Sébastien Hardy

Water supply alternatives in the urban agglomeration of La Paz: An efficient tool for social integration and resilience of urban water management

Keywords: water, crisis, discourse, decision makers, integration

The drinking water supply model of the urban agglomeration of La Paz is based on a large system; it is a structure that captures, produces and distributes large amounts of drinking water through a network of pipelines. This model developed in European cities by hygienists has limited efficiency in an urban agglomeration located on the edge of the semiarid zone of the Bolivian highlands where urban growth continues at a rapid pace. In a context with an increasing frequency and magnitude of water supply crises, this article seeks to show the alternatives developed by the low-income population to have access to water and suggests that these alternative forms practiced by these sectors may contribute to resilience to water scarcity.

Introduction

The drinking water supply service of La Paz was privatized with the creation of company Aguas del Illimani and was therefore considered expensive until 2006, with many households unable to access this resource [START_REF] Botton | Water from the heights, water from the grassroots: The Governance of common dynamics and public services in La Paz-El Alto[END_REF]. As a result of this situation, Bolivian decision makers today justify the control of this resource by the public sector. At the same time, these decision makers are alarmed by the effects of climate change in the Bolivian highlands and its consequences, particularly the availability of water resources for supplying, in a fair manner, an urban agglomeration of over two million inhabitants, with a very high rate of demographic growth. For example, throughout the water crisis from November 2016 to February 2017, decision makers repeatedly stated that this crisis was evidence of the consequences of climate change in the Bolivian highlands (Le [START_REF] Gouill | La crise de l'eau à Paz: de la crise environnementale à la crise technique et politique[END_REF][START_REF] Perales | La crisis de agua en La Paz: cambios y racionamiento de agua[END_REF], and that water management in the urban agglomeration must incorporate this fact, which carries with it reduced availability and greater fluctuation of this resource, as well as a fairer distribution among the population. The wealthier population consumes most of the water, while the poorest do not yet have access.

The Ministry of Environment and Water (MMAyA) uses loans obtained from multilateral banks -such as CAF and IDB-to fund many programs for a clearer understanding of the availability of water resources, including groundwater and the quality thereof (pollution due to heavy metals, urban effluents, agricultural activities), with the aim of taking measures to make the water resources sustainable for human activities. This ministry is also in charge of implementing the National Plan for Access to Drinking Water for 100% of the Bolivian population. With this aim, it has undertaken an aggressive policy both in capturing drinking water and in expanding the drinking water distribution network in urban areas of the La Paz agglomeration which have not yet been served [START_REF] Gouill | La crise de l'eau à Paz: de la crise environnementale à la crise technique et politique[END_REF]. National authorities seek to fulfil the objective of water for all set forth in the 2016-2020 Economic and Social Development Plan, thus demonstrating that they have internalized the notion that a large system for drinking water production and distribution through pipelines must necessarily be the norm of modernity in water supply [START_REF] Hardy | Les coopératives comme alternative technique et sociale? Les services d'approvisionnement en eau dans l'agglomération de La Paz: entre discriminations socio-spatiales et recherche d'efficacité[END_REF].

However, observations and surveys conducted in the urban agglomeration of La Paz in the first half of 2019 indicate that although the drinking water supply by means of the large system for drinking water production and distribution through the mains network [START_REF] Hardy | L'auto-organisation de la gestion urbaine de l'eau. La fonction des coopératives dans le système d'approvisionnement en eau de La Paz et El Alto (Bolivie)[END_REF] improved between the censuses of 2001 and 2012 -with a coverage rate of over 90% of households in La Paz and almost 89% in El Alto in 2012-numerous households continue to be excluded from the service because of their location in areas without coverage and/or due to a lack of financial resources for accessing it. This situation forces those residents to develop alternatives to the piped network for obtaining water; these alternatives correspond to the different uses made of this resource in homes. Seeking water based on its different uses facilitates access, as it implies more dispersed efforts and less energy invested in the search: small quantities need to be found for a great variety of uses. The efficiency in the strategy of access to water is enhanced. Given these conditions, this article seeks to demonstrate that the alternatives to a large system for drinking water production and distribution constitute a powerful tool for integrating sectors considered marginal into the city, and may even be pioneering elements in resilient management of water for a city located in a semi-arid zone.

The aim is also to show that these alternatives, which respond to the rationale of infrastructures outside the network [START_REF] Coutard | Vers l'essor de villes "post-réseaux": infrastructures, innovation sociotechnique et transition urbaine en Europe[END_REF] enable -contrary to the conclusions reached by decision makers-a better integration of the population, facilitating what the La Paz centralized traditional network does not allow (Coutard, 2005).

The first part of the article will present the situation of access to water in the La Paz agglomeration, considering its evolution over a decade. The second part will show how many households of the agglomeration that do not have access to drinking water from the network of the large system develop alternatives based on the different uses of water. The third part will focus on the advantages provided by the alternatives of consuming water in a more resilient manner in an agglomeration located in a semi-arid zone, changing the notion of modernity of the large system for drinking water supply imported from the Western world and pursued by the national authorities. The coverage of access for the inhabitants of the urban agglomeration of La Paz to drinking water through the network increased sharply between the 2001 and 2012 Censuses. In 2012, the coverage rate reached 91.66% in the municipality of La Paz and 88.35% in El Alto (INE, 2012), displaying a growth from 85.7% and 81% respectively compared to 2001 [START_REF] Hardy | The social conditions of self-organized utilities: water cooperatives in La Paz and El Alto, Bolivia[END_REF]. These figures indicate that despite its shortcomings (Hardy, 2009), the large system for the supply of drinking water of the La Paz conurbation, managed by a public company [START_REF] Hardy | L'auto-organisation de la gestion urbaine de l'eau. La fonction des coopératives dans le système d'approvisionnement en eau de La Paz et El Alto (Bolivie)[END_REF], is capable, under normal operating conditions, to provide a large amount of good quality water for human consumption. Despite an increase of 30.1% in the urban population of El Alto from 2001 to 2012 (INE, 2012), the La Paz public company for the production and distribution of drinking water was able to expand the network and its coverage, which led to a greater production of drinking water, demonstrating the system's high degree of efficiency. This efficiency of the large system for drinking water production and distribution was made possible by very costly investments for collecting even more drinking water (Map 1) and for expanding the distribution network to connected homes without increasing connection tariffs or the cost of water per cubic meter sold to the households.

Map 1. Production and distribution of drinking water

Since 2006, the Bolivian governments have set the goal of 100% access to drinking water for the Bolivian population. To reach this objective, they have promoted a policy of expanding the networks of water supply companies (Ministry of Planning, 2015) and of the Public and Social Water and Sanitation Company (EPSAS S.A.) in the area comprising the urban agglomeration of La Paz. To support the water supply companies, in its 2016-2020 National Development Plan, the Bolivian government has planned an investment of 10% of the national budget in the water sector, amounting to approximately USD 4.8 billion (Ministry of Planning, 2015). Added to this amount are the investments in the water sector provided by the bilateral (Venezuela, etc.) and multilateral (European Commission, etc.) donor community. In April 2019, the Minister of Environment and Water, Carlos Ortuño, one of the chief ministers of Evo Morales's government, informed that close to 2 billion euros had been invested in access to water for the Bolivian people (Página Siete, 8 April 2019). These investments have been fruitful despite the fact that their effects take time to materialize due to the limited institutional and technical capacities of the main actors in the country [START_REF] Gouill | La crise de l'eau à Paz: de la crise environnementale à la crise technique et politique[END_REF][START_REF] Agramont | Transdisciplinary learning communities to involve vulnerable social groups in solving complex water-related problems in Bolivia[END_REF][START_REF] Perales | La crisis de agua en La Paz: cambios y racionamiento de agua[END_REF], i.e. the national (Ministry of Environment and Water) and local (EPSAS) entities: they have generated investment plans and delivered the initial achievements. For the urban agglomeration of La Paz, the main achievement has been the expansion of the drinking water distribution network in El Alto, and, following the water crisis from November 2016 to February 2017, new infrastructure, including new water catchments and reservoirs [START_REF] Gouill | La crise de l'eau à Paz: de la crise environnementale à la crise technique et politique[END_REF]. In April 2019, Evo Morales, President of the Plurinational State of Bolivia, inaugurated the Pampalarama reservoir, built in the basin of the Kaluyo stream, which will presumably ensure water for covering the needs of the municipality of La Paz for the next 30 years. In July 2018, the water purification plant of Chuquiaguillo was inaugurated (Página Siete, 14 July 2018). In El Alto, groundwater exploitation through the drilling of new wells helps to unobtrusively expand the large system for the distribution of drinking water to cover the demand of this city (Página Siete, 8 April 2019).

The strategy of these investments has been and continues to be to increase the volume of drinking water collected; it ignores the different scientific studies published, the results of which converge and warn about a reduction of the quantity and quality of the water available in the region of the urban agglomeration of La Paz in the medium term [START_REF] Hunt | Climate change impacts and adaptation in cities: a review of the literature[END_REF][START_REF] Paucara | Efectos del cambio climático sobre la disponibilidad de agua y los recursos hídricos en Bolivia: pronósticos para el 2030[END_REF][START_REF] Perales | La crisis de agua en La Paz: cambios y racionamiento de agua[END_REF].

