) ij∈G Q β ′ Jij -1 (6)
in which for large-Q we have

Q β ′ Jij ≫ 1, thus Z = G Q φ(G) , φ(G) = N tot (G) + β ′ ij∈G J ij , (7)
which is dominated by the largest term, φ * = max G φ(G).

Finally we arrive at

Z = n 0 Q φ * , (8)
where the degeneracy of the optimal set is n 0 = O(1).

The free-energy of the system, F is proportional to the mean value of φ * :

φ * = -β ′ F = S ′ -β ′ E (9)
where the reduced entropy of the system is S ′ = N tot (G * ) and the mean energy is given by: E = -ij∈G * J ij . Similarly we obtain for the reduced entropy of the system with a contour of fixed spins as:

S ′ fix = N tot (G * ) -N Γ (G * )
, consequently the excess entropy associated with the contour is given by the mean number of clusters crossed by Γ:

S ′ Γ = N Γ . (10)
Using this relation S ′ Γ can be calculated numerically, which will be performed in the following section.

On the other hand analytical results on S ′ Γ can be obtained from the difference of N tot and N tot -N Γ in Eqs.(3) and (5), which is the derivative of ln Z(Q)ln Z fix (Q). At the critical point this difference is given by: ∼ L Γ f s (Q) + C Γ ln L Γ , where L Γ is the linear size of the contour, f s (Q) is the surface free-energy density, which is non-universal and the second term represents the corner contribution 31 . Thus we obtain for the excess entropy:

S ′ Γ = -Qf ′ s (Q)L Γ + b Γ (Q) ln L Γ , (11)
in which the first term corresponds to the "area-law" and in the second term the prefactor is factorized as:

b Γ (Q) = ∂c(Q) ∂ ln Q A Γ = c ′ A Γ , (12)
where c(Q) = c ′ ln Q + cst. is the central charge of the RBPM for large Q. A Γ is a geometrical factor, which does not depend on Q and it follows from the Cardy-Peschel formula 31

A Γ = 1 24 k γ k π - π γ k + 2π -γ k π - π 2π -γ k , (13)
where γ k is the interior angle at each corner.

In the numerical calculation we have considered finite samples of size L × L with periodic boundary conditions and the couplings were taken from a bimodal distribution: P (β ′ J) = [δ(w + ∆w -β ′ J) + δ(w -∆w -β ′ J)] /2. Having w = 1/2 the critical point is given from self-duality 32 : β ′ c = 1. In most of the calculations we have used ∆w = 1/3, but to check universality we have also performed some calculations with ∆w = 1/4. The linear size of the systems were L = 32, 64, 128, 256 and 512, and the number of independent samples varied from 80000 at the smallest sizes to more than 1200 for L = 512.

The optimal set of a given sample has been calculated by the optimal cooperation algorithm 23 , which works in polynomial time and has already been used to solve the RBPM in two- [START_REF] Anglès | [END_REF]25 and three-dimensional 33 lattices, as well as in scale free networks 34 . If there are multiple optimal sets in the system, then both the intersection and union of any pair of them yields an optimal set. In order to show that our results are independent of the choice of the representing optimal set in a given system, all our studies are carried out for the two limiting cases, namely for the union and intersection of all the optimal sets. We illustrate the cluster structure of the optimal sets in these limiting cases in Fig. 1. It is seen, that the intersection consists of a large number of smaller clusters, which are partially merged to common clusters having larger masses in the union. Consequently we have for the averaged number of crossing clusters: N Γ (intersection) ≥ N Γ (union), however the corner contributions being dominated by the large clusters are expected to be asymptotically identical in the two cases.

Having the optimal sets of different samples we have calculated the excess entropy for different contours: sheared squares, line segments and crosses, which are illustrated in Fig. 2. To subtract the corner contribution from the data we have used the so called geometric approach 28,30,35 : for each sample N Γ is calculated in two different geometries, which have the same boundary term, but different corner ones. Thus the corner contribution is obtained from their difference. The average in Eq.( 10) is performed over i) different samples and ii) over different (∼ 1000) positions of the contour in a given sample. For technical details we refer to our previous investigations on the non-random model [28][29][30] .

We start with sheared squares, having an opening angle γ ≤ π/2 and both its base and altitude is given by L/2. In the numerical method we have calculated the corner contribution of the excess entropy for different sizes and then finite-size estimates are calculated for the prefactor b Γ in Eq.( 12) by two point fit. These are presented in Fig. 3 for the union and intersection optimal sets. For this contour the geometrical factor, A in Eq.( 13) reads as

A Γ = 1 12 4 -π 1 γ + 1 π -γ + 1 π + γ + 1 2π -γ , (14)
and we put also the conformal result in Fig. 3 with an estimated reduced central charge c ′ = 0.74, see Eq.( 16).

For not too small opening angles the finite-size estimates are close to the conformal results, for smaller angles the corrections become larger. Extrapolating the prefactors for the two largest angles, γ = π/2 and π/4 gives approximately the same estimate for the reduced central charge for the intersection and the union, see in Table I. For line segments with length ℓ = L/2 we have only the corner contributions of two exterior γ = 2π angles, so that A Γ = 1/8. In this case the corner contribution is simply half of the number of common clusters between two line segments. Finite-size estimates for c ′ are shown in Fig. 4 which are extrapolated to the same value (within the accuracy of the calculation) both for in-tersection and union data, see Table I. We have checked, that the effective central charges for the disorder parameter, ∆w = 1/4, are practically indistinguishable from the results in Fig. 4. For a cross-like subsystem the corner contributions are expected to cancel out completely. We have checked, that this is indeed the case, see Fig. 5, showing the validity of the Cardy-Peschel formula. We have also studied subsystems comprised of n = 1, 2, 3 or 4 crosses (see in Fig. 2 of Ref [30]), in which case we have obtained finite-size estimates for c ′ . For 1 and 3 crosses these are presented in Fig. 6 and the extrapolated values in Table I. In this case results for the disorder parameter ∆w = 1/4 leads to the same extrapolated value.

IV. CENTRAL CHARGE FROM THE FINITE-SIZE CORRECTION TO THE FREE-ENERGY

The traditional way of subtracting the central charge of a two-dimensional lattice model is to study the finite-size correction to the critical free-energy density in the strip geometry 36,37 . Having the critical RBPM in an infinite strip of width L with periodic boundary conditions the free-energy density scales as:

β ′ f (L) = β ′ f 0 + πc ′ 6L 2 + O(L -4 ) , (15)
where f 0 ≡ f (L = ∞) and c ′ is the reduced central charge in the large-Q limit. Since the free-energy is the same for all optimal sets, we do not make a distinction here between the union and intersection data. In practical adaptation of this method we have used finite stripes of size L × αL for L = 24, 32, 48, 64, . . . and at a fixed α ≥ 1 we have plotted the free-energy densities as a function of 1/L 2 , see Fig. 7. As expected, the limiting value, f 0 , does not depend on the aspect ratio, α, but the slopes are different for α = 1 and for α = 4 and 8. For the latter two values the slopes are very close, therefore we used the data at α = 8 to estimate effective finite-size reduced central charge, c ′ (L) from two-point fits. These are shown in the inset of Fig. 7. As expected the effective c ′ (L)-s have no noticeable size-dependence for L ≥ 32, since the correction terms are O(1/L 2 ). Therefore the extrapolated value given in Table I is simply the mean value of the estimates in the inset of Fig. 7. The random bond Potts model is a basic problem of statistical physics of disordered systems and its large-Q limit is of special interest, when the free energy of the system is dominated by a single diagram, the optimal set. Thermal fluctuations of the Q → ∞ model are negligible compared to disorder fluctuations, therefore its critical properties are controlled by an infinite disorder fixed point. In this paper we considered this model on the square lattice and studied the properties of the excess entropy, S Γ , associated to a contour of bonds, Γ. S Γ is found to be proportional to the number of clusters in the optimal set which are crossed by the contour. Using conformal field theory the excess entropy is shown to have a universal corner contribution at the critical point, which scales with the logarithm of the linear size of the contour and its prefactor is proportional to the central charge of the model. We have performed large-scale numerical calculations and confirmed the validity of the conformal prediction. We have also obtained estimates for the central charge of the model, which are collected in Table I for different forms of the contours, as well as for the two extreme forms of the optimal sets. Within the error of the calculation these all agree with each other, as well as with the results of an independent estimate calculated from the finite-size dependence of the free-energy density. Based on these data we conclude with the estimate:

c ′ = 0.74(2) , (16)
for the reduced central charge of the model. This value is close to (although somewhat larger than) the previous estimate 20 : c ′ = 0.72 obtained through transfer matrix calculations for large, but finite values of Q. c ′ in Eq.( 16) is expected to be universal for any form of the quenched disorder, for the bimodal distribution it holds for 0 < ∆ < 1/2. It is different from that at ∆ = 1/2, which corresponds to bond percolation having a reduced central charge 28

c ′ perc = 5 √ 3 4π ≈ 0.689. ( 17
)
It remains the subject of other research to study if the central charge of the model is related to some proper-ties of the exactly solved random transverse-field Ising chain 27 , as expected from the numerical values of the critical exponents in the two models.

