Ismaël Rafaï
email: ismael.rafai@gredeg.cnrs.fr

Sébastien Duchêne
email: sebastien.duchene@umontpellier.fr

Eric Guerci
email: eric.guerci@gredeg.cnrs.fr

Ariane Lambert-Mogiliansky

Fabien Mathy
email: fabien.mathy@unice.fr

A Dual Process in Memory: How to Make an Evaluation from Complex and Complete Information? -An Experimental Study

Keywords: Fuzzy Trace Theory, Memory, Dual Process, Cognitive reflection test, Bounded rationality JEL Classification: C91, D83, D89

In this paper, we will put forward an original experiment to reveal empirical "anomalies" in the process of acquisition, elaboration and retrieval of information in the context of reading economic related content. Our results support the existence of the memory dual process suggested in the Fuzzy Trace Theory: acquisition of information leads to the formation of a gist representation which may be incompatible with the exact verbatim information stored in memory. We give to subjects complex and complete information and evaluate their cognitive ability. To answer some specific questions, individuals used this gist representation rather than processing verbatim information appropriately.

Nowadays, economic agents are overwhelmed by information, either as textual or visual content, which is easily and cheaply available via the Internet or social media. 1 Criteria and descriptions by which individuals use to make decisions are often complex in real life. Indeed, they need to cope with uncertain, dispersed, incomplete, and incompatible sources of information. People memorize each source of information and combine them by creating a mental picture of the particular problem they are faced with.

In this paper, we investigate how people combine heterogeneous and complex information to make a decision. People in our study were requested to make a single evaluation based on complex and abundant information from different perspectives.

In this context, the memory-reasoning relation plays a crucial role. How do people combine the different pieces of information in their memory? Does each source of information simply add a piece to the mental picture (as in a puzzle, the more data, the more precise the picture) or can the pieces of information interact and interfere, thus disrupting the mental picture? Fuzzy Trace Theory (hereinafter FTT) provides a reference psychological theory of cognition to deal with the proposed experimental setting [START_REF] Reyna | Fuzzy-trace theory: An interim synthesis[END_REF][START_REF] Brainerd | Fuzzy-trace theory: Dual processes in memory, reasoning, and cognitive neuroscience[END_REF]Liberali et al., 2012;[START_REF] Reyna | How fuzzy-trace theory predicts true and false memories for words, sentences, and narratives[END_REF]. FTT is based on a dual cognitive process (gist and verbatim), which rests on the assumption that people form representations of an event both by identifying semantic features (gist traces) and by storing surface details (verbatim traces). The theory has successfully accounted for various phenomena such as the generation of false memory, risk perception and estimation, as well as general biases and fallacies in decision making. The psychological literature reports that people tend to reason with gist rather than verbatim traces, that is, by attempting to create meaningful mental pictures which are not the sum of an event's surface details [START_REF] Reyna | A new intuitionism: Meaning, memory, and development in fuzzy-trace theory[END_REF].

In this paper, we report and discuss the results of an experiment where we test the role of verbatim and gist memory traces in making a correct evaluation. It is worth noting that in our experiment, the description of the event is complete, in that we provide sufficient information, distributed in various pieces, to make a rational and objective evaluation. First, we test whether the subjects have stored the information required to provide the rational answer and then see if they use this information -as they should -to answer correctly. We find that only a small proportion of the subjects answer correctly, which requires a proper combination of the relevant memorized pieces of information. Instead, most of them seem to rely more on gist representation rather than combining their verbatim traces. Finally, we also provide original statistical evidence showing that cognitive ability explains, to some extent, the misalignment between correct memory storage and the ability to answer correctly. This paper is organized as follows. The experimental design is described in Section 2. The results are discussed in Section 3, and Section 4 concludes.

Experimental Design

The experiment was conducted from 14 to 18 December 2015 at the Experimental Economics Laboratory of Nice (LEEN, France). 321 students from a broad range of disciplines at the University Côte d'Azur were recruited with a web-based online recruitment system (ORSEE, [START_REF] Greiner | numeracy and cognitive reflection, with implications for biases and fallacies in probability judgment[END_REF], and nineteen one-hour sessions were run. Tasks were implemented on computers using the z-Tree software [START_REF] Fischbacher | z-tree: Zurich toolbox for ready-made economic experiments[END_REF]. Subjects were remunerated for participating in the experiment. The payment included a fixed amount of EUR 10, plus a performance-related amount of up to EUR 6.2

The task consisted in reading a description of a country (Australia) without giving its name to participants. The information was presented in three or four consecutive screens, each providing a different perspective of the country:

• an introductory perspective (IP ) screen, made up of 131 words, giving a kind of "touristic description" of the country;

• a demographic perspective (DP ) screen, consisting of 92 words and 1 graph, providing demographic data of the country, including the number of inhabitants;

• an economic perspective (EP ) screen with 81 words and 2 graphs focusing on the definition and value of the gross national income (GN I) of the country;

• and, depending on the treatment: either a climate perspective (CP ) screen, containing climate (e.g. temperature) and geographical (desert, etc.) information of 131 words or an empty information screen (N P ) with a written message asking subjects to wait.

Each screen lasted 70 seconds. We introduced an empty (N P ) screen to compare the effect of additional climate information (CP ) while keeping the total time constant. Apart from the IP screen, which was always displayed in the first position, each of the twelve possible screen orders was presented to subjects. 3 After reading all the information, subjects were asked to answer several questions successively.

First, we asked subjects to evaluate the welfare of the country described compared to their own country (France): Q1) "According to you, on average, does an inhabitant of this country earn more or less money than an inhabitant of France?" 4

Then, we asked them to reveal the country which fits the best the description, according to them: Q2) "According to you, which country was described in the text?" Secondly, we tested their ability to recall verbatim information by administrating a set of memory control questions, in particular, about the total gross national income (GN I 1 ) and the number of inhabitants (P OP 1 ) of the country.

We also elicited the subject's belief about the "French average income" (F), which was not included in the description. Nevertheless, we argue that students should know it approximately, and rational individuals should thus use this verbatim information to answer Q1 correctly. Indeed subjects can combine the information given and their belief to answer Q1 in a rational way: 1) The GN I was introduced in EP ; 2) The country's number of inhabitants was presented in DP . Using the calculator, and given the definition of the GNI as "the sum of all incomes earned in a year by the inhabitants of a country", subjects should divide the GN I by the population to calculate GN I per capita, which indicates how much an inhabitant of the country earns in average. Potential mistakes in Q1 might be explained by the fact that subjects either did not memorize, have wrong beliefs about French average income or did not combine the verbatim information given previously.

Indeed, much information was provided to subjects during the experiment. Since we elicited verbatim memories of the gross national income (GN I 1 ) and population (P OP 1 ), we could assess the impact of verbatim memory on the evaluation of the welfare of the country's inhabitants (through the probability of answering Q1 correctly). As a control of verbatim memory, we introduced the variable MEM, equal to 1 if the subject recalled both the population and GN I, and 0 otherwise. 5

Finally, at the end of the experiment, we tested the subjects' cognitive reflection abilities by means of the "Cognitive Reflection Test" (CRT hereafter), proposed by [START_REF] Frederick | Cognitive reflection and decision making[END_REF]. We adopted the 3 See Table 7 in the appendix for the complete list of screen order. Screen shot of the different slide are available in the appendix.

4 Subjects may answer either "More" or "Less" by clicking on the respecting button. We counterbalanced the spatial location of the two buttons between subjects.

5 Subjects were informed that they were entitled to a maximum 10% margin of error for these two control questions.

modified version of the CRT offered by [START_REF] Finucane | Developing a tool for measuring the decision-making competence of older adults[END_REF] to ensure that students had never been exposed to the questions before.6 Subjects were not allowed to take any notes during the whole experiment, but they had access to a basic calculator if they wished to process verbatim information.

Behavioral Hypothesis

The correct answer to Q1 is "More" because the average income in Australia ($64, 540) is greater than in France ($42, 960). We score this answer as Q1 = 1 (respectively Q1 = 0, if the subject answered "Less").

Since the information provided can be used to correctly compute the average income per inhabitant of the described country (Australia), we postulate that subjects who are able to recall perfectly the GNI information and the number of inhabitants are more likely to answer Q1 correctly. Therefore, we expect a correlation between M EM and Q1.

Moreover, we conjecture that this correlation will be larger for subjects with a higher CRT score.

Indeed, [START_REF] Toplak | The cognitive reflection test as a predictor of performance on heuristics-and-biases tasks[END_REF] have shown that the CRT score is correlated with the probability of recalling verbatim information.We argue that this effect may play a role when retrieving the stored information to make an evaluation.

H1: Subjects recalling the verbatim information had a higher probability of answering Q1 correctly.

Another source of potential mistake in evaluation is the heterogeneity of subjects' beliefs about French average income. For example, someone with exact verbatim memory of both GN I and population of the described country, should rationally answer "Less" to Q1 if he overestimates French average income.

