Thierno-Moussa Bah

Stanislav Didenko

Di Zhou

Tianqi Zhu

Hafsa Ikzibane

Stephane Monfray

Thomas Skotnicki

Emmanuel Dubois

Jean-François Robillard

A CMOS Compatible Thermoelectric Device made of Crystalline Silicon Membranes with Nanopores

Keywords: Silicon, Silicon-on-Insulator, CMOS, Thermoelectricity, Energy Harvesting, Phonon Scattering, Peltier cooling

Herein, we report the use of nanostructured crystalline silicon as a thermoelectric material and its integration into thermoelectric devices. The proof-of-concept relies on the partial suppression of lattice thermal conduction by introducing pores with dimensions scaling between the electron mean free path and the phonon mean free path. In other words, we artificially aimed at the well-known "electron crystal and phonon glass" trade-off targeted in thermoelectricity. The devices were fabricated using CMOScompatible processes and exhibited power generation up to 5.5mW/cm 2 under a temperature difference of 280 K. These numbers demonstrate the capability to power autonomous devices with environmental heat sources using silicon chips of centimeter square dimensions. We also report the possibility of using the developed devices for integrated thermoelectric cooling.

Introduction

The blooming of the Internet of Things (IoT) and wireless autonomous sensor nodes has been delayed owing to the lack of reliable, safe, and low-cost energy sources [START_REF] Nordrum | Popular Internet of Things Forecast of 50 Billion Devices by 2020 Is Outdated[END_REF]. Thermoelectric generators (TEG) have these advantages [START_REF] Haras | Thermoelectricity for IoT -A review[END_REF]. Silicon has the advantages of being abundant, non-toxic. Numerous facilities and CMOS technological processes for low-cost mass production exist for silicon while conventional thermoelectric materials (Bi, Sb, Te, Pb alloys) are far less developed. The research for Si-based TEG [START_REF] Dávila | Monolithically integrated thermoelectric energy harvester based on silicon nanowire arrays for powering micro/nanodevices[END_REF][START_REF] Wang | Characterization of a Bulk-Micromachined Membraneless In-Plane Thermopile[END_REF][START_REF] Xie | Fabrication, and Characterization of CMOS MEMS-Based Thermoelectric Power Generators[END_REF] has attracted increasing interest. However, silicon is an inefficient thermoelectric material owing to its high thermal conductivity of 150W/m/K [START_REF] Haras | Thermoelectric energy conversion: How good can silicon be?[END_REF], whereas bismuth telluride alloys exhibit hundred times lower conductivity (~1.5W/m/K) [START_REF] Pinisetty | Thermal conductivity of semiconductor (bismuth-telluride)-semimetal (antimony) superlattice nanostructures[END_REF]. Moreover, the dimensionless figure of merit zT which characterizes the thermoelectric efficiency, is approximately 1 for Bi2Te3 at room temperature [START_REF] Očko | Thermoelectric materials: problems and perspectives[END_REF] and below 0.01 for bulk silicon [START_REF] Hao | Theoretical studies on the thermoelectric figure of merit of nanograined bulk silicon[END_REF]. At temperature T, zT is defined as

𝑧𝑇 = 𝜎 • 𝑆² 𝜅 • 𝑇 (1)
σ, S and  being the electrical conductivity, Seebeck coefficient, and thermal conductivity, respectively. The thermal conductivity accounts for the performance gap between the Si and Bi2Te3 micro-TEGs (Figure 1). Efforts are oriented towards cutting the phonon part of heat transport, which is the dominant contribution in semiconductors [START_REF] Jin | Prediction of phonon and electron contributions to thermal conduction in doped silicon films[END_REF]. The phonon contribution is evaluated in the diffusive regime as

