Dr Constantine E Kyritsis
email: kyritsis@softlab.ntua.gr

ORDINAL REAL NUMBERS 2. The arithmetization of order types

§0 Introduction . In this second paper on ordinal real numbers are proved, the main (elementary) properties of them. It is proved that the ordinal real numbers R α of characteristic α, is the maximal field of characteristic α (maximality) and that it is , according to the theory of Artin-Screier, a real closedfield. (It turned out ,after the work was completed and by thinking aside, that they are also Archimedean complete (see [ Glayzal A. 1937]),formal power series fields with real coefficients ,Dedekind complete (see [START_REF] Massaza | On the comletion of ordered fields[END_REF]), and Pythagorean fields ).

It is also proved a classification theorem which is analogous to the Hölder theorem for the Archimedean linearly ordered fields.In particular it is proved that any linearly ordered field of characteristic α contains the field Q α of ordinal rational numbers of characteristic α, as a dense subfield and it is contained in the field R α of ordinal real numbers of characteristic α, as a subfield ..As it is known, the linear segments of elementary euclidean geometry can be defined as special order-types with Archimedean property, and Archimedean (Hilbert) completness through axioms (see e.g. for a not ancient approach the Hilbert axiomatisation in [ Hilbert D.1977] ch 1 ).It can be proved to be order isomorphic with subsets of the real number field R. This is well-known and it can be called , the elementary arithmetisation of the order-types of Euclidean linear segments . On this fact is based the Cartesian idea of analytic geometry. This was an important turning point in the developments of the ideas and techniques of mathematics , of the discrete nature of numbers and continuous nature of geometry. The basic principle is that the continuum is developed from the discrete and not vice versa! An instance of this principle is the development of images and animation in computers through pixels and bits! It is surprising that in one of the consequences of the theory of ordinal real numbers, it is proved a far more advanced and complete result for the whole category of order types that has as corollary the previous important and elementary arithmetisation. Although more advanced, the result remains in the context of elementary theory of ordinal real numbers .In this result any order type can be "discretised" or "arithmetised" through the ordinal numbers.

The process of definition of the maximal fields R α , from the minimal (double well ordered ) monoids N α =α, of principal ordinal numbers,we call K-fundamental densification. It is proved that any order-type is order isomorphic to a subset of some field R α . Thus any order-type is constuctible by K-fundamental densification from ordinal numbers .This is called the K-arithmetisation of the order-types. Although in the way it is presented, this result is softly obtained, throws new light to the relation of ordinal numbers and ordertypes ,this relation turns out to be similar to the elementary relation of numbers and line segments in geometry. Also it, holds a second kind of arithmetisation ,the binary arithmetisation which we state in the same paragraph . §1 On the topology of linearly ordered fields. Local deepness, α-sequences.

The ordering of any linearly ordered field F defines a well known topology : the ordertopology denoted by T <; In this topology, as it is known, the field F is a topological field. This topology has very good separation properties; it is a T 1 -T 5 topology, that is a completely normal topology (see for instance [ Lynn A.Steen-Seebach J.A. Jr 1970] § 39 p. 66-68, also see [ Munkress J.R. 1975] Chapters I, II)

The previously described order topology is also called the locally convex topology compatible with the order (see [START_REF] Nachbin | Topology and Order[END_REF]] Ch I, II). (The convexity defined by the order).

Definition 1. Let X be a topological space. Let p X. The least ordinal α such that it exists a (local) base denoted by Bp of open neighbourhoods of the point p which is an α-sequence such that if x<y<α, U x  U y , is called local deepness of X at p. We notice that the concept of local deepness is very close to the concept of local weight of a topological space, where instead of ordinal we have an initial ordinal that is a cardinal number (see [START_REF] Kuratowski | Topology v.I v[END_REF]] V-I p. 53-54).

Examples of topological spaces such that every point has local deepness, are the ξ * -uniform topological spaces as they are defined in [Cohen L.W.Goffman C. 1949] pp 66 conditions 1.2.3.4.

As in the case of fields that are classes, we may permit topological spaces that are classes and the open sets is a class of subclasses closed to union and finite intersection. For such spaces, the local deepness may be Ω 1 that is the class of all ordinal numbers. Proposition 2. Let X a topological space and α, a limit ordinal such that every point has local-deepness α let AX.

It holds that there is an βsequence {x s |s<β} from elements of A such that . In other words topological convergence in X can be treated with β-sequences where β=car(α) is the upper character of α (see [N.L.Alling 1987] ch 1 §1.30 pp 29)

The proof is almost direct and to save space we shall not give it here.

Proposition 3. Let a field denoted by F of ordinal characteristic α, where α is a limit ordinal. Then every point x F in the order-topology has local-deepness car(α), where car(α) is the upper character of α (see [ N.L.Alling 1987] ch 1 §1.30 pp 29 ).

The proof is again direct and outside the scope of the paper.

Corollary 4. Convergence in the order-topology of a field of ordinal characteristic α, can be treated with β-sequences βcar(α) Needless to say, that in the case in which the topological space is a class and the local deepness is Ω 1 , then convergence can be treated with Ω 1 -sequences. §2 The Holder-type classification . (y) and r 1 (x). Also we have that 0<y-x<r 2 -r 1 .

Then y-x is a Qα-infinitesimal and is a Qα-infinite element of Fα, thus >α, contradiction since Char Fα = α.

Then there are not two element y,x Fα x<y with no element of Qα in [x,y], and Qα is dense in Fα. Q E D Remark. Thus every field Fα of characteristic α is a Weil completion of the field Qα of ordinal rational numbers (see [ Weil A.] ChIII Definition 2 but applied not only to local fields).

Theorem 6 (Maximality or completness up-to-characteristic ).

The field Rα is the maximal field, of characteristic α. In the sense that every field of characteristic α is contained as subfield of Rα (more precisely Rαcontains an order preserving monomorphic image of the field).The field R α is the unique fundamentally complete field of characteristic α.

Remark. This theorem is analogous to the well-known Holders theorem theorem ςηιψη στατες that every linearly ordered Archimedean field is a subfield of the field of real numbers. In other words the field of real numbers is the maximal Archimedean linearly ordered field. The previous property of the ordinal real numbers R α relative to their characteristic ,we call maximality or completness up-to characteristic .

But as an erroneous application of terms R is also the minimal Cauchy complete field of characteristic  and this also applies for the fields Rα in the sense that a completion of a linearly ordered field of characteristic α must be the field Rα .

Proof. Let any field of ordinal characteristic α denoted by Fα. By theorem 17 of [ Kyritsis C. 1991], the field Qα is contained in Fα: QαFα. Let x Fα. Let (L(x), R(x)) be the cut that x defines on Qα (L(x) = {v|v Qα v<x}, R(x) = {v|v Qα x<v}). Since Qα is dense in Fα (Lemma 5). There is a Cauchy α-sequence {x n |n w(α)} of elements of Qα that converges in Fα to x (all topologies are the order-topologies). Hence QαFαRα and the field Rα is a maximal field of characteristic α ; but also the field Rα is actually a minimal Cauchy complete field of characteristic α in the sense that the (strong) Cauchy completion of any field F α of characteristic α contains the field R α :Q α F α has as a

concequence that R α  . Thus if F α is complete then R α F α , F α R α hence F α =R α Q.E.D.
The theory of Artin-Schreier of real closed fields has an excellant application to the ordinal real numbers .

Corollary 7. The fields of ordinal real numbers Rα are real closed fields.

Proof. Direct from Theorem 6 , and remark 5 of [ Kyritsis C.1991] Q.E.D.

Post written Remark A. The author developed the theory of ordinal real numbers during 1990-1992 He had used the name "transfinite real numbers" without being aware that this term had been introduced by A.Glayzal during 1937 for his theory of linearly ordered fields.

From the moment he fell upon the work of A.Glayzal (see [ Glayzal A. 1937 ]) in the bibliography of the Book of N.L alling (see [N.L.Alling 1987 ] ) he changed the title to "Ordinal Real Numbers" . After the work had been completed ,the author realised , by thinking aside, a quite unexpected and not unhappy fact :That the fields of ordinal real numbers are algebraically and order isomorphic to the fields of transfinite real numbers of Galyzal .This can be deduced by the fact that the fields of transfinite real numbers are exactly all the Archemidean complete fields (see [Glayzal A. 1937] theorems 4,8,9) and by the maximality of the ordinal real numbers (theorem 6). Thus if R α is a field of ordinal real numbers of characterisic α, any Archemidean (linearly ordered field ) extension of it ,it shall have the same characteristic with R α . It seems that it can be proved , that any cofinal (coterminal) linearly ordered field extension ,is of the same characteristic . By the maximality of R α (theorem 6) it shall have to coinside with R α . In other words the fields of ordinal real numbers are Archemidean complete fields (although they may be non-Archemidean ).But this is a characteristic property of the fields of transfinite real numbersb of Glayzal.

Thus they are order and field isomorphic with fields of transfinite real numbers .Conversely ,let any field R(λ) of transfinite real numbers of Archemidean base λ. Let us denote by α its ordinal characteristic .Let us suppose that there is an order and field extension of it with the same characteristic .Then it has to be an Archemidean extension of R(λ). By the Archemidean completness of the transfinite real numbers ,it has to coinside with the R(λ). Thus the transfinite real numbers are also complete up-tocharacteristic .

But this is a characteristic property of the fields of the ordinal real numbers.Hence they are order and field isomorphic with fields of ordinal real numbers .Thus the ordinal real numbers should be considered as a different technique ,nevertheless indispensable and more far reaching .It is the technique that everyone would like to work. For the applications and especially with measurment proceeses ,the fields R α,ω are of prime interest and indispensable .

Post written remark C .The facts of the previous remark ,have as a concequence that the fields of ordinal real numbers are formal power series fields with coefficients in the real numbers and exponents in some order types.Thus the n-roots of their positive elements are contained in them (see [START_REF] Neumann | On ordered division rings[END_REF]] pp 211 ,4.91 Corollary).In other words they are Pythagorean complete fields.

Theorem 8. (The Holder-type classification theorem).

Every field of ordinal characteristic α, denoted by Fα (where α is a principal ordinal) is contained between the fields Qα and Rα :Q α F α R α .

Proof. Contained in the proofs of the theorem 7 and lemma 5

Q.E.D.

Remark.9

The previous theorem gives that the hierarchy of ordinal real numbers has universal embedding property for the category of linearly ordered fields, that is every linearly ordered field has an monomorphic image in some field of the hierarchy.The hierarchy of transfinite real numbers is known to have, also, this property .Such hierarchies we call universal embedding hierarchies. Especially the hierarchy of ordinal real numbers after the classification theorem 8 ,we call also, universal classification hierarchy.