Thus, a rational individual should compare the ratio between Australian's GN I 1 and P OP 1 he recorded in memory, with his belief about French average income (F ). Based on the belief elicited about French average income and the information revealed, we construct the variable R1 which indicates the sense of the inequality:

R1 = 1 ⇔ GN I 1 P OP 1 > F and R1 = 0 ⇔ GN I 1 P OP 1 < F .
Thanks to this method, we were able to distinguish if the suject's answer is coherent with a rational combination of the memorized information and her belief. We therefore expected a positive correlation between R1 and Q1.

H2: Subjects rationally combined their verbatim memory and their beliefs to answer Q1.

The FTT suggests that individuals are using a gist representation rather than the verbatim memory to make an evaluation. To estimate the information contained in subjects' gist representation, we used the answers to Q2 about the country described in the experiment. We are aware that even if the answer to this question did not exactly reveal the subjects' gist representations, in many cases it should indicate the most likely representation.

In particular, we used World Bank data to extract GN I 2 and P OP 2 from the country revealed by subjects. Then we compared the GN I 2 P OP 2 ratio with subjects' beliefs about the average income in France (F). The variable R2 shows the sense of this inequality.

7 R2 = 1 ⇔ GN I 2 P OP 2 > F and R2 = 0 ⇔ GN I 2 P OP 2 < F
Indeed, it is impossible to be sure that the gist representation of the inhabitants' welfare of the described country corresponds to the per capita GN I (using World Bank data) of the country revealed in Q2. For example, 1) one can think that South Africa is most likely the described country in many aspects but not concerning inhabitants' welfare; 2) gist representation of South Africa welfare may be lower or greater than it is in reality. These two possibilities lead to conservative results due to a noisy proxy of gist representation.

Despite this, if the probability of answering Q1 correctly, given that R2 = 1, is greater than the probability of answering it correctly, given that R2 = 0, then it suggests that subjects were using their gist representation in the evaluation of inhabitant's welfare of the described country.

H3: Subjects with a "rich-country" gist representation had a higher probability of answering Q1 correctly.

Verbatim reasoning about GN I and P OP should not be affected by any other information. In contrast, the gist representation of inhabitants' welfare is built from all the received information (demographic, economic, climatic, and touristic descriptions).

To understand the formation of this dual process, we investigated the impact of additional information on both verbatim and gist representations. According to the treatments, subjects received either a blank screen (N P ) or one with climate information (CP ). Since subjects with the CP screen received more information, it should be more difficult for them to remember economic and demographic verbatim information, compared to subjects with the blank screen.

H4: Climatic Information impacts gist and verbatim memory.

We now study whether the climate information, that describes the country as hot ("up to 50 degrees Celsius") and "mostly desert", can interfere with the gist memory of the country, leading to 7 Therefore, GN I2 corresponds to the gross national income of the country answered in Q2 and P OP2 to the population of this country. For example, if someone answered "South Africa" to Q2 and thought that the average income in France was $25, 400 per year, we extracted the GN I2 and P OP2 (respectively 3.233 × 10 11 and 50.52 × 10 6 ) using the World Bank Data API and compared the ratio with his belief about French average income as follows:

GN I2 P OP2 = 6, 399 < 25, 400 = F ⇒ R2 = 0.
With the same answer to Q2 but with a lower belief about French welfare, one would obtain R2 = 1. Since the answer to Q2 could be freely input, we excluded from our analysis the 41 subjects who did not report a real country (e.g. "a country in Africa", or "Brazil or China", etc.).

a representation of a poorer country. 8

Results

In our sample, only 28% correctly answered Q1 and 9% found the right country. 9 According to the CRT score, we divided the subject pool among two groups: subjects with a CRT score of 0 or 1 are hereafter called the "LOW " group (n = 266) and those with a CRT score of 2 or 3 the "HIGH group (n = 55).

56.1% of the subjects recalled correctly the GN I of the country and 51.4% recalled its number of inhabitants. 32.4% of the subjects recalled both information.

H1: Subjects recalling the verbatim information had a higher probability of answering Q1 correctly.

We reported in We did not find any significant difference in the probability to answer Q1 correctly, depending on the correct recall of verbatim information (t-test: t = -0.911, p = 0.363). We conducted a logistic regression (Table 4) on the probability of answering Q1 correctly, as a function of M EM and CRT score. We did not find any effect of recalling verbatim information on the correctness of Q1 for LOW subjects (cf. Model 3, coeff = -0.01, p = 0.975) but found a statistically significant one for HIGH subjects (cf. Model 2, coeff = 2.01, p = 0.0027). Indeed, HIGH subjects who recall the right verbatim information have a higher probability to answer Q1 correctly. 10 H2: Subjects rationally combined their verbatim memory and their beliefs to answer Q1.

We reported in Table 2, the cross tabulation between R1 and answer to Q1, depending on CRT score.

8 Indeed, we deliberately chose a country with these characteristics because hot temperatures and deserts could be associated with poorer countries. 9 Top answered countries were Brazil (30), Australia (29), "Africa" (18), China (14), Morroco (13), Egypt (12), South Africa (12), USA (12), and India (11).

10 Model 2 and 3 are equivalent and statistically better than Model 1 (Likelihood ratio test: p = 0.012). If individuals rationally processed their verbatim memory to answer Q1, we should expect that the probability of answering Q1 correctly, given that R1 = 1, should be greater than the probability of answering Q1 correctly, given that R1 = 0.

LOW

We did not find any significant difference in the probability to answer Q1 correctly, depending on R1 (t-test: t = -0.805, p = 0.421). We conducted logistic regressions (Table 4, Models 4,5,6) on the probability of answering Q1, as a function of R1 and CRT score. We found that the results obtained in H1 are robust to heterogeneity of beliefs about French average income. LOW subjects did not seem to process verbatim information to answer Q1 (Model 6, coeff = 0.01, p = 0.972). Conversely, HIGH individuals did so (Model 5, coeff = 1.68, p = 0.0157).11 H3: Subjects with a "rich-country" gist representation had a higher probability of answering Q1 correctly.

We reported in We found a significant difference in the probability to answer Q1 correctly, depending on having a "rich-country" gist representation (t-test: t = -4.958, p = 2.6 × 10 -6 ).

We conducted a logistic regression (Table 4, Models 7 to 10) on the probability of answering Q1, as a function of R2, R1 and CRT score. We found that gist representation impacted the evaluation for both LOW (Model 8: coeff = 1.30, p < 0.001) and HIGH subjects (Model 7: coeff = 1.76, p = 0.021).

However, we found no statistical difference of this effect between the two groups (p = 0.58). 12 P (Q1 = 1)

H1 H2 H3

( Log Num We did not find any significant difference in the probability to recall verbatim information, depending on receiving the climate information (t-test: p = 0.1618). However we found a significant difference of the effect of receiving the climate information on the probability of recalling a "rich country" (t-test: p = 0.044).

We ran a logistic regression (Table 6, left columns) on the probability of recalling both GN I and P OP verbatim information (M EM ), as a function of the CRT score and receiving the climate information (CP ).

We did not find that providing climate information to LOW subjects impacted their probability of recalling verbatim economic and demographic information (Model 12: coeff = 0.16, p = 0.62), as opposed to the HIGH group for which it increases their recall rate (Model 13: coeff = 1.45, p = 0.045).

We anticipated that giving more information would reduce the ability to remember demographic and economic data. Nevertheless, climate information (compared to a blank screen) may help HIGH subjects to remain focused on the country data.

We also ran a logistic regression on the probability of identifying a richer country than France (R2), as a function of CP and CRT (Table 6, right columns). We found that the effect of climate information on the gist representation is more pronounced for the HIGH group (Model 16: coeff = -1.54, p = 0.022) than for the LOW one (Model 15: coeff = -0.37 , p = 0.222). This tends to support the idea that climate information more strongly impacts the gist representation of HIGH subjects.

P (M EM = 1) Log Num

P (R2 = 1) (11) (12) (13) (14) (15) (16) (Intercept) -1.03

Discussion

To summarize the main points, results of this experiment can be accounted for the dual memory process described in the FTT. Moreover, subjects rely mainly on their gist information to evaluate the described country's welfare. This is principally true for LOW subjects, while HIGH subjects also seem to consider verbatim information.

Our results provide new evidence of the existence of a dual memory process (gist and verbatim) in the specific context of making an evaluation with complete and complex information. In this experiment, subjects received several pieces of information on different aspects of an unknown country.

After receiving the description, they were asked to evaluate the welfare of the inhabitants compared to that of their country of residence (France). We provided them complete information since the description included all the relevant information to answer correctly. We controlled both verbatim and gist memories by asking subjects to reveal: 1) recorded information about economics and demographics, and 2) the country that best fits the description according to them.

This protocol allowed us to distinguish verbatim and gist traces of economic and demographic data and to investigate which information was in line with the final evaluation. Furthermore, we controlled for subjects' cognitive abilities using a Cognitive Reflection Test. We found that subjects with low cognitive abilities did not combine the verbatim information to make the evaluation. Indeed, although they correctly recalled the gross national income and the population of the country, they did not make a consistent assessment with an appropriate combination of this information. Conversely, individuals with higher cognitive abilities tended to exhibit consistency between their evaluation and verbatim memory. Furthermore, regardless of their cognitive abilities, subjects relied mainly on their gist representation.