Areas with no network-based water coverage

Without downplaying the progress mentioned above, according to the 2012 Census, a portion of the population does not yet have access to drinking water through the large system of production and distribution of the urban agglomeration. Several reasons explain this fact.

The pace of expansion of the urban agglomeration has at times exceeded that of the large system: the zones become urbanized and populated in an unplanned manner and faster than the technical possibility of installing drinking water distribution networks.

Map 2. Households with no connection to the public drinking water network Map 2 is based on the data of the 2001 Census due to the non-existence of an equivalent level of information at the neighborhood level with data from the 2012 Census. The map shows that in the peripheral areas of the urban agglomeration, both in El Alto and La Paz, the most recently developed neighborhoods are less connected to the drinking water distribution network. Although the rates of connection of homes to the drinking water distribution network improved from 2001 to 2012, reaching 91.66% in the municipality of La Paz in 2012 and 88.35% in El Alto (INE, 2012), it is extremely likely that the peripheral areas continue to be the least connected urban areas. National highway No. 2 linking El Alto with Batallas and then Lake Titicaca is an example of the present situation. The new buildings stretch for kilometers from El Alto to Batallas (along the highway) before the housing development becomes denser in depth (perpendicular to the highway). This unplanned urbanization is occurring at a pace quicker than network installation. And this pattern of urban expansion is arc-shaped going from west to south, from El Alto to the highlands, along the main highways. In a workshop organized by CIDES in March 2018, municipal authorities of El Alto admitted to an average rate of unplanned urbanization of approximately 16 hectares (0.16 km²) per week.

What is more, when the drinking water systems are able to keep up with the pace of urbanization, people must be able to free up resources in order to connect to them. The public company EPSAS offers three tariffs for connecting to the distribution network. Despite the criticism of Aguas del Illimani during the Water War of El Alto, the nationalized company continues to offer a regulated tariff in peripheral urban areas with a matching contribution from users in terms of the connection work. The rate is EUR 91.50 (BOB 698.52 2 ). And for more consolidated urban zones, the connection tariff ranges from EUR 101.10 (BOB 771.62) to EUR 133.37 (BOB 1,018.08), equivalent to between onehalf and one-third of the monthly minimum wage (BOB 2,122 or EUR 278 in 2019). Many households in the La Paz conurbation simply do not have the financial resources to pay for this connection or the drinking water consumption invoices. In such cases, the residents must seek alternative supplies of water (drinking water or not).

Residents seeking alternatives to the drinking water distribution network 2.1. Some examples of alternatives, providing drinking water savings

Although the state of household access to the drinking water distribution network in the urban agglomeration of La Paz has improved considerably over the past decade, many homes do not yet have connections or have poor connections, which is partly attributable to the pace of growth of urbanization and population. Given this situation, the residents get organized, seek and find alternative supplies of water for drinking and other uses. They use small amounts of potable water for drinking, through the purchase of bottled water -the market for which is growing considerably, keeping pace with urbanization-and large amounts of unsafe drinking water for domestic uses such as laundry, for which it is not necessary that the water be drinkable. Numerous studies have been done in several districts of La Paz and El Alto (Map 1) on the cooperatives that supply water as alternatives to the large system [START_REF] Hardy | Les coopératives comme alternative technique et sociale? Les services d'approvisionnement en eau dans l'agglomération de La Paz: entre discriminations socio-spatiales et recherche d'efficacité[END_REF], 2014, 2016, and 2017). Consequently, this article seeks to explore other alternatives.

New surveys have revealed that due to the lack of resources for having an own individual connection, some homes located in neighborhoods served by the drinking water distribution network continue resorting to the collective taps installed in their neighborhoods. This is the case, for example, of the district of Ovejuyo in La Paz. To illustrate this variety of alternatives, the present article paid particular attention to the mentioned neighborhood, where some analyses have already been performed in recent years [START_REF] Vettraino | aux confins de La Paz (Bolivie). Quelle intégration à la ville pour des périphéries urbaines lointaines?[END_REF]. This district was one of the zones with few connections to the large system for drinking water distribution in 2001 (Map 1). Although the situation in terms of connections to the large system has improved, the residents of this peripheral district belong to low-income population segments. Moreover, the distance from the district to the center of the city, where health services and employment are to be found, forces them to invest more in transportation, at the expense of other services such as access to an individual connection to the drinking water distribution network. They thus organize to pay for connections to the EPSAS drinking water distribution network through collective intake devices (Photo 1).

Photo 1: Residents of the peripheral district of Ovejuyo, La Paz, who are connected to the EPSAS drinking water network obtain drinking water from a public tap because they cannot afford an individual connection (S. Hardy, 2019) Sometimes the residents live in areas served by the drinking water distribution network, but the costs of connection and drinking water consumed invoiced by EPSAS exceed the household resources available, leaving these residents with no solution but to seek alternatives to the water supply network. It has been seen in the district of Ovejuyo that many households have rainwater harvesting strategies, storing this water in containers with different capacities, according to their financial possibilities. Some residents even have tanks to store several thousand liters (Photo 2).

Photo 2: Rainwater harvesting strategy as an alternative to the drinking water distribution network (S. Hardy, 2019) Water collected in reservoirs can in some cases cover the drinking water needs of the poorest households of the district, particularly during the rainy season, when water is abundant and is replenished on a daily basis; however, generally and with greater frequency, it is employed for domestic uses, such as cleaning the home's courtyard or laundry, for which good quality water is required, but it need not be safe drinking water.

Through observations and surveys done in the district of Ovejuyo, the conclusion is that the population of the urban agglomeration of La Paz is developing water access strategies outside the EPSAS large system of distribution. Evidently, the water obtained is not always drinkable, but this is not necessary, as it suffices that water for daily domestic uses be of good quality. Households that resort to these strategies save money and inadvertently participate in conserving the water resource of the urban agglomeration through the consumption of water available in the vicinity.

The variety of alternatives: between resourcefulness and limited resources 3.1. Alternative strategies of access to water for the poorest households

A study performed in the La Paz district of Ovejuyo brought to light the alternatives for access to water employed by the poorest households due to a lack of connection to the EPSAS large system for drinking water distribution or simply because they have no money to pay for the service of the water network.

It is possible to reverse the significance of these observations. The use of alternatives to the drinking water distribution network is the result of a specific perspective: that of the observer who has always had access to the drinking water distribution network. However, many groups of inhabitants of the recently urbanized urban agglomeration have never had access to the network. For these residents, the large system for drinking water distribution is the most convenient and safe alternative to what has always been their daily access to water. Nonetheless, these alternatives are interesting for the observer, from the angle that through them an urban agglomeration such as La Paz can become more resilient in its urban water management.

For example, some residents of the urban agglomeration use, since they have no access to the distribution network and/or no money, the water from the numerous streams crossing the city to do laundry. The quality of that water somehow leaves much to be desired; however, some adaptations, such as diversions of the water flow patterns to pockets that filter the water, allow having water of sufficient quality for some domestic uses such as washing clothes. This can be observed in many parts of the urban agglomeration of La Paz (Photo 3). Based on the quantities of laundry and the interviews conducted, we can conclude that there is considerable laundry being done in the city making use of these facilities as alternatives to individual washing machines that consume a great deal of drinking water.

Photo 3: Washerwomen using water from the Choqueyapu River, filtering the water into rocky pockets to obtain water which is not drinkable but of sufficient quality for washing large amounts of clothes (S. Hardy, 2019) The transportation sector is another important user of good quality water. The establishment of a massive transportation system in the urban agglomeration of La Paz is relatively recent; it took place in the years 2014-2015 with the introduction of the municipal bus systems: Pumakatari in La Paz and Wayna Bus in El Alto, and with the start of operations of the urban cable car network between El Alto and La Paz. This system is however far from sufficient for covering all the transportation needs of the urban agglomeration's population and has not replaced the previously existing system which continues to operate in parallel with the mass transport system. This private system is made up of a great number of minibuses, large buses, shared and individual taxis, managed by cooperatives covering routes negotiated with the municipal authorities [START_REF] Núñez-Villalba | El conocimiento de la vulnerabilidad de la red vial como herramienta de compresión y reducción de la vulnerabilidad territorial: el caso de La Paz (Bolivia)[END_REF]. This privately-owned automobile fleet is made up of tens of thousands of vehicles that travel several dozens of kilometers each throughout the day. Their maintenance demands daily cleaning, which means they are large water consumers. These washing operations do not require the use of drinkable or good quality water. Besides, such use of drinking water would make this maintenance task very costly, affecting the profitability of the transport activity. Hence, it is common to see the drivers of the private transportation system of the urban agglomeration grouped together around springs, washing their vehicles. They can be seen for example on the street connecting Avenida Libertador with Avenida Saavedra (Photo 4) and on adjacent streets.