  FIG. 1: (Color online) Cluster structure of the same sample in the optimal sets for the intersection (upper panel) and for the union (lower panel) at L = 256.

FIG. 3 :

 3 FIG. 3: (Color online) Finite size estimates (L = 32, 64, 128, 256 and 512, from bottom to top) of the prefactor b with sheared squares as a function of γ for intersection (top)and union (bottom) data. For larger sizes the results are approaching the conformal result in Eq.(14), which is indicated by the dotted (black) line. Inset: Finite size estimates of the reduced central charge c ′ with sheared squares as a function of the system size at γ = π/2 (γ = π/4), indicated by solid (dashed) lines. The estimated value c ′ = 0.74 is shown by the dotted horizontal line.

FIG. 4 :

 4 FIG. 4: (Color online) Estimates for the reduced central charge c ′ for line segments. The estimated value c ′ = 0.74 is indicated by the dotted horizontal line.

FIG. 5 :FIG. 6 :

 56 FIG.5:(Color online) Difference between the excess entropies for one cross with two different sizes: 2L and L. There is no size dependence, hence no corner contribution in agreement with the Cardy-Peschel formula.

  FIG.7:(Color online) Critical free-energy density of L × αL systems with periodic boundary conditions as a function of 1/L 2 for α = 1 (square) and for α = 4 and α = 8 (stripes). The asymptotic value, f0 = -1.03951(1), is independent of α. The prefactor of the correction term is 0.91(3) for squares and it agrees for both type of stripes. In the inset the effective reduced central charge is shown for α = 8 and the dashed horizontal line represents the estimated value in TableI.

	1.0402 1.0404	α=1 α=4 α=8
	-f	1.04				0.9
	1.0398			c'	0.8 0.7
	1.0394 1.0396	0	0.0005	0.001 1/L 2	0.6 0.5	0.0015 0 0.01 0.02 0.03 0.002 1/L

TABLE I :

 I Numerical estimates for the reduced central charge c ′ from the corner contribution of the excess entropy using different contour geometries and from the finite-size correction to the free-energy in stripes.

			union intersection
	excess entropy	squares 0.74(3) 0.79(6) lines 0.74(3) 0.77(5)
		crosses 0.72(2) 0.73(1)
	free-energy	0.75(2)
	V. CONCLUSION

Acknowledgments

This work has been supported by the Hungarian National Research Fund under grant No OTKA K75324 and K109577. The research of IAK was supported by the European Union and the State of Hungary, co-financed by the European Social Fund in the framework of T ÁMOP 4.2.4. A/2-11-1-2012-0001 'National Excellence Program'. J.-Ch. A.d'A thanks to the 'Theoretical Physics Workshop' and F.I. to the Université Joseph Fourier for supporting their visits to Budapest and Grenoble, respectively.


Kevin Crowston
email: crowston@syr.edu

Joseph Feller
email: jfeller@ucc.ie

Carl-Eric Mols
email: carl-eric.mols@sonymobile.com

Tony Wasserman
email: tonyw@sv.cmu.edu

The Future of Open Source Research: A Panel Discussion

Description

Over the past decade, FLOSS has become mainstream. It's the technological foundation for a huge percentage of software, including products and services from both startups and established companies, as well as the dominant approach for software development and deployment tools. More recently, FLOSS has become the preferred solution for a growing number of governments and non-profit organizations around the world. What was once a niche research area has similarly become mainstream; FLOSS-related research publications are now found across a variety of social, economic, and software-related publications and conferences This growing popularity poses an interesting challenge for FLOSS research. While there remains a core set of topics specifically related to FLOSS, e.g., software licensing, many other topics, such as collaborative distributed development, have much broader applicability. Some researchers who previously submitted their work to FLOSS-related conferences and publications are now seeing many other opportunities to present their work, describing the open source aspects of their research in a broader, domain-focused context.

In this panel, we hope to address not only the most likely future directions for FLOSS research, but also how FLOSS is related to other research fields and the broader context of modern society. The panelists bring a wide variety of academic and industrial experience with FLOSS and related subject areas.

Individual Position Statements Kevin Crowston

My suggestions for the next phase of FLOSS research starts with observation that FLOSS is a topic, not a field, meaning that while we are looking at the same phenomenon, we do so adopting diverse approaches. So, why might sticking together make sense? A shared conference provides an opportunity to learn about work that you would miss in a disciplinary conference. But such visibility is only valuable if we can find commonalities that bring us together, e.g. sharing common inputs (e.g., datasets), common methods or complementary outputs across disciplines.

However, achieving interdisciplinarity requires a shared sense of priorities about the work to be done. A good common project would be to put our knowledge to work, that is, to tell interested parties how to have successful FLOSS projects (which would also serve as a test of our knowledge). A particular need is in the setting of scientific software. Funding agencies want to see the software they pay for reused and maintained, and openness fits the needs of science for reproducibility and auditability. What advice might we give about how to structure development to achieve quality code and to create a sustainable community around a piece of scientific software?

Joseph Feller I will discuss the current and potential relationships between FLOSS research and emerging research areas, reflecting upon the activities of the AIS Special Interest Group on Open Research and Practice (SIGOPEN), and of other research communities.

In particular, I will discuss

Carl-Eric Mols

My proposal for the future of Open Source research focuses on scaling up the software business aspects, in particular scaling up the Open Source maturity of organizations that develop software. While Open Source is widely used in the ICT industry, it is often in a passive, consuming way. It's uncommon to find companies that have extensive Open Source contributions and community participation as their core product/service development strategy, and even more rare to find companies that fully adapt their business models when their offering is increasingly based on Open Source.

With emerging software phenomena such as IoT, Industry 4.0, and Automotive Software, Open Source will massively impact the "traditional", non-ICT industry in many ways, not least in the way software development will need to be organized, and the way business will need to be conducted in order to avoid being eradicated by the Open Source giants. What wisdom on scaling software business can academia bring to the Open Source-challenged industry?

Anthony I. (Tony) Wasserman From a broader societal perspective, great improvements in the quantity and quality of FLOSS software has led to its adoption in a broad range of settings. As software has more and more societal impact, there is likely to be greater public demand for transparency and openness, especially for life-critical systems, such as those found in medical devices and autonomous vehicles. People will justifiably want to see the source code of heart pacemakers and vehicle control systems, which are now guarded as intellectual property by their makers; similarly people will expect governments to make extensive use of FLOSS and to make any software developed under government contracts to be released under an OSI-approved FLOSS license. These changes will, in turn, open up new areas for FLOSS-related research.

I
The Future of Open Source Research: A Panel Discussion

•

  the proliferation of open phenomena (e.g. open innovation, open design, crowdsourcing, crowdfunding, volunteer computing, collective intelligence, etc.) and the challenges this proliferation creates for the research community, • the important intellectual contribution that FLOSS research can make to these emerging fields of inquiry, and vice-versa, • the implications of FLOSS and related research findings for the creation of a more open practice of scholarly inquiry and dissemination (e.g. open data, open access publishing, etc.), and • the opportunity and need for greater integration, knowledge sharing, and intellectual cross-pollination between the global open research communities.

  . Likewise, open source software is increasingly discussed in connection with open innovation, open government, open data, and related areas.

  will discuss some implications of FLOSS becoming mainstream, many of which are already apparent. From a research perspective, research results that were previously published in open source research conferences and publications are now appearing in software engineering, economics, and political science conferences and publications. Open source-related research has diffused across various broader interest categories, making it more difficult to keep up with relevant research.