In addition, investigating the impact of additional information on verbatim and gist memory leads to novel results. For individuals with low cognitive abilities, we did not find that climate information impacted gist and verbatim traces of economic and demographic information. However, it helped high cognitive subjects to recall verbatim information and changed their gist representation about inhabitants' welfare. These results suggest further research to better understand both the link and balance between cognitive abilities and the coexistence of gist and verbatim reasoning. Ismaël Rafaï, Sébastien Duchêne, Eric Guerci, Ariane Lambert-Mogiliansky & Fabien Mathy « A dual process in memory: how to make an evaluation from complex and complete information? -An experimental study »

A Appendix

A.1 Table of Treatments

  a ë l R a f a ï , S é b a s t i e n D u c h ê n e , E r i c G u e r c i , E r i c G u e r c i , A r i a n e L a m b e r t -Mo g i l i a n s k y

Table 1 :

 1 Table 1, the cross tabulation between verbatim memorization and answer to Q1, depending on CRT score. Q1 and MEM contingency

		LOW	HIGH	TOTAL
		MEM=0 MEM=1 MEM=0 MEM=1 MEM=0 MEM=1
	Q1 = 0	135	59	24	12	159	71
	Q1 = 1	52	20	6	13	58	33
	mean	27.81%	25.32%	20.00%	52.00%	26.73 %	31.73%

  Table 3, the cross tabulation between R2 and answer to Q1.

		LOW	HIGH	T OT AL
		R2=0	R2=1	R2=0	R2=1	R2=0	R2=1
	Q1 = 0	114	48	26	6	140	54
	Q1 = 1	27	41	8	10	35	50
	mean	19.15% 46.07% 23.52% 62.50% 20.00% 48.08%
		Table 3: Q1 and R2 contingency

Table 4 :

 4 Impact of verbatim and gist memory on Q1

H4: Climate Information impacts gist and verbatim memory.

Table 5

 5

		LOW	HIGH	T OT AL
		CP=0	CP=1	CP=0	CP=1	CP=0	CP=1
	MEM = 0	48	139	11	19	59	158
	MEM = 1	18	61	3	22	21	83
	mean	27.27% 30.5% 21.42% 53.66% 26.25% 34.44%
	R2 = 0	34	107	6	28	40	135
	R2 = 1	28	61	8	8	36	69
	mean	45.16% 36.31% 57.14% 22.22% 47.37% 33.82%

displays the frequencies of recalling both GN I and P OP verbatim information (i.e. M EM = 1), as well as the probability of recalling a "richer country than France" (i.e. R2 = 1), according to CP .

Table 5 :

 5 MEM, R2 and CP contingency, conditional to CRT

  * * * -0.98 * * * -1.30 *

					-0.11	-0.19	0.29
		(0.25)	(0.28)	(0.65)	(0.23)	(0.26)	(0.54)
	CP	0.39	0.16	1.45 *	-0.57 * -0.37 -1.54 *
		(0.29)	(0.32)	(0.72)	(0.27)	(0.30)	(0.67)
	HIGH		-0.32			0.48
			(0.71)			(0.60)
	CP × HIGH		1.29			-1.17
			(0.79)			(0.74)
	LOW			0.32			-0.48
				(0.71)			(0.60)
	CP × LOW			-1.29			1.17
				(0.79)			(0.74)

Table 6 :

 6 Impacts of Climate Information on Gist and Verbatim Memory

	. obs.	321	321	321	280	280	280
	* * * p < 0.001,

* * p < 0.01, * p < 0.05

Table 7 : Table of Treatments A.2 Text of the introductory perspective (IP) in French Le

 7 pays que nous décrivons maintenant et dans les pages suivantes recèle de trésors culturels et naturels et offre différentes saisons idéales au voyage : des sommets enneigés, des phénomènes géologiques au coeur du désert, des prouesses architecturales, des plages rouges, des montagnes rugueuses et des forêts luxuriantes...

	Treatment Slide 1 Slide 2 Slide 3 Slide 4 #Subj
	T1	IP	EP	DP	CP	41
	T2	IP	EP	CP	DP	40
	T3	IP	DP	EP	CP	40
	T4	IP	DP	CP	EP	40
	T5	IP	CP	EP	DP	40
	T6	IP	CP	DP	EP	40
	T7	IP	NP	EP	DP	13
	T8	IP	EP	NP	DP	13
	T9	IP	EP	DP	NP	14
	T10	IP	NP	DP	EP	13
	T11	IP	DP	NP	EP	13
	T12	IP	DP	EP	NP	14
	Son patrimoine riche en spectacle grandiose et en diversité nous invite au dépaysement, à l'aventure,
	à la découverte d'un monde à part, immense, insolite et fascinant. Le relief de ce pays a été lentement
	dessiné au fil du temps par l'érosion et nous parvient intact et phénoménal.
	L'âge du paysage de ce magnifique pays se mesure en millions d'années. Il réunit dans son im-
	mensité une surprenante diversité de paysages et de climats, ainsi qu'une faune et flore originelles et
	uniques au monde.

A.

3 Screenshot of the demographic perspective (DP) in French A.4 Screenshot of the economic perspective (EP) in French CEE-M Working Papers 1 -2018

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Correct answers were paid EUR2 for Q1; EUR1 for Q2, and EUR3, for the control questions.

Indeed, the CRT is a well-known test. The modified CRT consists of three consecutive questions, each of which has an intuitive but false answer and a reflexive true answer. Scores were given on the basis of a number between 0 and 3 according to the number of correct answers.

We ran a likelihood ratio test, and Model 6 is statistically better than Model 4 (p = 0.038).

The latter regression also confirm H2.These results are robust for the use of M EM (see appendix) instead of R1 as a control of verbatim memory, which provides support to results presented in H1.

A.5 Text of the climatic information (CP) in French

La majeure partie du pays est désertique ou semi-aride. Seuls les coins sud-est et sud-ouest du pays bénéficient d'un climat tempéré et d'un sol moyennement fertile. Le nord du pays, qui connaît un climat tropical, est couvert soit de forêts tropicales humides, soit de prairies, soit de déserts.

Les précipitations sont extrêmement variables, avec de fréquentes sécheresses qui durent plusieurs saisons. Une tempête de poussière est parfois capable de s'étendre sur toute une région et il y a des cas de tornades de grande ampleur. Dans certaines zones, l'élévation de la salinité et la désertification causent des dégâts importants au paysage. Dans l'année, les températures les plus hautes peuvent atteindre les 50 • C, tandis que les minimums peuvent descendre en dessous de 0 • C.

A.6 Screenshot of the control questions in French


Urban isolation and daytime neighborhood social composition from twitter data

Julie Vallée

CNRS, Géographie-cités laboratory, Paris, France. Email: julie.vallee@parisgeo.cnrs.fr In a recent paper in PNAS, Wang et al. [START_REF] Wang | Urban mobility and neighborhood isolation in America's 50 largest cities[END_REF] investigate the key question of urban isolation in light of daily mobility. Urban isolation may actually be mitigated by individuals spending much of their everyday lives in neighborhoods where social composition and urban opportunities widely diverge from their home neighborhoods. From a large dataset composed of 128 million geotagged micromessages called tweets, sent by 392,000 Twitter users living in the 50 most populous American cities, the authors overcome the pervasive spatial assumption that social interactions are limited to home neighborhoods. However, they made one temporal assumption that merits further discussion. They classified neighborhoods into poor and nonpoor based on the proportion of residents living on incomes below the federal poverty line, and they similarly characterized neighborhood racial status into three groups: majority of white, black, or Hispanic residents. They used residential neighborhood composition to characterize both the neighborhoods in which people reside and those into which people travel on a daily basis. Through this approach, the authors assumed that social structures of neighborhoods are necessarily frozen over a 24-h period. Omission of daily changes in neighborhood social composition is frequent in the literature about neighborhood effects; however, in the case of this research, it is especially unfortunate because the study precisely aims to explore daytime contact and copresence between (poor and nonpoor) inhabitants. With Twitter messages [which are not only geographically informative but also temporally informative (hours and minutes)], it would still have been possible to move beyond nighttime residential-based estimates. Even if no information was available to define the individual social status of Twitter users, the authors could have qualified the neighborhoods visited daily from daytime concentrations of Twitter users living in poor/nonpoor neighborhoods. By doing so, the authors' conclusions about the urban isolation in American cities would have been even more convincing.

In the emerging literature concerning people's daily mobility and their exposure to nonresidential contexts, there is a frequent discrepancy between the high accuracy in measuring people's daily trajectories in cities and the low accuracy in measuring daily rhythms of neighborhood social composition. However, daily rhythms constitute an important timescale not only for people but also for places. The social determination of work and free time impacts the temporal signatures of places. Moreover, when planning public transportation, employment areas, or public events, public authorities frame and coordinate daily population flows. Some scholars, inspired by the time-geography approach, have highlighted the theoretical importance of considering time and space in tandem (2, 3) and have accounted for daily variations in neighborhood social mix to explore opportunities for social contact between members of different social groups (4-7). The recent "daycourse of place" framework (8) could be used to consider the rhythms of places more explicitly and to strengthen our understanding of urban isolation around-the-clock. Variation or stability in neighborhood social composition over a 24-h period is actually more than a phenomenon that may bias findings: It is a dynamical expression of urban isolation and neighborhood effects.