Remark.10

We notice that since every order type λ is order-embeddable in some transfinite real number field R(λ) (see [Glayzal A. 1937] )as Archemidean base which in its turn is embeddable in some ordinal real number field R α ,the above two hierarchies as hierarhies of order-types are universal embedding hierarchies for the category of order-types .Let an order type λ ; the least principal ordinal number α such that λ is order-embeddable (by a monomorphism) in the order-type and field R α , is called the fundamental density of the order type λ and is denoted by df(λ).

Remark.

In the [START_REF] Massaza | On the comletion of ordered fields[END_REF]] Definition I , is defined which cuts are the Dedekind cuts in linearly ordered fields .It is proved also that the Dedekind completion D(F) of a linaerly ordered field F is also its Cauchy completion (in the order topology ).If we take the Dedekind completion D(R α ) of a field of ordinal real numbers R α , it has to be its Cauchy completion which is again the R α . Thus the fields of the ordinal real numbers are alsoDedekind complete . Conversely ,let any Dedekind complete linearly field F .Let us denote with α its ordinal characteristic .Then by the Holder type classification (theorem 8 ) it is a subfield of the field R α of ordinal real numbers of characteristic α .Since the Dedekind completion D(F) =F coincides with the F and also with its Cauchy completion ,we get that F=R α , because the Cauchy completion of F is the R α . In other words the class of Dedekind complete fields coincides with the class of the fields of ordinal real numbers .

Summarising we mention that the fields of ordinal real numbers have at least four kinds of completnesses that characterise them : Cauchy completness ,Dedekind completness,completness up-to-characteristic, Archemidean completness .It seems that he previous four completnesses can be summarised by saying that there is no cofinal (coterminal ) order field extensions of them ;in short they are cofinally complete ,or cofinally maximal .They are also realcomplete (closed ,Artin-Shreier ) and Pythagorean complete.

Remark. By corollary 7 we get that the field C α is the algebraic closure of R α :C α = .

We close this paragraph by mentioning that an axiomatic definition of the field Rα (α is a principal ordinal) would be the following:

First axiomatic definition of Rα.

The field of ordinal real numbers Rα is the unique Funtamental (Canchy)complete, in the order-topology, field of characteristic α.

Second axiomatic definition of R α .

The field of ordinal real numbers R α is the unique complete (up-to-characteristic) field of characteristic α . These definitions apply even in the case of the field of real numbers (a = ω). § 3 The arithmetisation of order-types .

Remark.As it is known the linear segments of elementary Euclidean geometry can be defined as special order-types with Archimedean property and Hilbert completness through axioms (see e.g. for a not ancient approach the Hilbert axiomatisation in [START_REF] Hilbert | Grundlagen der Geometry Teubner Studienbucher[END_REF] ch 1 ).Then ,they can be proved to be order isomorfic with subsets of the real number field R. This is known as the elementary arithmetisation of the order-types of Euclidean linear segments.

Proposition 10.(the K-funtamental arithmetisation theorem of order-types.)

Every order-type λ is K-arithmetisable with ordinal numbers and has a fundamental density df(λ) which is a principal ordinal number .

In the next paper ,after the unification theorem of the transfinite real ,surreal ,ordinal real numbers ,a second arithmetisation theorem shall be proved. Two more universal hierarchies of formal power series fields shall be, also, proved that they are universal embedding hierarchies .We state these results here. For the definition of tree ,height of a tree, level of a tree, binary tree e.t.c.see [START_REF] Kuratowski | Set Theory North -Holland[END_REF]] ch ii § 1, § 2 . The binary tree of height the ordinal α we denote with D α . After the previously mentioned unification theorem 17 of the next paper we get that the hierarchy of binary trees is a universal embedding hierarchy for the order-types . Since the binary trees are subsets of linearly ordered fields and their elements consisting exclusively from 1's in the binary sequence, correspond to the ordinal numbers with the Hessenberg operations (see also [Conway J.H. 1976] ch 3 note pp 28 and also [ Kyritsis C. 1991Alt] the characterisation theorem ) this universal embedding property we call also binary arithmetisation .The least ordinal αsuch that an order-type λ is order embeddable in the binary tree D α ,we call the binary density of λ ,and we denote it by db(λ).

Theorem 11 ( The binary arithmetisation theorem of order-types )

Every order-type λ is binary arithmetisable and has a binary density db(λ) which is an ordinal number .

From the previous theorem ,by denoting a level of height α of a binary tree ,by T α ,and giving to the Cartesian product the lexicographical ordering ,we also get the next :

Corollary 12. The formal power series hierarchies R((D α )), , are universal embedding hierarchies for the linearly ordered fields . § 4 Some general results on linearly ordered fields .

In this paragraph we give some results generally for the category of linearly ordered fields. To save space we shall not give the proofs, since they do not have serious dificulties,nevertheless we shall indicate how they can be obtained .

Lemma 13 (On the rank and characteristic)

Let us suppose that the characteristic of the field F is where α, is α limit ordinal. It holds that the rank of the extension F/K is a cofinal order-type with the characteristic of the field F. That is cf(r(F/K))= cf(charF)=cf(char F-char K).

Remark.For the definition of the rank of an extension see [ Kyritsis C. 1991]

5.

The previous identities show also that the definition of operations used to define the surreal number fields are not something peculiar to these fields but hold in any linearly ordered field . Remark. For the definition of the transandental degree, base e.t.c see for instance [START_REF] Zariski | Commutative Algebra V.I[END_REF] vol. I pp. 95-102 also [Kyritsis C. 1991 ] § 4 ). The proof is obtained by using the Holder-type classification for F :Q α  F  R α where α=char(F).

The next proposition shows that all the information of an extension of linearly ordered fields is to be found in the ideal of infinitesimals (or in the infinite elements). Proposition 16. Let F/k, F'/k two (ordered) extensions of the same linearly ordered field k. If the ideals of K-infinitesimals of the extension denoted respectively by m F and m F ' are isomorphic as ordered integral domains ,then this isomorphism is extendable to an algebraic and order isomorphism of the fields F, F'.

Remark .The proof is direct from the definitions.

Remark. An extension F/k of the linearly ordered field k to F, is transcendental if Char F>char k and then the field F is an infinite dimensional vector space over k. Proposition 17 . Let F be a linearly ordered field of characteristic char(F)= where α is a limit ordinal . It holds that the field F in the order topology is totally disconnected .

Remark. The proof uses the existence, for every principal ordinal , of the fields of ordinal real numbers R .

Theorem 18 The classes of transfinite real numbers CR, and of ordinal real numbers Ω 1 R, coinside.

Proof. Since both Hierarchies of transfinite real and ordinal real numbers have the universal embedding property (see remark 9 ) ,every transfinite real number-field is contained in some ordinal real number-field and every ordinal real-number field in some transfinite real number-field.Thus CR Ω 1 R and Ω 1 R  CR, and CR = Ω 1 R .

Q.E.D. § 5 The A-Archimedeanity

The, at least two different, definitions of archemideanity, that can be found for instance in [Glayzal A. 1937] and in other authors as in [ Conway J.H. 1976 ] or [ Arin E. Schreier O. 1927] give us the opportunity to treat them in unified way through the concept of archemideanity relative to a monoid.

The fact that the linearly ordered field F has characteristic ω (the least infinite ordinal) is equivalent with the statement that the field F is Archimedean according to any (classical) known definition.

Let us denote by G a linearly ordered group and by A a monoid of endomorphisms of G as a group.

It is said that x is A-Archimedean to y where x,y G iff there are a,b A with a(x)y and b(y)x. If A is the domain Z of integers (the endomorphisms are multiplication with an integer )we simply say that x is Archimedean to y. If for every pair x,y of elements of G holds that x is A-Archimedean to y, it is said that G is A-Archimedean Let F be a linearly ordered field .If we consider it as an additive group, and we denote by A 1 a monoid of endomorphisms of the additive group , we get the concept of x being Aadditively Archimedean to y. If we consider the multiplicative group F * and we take a monoid, denoted by A 2 , of edomorphisms of the multiplicative group, we get the concept of x being A-multiplicatively Archimedean to y.

Let A=A 1 VA 2 be the monoid of mappings from F to F generated by the previous monoids . It is said that x is A-field-Archimedean to y iff there are a, b A such that a(x)y, b(y)x.

In any extension F/k of a field K by a field F, where F,k are fields of ordinal characteristic with char F>Char K, if we take as A 1 , to be the multiplication with elements from the field K ( considering the field F as a linear space over K), we get the concept of x being K-additively Archimedean to y.( For K=R this is also known as "x is commensurate to y " see [Conway J.H. 1976] ch 3 pp 31 ).

If A 1 is the multiplication with integers and A 2 is power with integral exponents ,then it is simply said that x is field Archimedean to y (Known also from the A. Gleyzal's definition of Archimedeanity) A non-Archimedean linearly ordered field denoted by F is simply a linearly ordered field for which not all pairs (x,y) of its elements are mutually additively Archimedean. (Thus charF>ω ) But it can be very well A-additively Archimedean for other monoids A.In particular if charF=α and A is the monoid of endomorphisms of the additive group of F defined by (field ) multiplication with ordinals less than α, then it is A-additively Archimedean and we denote it by writing that it is α-additively Archimedean

Lemma 5 .

 5 In every field of characteristic the field Qα is a dense subset.Proof. Let a field of characteristic α, which we denote by Fα. By the theorem 17 of[Kyritsis C. OR1] the field Qα is a subfield of Fα. Let us suppose that it is not dense in Fα. Then there are two elements xin the internal [x,y]. Then the element z = y-x is Qα-infinitesimal. of Qα can be written as r 2 -r 1 where r 2

  a field R α of ordinal real numbers of ordinal characteristic α. It is also a field of transfinite real numbers of archemidean base λ. The set of all elements of R α that as formal power series have support of ordinality less than βo(λ)=maximum ordinality of well ordered set of λ, and which we denote by R α,β is a field ,subfield of R α .Indeed R α,ο(λ) =R α .

