Carole Adam
email: carole.adam.rmit@gmail.com

Andreas Herzig

Dominique Longin

Vincent Louis

Unifying the intentional and institutional semantics of speech acts

Research about the semantics of agent communication languages traditionally sees the opposition between the mentalist and social approaches. In this paper we adopt a mixed approach since we propose a logical framework allowing us to express both the intentional and institutional dimensions of a communicative action. We use this framework to give a semantics for some speech acts representing each of Searle's categories except expressives. This semantics relaxes the criticized constraints imposed in FIPA-ACL and also extends this standard with new speech acts and new institutional features to characterise them. It has been implemented in an extension of the Semantic Add-on for the JADE agent development platform, and used in an industrial application in the context of automated B2B exchanges.

Introduction

Designing efficient Agent Communication Languages is an essential issue in Multi-Agent Systems in order to standardise exchanges between the agents. Research about the semantics of agent communication languages sees the subscribers of the social approach [START_REF] Fornara | A commitment-based approach to agent communication[END_REF][START_REF] Chaib-Draa | Diagal : An agent communication language based on dialogues games and sustained by social commitments[END_REF][START_REF] Singh | A social semantics for agent communication languages[END_REF] criticize mentalist approaches [START_REF] Sadek | A study in the logic of intention[END_REF][START_REF] Finin | KQML as an agent communication language[END_REF] for only grounding on the agents' private mental attitudes. But one can similarly reproach to social approaches to provide a semantics only based on the agents' public commitments, independently of their mental attitudes. Now these "social attitudes" are mainly descriptive, while mental attitudes allow one to predict the agents' behaviour. Moreover mental attitudes allow agents to reason about social notions. It is thus essential to consider both mental and social attitudes. Some researchers thus propose a mixed approach based both on public and private aspects [START_REF] Gaudou | A new semantics for the fipa agent communication language based on social attitudes[END_REF]. But they do not formalise institutional speech acts like declarations. Now such speech acts are essential in new application fields involving communication about norms, roles or powers of agents, for instance in electronic commerce or automated business to business exchanges.

In this paper we thus want to propose an alternative to the well-known standard of agent communication language FIPA-ACL [START_REF]The foundation for intelligent physical agents[END_REF] through the following changes: relaxed feasibility preconditions to allow a more flexible utilisation of the speech acts in various contexts; new institutional speech acts like declarations and promises; and an institutional interpretation of speech acts coupled with their classical intentional interpretation. Therefore we adapt an existing logical framework for the formalisation of institutional notions like roles, powers and norms [START_REF] Demolombe | Norms, institutional power and roles: towards a logical framework[END_REF]. We then formalise in this logical framework the institutional interpretation of some specific communicative actions, each one representing one of Searle's categories of speech acts (except expressive ones) [START_REF] Searle | Speech acts : An essay in the philosophy of language[END_REF]. Our notion of institution is very large (it is a set of rules and facts adopted by a group of agents, like the rules of a game, or the laws of a country) and covers formal, legal institutions as well as informal ones (social rules in a group...).

The paper is structured as follow. Section 2 discusses some other social semantics of speech acts. Section 3 briefly describes the syntax, semantics and axiomatics of our logical framework. The core of the paper (Section 4) is dedicated to the unified semantics of speech acts. We are then able to compare our semantics of ACL with some related ones in more details (Section 5). Finally we conclude about the future prospects opened by this work (Section 6).

State of the art

The mentalist approach consists in grounding the semantics of speech acts on the agents' internal mental attitudes. These are represented by belief, desire and intention modalities provided by BDI logics, that are classically used to formalise the reasoning of autonomous agents [START_REF] Rao | Modeling rational agents within a BDI-architecture[END_REF][START_REF] Wooldridge | Reasoning about rational agents[END_REF]. This resulted in the design of several standards of agent communication languages like KQML [START_REF] Finin | KQML as an agent communication language[END_REF] or FIPA [START_REF]The foundation for intelligent physical agents[END_REF], this latter one grounding on Sadek's rational interaction theory [START_REF] Sadek | A study in the logic of intention[END_REF].

These approaches were criticised a lot for being only based on private concepts (mental attitudes) instead of public verifiable notions (like commitments). Therefore some work exist aiming at enriching BDI logics with deontic operators like obligation [START_REF] Dignum | Towards socially sophisticated bdi agents[END_REF][START_REF] Broersen | Beliefs, obligations, intentions, and desires as components in an agent architecture[END_REF] or with institutional operators like count as or institutional power [START_REF] Jones | A formal characterisation of institutionalised power[END_REF], in order to formalise the institutional interpretation of speech acts exchanged by the agents. In previous work we used such an extended BDI framework to express the semantics of speech acts with institutional effects [START_REF] Demolombe | Speech acts with institutional effects in agent societies[END_REF] but we were limited to declarative speech acts, and the intentional and institutional dimensions were quite blended.

Various other work aims at providing an institutional semantics for speech acts. For example Dignum and Weigand [START_REF] Dignum | Communication and deontic logic[END_REF] propose a logical framework combining illocutionary and deontic logic to study and model the norms resulting from communication between agents; however, they only consider directive speech acts. Boella et al. [START_REF] Boella | Role-based semantics for agent communication: embedding of the 'mental attitudes' and 'social commitments' semantics[END_REF] propose a role-based semantics allowing them to combine social commitments and mental attitudes to express the semantics of speech acts in the context of persuasion dialogues. Actually they rewrite the FIPA feasibility precondition and rational effects of speech acts but replace the private mental attitudes involved by public mental attitudes attributed to the agents' roles instead of the individual agents. This solves the flaw of mentalist approaches, criticised for grounding on unverifiable mental attitudes, but finally there is no distinct institutional interpretation of speech acts, that could differ from one institution to another. In the following subsections we give some details about two approaches: Fornara and Colombetti's approach based on the notion of commitments, and Lorini et al. 's approach based on the notion of acceptance.

Fornara and Colombetti: semantics in terms of social commitments

As opposed to the mentalist approach, the social one [START_REF] Singh | An ontology for commitments in multiagent systems: Towards a unification of normative concepts[END_REF][START_REF] Singh | A social semantics for agent communication languages[END_REF][START_REF] Chaib-Draa | Diagal : An agent communication language based on dialogues games and sustained by social commitments[END_REF] assumes that private mental attitudes are not verifiable and thus grounds on the concept of public (thus verifiable) commitments [START_REF] Castelfranchi | Commitments : From individual intentions to groups and organizations[END_REF] to express the semantics of speech acts.

All the commitments taken by the agents are stored for possible future reference. The semantics of speech acts is expressed only in terms of such commitments. For example Fornara and Colombetti [START_REF] Fornara | A commitment-based approach to agent communication[END_REF] ground on Castelfranchi's notion of commitment [START_REF] Castelfranchi | Commitments : From individual intentions to groups and organizations[END_REF] to define a library of communicative acts. From the classification of speech acts into four categories (assertives, directives, commissives and declaratives) inspired from Searle's work [START_REF] Searle | Speech acts : An essay in the philosophy of language[END_REF], they redefine for each category the semantics of its speech acts in terms of social commitments. Thanks to this library, they provide a communication tool based on social commitments, alternative to the FIPA-ACL standard. This tools allows rational agents to reason about the underlying rules of communication and to respect them in order for the system to behave well.