 = 𝑐 • 𝑣 •  3 (2)
where c, v and  are the specific heat, group velocity (average elastic wave velocity), and phonon mean free path, respectively. The heat capacity depends on the atomic density of the material, and thus cannot be modified. Group velocity modulation would require modifying the atomic structure of the crystal to induce coherent effects, unexpected at room temperature. Suppression of the lattice contribution can be achieved by reducing the phonon mean free path through diffusion through artificial defects and boundaries. This process can be obtained by the use of thin films [START_REF] Haras | Thermoelectric energy conversion: How good can silicon be?[END_REF], nanowires [START_REF] Boukai | Silicon nanowires as efficient thermoelectric materials[END_REF], the addition of impurities or polycrystalline silicon [START_REF] Lee | Nanoporous Si as an Efficient Thermoelectric Material[END_REF], and silicon nanopatterning [START_REF] Haras | Fabrication of Thin-Film Silicon Membranes With Phononic Crystals for Thermal Conductivity Measurements[END_REF][START_REF] Tang | Holey Silicon as an Efficient Thermoelectric Material[END_REF]. Previous studies have shown that holey structures can further decrease the thermal conductivity with a minor impact on the electrical conductivity [START_REF] Yu | Reduction of thermal conductivity in phononic nanomesh structures[END_REF]. The combination of thin films and phonon scattering by nanometric pores lowers the thermal conductivity to 34 W/m/K [START_REF] Haras | Fabrication of Thin-Film Silicon Membranes With Phononic Crystals for Thermal Conductivity Measurements[END_REF] and even to 2 W/m/K [START_REF] Tang | Holey Silicon as an Efficient Thermoelectric Material[END_REF][START_REF] Yu | Reduction of thermal conductivity in phononic nanomesh structures[END_REF] depending on the specific design. The first value was for samples with partially perforated patterns and thus constitutes a conservative hypothesis. The lowest value represents the amorphous limit of silicon thermal conductivity [START_REF] Wada | Thermal Conductivity of Amorphous Silicon[END_REF], which makes it a very ambitious limit that can probably be obtained only at the cost of substantial material defects.

In addition to reducing the thermal conductivity of silicon, research in recent years has dealt with the development of silicon-based (mainly polysilicon and silicon nanowires) thermoelectric micro-harvesters or generators. Figure 1 reports the main results [START_REF] Dávila | Monolithically integrated thermoelectric energy harvester based on silicon nanowire arrays for powering micro/nanodevices[END_REF][START_REF] Xie | Fabrication, and Characterization of CMOS MEMS-Based Thermoelectric Power Generators[END_REF][START_REF] Ziouche | Silicon-Based Monolithic Planar Micro Thermoelectric Generator Using Bonding Technology[END_REF][START_REF] Li | Chip-Level Thermoelectric Power Generators Based on High-Density Silicon Nanowire Array Prepared With Top-Down CMOS Technology[END_REF][START_REF] Tomita | 10μW/cm2-Class High Power Density Planar Si-Nanowire Thermoelectric Energy Harvester Compatible with CMOS-VLSI Technology[END_REF][START_REF] Perez-Marín | Micropower thermoelectric generator from thin Si membranes[END_REF] from state-of-art siliconbased micro-harvesters with respect to state-of-art bismuth telluride harvesters [START_REF] Bottner | Micropelt miniaturized thermoelectric devices: small size, high cooling power densities, short response time[END_REF]. The silicon TEGs exhibit output power in the µW/cm 2 range against mW/cm 2 for vertical Bi2Te3 [START_REF] Bottner | Micropelt miniaturized thermoelectric devices: small size, high cooling power densities, short response time[END_REF][START_REF] Tainoff | Network of thermoelectric nanogenerators for low power energy harvesting[END_REF] TEG for same temperature differences. The thermoelectric properties of these materials primarily explain this gap.

In this study, we demonstrate a proof-of-concept thermoelectric device using patterned silicon membranes. The device consists of a silicon frame equipped with planar configuration thermopiles. For the sake of demonstration, the hot source is emulated by resistive heaters. A thermoelectric converter could be made from this device, by adding a silicon cap, mounted onto the hot source and etched in order to distribute the heat flow at the locations of the resistive heaters. An example of such configuration is given by Ziouche et al. [START_REF] Ziouche | Silicon-Based Monolithic Planar Micro Thermoelectric Generator Using Bonding Technology[END_REF]. In the following section, we present the fabrication process and discuss its performance with respect to plain membranes devices and especially according to the state-of-art presented in Figure 1. We also mention modeling results reported in [START_REF] Bah | Performance Evaluation of Silicon Based Thermoelectric Generators Interest of Coupling Low Thermal Conductivity Thin Films and a Planar Architecture[END_REF] that suggests that the performance of PE Sibased TEG compares favorably with that of Bi2Te3 based TEG when the heat flow is not controlled by a bulky heat sink (i.e., in the specific use case for IoT, which is deeply constrained by compactness and low form factor requirements).