  § 4. For the proof of the previous theorem we use the existence for any principal ordinal of the ordinal real numbers fields of characteristic . Let F be a linearly ordered field. If x F by L(x) we denote the set L(x) = {y| y F y<x} and by R(x) the set R(x) ={y| y F x<y}. By elementary arguments on linearly ordered fields the following identities can be proved. Lemma 14 Let x, y F. The following hold 1. L(-x) = -L(x) R(-x) = -R(x) 2. L(x+y) = L(x)+y = x+L(y) R(x+y) = R(x)+y = x+R(y) 3. L(x.y) = L(x).y + xL(y) -L(x).L(y) = R(x).y + xR(y) -R(x)R(y) R(x.y) = L(x).y+xR(y) -L(x).R(y) = R(x).y + xL(y) -R(x).L(y) 4.

  work we will understand the true peculiarity of the technique of the surreal numbers. Lemma 15 If F/k is an extension of two linearly ordered fields , it holds that tr.d.(F/k)  2 (Char.F) where tr.d.is the transandental degree of the extension .

Acknowledgments. I would like to thank professors W.A.J.Luxemburg and A. Kechris (Mathematics Department of the CALTECH) for the interest they showed and that they gave to me the opportunity to lecture about the ordinal real numbers in CALTECH. Also the professors H. Enderton and G.Moschovakis (Mathematics Department of the UCLA) for their interest and encouragement to continue this project.


Dr Constantine E Kyritsis
email: kyritsis@softlab.ntua.gr

ORDINAL REAL NUMBER 3. The techniques of transfinite real , surreal, ordinal real, numbers ; unification

§ 0 Introduction . In this third paper on ordinal real numbers, it is proved that the three different techniques and Hierarchies of transfinite real number-fields, of surreal numbers, and of ordinal real numbers , give by inductive limit ,or union, the same class of numbers ,already known as the class No .It can be characterized, simply, as the smallest (linearly ordered field which is a ) class and contains every linearly ordered set-field as a subfield . Τhis class ,and also the category of linearly ordered set-fields, we call " the linearly ordered Newton-Leibniz realm of numbers". It is obvious that without the set theory of G. Cantor as it is formalized, for instance, by Zermelo-Frankel and a correct thinking about the infinite, this "realm of numbers" would not be definable.

Nevertheless we could follow a still different approach. If in the completion of rational numbers to real numbers we ramify the equivalence relation of convergent sequences to others that include not only where the sequences converge (if they converge at the same point) but also how fast (if they converge in the same way e.g finally equal ,an attribute related also to computer algorithms complexity) ,then we get non-linearly ordered topological fields that contain ordinal numbers (certainly up to ω a , a= ω ω ) and are closer to practical applications. This approach does not involve the random variables at all, but involves directly sequences of rational numbers as "Newtonian fluxions". This creation can be considered as a model of such linearly ordered fields (up-to-characteristic ω a , a= ω ω ) ,when these linearly ordered fields are defined axiomatically. This gives also a construction of the real numbers with a set which is countable. This does not contradict that all models of the real numbers are isomorphic as the field-isomorphism is not in this case also an isomorphism so the Cauchy-real numbers and such a model still have different cardinality.

If we want to define in this way all the Ordinal real numbers , then it is still possible but then this would give also a device for a model of all the ZFC-set theory! And such a model is indeed possible: By taking again sequences of non-decreasing (in the inclusion) finite sets of ZFC , and requiring that any property ,relation or operation if it is to hold for this setsequence it must hold finally for each term of the sequence and finite set. In other words we take a minimality for every set of it, relative to the axiom of infinite. It is easy to prove that the (absolute) cardinality of such a model is at most 2 ω , that is at most the cardinality of the continuum. We could conceive such a model as the way a computer with its algorithms , data bases tables etc would represents sets of ZFC in a logically consistent way. Needles to say that a similar model of ZFC set theory could be given within Euclidean geometry! (e.g. after the Hilbert's axiom system, and in addition accepting only the natural numbers). There is no contradiction with the 2nd-incompletness theorem of Gödel as the argument to prove that it is a model of ZFC-set theory is already outside ZFC-set theory (as are also the arguments of Gödel ,or of Lowenheim-Skolem that gives a countable model of ZFC-set theory).

It is not directly apparent that so different techniques and ideas would have such an underlying unity. It is, also , surprising that, although the Hessenberg operations were very early known in the theory of ordinal numbers, (at least since 1906, see [START_REF] Gleyzal | Transfinite real numbers[END_REF]) no one went far enough to define through them, fields in a way similar to the way that the real numbers are defined from the natural numbers. Although G.Cantor, himself was conceiving the ordinals as a natural continuation of the natural numbers (see [Frankell A. A. 1953 ] introduction pp 3 ) ,as it is kmown, he rejected the attempts to define infinitesimals through them . (see [Frankell A. A. 1953] ch ii § 7.7 pp 120). We could speculate that un underlying reason for this, might be that, his set-theory was already strongly attacked and was facing the danger of final rejection ,and these were good enough reasons to avoid the additional charge that his theory " opened the door" to infinitesimals . In spite of this, there are many who might consider that although the present results are coming now, nevertheless it is too late and ,they might speculate ,for this long delay (more than eighty years) and diversion of ideas and technique, nevertheless on the same subject, we could suspect systematic obstructions, that came outside the mathematics. Nevertheless, there are others who consider that it is too early for such a development ,and especially for an analysis on such numbers. It seems that it has never been published any "partially ordered Newton-Leibniz realm of numbers" (in other words a category of transcendental extensions of the real numbers ,that are partially ordered fields and complete in the order topology ) with reasonably "good" properties for a classification.

In this paper we use the surreal numbers, as they are definable in the Zermelo-Frankel set theory, through the binary trees, directly as a class, and not as union of some set fields.(The original technique of J.H.Conway).I met J.H.Conway during 1992 at Philadelphia in the USA, I talked to him about the new developments in this area of research and I gave to him the present work but as he told me he had more than a decade that for the last time he had active interest in the subject. I is somehow necessary to make use of classes instead of sets; since, for the kind of "induction" that the J.H.Conway uses, we prove that it is reduced to the usual transfinite induction on the height of the elements of the trees; but in their union as a class and not for each one of them separately as a set; in the latter case in which the trees are sets the induction fails .The key-point is to prove that for every cut that J.H.Conway uses it does really exist a unique element of the trees of least height . "simplest number" as it is used to be called ). This is a very crucial point, for the whole technique of the surreal numbers, to work, and it seems that it has been obscured, by not paying sufficient attention to it

The author has initially included also the non-standard real numbers in the classification. As they are also linearly ordered fields and the present classification is of all linearly ordered fields it was natural to include them. There were experts in non-standard analysis that were glad about it. Nevertheless there were experts that insisted that according to the initial definition of A.Robinson and not of later definitions, it was not claimed that the non-standard real numbers were sets inside Zermelo-Frankel system. Only if Zermelo-Frankel system was used to model meta-mathematics also the they would be also sets. This was nevertheless different as such sets would models of meta-mathematical entities different than the sets that are models of mathematical and not meta-mathematical entities. Because of their arguments and in spite the fact that this made some other researchers of non-standard mathematics unhappy, the author prefers in this first publication about ordinal real numbers not to include the non-standard real numbers in the unification. Any definition nevertheless that has the non-standard real numbers as ordinary sets of Zermelo-Frankel set theory, would naturally lead to a straightforward proof that such fields are always subfields of some field of ordinal real numbers! The author has already produced pages with this proof that is based on the premise that I mentioned. § 2. The surreal numbers .

In this paragraph we define the class No of surreal numbers inside the ZF-set theory.We use the binary trees (see [START_REF] Conway | On numbers and games Academic press[END_REF]] appendix to part zero pp 65 and [Kuratwski K.-Mostowski A. 1968] Ch IX §1, §2).The crucial point is to prove that for the cuts defined by J.H.Conway in these trees it does really exist a unique element strictly greater than all the elements of the left section and strictly smaller than all the elements of the right section (the "simplest number" ).Through this the Conway-induction us reduced to the usual transfinite induction on the height of the elements of the tree .As we shall see this works for the union of all trees as a class but fails for each one set-tree .For the definition of the tree, binary tree, height, levels of the tree ,H ξ -set see [Kuratwski K.-Mostowski A. 1968] Ch IX §1, §2 Theorem 2, . The binary tree of height α we denote by D α . More precisely we are interested for the trees of the next definition.

Definition 1. Let α be an ordinal . We define = {x|x Da such that there is β<α such that for the element x as a zero-one sequence x = {x ξ |ξ<a} holds that x β = 1 and x ξ = 0 for ξ>β}.

We call the set

the open full-binary tree of height α.

We also remind that if for the height α, holds that (α) is a cofinal to α regular aleph: and for every  l D(L), r I(R) it holds that l < x < r}. By the H ξ property of it holds that M   . Let A = {β|β is an ordinal number such that there is x M with x T β where T β is the β-level of Dα in other words there is x M of height β}. Let α 0 = min A. Let Dα 0 (L) Iα 0 (R) the subsets of D(L) R(L) of elements of height less than α 0 , and let Mα 0  M the subset of M that consists of elements of height α 0 . Suppose that the set Mα 0 contains two elements x, y with e.g. x  y. We will prove that Mα 0 contains only one element. Let x'={x β |β<α 0 } that is that part of the α 0 -sequence x with terms of indifes less than α 0 .

(α) =  cf(α) =  ξ
And the same also with y' = {y β |β < α 0 }. Then there is l x or r x and l y or r y respectively in Dα 0 (L), Iα 0 (R) such that they are equal with x', y'. If x=r x then, if the α 0 -term of x is 0 or 1, in both cases x > r x , contradiction. Hence there is no such r x and also such r y . Then l x =x' l y =y' and l x l y . The α 0 -term of x and y might be 0 or 1. The only possible cases are {x = (l x ,0), y = (l y ,0)}, {x = (l x ,0), y = (l y ,1} {x = (l x ,1), (l y ,1)}, {x = (l x ,1), y = (l y ,0} where with the parenthesis we symbolize the α 0sequence which is the elements x, y. Let us suppose that x  y and , the part of the α 0 -sequence with terms with indices less than δ, with δ  α . The next definition is the definition of the class of surreal numbers in the ZF-set theory and it depends as we mentioned on the lemma 2 .As it is seen ,in the hypotheses of the lemma 2 the cardinality of halfs of the cut is bounded by (α). If it is to include all possible cuts of the tree then the lemma 2 will give the element x ο in some tree , of sufficient greater height,thus outside the original tree D α . This is why we mentioned that the definition of surreal numbers (with the original technique of J.H.Conway ) does not apply to the trees separately .

Definition 4. Let U =No be the union of all regular open full-binary trees. It is a class (after axiom A2.(see [START_REF] Cohn | Universal Algebva Harper -Row[END_REF]] p1-36)) Operations may be defined in this linearly ordered class according to the formulae of Lemma 2 in [Kyritsis C.1991 Alt. or Free etc.)] II, that hold for every linearly ordered field that is:

1. let α be an ordinal with (α)=  cf(α) and L,R subsets of such that for every l  L, r  R holds that l < r. Then there exists a regular aleph β such that and (L), (R)< (β). Then there is by lemma 2 a unique element of least height such that l < x 0 < r for every l  L, r  R, we denote this element by {L|R} and we write x 0 = {L|R}. We note that although , it holds that and α <β.