However they are limited to the institutional dimension of speech acts and neglect their relations with the agents' mental attitudes. Yet agents must be able to reason autonomously about the institution before making their decision to perform a given speech act. Moreover no specific institution is explicit in their commitments, making it impossible to have various commitments in various institutions; therefore it is also impossible for speech acts to have different effects depending on the institution within which they are interpreted. For example the action of nodding one's head is interpreted in the context of French gestural language as meaning "yes", while in the context of Bulgarian gestural language it is interpreted as meaning "no".

Lorini et al. : semantics in terms of group acceptance

Lorini et al. [START_REF] Lorini | The institutional dimension of speech acts: a logical approach based on the concept of acceptance[END_REF] define a new semantics for speech acts using Gaudou et al. 's Acceptance Logic [START_REF] Gaudou | Anchoring institutions in agents' attitudes: towards a logical framework for autonomous multi-agent systems[END_REF]. A L is a modal logic extended with the notion of acceptance, representing what a group of agents willingly accept to consider as true (even if some (or all) members of the group believe the opposite) in a given institutional context (and that they can refuse in another context). Acceptances influence the agents' behaviour and utterances in this institutional context. They are represented with the operator [C : x ]φ reading "agents in group C accept that φ while functioning as members of this group in the institutional context x".

Institutional notions are not primitive but defined from this notion of acceptance. Thereby institutional facts (facts that are only valid in an institutional context, but not objectively valid) are considered to be equivalent to a group acceptance in all groups of agents while they function in the considered institutional context. This strong link assumed between acceptance and institutional facts may be a particular rule of the specific "ordinary communication" institution but cannot be generalised to all institutions, particularly legal ones.

The authors then consider the speech act Promise in the institutional context of Ordinary Communication (OC). According to them, if i informs j that he is going to perform action α for him, and j intends i to perform this action for him, this counts as a promise at the next instant. The consequence of this promise is that i is obliged to perform action α for j. Moreover the acceptance by these two agents i and j while functioning as a group in institution OC that i has promised to perform action α for j and that j intends him to do so implies a social commitment of i towards j to perform α for him. This framework is interesting but Lorini et al. have only formalised the promise yet. Moreover they do not seem to make a clear distinction between the intentional and institutional preconditions to perform a speech act.

Our logical framework

We adapt here an existing logical framework for norms, institutional powers and roles defined in [START_REF] Demolombe | Norms, institutional power and roles: towards a logical framework[END_REF]. It is a multi-modal logic with modal operators of belief, intention, obligation, institutional facts and consequences, and action.

Syntax

Let AGT = {i, j, ...} be a finite set of agents. Let ACT = {α, β...} be the set of actions. We suppose that some actions in ACT are of the form i:α, where i is the author of action α (the agent who performs it). Let AT M = {p, q, ...} be the set of atomic formulas. Let IN ST = {s, t, ...} be the set of institutions. Complex formulas are denoted by φ, ψ... The language of our logic is defined by the following BNF grammar: φ ::= p|¬φ|φ ∨ φ|B i φ|Ch i φ|I i φ|D s φ|φ ⇒ s φ|Oφ|before α φ|after α φ where p ranges over AT M , α over ACT , i over AGT , and s over IN ST . The classical boolean connectives ∧, →, ↔, ⊤ (tautology) and ⊥ (contradiction) are defined from ∨ and ¬ in the usual manner. The operators done α φ, happens α φ, P φ, F φ and power (i , s, φ, α, ψ) will be defined as abbreviations.

Semantics and axiomatics

We only give here the informal meanings of our operators. It is sufficient to know that they have a Kripke semantics in terms of possible worlds. We also give some useful axioms. This framework is adapted from Demolombe and Louis' logic of norms, roles and institutional powers [START_REF] Demolombe | Norms, institutional power and roles: towards a logical framework[END_REF]. But please notice that actually, the details of the semantics of operators is not important, and any other institutional logic would work.

Belief, intention and action B i p means that agent i believes that p. Ch i p means that agent i prefers p to be true. These two normal operators have a standard KD45 axiomatics. I i p means that agent i intends that p. Its axiomatics is that defined for FIPA by Sadek [START_REF] Sadek | A study in the logic of intention[END_REF]. In particular intention is linked with belief by the following mix axioms:

introspection: I i p ↔ B i I i p automatic dropping of achieved intentions: I i p → ¬B i p before α and after α are normal modal operators defined in standard tense logic in linear time version [START_REF] Burgess | Handbook of philosophical logic, volume 7, chapter Basic tense logic[END_REF]. done α φ = ¬before α ¬φ means that action α has just been performed, and φ was true before. happens α φ = ¬after α ¬φ means that action α is about to be performed and φ will be true just after.

Institutional modalities Finally this framework also provides some specific operators to formalise institutional concepts. These operators have a parameter s specifying the institution within which they are valid. Here we consider an institution as a set of institutional facts and rules that a group of agents (the "members" of this institution) adopt. This is a general view that can account for various institutional contexts, be they formal institutions or informal ones: the law of a country, a contract between two parties in a business relationship, a social structure, the rules of a game... An institutional fact is a fact that is recognised to be valid in the context of a given institution, but that can make no sense in itself; i.e. it is not a physically observable fact (what Searle calls a "brute fact") but something written in the registry of this institution. For example the fact that two people are married, or that one is authorised to drive a truck, is only valid w.r.t. the law of a country; all deontic fact should also be encapsulated in an institutional fact to make the institution in which they hold explicit. We represent these institutional facts with the operator D s φ meaning that for institution s, it is officially established that φ holds. In particular if φ is an agent's mental attitude, then D s φ can be understood as this agent's commitment (either a propositional commitment if φ is a belief, or a commitment in action if φ is an intention).

Institutional facts can be deduced from other facts thanks to the rules of the institution. For example the presentation of an invoice by a provider to his client counts as an obligation for the client to pay it. The existence of the invoice is physically observable, while the obligation is only valid in an institutional context. We represent these normative consequences with the primitive operator p ⇒ s q, meaning that according to the norms holding in institution s, p entails q. This operator is known in the literature as count as, and has been first formalised by Sergot and Jones [START_REF] Jones | A formal characterisation of institutionalised power[END_REF]. The following mix axioms explicit the link between institutional facts and normative consequences:

(φ ⇒ s ψ) → D s (φ → ψ) (SD) (φ ⇒ s ψ) → (φ → D s φ) (SC)
From these axioms and the properties of D s (see [10, p.8] for details) we can deduce:

(φ ⇒ s ψ) → (φ → D s ψ) (SP)
A particular case of normative consequence concerns the consequences of the performance of an official procedure. Actually some agents can have the power when performing a given procedure under some conditions to create new institutional facts. We represent these institutional powers as an abbreviation power (i , s, cond , α, φ) = ((done i:α ⊤ ∧ cond) ⇒ s φ). Intuitively this means that i has the power in institution s, by performing action α and if condition cond holds, to see to it that φ becomes officially true in institution s. For example a mayor has the power in the law of the French Republic, by performing a declaration, and on condition that the two people agree, to marry them. Obviously these powers result from the agent's role in the institution, but this is not the focus of this paper so we will not remind how roles are formalised in the original framework (the interested reader can refer to [START_REF] Demolombe | Norms, institutional power and roles: towards a logical framework[END_REF] for details on this point).

Deontic modalities

We have a modality for impersonal obligation to be: Oφ reads "it is obligatory that φ", and its axiomatic is that of the Standard Deontic Logic [START_REF] Jones | Handbook of philosophical logic, volume 8, chapter Deontic Logic and Contrary-to-duties[END_REF], that is KD. Obligations to do can be expressed as obligations to be in a state where the obliged action has been performed. Obligations are impersonal since no agent is explicitly responsible for their fulfilment, but such an agent can implicitly appear in their content. For instance Odone i:α ⊤ means that it is obligatory (for no one in particular) to be in a state where i has just performed action α; this can be understood as "i has the obligation to perform action α".