Amaury Delamarre

Daniel Suchet

Nicolas Cavassilas

Yoshitaka Okada

Masakazu Sugiyama

Jean-Francois Guillemoles

Jean-François Guillemoles

An electronic ratchet is required in nanostructured intermediate band solar cells

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I. INTRODUCTION

The single junction solar cell efficiency, under one sun illumination, has been shown to be limited by the Shockley-Queisser (SQ) limit, of about 33% [START_REF] Shockley | Detailed Balance Limit of Efficiency of p-n Junction Solar Cells[END_REF]. Several strategies have been suggested to overcome this limit, one of them being the concept of Intermediate Band Solar Cell (IBSC) [START_REF] Luque | Increasing the efficiency of ideal solar cells by photon induced transitions at intermediate levels[END_REF], [START_REF] Guillemoles | Limiting efficiencies of LDS intermediate band solar cells[END_REF], where energy levels are introduced within a semiconductor bandgap, to allow the sequential absorption of sub-bandgap energy photons that would be lost otherwise. Among the different strategies to introduce intermediate levels [START_REF] Okada | Intermediate band solar cells: Recent progress and future directions[END_REF]- [START_REF] Ramiro | Review of Experimental Results Related to the Operation of Intermediate Band Solar Cells[END_REF], a promising one is to take advantage of nanostructures such as Quantum Dots (QDs) or Quantum Wells (QWs).

Nevertheless, to date, no IBSC overcoming the most efficient single junction cells has been demonstrated. Two reasons can be inferred from previous practical realizations. Firstly, using nanostructures, a severe open-circuit voltage (VOC) drop is commonly observed when sub-gap states are introduced [START_REF] Luque | Understanding intermediate-band solar cells[END_REF], [START_REF] Ramiro | Review of Experimental Results Related to the Operation of Intermediate Band Solar Cells[END_REF], which denotes strong recombinations. For intrinsic (Auger recombination [START_REF] Tomić | Intermediate-band solar cells: Influence of band formation on dynamical processes in InAs/GaAs quantum dot arrays[END_REF], efficient thermalization in QWs [START_REF] Milde | Light absorption and many-particle electron-phonon interactions in intermediate band quantum well solar cells: Light absorption and el-phon interactions in QW solar cells[END_REF] or via multiple states in QDs [START_REF] Luque | New Hamiltonian for a better understanding of the quantum dot intermediate band solar cells[END_REF]) or extrinsic (growth issues when going towards large dot densities [START_REF] Okada | Intermediate band solar cells: Recent progress and future directions[END_REF]) reasons. Secondly, low sub-gap currents are observed, due to the monochromatic nature of the inter-sub-band transitions, or to the difficulty of obtaining large dot densities [START_REF] Okada | Intermediate band solar cells: Recent progress and future directions[END_REF]- [START_REF] Ramiro | Review of Experimental Results Related to the Operation of Intermediate Band Solar Cells[END_REF]. Although nanostructured devices have been widely studied since the theoretical description of the IBSC [START_REF] Okada | Intermediate band solar cells: Recent progress and future directions[END_REF]- [START_REF] Ramiro | Review of Experimental Results Related to the Operation of Intermediate Band Solar Cells[END_REF], no systematic study on the interplay of those two parameters on the efficiency has been proposed. We derive in this letter the corresponding minimal requirements in order to overcome the SQ limit, which can be used as guidelines for screening candidate nanostructures. We highlight that currently investigated systems, and especially the widely studied InGaAs QDs in GaAs (e.g. [START_REF] Okada | Intermediate band solar cells: Recent progress and future directions[END_REF]- [START_REF] Ramiro | Review of Experimental Results Related to the Operation of Intermediate Band Solar Cells[END_REF], [START_REF] Guimard | Fabrication of InAs/GaAs quantum dot solar cells with enhanced photocurrent and without degradation of open circuit voltage[END_REF]- [START_REF] Blokhin | AlGaAs/GaAs photovoltaic cells with an array of InGaAs QDs[END_REF]), are far from meeting those minimal requirements.

Since nanostructures compatible with high efficiencies appear challenging to fabricate, we consider in addition the possibility to introduce a ratchet mechanism, a concept that can be applied to any energy converter, and in particular to mesoscopic systems [START_REF] Linke | Experimental Tunneling Ratchets[END_REF]. When implemented in an IBSC, a ratchet has been shown to result in an efficiency increase in the radiative limit (from 46.7 to 48.5%) [START_REF] Yoshida | Photon ratchet intermediate band solar cells[END_REF]. In addition, it is expected to reduce the harmful effects of absorption reduction [START_REF] Pusch | Limiting efficiencies for intermediate band solar cells with partial absorptivity: the case for a quantum ratchet: Limiting efficiencies for intermediate band solar cells[END_REF]. However, its impact in the presence of non-radiative recombination has not been thoroughly investigated [START_REF] Olsson | Ferromagnetic Compounds for High Efficiency Photovoltaic Conversion: The Case of AlP:Cr[END_REF], despite the fact that the issue of VOC preservation is a common issue of most practical realization [START_REF] Luque | Understanding intermediate-band solar cells[END_REF], [START_REF] Ramiro | Review of Experimental Results Related to the Operation of Intermediate Band Solar Cells[END_REF]. We will show that a ratchet mechanism strongly relaxes the requirements on the absorption and radiative efficiency, to the point that any combination of those two parameters is compatible A part of this paper is supported by the Research and Development of Ultra-High Efficiency and Low-Cost III-V Compound Semiconductor Solar Cell Modules project under the New Energy and Industrial Technology Development Organization. A part of this paper is supported by a Grant-in-Aid from the Japan Society for the Promotion of Science (JSPS).

Amaury Delamarre, Yoshitaka Okada and Masakazu Sugiyama are with NextPV, LIA CNRS-RCAST/U. Tokyo-U. Bordeaux, Tokyo 153-8904, Japan, and also with the Research Center for Advanced Science and Technology, The University of Tokyo, Tokyo 153-8904, Japan (e-mail: delamarre@hotaka.t.utokyo.ac.jp, okada@mbe.rcast.u-tokyo.ac.jp, sugiyama@ee.t.u-tokyo.ac.jp).

Daniel Suchet is with NextPV, LIA CNRS-RCAST/U. Tokyo-U. Bordeaux, Tokyo 153-8904, Japan and also with LPICM, CNRS/Ecole Polytechnique

Universite

Paris-Saclay, Palaiseau 91128, France (e-mail: daniel.suchet@polytechnique.org).

Nicolas Cavassilas is with NextPV, LIA CNRS-RCAST/U. Tokyo-U. Bordeaux, Tokyo 153-8904, Japan and also with Aix Marseille Université, CNRS, Université de Toulon, IM2NP -UMR 7334, Marseille, France (e-mail: nicolas.cavassilas@im2np.fr).

Jean-François Guillemoles is with NextPV, LIA CNRS-RCAST/U. Tokyo-U. Bordeaux, Tokyo 153-8904, Japan and also with IPVF, CNRS, UMR 9006, 30 route départementale 128, 91120, Palaiseau, France (e-mail: jf.guillemoles@cnrs.fr).

with efficiencies in excess of the SQ limit. Moreover, this perspective allows us to conclude that QWs present an advantage over QDs for realizing IBSCs, due to their increased ability to form mini-bands that enlarge the sub-gap absorption width. This compensates the increased non-radiative recombination of QWs, when compared to QDs, which is the reason why they have been discarded [START_REF] Araújo | Absolute limiting efficiencies for photovoltaic energy conversion[END_REF] and much less investigated so far.

II. THEORETICAL MODEL

The model developed in this paper is similar to those used in previous publications [START_REF] Yoshida | Photon ratchet intermediate band solar cells[END_REF], [START_REF] Pusch | Limiting efficiencies for intermediate band solar cells with partial absorptivity: the case for a quantum ratchet: Limiting efficiencies for intermediate band solar cells[END_REF], to which non-ideal behaviors, expected from nanostructures, will be introduced. We consider the IBSC represented in Figure 1, in which the CB is split into two distinct bands CB1 and CB2, separated by the ratchet energy ΔE. We assume that the relaxation from CB1 to CB2 is fast, so that both bands share a common quasi-Fermi level. We note Eg1 the bandgap from VB to CB2, Eg2 the bandgap from VB to IB, and Eg3 the bandgap from IB to CB1. We assume Eg1>Eg2>Eg3. Transitions from CB2 to IB are forbidden. We assume that Eg3 is related to intersubband transitions, so that non-idealities will be introduced relatively to this bandgap. In the following, an IBSC including a ratchet mechanism will be referred to as a RBSC, whereas we will continue to refer to the classical device, without ratchet (i.e. for ΔE = 0 eV), as an IBSC. We note that including a ratchet on the conduction band, valence band or intermediate band is strictly equivalent when calculating the device efficiency limit, which only depends on the three bandgap values [START_REF] Pusch | Limiting efficiencies for intermediate band solar cells with partial absorptivity: the case for a quantum ratchet: Limiting efficiencies for intermediate band solar cells[END_REF].