Photo 4: A taxi and a fleet of minibuses on a La Paz street using water from a spring for their daily vehicle washing operation (S. Hardy, 2019) The urban agglomeration is also a national transportation hub, generating a considerable amount of transport activity via inter-departmental and international highways. Trucks crossing Bolivian and international highways (such as the one connecting to the port of Arica) also require washing, and an activity of washing heavy trucks has been structured in response to this demand. This activity depends less on water from springs and more on the rivers from which non-drinking water is pumped for industrial cleaning, as is the case of the Seco River in El Alto or the Irpavi River [START_REF] Pellicer | Notes de travail de terrain[END_REF] 3 . This free water generates income for those exploiting it, and because the services offered by these washing stations have a better quality-price relationship than those offered by stations employing water from the network, they benefit users. Tens of thousands of vehicles consume water from these sources of non-drinking water on a daily basis and, therefore, do not affect the drinking water supply of public company EPSAS, which then has more water for drinking water purposes.

The municipality of La Paz, a massive promoter of alternatives to the network

The municipality of La Paz is an institutional network user. It requires good quality water for its office buildings, schools and health centers. That water is obtained, for example, directly through the transfer of central powers of the State to the municipalities. But there is also a broad range of uses that do not require drinking water. And the alternatives to the network reduce the amounts paid by the municipality to EPSAS.

Thus, the municipality is making increasing use of spring water. The many water supply crises that have affected the urban agglomeration of La Paz (Hardy, 2009 and[START_REF] Hardy | Évolution des risques d'origine naturelle et de leur gestion à La Paz (Bolivie)[END_REF][START_REF] Gouill | La crise de l'eau à Paz: de la crise environnementale à la crise technique et politique[END_REF][START_REF] Perales | La crisis de agua en La Paz: cambios y racionamiento de agua[END_REF] in recent years have undoubtedly contributed to this change in water supply practices in the area of the municipality of La Paz.

Every day, including Sundays, a large number of tank trucks belonging to the municipal government of La Paz or rented by it collect water using electric pumps at springs. One of these springs is located, for example, at Avenida Hernán Siles Zuazo, south of the La Salle school (Photo 5). There is another such location on Avenida Del Poeta, next to the spring at the Grotto of the Virgin of Lourdes. Having collected the water, these tank trunks then distribute this resource in the city, watering parks and gardens, providing water for municipal public works sites, etc. For these uses, the water must be of good quality, but it must not necessarily be safe drinking water.

Photo 5: A motor pump used to fill a tank trunk with non-drinking water from a spring which is then used by the municipal government of La Paz for city uses requiring good quality water (S. Hardy, 2019).

The municipal authorities of La Paz are aware of the issues related to the alternatives to the drinking water distribution network. In the first place, because of the lower cost, and in the second place, and increasingly so, because of the exploitation of good quality resources for uses not requiring drinking water. Thus, in 2019, a municipal ordinance was passed to improve the water collection from the spring at the Grotto of the Virgin of Lourdes and to prohibit the private transport sector from exploiting this resource. The importance of this alternative for managing a city in a semi-arid zone seems to be slowly finding its way into people's minds [START_REF] Lorrain | Villes sobres: Nouveaux modèles de gestion des ressources[END_REF].

Conclusion

Bolivian national authorities have incorporated the notion that the large system for drinking water production and distribution through the mains network -a Western model inherited from the hygienist movement-represents modernity and is the model to be followed to grant 100% of the Bolivian population access to drinking water. Given this conception, the national authorities do not consider alternatives to this model, which would be interesting in order to promote more resilient water management in the Bolivian cities. Water uses differ, and certain alternatives are sometimes innovating solutions adapted to local conditions on the ground, to the limited ability to pay on the part of certain users, and to the lack of property deeds [START_REF] Criqui | Attention! Travaux en cours: L'extension des réseaux de services essentiels dans les quartiers irréguliers de Delhi et Lima[END_REF]. Often these alternatives are perceived as second-class solutions for second-class citizens, ignoring that on the contrary, they reflect the resourcefulness of residents to preserve a resource and make cities more resilient.

Added to this, in the context of La Paz, where water quantity and quality problems, the exploitation of certain alternatives, for uses in which the water quality should be good, could have a positive effect on the general quality of this resource by making its production less costly.

Taking into consideration the water resource alternatives to the present network, which are already being partially used by some residents for certain uses, in the context of climate change, could open up a range of water supply solutions for all.

1.

  Who are those accessing drinking water, and where, in the urban agglomeration of La Paz? 1.1. Better access to drinking water in the urban agglomeration of La Paz

Research officer at the Institute of Research for Development (IRD), UMR 8586 PRODIG. sebastien.hardy@ird.fr

At August 2019, BOB 1 equalled EUR 0.13.


Karine Altisen

Stéphane Devismes

Anaïs Durand

Colette Johnen

Franck Petit

Brief Announcement: Self-stabilizing Systems in Spite of High Dynamics

Keywords: Self-stabilization, time-varying graphs, leader election, speculation

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Starting from an arbitrary configuration, a self-stabilizing algorithm [START_REF] Edsger | Self-stabilizing systems in spite of distributed control[END_REF] makes a distributed system reach within finite time a configuration from which its behavior is correct. Essentially, self-stabilizing algorithms tolerate transient failures, since by definition such failures last a finite time (as opposed to crash failures, for example) and their frequency is low (as opposed to intermittent failures). Even though self-stabilization is not inherently suited to handle intermittent and permanent failures, several works show that in many cases self-stabilization can still be achieved despite such faults occur. Indeed, strong forms of self-stabilization have been proposed to tolerate permanent failures, e.g., fault-tolerant selfstabilization [START_REF] Beauquier | On FTSS-solvable distributed problems[END_REF] to cope with process crashes, and strict stabilization [START_REF] Nesterenko | Dining philosophers that tolerate malicious crashes[END_REF] to withstand Byzantine failures. Furthermore, several self-stabilizing algorithms, e.g., [START_REF] Delaët | Self-stabilization with r-operators revisited[END_REF], withstand intermittent failures such as frequent lost, duplication, or reordering of messages. All these aforementioned works assume static communication networks. Nevertheless, self-stabilizing algorithms dedicated to arbitrary network topologies tolerate, up to a certain extent, some topological changes. Precisely, if topological changes are eventually detected locally at involved processes and if the frequency of such events is low enough, then they can be considered as transient faults. Actually, a number of works, e.g., [START_REF] Ajoy | Self-stabilizing leader election in dynamic networks[END_REF], use this kind of argument to claim that they are suited for the dynamic context. Furthermore, several approaches, like superstabilization [START_REF] Dolev | Superstabilizing protocols for dynamic distributed systems[END_REF] and gradual stabilization [START_REF] Altisen | Gradual stabilization[END_REF], aims at additionally providing specific countermeasures to efficiently treat topological changes when they are both spatially and timely sparse. However, all these aforementioned approaches, e.g., [START_REF] Altisen | Gradual stabilization[END_REF][START_REF] Ajoy | Self-stabilizing leader election in dynamic networks[END_REF][START_REF] Dolev | Superstabilizing protocols for dynamic distributed systems[END_REF], become totally ineffective when the frequency of topological changes drastically increase, in other words when topological changes are intermittent rather than transient. Actually, in the intermittent case, the network dynamics should be no more considered as an anomaly but rather as an integral part of the system nature.

Contribution. We study self-stabilizing leader election in highly dynamic identified message-passing systems where the dynamics is modeled using Time-Varying Graphs [START_REF] Casteigts | Time-varying graphs and dynamic networks[END_REF] (TVGs, for short) to obtain solutions tolerating both transient faults and high dynamics. We consider three wide classes of TVGs, respectively denoted by T C B (∆), T C Q (∆), and T C R , where

T C B (∆) ⊆ T C Q (∆) ⊆ T C R : T C B (∆)
is the class of TVGs with temporal diameter bounded by ∆ [START_REF] Gómez-Calzado | A connectivity model for agreement in dynamic systems[END_REF], T C Q (∆) is the class of TVGs with temporal diameter quasi-bounded by ∆ (introduced here), and T C R is the class of TVGs with recurrent temporal connectivity [START_REF] Casteigts | Time-varying graphs and dynamic networks[END_REF].

We first study conditions under which our problem can be solved. Actually, our results show that the assumption on the knowledge of the number n of processes is central. To see this, we introduce the notion of size-ambiguity, which formalizes the fact that some subsets of processes do not share enough initial knowledge on n to detect that the system is not limited to themselves. In other words, such an ambiguity comes from the fact that n is only partially known by the processes. Our results show that, despite the existence of unique process identifiers, any deterministic self-stabilizing leader election algorithm working in the class T C Q (∆) or T C R cannot be size-ambiguous. Hence, to make the problem solvable in those classes, we assume each process knows exactly n.