Experimental Methods

The devices were fabricated from SOI wafers using the conventional CMOS processes (Figure 2). The process is mainly based on previous work carried out to reduce the thermal conductivity of silicon by means of silicon thinning coupled to pore lattices [START_REF] Haras | Fabrication of Thin-Film Silicon Membranes With Phononic Crystals for Thermal Conductivity Measurements[END_REF][START_REF] Lacatena | Phononic engineering of silicon using "dots on the fly" e-beam lithography and plasma etching[END_REF]. The departure point is an SOI wafer with a 70 nm thick active layer (a) high-resolution ebeam lithography and Cl2/Ar reactive ion etching (RIE) was performed to define the pore patterns (b). Then, the silicon is p-boron and n-phosphorus doped by ion implantation with a target of 10 19 cm -3 . Sheet resistivity measurements give 2.3410 -2 .cm and 6.3810 -3 .cm for p and n-type respectively corresponding to an effective doping level of 610 18 cm -3 and 910 18 cm -3 (c). A low-stress silicon nitride was then deposited (d) to insulate the thermopiles from the central platinum resistive heater emulating the devices' hot source. Silicon nitride also aims at improving the mechanical strength of structures. Cavities are etched around the thermopiles as apertures for subsequent suspension (e). Metallic sensors and leads are obtained by platinum and gold evaporation (f). Before metallization, the Si membranes a thermal oxide layer is grown to protect them from xenon-difluoride (XeF2) etching (g). This last step of the process ensures thermal insulation of the Si membranes from the wafer. The device was made of five thermopiles connected electrically in series and thermally in parallel (Figure 3-a). It was equipped with four pads to supply heating power and I-V measurements.

Each thermopile (50µm long, 10µm wide and 205 nm thick) was composed of two n-and p-doped membranes, as shown in Figure . 3-b. The membrane stack is composed of the following components: 15nm SiO2/62 nm Si/ 145 nm SiO2. At the center, two platinum straps electrically short-circuit the p-n boundary (Figure 3-c). Platinum resistive heaters were deposited over the silicon nitride layer to avoid current leakage (Figure c,d). On half of the devices, a lattice of 40 nm diameter pores and 100 nm pitch patterns was etched (Figure 3-e, f). These dimensions are chosen as a compromise between technological capabilities and minimizing the pitch size, neck size and maximizing the porosity in order to enhance the effect of thermal conductivity reduction [START_REF] Yanagisawa | Impact of limiting dimension on thermal conductivity of one-dimensional silicon phononic crystals[END_REF][START_REF] Dechaumphai | Thermal transport in phononic crystals: The role of zone folding effect[END_REF].

The electrical continuity of the thermopiles was ensured using a metallic Pt strap between adjacent thermopiles. The devices were studied both in terms of energy harvesting under a temperature gradient (Seebeck effect) and thermoelectric cooling when current flows across thermopiles (Peltier effect). Regarding Seebeck effect, the devices were characterized by a thermo-resistive methodology [START_REF] Tang | Holey Silicon as an Efficient Thermoelectric Material[END_REF][START_REF] Yu | Reduction of thermal conductivity in phononic nanomesh structures[END_REF][START_REF] Maire | Reduced thermal conductivities of Si one-dimensional periodic structure and nanowire[END_REF]. Indeed, Pt is known to have a significant temperaturedependent coefficient of resistance 2.5×10 -3 /K in this case. This value differs from the bulk value because the Pt film is 30 nm thin and was systematically measured for each sample before the thermometry measurement. Infrared imaging (Figure 4-b) demonstrates the functionality of these heaters.

The electrical power (PH) produced at the center of the thermopiles is converted into heat flux (QH) and, in the first approximation, equally diffuses across the thermopiles (Figure 4-c) to the cold ends. Measurements were performed under vacuum to eliminate conducto-convective fluxes on in air and guide heat flux into the membranes. During the measurements, the frame wafer was maintained at a constant temperature of 25°C. Because the thermopiles were etched from the SOI substrate, efficient heat coupling was achieved at the cold ends. This assumption is supported by the high thermal conductance of the frame wafer and the infrared thermometry. It follows that the cold sides of the membranes are at the same temperature as the substrate, which was verified by IR imaging. The characterization was performed in two steps: i) by variation of the Pt heating current, a V-T characteristic enables measurement of the Seebeck voltage, and ii) at a constant Pt heating current, the I-V curves characterize the power generation. To capture the Peltier effect, a current was applied to the thermopiles and imaged using IR thermometry in ambient air. This electrical current should produce heat absorption or generation at the center of the thermopiles depending on its direction.