Permissions and interdiction are defined from obligations in a standard way: P φ = ¬O¬p means that it is permitted that φ, and F φ = O¬φ means that it is forbidden that φ.

Please notice that no institution is explicit as a parameter of this obligation modality. But such obligations will be encapsulated in institutional facts to express the institution in which they are valid. For example D s Oφ means that "in institution s, it is obligatory that φ".

Semantics of speech acts

Preliminary remarks

Intentional and institutional dimensions The FIPA-ACL standard [START_REF]The foundation for intelligent physical agents[END_REF] defines features allowing one to give an intentional dimension to the observation and interpretation of a communicative action: the feasibility precondition (the appropriate mental attitudes to perform the speech act) and the rational effect (this is a formula φ representing the content of the speaker i's intention that he intends the receiver j to know; so the performance of the speech act allows any observer k to deduce this corresponding intentional effect: B k I i B j I i φ). Please notice that the performance of the speech act does not automatically allow one to deduce its rational effect, but only its intentional effect, meaning that any agent k believes that the speaker i intends the hearer j to recognize its (i's) intention to achieve the rational effect φ. However, nothing ensures that i indeed achieves φ, his speech act may fail, for example the hearer may not obey an order, or may not believe an assertion. Thus the rational effect can only be deduced under some constraining hypotheses such as the sincerity and competence hypotheses used in FIPA.

In a similar way, we want to provide here the institutional dimension of the observation and interpretation of a communicative action relative to one or several institutions. This institutional interpretation is composed of the following features:

a permission condition that is necessary and sufficient for the speaker to be allowed to perform this speech act; a power condition that also needs to be true for the speech act to have an institutional effect; an explicit institutional effect that is obtained when the speech act is performed while permission and power conditions were true.

We will thus be able to combine the intentional and institutional dimensions of communicative actions (formalised as speech acts [START_REF] Searle | Speech acts : An essay in the philosophy of language[END_REF]), both essential to fully characterise their interpretation. Lorini et al. have also investigated such a unified approach but they have only formalised the interpretation of a promise in the context of ordinary communication; we aim at being much more generic.

In particular we formalise one speech act from each of Searle's categories of illocutionary forces, except the expressive one.

Actually we have relaxed some of the (widely criticised) strong constraints imposed by FIPA-ACL semantics on the appropriate context of performance of speech acts. Instead of imposing these conditions as strong constraints, we have moved them into the permission preconditions of the speech act. The agents are thus physically able to disobey these constraints, but it is forbidden by the interaction norms, and they may incur sanctions for such violations.

Notations In the sequel we use the following abbreviations:

-FP = feasibility preconditions -RE = rational effect -PermC = (institutional) permission condition -PowC = power condition -EE = institutional explicit effect Speech acts are actions of the form Force(sp, ad , inst, content) where sp ∈ AGT is the speaker, ad ∈ AGT is the addressee, inst ∈ IN ST is the institutional context, content is the propositional content and can be any formula of our language, and F orce ∈ {inf orm, promise, command, declare} is the illocutionary force.

Action laws We now explain how the intentional and institutional dimensions of actions interact by providing the action laws governing the performance of speech acts.

We notice that F P is a factual executability precondition, while P ermC is an ideal one. But even ideal worlds are submitted to physical world laws, i.e. P ermC is not sufficient for the action to be executable, F P also has to be true. For example a mayor has the permission to marry people by making a declaration, but the declaration must be executable; thus if he is voiceless one day, he will be unable to marry anyone.

We thus have the following executability laws. The factual executability law (F EL α ) means that an action happens iff its feasibility precondition is true and the agent chooses to perform it. The ideal executability law (IEL α ) means that ideally, an action should happen iff it is permitted and feasible.

happens α ⊤ ↔ (FP (α) ∧ Ch i happens i:α ⊤) (F EL α ) O(happens α ⊤ → PermC (α)) (IEL α )
We also have the following effect laws. The rational effect law (REL α ) means that if the power precondition of an action is false, then only its rational effect can be deduced after its performance. The power effect law (P EL α ) means that if the power condition of an action is true, then both its rational and institutional effects can be deduced after its performance.

¬PowC (α) → after α RE (α) (REL α ) PowC (α) → after α RE (α) ∧ EE (α) (P EL α )
From these laws we can deduce the following theorems clarifying the factual executability and effects of α depending on the different combinations of its feasibility and power preconditions. If FP (α) is false then α is not executable.

¬FP (α) → after α ⊥

If FP (α) is true but PowC (α) is false, α is about to happen after which its rational effect will be true.

(FP (α) ∧ ¬PowC (α)) → (happens α ⊤ ∧ after α RE (α))
Finally if both FP (α) and PowC (α) are true, α is about to happen after which both its rational and institutional effects will be true.

(FP (α) ∧ PowC (α)) → (happens α ⊤ ∧ after α (RE (α) ∧ EE (α)))

Assertives: inform

The assertive speech act Inform commits the speaker to the truth of a proposition. The notation inform(i , j , s, φ) reads "agent i informs j in institution s that φ is true".

Intentional interpretation As we said before we have relaxed FIPA constraints on the executability of speech acts. We thus impose no feasibility precondition here.

F P (inform(i , j , s, φ)) = ⊤

The rational effect (the content of the speaker i's intention that he intends the receiver j to know) is that j believes the promised proposition φ to be true:

RE(inform(i , j , s, φ)) = B j φ
Institutional interpretation The permission precondition to inform j that φ in institution s includes the constraints removed from the factual feasibility preconditions: the speaker should not believe that the hearer already knows if φ, and he should not be already committed on ¬φ in the same institution.

PermC (inform(i , j , s, φ)) = ¬D s B i Bif j φ ∧ ¬D s B i ¬φ
Now the institutional effect of Inform is to retract possible opposite commitments contracted before and to assert a new commitment on φ. Indeed, even if agent i was previously committed on ¬φ (and therefore was not permitted to inform anyone that φ), he may violate that obligation. But these two commitments are inconsistent so the previous one must be retracted while asserting the new contradictory one. Though one can still detect that the opposite commitment was true when i performed the action and that he has thus violated the rules of the institution. Actually due to the seriality of D s we have that D s B i φ → ¬D s B i ¬φ. So the explicit institutional effect of inform is the new commitment:

EE (inform(i , j , s, φ)) = D s B i φ
This effect is always obtained and does not depend on particular powers of i, so the power condition is trivial.

PowC (inform(i , j , s, φ)) = ⊤ Example For example in the context of B2B exchanges, if a provider sends his catalogue to a client, this counts as information about the prices given in this catalogue. As an effect of this action, the provider is thus committed to these prices during the validity of his catalogue. In the specific institution s constituted by the contract between the provider and the client, we assume that we have a specific rule forbidding to contradict one's commitments, which takes the form D s O(D s B i p → after α D s B i p), for every speech act a, where p is the proposition denoting that the price is 100. This means that in institution s, it is obligatory that if an agent i is committed to believe that the price of an item is 100, then after any speech act he is still committed to this (in other words it is forbidden to retract this commitment by any speech act). From this we can deduce that the provider is obliged to respect the given prices, i.e. D s O(D s B i p → after Inf orm(i,j,s,¬p) ⊥) (it is obligatory that if i is committed to p, then the action of informing any agent j that ¬p is not feasible).