We then propose self-stabilizing leader election algorithms for the three considered classes. In more detail, we present a self-stabilizing leader election algorithm for Class T C B (∆) with a stabilization time of at most 3∆ rounds, assuming every process knows ∆, yet using no information on n. This in particular shows that our necessary condition is tight. Then, we propose a self-stabilizing leader election algorithm for Class T C Q (∆) assuming every process knows ∆ and n. In general, stabilization time cannot be bounded in T C Q (∆); nevertheless its stabilization time in T C B (∆) is at most 2∆ rounds. In other words, this algorithm is speculative (in the sense of [START_REF] Dubois | Introducing speculation in self-stabilization: an application to mutual exclusion[END_REF]) since it exhibits better performances in a subset of more probable executions. Finally, we propose a self-stabilizing leader election algorithm for Class T C R , where only n is known, yet requiring unbounded local memories. Again, in general, stabilization time cannot be bounded in T C R , yet the algorithm is speculative since its stabilization time in T C B (∆) is at most ∆ + 1 rounds.

Roadmap. Due to the lack of space, we only present here our three algorithms. All our results are available in an online technical report. 1 In Section 2, we briefly present the model. In Section 3, we outline our algorithms. We give perspectives in Section 4.

Preliminaries

Time-varying Graphs. A time-varying graph [START_REF] Casteigts | Time-varying graphs and dynamic networks[END_REF] (TVG for short) is a tuple G = (V, E, T , ρ) where V is a set of nodes, E is a set of arcs between pairwise nodes, T is an interval over N * , and ρ : E × T → {0, 1} is the presence function that indicates whether or not a given arc exists at a given time. We denote by o T = min T the first instant in T . The snapshot of G at time t ∈ T is the graph G t = (V, {e ∈ E : ρ(e, t) = 1}). A journey is a sequence of ordered pairs J = (e 1 , t 1 ), (e 2 , t 2 ), . . . , (e k , t k ) where ∀i ∈ {1, . . . , k}, e i = (p i , q i ) ∈ E satisfies ρ(e i , t i ) = 1 and i < k ⇒ q i = p i+1 ∧ t i < t i+1 . The temporal length of a journey J = (e 1 , t 1 ), (e 2 , t 2 ), . . . , (e k , t k ) is equal to t k -t 1 + 1. The temporal distance from p to q at time t is the first arrival time of a journey from p to q whose starting time is at the earliest time t. The temporal diameter of G, is the maximum temporal distance between any two nodes of G at any time.

TVG Classes. We consider the following TVG classes. Class T C R (Recurrent Temporal Connectivity): at any point in time, every node can reach all the others through a journey. Class T C Q (∆) with ∆ ∈ N * (Quasi Bounded Temporal Diameter): at any point in time, every node can eventually reach each other node through a journey of temporal length at most ∆. Class T C B (∆) with ∆ ∈ N * (Bounded Temporal Diameter): at any point in time, the temporal distance between any pair of two nodes is at most ∆, i.e., the temporal diameter is bounded by ∆.

Notice that,

∀∆ ∈ N * , T C B (∆) ⊆ T C Q (∆) ⊆ T C R .
Computational Model. We assume a distributed system made of a set of n processes, denoted by V . Each process p has a unique identifier denoted by id(p) and taken in an arbitrary domain IDSET totally ordered by <. We denote by the process of minimum identifier. A distributed algorithm A is a collection of n local algorithms A(p), one per process p ∈ V . The state in A of each process p ∈ V is defined by the values of its variables in A(p). Some variables may be constants in which case their values are predefined. A configuration of A for V is a vector of n components (s 1 , s 2 , . . . , s n ), where s 1 to s n represent the states of the processes in V .

Processes execute their local algorithms in synchronous rounds. For every i > 0, the communication network at Round i is defined by G o T +i-1 , i.e., the snapshot of G after i -1 instants elapse from the initial time o T . For any (synchronous) round i ≥ 1, the system moves from the current configuration γ i-1 to some configuration γ i . Such a move is atomically performed by every process p ∈ V according to the following three steps, defined in its local algorithm A(p): (1) p sends a message consisting of all or a part of its local state in γ i-1 , (2) p receives all messages sent by its neighbors in G o T +i-1 , and (3) p computes its state in γ i .

Self-stabilizing Leader Election. In identified networks, leader election usually consists in making the processes agree on one of the identifiers held by processes. The identifier of the elected process is stored at each process p in an output variable, denoted here by lid(p). In the self-stabilizing context, variables lid may be initially corrupted; in particular some of them may be initially assigned to fake IDs, a value v ∈ IDSET such that v is not assigned as a process identifier in the system. Despite such fake IDs, the goal of a self-stabilizing algorithm is to make the system converge to a configuration from which a unique process is forever adopted as leader by all processes. Our solutions always elect the process of lowest ID, . To that goal, our algorithms implement two main mechanisms: one for removing all fake IDs from the system, and the other for computing the minimum value among the remaining IDs.

Algorithms

Algorithm for T C B (∆). Recall that we assume every process knows ∆. Then, each process p maintains two variables: the output lid(p) eventually contains the ID of the leader; ttl(p) represents the degree of mistrust of p in lid(p) and allows to eliminate messages containing fake IDs. The value ttl(p) increases at each round if p does not receive a message; otherwise it is updated thanks to the received messages. ttl(p) can increase up to 2∆ -1. Process p never increases ttl(p) from 2∆ -1 to 2∆; instead it locally resets and declares itself as the leader: lid(p) := id(p) and ttl(p) := 0.

At each round i, p first sends its leader ID together with its degree of mistrust. Then, p selects the received message id, ttl which is minimum using the lexicographic order (i.e., the message with the lowest ID and with the lowest ttl to break ties, if any). If id is smaller than lid(p), p updates its leader lid(p). If id = lid(p), it updates the ttl(p) by taking the smallest value between ttl(p) and ttl (in this way, p may decrease its mistrust in lid(p)). In either case, ttl(p) is then incremented if lid(p) = id(p). Finally, if lid(p) ≥ id(p), p systematically resets. So, if p believes to be the leader at the end of Round i (i.e., lid(p) = id(p)), then it sends its own ID together with a degree of mistrust 0 at the beginning of the next round, i + 1.

The resets allow to remove all fake IDs within at most 2∆ rounds. From that time, (lid( ), ttl( )) = (id( ), 0) forever. So, after 2∆ rounds, sends id( ), 0 at each round and all processes will receive messages id( ), d , with d ≤ ∆ < 2∆ (since ∆ ∈ N * ), at least every ∆ rounds since the temporal diameter is upper bounded by ∆. Thus, within at most ∆ additional rounds, they will all adopt as leader and never more reset, ensuring that will remain the leader forever. Hence, this algorithm is a self-stabilizing leader election algorithm for T C B (∆) and its stabilization time is at most 3∆ rounds.

Algorithm for T C Q (∆). We assume now that every process knows both ∆ and n. Then, each process p uses a variable members(p) to collect IDs. Actually, members(p) is a FIFO queue containing at most n pairs id, t , where id is an identifier and t is a timestamp, i.e., an integer value less than or equal to ∆. At each round i, p sends all pairs id, t of members(p) such that t < ∆ at the end of Round i -1. (The timestamps allow to eventually remove all fake IDs.) Then, p updates members(p) by calling function insert on each received pair id, t such that id = id(p).

The insertion function insert works as follows: if id already appears in members(p), then the old pair tagged with id is removed first from the queue and in either case, id, t is appended at the tail of the queue. In particular, since the size of members(p) is limited, if the queue is full, its head is removed to make room for the new value. Using this FIFO mechanism, initial spurious values eventually vanish from members(p).

After all received pairs have been managed, the timestamps of all pairs in the queue are incremented and then, id(p), 0 is systematically inserted at the tail of the queue. This mechanism ensures two main properties. First, every timestamp associated to a fake ID in a variable members is eventually forever greater than or equal to ∆; and consequently, eventually no message containing fake IDs is sent. Second, by definition of T C Q (∆), for every two distinct processes p and q, there are journeys of length at most ∆ infinitely often, so each process p regularly receives messages containing id(q) with timestamps smaller than ∆. Thus, eventually members(p) exactly contains all IDs of the networks. Now, at the end of each round, p updates its leader variable with the smallest ID in members(p). Hence, the process of lowest ID, , is eventually elected. Thus, this algorithm is a self-stabilizing leader election for T C Q (∆).

In general, stabilization time cannot be bounded in T C Q (∆). Yet, if our algorithm is deployed on a TVG of class T C B (∆), then after ∆ rounds, processes no more receives messages with fake IDs. For that point, the identity of every process will be inserted into queue of all processes within at most ∆ rounds, which ensures stabilization time of at most 2∆ rounds. Therefore, this algorithm is speculative.

Algorithm for T C R . Similarly to the previous algorithm, each process p uses a variable members(p) to collect IDs. However, this time, only n is known and members(p) is a map that can contain up to n IDs, each of them being associated with a timestamp.

At each round i, p sends the content of members(p). Then, p updates members(p) by calling function insert on each received pair id, t such that id = id(p). The function insert works as follows: if id already appears in members(p), then the associated timestamp is updated by keeping the smallest value. Otherwise, p tries to insert id, t in the map. Actually, id, t is inserted in the map if the map is not full or t is smaller than the greatest timestamp in the map. In this latter case, id, t overwrites any value having this timestamp in members(p). This overwriting mechanism allows to eventually remove all fake IDs from members(p), since their timestamps will regularly increase. After members(p) has been updated, all timestamps of members(p) are incremented and then, id(0), 0 is systematically inserted in the map.