2. If and we denote the height of x, y by h(x), h(y) and by L(x), L(y), R(x), R(y) the sets Then the operations are defined through simultaneous two-variable transfinite induction in the form of the lemma 2,3 in [ Kyritsis C. 1991 Free etc.], for the heights of the trees where for the initial segments of ordinals we substitute the corresponding trees of No (For every ordinal β<α such that N(β)=N cf(β) corresponds a tree

). Thus the function of operation is defined not on w(α) 2 but on 2 . For the addition, the next rule is used x+y={L(x)+y  x+L(y)| x+R(y)}  R(x)+y}. This definition presupposes the definition of addition.

Inverse is defined by

As it is proved in [START_REF] Conway | On numbers and games Academic press[END_REF]] Ch0, 1 the set No is a linearly ordered c-field. The characteristic of No is easily proved to be Ω 1 , we call this c-field, c-field of surreal numbers. According to Definition 3 No is an H ξ -leveled Dedekind complete field. § 3 The unification .

In this paragraph we prove that all the three different techniques and hierarchies of transfinite real ,of surreal ,of ordinal real numbers give by inductive limit or union the same class of numbers .We have already proved that CR=Ω 1 R=C * R. (see corollary 10) and it remains to prove No=CR. [START_REF] Conway | On numbers and games Academic press[END_REF]] ch 2 § ""containment of the ordinals "note pp 28 and also [Kyritsis C.1991 From the previous lemma we get that  = No  Ω 1 R ,thus :

The unification theorem 7

It holds that the classes of transfinite real numbers CR , of surreal numbers No, of ordinal real numbers Ω 1 R ,coincide ,and it is the smallest class (and linearly ordered c-field ) ,that contains all linearly ordered set-fields as subfields.

We can have obviously analogous statements for the other classes of numbers (complex , quaternion e.t.c.). After the previous theorem, the binary arithnetisation of the order-types, stated in [ Kyritsis C. 1991] II ,theorem 11, is directly provable. We remark that because the levels of the open full binary trees have the property that any upper (lower bounded set has supremum (infimum ) ,(see [START_REF] Kuratowski | Set Theory North -Holland[END_REF]] ch ix §1, § 2 theorem 2 ),and after the Hilbert and fundamental (Cauchy) completness of the ordinal real numbers, and remark after definition 13 and ω-normal form according to [START_REF] Frankel | Abstract set Theory[END_REF]] ch 3 theorem 21 ,and after corollary 21 in [Kyritsis C. 1991] ,II , we also get:

  the open full-binary tree is an H ξ set, (see[Kuratwski K.-Mostowski A. 1968] ChIX §2 Theorem 2, the proof works also for trees where (α)=  cf(α) )Lemma 2. For every pair of subsets L, R of the open-full-binary tree of height the ordinal α, such that (α) is a regular aleph, and holds that: for every l L, r R, l < r, and (L), (R) < (α), there is exactly one element x 0 of least height in such that l < x 0 < r for every l L, r R. Proof. Let D(L) = {x|x such that there exists l L with x  l} and I(R) = {x|x such that there exists r R with r x} that is D(L), I(R) are the decreasing and increasing lower and upper half subsets of determined by L, R, in the linear ordering of as a tree (see [Kuratwski K.-Mostowski A. 1968] Ch IX §1 Lemma A). Let the set M = {x|x e

  Let the least value of δ, be denoted by δ 0 such that . If holds that because x<y. In the sequent, let z=(Dβ(x)=Dβ(y) β< δ 0 , 1). Then x < z  y. If δ 0 = a 0 then x=y because Xα 0 = Yα 0 = 1. Then δ 0 < α 0 and also z < y and x < z< y and the height of z is δ 0 < a 0 contradiction. Hence x= y, and Mα 0 contains only one element. , R have height <c), then a 0  c by the H ξ -property of if c is also such that (α)=  cf(α) Q.E.D. Definition 3. The open full binary tree of height α, such that (α) is a cofinal to α, regular aleph , I call regular open full-binary tree. The property of the previous lemma of a regular open full-binary tree I call H ξ -leveled Dedekind completness. We remark that the class of regular alephs is unbounded (see [Kuratwski K.-Mostowski A. 1968] p. 275 relation 5 ) Thus the class of ordinals α such that (α)=  cf(α) is unbounded.

  3. The opposite is defined by:-x = {-R(x)|-L(x)}4. Multiplication is defined by x.y={L(x).y+xL(y)-L(x).L(y)R(x).y+xR(y)-R(x)R(y) |L(x).y+x.R(y)-L(x).R(y)  R(x).y+x.L(x)-R(x).L(x)}.

Lemma 5 .

 5 It holds that CR=Ω 1 R=C * R No. Proof .Let an open full binary tree of height the principal ordinal a .Then  No ,and the field-inherited operations in the initial segment W(α) are the Hessenberg operations (see

  Alt] the characterisation theorem ).If α was not a principal ordinal, the W(α) would not be closed to the Hessenberg operations .Thus the N α , Z α , Q α are contained in No ,since what it is used to define them from W(α) is only the field operations .The Q α is a field and from the fact that No is closed to extensions of its set-subfields (see[ Conway J.H. 1976] ch 4 theorem 28 )we deduce that the field of ordinal real numbers R α is contained in No, for every principal ordinal number α .Thus R α =Ω 1 RNo .Q.E.D. Lemma 6 . For every regular open full binary tree , it holds that  R β , for some sufficiently big principal ordinal number β . (With the inclusion is meant that the restriction of ordering of R α in the tree, coincides with the ordering of the tree). Proof . We shall prove it by transfinite induction .It holds for the trees of finite height. The transfinite induction shall be on the transfinite sequence of all ordinal numbers such that (α)=  cf(α) and (α) is a regular aleph. Let us suppose that it holds for all such ordinal numbers of W(α), and (α)=  cf(α) and (α)

  representation, leveled Hilbert completeness, leveled fundamental (Cauchy) completeness, leveled H ξ Dedekind completeness ,leveled supremum completeness and representation with ω-normal forms.

	Theorem	8 . The	class	of	numbers	CR=Ω 1 R=No	has	leveled
	formal power series

Counterexamples in Toplogy Springer 1970 . Seebach J. A. Jr.1970] [Viswanathan T.M.1977] Ordered fields and sign-changing polynomials. Journal fur reine und angewante Mathematik, 296 pp 1-9,1977.

[ Van der Waerden B.L.1970

LINEAR NUMBERS 3.

The technique of "free operations-fundamental completion" for the definition of a wellordered classifying Hierarchy of Linearly ordered associative commutative number-fields. The transfinite real , the non-standard real, the surreal numbers; unification . by Dr.Konstantine E.Kyritsis

Computer Technology Institute

Patra Greece

Abstract

In this paper is introduced a forth new technique of "the enlargment real numbers",that in some sense includes the non-standard real numbers of A.Robinson.It is proved ,that the three different technics and hierarchies of transfinite real numbers , of the surreal numbers ,of the ordinal real numbers, give by inductive limit or union the same class of numbers .

Key words

linearly ordered commutative fields transfinite real numbers non-standard enlargment fields of the real numbers surreal numbers formal power series fields


Juan C Muñoz-Fernández

Alessia Knauss
email: alessia.knauss@chalmers.se

Lorena Castañeda

Robert Heinrich
email: heinrich@kit.edu

Matthias Becker
email: matthias.becker@iem.fraunhofer.de

Nina Taherimakhsousi
email: ninata@uvic.ca

Capturing Ambiguity in Artifacts to Support Requirements Engineering for Self-Adaptive Systems

Keywords: runtime requirements, self-adaptive systems, artifacts

Self-adaptive systems (SAS) automatically adjust their behavior at runtime in order to manage changes in their user requirements and operating context. To achieve this goal, a SAS needs to carry knowledge in artifacts (e.g., contextual goal models) at runtime. However, identifying, representing, and refining requirements and their context to create and maintain such artifacts at runtime is a challenging task, especially if the runtime environment is not very well known. In this short paper, we present an early concept to requirements engineering for the implementation of SAS in the context of uncertainty. Especially the wide variety of knowledge materialized in artifacts created during software engineering activities at design time is considered. We propose to start with a list of ambiguous requirements -or under-specified requirements -, leaving the ambiguity in the requirements, which will in the later steps be resolved further as more information is known. In contrast to conventional requirements engineering approaches, not all ambiguous requirements will be resolved. Instead, ambiguities serve as key input for self-adaptation. We present five steps for the resolution of the ambiguity. For each step, we describe its purpose, identified challenges, and resolution ideas.

Introduction

Requirements engineering (RE) activities for self-adaptive systems (SAS) trespass the limits of requirements analysis phases taking place also at design time and runtime [START_REF] Qureshi | Towards a Continuous Requirements Engineering Framework for Self-Adaptive Systems[END_REF]. When developing a SAS, designers need to define structural and behavioral adaptation points explicitly (where, when, what, how). In traditional RE approaches, a particular set of artifacts (e.g., requirements lists, goal models, feature models, and statecharts) is created and managed by requirements engineers. To achieve adaptation, a SAS must be aware of this knowledge gathered at design-time to be able to adapt itself and carry the knowledge, e.g., in the form of models at runtime [START_REF] Bencomo | Requirements Reflection: Requirements as Runtime Entities[END_REF]. Furthermore, based on this knowledge, a SAS can decide about the best suitable adaptation at runtime. In related work, adaptive requirements have been introduced to explicitly define variation points in requirements to leave room for adaptation [START_REF] Qureshi | Engineering Adaptive Requirements[END_REF].

We propose to focus on the reuse of the knowledge contained by the set of artifacts usually created and managed at design-time and to enable their facilitation at runtime systematically. We define an artifact as a machine-readable document relevant to requirements and context. A draft of the artifacts are created by requirements engineers at design time, and they will be updated at runtime.