Jesús Moreno-León

Terhi Kilamo
email: terhi.kilamo@tut.fi

Gregorio Robles

FLOSS Education and Computational Thinking Workshop @ OSS 2016

Introduction

The presence of FLOSS in education has not stopped growing in the last years. The trend has been clear both in K-12 and higher education. While using FLOSS can support teaching computer science and other disciplines, its benefits lie in teaching FLOSS itself as part of the curriculum.

An example that can illustrate this situation is the teaching of computational thinking skills through computer programming, which is one of the latest trends in education -for instance, Finland has just added coding and computational thinking as part of the national core curriculum for primary education. This field has been globally addressed almost exclusively with FLOSS technologies, both by using FLOSS platforms and programming languages, such as Scratch or Alice, but also by including in the curriculum the social aspects of software development that characterize FLOSS movements, like sharing and contributing to the community.

The purpose of this workshop is to bring together free software experts and educators to discuss challenges that we face in the educational world at present and that we will face in the future and how they can be undertaken from a FLOSS perspective.

Topics of Interest

The topics of interest of this workshop include but are not limited to:

-Teaching experiences with FLOSS/free content -FLOSS in higher education -FLOSS in K-12 -FLOSS practices in education -FLOSS in the curriculum -Computational thinking teaching and FLOSS

  c IFIP International Federation for Information Processing 2016Published by Springer International Publishing Switzerland 2016. All Rights Reserved K.Crowston et al. (Eds.):OSS 2016, IFIP AICT 472, pp. 204-205, 2016. DOI: 10.1007/978-3-319-39225-7 Specific Research Questions Specific questions that are of special interest to this workshop are:-Which FLOSS approaches have proven beneficial to education? -What experiences do you have in collaboration with FLOSS communities in education contexts? -How do you produce and share your educational materials? -What assessment and certification models did you apply? Why have those models been chosen? -What models for sustainability and revenue generation worked successfully? -What efforts are undertaken to come towards a compatible or standardised curriculum? -What indicators are measured to show strengths and weaknesses of the initiative? -How is copyright and licensing managed in your institution/initiative? What (potential) impact does this policy have on sustainability? -How can curricula be designed to foster the spirit of sharing? -What FLOSS technologies do you use to teach computational thinking? -How do you promote FLOSS social aspects in your lessons? Programme Committee -Yasemin Allsop, University of Roehampton (United Kingdom) -Alessandro Bogliolo, University of Urbino (Italy) -Jordi Freixenet, University of Girona (Spain) -Petri Ihantola, Tampere University of Technology (Finland) -Oystein Imsen, Laer Kidsa Koding (Norway) -Terhi Kilamo, Tampere University of Technology (Finland) -Simon Marsden, University of Portsmouth (United Kingdom) -Joek van Montfort, Scratchweb Foundation (Netherlands) -Jesús Moreno, Programamos.es (Spain) -Eduard Muntaner, Inventors for Change (Spain) -Peter Parnes, Luleå University of Technology (Sweden) -Gregorio Robles, Universidad Rey Juan Carlos (Spain) Sponsorship This workshop is sponsored in part by the Region of Madrid under project "eMadrid -Investigación y Desarrollo de tecnologías para el e-learning en la Comunidad de Madrid" (S2013/ICE-2715).


A A Klochikhin

V P Kochereshko

S Tatarenko

Influence of free carriers on exciton ground states in quantum wells

Keywords: number(s): 71.10.Ca, 71.45.Gm, 73.21.Fg, 78.55.Et

The influence of free carriers on the ground state of the exciton at zero magnetic field in a quasi-two-dimensional quantum well doped with electrons is considered in the framework of the random phase approximation. The effects of the exciton-charge-density interaction and the inelastic scattering processes due to the Hartree-Fock electron-electron exchange interaction are taken into account. The effect of phase-space filling is considered using an approximate approach. The results of the calculation are compared with the experimental data available.

I. INTRODUCTION

The presence of free electrons in quasi-two-dimensional semiconductor structures strongly alters their physical characteristics. In optical investigations of these objects, the photon excites a conduction electron -valence hole pair. To understand the optical properties, it is crucially important to take into account the interactions between this pair and the free carriers.

At low and intermediate electron concentrations, the Coulomb interaction between the electron and the hole leads to formation of the bound exciton state. This shrinks the observable band gap E G by the exciton binding energy E B . Similarly to degenerate 3D semiconductors, the electron-electron interaction causes a static screening of the electron-hole Coulomb potential. The screening as well as the filling of phase-space up to the Fermi energy leads to a strong decrease of the exciton binding energy as free electron concentration increases. At some concentration of free electrons the exciton bound state disappears and the so-called Mahan exciton can be expected. Considerable attention has been paid in the literature to experimental and theoretical investigations of the 2D-systems at high concentration of free carriers. The theory of this phenomenon was developed in Ref. 1 for the 3D crystal and in Ref. [2][3][4][5] for 2D systems.

Experimental data [6][7][8][9][10] give evidence that, within a relatively narrow interval of free electron concentrations below the metallic phase, the optical absorption band associated with the bound exciton shifts and broadens, and its oscillator strength decreases as carrier concentration is increased. The dependence of these characteristics on the electron concentration allows one to relate these processes with exciton-electron interactions.

In addition, starting from very low concentrations of free electrons in the quantum well, a bound trion (negatively charged exciton) state [11][12][13][14] is observed in experiments [6][7][8][9][10] . The present article will not be con-cerned with properties related to the trion state, but its existence provides another channel for exciton-electron interactions.

The observed broadening of the exciton line [6][7][8][9] clearly shows that not only static screening but also inelastic dynamical processes should be considered for the explanation of the exciton spectrum.

The exciton created by a photon has almost zero wavevector. Further interaction of the exciton with free electrons followed by scattering of a conduction electron near the Fermi level with p ≈ p F creates a conduction electron -conduction hole pair. This pair has an arbitrary wave-vector of the center of mass motion q and an energy h2 q 2 /2m + h2 pq/m which is the difference between the conduction electron's energy in the final and initial states, (h 2 (p + q) 2 /2m and h2 p 2 /m, respectively).

The threshold energy of this excitation at q = 0 and temperature T = 0 K is zero. Therefore, the lower boundary of the continuum spectrum of the combined excitation consisting of the free exciton plus an electron -conduction hole pair coincides with the energy of the free exciton. As a consequence, this inelastic scattering process results in a homogeneously broadened absorption spectrum with its maximum shifted toward high energies with increasing free electron concentration.

The mechanism responsible for the creation of an electron -conduction hole pair can be either the Coulomb interaction or the exchange interaction between the electron bound in the exciton and the free electrons. For the negatively charged trion, which has two bound electrons coupled in a spin singlet state, the Coulomb interaction seems to be the only important mechanism of interaction with free electrons 15 . In the case of the electrically neutral exciton the strong cancellation of the electron and valence hole self-energy corrections and vertex correction decreases the role of the Coulomb interactions in the scattering processes. This means that the Hartree-Fock exchange interaction between the electron belonging to the exciton and a free electron should be considered as the most probable reason for the broadening of the exciton absorption band.

We consider in detail the behavior of the bound exciton absorption band as a function of free electron concentration in a quantum well doped by electrons at low and intermediate concentrations, and we extrapolate the results up to the concentration at which the bound exciton state disappears. We will show in this paper that a second order exchange process strongly influences the bound exciton state and the optical spectra of the doped quantum wells, in addition to the nearly rigid band gap renormalization produced by the first order Hartree-Fock exchange interaction. The second order process broadens the exciton absorption band and shifts the energy of its maximum.

The paper is organized as follows. In Section II we formulate the exciton equation in a quasi-two-dimensional quantum well containing free electrons using the known results for the 3D case. We take into account in this Section the static screening of the Coulomb interaction and the effect of phase-space filling. Section III is devoted to the Hartree-Fock exchange interaction between the electron constituting the exciton and the free electrons. In Section IV we describe the calculation of the absorption coefficient as a function of the free electron concentration and give a comparison with experimental data.

II. EXCITON IN A QUANTUM WELL

We consider only temperature T → 0 K assuming that k B T is much less than the Fermi energy. Despite the reduced dimensionality there exists a deep analogy with the 3D Fermi systems in the methods of calculation of multi-particle effects.