Commissives: promise to

This commissive speech act commits the speaker on a course of action. The notation promise-to(i , j , s, α) reads "i promises to j in institution s to perform action α".

Intentional interpretation

We begin with specifying the intentional dimension of this speech act, that is not given in FIPA-ACL. A promise-to is feasible if the speaker believes that the hearer intends him to perform the concerned action 3 . For example it makes no sense that a child promises to his father to play, while it makes sense to promise him to make his schoolwork. So:

F P (promise-to(i , j , s, α)) = B i I j done α ⊤
The rational effect pursued by the speaker is that the hearer be aware of his intention to perform the promised action:

RE(promise-to(i , j , s, α)) = B j I i done i:α ⊤

Institutional interpretation

In an institutional context s, this promise to perform an action α is permitted on condition that the action i :α is not explicitly forbidden itself. So the permission precondition is the following:

PermC (promise-to(i , j , s, α)) = ¬D s O¬happens i:α ⊤
The institutional effect consists in ratifying in institution s the speaker's intention to perform action α; so after promise-to(i , j , s, α) the speaker has stored in the registry of s its intention to perform α, which is similar to him being committed in s to this course of action. EE (promise-to(i , j , s, α)) = D s I i done i:α ⊤ This is thus similar to the inform(i , j , s, φ) speech act except that a promise stores a commitment in action while an inform stores a propositional commitment.

There is no power precondition, so the institutional effect of a promise is always reached.

PowC (promise-to(i , j , s, α)) = ⊤

Example A client c promises to pay his provider p once the ordered goods have been delivered. The action to pay is denoted by α pay This promise is valid in the context of a B2B exchange contract, that is a particular institution denoted b2b here. So this promise is formalised as: promise(c, p, b2b, α pay ). This promise is permitted since obviously the promised action to pay is not forbidden: ¬D b2b O¬happens αpay ⊤. So when the client receives the delivery, his promise allows to deduce his commitment (ratified intention) to pay: D b2b I c done c:αpay ⊤, that is the institutional effect of this speech act.

Directives: command

This directive speech act is an attempt from the speaker to make the hearer perform some action. The notation command (i , j , s, α, cond ) reads "i orders to j in institution s, in virtue of condition cond, to perform action α".

Intentional interpretation According to the FIPA-ACL semantics, a request is feasible only if the speaker does not believe the hearer to already intend to perform the commanded action, and does believe that the part of the feasibility preconditions of the commanded action that concerns him (i.e. that are his mental attitudes) are valid. Here we consider that when α is an action of agent j then FP (α) is of the form FP i (α) ∧ FP ̸ =i (α) where the former is "i's part of FP (α)" (similar to FIPA-ACL notation FP (α) [i\j], that is the part of FP (α) that are mental attitudes of agent i). But we do not impose this constraint on the feasibility of α as a feasibility precondition of the command, but rather as a permission precondition. So:

FP (command (i , j , s, α, cond )) = ⊤
The rational effect of a command (i.e. the effect that i intends j to believe that i intends to achieve) is that j has performed the commanded action:

RE (command (i , j , s, α, cond )) = done j :α ⊤

Institutional interpretation

The permission precondition to command someone to perform an action is to be empowered to do so, i.e. to dispose of the institutional power to create the obligation to perform the commanded action by commanding it, and to validate the condition of this power, that is an explicit attribute cond of this directive speech act. An additional permission precondition is the constraint coming from FIPA feasibility precondition that we relaxed, that is that th part of the feasibility preconditions of α that depends on i hold (one should not command someone to perform an action whose preconditions are made false by his own mental attitudes).

PermC (command (i , j , s, α, cond )) = cond ∧ power (i , s, cond , command (i , j , s, α, cond ), Odone j :α ⊤) ∧ FP i (α)
The explicit institutional effect of this power is to create two new institutional facts, corresponding to the obligation for j to perform α, and the recording of j's knowledge of his obligation. Actually this obligation could exist before, and in this case the command corresponds to a notification; but it can also be created from scratch by the command (see the examples in the next paragraph for clarification).

EE (command (i , j , s, α, cond )) = D s Odone j :α ⊤ ∧ D s B j Odone j :α ⊤
Example For example a parent can command his children to clean his room. In this case, the action becomes obligatory through the command, because of the parent's authority over his son (that is the condition of his power). Similarly a professor commanding his students to make some schoolwork creates the obligation for them to do so, on the strength of his role of professor.

But an order does not necessarily create an obligation, and may just put in focus an existing one. For example a bailiff can be sent to officially command an uncooperative client to pay an invoice. In this case the obligation already exists (and is attested by the invoice) so the bailiff only reminds the client of it 4 . He is permitted to perform such a command in virtue of his role of bailiff and because he is sent by the provider.

Declaratives: declare

This declarative speech act changes the institutional reality by creating a new institutional fact. The notation declare(i , j , s, cond , φ) reads "i declares to j in institution s that given condition cond, the fact φ is now established". The condition usually bears upon the speaker's role that empowers him to perform such a declaration.

Intentional interpretation This intentional interpretation is partly inspired from the intentional interpretation of an inform(i , j , s, D s φ). The feasibility precondition of a declaration is that the speaker does not believe the declared fact to be already established (indeed a declaration must create a new institutional fact). The rational effect (i.e. the intended effect) is the same as an inform about D s φ, i.e. to make the hearer aware of this information. So:

FP (declare(i , j , s, cond , φ)) = ¬B i D s φ RE (declare(i , j , s, cond , φ)) = B j D s φ
Institutional interpretation The permission precondition to perform declare(i , j , s, cond , φ) is that i really has the power to establish the declared fact φ by declaring it under the announced conditions cond. This power is locally granted by each specific institution to some agents depending on their role. For example the French republic grants the mayors the right to pronounce two people husband and wife, under the condition that they both consent to it. Thus an ordinary agent who is not mayor does not have this power, so that he is not allowed to pronounce marriages.

PermC (declare(i , j , s, cond , φ)) = power (i , s, cond , declare(i , j , s, cond , φ), φ)

The explicit effect of a declaration is to store the declared fact in the institution, as well as the fact that the hearer is officially aware of this fact.

EE (declare(i , j , s, cond , φ)) = D s φ ∧ D s B j D s φ
This explicit effect is only obtained under the additional condition that cond is valid:

PowC (declare(i , j , s, cond , φ)) = cond

Example For example a country can declare war to another one, by the voice of its representative that is empowered to do so, and under some conditions like the agreement of some counsellors. A mayor is empowered by its country to pronounce weddings under some conditions that the people are of age and consenting.

Citizens have to declare their income to the public treasury in order to calculate the amount of tax that they will pay. This is a declaration since the effect is a new institutional fact officially establishing one's declared income as being believed by him. Any citizen is empowered to do so. Moreover the law imposes a constraint on the generated commitment, that is an obligation to believe this income to be true. Thereby if the declared income was false the citizen is liable for prosecution and sanctions.

Example of reasoning with our action laws

This example is situated in the context of a B2B exchange (in institution b2b) between a buyer b and a seller s. The seller intends potential clients to know the prices of his products, e.g. I s B b p. With our relaxed feasibility precondition, he can use an assertive speech act whatever the context. Though if the buyer has already been informed of the prices before (D b2b B s B b p), the seller is not permitted to inform him again. Thus if he informs him anyway, according to IEL α he violates an obligation. This can be detected by other agents, and specific rules of the institution may specify sanctions to compensate this. Being aware of such pre-specified sanctions, an agent can deliberately choose to violate an obligation if the intended outcome (here, that clients be aware of the seller's offer) is more important than the incurred sanction. This shows the importance of having both intentional and institutional semantics of speech acts, to allow agents to reason about the relative importance of their goals and their obligations, in order to make an appropriate decision.