We start from a set of elicited requirements tolerating some degree of ambiguity. We propose to keep on purpose the potentially contained ambiguity captured in the requirements artifacts and to design and implement the software without (entirely) resolving the ambiguity. We define the ambiguity as of under-specification of requirements. Usually, resolving ambiguities is an essential part of the traditional RE activities. In a SAS, however, we claim that embracing ambiguity throughout software design and implementation is a major step towards defining and extending relevant context attributes for self-adaptation. Nevertheless, only the ambiguity that leads to feasible variability points should be maintained. For example, for autonomous vehicles, the requirements will be refined after the vehicle is developed (e.g., using continuous experimentation). A SAS shall resolve contextually relevant ambiguity itself or with human help at runtime. Runtime testing will then serve to guarantee that the detailed requirements are implemented properly. This resolution requires an implementation of the system without fully hard coding all choices, thereby leaving space for runtime variability.

Two aspects enable a SAS to adapt to emerging situations at runtime within safe and secure bounds. On the one hand, the combination of knowledge about ambiguity in requirements and its context (problem space). On the other hand, foreseen adaptation points in the software itself (solution space).

Guiding Artifact Centric RE Activities for SASs

To guide software engineers during requirements engineering and design of SAS, we propose an iterative and incremental approach comprising five steps. Table 1 summarizes the steps and its associated artifacts. The steps are described in the rest of this section.

Step 1: Capture Requirements This first step starts with a traditional requirements elicitation activity at design time resulting in a set of requirements. Some of these requirements can be

ambiguous.

The assumption here is that such ambiguities will be either resolved by the SAS or with human participation at runtime. Therefore, it is crucial that the requirements elicited in this step be documented or mapped to a machinereadable notation (e.g., goal models). In later steps when more information on the context is available, goal models can be extended to contextual goal models [START_REF] Raian | Modeling and Reasoning about Contextual Requirements: Goal-Based Framework[END_REF] and use cases can be extended to adapt cases [START_REF] Luckey | Adapt Cases: Extending Use Cases for Adaptive Systems[END_REF].

Challenges: The consideration of leaving ambiguity in requirements: Some types of ambiguity might need to be resolved, as they will not lead to any adaptation needs (e.g., logical inconsistencies, stakeholders' interpretations) while other types of ambiguity are necessary for the SAS variability at runtime.

Ideas: Define different categories of ambiguity and their effect on adaptation capabilities of a SAS through empirical investigations.

Step 2: Identify Variability Points The identification of ambiguities is an iterative process comprising three cases:

(1) If a requirement is ambiguous in our accepted sense, a new variation of the requirement is produced for each interpretation and is evaluated again. (2) If a requirement is not ambiguous, then its variabilities are identified, and all considerable behaviors are defined as an alternative to the requirement. (3) If there is no variability, then the requirement is considered as solid with only one expected behavior. That behavior determines a system's state, and it is marked as valid or not. It is important to store the older versions of a requirement as well as invalid behaviors as system's knowledge. The initially defined requirements can be associated with a special type of requirements called architecturally significant requirements (ASRs). ASRs should provide relevant information for designers based on quality attributes, architecting family of related products and specific technologies required, as proposed by Bass et al. [START_REF] Bass | A Comparison of Requirements Specification Methods from a Software Architecture Perspective[END_REF]. The identification of ASRs is important to satisfy all requirements adequately.

Existing work on variability in software engineering focuses on software product lines (SPL) [START_REF] Tawhid | Product Model Derivation by Model Transformation in Software Product Lines[END_REF], in which variability is used to produce new versions of a software product whereas goal-oriented modeling use variability to define behaviors of the system [START_REF] Gonzales-Baixauli | Visual Variability Analysis for Goal Models[END_REF]. A Dynamic software product line (DSPL) target a single system with multiple and dynamic bindings [START_REF] Capilla | An overview of dynamic software product line architectures and techniques: Observations from research and industry[END_REF]. REFAS [START_REF] Muñoz-Fernández | Towards a Requirements Specification Multi-View Framework for Self-Adaptive Systems[END_REF] includes support the sub-specification with concern levels and aggregation relations. The specification is centered on constraints that are applicable at design time and runtime.

Challenges: Techniques and methods are required to provide (1) runtime variability assurance (i.e., reaction to a variability at runtime), (2) unexpected variability and ambiguities management (i.e., not being prepared for certain variability), and (3) alternatives to model the sub-specification of variability and adaptation requirements, including the horizontal and hierarchical relations to be evaluated only at runtime.

Ideas: (1) The development of an evaluation framework with techniques and metrics to assess the variability of the requirements. This framework should consider the restrictions of a runtime environment such as response time, performance, and availability. [START_REF] Bencomo | Requirements Reflection: Requirements as Runtime Entities[END_REF] The development of runtime techniques to validate the variabilities of the requirements according to the states of the running system. (3) The development of runtime management infrastructures to sense, analyze and act upon unforeseen variabilities (i.e., discard or accept) and ambiguities (i.e., generate its variabilities) on the requirements such as the DYNAMICO reference model. [START_REF] Villegas | DYNAM-ICO: A reference model for governing control objectives and context relevance in self-adaptive software systems[END_REF]. [START_REF] Gonzales-Baixauli | Visual Variability Analysis for Goal Models[END_REF] The runtime consideration for the definition of relevant relations that constraint the adaptation (e.g., constraints to adapt between two scenarios).

Step 3: Identify Context which Influences System Behavior After having identified variation points, the requirements engineer has to understand the influence factors for variation points. We propose to represent these factors through context-attributes [START_REF] Knauss | Eliciting Contextual Requirements at Design Time: A Case Study[END_REF], which means that ambiguous parts of requirements can be refined further through context attributes (i.e., relevant information for the system that can be sensed and monitored). The context data can be obtained from the system and information about the external environment.

Challenges: (1) the identification of context attributes that are relevant to the variation points -this is partially done by the system and requires techniques to support this process -and (2) how to reduce invalid or not desired adaptation possibilities from an initial situation.

Ideas: Modeling constraints between variables for particular values and evaluate them at runtime. With the results of this evaluation, the system can reduce the adaptation possibilities of the model.

Step 4: Identify Context Situations and Variabilities During this step, context is analyzed with the purpose of identifying the situations that are relevant to runtime events. For this purpose, the context already identified is traced to each requirement while specifying the meaning for each of them. Changes in the context have different reactions and implications for requirements depending on the interpretation of the situations and requirements.

Existing work on context situations in software engineering include techniques to manage context information, such as context models [START_REF] Villegas | Context Management and Self-Adaptivity For Situation-Aware Smart Software Systems[END_REF][START_REF] Sheng | Contextuml: a uml-based modeling language for model-driven development of context-aware web services[END_REF] and using machine learning techniques to keep context information up-to-date [START_REF] Knauss | ACon: A learning-based approach to deal with uncertainty in contextual requirements at runtime[END_REF].

Challenges: (1) Information on the scenario that might not be sensed by the system is not available. ( 2) Situations (or scenarios) can be subjective. [START_REF] Capilla | An overview of dynamic software product line architectures and techniques: Observations from research and industry[END_REF] The generation of all possible situations poses a risk to the system's performance. [START_REF] Gonzales-Baixauli | Visual Variability Analysis for Goal Models[END_REF] The influence of values from context variables is oversimplified. They are directly linked to some of the requirements.

Ideas: (1) The implementation of runtime models to represent situations, as well as required infrastructures to support the evolution of such representations.

(2) Strategies to guarantee the system's performance while generating diverse situations, including mechanisms and metrics to stop the system before an explosion of scenarios and computing possible scenarios offline or when the system is idle. [START_REF] Capilla | An overview of dynamic software product line architectures and techniques: Observations from research and industry[END_REF] The definition of influence from context variables mediated by scenarios or other runtime context conditions to support more complex situations. Thus, the semi-automatic or automatic derivation of possible situations only from those influence factors. (4) The support of complex expressions to relate the context with the requirements closer to the real system configuration.

Step 5: Design Variation Points in the Solution Space The intention of this step is to provide a software solution that (i) can satisfy the goals according to a given variability definition and (ii) is extendable when new goals emerge. The realization must especially support adaptation at runtime.

Existing work includes external and internal approaches [START_REF] Pohl | Software Product Line Engineering: Foundations, Principles and Techniques[END_REF]. External approaches use explicit feature models linked to artifact elements using (1) hard links/references or (2) an expression language where elements in artifacts are annotated using Boolean expressions over features. Internal approaches present the expressiveness of an artifacts language used to express variability (e.g., dynamic binding in Java, preprocessor in C and conditional "tags" in XML). Other techniques for dealing with variability include tailoring of artifacts (e.g., scripts in ANT or MAKE) and model transformation languages (e.g., ATL or QVT).

Challenges: Knowledge about variability is not fully (re)used at runtime. DSPL approaches support this reuse, but the relation between design-time and runtime variability is not well established.

Ideas: To have different views on the variability data: some for design-time (suited for requirements engineers) and some for runtime (suited for software: adaptation manager components). These views imply a language that requires the integration of knowledge (meta-data) from both -design time and runtime, i.e., make explicit why a variation at a certain variation point is an alternative. Such knowledge can also facilitate automated decision-making at runtime.

Concluding Remarks

SAS are expected to deal with uncertainty from their requirements and operating context, resulting in situations that require adaptations of the system. Some of these situations already occur at requirements level.

In this short paper, we sketched an artifact-centric approach for RE in the domain of SAS comprising five guiding, incremental, and iterative steps. For each step, we described its purpose, estimated associated challenges and potential ideas on how to resolve these challenges. The overall goal of the presented concept is to bridge design-time and run-time RE activities to manage ambiguities better.

Future work has to investigate the presented challenges and corresponding evaluation of results. As for the methodology, our proposal is to use empirical research methods, but combining quantitative and qualitative experiments with selected project cases from industry.

Table 1 .

 1 Guiding Activities and Excerpt of Related Artifact Types and Examples

	Step	Activity	Artifact
	1	Capture requirements	Specification (e.g., early goal models)
	2 (Iter.) Identify variability points	Model (e.g., late goal models)
	3 (Iter.) Identify context	Context model (e.g., ontologies)
	4 (Iter.) Identify situations	Situations model (e.g., reasoning support)
	5 (Iter.) Design variability (solution space) Architecture models (e.g., component models)

Acknowledgments

We thank T. Vogel, M. Tichy, and A. Gorla, organizers GI-Dagstuhl Seminar 14433 in which this paper was conceived. This work was supported by Vinnova grant 2014-06229.