The carrier generation 𝐺 𝑖 is given by integrating the solar flux multiplied by the absorption of each subgap:

𝐺 𝑖 = 𝑓 4𝜋 3 ℏ 3 𝑐 0 2 ∫ 𝑎 𝑖 (𝐸) 𝐸 2 𝑒𝑥𝑝 ( 𝐸 𝑘𝑇 𝑠𝑢𝑛 ) -1 . 𝑑𝐸 ∞ 0 (1)
𝑓 is a geometrical factor, that we choose equal to 6.79*10 -5 for an unconcentrated generation. 𝑎 𝑖 (𝐸) is the absorption for each bandgap. We assume an ideal absorption for Eg1 and Eg2, i.e. 𝑎 1 (𝐸) = 1 only above Eg1, and 𝑎 2 (𝐸) = 1 strictly between Eg2 and Eg1. In an ideal case, 𝑎 3 (𝐸) = 1 strictly between Eg3 and Eg2. We will study in the following the impact of narrow absorption, in which 𝑎 3 (𝐸) = 1 is limited to the energy range from 𝐸𝑔 3 to 𝐸𝑔 3 + 𝛿𝐸.

The radiative recombination 𝑅 𝑖 for each bandgap 𝑖 is given by a generalized Planck's law [START_REF] Würfel | The chemical potential of radiation[END_REF]:

𝑅 𝑖 = 1 4𝜋 2 ℏ 3 𝑐 0 2 ∫ 𝑎 𝑖 (𝐸) 𝐸 2 𝑒𝑥𝑝 ( 𝐸 -∆𝜇 𝑖 𝑘𝑇 𝑐𝑒𝑙𝑙 ) -1 . 𝑑𝐸 ∞ 0 (2)
The free energy ∆𝜇 𝑖 is equal to the quasi-Fermi level splitting in each bandgap. We use the temperatures 𝑇 𝑠𝑢𝑛 = 6000𝐾 for the sun spectrum and 𝑇 𝑐𝑒𝑙𝑙 = 300𝐾 for the cell. In order to investigate the influence of non-radiative recombinations, we introduce a radiative recombination ratio 𝑟 𝑟𝑎𝑑 that defines the global recombination rate for Eg3 following:

𝑅 3 * = 1 𝑟 𝑟𝑎𝑑 𝑅 3 (3)
We will assume that the recombinations in Eg1 and Eg2 remain radiative. The quasi-Fermi level splittings are related to the voltage V following:

𝑞𝑉 = ∆𝜇 1 = ∆𝜇 2 + ∆𝜇 3 (4)
The continuity equation in the IB reads:

𝐺 2 -𝑅 2 -𝐺 3 + 𝑅 3 * = 0 (5)
The extracted current is the difference between the carrier generation and recombination in the conduction band:

𝐽 𝑞 = 𝐺 1 -𝑅 1 + 𝐺 3 -𝑅 3 * (6)
The above equations permit to compute IV curves and efficiencies for given sets of bandgaps Eg1, Eg2, Eg3 and the ratchet energy ΔE.

III. RESULTS AND DISCUSSION

The simulation results, with varying absorption width 𝛿𝐸 and radiative efficiencies 𝑟 𝑟𝑎𝑑 are presented in Figure 2 and Figure 3. For a first understanding of the results, the efficiencies of ideal devices (RBSC, IBSC and single junction) are compared in Figure 2 as a function of Eg1. For each value of Eg1; Eg2, Eg3 and ΔE are optimized. As pointed out in [START_REF] Yoshida | Photon ratchet intermediate band solar cells[END_REF], the ratchet allows for some improvement compared to the IBSC configuration in the radiative limit, from 46.7% to 48.5% with ∆𝐸 = 260 𝑚𝑒𝑉 . This improvement is gradually decreased when Eg1 increases, and becomes negligible for 𝐸𝑔 1 > 2.5 𝑒𝑉. We also observe that for 𝐸𝑔 1 < 1.1 𝑒𝑉, without a ratchet, no performance enhancement can be expected by insertion of an IB compared to the single junction. By contrast, the RBSC always exhibits an improvement in our range of study.

The impact of non-idealities, found in nanostructures, is then investigated. First, we study the effect of an absorption width for Eg3 reduced to 𝛿𝐸 = 250 𝑚𝑒𝑉. Such broadenings can be obtained by the formation of minibands in coupled nanostructures. In QWs, widths of about 200 meV have been observed for intersubband transitions [START_REF] Georgiev | Short-wavelength intersubband absorption in strain compensated InGaAs/AlAs quantum well structures grown on InP[END_REF]. For both the IBSC and the RBSC, an efficiency decrease is observed. This decrease being more pronounced for large Eg1, and negligible for 𝐸𝑔 1 < 1.2 𝑒𝑉. In the RBSC case, the optimum is found for a ratchet energy ∆𝐸 = 380 𝑚𝑒𝑉. Second, a radiative efficiency 𝑟 𝑟𝑎𝑑 = 10 -3 is assumed in addition to the narrow absorption. Such a value can be considered as an upper limit for intersubband transitions at energies larger than the optical phonon energy in QWs [START_REF] Yang | Infrared laser based on intersubband transitions in quantum wells[END_REF]. On the one hand, an efficiency reduction is observed for the IBSC, so large that it does not provide any benefit when compared to the best single junction performance. On the other hand, the efficiency in the ratchet configuration is only slightly affected over the whole range of study, the optimum performance being obtained for a ratchet energy increased to ∆𝐸 = 540 𝑚𝑒𝑉. This observation indicates that an RBSC, in this configuration, is relatively insensitive to nonradiative recombinations. Because such detrimental recombinations are important in nanostructured IBSC, this constitutes a significant result.

Figure 2 Efficiency as a function of Eg1 for the single junction (green line), IBSC (yellow lines), and RBSC (blue lines). For the IBSC and the RBSC, the full lines correspond to ideal absorptions and recombinations in the radiative limit. The doted lines correspond to an absorption limited to 𝛿𝐸 = 250 𝑚𝑒𝑉 , and recombinations in the radiative limit for Eg3. The dashed lines correspond to 𝛿𝐸 = 250 𝑚𝑒𝑉, and a radiative recombination rate of 10 -3 for Eg3.

For more insight into this remarkable result, we study the interplay of reduced absorption width and radiative recombination ratio in Figure 3(a) and (b). In Figure 3(a), the optimum efficiency is displayed as a function of the absorption width, in the IBSC and RBSC configurations, in the radiative limit and for 𝑟 𝑟𝑎𝑑 = 10 -3 . In the radiative limit, we observe that the IBSC efficiency remains constant until the absorption width is reduced to 550 meV, before decreasing linearly until 𝛿𝐸 = 150 𝑚𝑒𝑉. Below this value, the implementation of an intermediate level is only detrimental compared to the single junction. No effect is observed for an absorption width above 550 meV, since in that range the optimal bandgap arrangement is obtained for 𝐸𝑔 2 -𝐸𝑔 3 < 550 𝑚𝑒𝑉. Remaining in the radiative limit, we observe that the RBSC is more tolerant to the absorption narrowing than the IBSC, the performance being constant until 𝛿𝐸 = 450 𝑚𝑒𝑉 before a linear decrease. Nevertheless, we note that an improvement relative to the SQ limit is possible as soon as the absorption width is non-zero. When combining the absorption width narrowing to a non-radiative recombination ratio of 10 -3 , we observe that the efficiency of the IBSC is reduced for any absorption width. Below 𝛿𝐸 = 250 𝑚𝑒𝑉, no improvement compared to the single junction is expected. On the opposite, the RBSC performance is only impacted for an ideal absorption, while the variation is negligible below 𝛿𝐸 = 300 𝑚𝑒𝑉. As a consequence, the efficiency gain thanks to the ratchet is even more apparent, growing from 2.9% absolute for a wide absorption up to 10.8% for 𝛿𝐸 = 300 𝑚𝑒𝑉.

In Figure 3(b), the optimum efficiency is displayed as a function of the radiative recombination ratio, in the IBSC and RBSC configurations, for an ideal absorption and for 𝛿𝐸 = 250 𝑚𝑒𝑉 . For an ideal absorption, the IBSC efficiency decreases monotonously until 𝑟 𝑟𝑎𝑑 = 10 -12 where it reaches the SQ limit. The RBSC is more tolerant to the radiative efficiency deterioration, so that the efficiency gain against the IBSC grows from 1.8% absolute at 𝑟 𝑟𝑎𝑑 = 1 up to 5% at 𝑟 𝑟𝑎𝑑 = 10 -12 . When combining the 𝑟 𝑟𝑎𝑑 reduction to an absorption range limited to 250 meV, the IBSC efficiency is strongly reduced: it amounts to 34.4% in the radiative limit, and falls below the SQ limit as soon 𝑟 𝑟𝑎𝑑 is lower than 10 -2 . By contrast, the ratchet structure drastically reduces the cell sensitivity to the non-radiative recombinations, the efficiency only suffering from a 1% reduction between 𝑟 𝑟𝑎𝑑 = 1 and 𝑟 𝑟𝑎𝑑 = 10 -6 , from 41.0% to 40.0%.