The exciton equation can be formulated by summing the ladder diagrams for the exciton Green function similarly to the procedure described by Mahan 1 . Accepting this paper as a guide we present the exciton equation in a quantum well with finite concentration of free electrons as

- ∇ 2 2µ ψ λ ( -→ ρ ) + d 2 ρ ′ V 2D (ρ ′ )K( -→ ρ - -→ ρ ′ )ψ( -→ ρ ) = (1) = E λ ψ λ ( -→ ρ ),
where µ = m e m h /(m e + m h ) is the reduced mass of the electron and valence hole, -→ ρ is the two-dimensional radius-vector between the electron and hole positions in the exciton. V 2D ( -→ ρ ) is the quasi-two-dimensional Coulomb potential at zero temperature (in the calculation of which the static screening should be taken into account), and

K(ρ) = - 1 (2π) 2 d 2 q e iq -→ ρ (1 -Θ(p F -q)), ( 2
)
where pF is the Fermi wave vector, Θ(p F -q) is the step-like theta-function, and (1 -Θ(p F -q)) is the Fermi distribution function at zero temperature restricting the integral region in two-dimensional reciprocal space to q ≤ pF . We present K(ρ) as two items

K(ρ) = -δ(ρ) + 1 (2π) 2 d 2 q e iq -→ ρ Θ(p F -q). (3)
After substitution of Eq.(3) into Eq.( 1) we obtain two items for the exciton potential energy. The first item is an attractive potential V 2D ( -→ ρ ) and the second is a repulsive term that arises due to phase-space filling

δV 2D (ρ) = 1 (2π) 2 d 2 q d 2 -→ ρ ′ V 2D (ρ ′ )e iq( -→ ρ - -→ ρ ′ ) Θ(p F -q) (4)
The matrix element of this equation responsible for the correction to the exciton energy is

d 2-→ ρ ψ λ ( -→ ρ )δV 2D (ρ)ψ λ ( -→ ρ ) = (5) 1 (2π) 2 2 d 2 p d 2 q ψ λ (p)V 2D (q)ψ λ (|p -q|)Θ(p F -q)
In further calculations we find the ground state energy E ex of the Schrödinger equation

- ∇ 2 2µ ψ λ ( -→ ρ ) + V 2D (ρ)ψ( -→ ρ ) = E ex ψ ex ( -→ ρ ), (6)
where V 2D (ρ) is the quasi-two dimension Coulomb potential taking into account the static screening. Additionally, we should calculate the phase-space filling correction according to Eq.( 5).

A. Coulomb potential in a quasi-two-dimensional quantum well

The Coulomb and exchange interactions in a quantum well should be transformed to take into account the fact of the reduced dimensionality due to the confinement.

The effective Coulomb potential in the well depends on the thickness and shape of the quantum well, and on the concentration of free electrons.

We find the effective Coulomb potential assuming that all the particles occupy their lowest confined state in a quantum well with infinitely high barriers. In this case the wave functions of all the particles can be characterized by the quantum well thickness L.

As the first step, we obtain an effective "quasi-2D" Coulomb potential without static screening V 2D 0 ( -→ ρ ) (see Eq.( 11) below)by using the Coulomb interaction in 3D systems as a starting point, similarly to Ref. [15]. The Coulomb Hamiltonian in a 3D system is

H Coul = 1 2 i,j,σ,σ ′ d 3 r i d 3 r j V Coul (r i -r j )|ψ σ (r i )| 2 |ψ σ ′ (r j )| 2 , (7)
where σ denotes the spin quantum number, and

V Coul (r 1 -r 2 ) = ± 1 (2π) 3 d 3 q 4πe 2 εq 2 exp(iq(r 1 -r 2 )), (8)
where ε is the static dielectric constant, and the r i , i = 1, 2 are the radius-vectors of the two interacting particles. We may express the r i as coordinates in an arbitrary plane and perpendicular to this plane r i → ( -→ ρ i , z) and, similarly, components of the wave-vector q in the reciprocal space are q and q z . The signs ± are related to the electron-electron and electron -valence band hole interactions respectively. In writing Eq. ( 7) and below, we are using CGS units.

We exclude the 3D motion in the perpendicular direction by calculating the matrix elements of exponentials with wave-functions ψ(z) = 2/L cos(πz/L) assuming that the wave functions of the exciton and conduction electron -conduction hole pairs are completely confined within the quantum well of thickness L restricted by infinite barriers. The Coulomb vertex contains the product of two matrix elements having the form

M z,z = [exp(izq z ] ψ(z),ψ(z) = ∞ k=2 [1-(q z L) 2 /(2πk) 2 ]. (9) The integral over q z ∞ 0 dq z 2π 4πe 2 [M z,z (q z )] 2 ε[q 2 + q 2 z ] (10)
can be calculated numerically or represented with good accuracy by the function [ 4e 2 εq arctan qL q ] where q L = 2π/ L and

L L = ∞ 0 Ldq z 2π [M z,z (Lq z /2π)] 2 ≃ 0.75.
As a result we can present the effective quasi-2D Coulomb interaction without screening as

V 2D 0 (q) = 4e 2 εq arctan q L q . ( 11
)
The screened effective Coulomb potential in the random phase approximation has the form

V 2D (q) = 4e 2 arctan qL q q ε(q, 0) (12)
where ε(q, 0) is the real part of the dielectric function that takes into account free electrons

ε(q, 0) = ε 1 - 4e 2 εq arctan q L q Reχ(q, 0) (13)
In the limit ω → 0 the real part of the dielectric susceptibility can be presented as

Reχ(q, 0) = - m * πh 2 {1 -Θ(q -2p F ) 1 -(2p F /q) 2 } (14)
And, finally, the screened effective Coulomb potential in the random phase approximation is

V 2D (q) = 4e 2 arctan qL q q × (15) 1 + 4 π qa 0 arctan qL q {1 -Θ(q -2p F ) 1 -(2p F /q) 2 -1
, where a 0 = h2 ε/m * e 2 is the radius of the 3D donor state. Screening in quasi-2D systems was considered in early publications by using perturbation theory 3 . However, new experimental data 6,9 were interpreted as showing that the exciton binding energy decreases very sharply with an increase in the concentration of two-dimensional electrons for CdTe quantum wells and for GaAs/AlGaAs quantum wells 10 .

We have calculated the absolute value of the exciton binding energy as a function of free electron concentration in the random phase approximation for 10 nm thick CdTe quantum wells, of the type studied experimentally in Refs. 6-9, assuming infinite-height barriers. The result is presented as Curve 1 in Fig. 1a. The phase-space filling correction of Eq.( 5) is given in Fig. 1a by Curve 2. The parameters used in our calculations are presented in Table I.

An effective Rydberg energy of E B = 18 meV for electron concentration of about 10 10 cm -2 was estimated by Teran et al 9 from the energy distance between the ground and first excited exciton states observed in their PLE spectrum. Such a value of E B is obtained in our calculation at 10 9 cm -2 by using the parameters presented in Table 1. We calculate the quasi-2D Rydberg at zero concentration to be 22 meV. This is twice the 3D value of 11 meV. The relatively strong decrease of the effective Rydberg in the low concentration range from 0 to 10 9 cm -2 is explained by the strong screening of the Coulomb potential at q ≤ 1/a 0 in the random phase approximation as follows from Eq. (15).

From concentrations about 10 10 cm -2 , the phase-space filling correction significantly decreases the binding energy. Extrapolating Curve 2 up to the point where it crosses Curve 1 in Fig. 1a, we can estimate the electron concentration (≃ 3 • 10 11 cm -2 ) above which the bound exciton states should disappear.

The decrease of the exciton binding energy due to screening and phase-filling correction is followed by decrease of its oscillator strength. At zero electron concentration the exciton oscillator strength is proportional to

[ d 2 q/(2π) 2 ψ ex (q)] 2 = [ψ ex (ρ = 0)] 2 ∼ 1/a 2
ex where a ex is the effective exciton radius. For the oscillator strength at finite electron concentration we have

A(n e ) = d 2 qΘ(q -pF )ψ ex (q) (2π) 2 ψ ex (ρ = 0). ( 16
)
In this case two mechanisms decrease the oscillator strength: the increase of the exciton radius and the phase-space filling restriction. The major effect is the increase of the exciton radius resulting from the decrease of the exciton binding energy. We should mention that some underestimation of the phase space filling at high electron concentrations occurs in our calculation of the exciton binding energy because we consider this effect as a perturbation only.