Abdoulwahab Adaine

Sidi Ould

Saad Hamady

Nicolas Fressengeas

Effects of structural defects and polarization charges in InGaN-based double-junction solar cell

Keywords: Simulation, Optimization, Solar cell, InGaN, Heterojunction, Tunnel, Polarization, Defects

The performance of a double heterojunction solar cell based on Indium Gallium Nitride (InGaN) including a tunnel junction was simulated. The most challenging aspects of InGaN solar cells development being the crystal polarization and structural defects detrimental effects, their impact on the solar cell performances has been investigated in detail. The solar cell simulation was performed using physical models and InGaN parameters extracted from experimental measurements. The optimum efficiency of the heterojunction solar cell was obtained using a multivariate optimization method which allows to simultaneously optimize eleven parameters. The optimum defect free efficiency obtained is 24.4% with a short circuit current J SC = 12.92mA/cm 2 , an open circuit voltage V OC = 2.29V and a fill factor FF = 82.55%. The performances evolution as functions of the polarization and the defects types and parameters was studied from their maximum down to as low as a 2% efficiency.

Introduction

Among the materials currently being studied for high efficiency solar cells lies the Indium Galium Nitride (InGaN) alloy. Indeed, InGaN possesses unique properties, such as its wide band-gap that is tunable from 0.7 eV to 3.4 eV [START_REF] Zhang | Design of Multijunction Photovoltaic Cells Optimized for Varied Atmospheric Conditions[END_REF]. It also presents strong interatomic bonds and high thermal conductivity [START_REF] Bhuiyan | InGaN solar cells: present state of the art and important challenges[END_REF], which makes it an ideal material for photovoltaic solar panels resistant to high energy radiation.

Several p-n and p-i-n homojunction and heterojunction structures were studied. The structures with low Indium composition have conversion efficiencies below 5% [START_REF] Matioli | High internal and external quantum efficiency InGaN/GaN solar cells[END_REF][START_REF] Islam | MOVPE growth of InGaN and fabrication of homo-junction solar cells[END_REF]. Indeed, in practice, the InGaN layers with a high concentration of Indium are very difficult to achieve today. For relatively high-Indiumcontent structures, Doolittle et al. [START_REF] Fabien | Guidelines and limitations for the design of high-efficiency InGaN single-junction solar cells[END_REF] theoretically showed that a p-i-n structure having an Indium composition of 60% exhibits * Corresponding author Email addresses: abdoulwahab.adaine@univ-lorraine.fr (Abdoulwahab Adaine), sidi.hamady@univ-lorraine.fr (Sidi Ould Saad Hamady), nicolas@fressengeas.net (Nicolas Fressengeas) a maximum efficiency of about 19%. Our previous studies [START_REF] Saad Hamady | Numerical simulation of InGaN Schottky solar cell[END_REF][START_REF] Adaine | Simulation study of a new InGaN p-layer free Schottky based solar cell[END_REF] have shown that single junction structures have conversion efficiencies not exceeding 20%. The p-n structure presents an optimal efficiency of 17.8% while that of a p-i-n structure is 19.0%. These previous works, using empirical parameters and models, demonstrated that the theoretical maximum efficiency of a single junction is limited to about 20%.

In this paper, we simulate and optimize an InGaN-based dual-junction solar cell including a tunnel junction. This allows the conversion of different parts of the solar spectrum, thus enhancing the solar light absorption efficiency and the photocarriers transport. Photovoltaic panels using such solar cells can potentially achieve efficiencies exceeding those of systems that use conventional single junction solar cells.

In the past, only two papers on InGaN-based double-junction solar cells have been published. A first study published in 2008 [START_REF] Zhang | Theoretical design and performance of ingan two-junction solar cells[END_REF] reports a dual-junction InGaN solar cell with a conversion efficiency of 35.1%. Another similar study published in 2016 [START_REF] Mesrane | Theoretical Study and Simulations of an InGaN Dual-Junction Solar Cell[END_REF] confirms these results with a roughly similar efficiency of 34.9%. In these works the two subcells of the solar cell are studied separately. The tunnel junction between the two subcells is thus not included in the simulations. The effects related to the junction, such as the electrical resistance, the discontinuity of the bands, are thus ignored. Furthemore, the physical parameters of the material used, as well as the optical properties, notably the refractive index, are not reported. Furthermore, the authors did not take into account the structural defects in the InGaN layers and their polarization.

Moreover, the optimum efficiency obtained results from a parametric analysis which obviously does not allow a rigorous optimal efficiency of the solar cell. Indeed, parametric optimization can optimize only one parameter at a time, it thus does not allow to take into account the dependencies of optimized parameters.

We thus propose here a new approach which consists in performing a global simulation of the complete InGaN-based double-junction solar cell, including the tunnel junction, using rigorous physical models and empirical material parameters, and including structural defects and polarization effects.

Indeed InGaN is usually grown on lattice-mismatched Sapphire (Al2O3) or GaN/Sapphire substrates by MOVPE [START_REF] Sánchez | V-defects and dislocations in InGaN/GaN heterostructures[END_REF].

Thus, during the growth, a relaxation occurs beyond a certain thickness called "critical thickness" [START_REF] Holec | Critical thickness calculations for InGaN/GaN[END_REF][START_REF] Reed | Critical layer thickness determination of GaN/InGaN/GaN double heterostructures[END_REF][START_REF] Parker | Determination of the critical layer thickness in the InGaN/GaN heterostructures[END_REF], resulting in a high density of dislocations and other structural defects in the InGaN layers, which impacts the solar cell performances. In addition to the defects, InGaN presents a spontaneous polarization due to the intrinsic asymmetry of the bonds in the wurtzite crystal structure, as well as a piezoelectric polarization mainly due to the lattice mismatch between the layers [START_REF] Li | Effects of polarization charge on the photovoltaic properties of InGaN solar cells[END_REF][START_REF] Zhang | Measurement of polarization charge and conduction-band offset at ingan/gan heterojunc-tion interfaces[END_REF].

The device is simulated in the framework of a drift-diffusion model using the Atlas device simulation framework from the Silvaco company. It is the most commonly used model to accurately describe the fundamental physics of semiconductorbased devices. For the carrier mobility, we use the classic model of Caughey-Thomas in which the carrier mobility is a function of doping and temperature [START_REF] Caughey | Carrier mobilities in silicon empirically related to doping and field[END_REF]. In order to model the losses in the structure, we take into account the Shockley-Read-Hall (SRH) [START_REF] Ryu | Rate equation analysis of efficiency droop in InGaN light-emitting diodes[END_REF] and direct Auger recombination models using Fermi statistics. An empirical model of the absorption of light using and interpolating experimental data was used [START_REF] Brown | Finite element simulations of compositionally graded InGaN solar cells[END_REF] and combined to a precise model for the refractive index initially developped for GaN [START_REF] Djurišić | Modeling the optical constants of hexagonal GaN, InN, and AlN[END_REF]. When the direct experimental data are not available, the InGaN parameters values are determined using Vegard's law linear interpolation between InN and GaN binaries values, except for the bandgap (E g ) and the electronic affinity (χ), where we used a quadratic modified Vegard's law with a bowing parameter of b = 1.43 eV for the bandgap and b = 0.8 eV for the electronic affinity [START_REF] Brown | Finite element simulations of compositionally graded InGaN solar cells[END_REF]. All these physical models and dependent material parameters are detailed in our previous work [START_REF] Saad Hamady | Numerical simulation of InGaN Schottky solar cell[END_REF][START_REF] Adaine | Simulation study of a new InGaN p-layer free Schottky based solar cell[END_REF] and summarized in table 1.

The optimization of the device is done with rigorous mathematical methods. The objective is to find the optimal efficiency of the solar cell by simultaneously optimizing a given set of physical and geometrical parameters of the cell. We have, for this solar cell, optimized simultaneously eleven parameters.

With that many parameters to optimize, a careful fine tuning of the previously used optimization algorithms was necessary in order to ensure their convergence while keeping a reasonable computing time, which can be as high as 30 hours per simulation.

This optimization work was realized with a Python package that we developed using the SciPy optimizers [START_REF] Jones | SciPy: Open source scientific tools for Python[END_REF]. This packaged source code will be released to the photovoltaic community soon.

The paper is organized as follows. In the next section 2, we describe the solar cell structure as well as the InGaN structural properties that can limit the performance of the solar cell, including structural defects and spontaneous and piezoelectric polarizations. In section 3, we present the optimal results obtained and discuss the effects of the polarization and structural defects on the performance of the solar cell, before concluding. The structure is studied under an AM1.5 illumination (about 100mW /cm 2 ). The high-energy photons are absorbed by the top subcell (subcell-1) while those of lower energies are transmitted through the junction to be absorbed by the bottom subcell (subcell-2). The solar cell structure therefore absorbs a larger solar spectrum and can thus produce more energy than a single junction cell, provided the current can be matched between the two subcells.

Indeed, the two subcells being connected in series, the subcell that produces the smallest current limits the maximum current that can flow through the solar cell. Therefore, the optimum performance of the cell strongly depends on the subcells electrical interconnection and their respective optimal currents.

To achieve that fine tuning, the two subcells are connected by a tunnel junction, consisting of yet another p-n junction built in the opposite direction to that of the two subcells wich are designed for photocurrent generation. We designed here a p-n junction whose Indium composition can be different from that of the subcells and that includes highly doped p++ and n++ layers. Therefore, This junction space-charge region is spatially narrow and its Fermi level is located both in the conduction and valence band: the bottom of the conduction band is on the same level as the top of the valence band.This way, the charge carriers can cross the potential barrier by tunneling.

Finally, the device proposed here can be realized in practice using the conventional InGaN growth on insulating substrates such as Sapphire (Al 2 O 3 ) or SiC via a GaN or similar heavily doped buffer layer.

Spontaneous and piezoelectric polarizations

The InGaN crystal structure belongs to the wurtzite group [START_REF] Bernardini | Spontaneous polarization and piezoelectric constants of III-V nitrides[END_REF]. It is a non-centrosymmetric structure having two types of polarities according to the arrangement of atoms of Gallium and Nitrogen during the epitaxial growth. They are called Ga-face polarity if the last epitaxially grown atomic layer is composed of Gallium atoms and N-face polarity if the last epitaxially grown atomic layer is composed of Nitrogen atoms [START_REF] Liu | Substrates for gallium nitride epitaxy[END_REF]. This polarity will generate a spontaneous polarization (PSP) in the structure, spatially separating electrons and holes and thus creating an accumulation of charges at the InGaN surface.

InN GaN e31(C/m 2 ) -0.41 -0.34 e33(C/m 2 ) 0.81 0.67 c13(GPa) 94 100 c33(GPa) 200 392 P sp (C/m 2 ) -0.042 -0.034
Table 2: Experimental data of piezoelectric parameters and spontaneous polarization values for InN and GaN [START_REF] Piprek | Semiconductor Optoelectronic Devices: Introduction to Physics and Simulation[END_REF].