In light of those simulation results, we are now able to draw guidelines for the realization of efficient IBSC with nanostructures, with respect to three parameters: the radiative efficiency and the absorption width of the inter-subband transition, and the possibility of using an electronic ratchet.

On Figure 2 and Figure 3, it was highlighted that below some critical values of 𝑟 𝑟𝑎𝑑 and 𝛿𝐸, an IBSC cannot overcome the Shockley-Queisser limit. This observation allows us to define minimum required (𝑟 𝑟𝑎𝑑 , 𝛿𝐸), which we plot in Figure 4. It is worth noting that any (𝑟 𝑟𝑎𝑑 , 𝛿𝐸) couple, for an RBSC, in the present range, allows expecting efficiencies higher than the single junction limit. As a consequence, this plot equivalently represents the region in which a ratchet mechanism is required for an IBSC to overcome the SQ limit. Nanostructures cells being non-ideal, it is likely that they will stand in that region. As an example, it was calculated in [START_REF] Tomić | Intermediate-band solar cells: Influence of band formation on dynamical processes in InAs/GaAs quantum dot arrays[END_REF] that for InAs QDs in a GaAs matrix, for 5 nm dot-to-dot spacing, the IB to CB absorption width was 14 meV, and the non-radiative recombination ratio would be around 10 -5 . Although it is agreed that this system is not ideal, the plot in Figure 4 highlights that it is far from overcoming the SQ limit, without a ratchet mechanism.

We also observe that even in the radiative limit, an absorption width of at least 170 meV is necessary to overcome the SQ limit. This observation leads us to the conclusion that some means to widen the absorption width are mandatory if a ratchet mechanism is not included. This may be obtained with various nanostructure dimensions in a single device, or minibands.

Figure 4 Plot representing the minimum required (𝑟 𝑟𝑎𝑑 , 𝛿𝐸) combinations to reach an efficiency larger than the SQ limit, for an IBSC. Within the graph range, any couple can be considered to overcome the SQ limit for an RBSC.

A remarkable feature of an RBSC is its relative insensitivity on the reduction of 𝑟 𝑟𝑎𝑑 , for narrow absorption, as presented on Figure 3(b). This consideration will have implication regarding the practical implementation of nanostructured IBSC. Indeed, QD have been mainly considered, arguably because they present a true zero density of states in the energy range between the IB and the continuum. Hence, they are expected to exhibit larger radiative recombination ratio compared to QW, in which a continuum of states is available, allowing efficient relaxation via phonon scattering. Nevertheless, QD are not free from non-radiative recombination as well, via intrinsic mechanisms (Auger [START_REF] Tomić | Intermediate-band solar cells: Influence of band formation on dynamical processes in InAs/GaAs quantum dot arrays[END_REF]), or defects [START_REF] Tomić | In-plane coupling effect on absorption coefficients of InAs/GaAs quantum dots arrays for intermediate band solar cell[END_REF]. Since both QD and QW will exhibit relatively narrow absorption widths, the radiative recombination ratio benefit of the former on the latter is no more critical in an RBSC. However, a benefit could be obtained by enlarging the absorption width, which can be obtained thanks to a miniband. The growth of packed nanostructures, necessary to build a miniband, seems more favorable in the QW case.

Following this observation, we suggest a structure that creates a ratchet on the conduction band with quantum wells, as represented in Figure 5. This structure uses two sections. The first section consists in large wells, that allow two confined levels, namely IB and CB1. Several coupled wells are necessary to create a miniband that enlarge the absorption width. The second section is made of thinner wells, allowing a single state CB2, that will be engineered to lie at an energy ΔE below CB1. In order to minimize the recombination from CB2 to IB, an intermediate section is included that will prevent the CB2 wavefunction to extend in section 1. Similarly to what is used in quantum cascade systems, the energy difference between the states connecting CB1 and CB2 should equal the phonon energy to allow efficient relaxation. For practical realization, material combinations exhibiting together large band offsets on the conduction band and reduced offsets on the valence band are required. As an example, InGaAs/AlAsSb lattice matched or strained on InP [START_REF] Georgiev | Optical properties of InGaAs/AlAsSb type I single quantum wells lattice matched to InP[END_REF], or InAs/AlAsSb [START_REF] Esmaielpour | Suppression of phonon-mediated hot carrier relaxation in type-II InAs/AlAs x Sb 1x quantum wells: a practical route to hot carrier solar cells: Suppression of phonon-mediated hot carrier relaxation in type-II InAs/AlAs x Sb 1x quantum wells: a practical route to hot carrier solar cells[END_REF] could be used. Other materials have been studied for intersubband transition in the conduction band, among III-V (InGaAs/InAlAs strain balanced on InP [START_REF] Smet | Electron intersubband transitions to 0.8 eV (1.55 μm) in InGaAs/AlAs single quantum wells[END_REF]), nitrides (GaN/AlN [START_REF] Hofstetter | Photodetectors based on intersubband transitions using III-nitride superlattice structures[END_REF]), or II-VI (ZnSe/BeTe [START_REF] Akimoto | Short-wavelength intersubband transitions down to 1.6 μm in ZnSe/BeTe type-II superlattices[END_REF], (ZnSe/CdSe)/MgSe [START_REF] Chen | MgSe/ZnSe/CdSe coupled quantum wells grown on InP substrate with intersubband absorption covering 1.55 μ m[END_REF]). Nevertheless, it should be confirmed that the hole transport is still possible in such structures.

Several other ways to implement a ratchet have been suggested in previous literature. Using a quantum cascade has been considered, for a ratchet on the intermediate band [START_REF] Curtin | Quantum Cascade Photon Ratchets for Intermediate-Band Solar Cells[END_REF]. Other strategies include using direct/indirect transitions [START_REF] Yoshida | Photon ratchet intermediate band solar cells[END_REF], ferromagnetic compounds [START_REF] Olsson | Ferromagnetic Compounds for High Efficiency Photovoltaic Conversion: The Case of AlP:Cr[END_REF], 2D materials [START_REF] Chen | A design of intermediate band solar cell for photon ratchet with multi-layer MoS 2 nanoribbons[END_REF] or organic materials [START_REF] Ekins-Daukes | A molecular approach to the intermediate band solar cell: The symmetric case[END_REF]. We also note that previous studies have considered the ratchet mechanism for the up-conversion of low energy photons [START_REF] Trupke | Improving solar cell efficiencies by up-conversion of sub-band-gap light[END_REF]- [START_REF] Sellers | Novel nanostructures for efficient photon upconversion and highefficiency photovoltaics[END_REF].

It is also worth noting that other requirements for efficient IBSC, as e.g. the necessity of a partially filled IB, are also required and were not considered in this paper [START_REF] Luque | Operation of the intermediate band solar cell under nonideal space charge region conditions and half filling of the intermediate band[END_REF]- [START_REF] Li | High saturation intensity in InAs/GaAs quantum dot solar cells and impact on the realization of the intermediate band concept at room-temperature[END_REF]. IV. CONCLUSION QDs and QWs have been considered for the realization of IBSC, a concept that has the potential to overcome the Shockley-Queisser limit. Nevertheless, those nanostructures present intrinsic unfavorable properties: intersubband absorptions are only allowed in narrow energy widths, and non-radiative intersubband relaxations are efficient. We have quantified the effect of those two parameters on the cell efficiency, considering in addition the possibility of introducing a ratchet mechanism for mitigating the performance degradation. This allows us to propose guidelines for the design of efficient IBSCs.

We find minimal values of absorption width and non-radiative recombination that are required for efficiencies exceeding the SQ limit when no ratchet is included. Although not theoretically impossible, currently considered nanostructures are not predicted to fulfill those minimal requirements. A key finding is that the introduction of a ratchet mechanism makes the cell highly resilient to the drawbacks of nanostructures, to the point that any combination of absorption width and non-radiative recombination becomes compatible with efficiencies overcoming the Shockley-Queisser limit. A ratchet system therefore appears as a required feature for the practical realization of nanostructured IBSCs.

We observed in addition that the ratchet cell becomes much less sensitive to non-radiative recombination when the absorption width is reduced, which is a remarkable behavior for a solar cell. As a consequence, despite their apparent radiative recombination ratio deficit compared to QDs, it turns out that QWs are good candidates for the realization of efficient IBSC. To take advantage of this observation, a structure realizing a ratchet on the conduction band using QWs is suggested.

Figure 1

 1 Figure 1 Schematic of an IBSC including a ratchet structure. Compared to a classical IBSC, the conduction band is divided into CB1 and CB2, separated by a ratchet energy ΔE. CB1 is radiatively connected to IB, CB2 is radiatively connected to VB, while no transition from CB2 to IB is allowed. The relaxation structure can be equivalently implemented on the intermediate or the valence band.