Results & Discussions

First, the heaters are calibrated in order to achieve a given temperature difference. We note that patterned thermopiles sustain a higher temperature difference for a given amount of heating power. There is a two-fold interest in this result. First, the use of a thin-film planar device significantly increases the thermal resistance to approximately 10 6 K/W. This value governs the temperature difference in a TEG and should be compared to the typical Bi2Te3 TE legs of 10 4 K/W [22] (10 2 K/W for bulk Si with typical leg dimensions of 115 mm 3 ). In addition to the thin-film geometry, patterning enhances thermal resistance by a factor of 1.8. The reduction in thermal conductivity [START_REF] Haras | Fabrication of Thin-Film Silicon Membranes With Phononic Crystals for Thermal Conductivity Measurements[END_REF][START_REF] Tang | Holey Silicon as an Efficient Thermoelectric Material[END_REF] with patterning explains this gain. Figure 5 shows the Seebeck voltage generated as a function of temperature difference across the thermopiles. Again, characterization was performed on plain and patterned samples. As expected, the Seebeck voltage increased linearly with the temperature gradient. Furthermore, the plain devices produced an electrical voltage close to the value expected by a Finite Element Method (FEM) software using a Seebeck coefficient Sp-Sn= 545µV/K. The FEM cannot model the physical effect of pore pattering, therefore, the modeling results can only be compared to those of plain devices. The measurements showed Seebeck coefficients per thermopile of 570µV/K and 590µV/K for the plain devices and 820µV/K and 840µV/K for the patterned devices. Power generation was investigated by studying the I-V characteristics of the thermopiles as a function of the applied temperature difference. Figures 6-a and-b report respectively the plain and patterned thermopiles I-V curves for T from 0 to 282 K temperature differences, respectively. Typical thermoelectric characteristics were obtained, and a generator regime (IV <0) was observed. The electrical current is given by

𝐼 = 𝑉 -𝑉 (Δ𝑇) 𝑅 ( 3
)
where VGbias is the thermopile bias voltage, VS(ΔT) is the Seebeck voltage at a given temperature difference across the thermopiles, and RG is the devices' internal resistance, which is approximately 130kΩ and 260kΩ for plain and patterned thermopiles, respectively. This increase of electrical resistivity with patterning is confirmed by independent fourprobe resistivity Van-der-Pauw measurements on dedicated areas of the wafer. Because the dimensions of the patterned lattice are much larger than the typical electron mean free path (a few nanometers [START_REF] Qiu | First-principles simulation of electron mean-free-path spectra and thermoelectric properties in silicon[END_REF]), it is unlikely that the resistivity is significantly affected by the pore patterns. We note that the change of resistivity between plain and patterned membranes can be evaluated as an effective change of the dopant concentration. (see Suppl. Material) A concomitant increase in the Seebeck coefficient and resistivity is then expected from the Mott relation [START_REF] Cutler | Observation of Anderson Localization in an Electron Gas[END_REF][START_REF] Fritzsche | A general expression for the thermoelectric power[END_REF] which order of magnitude is quite in agreement with the 1.4 times increase of the Seebeck coefficient.

From the I-V curves, we calculated the output power corresponding in the generation regime and extracted the maximum output power. Figures 6-c and -d present the maximum output power densities with respect to the temperature difference across the thermopiles (Figure 6-c) and according to the heating power of the Pt heaters (Figure 6-d). Figure 6-c shows that compared to the temperature difference across the thermopiles, the plain and patterned devices exhibit comparable performance. This observation reflects that the Seebeck effect is enhanced by patterning while the electrical resistance is increased, leading to a constant power factor. The maximum output power is given by ( 4), and at a given temperature difference, it depends only on the Seebeck coefficient and electrical resistance.

𝑃 = 𝑉 4 * 𝑅 = 𝑆 * ∆𝑇 4 * 𝑅 (4)
where S and ΔT are the Seebeck coefficient and the temperature difference across the thermopiles, respectively. Globally, patterning would therefore have no impact on generator performance. Figure 6-c also shows a comparison between the measurements and FEM modeling showing perfect agreement with the measurements.

The patterned and plain devices were compared again, but this time with respect to the heating power of the Pt heaters (Figure 6-d). Under these conditions, we can notice that, compared to the heating power, patterned devices exhibit better performance than plain thermopiles. This highlights the benefits of thermal gradient management for thermoelectric harvesting. Indeed, [START_REF] Xie | Fabrication, and Characterization of CMOS MEMS-Based Thermoelectric Power Generators[END_REF] gives the maximum output power, which depends not only on the Seebeck coefficient and electrical resistance that compensate each other, but also on the temperature difference across the thermopiles. The temperature difference across the PE thermopiles was higher for a given heating power (see Suppl. Material).