Actually, our algorithm guarantees two main properties. First, at the beginning of any round i, any timestamp associated to a fake ID is greater than or equal to i -1. Second, by definition of T C R , at any point in time, every process can reach all the others through a journey. The key property is then to show that if some broadcast initiated by process p reaches a process q at Round i, then the value of the timestamp in the message is small enough to ensure the insertion of id(p) into members(q). These two properties ensure that eventually members(p) exactly contains all IDs of the network. Now, at the end of each round, p updates its leader variable with the smallest ID in members(p). Hence, is eventually elected. Thus, this algorithm is a self-stabilizing leader election for T C R .

In general, stabilization time cannot be bounded in T C R . Yet, assume that the algorithm is deployed on a TVG of class T C B (∆). After the first round, every process p sends the pair id(p), 0 at each round. Thus, any process q receives a pair id(p), t with t < ∆ not later than Round ∆ + 1. Now, when this event occurs, each fake ID in members(q) has a timestamp greater than t. So, id(p) is definitely inserted into members(q). Hence, the algorithm is speculative since its stabilization time in T C B (∆) is at most ∆ + 1 rounds.

Perspectives

Beyond extending our results to other particular problems, our future work will focus on studying expressiveness of self-stabilization in TVGs. To that goal, we plan to first investigate broadcast problems, again in very general TVG classes. Indeed, coupled with our leader election solutions, they should allow to build generic transformers, following, for example, the approach proposed in [START_REF] Katz | Self-stabilizing extensions for message-passing systems[END_REF].

https://hal.archives-ouvertes.fr/hal-02376832/

* This study was partially supported by the French ANR projects ANR-16-CE40-0023 (DESCARTES) and ANR-16 CE25-0009-03 (ESTATE).


J P Connolly

C Rohr

Quantum Well Cells for Thermophotovoltaics

Keywords: numbers: 78.66.Fd, 68.65.+g, 85.60.Bk, 73.61.Ey, 81.05.Ea

Quantum well cells (QWCs) are p -i -n photovoltaic devices with multiple quantum wells (MQWs) inserted in the intrinsic region. They show the desirable characteristic for thermophotovoltaics (TPV) of an easily tuneable bandgap. This is highly relevant because the bandgaps of homostructures are restricted by the range of lattice matched substrates available. As a consequence bulk structures cannot be tuned to desirable narrow band emission spectra. QWCs, however, display tuneable bandgaps from several points of view. Firstly, the MQW band-edge is inherently tuneable by changing well depth and dimension. More significantly, strain compensated MQW systems allow lattice mismatched layers to be deposited without the formation of dislocations. This permits a much greater range of compositions and hence band-gaps than are obtainable with lattice matched systems. We first review the fundamental understanding of spectral response and dark current of the QWC and describe the initial proposal of QWCs for TPV before examining subsequent work on lattice matched InGaAsP/InP. Design limitations of the lattice matched materials can be eliminated by applying strain compensation techniques to strained materials. We show dark current measurements for lattice matched and strained systems with dark currents lower than other promising TPV designs despite longer wavelength absorption edges. This work has recently produced a QWC capable of reaching the Holmia narrow band emission peak at wavelength 1950 nm which shows the promise of this approach for TPV.

Introduction

The quantum well cell (QWC) is a photovoltaic device with a multiple quantum wells system (MQW) inserted in the intrinsic region of a p -i -n cell [START_REF] Barnham | A new approach to high-efficiency multi-band-gap solar cells[END_REF]. Carriers photogenerated in the quantum wells (QWs) escape with high efficiency thereby enhancing the short circuit current J sc compared to an identical cell without wells known as a barrier control [START_REF] Paxman | Modeling the spectral response of the quantum well solar cell[END_REF]. A bulk cell with the same bandgap as the quantum wells (known as a well control) can absorb the same range of wavelengths and produce a similar J sc . However the dark current in such a bulk cell is greater than that of the QWC. As a consequence studies of the dark current have suggested that the J sc increase in QWCs may more than compensate the increase in dark current giving rise to highly efficient cells.

QWCs have inherent advantages for TPV due not only to better voltage performance but also to a flexible bandedge applicable to long wavelength operation. Each period comprised of wells and barriers can be designed so that the strain in the former is balanced by strain of opposite sign in the latter in a technique known as strain compensation or strain balancing. This allows growth of cells made from material with a much wider range of compositions for which no lattice matched substrates exist.

QWCs are therefore unique in that they can be tailored for a particular incident light spectrum. Finally, these cells show better temperature coefficients of efficiency with temperature than bulk cells which is of particular interest for TPV [START_REF] Rohr | InGaAsP Quantum Well Cells for Thermophotovoltaic Applications[END_REF][START_REF] Ballard | The effect of temperature on the efficiency of multi quantum well cells for solar and thermophotovoltaic applications[END_REF].

Fundamental studies

Photocurrent enhancement

Early work on QWCs [START_REF] Paxman | Modeling the spectral response of the quantum well solar cell[END_REF] looked at solar applications in Al x Ga 1-x As with GaAs QWs and presented modelled experimental quantum efficiency (QE) of such devices showing the increase in J sc due to the presence of wells. The theory explains this by assuming 100% escape efficiency for carriers photogenerated in the wells. This assumption is reported valid as long as a field is maintained across the wells. The accurate modelling of measured QE confirms the theoretical description of the densities of states in the QWC. This model has been extended to allow for strain effects in the barriers and wells [START_REF] Rohr | Strain-balanced In 0.62 Ga 0[END_REF] using modelling work to be published.

Serdiukova et al [START_REF] Serdiukova | [END_REF] present a model specifically applied to In x Ga 1-x As MQWs for TPV looking at strained wells (x > 0.6) kept below their critical thickness. The work presents a QE model together with dark current calculations and does not include experimental data. The model predicts that J sc enhancements of up to a factor of 2 are possible over the bulk In 0.53 Ga 0.47 As baseline lattice matched to InP, with efficiency increasing from 4.5% to 9% as a result for a 1500K blackbody spectrum. It then predicts very high efficiency limits, for example 42% for a 2000K blackbody spectrum though details concerning this upper limit are incomplete.

The two models show essentially similar results for spectral response when compared for the available results published. The main conclusion is that QWs boost J sc significantly in a way which is well understood in lattice matched and strained materials.

Dark current

The QWC shows promise for photovoltaic applications if the increased J sc due to the QWs is not negated by an increase in dark current.

Experimental and theoretical analysis of the electroluminescence (EL) of GaAs/InGaAs and AlGaAs/GaAs single quantum well (SQW) devices has been presented by Nelson [7]. This analysis suggests that the electron and hole quasi-Fermilevel separation (∆E f ) in the wells is less than the applied bias by a few tens of meV, whereas in the barriers ∆E f is equal to the applied bias. In this model the carrier concentrations in the wells are lower than expected if carrier populations are described by the applied bias. As a result the reduced recombination and hence dark current suggests that the QWC is an efficient photovoltaic design.

Of particular relevance to TPV is a temperature dependent study for double-well AlGaAs/GaAs QWCs [START_REF] Kluftinger | Temperature-dependent study of the quasi-Fermi level separation in double quantum well p-i-n structures[END_REF]. Significantly, ∆E f was found to rise with temperature. It is suggested that this is due to thermally assisted escape from the wells under an applied field, and that the dark current will rise less quickly with increasing temperature than bulk cells. Experimental measurement and analysis of photocurrent and photoluminescence data provide evidence for temperature dependent escape in InGaAs/InP [START_REF] Zachariou | A carrier escape study from InP/InGaAs single quantum well solar cells[END_REF].

Nelson et al [START_REF] Nelson | Effect of QW location on quantum well solar cell dark currents[END_REF] have successfully modelled the dark currents of SQW cells with dark currents dominated by non-radiative recombination in the intrinsic region. The approach outlined in Ref. [START_REF] Nelson | Effect of QW location on quantum well solar cell dark currents[END_REF] has been extended to MQWs subject to the depletion approximation [START_REF] Connolly | Multiple quantum well solar simulator[END_REF] where the authors suggest a reduced ∆E f in the wells to explain the dark currents. Similar results were subsequently found in In 0.53 Ga 0.47 As QWCs lattice matched to InP [START_REF] Connolly | Modelling multi quantum well solar cell efficiency[END_REF]. The value of ∆E f in these MQW cases was higher than previous work on SQW cells though the authors suggest this is an overestimate due to the depletion approximation.

Lattice matched QWCs applied to TPV

The application of QWCs to TPV was first proposed by Griffin et al [START_REF] Griffin | Advantages of quantum well solar cells for TPV Thermophotovoltaic Generation of Electricity: Third NREL Conf[END_REF] and independently by Freundlich [START_REF] Freundlich | Quantum well thermophotovoltaic energy converter[END_REF].