Figure 3

 3 Figure 3 Optimum efficiency of an IBSC (yellow squares) and an RBSC (blue circles): (a) as a function of the absorption width, assuming recombinations at the radiative limit (closed symbols) and a radiative recombination ratio of 10 -3 (open symbols). (b) As a function of the radiative recombination rate, assuming an ideal absorption (closed symbols) and an absorption width reduced to 250 meV (open symbols). In both (a) and (b) the green dashed line represents the Shockley-Queisser limit.

Figure 5

 5 Figure 5 Generic quantum well structure to realize the ratchet mechanism on the conduction band. The left section allows two quantized levels, namely IB and CB1, whereas thinner wells on the right-hand side only allow a single level CB2, at an energy ΔE lower than CB1. In order to prevent the recombination between CB2 and IB, intermediate wells can be inserted.


Amaury Delamarre

Daniel Suchet

Nicolas Cavassilas

Zacharie Jehl

Yoshitaka Okada

Masakazu Sugiyama

Jean-Francois Guillemoles
email: jf.guillemoles@cnrs.fr

Jean-François Guillemoles

Electronic ratchets as necessary stepping stones for new PV concepts

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INTRODUCTION

A ratchet is a concept popularized by Richard Feynman [START_REF] Feynman | Ratchet and pawl[END_REF], in which a dented wheel is designed in such a way that it can rotate in one direction only. The allowed rotation involves a dissipation of E so that the pawl can block the reverse rotation. The wheel can be coupled to blades, which, being sensitive to gas pressure fluctuations, could rotate the wheel so as to lift a weight, thereby producing work from a single heat source (Fig. 1).

Figure 1:

The principle of a Feynman Ratchet is illustrated. A dented wheel is designed in such a way that it can rotate in one direction only. The allowed rotation involves a dissipation of E so that the pawl can block the reverse rotation. The wheel can be coupled to blades, which, being sensitive to gas pressure fluctuations, could rotate the wheel so as to produce work. This apparent paradox, which opposes the second law of thermodynamics, can be lifted by a closer look into the details of the device. The pawls has to be loose enough to allow the wheel to rotate at all; but then temperature induced fluctuations, can also lift the pawl, and allow the wheel to rotate backwards. At the end of the day, at constant temperature, the device works as an electrical rectifier where the mechanical torque plays the role of the electric field and the angular velocity that of the electrical current. It is therefore not surprising to see that such a ratchet system can bring significant insights in the context of solar cells, where the same rectifying behavior allows charge separation and thereby energy conversion. And just as in the case of Feynman's ratchet, a closer look on the details of charge separation in a solar cell has led (and still leads) to lively debates concerning the behavior of the ratchet mechanismin this case, the built-in potential [START_REF] Wurfel | Charge Carrier Separation in Solar Cells[END_REF], [START_REF] Würfel | Physics of solar cells: from basic principles to advanced concepts[END_REF].

In this work, we highlight the role of a ratchet system for advanced concepts in PV. We will mainly focus on Intermediate level Solar Cell (IBSC), an advanced concept [START_REF] Wolf | Limitations and Possibilities for Improvement of Photovoltaic Solar Energy Converters: Part I: Considerations for Earth's Surface Operation[END_REF]- [START_REF] Guillemoles | Limiting efficiencies of LDS intermediate band solar cells[END_REF], that has been predicted to result in efficiencies of about 46.7% under one sun illumination [START_REF] Luque | Increasing the efficiency of ideal solar cells by photon induced transitions at intermediate levels[END_REF]. We show that adding a ratchet system to such a device appears not as a mere improvement, but as a requirement to overcome the Shockley-Queisser limit [START_REF] Shockley | Detailed Balance Limit of Efficiency of p-n Junction Solar Cells[END_REF] with materials currently available. Considering electrical figures of merit, we offer a qualitative and quantitative interpretation of the influence of the ratchet on such systems. This analysis can contribute to a better understanding of the physics at stake behind the ratchet mechanism.

II. THEORETICAL MODEL

The IBSC is a concept of device where, in parallel to the direct generation of carriers by absorption of photons, these can be generated in two absorption steps via an intermediate level (Figure ). Nevertheless, photogenerated carriers can also recombine back using the intermediate level, that acts as a catalyst for this reaction. The theoretical performance of IBSC is close to that of triple junctions, series connected, but efficient devices have not been demonstrated yet because of current material limitations in absorption efficiency and bandwidth, and because of non-radiative recombination.

In nanostructures (quantum dots (QD) and quantum wells (QW)), which are one of the most considered path to the practical implementation of an intermediate band [START_REF] Okada | Intermediate band solar cells: Recent progress and future directions[END_REF], this can be explained by two intrinsic properties: the reduced absorption energy width between the quantized states and the continuum, as well as increased non-radiative recombinations. As a consequence, a Voc drop and a limited current increase are commonly observed. We have quantified [START_REF] Delamarre | An Electronic Ratchet Is Required in Nanostructured Intermediate-Band Solar Cells[END_REF] the impact of those two properties on the expected efficiencies, and highlighted that currently considered structures are not compatible with efficiencies in excess of the SQ limit. It has been suggested that a ratchet mechanism can increase the efficiency of an IBSC [START_REF] Delamarre | An Electronic Ratchet Is Required in Nanostructured Intermediate-Band Solar Cells[END_REF]- [START_REF] Green | Energy conversion approaches and materials for high-efficiency photovoltaics[END_REF]. When implemented on the intermediate band (as in Figure (b), although implementation on the conduction or valance band can be also considered [START_REF] Pusch | Limiting efficiencies for intermediate band solar cells with partial absorptivity: the case for a quantum ratchet: Limiting efficiencies for intermediate band solar cells[END_REF]), a ratchet consists in an additional state at a slightly lower energy, from which recombination to the valence band is forbidden. In case of reduced absorption strength, it has been further demonstrated that the efficiency reduction becomes less pronounced [START_REF] Pusch | Limiting efficiencies for intermediate band solar cells with partial absorptivity: the case for a quantum ratchet: Limiting efficiencies for intermediate band solar cells[END_REF]. However, the effect on the non-radiative recombination has been given less consideration so far [START_REF] Olsson | Ferromagnetic Compounds for High Efficiency Photovoltaic Conversion: The Case of AlP:Cr[END_REF], despite a Voc loss being regularly reported in practical realization of IBSC.

(c) Efficiencies of the single junction (full green line), IBSC and RBSC, as a function of the largest gap value (Eg1), in the radiative limit and for an absorption (respectively full orange and blue lines), with a narrow absorption (δE = 250 orange and blue dotted lines), with a narrow absorption and non-radiative recombination ratio of 10 -3 for the transition Eg3 (dashed orange and blue lines).

To account for these effects, the bandgap configuration and the notations of Figure 2 (a) are considered. The limit efficiencies are calculated following a method similar to [START_REF] Yoshida | Photon ratchet intermediate band solar cells[END_REF], to which the non-ideal behavior of the nanostructures will be added. For each transition i, the absorption Gi is given by the convolution of the absorption ai and the sun spectrum, approximated by the black body radiation flux at a temperature Tsun = 6000 K. We consider the efficiencies in non-concentrated light. The radiative emission R i for each transition i is given by the generalized Planck's law, for Tcell = 300 K and using the quasi-Fermi level splitting for each bandgap, Δµi.

The transitions associated with the bandgap Eg1 and Eg2 are considered ideal, that is, a1=1 strictly above Eg1, and a2=1 strictly between Eg2 and Eg1, whereas their recombination is purely radiative. Non-idealities will be considered for Eg3. This will mimic the nanostructured devices in which the intermediate band is obtained thanks to quantized states in the conduction band, such as the widely studied InAs QDs in GaAs [START_REF] Okada | Intermediate band solar cells: Recent progress and future directions[END_REF]. For this transition, both absorption and recombination are assumed only between Eg3 and Eg3+δE where δE is the absorption energy width. Moreover the recombination is increased by considering a radiative recombination ratio rrad such that

𝑅 3 * = 1 𝑟 𝑟𝑎𝑑 𝑅 3 .
The full model can be found in [START_REF] Delamarre | An Electronic Ratchet Is Required in Nanostructured Intermediate-Band Solar Cells[END_REF], [START_REF] Yoshida | Photon ratchet intermediate band solar cells[END_REF], [START_REF] Suchet | Analytical optimization of intermediate band systems: Achieving the best of two worlds[END_REF]. It is firstly used to numerically compute I-V curves and efficiencies for given sets of bandgaps, which allows finding the best configurations. Another solution is to solve the system analytically, in the framework of Lagrange multipliers, which provides additional equations by explicitly considering the system constraints (matching of the currents through Eg2 and Eg3, Δµ1= Δµ2+Δµ3).