III. HARTREE-FOCK EXCHANGE CORRECTIONS

We should say that the inelastic 2D plasmon interaction process due to Coulomb interaction with free electrons is significantly weaker for the electro-neutral exciton than for charged particles. This process does not give a notable additional shift or broadening of the exciton line, in contrast to its effects for uncorrelated electrons and holes 16 or for the charged trion 15 . For the exciton, the cancellation of the electron and hole self-energy and vertex corrections leads to a decrease of this interaction. We will show that it is the Hartree-Fock exchange interaction that can describe the shift and broadening of the exciton band.

The early calculations of the band gap renormalization in quantum wells were done in Refs. [17][18][19] It is convenient to illustrate the calculations of the exchange self-energy corrections by Feynman diagrams starting from the corrections to the single electron interacting with the Fermi sea of electrons neglecting the electron-valence-band hole exchange interaction for simplicity.

We consider the first order self-energy correction to the electron state due to Hartree-Fock exchange interaction, presented in Fig. 2(a), in the framework of the rigid shift approximation, taking the external wave vector equal to zero. This leads to a wavevector-independent shift of the conduction band and, therefore, of the exciton states.

The diagram illustrating the second order correction (Fig. 2(b)) can be obtained, for instance, from the Feynman diagrams in Fig. 37(b) of Ref. [START_REF] Pines | The theory of quantum liquids[END_REF] the electron lines. Taking into account that we are interested in broadening of the exciton created by a k ≈ 0 photon, we put the value of the wave-vector of the renormalized particle equal to zero. The correction to the exciton energy at zero wave-vector of the center of mass is given in Fig. 2(c). It can be seen that this diagram is obtained by putting the self-energy diagram of Fig. 2(b) at Q = 0 into the electron line of the exciton. Finally, in order to describe the self-energy corrections to the exciton state, we have to sum the series of multiple scattering processes. In calculations of the exciton Green function we consider only the ground state. This allows us to transform the series into a geometrical progression. Therefore, the major part of the problem is to calculate the self-energy correction to the ground state.

by breaking one of

q

-q E ex q a b q -q c -p-q -p-q -p p+q Q Q Q-q -p Q+p+q 0 0 0 0

A. The Hartree-Fock exchange interaction

First of all we have to express the exchange self-energy correction to the exciton through the 2D Coulomb-like V (q) 2D exch Hamiltonian taken in the form

H exch = 1 2 i,j,σ =σ ′ d 3 r i d 3 r j V exch (r i -r j )× ψ σ (r i )ψ σ (r j ) * ψ σ ′ (r j )ψ σ ′ (r i ) * . ( 17
)
where the diagram of Fig. 2(a) contains the vertex

V exch (r 1 -r 2 ) = - 1 (2π) 3 d 3 q 4πe 2 εq 2 exp(iq(r 1 -r 2 )). ( 18
)
The diagram of Fig. 2(c) contains the vertex depending on the wave-vector q and the similar vertex depending on (p + q).

The integrals over z 1 and z 2 for both diagrams we calculate similarly to Eqs. (9,10). Then the first integral over dq z gives the same expression as Eq.( 11).

Rigid shift of the conduction band

This leads to a rigid shift of the conduction band states

δE R δE R = 1 (2π) 2 d 2 q 2e 2 εq arctan q L q . (19)

Approximately, the dependence of this equation on the electron concentration is

√ n e . The result of the numerical calculation of the band edge shift is presented in Fig. 1b by Curve (1). The exciton binding energy with respect to the shifted band edge is presented in Fig. 1(b) by Curve (2).

Shift and broadening of the exciton band due to the second order correction

The expression corresponding to the diagram of Fig. 2(c) contains two matrix elements over the internal wave function of the ground state exciton. The first of them has the form

M 11 (q) = d 2 ρ|ψ 1 (ρ)| 2 exp(iqρ) = (20) 1 [1 + (qa B /2) 2 ] 3/2
and a similar equation can be written for M 11 (|q + p|).

The structure of the exchange loop in Fig. 2(c) P exch q (hω) differs from the Coulomb contribution 15 and has the form As compared to Fig. 3, the threshold energy of each exciton band is shifted upwards : Here, in addition to the second order shift and broadening due to exchange interaction, the static screening, the first-order rigid shift due to exchange interaction, the phase-space filling effect, and the decrease of the oscillator strength are all taken into account. The ten electron concentrations expressed in units of 10 10 cm -2 are equal to 0.5, 1.0, 1.5, 2.5, 3.75, 5.0, 7.5, 10.0, 15.0, and 25.0.

P exch q (hω) = d 2 q (2π) 2
∞ 0 dq 1 z 2π 4πe 2 M 2 zz (q 1 z )P exch q,q 1 z (hω), (21)
where the variable q 1 z = (p + q) z is introduced. The function P exch q,q 1 z (hω) is

P exch q,q 1 z (hω) = (22) - d 2 p 2π M 11 (q)M 11 (|q + p|) ε|q + p| 2 + (q 1 z ) 2 η(p, p + q, hω).
and η(p, p + q, hω) = (23)

1 hω + iδ -ω pq - 1 hω + iδ + ω pq .
We have neglected the exciton kinetic energy in the final state (recoil effect) which is small compared to the energy of the electron-conduction hole pair.

The wave-vector dependence of both matrix elements M ex (q) and M ex (|q + p|) arises at values of the order of the reciprocal Bohr radius of the exciton 1/a 2D B . We will assume this quantity to be essentially larger than the Fermi wave-vector p F which is the characteristic value for p. Therefore, we approximate the product of these matrix elements by the expression M ex (q)M ex (|q + p|) ≈ M ex (q) 2 . Taking into account that the integral over dq 1 z is restricted by q 1 z ≈ q L = 2π/L we take the function P exch q,q 1 z (hω) out of the integral of Eq.( 24) at q 1 z = q L . After these simplifications the integral over dq 1 z gives L -1 and the remaining integral over d 2 p in Eq.( 22) can be calculated analytically.

Finally, we obtain the exciton self-energy correction due to exchange interaction in the form

δE ex (hω) = 1 (2π) 2 d 2 q 4e 2 M ex (q) 2 εq arctan q L q ×
(24)

d 2 p e 2 ε L[|q + p| 2 + (q L ) 2 ] η(p, p + q, hω).
The broadening of the exciton absorption band and the upward shift of its peak with respect to the exciton line at zero electron concentration, resulting from this correction, are shown in Fig. 3 for a set of ten electron concentrations ranging up to n e = 2.5 • 10 11 cm -2 . The absorption curves have the form:

α(ω) ∼ Γ(ω) (hω -E ex (n e = 0) -∆(ω)) 2 + Γ 2 (ω) , (25)
where Γ(ω) and ∆(ω) are the imaginary and real parts of the exciton self-energy correction of Eq.( 24). For convenience the absorption band profiles are normalized to unity in Fig. 3. We see that the values of the upward shifts are comparable with the effect of the rigid downward shift due to the first order exchange interaction in Fig. 1b.

In Fig. 4 we present the transformed spectra of Fig. 3 after the first-order rigid shift due to exchange interaction, the phase-space filling effect, as well as the decrease of the oscillator strength are all taken into account. The shape of the exciton absorption band can be represented in this case as

α(ω) = 2A(n e ) Γ(ω) (hω -E ex (n e ) -∆(ω)) 2 + Γ 2 (ω) , (26)
where A(n e ) is given by Eq.( 16). The position of the exciton ground state at zero temperature and electron concentration equal to 0.5 • 10 10 cm -2 is taken as the point of reference. The parameters used in our calculations are presented in Table I. -4) are the experimental data on absorption of a 10 nm CdTe(0.3%Mn) single quantum well in zero magnetic field at different free electron concentrations. A resonance peak (T) corresponds to creation of the singlet trion appearing below the exciton resonance (X). Solid curves are the calculated exciton line profiles at free electron densities of 1.5, 3.75, 7.5, and 15.0 10 10 cm -2 . The dashed curve is a background added to the calculated spectra.

IV. COMPARISON WITH EXPERIMENTAL DATA

We have compared the above theory with optical absorption data at 2K for a 10 nm CdTe/CdMgTe single quantum well at zero magnetic field. This sample was grown by Molecular Beam epitaxy on a CdTe 0.88 Zn 0.12 Te substrate. The alloy substrate is transparent to about 1640 meV, allowing direct measurements of the optical transmission spectrum in the energy region (1610-1630 meV) of the heavy-hole exciton in the CdTe quantum well. Also, the substrate imposes a large biaxial strain on the CdTe well layer, splitting off light-hole transitions to high energies, which simplifies the optical spectrum. The upper CdMgTe barrier was doped with donors so as to give n e = 1.5 • 10 11 cm -2 in the well in equilibrium. 21 Optical data for this sample were provided by Ronald Cox of Néel Institute. During the optical transmission measurements, done using a tungsten lamp source, the electron concentration could be varied down to a low value by pumping the sample with blue laser light. The precise dependence of n e on pumping light intensity was not known but the minimum concentration was estimated to be in the range (0.2 -0.4) • 10 11 cm -2 as judged from the intensity of the X -(trion) resonance.