Griffin has noted that available bulk materials cannot be perfectly matched to available spectra and the application of QWCs is suggested, since the bandgap is readily tuneable by adjusting the dimensions of the QWs and their depth. Two QWC systems were studied. The first (QWC-A) comprised 15 In 0.53 Ga 0.47 As wells, the second (QWC-B) 60 quaternary In x Ga 1-x As y P 1-y wells, both lattice matched to InP. When compared to a double-heterostructure control with an In 0.53 Ga 0.47 As i layer under a simulated Ytterbia spectrum, QWC-A showed an increase in V oc by a factor of 1.7. QWC-B was compared to QWC-A and showed an increase in J sc by a factor of 2.4, and no significant loss in V oc .

Further work has examined QWCs for TPV in quaternary In x Ga 1-x As y P 1-y lattice matched to InP [START_REF] Rohr | Optimisation of InGaAsP quantum well cells for hybrid solar-thermophotovoltaic applications Thermophotovoltaic Generation of Electricity: Fourth NREL Conf[END_REF]. The QWC is compared to another promising technology which is the monolithic-interconnected-module (MIM) design. Although the QWC bandgap is slightly greater than the InGaAs MIM cell, it is found that the QWC dark current is orders of magnitude lower (see figure 1).

This leads to predicted efficiencies of the QWC being better than that of the bulk InGaAs MIM and bulk GaSb under many spectra as indicated in table 1 assuming a back surface reflectivity of 100% (see below).

As these QWCs also perform well under solar illumination it is suggested they could be used in hybrid solar-TPV systems [START_REF] Rohr | Optimisation of InGaAsP quantum well cells for hybrid solar-thermophotovoltaic applications Thermophotovoltaic Generation of Electricity: Fourth NREL Conf[END_REF].

Strain compensated cells

The longest wavelength absorption edge possible with the lattice matched system is about 1700 nm corresponding to In 0.53 Ga 0.47 As. However smaller bandgaps are required for lower temperature operation with selective emitters such as Thulia or Holmia having a peak emission at about 1750 nm and 1950 nm respectively. Figure 2 shows the progress in extending the spectral response of the QWC compared with bulk lattice matched In 0.53 Ga 0.47 As as described in this section.

Strain balancing

A subsequent approach to reach longer wavelengths than obtainable with lattice matched material has been investigated. This is the concept of strain compensation or strain balancing [START_REF] Ekins-Daukes | Strainbalanced GaAsP/InGaAs quantum well solar cells[END_REF][START_REF] Ekins-Daukes | Strain-balanced criteria for multiple quantum well structures and its signature in X-ray rocking curves Crystal[END_REF] and should take elastic constants into account, particularly if the strain is high. In a multilayer structure, lattice mismatched layers can be grown with alternating tensile and compressive strain. If the growth is such that the stress at successive interfaces is opposite in sign and equal then an arbitrary number of periods can be grown. For highly strained materials the strain energy must be kept below a critical value, however, to avoid the onset of three-dimensional growth [START_REF] Nasi | Extended defects in InGaAs/InGaAs strain-balanced multiple quantum wells for photovoltaic applications[END_REF]. This constraint and its consequences for well composition and dimensions limit the extension of absorption to about 2100 nm.

In practice the stress balancing puts stringent requirements on growth. Although these practical reasons set a limit on the number of multilayer periods, the structure is primarily constrained by background doping which limits the maximum i region width which can sustain a built in field.

InGaAs bulk

Development

Rohr et al [START_REF] Rohr | Potential of InGaAsP quantum well cells for thermophotovoltaics[END_REF] have presented a successful MQW grown in this manner but balancing the strain without accounting for the different elastic constants in barriers and wells. This work presents modelling comparing the strain compensated QWC with the In x -Ga 1-x As y P 1-y QWC reviewed above, a GaSb and again a MIM device. Comparisons under a wide range of spectra including solar, black body and narrow band emission spectra are given in table 1. In all cases the predicted QWC result outperforms the bulk devices. Under some spectra the earlier lattice matched design shows efficiencies higher than the strain compensated design because of the bandgap of the latter being lower than that required to match the relevant spectrum. Overall this work suggests that a strain balanced QWC with appropriately designed band structure is equal or superior to lattice matched QWCs and to bulk devices.

Experimental results [START_REF] Rohr | Strain-balanced In 0.62 Ga 0[END_REF] support this and extend the balancing by taking account of material elastic stiffness constants. The device grown extends response to slightly longer wavelength than GaSb and significantly further than the InGaAs MIM design. In this case however the QWC result is extrapolated for optimisations. These are reasonable, consisting of using the model to fit the experimental QE and applying an anti-reflection coat and a planar reflecting back surface subsequently seen in other samples [START_REF] Abbott | A comparative study of bulk InGaAs and InGaAs/InGaAs strain-compensated quantum well cells for thermophotovoltaic applications[END_REF][START_REF] Abbott | Characterisation of strain-compensated InGaAs/InGaAs quantum well cells for TPV applications The Fifth Conf. on Thermophotovoltaic Generation of Electricity[END_REF] as discussed below.

The extended response of the strain compensated QWC again outperforms MIM and GaSb cells in all spectra (see table 1). This is mainly due to the lower measured dark current [START_REF] Rohr | Strain-balanced In 0.62 Ga 0[END_REF]. The lattice matched QWC is preferable however in solar and higher energy spectra such as Ytterbia because of its higher bandgap, consistent with the original aim of extending the spectral response of the QWC.

Optimisation for Thulia (1750nm)

Device optimisation has been considered with the initial requirement that the spectrum should be at a moderate temperature which is more desirable from a systems point of view. This means that QWC design has concentrated on extending the long wavelength response to match long wavelength narrow band emitter spectra.

Whereas the early work concentrated on black body, Ytterbia and Erbia spectra, a desire to allow lower temperature combustion has moved attention to Thulia (emission centred on 1750nm) and Holmia (1950nm) spectra.

The strain compensation methodology has been extended to account for the different elastic properties of barriers and wells [START_REF] Ekins-Daukes | Strain-balanced criteria for multiple quantum well structures and its signature in X-ray rocking curves Crystal[END_REF]. Comparisons presented by Abbott [START_REF] Abbott | Characterisation of strain-compensated InGaAs/InGaAs quantum well cells for TPV applications The Fifth Conf. on Thermophotovoltaic Generation of Electricity[END_REF] examine the performance of bulk InGaAs and of QWCs with progressively lower QW bandgaps.

The successful growth and characterisation of a stress balanced QWC comprising 40 wells and extending absorption to 1950nm which covers the Thulia spectrum adequately is reported [START_REF] Abbott | Characterisation of strain-compensated InGaAs/InGaAs quantum well cells for TPV applications The Fifth Conf. on Thermophotovoltaic Generation of Electricity[END_REF]. Transmission electron microscopy (TEM) measurements reveal no visible dislocations originating from the highly strained barriers and wells showing that the stress compensation technique is applicable to a large number of QW periods.

Concerning dark currents, a more moderately strained QWC comprising 30 strain compensated In 0.62 Ga 0.38 As wells with an absorption edge at 1800 nm has a dark current below that of the lattice matched bulk In 0.53 Ga 0.47 As p -i -n cell [START_REF] Rohr | Strain-balanced In 0.62 Ga 0[END_REF][START_REF] Abbott | Characterisation of strain-compensated InGaAs/InGaAs quantum well cells for TPV applications The Fifth Conf. on Thermophotovoltaic Generation of Electricity[END_REF]. While the dark current of the highly strained QWC is higher the authors suggest that this may be partly due to poor quality overgrowth and surface morphology despite the good QW interface quality as shown by the TEM measurements.

Furthermore, reflection from the metallic contact on the back surface plays a significant role in the spectral response of these cells [START_REF] Abbott | A comparative study of bulk InGaAs and InGaAs/InGaAs strain-compensated quantum well cells for thermophotovoltaic applications[END_REF]. The effect is very significant for these structures, since the thin MQW layers cannot absorb all the incident light with a single pass and the number of MQW periods is constrained by the limits imposed on practical i region widths due to unintentional background doping. It follows that light trapping techniques are particularly relevant to these structures. The effectiveness of this technique can easily be improved, and should readily attain reflectivities approaching 100% instead of the 65% observed with appropriate choice of substrates and processing steps to ensure a specular back surface.

Three different indications suggest that the back surface reflectivity is about 65% in the case of Ref. [START_REF] Abbott | A comparative study of bulk InGaAs and InGaAs/InGaAs strain-compensated quantum well cells for thermophotovoltaic applications[END_REF]. Firstly, direct reflectivity measurements appear significantly higher than expected for InP below the bulk and QW bandgaps. Secondly, a multilayer analysis was applied and requires the same back reflectivity to reproduce experimental results. Finally, modelling of the QE including internal reflections requires the same value to reproduce the measured QE in all three samples.