In addition to the spontaneous polarization, the lattice mismatched between different composition layers induces strain and thus piezoelectric polarization. This polarization also results in a displacement of electrons and holes at the surface of the layers and therefore may either reduce or enhance the total polarization [START_REF] Liu | Substrates for gallium nitride epitaxy[END_REF]. Thus, without external polarization, the total polarization of the structure is equal to the sum of the spontaneous and piezoelectric polarizations.

Charge accumulation at the extremities of the structure can then change the electrical properties of the solar cell and the resulting photovoltaic performances. Indeed, if the electric field associated with the total polarization is in the same direction as the field generated in the space-charge region of the PN junction, the polarization could improve the performance of the cell, or reduce it otherwise.

The spontaneous polarization (P sp ) is quadratically interpolated from the modified Vegard's law with a bowing factor of b = 0, 038C/m 2 [START_REF] Bernardini | Nonlinear behavior of spontaneous and piezoelectric polarization in III-V nitride alloys[END_REF][START_REF] Fiorentini | Evidence for nonlinear macroscopic polarization in III-V nitride alloy heterostructures[END_REF]:

P InGaN sp = xP InN sp + (1 -x)P GaN sp -bx(1 -x), ( 1
)
where x is the Indium composition of In x Ga 1-x N, P InGaN sp is the spontaneous polarization of In x Ga 1-x N while P InN sp and and P GaN sp are the spontaneous polarizations of InN and GaN binaries. Their values are given in Table 2.

The piezoelectric polarization (P pi ) is linearly interpolated by Vegard's law [START_REF] Bernardini | Nonlinear behavior of spontaneous and piezoelectric polarization in III-V nitride alloys[END_REF]:

P InGaN pi = xP InN pi + (1 -x)P GaN pi , (2)
where P InGaN pi is the piezoelectric polarization of In x Ga Critical Layer Thickness (nm)

Figure 2: The critical layer thickness for In x Ga (1-x) N/GaN layer as a function of the Indium composition using a Fischer model [START_REF] Fischer | New approach in equilibrium theory for strained layer relaxation[END_REF]. Blue circles represent experimental measurements from [START_REF] Parker | Determination of the critical layer thickness in the InGaN/GaN heterostructures[END_REF] and the red diamonds the experimental measurements from [START_REF] Reed | Critical layer thickness determination of GaN/InGaN/GaN double heterostructures[END_REF].

P InN pi and P GaN pi are the piezoelectric polarizations of InN and GaN binaries. The latter are evaluated from piezoelectric constants and lattice parameters [START_REF] Bernardini | Spontaneous polarization and piezoelectric constants of III-V nitrides[END_REF]:

P pi = 2 × a s -a 0 a 0 × (e31 - c13 c33 × e33), (3)
where e31 and e33 are piezoelectric constants; c13 and c33 are elastic constants; a s and a 0 are the lattice parameters of the substrate and of the material layer respectively. Their values are summarized in Table 2.

Critical thickness

In our modeling conditions, the spontaneous polarization is applied throughout the structure, while the piezoelectric polarization is applied only to the strained layers. In order to determine if a layer is strained or relaxed, we refer to the experimental measurements of critical thicknesses as a function of Indium composition. Since no measurements has been done for high Indium content layers, we have fitted the experimental data reported in [START_REF] Reed | Critical layer thickness determination of GaN/InGaN/GaN double heterostructures[END_REF] and [START_REF] Parker | Determination of the critical layer thickness in the InGaN/GaN heterostructures[END_REF] by a Fischer model [START_REF] Fischer | New approach in equilibrium theory for strained layer relaxation[END_REF] to extrapolate the critical thicknesses for high Indium content layers (Figure 2). For the extrapolation based on the measurements reported in [START_REF] Parker | Determination of the critical layer thickness in the InGaN/GaN heterostructures[END_REF], the critical thickness corresponding to an Indium composition of x = 0.6 would be about 30nm, while it would be closer to 10nm for that based on [START_REF] Reed | Critical layer thickness determination of GaN/InGaN/GaN double heterostructures[END_REF]. Therefore, for our simulations, we considered that all layers with a thickness less than 10nm are strained and thus that layers having a thickness greater than 10nm are relaxed. This layer relaxation induces a high density of dislocations and other structural defects in the epitaxial layer.

Structural defects

Among the defects in III-Nitride alloys, and particularly evidenced in GaN, one can cite point antisites and interstitials defects, threading dislocations [START_REF] Kapolnek | Structural evolution in epitaxial metalorganic chemical vapor deposition grown GaN films on sapphire[END_REF], stacking faults [START_REF] Gloux | A comparative structural investigation of GaN implanted with rare earth ions at room temperature and 500 c[END_REF], stacking mismatch boundaries [START_REF] Sverdlov | Formation of threading defects in GaN wurtzite films grown on nonisomorphic substrates[END_REF], grain boundaries [START_REF] Xu | Effect of in situ thermal treatment during growth on crystal quality of GaN epilayer grown on sapphire substrate by MOVPE[END_REF], nanopipes and inversion domains [START_REF] Ponce | Nanopipes and Inversion Domains in High-Quality GaN Epitaxial Layers[END_REF]. All these defects cause perturbations in the periodicity of the crystal and affect the electronic properties of the devices. The defect whose associated energy lies in the bandgap, exchange with the conduction and valence bands through the emission and capture of electrons or/and holes. Garni et al. [START_REF] Garni | Scanning tunneling microscopy and tunneling luminescence of the surface of GaN films grown by vapor phase epitaxy[END_REF], showed that threading dislocations in GaN act as nonradioactive centers and diffusion centers in electron transport [START_REF] Ng | In-situ reflectance monitoring during MOCVD of AlGaN[END_REF]. Moreover, Garni et al. also showed that these dislocations defects cause nonradioactive recombination which causes crystal heating up and thus significantly influences the performances of the device.

According to calculations by Neugebauer and Van de Walle [START_REF] Neugebauer | Atomic geometry and electronic structure of native defects in GaN[END_REF], the formation energy of GaN point defects depends on the position of the Fermi level. The type of the defects usually observed in GaN thus depends on the doping type of the layer. This study shows that, for p-type doping, the Nitrogen vacancy have the lowest energy and are therefore most likely to be present in this layer while for n-type doping, Gallium vacancy are dominating. The Nitrogen vacancies being n-type and the Gallium vacancies being p-type, one can conclude that on p-type layers, the donor-type defects will be dominant, while, in n-type layers, the acceptor-type defects will.

Stampf et al. [START_REF] Stampfl | Native defects and impurities in InN: First-principles studies using the localdensity approximation and self-interaction and relaxation-corrected pseudopotentials[END_REF] also showed that, for InN, the Nitrogen vacancies have a lower energy than other defects in the p-type layers and thus are more likely to be present. It is then assumed that InGaN has the same types of defects as GaN and InN. We therefore consider in our simulations the donor-type defects in p-type layers and acceptor-type defects in n-type layers.

Furthermore and to take into account the disorder due to dislocations, the defect density of states is defined as a continuous function of the energy. It is a combination of exponentially decaying band tail states and Gaussian distributions states within the bandgap. Band tail states represent the shallow states in the bandgap and Gaussian distributions represent the deep ones.

They both impact the effective charge mobility through the capture and recombination of opposite charges. The total density of states is thus composed of four bands: two band tails and two bands of deep levels.

Influence of defect and polarization on efficiency

Solar cell optimization

The simulated and optimized cell structure is shown Finally, for the optimizing algorithm to work as expected, the sought parameters should have similar variation ranges. Therefore, the Indium composition and the layer thicknesses are normalized to 1, their maximum value. For doping, the normalization is logarithmic (log 10 ) and normalized to 19, its maximum value. The mathematical optimization methods used for optimization are detailed in our recent published work [START_REF] Saad Hamady | Numerical simulation of InGaN Schottky solar cell[END_REF][START_REF] Adaine | Simulation study of a new InGaN p-layer free Schottky based solar cell[END_REF].

Optimization is run several times with several starting points randomly chosen to ensure that the optimum obtained is absolute.

Using this procedure, a global maximum cell efficiency of 24.4% is found. Associated to this maximum efficiency, the dual junction solar cell main electrical characteristics at the optimum are the short circuit current J SC = 12.92mA/cm 2 , the open circuit voltage V OC = 2.287V and the fill factor FF = 82.55%. The optimal values obtained for the free parameters are given in Table 3. Figure 3 shows the current-voltage and power-voltage characteristics of the this optimal solar cell. By comparison, previous simulation works for InGaN-based double-junction solar cells obtained a higher efficiency [START_REF] Zhang | Theoretical design and performance of ingan two-junction solar cells[END_REF][START_REF] Mesrane | Theoretical Study and Simulations of an InGaN Dual-Junction Solar Cell[END_REF] of 35.1%. This is due to the fact that this work did not take into account the tunnel junction as itself. The two subcells were simulated separately and the effects of the series resistance and tunnel junction ignored.

In addition, our work is based on an empirical conduction and absorption model directly deduced from experimental data, associated to a rigorous optimization procedure. Therefore, we expect the obtained optimal efficiency to be on one side actually optimal and, on the other side, not overestimated.

Spontaneous and Piezoelectric polarization

Once the optimal solar cell found, the effects of the polarization charges on its performances have been analyzed.

As pointed out above, the spontaneous polarization is applied throughout the structure while the piezoelectric polarization is applied only to the constrained layers. subcell-2 has an optimum thickness of 0.56µm, and is thus relaxed, as deduced from figure 2 and previously discussed in section 2.3. The tunnel junction, at the bottom, has a thickness of 10nm, so it is a constrained layer. The subcell-1 has a lattice parameter very close to that of the tunnel junction, it is also constrained. In short, the piezoelectric polarization is considered for the tunnel junction and subcell-1 only. 1) and (2). To evidence the impact of a strong polarization, it is gradually increased up to 3 times the total polarization. subcell-1 is on the left of the figure, while subcell-2 is on its right, the tunnel junction being sandwiched between the two. 2). For a small polarization, a normalized value of 1.0, the solar cell performances barely change, the optimal efficiency dropping from 24.4% to 23.3%. However, should the polarization charges be higher, a sharp decrease in the open circuit voltage V OC would be seen while the short circuit current J SC would remains constant. In addition, for high polarizations, a change in the shape of the curve would be observed, considerably degrading the fill factor.