III. RESULTS

On Already in that configuration, it is observed that the implementation of a ratchet can result in an efficiency improvement. When the absorption width is reduced (δE = 250 meV), an efficiency reduction is observed for both the IBSC and the RBSC. When considering in addition a nonradiative recombination ratio of rrad=10 -3 for Eg3, we observe that the IBSC gives efficiencies lower than what can be achieved with the best single junction. Oppositely, we find that the introduction of the reduced radiative efficiency has a negligible impact on the RBSC performances.

A more detailed study [START_REF] Delamarre | An Electronic Ratchet Is Required in Nanostructured Intermediate-Band Solar Cells[END_REF] shows that a ratchet makes the cell tolerant to non-ideal characteristics, to that point that any value of the absorption width and radiative recombination ratio is compatible with efficiencies in excess of the Shockley-Queisser limit. This result gives new opportunities to realize IBSC with quantum structures. So far, QD have been preferred to QW for the practical implementation of IBSC, since the former are expected to exhibit larger radiative recombination ratios. We show here that when a ratchet system can be included, the reduced radiative efficiency of the QW can be acceptable to overcome the SQ limit, thanks to their larger expected absorption. For a better understanding of the mechanisms at stake, the Voc, Jsc and FF values are compared in Figure , instead of the aggregated efficiency value. As could be seen on Fig 2 .c, for too small values of Eg1, the optimal situation in the absence of ratchet aborts the intermediate band and behaves as a single junction. Above this threshold, we observe that the V OC of the cell including a ratchet keeps values close to that of the single junction, whereas the standard IBSC shows values reduced by about 10 %. Conversely, the RBSC gives a Jsc close to that of the IBSC, much above that of SJ, therefore taking advantage of the additional generation from the IB.

Figure 3 Top: Open-circuit voltage, as a function of the Eg1 bandgap, for an IBSC with a ratchet (filled blue symbols) and an IBSC w/o ratchet (open yellow symbols). A narrow absorption (δE = 250 meV) and non-radiative recombination ratio of 10-3 are considered for the transition Eg3

Middle: same for the short circuit current. Bottom: same for the FF.

These numerical observations have been supported analytically It has been shown that the introduction of a ratchet, by offering one additional degree of freedom to the system thanks to the ΔE adjustable parameter, allows for the simultaneous preservation of the open-circuit voltage and the current matching of the transition associated to Eg2 and Eg3. Oppositely, a standard IBSC cannot fulfill those two criteria, resulting in a trade-off providing lower performances. This is illustrated in Figure , where the global open-circuit voltage, the generation through the sub-Eg1 transitions (Eg2 and Eg3) and the energy conversion are represented as a function of the ratchet energy. the energy conversion graph, we have differentiated the energy provided by the largest gap (Eg1) and the energy provided by the two lowest gaps (together Eg2 and Eg3). In those graphs, we fixed δE = 200 meV and rrad=10 -3 . Eg1 is also fixed at 1.86 eV, which is the value that optimizes the IBSC efficiency. We observe that the ratchet is at the same time able to achieve current matching (G2 equaling G3) and to optimize the recombination, the V oc reaching a plateau, equivalent to the single junction at a similar Eg1.

Figure 4 Evolution of recombination, generation and conversion

efficiency as a function of the ratchet value, for Eg1 fixed at 1.86 eV, which is the optimal value for an IBSC when the absorption width is 200 meV and the radiative efficiency 10 -3 . Eg2 is optimized for each ratchet value. Top: Voc as a function of ratchet energy E.

The green line represents the value of the single junction for the same Eg1. Middle: photogenerated current via transition 2 (middle) and 3 (small gap) as a function of E. Bottom: fraction converted by the Eg1 transition and fraction converted with the help of the intermediate band (i.e. converted by Eg2 and Eg3). The green dashed line represents the efficiency of the single junction with the same Eg1.

For an implementation of a ratchet in real world, it is needed that some transitions must be forbidden (Figure ), and this calls for specific selection rules. They may be spatial, in momentum space, or spin related [START_REF] Okada | Intermediate band solar cells: Recent progress and future directions[END_REF]. Structure with quantum cascade, exhibiting spatial separation have been demonstrated experimentally recently [START_REF] Vaquero-Stainer | Semiconductor nanostructure quantum ratchet for high efficiency solar[END_REF] . Structures using spin selection rules could be realized in principle with dilute magnetic semiconductors [START_REF] Olsson | Ferromagnetic Compounds for High Efficiency Photovoltaic Conversion: The Case of AlP:Cr[END_REF], albeit in practice it is hard to obtain long minority carrier lifetimes and high Curie temperatures in these materials. Finally, another unexplored possibility is some 2D materials (MoS2 nanoribbons) that could exhibit relevant selection properties in reciprocal space [START_REF] Chen | A design of intermediate band solar cell for photon ratchet with multi-layer MoS2 nanoribbons[END_REF]. These conclusions were obtained with detailed balance calculations using the generalized Planck law. When more elaborate models are used, such as NEGF are used [START_REF] Cavassilas | Beneficial impact of tunneling in nano-structured intermediate-band solar cell[END_REF], an additional Voc loss is found due to broadening of the final states that leads to increased recombination. Moreover, Jsc losses from lower optical absorption were also found.

IV.

DISCUSSION

Several interpretations are often considered to explain the influence of a ratchet. It is sometimes argued that the ratchet induces an energy dissipation ΔE as carriers relax from the top to the bottom of the shifted intermediate band. However, this picture does not hold when considering the full process driving a carrier from the valence to the conduction band. In an IBSC, when absorbing two photons with energies 𝐸 𝛾1 and 𝐸 𝛾2 , the carrier will dissipate (𝐸 𝛾1 -𝐸 𝑔2 ) when relaxing after the first transition from VB to IB and ( 𝐸 𝛾2 -𝐸 𝑔3 ) when relaxing after the second transition from IB to CB. In an RBSC, the carrier would first dissipate (𝐸 𝛾1 -𝐸 𝑔2 ), then ΔE through the ratchet, then (𝐸 𝛾2 -(𝐸 𝑔3 + ΔE)) over the second transition. The total amount of dissipated energy is unchanged; what changes is the band in which this energy is dissipated, and the spectral range of absorbed photons.

It is also worth noting that, in some conditions, the optimal ratchet can be negative (see Figure 5). In this case, carriers brought to the intermediate band have to climb up a potential barrier, gathering energy from the ambient temperature, before they undergo a second transition to the conduction band. This situation occurs when photogeneration G3 is initially so low that its increase compensates the additional recombination induced by ΔE<0. While this situation is purely conceptual (an appropriate treatment would require considering the influence of populations on the absorptivity), it highlights the limits of an interpretation based on relaxation. Ratchet have also been suggested to reduce the "Boltzmann" entropy creation [START_REF] Yoshida | Photon ratchet intermediate band solar cells[END_REF], which is due to the geometrical mismatch between the absorption and emission optical étendues. As concentration is increased, it appears indeed that the role of the ratchet becomes more and more limited in the ideal case [START_REF] Yoshida | Photon ratchet intermediate band solar cells[END_REF], even if a direct link with the absorption and emission étendues mismatch is not obvious. This numerical observation on the impact of concentration can be expanded with the other non-idealities considered above. We have found that the ratchet remains useful even under full concentration when non-radiative recombination (which produce additional entropy) are included, but not when considering a narrow absorption (which does not produce additional entropy). While there is a clear role of the ratchet energy to improve net current photogeneration through the IB while keeping current matching, the numerical observation above may also support the idea that the ratchet helps mitigating entropy production in the system.

V. CONCLUSION

We have discussed the ratchet analogy that has been proposed in the context of intermediate band solar cell. In this course, we have quantified the impact of non-radiative recombination and reduced absorption on the efficiency of nanostructured IBSC. The impact of those two intrinsic deficiencies result in strong performance reductions, unless an electronic ratchet is introduced, in which case efficiencies in excess of the SQ limit can be expected.

We have analyzed in depth the origin of the action of the ratchet on efficiency and concluded after careful examination of the proposed explanations that the main actions are (i) matching of the currents in the 2 step photogeneration process and (ii) reduction of the recombination losses.

These conclusions were obtained with detailed balance calculations using the generalized Planck law. When more elaborate models are used, such as NEGF, other factors affecting recombination and generation were found that also serendipitously pointed out to additional beneficial action of the ratchet.

Figure 2

 2 Figure 2 Band diagram of (a) an IBSC and (b) an RBSC with a ratchet implemented in the IB. (c) Efficiencies of the single junction (full green line), IBSC and RBSC, as a function of the largest gap value (Eg1), in the radiative limit and for an absorption (respectively full orange and blue lines), with a narrow absorption (δE = 250 orange and blue dotted lines), with a narrow absorption and non-radiativerecombination ratio of 10 -3 for the transition Eg3 (dashed orange and blue lines).

  Figure (c) are represented the efficiency as a function of Eg1, for the single junction, the IBSC and the RBSC. The position of the intermediate level and of the ratchet is optimized for highest efficiency. The full lines represent the efficiencies in the ideal configuration (rrad=1, unlimited δE).

Figure 5

 5 Figure 5 Power conversion efficiency as a function of the ratchet ΔE in the ideal case with Eg1 = 2 eV and Eg2 = 1.1 eV.