All three cells have similar bulk response although the 30 well QWC is slightly lower. The efficiency under a Thulia spectrum increases from 11% for the lattice matched bulk InGaAs p -i -n to 13.3% for the 1800 nm 30 QW sample and then to 18.1% for the 40 QW sample with a bandedge at 1975 nm despite its relatively poor dark current.

Optimisation for Holmia (1950nm)

The response of the QWC has been extended to wavelengths suitable for systems based on the Holmia emitter which has an emission peak at about 1950nm. In this case the barriers and wells are more highly strained laying higher demands on growth accuracy. A two well sample has been successfully grown and characterised [START_REF] Rohr | Strain-compensated ingaas/ingaas quantum well cell with 2µm band-edge The Fifth Conf. on Thermophotovoltaic Generation of Electricity[END_REF] with a composition of about 74% indium in the wells resulting in strain at about +1.44%.

TEM measurements again show sharp MQW interfaces, and atomic force microscopy measurements show surface morphology within tolerances. The MQW widths are shown to be above their nominal widths but within their critical thicknesses.

Cathodoluminescence measurements show a peak at 2040nm which is consistent with the modelled QE, confirming the accurate description of the MQW band-edges in the modelling. It has therefore been shown that strain compensation can extend the absorption edge beyond 2000 nm.

Carrier escape studies have been performed on lattice matched InGaAs QWCs [START_REF] Zachariou | A carrier escape study from InP/InGaAs single quantum well solar cells[END_REF]. For this highly strained 2 QW sample similar studies show that not all the photogenerated carriers are collected, contributing to poor electrical performance under illumination [START_REF] Rohr | Strain-compensated ingaas/ingaas quantum well cell with 2µm band-edge The Fifth Conf. on Thermophotovoltaic Generation of Electricity[END_REF]. The carrier collection improves with increasing temperature, although it is shown to remain incomplete at 100 • C. The authors speculate that transport problems could be partly responsible. By contrast, in the strain compensated QWC with 40 QWs and an absorption edge of 1975 nm described above, efficient carrier escape has been observed at room temperature and around zero to modest forward bias [START_REF] Abbott | Characterisation of strain-compensated InGaAs/InGaAs quantum well cells for TPV applications The Fifth Conf. on Thermophotovoltaic Generation of Electricity[END_REF].

Conclusions

The work reviewed has shown that the QWC shows desirable characteristics for TPV applications.

Firstly, two methodologies have shown that QWCs show promising voltage behaviour, and studies have shown that carrier escape from the wells is in general good. Of relevance here is the temperature studies carried out to date detailing advantages in the QWC design for operation at higher temperatures.

Secondly, and perhaps more importantly, the QWC design is unique in allowing the bandgap to be tailored and hence making it possible to optimise the spectral matching between TPV cell and light spectrum. This is because the well/barrier lattice opens the possibility of growing strained layers on available substrates.

Work on this design has shown promising results. QWCs have demonstrated dark currents lower than relevant bulk cells with comparable or higher bandgaps. As a result, efficiencies projected for the QWC in a range of spectra are higher than available bulk alternatives.

Figure 1 .

 1 Figure 1. Dark current comparison for lattice-matched and strain-balanced QWC, bulk InGaAs, and GaSb

Figure 2 .

 2 Figure 2. Quantum efficiency data for a bulk In 0.53 Ga 0.47 As cell, lattice matched QWC and two strain compensated QWCs showing the progressively longer wavelength bandedges achieved.

Table 1 .

 1 Comparison of predicted efficiencies (in %) of lattice-matched bulk InGaAs MIM, GaSb, a lattice-matched InGaAsP QWC, and a 1770 nm and a modelled 2100 nm strain-compensated QWC with simulated back-mirror and zero front surface reflectivity, under various spectra normalised to 100 kW/m 2 , and 5 % grid shading[START_REF] Rohr | Strain-balanced InGaAs/InP quantum well cells for thermophotovoltaics[END_REF][START_REF] Rohr | InGaAs/InGaAs strain-compensated quantum well cells for thermophotovoltaic applications[END_REF].

	InGaAsP 1770 nm 2100 nm

The J sc of the QWC to date is constrained by the limited number of periods that can be grown because of restrictions on the i region width. Evidence of light reflections suggest a solution to this issue through having more than one light pass. This phenomenon should be easily optimised by a combination of choice of substrates and processing steps ensuring back reflectivities approaching 100% and structural light trapping schemes ensuring multiple light passes through the structure.

We conclude that the application of the QWC to TPV applications shows promise because of the temperature, voltage, and variable absorption edge characteristics of this design.


James P Connolly

Jenny Nelson

Keith W J Barnham

Ian Ballard

C Roberts

J S Roberts

C T Foxon

Simulating multiple quantum well solar cells

Keywords: Quantum well solar cell, dark current, light current, efficiency

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abstract

The quantum well solar cell (QWSC) has been proposed as a route to higher efficiency than that attainable by homojunction devices. Previous studies have established that carriers escape the quantum wells with high efficiency in forward bias and contribute to the photocurrent. Progress in resolving the efficiency limits of these cells has been dogged by the lack of a theoretical model reproducing both the enhanced carrier generation and enhanced recombination due to the quantum wells.

Here we present a model which calculates the incremental generation and recombination due to the QWs and is verified by modelling the experimental light and dark current-voltage characteristics of a range of III-V quantum well structures. We find that predicted dark currents are significantly greater than experiment if we use lifetimes derived from homostructure devices. Successful simulation of light and dark currents can be obtained only by introducing a parameter which represents a reduction in the quasi-Fermi level separation.

Introduction

The quantum well solar cell (QWSC) [START_REF] Barnham | A New Approach to High Efficiency High-Bandgap Solar Cells[END_REF] consists of a pin structure with nm-wide regions of low bandgap (quantum wells) in the nominally undoped intrinsic region where carriers occupy discrete energy levels. Experiment has shown ( [START_REF] Paxman | Modelling the Spectral Response of the Quantum Well Solar[END_REF], [START_REF] Barnes | Characterisation of GaAs/InGaAs quantum wells using photocurrent spectroscopy[END_REF]) that photo-excited electrons and holes in these quantum wells escape with high efficiency in forward bias and contribute to the photocurrent. If the photocurrent enhancement exceeds the dark current increase this may be a highly efficient solar cell design. Detailed balance arguments in the radiative limit [START_REF] Araujo | Proc. 12th European Photovoltaic Solar Energy Conference[END_REF] have suggested the Voc is determined by the lowest absorption energy which is the well bandgap. Further work [START_REF] Barnham | Jane Osborne voltage enhancement in Quantum Well Solar Cells[END_REF] however indicates that efficiency enhancements are possible over single bandgap designs in non-radiative dominated structures. This has been confirmed by work ([6] [START_REF] Tsui | Determination of the Quasi Fermi level separation in single quantum well p-i-n diodes[END_REF]) which showed smaller quasi-Fermi level (QFL) separation in the wells than in the barriers. This suggests that significant efficiency enhancements are possible with this solar cell design. This paper describes QWSC photocurrent and dark current calculations with a view to establishing whether the effect of enhanced photocurrent overcompensates the dark current losses and secondly modelling the overall device performance. The photocurrent is calculated from the diffusion equation applied to photogenerated minority carriers in bulk and well material and is well understood. The dark current aspect is a 1 Electronic mail: j.connolly@ic.ac.uk Shockley-Read-Hall (SRH) calculation including an ideal diode component. The model examines the extent to which the dark current behaviour of QWSCs can be characterised in terms of the quantum well density of states without considering escape and capture rates from the quantum wells. Furthermore, it explores the extent to which the QWSC dark current is determined by the SRH recombination rates at the centre of the intrinsic region, and for many wells where this methodology is expected to be accurate. This is explored by modelling a range of experimetally measured dark currents in the well characterised GaAs/AlxGa1-xAs system.

QWSC current voltage model

The program "SOL" calculates the incremental light and dark currents due to the quantum wells. It will be shown that the changes in photocurrent and dark current can both be reduced to functions of the quantum well densities of states. The photocurrent model of the QWSC consists of a spectral response calculation as a function of bias, cell dimensions and quantum well energy spectrum. Cell photocurrents can then be calculated for different incident light spectra and power levels. This has been described for the GaAs/AlxGa1-xAs GaAs/InxGa1-xAs InGaPx GaxAsP and InGa0.53xAsxP materials systems [START_REF] Paxman | Modelling the Spectral Response of the Quantum Well Solar[END_REF], [START_REF] Barnes | Characterisation of GaAs/InGaAs quantum wells using photocurrent spectroscopy[END_REF]. The dark currents measured in QWSC structures in these materials show idealities close to 2. This is characteristic of SRH recombination in the depletion layer. The dark current model applied here is a simplified implementation of a self consistent model by Nelson et al [START_REF] Nelson | Effect of quantum well location on single quantum well p-i-n photodiode dark currents[END_REF] which has succesfully described the dark currents of single quantum well cells for a range of single well positions and i region background doping levels. The carrier generation rate G at wavelength λ, position x, incident photon flux F , R the surface reflectivity and absorption coefficient α is defined as

G(x, λ) = F (λ)(1 -R(λ))α(x)e - x 0 αxdx (1)
In the quantum wells the generation is calculated from Fermi's golden rule governing transition probabilities between electron and hole well states. These are found by solving the effective mass equation for electrons and holes confined in the QW within the envelope function approximation. Omitting constants for brevity it is written as

αqw ∼ Nstates jn=1,jp=1 M 2 H(Ejn -Ejp -hω) (2)
where the double sum is over initial and final valence and conduction band states labelled jn and jp, M is the transition Hamiltonian, H is the Heaviside step function delta, Ejn and Ejp are the energy levels of the initial and final quantum well states.