Detailed comparison with other work

In this section we compare our semantics of speech acts with those proposed by Fornara and Colombetti, and by Lorini et al. (that we have presented above).

Concept of commitment

We have shown before that what we mean by commitment in this work is a ratified mental attitude, i.e. a mental attitude (belief or intention) stored in the institution. This notion is similar to Fornara and Colombetti's commitment that is also a public concept, except that we have not made explicit its creditor. Actually the debtor is committed towards the whole institution, but an implicit creditor can sometimes be found in the content φ of the commitment. For example if agent i promises to j to pay him, he commits himself to a proposition involving agent j, expressing that j will be payed at some future instant. The creditor can sometimes be found in the sanction associated with the violation of the commitment, too; for example the obligation to pay damages to an agent.

Our notion of ratified mental attitude is also similar to Gaudou et al. 's notion of acceptance, because it must influence the agent's behaviour and utterances. Indeed, the agent's ratified mental attitudes are mental attitudes that he has expressed, that are stored in the institution, and to which he must conform while subsequently acting and speaking, even if they are not consistent with his real mental attitudes. For example an agent who promises that he has seen a given movie must then be able to talk about it in order to be consistent with his promise; if he is unable to narrate the end of the movie one can notice that he is contradicting his commitment.

Notion of institution

By institution we mean a set of rules and facts that are adopted by a group of agents (the members of the institution). This seems to be a more generic notion than Lorini et al. 's concept of informal institution, since it accounts for this particular kind of institutions but also for various other ones: laws of a country, rules of a game, contract between businesses, social norms of a culture... In particular it allows to have institutional rules that are ignored by the members of the institution, what is the case for law for example, since one cannot be aware of the whole set of laws of his country, while he is one of its citizens. Fornara and Colombetti do not make explicit the institutional context in which their speech acts are interpreted, so we believe that they also consider a kind of "ordinary communication" institutional context.

In our view informal institutions are described by a specific set of facts and rules, determining their specific functioning. In particular the fact that all agents must accept a fact for it to become institutional is a particular institutional law. In other kinds of institutions, facts must be adopted by a majority of members (voting to create a law or to elect the president for example), or the opinion of one single member can suffice (the referee is always right). Thus we cannot adopt such an hypothesis in our account. Indeed on the contrary we consider the generic interpretation of speech acts in any institution s. More specific rules can be additionally specified in each particular interpretation, but the object of this paper is to identify for each category of speech acts the features that are common to their institutional interpretations whatever the institutional context.

In this work we have provided an expressive logical framework blending the agents' mental attitudes (beliefs, intentions) with their social attitudes (obligations, institutional facts and powers...). To illustrate its expressivity, please notice that our framework allows to represent some forms of contrary-to-duty obligations-to-do. Such obligations take the form:

Oafter α ⊥ → after α Odone repairα ⊤
where repair α is the contrary-to-duty obligation associated to the violation of the obligation to refrain from doing α. This means that if it is forbidden to perform α, then after α it is obligatory to perform a repairing action repair α .

We have then used this framework to provide a semantics for an agent communication language based on FIPA-ACL but relaxing its widely criticised too constraining feasibility conditions, and adding permission preconditions. This way, agents can choose to perform forbidden speech acts but would then be liable to sanctions in the corresponding institution. Our ACL semantics also includes new speech acts (commissives and declaratives). It generalises existing approaches by unifying the intentional and institutional dimensions in one single framework, while strongly distinguishing them; moreover it allows to consider various kinds of institutional contexts; finally it provides action laws taking both dimensions into account.

In future work we intend to improve the institutional and intentional semantics of speech acts by accounting for deadlines. Various researchers [START_REF] Broersen | Designing a deontic logic of deadlines[END_REF][START_REF] Dignum | Obligations and dense time for specifying deadlines[END_REF][START_REF] Demolombe | Norms with deadlines in dynamic deontic logic[END_REF] have shown that an important feature of obligations to perform an action is the deadline before which this action must be performed, that is essential to be able to assess the violation or fulfillment of such obligations. Though for the sake of simplicity we have omitted deadlines in this paper. An idea to manage them in future work could be to use existing formalisations of norms with deadlines, or to ground on linear temporal logic with until and since operators [START_REF] Burgess | Logic and time[END_REF].

Finally we would like to mention that our framework for the institutional interpretation of speech acts has been successfully implemented into institutional agents that have been used in a prototype of industrial application: a multi-agent mediation platform for automated business to business exchanges [START_REF] Bourge | A multi-agent mediation platform for automatic b2b exchanges[END_REF].

Please notice that threats such as "I promise that I will kill you" cannot be considered as promises in the sense of Searle.

Actually this seems to be a notification rather than a command, but the aim is to make the client behave, while the aim of a notification is only to make the receiver officially aware of what is notified. In further work we expect to study into more details the links between declarations, commands and notifications.


Stéphane Bellon

Ole Ostermann

Agrarökologie in der Praxis

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


M Bescond

D Logoteta

F Michelini

N Cavassilas

T Yan

A Yangui

M Lannoo

K Hirakawa

Thermionic cooling devices based on resonant-tunneling AlGaAs/GaAs heterostructure

We study by means of full quantum simulations the operating principle and performance of a semiconductor heterostructure refrigerator combining resonant tunneling filtering and thermionic emission. Our model takes into account the coupling between the electric and thermal currents by self-consistently solving the transport equations within the non-equilibrium Green's function framework and the heat equation. We show that the device can achieve relatively high cooling power values, while, in the considered implementation, the maximum lattice temperature drop is severely limited by the thermal conductivity of the constituting materials. In such out-ofequilibrium structure, we then emphasize the significant deviation of the phonon temperature from its electronic counterpart which can vary over several hundred Kelvin. The interplay between those two temperatures and the impact on the electrochemical potential is also discussed. Finally, viable options toward an optimization of the device are proposed.

I. INTRODUCTION

Handling the rapid increase of the heat power densities associated to the electronic miniaturization is a major scientific and industrial issue. In microelectronics, self-heating effects result, indeed, in significant reductions of both transistor efficiency 1,2 and lifetime [START_REF] Srinivasan | Dependable Systems and Networks, 2004 International Conference on[END_REF] . Similar difficulties arise also in the case of optoelectronic devices, particularly those operating with intense radiation, as light-concentrating solar cells [START_REF] Sun | [END_REF] . The development of embedded systems in a Internet of Things perspective will, moreover, require local, and possibly on-chip, thermal management capabilities [5][6][7] .

Temperature stabilization by means of electronic refrigerators has attracted much interest, as a scalable, reliable and "green" option. The researches in this field have mainly focused on two approaches, which rely either on thermoelectricity or thermionic emission of hot carriers.