Q and rTEG are the heat gradient across the thermopiles (or heating power) and the thermal resistance of the thermopiles, respectively. Figures 6-c and -d highlight then the fact that the main benefit of pore patterning is better thermal gradient management, especially owing to a thermal conductivity reduction [START_REF] Haras | Fabrication of Thin-Film Silicon Membranes With Phononic Crystals for Thermal Conductivity Measurements[END_REF][START_REF] Tang | Holey Silicon as an Efficient Thermoelectric Material[END_REF][START_REF] Yu | Reduction of thermal conductivity in phononic nanomesh structures[END_REF]. The gap between the patterned and plain thermopiles in figure 6-d can be improved with a better patterns design. From the range of output power, we can easily imagine the use of a few cm 2 of such generators to supply power to autonomous sensor nodes [START_REF] Vullers | Micropower energy harvesting[END_REF]. Moreover, compared to the state-of-the-art (Figure 1), with respect to the silicon-based thermoelectric microharvesters, the developed devices mainly exhibit better performance for the same temperature differences. The material crystalline structure and harvester architectures can explain this difference in performance. Polysilicon has lower electrical resistivity than crystalline silicon. Low-dimensionality systems, such as silicon nanowires, feature improved thermoelectric efficiency. Tomita et al. [START_REF] Tomita | 10μW/cm2-Class High Power Density Planar Si-Nanowire Thermoelectric Energy Harvester Compatible with CMOS-VLSI Technology[END_REF] managed to develop Si nanowire (NWs)-based harvesters exhibiting state-of-the-art performance. As expected, the performance of the vertical bismuth telluride micro harvesters [START_REF] Bottner | Micropelt miniaturized thermoelectric devices: small size, high cooling power densities, short response time[END_REF] is four orders over the developed devices. This is related to several causes. First, bismuth telluride legs feature lower electrical resistance, less than 1Ω, against tens of kΩ per leg for the developed devices. Secondly, the bulk Bi2Te3 parts exhibit low thermal conductivity. However, these performances are greatly diminished when the heat cooling capacity is low. This cooling capacity is quantified by an effective conductoconvective transfer rate h (W/m²/K) (see Suppl. Material).

The characterization of the devices is completed with the observation of Peltier cooling. Figure 7 shows infrared (IR) images of a device biased at 7.5V. The left picture represents the result of current propagation from p to n end, and the right picture represents the current propagation from n to p end.

The figure shows that, depending on the current direction, heat generation or absorption is observed at the center of the thermopiles, which is characteristic of the Peltier effect. The heat generation and absorption were only observable at the center of the thermopiles because, first, the sample was maintained at 75°C during all the experiments, and second, the silicon membranes were too thin to absorb IR waves. The Peltier effect and, in particular, Peltier cooling, is then possible on silicon technologies, opening the way to a possible integration of thermoelectric coolers. However, it is worth noting that the TEG device was not designed for Peltier cooling, which is naturally much lower than Joule heating.

Conclusion

In summary, a phonon scattering approach was used to reduce the heat conduction in crystalline silicon thin membranes. The fabrication and demonstration of thermoelectric conversion using crystalline silicon was shown. The device principle relies on better use of the available thermal difference in a planar thin membrane architecture and additional pore patterning. The results demonstrated the benefits of patterning. Moreover, phonon scattering induces an increase in the Seebeck coefficient compensated by an increase in electrical resistance and better thermal gradient management through the thermopiles. The net gain remains positive and we report 400nW/cm 2 under T = 3K up to 5.5mW/cm 2 under T = 282K according to the temperature difference across the thermopiles. This CMOS-compatible device opens perspective in the relevant range (cm² footprint, mW power) for IoT power supply. In addition, the studied devices can be considered as state-of-the-art silicon micro-harvesters. This study also highlights the feasibility of using integrated crystalline silicon thermoelectric coolers.

Figure 1 :

 1 Figure1: State-of-art micro-harvesters' maximum output power per footprint area as a function of the temperature difference across the generator[START_REF] Dávila | Monolithically integrated thermoelectric energy harvester based on silicon nanowire arrays for powering micro/nanodevices[END_REF][START_REF] Xie | Fabrication, and Characterization of CMOS MEMS-Based Thermoelectric Power Generators[END_REF][START_REF] Ziouche | Silicon-Based Monolithic Planar Micro Thermoelectric Generator Using Bonding Technology[END_REF][START_REF] Tomita | 10μW/cm2-Class High Power Density Planar Si-Nanowire Thermoelectric Energy Harvester Compatible with CMOS-VLSI Technology[END_REF][START_REF] Perez-Marín | Micropower thermoelectric generator from thin Si membranes[END_REF][START_REF] Bottner | Micropelt miniaturized thermoelectric devices: small size, high cooling power densities, short response time[END_REF][START_REF] Tainoff | Network of thermoelectric nanogenerators for low power energy harvesting[END_REF][START_REF] Li | Microfabrication of thermoelectric materials by silicon molding process[END_REF]

Figure 2 :

 2 Figure 2: Patterned silicon membranes thermoelectric devices' fabrication process.

Figure 3 :

 3 Figure 3: (a) Scanning Electron Micrograph of a thermoelectric device made of 5 suspended thermopiles. (b) Close-up view of three thermopiles. (c) Platinum resistive heater. (d) Detail of the membranes pore patterns. (e) Cross section Transmission Electron Micrograph of the membrane.