To explain these behaviors, we have analyzed the conduction band in the tunnel junction with various polarization charges, as is shown in figure 4 Moreover, the presence of this induced field reveals a reverse biased diode, which opposes the efficient collection of the carriers. A threshold effect appears: when the induced field is less than the electric field of the cell, a deformation of the I-V curve is observed which severely degrades the fill factor and thus decreases the solar cell efficiency though no change on the short circuit current is J SC is observed; on the contrary, if the induced field is largely greater than that of the cell, the deformation of the I-V curve also results a decrease of the J SC . Indeed, the induced field opposes efficient collection of the carriers. This leads to a decrease of the current in the subcell-2. This latter then imposes its low current to the whole structure.

Deep and band tail defects

As for polarization, structural defects play a central role in an InGaN solar cell operation. The InGaN layer composing subcell-2 being relaxed, the dislocations generated are similar to line charges and are an important source of defects, considerably reducing the mobility of the charge carriers and the efficiency of the solar cell.

In figure 5, the impact of the Urbach band tail energy on the cell performances was studied. In figure 6, it is the impact of the deep defects Gaussian distribution that was studied. Figures 5(a Indeed, an increase of the Urbach energy implies a reduction in the optical gap owing to the presence of states in the InGaN bandgap, leading to a higher probability of carrier recombination. In the same way, for the deep states, the increase of the decay energy also leads to an increase of the state density in the bandgap and therefore to a higher probability of recombination of the carriers. Both these effects thus result in the decrease of the short circuit current J SC .

However, it can be noted the an increase in the disorderin the Urbach energy-degrades the solar cell performances much more than a deep states decay energy increase. Indeed, for an Urbach energy of 60meV, the short circuit current J SC drops from more than 10 to 6.65mA/cm 2 , while the variation of the decay energy, between 20meV and 160meV only induces a slight decrease in the same short circuit current J SC from 12.8 down to 10.5mA/cm 2 . In InGaN solar cells, material disorder is therefore more detrimental to the performances than deep defects states. Figure 6: Impact of the Gaussian distribution decay energy on the performances of the solar cell. The efficiency is slightly degraded from 22.90% down to 19.45% for a defect decay energy varying from 0.02eV up to 0.20eV, the fill factor remaining unchanged around 81%. The capture cross-sections are taken equal to 1 × 10 -14 cm 2 . The energies that correspond to the Gaussian distribution peaks are fixed at 0.6eV, the Urbach energy of the band tail is set at 0.05eV, the defect density of donor-type is equal to 1 × 10 18 cm 2 . The defect density of acceptor-type is equal to 1 × 10 17 cm 2 . Figure 7: Impact of the defect density of donor-type states in a Gaussian distribution on the performance of the solar cell. The defect density of acceptor-type states is constant and equal to 10 17 cm -3 . The capture cross-sections are taken equal to 10 -14 cm 2 . The energies that correspond to the Gaussian distribution peaks are set to 0.6eV.

Finally, in Figure 7, the effect of the density of donor-type defects was studied. Figure 7(a) shows its impact on the efficiency whereas figure 7(b) shows its impact on the short circuit current J SC and the open circuit voltage V OC . For donor-like defects, that are mostly found in the p-type layer, it can be observed that, when the defects density is less than the doping density in the layer, the efficiency remains close to its optimal value. On the contrary, as soon as the defects density exceeds the doping density, the efficiency of the cell dramatically decreases as shown in Figure 7(a). Indeed, by their ability to emit or capture free carriers, defects can compensate for intentionally introduced donor or acceptor levels. This reduces the mobility of the photogenerated carriers, thus reducing the short circuit current J SC (Figure 7(b)) and the solar cell overall efficiency.

Conclusion

In this work, a detailed investigation on the performance of an InGaN-based double-junction solar cell was carried out. We have globally simulated the solar cell using empirical InGaN material parameters, to avoid any overestimation in the solar cell performances. In order to take into account the interdependence of the solar cell physical and geometrical parameters, ensuring the absoluteness of the optimized photovoltaic performances, the cell was optimized using a multivariate optimization algorithm that simultaneously optimizes eleven physical and geometrical parameters. We obtained an optimal efficiency of 24.4%, with a short circuit current J SC = 12.92mA/cm 2 , an open circuit voltage V OC = 2.287V and a fill factor FF = 82.55%.

We quantitatively investigated the impact on the solar cell performances of the internal polarization and structural defects in InGaN. We have shown that the internal polarization reduces the performance of the cell by inducing an electric field which does not favor an efficient collection of photo-generated carriers; showing the importance of controlling this polarization when designing the lattice-mismatched solar cell heterostructure. We have also investigated the impact of structural defects in InGaN, including disorder and deep defects, and correlated their effect to the InGaN doping concentration.

2 .

 2 Solar cell, polarization and defect modeling 2.1. Solar cell structure The modeled solar cell is based on an InGaN-made double junction structure shown on Figure 1. This structure is composed of two vertically stacked p-n junctions with a top wide bandgap cell, named subcell-1 in the following, and a bottom narrower bandgap cell, named subcell-2, both being connected by a tunnel junction. The difference in the bandgaps is obtained by changing the Indium composition x of InGaN.

Figure 1 :

 1 Figure 1: Solar cell structure with two p-n junctions (subcell-1 and subcell-2) connected by a tunnel junction. The cell is realized on a substrate via a heavily doped GaN buffer layer. x 1 represents the composition of the subcell-1, x 2 the Indium composition of subcell-2 and x 3 the induium composition of the tunnel junction. The structure is under AM1.5 illumination.

schematically in Figure 1 .

 1 It includes three different Indium compositions: subcell-1 has an x 1 Indium composition and subcell-2 has an x 2 one while the junction tunnel can exhibit a yet different x 3 Indium composition. The 11 parameters for which we sought the optimal values are the the 2 layer thicknesses and dopings for each PN junction along with their compositions. These 10 parameters are included globally in the optimization process along with the tunnel junction Indium composition. The thicknesses and dopings of the tunnel junction have not been included in the optimization process because our simulations have shown that they need to be as highly doped and thin as the technology allows it. All the 11 optimization parameters have been constrained within a variation domain, which is the interval in which they are meaningful in physical and technological terms. The Indium compositions are allowed within the full range: x i ∈ [0, 1], ∀i ∈ {1, 2, 3}. The InGaN layer n-type and p-type dopings (N d and N a ) minimum allowed values are chosen to be the minimal residual doping of 10 16 cm -3 , as usually measured in this material. Their maximum values are, on their side, set to 10 19 cm -3 , which is the highest doing value used. To put it shortly: N d ∈ [10 16 , 10 19 ]cm -3 and N a ∈ [10 16 , 10 19 ]cm -3 . For Subcell 1

2 )Figure 3 :

 23 Figure 3: Current-voltage and power-voltage charateristics of the solar cell at the optimum.

  Current-voltage characteristics with different polarization charges. Conduction band diagrams in the tunnel junction with different polarization charges.

Figure 4 :

 4 Figure 4: Impact of total polarization on the performances of the solar cell. The normalized value of 1.0 corresponds to the value in C/m 2 , obtained by the sum of the spontaneous and piezoelectric polarization given respectively by equations (1) and (2). To evidence the impact of a strong polarization, it is gradually increased up to 3 times the total polarization. subcell-1 is on the left of the figure, while subcell-2 is on its right, the tunnel junction being sandwiched between the two.

Figure 4 (

 4 Figure 4(a) shows the I-V curves for solar cells with different polarization charge. The normalized value of 1.0 corresponds to the value in C/m 2 , obtained by the sum of the spontaneous and piezoelectric polarization given respectively by equations (1) and[START_REF] Bhuiyan | InGaN solar cells: present state of the art and important challenges[END_REF]. For a small polarization, a normalized value of

  (b). The polarization charges bend the conduction band at the interface between the tunnel junction and subcell-2; the bend curvature increasing with the polarization. This causes charges to accumulate at this interface, which in turn create an electric field. This field, induced by the polarization, is in the direction opposite to that of the PN junction internal field. It thus tends to lower the total integrated electric field in the cell, which explains the decrease in the open circuit voltage V OC .

  ) and 6(a) show the different current-voltage characteristics of the solar cell, while figures 5(b) and 6(b) show the variations of J SC and V OC . Band tails reflect the state of disorder of the material. The characteristic energies of the band tails and deep defects usually increase with the Indium composition. This is the reason why their influence on the solar cell performances was studied. The increase of the Urbach energy as well as that of the decay energy (Figures 5(a) and 6(a)) leads to a decrease of the J SC while the V OC remains almost constant (Figures 5(b) and 6(b)).

  Current-voltage characteristics with various Urbach energy of the band tail. Variation of J SC and V OC as a function of the Urbach energy of the band tail.

Figure 5 :

 5 Figure5: Impact of the band tail Urbach energy on the performances of the solar cell. The efficiency drops from 20.54% down to 4.89% for an Urbach energy varying from 0.02eV up to 0.16eV. The fill factor remains unchanged around 82%. The capture cross-sections are taken equal to 1 × 10 -14 cm 2 , the band tail state densities are taken equal to 2.3 × 10 18 cm 2 for the valence band and 4.6 × 10 19 cm 2 for the conduction band.

  Solar cell efficiency as a function of the defect density of donor-type states in a Gaussian distribution.

  Variation of J SC and V OC as a function of the defect density of donor-type states in a Gaussian distribution.

Table 1 :

 1 Parameters of InN an GaN and significant physical model used in the simulations of the solar cell structure

	Parameters [18]	InN	GaN
	Bandgap E g (eV )	0.7	3.42
	Electron affinity χ (eV )	5.6	4.1
	Hole and electron lifetime (ns)	1	1
	Dielectric constant	15.3	8.9
	N c (cm -3 ) N v (cm -3 )	9.1 × 10 17 5.3 × 10 19	2.3 × 10 18 4.6 × 10 17
	Physical model
	Charge carrier mobility	Caughey-Thomas Model [21]
	Absorption Coefficient	Phenomenological Model [18]
	Refraction Index	Adachi Model [19]

Table 3 :

 3 Optimized parameters of the heterojunction InGaN solar cell. The parameters marked with an asterisk are the eleven simultaneously optimized parameters, the other parameters being fixed. x 1 , x 2 and x 3 are the Indium compositions for subcell-1, subcell-2 and tunnel junction respectively (see figure1). L p and L n are the InGaN layer thicknesses of the P and N layers respectively. N a and N p are the dopings of the P and N layers respectively. The optimum efficiency found is 24.4%, associated to a J SC = 12.92mA/cm 2 short circuit current, a V OC = 2.287V open circuit voltage and a FF = 82.55% fill factor.

the InGaN layer thicknesses (L p and L n ), we chose to set the minimum values to 10nm and the maximum values to 1µm. Therefore: L p ∈ [0.01, 1.00]µm and L n ∈ [0.01, 1.00]µm.