Thermoelectric devices are based on the Peltier effect. Their performance can be quantified through the dimensionless figure of merit ZT = σS 2 T /(κ e +κ l ), where σ is the electronic conductivity, S is the Seebeck coefficient, T is the average temperature of the hot and cold side of the device, and κ e and κ l are the electronic and lattice thermal conductivities, respectively. An efficient thermoelectric system must therefore have a high σ (to minimize the Joule effect), low κ e and κ l (to avoid thermal short-circuit), and a high S (to maximize the conversion of electrical energy to temperature variation, or vice versa). Despite extensive research efforts, for over fifty years the best thermoelectric materials had ZT ≈ 1. In 1993, the pioneering work of Hicks and Dresselhaus 8 paved the way to the exploitation of nanoscale effects to improve the thermoelectric device efficiency. Since then, experimental studies, e.g. on thin-film superlattices 9 , quantum-dot superlattices 10 and nanowires 11,12 , reported higher values of ZT , mostly due to the substantial reduction of the phonon thermal conductivity in nanostructures [13][14][15] . Thermionic devices were proposed as an alternative to thermoelectric devices, able to reduce the effects of carrier scattering and lattice thermal conduction. The operation of these devices is based on the extraction of thermally excited electrons from an emitter (the cold side of the device), via thermionic emission over a potential barrier. The carriers, then, almost ballistically reach the collector (the hot side), where they thermalize at lower energies. A voltage generator sustains a current in the device, by supplying cold carriers from the collector to the emitter. It has been pointed out that the working principle of thermionic devices can be described as well in terms of Peltier cooling and heating at the emitter-barrier and barrier-collector interfaces, respectively, and that the physical quantities defining their performance are the same as those entering the expression of ZT 16 . Nevertheless, the performance of thermionic devices is traditionally assessed by referring to the more device-oriented Coefficient Of Performance COP = J Q /(JV ), where J Q is the heat power density per unit area extracted from the cold side, and JV is the the density of power supplied to the device (J is the electrical current density and V the voltage bias).

Within the same miniaturization trend as for their thermoelectric counterparts, solid state implementation of these devices based on heterostructures have been envisaged. The first solid state thermionic refrigerator (analogous of a vacuum diode) has been proposed by Mahan [17][18][19] . Currently, other options are being investigated, as it has been pointed out that simple single-barrier thermionic devices cannot exhibit performance higher than their bulk thermoelectric counterparts 16,20,21 . In particular, on the basis of the theoretical predictions in Refs.22-29, the energy filtering properties of multibarrier structures (typically in the form of superlattices) have been successfully exploited. With respect to thermoelectric coolers, these devices are expected to offer advantages in terms of compatibility with standard integration processes, high cooling power density and fast transient response 30 .

In this paper, we study a AlGaAs/GaAs heterostructure-based thermionic refrigerator first proposed by Chao et al. 31 , which exploits resonant tunneling filtering to enhance the cooling at the interface with a thermionic barrier. Our numerical full quantum simulations are able to take into account the thermal effects by self-consistently coupling the electron transport equations within the non-equilibrium Green's function (NEGF) formalism with the heat equation 32,33 . In our simulations, this coupling is obtained by defining local effective temperatures for the acoustic and polar optical-phonon baths. However, in out-of-equilibrium conditions, the electron temperature is expected to deviate from that of phonons. To complete the physical picture of the system, we therefore also compute a non-ambiguous and thermodynamically-consistent values of electron temperature and electrochemical potential 34,35 .

The paper is organized as follows. Section II describes the electronic quantum transport and heat transport models. In Section III we illustrate and discuss the simulation results in terms of current characteristics, coefficient of performance, phonon and electron temperatures, and variation of electrochemical potential. Finally, in Section IV we draw our concluding remarks. The thermally excited electrons are then removed from the well by thermionic emission over a 150 nm thick AlGaAs barrier, that serves as a thermal wall to reduce the heat backflow.

II. DESCRIPTION OF THE MODEL

The left and right GaAs access regions are 150 nm thick and n-doped with concentration 10 18 cm -3 .

A. Electronic quantum transport model

We use the NEGF transport equations expressed within the effective mass approximation to implement a one-dimensional (1D) quantum simulator along the heterostructure growth direction (x). The single band effective mass Hamiltonian describes the Γ-valley of the conduction band of the III-V semiconductors. As the considered structure is translationally invariant in the perpendicular y and z directions, Born-Von-Karman periodic boundary conditions are applied on the transverse wave vector component k t such that k t = n kt ×2π/L t , with L t = 50 nm and n kt an integer indexing the transverse modes whose degeneracy is equal to π(2 × n kt + 1) 36 . In the following, we briefly summarize the main features of the NEGF approach in matrix notation. We first define the retarded Green's function at the energy E for each transverse mode k t , The Al mole fractions are y = 0.3 and x = 0.12. The GaAs QW thickness is 4.8 nm; b) Current density as a function of the voltage V RL applied between the left and right contacts, for a contact temperature T = 300 K.

G r kt = (E -V )I -H kt -Σ r L,kt -Σ r R,kt -Σ r S,kt -1 , ( 1
)
where I is the identity matrix, H kt represents the effective mass Hamiltonian for the transverse mode k t and V is the electrostatic potential energy. Σ r L/R and Σ r S are the retarded self-energies for the left/right semi-infinite device contacts 37 and scattering mechanisms, respectively.

From the retarded Green's function, the lesser/greater Green's functions are then obtained as

G ≶ kt = G r kt Σ ≶ L,kt + Σ ≶ R,kt + Σ ≶ S,kt G r † kt , (2)
where the Σ ≶ are the lesser/greater self-energies, related to their retarded counterpart by

Σ r = 1 2 [Σ > -Σ < ] . (3)
Only acoustic-and polar optical-phonon interactions are considered, since non-polaroptical phonons turn out to be negligible in the materials constituting the device [START_REF] Price | [END_REF] . Interactions self-energies are calculated within the self-consistent Born approximation (SCBA) 39-41 .

In our approach, the acoustic and polar optical-phonon baths locally follow a Bose-Einstein distribution, and are therefore assumed at equilibrium. On the other hand, acoustic and optical phonons can not be assumed at equilibrium with respect to each other, as the net anharmonic decay of optical phonons into acoustic, driven by an unbalance in their respective energy densities, plays a fundamental role in the thermal transport 42 . This is taken into account in our model by defining, at each position of the domain, two different temperatures, T AC and T P OP for the acoustic and polar optical-phonons, respectively. As detailed in the next subsection, these temperatures are self-consistently computed by coupling the electron transport equations with the heat equation. Within the elastic assumption, the SCBA self-energy for acoustic-phonons is independent of k t and can be expressed as 39,43

Σ ≶ AC (j, j, E) = k t π(2n k t + 1) Ξ 2 k B T AC (j) ρu 2 s G ≶ k t (j, j, E), ( 4
)
where Ξ is the deformation potential, ρ is the mass density and u s is the sound velocity.

In Eq.( 4) index j indicates the x position along the discretized domain. Interactions with acoustic phonons are here assumed to be local and we then consider only the diagonal part of the Green's function. Such approximation has been demonstrated to be valid for acoustic phonons down to very low temperatures 44 . Concerning polar optical-phonons (POP), we adopt the diagonal expression of the scattering self-energy recently proposed as an effective description of their long range interactions 45 . For a given wave vector k t , it can be shown that:

Σ ≶ P OP,kt (j, j, E) = λM 2 2πS k t (n L (j) + 1)G ≶ k t (j, j, E ± ω L ) +(n L (j))G ≶ k t (j, j, E ∓ ω q L ) × π π/Lt π(2n k t + 1) (k t -k t cos θ) 2 + (k t sin θ) 2 dθ, (5)
with n L (j) = (e

( ω L )/(k B T P OP (j)) -1) -1 , M 2 = 2π ω L e 2 ( 1 ∞ -1 0 )
, θ is the angle between k t and k t , S = πL 2 t , and ω L =35 meV. M is the Fröhlich factor in which 0 and ∞ represent the static and high frequency dielectric permittivity respectively. Finally λ is a scaling factor which takes into account for the diagonal approximation. The value λ=8 used in this paper has been obtained within the comprehensive and physically-based analytical model proposed in Ref. 45.