Figure 4 :

 4 Figure 4: SEM view of a 5 thermopiles device (a) IR imaging (performed in ambient air) after Pt resistive serpentines voltage bias (b) and cross-sectional scheme of a thermopile.

Figure 5 :

 5 Figure 5: 5 plain thermopiles (black squares & red dots) and 5 patterned thermopiles (green & blue triangles) output voltage with respect to the temperature difference across thermopiles

Figure 6 :

 6 Figure 6: 5 plain (a) and PE (b) thermopiles I-V curves with temperature gradient and measured maximum output power density versus temperature difference across thermopiles (c) and versus the Pt heaters' heating power (d)

Figure 7 :

 7 Figure 7: IR imaging pictures of the device with respect to the electrical current direction in the thermopiles in Peltier mode. Left: current for p end to n end and right: current from n end to p end.

Acknowledgments

This work has received: i) funding from STMicroelectronics-IEMN common laboratory; ii) funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013) ERC Grant Agreement no. 338179, iii) support from the French RENATECH network, iv) support from the NANO2017 program, and v) support from the French government through the National Research Agency (ANR) under program PIA EQUIPEX LEAF ANR-11-EQPX-0025 and project TIPTOP ANR-16-CE09-0023.

Data availability statement

All data that support the findings of this study are included within the article. More details regarding the fabrication process and characterization methodology can be found in the supplementary material.


Minjia Shi

Yaya Li

Wei Cheng
email: wei.cheng@telecom-paris.fr

Dean Crnković
email: deanc@math.uniri.hr

Denis Krotov
email: krotov@math.nsc.ru

Patrick Solé
email: sole@enst.fr

Self-dual bent sequences for complex Hadamard matrices

Keywords: PUF functions, Bent sequences, Hadamard matrices, regular Hadamard matrices, Bush-type Hadamard matrices AMS Classification (MSC 2010): Primary 94D10, Secondary 15B34

. In this paper we introduce the analogous notion for complex Hadamard matrices, and we study the self dual class in length at most 98. We use three competing methods of generation: Brute force, Linear Algebra and Groebner bases. Regular Hadamard matrices and Bush-type Hadamard matrices provide many examples. We introduce the strong automorphism group of complex Hadamard matrices, which acts on their associated self-dual bent sequences. We give an efficient algorithm to compute that group.

Introduction

Complex Hadamard matrices are matrices C of order n with entries in the fourth roots of unity Ω 4 = {±1, ±i} satisfying

CC * = nI,
where * denotes the transpose conjugate, and I is the identity matrix of order n. They were introduced by Turyn and studied by Seberry [10], and Kharaghani [START_REF] Kharaghani | Regular complex Hadamard matrices[END_REF][START_REF] Kharaghani | The excess of complex Hadamard matrices[END_REF], among others. A survey is [9]. A webpage is [14]. Complex Hadamard matrices are conjectured to exist for all even n [10].

If n is even and the sum of two squares n = a 2 + b 2 , then at least one regular C is conjectured to exist.

Recently, a notion of bent sequences attached to Hadamard matrices was introduced in [START_REF] Solé | Bent Sequences over Hadamard Codes for Physically Unclonable Functions[END_REF] from a motivation of security. In a companion paper [START_REF] Shi | Self-dual Hadamard bent sequences[END_REF] the self-dual subclass bent sequences for Hadamard matrices is studied. In the present paper we conduct the analogous study for complex Hadamard matrices. The main hurdle in this generalization was in the definition, as explained in the Preliminaries section.

Preliminaries

A self-dual bent sequence attached to the complex Hadamard matrix C is defined as X ∈ Ω n 4 such that

CX = λX,
where λ is an eigenvalue of C.

Proposition 1

If there exists at least one self-dual bent sequence of length n, then n is the sum of at most two squares.

Proof.

By the Hadamard property we see that |λ| 2 = n. By eigenvalue definition, we see that

λ = a + ib ∈ Z[i]. Taking squared norms we get n = a 2 + b 2 . If one of a, b is zero then n is a square. If both are non zero then n is a sum of two squares.
An equivalent definition is thus:

let n = a 2 + b 2 , with a, b ≥ 0. A self-dual bent sequence attached to C is defined as X ∈ Ω n 4 such that CX = (±a + ±ib)X,
where (±a+±ib) is an eigenvalue of C. Note that b+ia = i(a-bi) = i(a+bi) * , so that swapping a and b amounts to simple changes in C and X.