The photocurrent Ji due to the QW at any λ is found by integrating G across the QW layers and assuming 100% escape.

The QW spectral response is added on to that of the bulk regions of the device calculated as in [START_REF] Paxman | Modelling the Spectral Response of the Quantum Well Solar[END_REF].

Figure 1 shows the calculated and measured QE for an Alx-Ga1-xAs QWSC comprising 30 GaAs wells together with the same for a control sample which is identical but without wells. Under an standard AM1.5 spectrum the theoretical short circuit current (Jsc) for this sample is 108Am -2 versus 116Am -2 calculated from the experimental SR. These results agree with calibrated results from NREL to within 3% in the worst case. The SRH [9] recombination rate U (x) in the i region is given by

U (x) = pn -n 2 i τn(p + pt) + τp(n + nt) (3)
where where pt and nt are hole and electron populations in trap levels in the bandgap, p and n are hole and electron populations in the valence and conduction bands and τp and τn are their respective lifetimes.

Electron and hole populations in barrier regions of the intrinsic region are expressed in terms of the intrinsic population ni, Fermi levels E f and the intrinsic energy Ei

n b = nie E f n -E i KT p b = nie E i -E f p KT ( 4
)
where ni is given by the effective masses of electrons and holes via the conduction and valence band effective densities of states and the law of mass action in equilibrium, and for n b the subscript b refers to barrier material. It has been shown [START_REF] Nelson | Effect of quantum well location on single quantum well p-i-n photodiode dark currents[END_REF] that the variation in electron and hole populations in well and barrier regions can be reduced to a difference in effective mass within the parabolic band approximation and a constant factor modifying the conduction and valence band densities of states which is determined by a sum over quantum well allowed states. These are written in terms of two functions θp and θn. Electron and hole populations nq and pq in the wells are then given as follows in terms of the barrier concentrations

nq = n b e θn pq = p b e θp (5)
The θ functions are as follows

θn = kT ln 1 L ( 2 N C ) 1/3 mnq mn N jn=1 e ( Ecq -E jn KT ) θp = kT ln 1 L ( 2 N V ) 1/3 mpq mp N jp=1 e ( E jp -E jq KT ) ( 6
)
where L is the quantum well width. mn,p are electron and hole masses in the barrier and mnq,pq are the masses in the well. These functions contain all information necessary about the quantum wells with the sole exception of carrier lifetimes. We note that they are symmetrical to the well photocurrent enhancement term of equation 2. Both involve a sum over well states and the necessary information is the same as in equation 2 and reduces to band structure data about the well relative to the barriers. The functions will not be symmetrical for electrons and holes because of different valence/conduction offsets and effective masses. The contributions from neutral layers are included in terms of the ideal diode formalism The electron and hole Fermi-levels are set equal to the valence and conduction band edges. The depletion approximation is made with the result that photogenerated carriers are omitted in the dark current calculation. The dark current of the device in this picture is given by the integral of the SRH recombination rate over the depletion region including p and n layer depletion layers if present, incrememented by the ideal diode contribution from the charge neutral regions. An example SRH profile is shown in figure 2.

The quantum wells show are step-like peaks in the profile, and clearly dominate the dark current of a QWSC. Samples were grown by MBE and MOVPE in the Centre for Electronic Materials and Devices in Imperial College, Phillips Laboratories (Redhill) and the Sheffield III-V Semiconductor Facility. They take the form of pin diodes. Structures having GaAs wells in the i region will be referred to as QWSCs and identical pin structures of homogeneous composition grown in the same growth run are termed controls. The experimental configuration has been described previously [START_REF] Paxman | Modelling the Spectral Response of the Quantum Well Solar[END_REF].

The fitting parameters for dark currents are the non radiative lifetimes. Estimates for these are drawn from studies of barrier and well controls which are GaAs/AlxGa1-xAs and GaAs controls. These lifetimes are then applied to QWSC structures and the resulting prediction compared with experiment. Figure 3 shows modelled and calculated IV for GaAs and Alx-Ga1-xAs controls. The upper curves shows a GaAs control.

The model shows SRH and ideal dark currents and their sum.

The onset of radiatively dominated behaviour is visible below the built in voltage of 1.44V and is well described. The parameters for the ideal Shockley fit are the same as those used for the SR leaving the non radiative lifetimes as the only parameters. They are chosen equal in this case. Making them different shifts the SRH peak with position and changes the ideality of the dark current in disagreement with experiment.

The lifetime of 10ns is in close agreement with lifetimes used in previous work [START_REF] Nelson | Effect of quantum well location on single quantum well p-i-n photodiode dark currents[END_REF].

The lower curves shows similar data and theory for an Alx-Ga1-xAs control with a much shorter SRH lifetime of 0.6ns. Again this is in agreement with previous work by the same factor in a different sample. Modelling an identical control with aluminium mole fractions x = 20% yields a lifetime of 0.05ns. Figure 6 shows the same data for a 50 well QWSC with Al fractions 33%. Fitting the data can again be achieved by reducing the QFL separation by the same shift.

We have observed that experiment systematically records a lower dark current than predicted by the control lifetimes. This can be explained in a variety of ways by the model. Longer lifetimes in the wells linearly decrease the dark current until the well lifetime is much longer than in the barrier. However there is no reason why the lifetimes in the QW should be greater than in bulk material. The approach we have investigated is the possibility of a lower QFL separation δE f in the quantum wells. We find that a narrowing of the QWFL separation in the wells by a factor symmetrical for holes and electrons models all samples well, as shown by the dashed lines in graphs 4 to 6 despite the fact that the barrier bandgap varies significantly, and that one sample has 50 wells. This reaffirms work [START_REF] Nelson | Observation of suppressed radiative recombination in single quantum well pin photodiodes[END_REF] on the QFL separation which was found to be significantly smaller in SQW samples at 10meV Fig. 6. 50 well QWSC Al 33% (dashed line modified δE f see text)

Conclusions

The QWSC benefits from an increase in photogeneration in the wells which is counterpointed by recombination in the lower bandgap well regions. In order to study which effect is greater we study photocurrent and dark current and express the modifications to dark and photocurrent in terms of the quantum well density of states. The photocurrent from the wells is determined with no free parameters to good accuracy. The dark current for the control homojunctions is well understood. Modelling these gives us an estimate of carrier lifetimes in the SCR, together with verifying the transport parameters used in the QE calculation at the onset of ideal diode behaviour, if present.

The QWSC dark current depends on four lifetimes. We reduce this to two by assuming that hole and electron non radiative lifetimes are equal. In the absence of direct measurements we derive the barrier lifetimes from controls with the barrier composition. Similarly, controls with the well composition set an upper limit on the well lifetimes employed. We see a systematic overestimation of the dark current, indicating that the well lifetimes in a QWSC system are longer than expected by one order of magnitude to a first approximation. This can be explained in terms of longer well lifetimes or a smaller QFL separation in the wells. This strongly suggests that these structures have the potential to be efficient photoconverters.

Finally we note that this formulation of the MQW dark current problem shows shortcomings because of the approximations made. This is particularly noticeable in the case of single well samples. A full treatment needs a self consistent approach to the Poisson equation. This would be noticeable for single quantum well samples where the position of the quantum wells and the background doping is important. We note however that the treatment applies reliably to MQW systems studied here since dark current is determined mainly by the balance between QW and barrier material, in that on average the position of the wells is not critical. This is borne out by the range of GaAs/AlxGa1-xAs samples examined.

Fig. 1 .

 1 Fig. 1. Experimental and calculated SR of a 30well AlxGa 1-x As sample with aluminium x = 30%

Fig. 2 .

 2 Fig. 2. SRH recombination profile for the 30 well QWSC of figure

Fig. 3 .

 3 Fig. 3. GaAs control dark current and lifetimesFigures 4 and 5 show dark current modelling for 30 well MBE grown QWSCs with Al composition of 20% and 33% and with lifetimes derived from relevant controls assuming similar material quality.Figure6shows the same data for a 50 well QWSC with Al fractions 33%. Fitting the data can again be achieved by reducing the QFL separation by the same shift. We have observed that experiment systematically records a lower dark current than predicted by the control lifetimes. This

Fig. 4 .

 4 Fig. 4. 30 well QWSC Al 20% (dashed line modified δE f see text)

Fig. 5 .

 5 Fig. 5. 30 well QWSC Al 33% (dashed line modified δE f see text)

for a biasses 0.8V, 1.15 and 1.5V, showing different θn and θp and the reduction of SCR width at high bias.