The total phonon self-energy Σ ≶ S,kt for a given mode k t is then:

Σ ≶ S,kt = Σ ≶ AC + Σ ≶ P OP,kt . (6)
Once the lesser/greater Green's function G ≶ kt of each mode k t is determined, physical quantities can be straightforwardly calculated 39 . The electron density n along the transport direction reads:

n j = -2 × i 2π kt π(2n kt + 1) +∞ -∞ G < kt (j, j; E)dE, (7)
= -i +∞ -∞ G < (j, j; E)dE, (8)
with G < (j, j; E) = kt (2n kt + 1)G < kt (j, j; E). The carrier current density (in A/m 2 ) flowing from position j to j + 1 along the x direction is calculated from the off-diagonal elements (j, j + 1) of G < kt (i, j; E) as

J j→j+1 = +∞ -∞ dE e kt (2n kt + 1) S H j,j+1 G < kt (j + 1, j; E) -G < kt (j, j + 1; E)H j+1,j , (9)
= +∞ -∞ J j→j+1 (E)dE. ( 10
)
where H j,j+1 corresponds to the nearest neighbors hopping terms in the discretized tightbinding like Hamiltonian and J j→j+1 (E) is the current density spectrum (in A/(m 2 • eV)).

From Eq.( 10) we can deduce the corresponding electronic energy current 46 :

J E j→j+1 = +∞ -∞ EJ j→j+1 (E)dE. (11)
In practice, the set of Eqs. ( 1)-( 6) is solved self-consistently using a recursive algorithm [START_REF] Ferry | Transport in Nanostructures[END_REF]47 until the criteria of convergence for both electron density and carrier current density are reached. The potential energy V is self-consistently determined by nonlinearly coupling the transport equations ( 1)-( 6) with the Poisson equation through the electron density.

B. Heat transport model

We compute the lattice temperature by solving the 1D heat equation along the x direction.

The discretized heat equation on the site j reads:

- ∂ ∂x [κ th (x) ∂ ∂x T AC (x)] j = Q j , (12)
where κ th is the thermal conductivity. It is taken equal to the GaAs bulk value (46 W/(m•K)) in all the device, except in the QW region, where it is set to 4 W/(m•K) in order to take into account the thermal resistance associated to the interface between different layers 48,49 . The temperature entering the heat equation is the one associated to the acoustic phonons, since they have a larger velocity than their polar optical counterparts and are mainly responsible of the heat transport [START_REF] Pop | SISPAD Tech. Dig[END_REF] . Left and right reservoirs are assumed to be at the thermodynamic equilibrium at T AC = 300 K. This value is set by enforcing Dirichlet's boundary conditions at the left and right contact. This assumption corresponds to consider massive contacts with a sufficiently high thermal capacitance. Q j is the volumetric source term which corresponds to the heat power density (in W/m 3 ) generated by electron-phonon interactions. In the framework of the previously described electron transport formalism, it can be computed as 33,46 :

Q j = -∇ j • J E . (13)
A negative value of Q j corresponds to an energy transfer from the lattice to electrons, while a positive one describes the reverse phenomenon. In the absence of electron-phonon interactions, the charge conservation entails ∇ • J (E) = 0 for each E, and, therefore, Q = 0.

From a physical point of view, electrons loose or increase their energy by scattering with polar optical-phonons. In turn, optical phonons decay into acoustic phonon modes, which sustain the thermal energy propagation along the device. In stationary conditions, the power transfer from optical to acoustic phonons must be equal to the heat power density Q j defined above. Within a relaxation time approximation 42 , we can thus write

(T P OP (j) -T AC (j))C P OP τ P OP →AC = Q j , (14)
where τ P OP →AC is the relaxation time of polar optical-phonons into acoustic phonons (τ P OP →AC = 4.1610 -12 s) [START_REF] Luo | [END_REF] and C P OP is the thermal capacitance of the polar opticalphonons per unit volume (C P OP = 1, 72.10 6 J/(m 3 •K)) [START_REF] Luo | [END_REF] . The numerator of the left side expresses the average energy per unit volume exchanged between the polar optical and acoustic phonon baths in an interval τ P OP →AC . Eq.( 14) allows us to compute T P OP (j) from the knowledge of Q j and T AC (j).

The computed values of T AC and T P OP are injected in Eq.( 4) and Eq.( 5), respectively.

This establishes the coupling between the heat equation and the electron transport equations.

The heat equation is iteratively solved together with the transport equations and the Poisson equation, until a global self-consistency is achieved. accounts for most of the power extracted from the lattice, corresponds to the QW, where the absorption of polar optical-phonons mostly occurs. scattering. This results in an enhanced cooling effect. In the considered implementation, the maximum temperature difference that the refrigerator is able to establish is around 2 mK. Such a small value is to be attributed to the high lattice thermal conductivity of AlGaAs and GaAs, which effectively level out the temperature inside the device.

Considering the present structure, the cooling power density of the device J Q , obtained by integrating over x the negative part of the heat power density (Eq.( 13)), increases with the applied bias. It varies from 2,26•10 6 W/m -2 at V RL =0.1 V to 7,5•10 6 W/m -2 at V RL =0.6

V. Interestingly, the related coefficient of performance (COP), defined as the ratio of J Q by the applied power (P Elec =J × V RL )) follows the opposite trend. It goes from 11,7 % at V RL =0.6 V up to 120 % at V RL =0.1 V. Such high COP values for low bias is physically sound as long as the integration of the heat power density over the whole device remains equal to P Elec . It also confirms the real capacity of the considered device to cool the lattice.

B. Local electron temperature

As previously mentioned, temperature of phonons in a non-equilibrium system can significantly differ from its electronic counterpart. The comparison between the electron temperature and the one of optical phonons could provide information about the physics of the device and its possible optimizations. In this section, we calculate the local electronic temperature based on the virtual probe approach. Such a method can determine the local electronic temperature and electrochemical potential by cancelling the particle and energy currents between a floating probe and the SHR device. The probe is then in local thermodynamic equilibrium with the non-equilibrium structure. Stafford and co-workers 34,35,52 recently showed that temperature and electrochemical potential determined within this approach are physically-consistent, as they are unique and fulfill the four laws of thermodynam-ics. More precisely, temperature and electrochemical potential are completely determined when the probe-system coupling is: i) maximally localized, to provide a good spatial resolution, ii) weak enough to generate a non-invasive measurement, iii) broadband, "to ensure that the measured physical properties depend on the energy spectrum of the structure and not on the one of the probe".

We then consider a thermoelectric probe at the position j along the x-axis defined by the following self-energy (similar to the Büttiker probes [53][54][55] ):

Σ > (j; E) = -i[1 -f F D (E, µ j , T e j )]LDOS(j; E)ν coup , (15)
Σ < (j; E) =if F D (E, µ j , T e j )LDOS(j; E)ν coup , (16)
where f F D is the Fermi-Dirac distribution of the electrons in the probe, µ j and T e j are respectively the local electrochemical potential and electronic temperature at the position j;

LDOS(j; E) = i [G > (j,j;E)-G < (j,j;E)] 2π
is the local density of states of the probe (taken equal to the one of the device) and ν coup is the energy independent coupling strength between the probe and the system. In the considered case, the exact value of ν coup is not important, as it will cancel out in the following computations.