For comparison with this experimental data we will use spectra of Fig. 4 fitting the absolute exciton energy. Results are presented in Fig. 5. As it is seen, the shift of the exciton maximum agrees well with experimental data if we take the electron concentration for the lowest band equal to 1.5 • 10 10 cm -2 and for the highest one to 1.5 • 10 11 cm -2 . The lowest value appears to be slightly less than the value estimated in experiment (2 -4 •10 10 cm -2 ) while the upper value agrees well with experimental data.

However, the experimental broadening values exceed notably the calculated values for all the samples. This may be attributed to the existence of some additional broadening mechanism which was not taken into account in our calculations.

The existence of an additional mechanism of exciton state decay might be assumed because the ground exciton state is not the lowest bound state in the presence of the Fermi sea. At least the trion state might serve as such a state. Another reason for the deviation could be the forced simplifications admitted in our calculations.

Conclusion

The shift and broadening of the exciton absorption bands in CdTe/CdMgTe quantum wells as a function of the free carrier concentration at zero magnetic field is investigated. It is shown that the exciton's shift and broadening behavior is related to the Hartree-Fock exchange interaction between the electron bound to the valence band hole and the free carriers. This is in contrast to the case of the charged trion, where it is the Coulomb interaction with the free electrons that plays the leading role. The calculations performed have shown that reasonable agreement can be obtained with experimental data.

We wish to thank Dr. R. Cox for the experimental data used in the paper and for many helpful discussions.

Figure 1 :

 1 Figure 1: (Color online) (a) Curve (1) presents the dependence of the absolute value of the exciton binding energy on the free electron concentration for a 10 nm wide CdTe quantum well, taking into account only the static screening. Curve (2) presents the repulsive phase-space filling correction decreasing the exciton binding energy, and Curve 3 is the absolute value of the Fermi energy. (b) Curve (1) is the rigid shift of the band gap EG corresponding to diagram (a) of Fig.2. Curve (2) is the exciton binding energy with respect to EG as a function of the electron concentration taking into account the static screening and phase-space filling effects. Curve (3) is the Fermi energy with respect to EG.

Figure 2 :

 2 Figure 2: (a) Feynman diagram which describes the rigid shift of the electron energies. (b) The second order exchange selfenergy correction to the electron state. (c) The second order self-energy correction to the exciton state (double horizontal line connected by the wavy Coulomb lines) due to exchange interaction, obtained from diagram (b) at Q = 0. Solid lines are the electron and hole Green functions, the wavy lines are the exchange vertexes.

Figure 3 :

 3 Figure3: (Color online) Broadening and shift of the normalized to unity, spectral density of the exciton ground state caused by the second order exchange interaction for ten electron concentrations in a 10 nm CdTe/CdMgTe quantum well.The zero of energy corresponds to the exciton ground state at zero temperature and zero electron concentration. The ten electron concentrations expressed in 10 10 cm -2 are equal to 0.5, 1.0, 1.5, 2.5, 3.75, 5.0, 7.5, 10.0, 15.0, and 25.0.

Figure 4 :

 4 Figure4: (Color online) Spectral density of the exciton ground state for a 10 nm CdTe/CdMgTe quantum well parameters at increasing electron concentration. The zero of energy corresponds to the exciton ground state at zero temperature and electron concentration equal to 0.5 • 10 10 cm -2 . As compared to Fig.3, the threshold energy of each exciton band is shifted upwards : Here, in addition to the second order shift and broadening due to exchange interaction, the static screening, the first-order rigid shift due to exchange interaction, the phase-space filling effect, and the decrease of the oscillator strength are all taken into account. The ten electron concentrations expressed in units of 10 10 cm -2 are equal to 0.5, 1.0, 1.5, 2.5, 3.75, 5.0, 7.5, 10.0, 15.0, and 25.0.

Figure 5 :

 5 Figure 5: (Color online) Curves(1-4) are the experimental data on absorption of a 10 nm CdTe(0.3%Mn) single quantum well in zero magnetic field at different free electron concentrations. A resonance peak (T) corresponds to creation of the singlet trion appearing below the exciton resonance (X). Solid curves are the calculated exciton line profiles at free electron densities of 1.5, 3.75, 7.5, and 15.0 10 10 cm -2 . The dashed curve is a background added to the calculated spectra.

Table I :

 I Parameters used in calculations: electron and hole effective masses, the binding energy of the 3D exciton in CdTe

	E 3D ex , the quantum well thickness tQW , the effective exciton radius a 3D B , and the static dielectric constant ε.
	me	m h E 3D ex meV tQW Å a 3D B Å ε
	0.107m0 0.45m0	11.0	100	63 10.4


Raoudha Gaha
email: raoudha.gaha@gmail.com.

Bernard Yannou

Abdelmajid Benamara

Selection of a green manufacturing process based on CAD features

Keywords: Green manufacturing process / CAD model / LCA / Feature technology / Scenarios, Nomenclature CAD = Computer Aided Design CAPP = Computer Aided Process Planning, CAM =

come   L'archive ouverte pluridisciplinaire

Introduction

Face to the growing awareness of environmental concerns issued from human activities, eco-design aims at offering a comprehensive approach for the products and services development field, especially during the design process [START_REF] Brezet | Dynamics in Ecodesign prac tice[END_REF][START_REF] Janin | Démarche d'éco-conception en entr eprise, un enjeu: construire la cohérence entre outils et processus[END_REF]. However in the late detailed design phase, the designer action field is seriously limited as he/she may handle few degrees of freedom. Manufacturing process is one of these degrees of freedom for CAD designer. In the literature environmentally conscious manufacturing process is widely studied; Ilgin and Gupta classified more than 500 researches on that domain [START_REF] Ilgin | Environmental ly conscious manufacturing and product recovery (E CMPRO): a review of the state of the art[END_REF]. However, fewer methods have been developed for an environmental evaluation in CAD phase of possible manufacturing process plan scenarios to facilitate decision making. In this phase, features are the core of a geometric model. According to Bidarra et al. [START_REF] Bidarra | Semantic fea ture modeling-Computer Aided Design[END_REF], modeling features compared to conventional geometric modeling enables the combination of functional and engineering information on the form in a product model. For Ghodous [START_REF] Ghodous | Modélisation Intégrée de Donné es de Produit et de Processus de Conception[END_REF], this Feature technology (FT) facilitates the integration between different phases of product development such as design and manufacturing. FT serves also to integrate CAD systems and Ecodesign tools such as Life Cycle Assessment (LCA) software. Hence we propose a new methodology based on features to generate possible process plans from one side and to evaluate environmentally each scenario in order to select the greenest one on the other side.

In this paper we start by a state of the art of benefits attempted by the integration between CAD and Computer aided process plans (CAPP) based on FT. Secondly, we present a literature review of works integrating environmental evaluation into manufacturing process. Thirdly, we propose our approach. Finally we apply our methodology into a case study for validation.

Feature technology and environmentally conscious manufacturing process: Overview

A feature is a local shape of a product directly related to the manufacturing process. Smith and Dagli [START_REF] Smith | Manufacturing f eature identification for intelligent design -Intellige nt Systems in Design and Manufacturing[END_REF] define features as high-level design primitives with their attributes, their qualifiers and restrictions which affect the functionality and/or the ability for the product to be manufactured. It was used to reduce the environmental impact playing with different parameters in the production phase. Works, evaluating Environmental Impacts (EIs) of production processes, found in the literature can be divided into three group; (1) selection of green process planning from different alternatives (2) selection of green process parameters such as cutting fluid or machining tool. The integration of Computer Aided Design (CAD) and Computer Aided Process Planning (CAPP), as realized by Yifei et al. [START_REF] Yifei | A feature-extraction-based process-planning system[END_REF], represents a success in generating different possible process plans in real time and their corresponding EIs. It becomes possible to select the most appropriate machining tools with low EIs and also the optimum machining scenario. These benefits are possible thanks to features models that provide the necessary data for a Life Cycle Inventory (LCI).