In the case n = 2 2m = (2 m ) 2 , and C the Sylvester Hadamard matrix of order n such sequences were studied in [START_REF] Sok | Patrick Solé Classification and Construction of quaternary self-dual bent functions[END_REF]. The case of n a square and C an arbitrary real Hadamard matrix is treated in [START_REF] Shi | Self-dual Hadamard bent sequences[END_REF].

The even integers ≤ 98 and sum of at most two squares are

C = H t -iH t -iH t H t .
If the row sum of H is σ, then C is regular of constant row sum (1 + i)σ.

Remark:

The matrix C in this Theorem has an order 4 or a multiple of 8. This is a special case of Theorem 1 of [10]: The Kronecker product of a Hadamard matrix of order n by a complex matrix of order h is a complex Hadamard matrix of order hn. The Magma command is KroneckerProduct(A, B) for the Kronecker product of A by B.

Example: The following program constructs 5 complex Hadamard matrices of order 32 from the 5 non-equivalent Hadamard matrices of order 16.

R<i>:=CyclotomicField(4); C2:=Matrix(R,2,2,[1, i,i,1]); D:=HadamardDatabase();Q:=RationalField(); for j:=1 to 5 do H:=Matrix(D,16,j);H:=ChangeRing(H,R); C32:=KroneckerProduct(C2,H); Eigenvalues(C32); end for; Note that the eigenvalues of these matrices all have squared norm 32. So there are more eigenspaces to consider.

Bush type

A complex analogue of the Bush-type Hadamard matrix is the following result, inspired by [START_REF] Kharaghani | On the twin designs with the Ionin-type parameters[END_REF]Th. 1]. Similar constructions appear in [4, §5].

Theorem 1 Let there exists a complex Hadamard matrix of order 2n. Then there exists a Bush-type complex Hadamard matrix of order 4n 2 . This matrix is regular of row sum 2n. Proof. Let K be a normalized complex Hadamard matrix of order 2n, and let r 1 , r 2 , . . . , r 2n be the row vectors of K. Let C i = r t i r i , for i = 1, 2, . . . , 2n. Then the following properties are easy to check:

1. C t i = C i , for i = 1, 2, . . . , 2n. 2. C 1 = J 2n , C i J 2n = J 2n C i = 0, for i = 2, 3, . . . , 2n. 3. C i C * j = 0, for i = j, 1 ≤ i, j ≤ 2n. 4. C 1 C * 1 + C 2 C * 2 + • • • + C 2n C * 2n = 4n 2 I 2n . Let H = circ(C 1 , C 2 , . . . , C 2n
), the block circulant matrix with the first row C 1 , C 2 , . . . , C 2n . Then H is a Bush-type complex Hadamard matrix of order 4n 2 . The regularity follows by property 2.

Example:

The following matrix K is a complex Hadamard matrix of order 4

K =     1 1 1 1 1 i -1 -i 1 -1 1 -1 1 -i -1 i     , The matrix C 1 = J 4 . The matrices C 2 , C 3 and C 4 are C 2 =     1 i -1 -i i -1 -i 1 -1 -i 1 i -i 1 i -1     , C 3 =     1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1     , C 4 =     1 -i -1 i -i -1 i 1 -1 i 1 -i i 1 -i -1     .
The matrix

H = circ(C 1 , C 2 , C 3 , C 4
) is a Bush-type complex Hadamard matrix of order 16.

Another construction of Bush-type complex Hadamard matrix is as follows.

Proposition 3 Let there exist a Bush-type Hadamard matrix of order n 2 . Then there exists a Bush-type complex Hadamard matrix of order n 2 having the entries belonging to the set Ω 4 .

Proof.

Let H = [H i j] be a a Bush-type Hadamard matrix of order n 2 , where H i j, 1 ≤ i, j ≤ n, are blocks of order n. By multiplying the off-diagonal blocks with i, we obtain a Bush-type complex Hadamard matrix.

Conference matrices

A construction indicated in [2, p.67] and in [10, Theorem 3] is connected to Paley II. The Jacobsthal matrix is the matrix C q defined in [7, chap. 2, §3] by C q (x, y) = χ(y -x), for x, y ∈ F q . Here χ denotes the quadratic character defined by the three following mutually exclusive cases:

χ(z) =      0 if z = 0, 1 if z = , -1 if z = .
Note that C q is symmetric if q ≡ 1 (mod 4), since then -1 is a quadratic residue. Its extended version S q is obtained by adding a border of ones according to the rule in [START_REF] Van Lint | A course in Combinatorics[END_REF][START_REF] Solé | Bent Sequences over Hadamard Codes for Physically Unclonable Functions[END_REF].