By enforcing the simultaneous cancellation of the electron charge and energy currents between the device and the probe, we obtain a system of two coupled nonlinear equations in the unknowns µ j and T e j :

∆J(j) = +∞ -∞ Σ > (j; E)G < (j, j; E)dE - +∞ -∞ G > (j, j; E)Σ < (j; E)dE = 0, (17)
∆J E (j) = +∞ -∞ EΣ > (j; E)G < (j, j; E)dE - +∞ -∞ EG > (j, j; E)Σ < (j; E)dE = 0. ( 18
)
The system is iteratively solved at each position j through a Newton-Raphson algorithm 56 .

We remark that the model adopted for the probe fulfills the above-mentioned conditions ensuring the possibility to unambiguously define a local electron temperature and electrochemical potential. The system of Eqs. ( 17) and ( 18) is therefore expected to have a unique solution. These calculations are done as a post-processing step once the self-consistent lesser and greater Green's functions of the system are obtained. in the thin AlGaAs barrier, as a results of the energy filtering (which selects electrons close to the bottom of the conduction band). It then rises by ∼12 K in the quantum well, where electrons increase their kinetic energy due to a combined effect of polar optical phonons absorption and thermionic emission. At the onset of the thick AlGaAs barrier, the kinetic energy of electrons is mostly converted in potential energy, and the electron temperature drops again, until a minimum value of 236 K. In the region between the quantum well and the right contact, the constant electric field associated to the linear bending of the AlGaAs barrier induces an increase of the average electron kinetic energy. This results in an increase of the electron temperature up to 337 K. By moving further toward the right contact, electrons start thermalizing and the temperature decreases. Electrons participating in the transport correspond to a much smaller charge density with respect to the total mobile charge in the doped access regions (10 15 against 10 18 cm -3 ). Therefore, the hot electrons reaching the right access region are screened along few nanometers and the global electronic temperature recovers its contact equilibrium value of 300 K. In order to validate this approach of measurement of the electronic temperature and electrochemical potential in non-equilibrium regime, we consider the following electron density expression:

n eq j = +∞ -∞ LDOS(j; E)f F D (E, µ j , T e j )dE. (19)
A consistency check can rely on the comparison between the charge density values obtained from Eq.( 19) and those obtained by solving the NEGF transport equations (Eq.( 8)). In the two computational approaches definitely supports the values of T e and µ.

IV. CONCLUSION

We numerically investigated the cooling properties of a GaAs/AlGaAs SHR based on a combination of resonant-tunneling and thermionic processes. By adopting a full-quantum approach, we have resolved the electric and thermal effects at play and described their reciprocal influence. The physical picture provided has been further complemented by introducing, through a virtual probe approach, a thermal characterization of electrons. In the considered implementation of the device, we have found power cooling densities of the order of 10 2 W/cm 2 and a maximum COP of 120% at low bias (V RL =0.1 V). The difference of lattice temperature the device is able to maintain is, however, limited to few mK, due to the high thermal conductivity of GaAs and AlGaAs. Unlike the phonon case, electronic temperature can vary up to several hundred of kelvins. At high applied bias (i.e. V RL >0.4 V) the electronic temperature can be reduced by 90 K in the QW region, while exceeding 700 K in the hot region. Besides the obvious advantage that could derive by considering materials with lower thermal conductivities, a viable option to improve the performance of the refrigerator consists in increasing the doping of the access regions. For a given supply power, this is expected to increase the electron density in the quantum well, and therefore the rate of phonon absorption. An optimization of the process of heat extraction from the lattice could be also performed by finely tuning the energy of the resonant level. A higher thermal isolation between the hot and cold side of the device could be obtained by tailoring the shape of the wider AlGaAs barrier, particularly by flattening its top in order to ballistically guide electrons to thermalize farer from the quantum well. Finally, looking at a suitable trade-off between a decrease of the thermal conductivity by including superlattice sections and the concurrent degradation of electron mobility can be also envisaged.

V. ACKNOWLEDGEMENT

This work was granted access to the HPC resources of Aix-Marseille Université financed by the project Equip@Meso (ANR-10-EQPX-29-01).

Figure ( 1

 1 Figure (1.a) represents the band diagram and the working principle of the studied semiconductor heterostructure refrigerator (SHR). The cold electrons are injected into 4.8 nm wide GaAs quantum well (QW) by resonant tunneling through a 2.4 nm thick AlGaAs potential barrier. The cooling effect is driven by the transfer of energy from the lattice to electrons in the resonant state, mediated by the electron polar optical-phonon scattering.

FIG. 1

 1 FIG. 1. a) Schematic band diagram of the studied semiconductor heterostructure refrigerator (SHR) device. The left and right n-GaAs access regions are 150 nm wide and doped at a concentration of 10 18 cm -3 . The thin and thick AlGaAs barriers are 2.4 nm and 150 nm wide, respectively.

Figure ( 1 FIG. 2 .FIG. 3

 123 Figure(1.b) shows the electron current density as a function of the bias V RL between the right and left contacts, when the temperature in the reservoirs is set to 300 K. The resonant tunneling can be easily identified. This physical phenomenon together with the thermionic emission above the thick barrier, are also visible in the current spectrum of Figure(2).Electrons are mainly injected through the thinner AlGaAs barrier at energies close to the resonant state in the GaAs QW. This filtering results in a charge accumulation in the well, which enhances the electron-phonon scattering rates 39 . On the other hand, no available states are present in the well at energies below the resonant state, and therefore electrons can only absorb phonons. If the energy difference between the resonant state and the top of the wider AlGaAs barrier is comparable with the energy of the polar optical-phonons (as it is the case in Fig.(2)), after absorbing a phonon, electrons can overcome the barrier and flow toward the right end of the device. As the potential of the wider AlGaAs barrier decreases, states at lower energies are made available for electrons. They then start thermalizing toward the Fermi-Dirac distribution by emitting polar optical-phonons and transferring heat power to the lattice. Figure (3.a) shows the corresponding electron energy current density (Eq.(11)) along the device. The current smoothly increases in the left access region, where phonon absorption dominates over emission 33 , increases more rapidly in the quantum well, where absorption is enhanced, and finally monotonically decreases as a consequence of electron thermalization.The energy transferred between electrons and the lattice along the device can be exactly evaluated by the heat power density (Eq.(13)) as shown in Figure(3.b). The heat power density is negative on the left side, where a net transfer of energy from the lattice to electrons takes place, and positive on the right side. A negative peak of the heat power density, which

Figure ( 4 )

 4 Figure (4) shows the self-consistent polar optical-phonon temperature along the device for several values of V RL . As expected, it decreases in the left region, while it increases on the right side. The temperature reaches its minimum in the QW. An increase of V RL entails an increase of the current (see Fig.(1-a)), and as a consequence, an increase of electron-phonon

FIG. 4 .

 4 FIG.4. Polar optical-phonon temperature profiles for different applied V RL .

Figure ( 5 FIG. 5

 55 Figure(5.a) shows the electron temperature along the device at V RL =0.1 V. We first see that temperature varies over a much wider scale than in the polar optical-phonon case. The coldest temperature is still reached in the QW region. The temperature first drops by ∼22 K

ForFIG. 6

 6 FIG.6. a) Electronic temperature along the device obtained at V RL = 0.4 V (solid line) and b) corresponding local electrochemical potential profile (solid line). In both panels the conduction band edge is plotted as reference in dashed line.

Figure ( 7 )FIG. 7 .

 77 Figure (7) such a comparison is proposed for V RL =0.4 V. The excellent agreement between