We present below the most important works that guided us.

Analysis of CAD/CAPP integration with features

The integration between CAD and CAPP (Computer Aided Process Plan) systems is based on Feature Technology (FT). Many researches are focused on this integration.

Based on feature technology, Jiang et al. propose an automatic process planning system for the quick generation of manufacturing process plans directly from CAD drawings [START_REF] Jiang | An automatic process planning system for the quic k generation of manufacturing process plans directl y from CAD drawings[END_REF]. Features are also used to identify the finish-cut machinable volume by getting the rough-machined part from the final part [START_REF] Arivazhagan | A STEP AP 2 03/214 based Machinable Volume Identifier for Ide ntifying the Finish Cut Machinable Volumes from R ough Machined parts[END_REF]. Moreover we find in literature different researches which are oriented to elaborate a process sequence in a CAPP for a component based on features. For examples, Yusof and Case [START_REF] Yusof | STEP compliant C AD/CAPP/CAM system for turning operations[END_REF] established a STEP compliant CAD/CAPP/CAM (Computer Aided Manufacturing) system for turning operations. Features are explored to select an optimal Setup plan machining features of prismatic parts [START_REF] Hebbal | Setup plann ing for machining the features of prismatic parts[END_REF]. Kumar presents a methodology that can model any given component using cut features (mimicking machining operation) and produce an optimal choice of machine tools for machining that component [START_REF] Kumar | Optimal choice of a machine tool for a machining job in cae environment[END_REF]. Deja and Siemiatkowsk [START_REF] Deja | Feature-b ased generation of machining process plans for opti mised parts manufacture[END_REF] present a conceptual scheme which generates alternative process plans as a network.

EIs evaluation of process plans and selection of the greenest one

The evaluation of manufacturing environmental impacts is studied in most research works. It is aimed at limiting or avoiding the use of processes which have an impact on ecosystems, human health, etc. Various proposed approaches have been utilized to evaluate the environmentally oriented metrics of manufacturing processes, ranging from experimental investigations to the development of mechanical models. Jiang et al [START_REF] Jiang | D evelopment of an environmental performance asses sment method for manufacturing process plans[END_REF] present in their work an overview of these approaches. Then, they propose an environmental performance assessment method for manufacturing process plans. The method developed uses actual values for various environmental measures for the different operations in the plan to form an impact matrix. Le Borhis et al. [START_REF] Bourhis | Sustainable manufacturing: evaluation and modeling of environmental impacts in additive manufacturing[END_REF] developed a methodology based on a predictive model of flow consumption that is defined from the manufacturing path and the part CAD model. analyze, in their respective work, the environmental impact of dissimilar waste streams. They apply a scoring system which evaluates factors such as toxicity, carcenogenesis, irritation, flammability, and reactivity, reducing complexity of processing alternatives through a feature-based approach. A hierarchical part planning strategy for environmentally conscious machining is developed [START_REF] Srinivasan | Feature-based process planning for environmentally conscious ma chining-Part 1: microplanning[END_REF], and it is shown that a feature can be machined with different scenarios. The purpose is to choose the least impactful set of machining sequences. Figure 1 shows the example considered by Sheng [START_REF] Sheng | Multi-objective process planning in environmental ly conscious manufacturing: a feature-based approa ch[END_REF] to demonstrate the influence of changing the order of machining sequences on environmental impacts of a production process. In [START_REF] Sheng | Multi-objective process planning in environmental ly conscious manufacturing: a feature-based approa ch[END_REF], Sheng decomposes the environmental impact component into "micro" analyses of individual features [START_REF] Srinivasan | Feature-based process planning for environmentally conscious ma chining-Part 1: microplanning[END_REF] and "macro" analyses of feature interactions [START_REF] Srinivasan | Feature based process planning in environmentally conscious mac hining-Part 2: macroplanning[END_REF]. To evaluate the resources and environment attributes of manufacturing process, Zhang, H.et al. [START_REF] Zhang | T he Evaluation of Resources and Environment Attrib utes of Manufacturing Process Based on Scatter De gree Combination Evaluation Method[END_REF] present an IPO (Input-Process-Output) model and an evaluation index system in order to select the greenest process plan.

More than the evaluation of EIs, features are useful for selecting green process planning, such as the multi-objective optimization decision-making framework model of process planning for green manufacturing established by Cao et al. [START_REF] Cao | St udy on model set based process planning strategy fo r green manufacturing[END_REF] to put forward a new process planning strategy for green manufacturing and to develop a process planning software tool. In 2006, Tan et al [START_REF] Tan | Improved methods for process ro uting in enterprise production processes in terms of[END_REF] developed a new production process which determines a method focusing on sustainable manufacturing. It is based on case-based reasoning, expert systems, and feature technology for designing a new component's process flow. A sustainable development assessment is the basis of this process flow.

Fig.1 Influence of machining sequences order on the EIs of a part consisting of three types of features: a pocket, a hole and a finished planar face [16]

Zhao et al. [START_REF] Zhao | Toward the development of process plans with reduced environmental impacts[END_REF] also propose a method for environmentally conscious process planning. The method begins with an existing process plan, and then identifies impactful process steps, and associated design features, in terms of manufacturing cost and environmental impact. Alternative processes that can achieve these features are then considered to generate alternative process plans. This might be useful to integrate into CAD-LCA integrated systems to help a designer comparing different alternatives till features attribution. Nawata et al. [START_REF] Nawata | Life-cy cle design system for machined parts-linkage of LC I data to CAD/CAM data[END_REF] propose a new lifecycle design system specifically applicable to machined parts. The system automatically generates life-cycle assessment (LCA) feedback for the design process through the effective linkage of life-cycle inventory data with computer-aided design/computer-aided manufacturing (CAD/CAM) data by feature-based modeling. Hence, a determining idea coming out of CAD/CAM/CAPP integration is the possible generation of different machining scenarios.

Each scenario has environmental impacts related to the parameters of the process plan attributed. Feature technology, in CAD phase, can help the designer select, from possible alternatives, the greenest process.

Selection of green process parameters

Manufacturing process planning is based on a set of parameters such as cutting tool, cutting fluid, or machining tool. Tan, et al [START_REF] Tan | A decision-making framework model of cutting fl uid selection for green manufacturing and a case stu dy[END_REF] used feature technology in selecting the most ecological cutting fluid, and machining tool because they are important parameters in developing green products. In a similar vein, the selection of an ecological machine tool is the aim of Jiang and Zhang's study [START_REF] Jiang | A vecto r projection method to evaluating machine tool alter natives for green manufacturing[END_REF], where they develop the model called Vector Projection Method. This method, based on feature technology, creates a reference index system to justify machine tool selection, by considering resource consumption and environmental impact due to part features which needs an energy input/expenditure (electric mechanic, pneumatic, etc.) to be realized. Estimating the required energy and optimizing it presents a challenge for green manufacture.

Possible production energy consumption scenarios in a feature level are studied by Deshpande [START_REF] Deshpande | Feature level energy assessments for discrete p art manufacturing[END_REF], and assessed through MT Connect standard methodology. Other process parameters, such as finish cut and feed rate generate EIs due to their relation to the part quality. Helu et al. [START_REF] Helu | Impact of green ma chining strategies on achieved surface quality[END_REF], using feature technology, develop a methodology showing that the impact of green machining strategies such as part quality on achieved surface quality is most influenced by the finish cut(s) and feed rate. Improving part quality can also reduce life cycle EIs. Zhang et al. [START_REF] Zhang | Stud y on optimized principles of process parameters for environmentally friendly machining austenitic stain less steel with high efficiency and little energy cons umption[END_REF], present a study on optimized principles of process parameters (such as cutting depth, cutting velocity and feed rate) for environmentally friendly machining austenitic stainless steel with high efficiency and little energy consumption.

CAD and CAPP integrations may also allow to define more Life Cycle scenarios during the detailed product-process definition of a part or assembling. It is, therefore, necessary to integrate these benefits into CAD-LCA integrated tools to perform environmental assessment in real time and also choose the greenest process plan from existing possible scenarios. These contributions can be accompanied by a green end of life with promotion of design phase disassembly and recyclability.

Figure 2 summarizes benefits achieved from CAD/CAM/CAPP integration by using FT. These benefits confirm the ability to enrich features with required LCA data. In this paper, we are interested in the manufacturing phase of a part life cycle.

Proposed approach

In our methodology we explore feature technology to evaluate the environmental impact of a product manufacturing process from the geometric modeling phase by an LCA tool. In fact, it consists in giving the designer the machining environmental impact of the current CAD model once a feature is attributed. The methodology presented is an algorithm that allows the designer to see the machining environmental impact of each feature chosen in a real time and presents him an optimal scenario in terms of environment by using CAD, CAM, CAPP and PLM systems. These systems can provide many types of product information such as geometric and non geometric (material, tolerances or engineering constraints). CAM and CAPP systems can provide information about manufacturing processes and PLM systems can include the requirements phase, analysis and design stages, manufacturing, product launch, distribution, quality assurance, in-service maintenance and spare parts provisions. Some missing data are included in databases such as those of EcoInvent [START_REF] Frischknecht | Overview and methodology[END_REF] and "Bilan Produit" [START_REF]Bilan Produit[END_REF] where impacts such as earlier manufacturing operations and raw materials processing are already included.

Hence, the extracted and included data are enriching features, in CAD phase, with different types of information in order to practice and LCA, which consists in four stages defined by standard organizations (ISO14040) [33]; goal and scope, life cycle inventories, life cycle analysis and result interpretation as shown in Figure 3.  In the fourth step, Environmental Evaluation ( EE) is realized and the designer gives only the Func tional Unit (FU), which is established from the main technical characteristics of the product studied (we p ropose the FU related to the whole product) and the necessary data for realizing Simplified Life Cycle A ssessment (SLCA) is extracted automatically from C AD/CAM/CAPP systems. Data extraction is done th roughout generic algorithm that will be described in future works. The environmental impact calculus is done based on Morbidoni's Work [START_REF] Morbidoni | CA D-Integrated LCA Tool: Comparison with dedicate d LCA Software and Guidelines for the improveme nt[END_REF] where the m anufacturing part environmental impact can be calc ulated by the proposed simplified approach given by Equation 1: LCA Part. manuf= LCA Energy consumption+LCAWaste material LCA cutting fluid+...+  n i=1 LCA feature.i (eq.1)

Where; LCA Prod. manuf =environmental impact of part manufacturing phase LCA Energy consumption =environmental impact of part energy consumption during manufacturing process LCA waste material =environmental impact of part waste material LCA cutting fluid environmental impact of lubricant used for machining the part LCA feature.i=total environmental impact of feature i. The method used for calculus is the CML method [START_REF] Guinée | Environmental life c ycle assessment of products: Guide[END_REF]. It is a method which respects the standardized procedure of environmental assessment. In this met hod, environmental impacts are calculated along wit h four steps: classification, characterisation, normati on and assessment. Add a new feature mum one). Therefore, a feasibility test is practiced; i f the selected scenario is feasible then step six is rea ched, else another scenario is selected and tested.  The sixth step consists in saving the selected m achining scenario if the feasibility test is positive.

 The seventh step is reached if the test of finish ing designing part is positive; otherwise the last fou r steps will be redone (Steps 3, 4, 5 and 6).

The calculus, in this paper, is realized on global machining processes, in order to approve the feasibility of our approach. However, we propose, in our work [START_REF] Gaha | An Environmental Impact/Cost Model for Evaluating Multiple Feature-Based Machining Methods[END_REF][START_REF] Gaha | A New Eco-design Approach on CAD systems[END_REF] a methodology based on process plan activities aiming at selecting the optimum scenario having the best compromise environmental impact versus cost.

Figure 5 represents a framework of a new Ecodesign tool, oriented green manufacturing, applying the approach proposed previously, which we name "Green-CAD". It is composed of a connector which collects necessary data for LCA tool from CAD, CAM, CAPP and PLM systems which collaborate together on the same part. Once the environmental evaluation is done, graphical representation is shown to the designer. The graphic allows to see, at the user interface, the optimal scenario from proposed ones.

Case study

To reach the final part, the designer assigns features. For each assigned feature, manufacturing scenarios can be generated, each scenario being associated with its own EIs. Our methodology, as shown before, consists of; firstly proposing alternative manufacturing scenarios after each feature selected, secondly evaluating environmentally proposed scenarios and finally comparing between the obtained results in order to help the designer choosing the most ecological one. We further illustrate the proposed methodology with a case study. We consider the part shown in the right side of figure 4 which is a part made of brass and weighting 1kg. Based on features, there are two models that can be used for the sample part; the design model and the process model. In our study we consider the process model, where each feature or a set of features requires a manufacturing process. By combinatorial rules and data extracted from integrated systems (CAM, CAPP and PLM) to CAD system, possible manufacturing scenarios are proposed. Then, based on the integrated CAD/LCA system, that we proposed previously, EIs of each scenario are assessed.

Based on features, the identification of geometric forms of the considered part allows to generate four possible scenarios to realize it, as shown in figure 6.

 Scenario 1: the part can be made by Molding, then Surfacing, then Inferior Milling and Superior Milling processes.

 Scenario 2: the part can be obtained only by Molding process.

 Scenario 3: the part can be made by "Molding", then Surfacing and Welding processes.

 Scenario 4: the part can be made by Molding, then Surfacing, then Drilling and Thread processes.

User interface

A Simplified LCA is performed, by using the tool "Bilan Produit", which is a software developed by ADEME institution [START_REF] Zhang | Stud y on optimized principles of process parameters for environmentally friendly machining austenitic stain less steel with high efficiency and little energy cons umption[END_REF]. The database used was established in collaboration with EcoInvent Centre and the Swiss Centre for life Cycle inventories [START_REF] Helu | Impact of green ma chining strategies on achieved surface quality[END_REF], in order to assess the environmental impact of each scenario and compare them. By using the CML method [START_REF] Guinée | Environmental life c ycle assessment of products: Guide[END_REF], calculated impacts are presented in the following categories considered for each step: Energy consumption NR (MJ eq), Use resources (kgSbeq), Greenhouse GWP 100 mod (kg CO2eq), Acidification (kg SO2 eq), Eutrophication (air ground water) (kg PO4eq), Photochemical pollution (kg C2H4), Aquatic Ecotoxicity (kg1.4-DBeq) and Human toxicity (kg 1.4-DB eq). These impact indicators are expressed in units incomprehensible by non environmental expert designers. The normation occurs by taking the ratio between the value of product's impact and the reference value of the normation. The normation chosen in this software takes as reference the daily consumption of an average European.

The results are expressed in points, in order to allow the designer to globally apprehend which is the most ecological scenario. For the final part, four machining scenarios are proposed, the comparison between their environmental impacts are shown in figure 7 which highlights that scenario 4 is the most ecological one.

Conclusion

This paper presents the necessity of lowering EIs of manufacturing processes. It recalls some researches aiming at helping designers choosing the ECMP in CAD phase. Feature technology is used, in generating scenarios in CAPP and for the choice of greenest ones in terms of environmental impacts. Our proposed approach, also, explores feature technology to evaluate environmentally possible manufacturing processes of a product from the geometric modeling phase by an LCA tool in real time. A case study is developed to show the manufacturing scenarios generation and their related environmental impacts, which are calculated with the CML method. From this work, we deduce that FT enables the efficient retrieval of necessary Life Cycle Inventory (LCI) data. Hence, as a continuation to be published, we are developing the concept of "eco-feature" to model design data integration in the detail design phase and prepare a mechanical product to real-time simplified LCAs.

Fig . 2

 2 Fig .2 Benefits of CAD/CAM/CAPP integration based on FT

Fig. 3

 3 Fig. 3 LCA stages (ISO 14040) Enriching feature make it a Life Cycle Inventory (LCI) model. This model can be used for an Environmental Evaluation during CAD phase.Our approach is composed of seven steps as shown in figure4: The first step is done by the designer in selecti ng a feature from CAD system data base. In the second step, the designer designates a m

Fig. 4 7 .

 47 Fig.4 Proposed methodology for selecting green manufacturing process based on feature technology After an environmental impact calculus of manufa cturing processes, a comparative graphic between sc enarios is shown to the designer. The graphic repres entation is with End-points Impacts (Resources, Hu man Health and Eco-system damages) in a Single S

Fig. 5

 5 Fig.5 CAD/CAM/CAPP/PLM Integration Framework: "Green-CAD" Framework

Fig. 6 A

 6 Fig.6 A model for generation and comparison of possible manufacturing scenarios for a part case study