S q = 0 j j t C q ,
with j being an all one row vector of length q.

Proposition 4 If q is a prime power ≡ 1 (mod 4), and S q denotes the extended Jacobstahl matrix, then iI +S q is a complex Hadamard matrix of order q + 1.

Proof.

It is known that S q is a so-called conference matrix [7, chap. 2, ( 16)], and therefore satisfies S q S t q = qI. Hence (iI + S q )(iI + S q ) * = (iI + S q )(-iI + S q ) = (q + 1)I, where the second equality follows by S q = S t q = S * q . The calculation in the proof extends to the situation when we replace S q by conference matrices with zero diagonal [START_REF] Balonin | A Review and New Symmetric Conference Matrices[END_REF]. In particular this constructs complex Hadamard matrices of orders {10, 18, 26, 50, 58, 74, 82, 90, 98}.

Unfortunately, the spectrum of a matrix in that family is not favorable to the existence of self-dual bent functions.

Proposition 5 Let q be a prime power ≡ 1 (mod 4), and with S denoting the extended Jacobstahl matrix, write C = iI + S q . The minimal polynomial of C is x 2 -2ix -(q + 1).

Proof. Since S is real and symmetric, we get C * = -iI + S q = C -2I. The Hadamard relation entails then C(C -2iI) = (q +1)I. the result follows.

Given that the roots of the quadratic are i ± √ q, they belong to Q(i) iff q is a perfect square. That leaves the following orders to test for that construction: {10, 26, 50, 82}.

Williamson type

A Hadamard matrix H of order 4m is said to be quaternionic if there are four matrices A, B, C, D of order m such that

H = A ⊕ I + B ⊕ i + C ⊕ j + D ⊕ k,
where i, j, k are quaternionic units given by

i =     0 1 0 0 -1 0 0 0 0 0 0 -1 0 0 1 0     , j =     0 0 1 0 0 0 0 1 -1 0 0 0 0 -1 0 0     , k = ij.
If furthermore, we assume A, B, C, D to be symmetric and circulant, we shall say that H is Williamson type.

By Lemma 3 of [START_REF] Kharaghani | The excess of complex Hadamard matrices[END_REF], we know that the existence of such a matrix entails that of a complex Hadamard matrix of the form S T -T S where the overline denotes complex conjugation. One may take S = X + iY and T = V + iW, where X = (A + B)/2, and Y = (A -B)/2. Similarly V = (C -D)/2, and W = (C + D)/2. Lemma 6 of [START_REF] Kharaghani | Regular complex Hadamard matrices[END_REF] exploits this correspondance to construct regular complex Hadamard matrices. In the next result, we use a similar construction. To construct B k we apply a greedy algorithm. We construct the list J of the indices of the columns of B k as follows.

1. Initialize J at J = [START_REF] Balonin | A Review and New Symmetric Conference Matrices[END_REF] 2. Given a column of index we compute the ranks over the complex of r and r of the submatrices of B with k rows and columns defined by the respective lists J and J = Append(J, )

3. If r < r then update J := J

Repeat until |J| = rank(B)

Remark: If the first column of B is zero, step 1 does not make sense, but then there is no self-dual bent sequence in that situation, as all eigenvectors have first coordinate zero.

Complexity: Roughly of order v 3 2 k . In this count v 3 is the complexity of computing an echelonized basis of the eigenspace of C attached to a + bi. The complexity of the invertible minor finding algorithm is of the same order or less.

Numerical examples 7 Conclusion 8 Appendix on Hadamard matrices

In this appendix, we indicate how we constructed the matrices used in our computer experiments.

Order 16

One matrix is obtained applying our Theorem 1 . The other matrix is obtained from the real Bush-type Hadamard matrix from a matrix in [START_REF] Kharaghani | On the twin designs with the Ionin-type parameters[END_REF], by multiplying all off-diagonal blocks by i.

Order 36

One matrix is obtained from the complex Hadamard matrix of order 6 using our Theorem 1. Two matrices are obtained from the real Bush-type Hadamard matrices of order 36 given in [START_REF] Janko | The existence of a Bush-type Hadamard matrix of order 36 and two new infi nite classes of symmetric designs[END_REF][START_REF] Janko | A block negacyclic Bush-type Hadamard matrix and two strongly regular graphs[END_REF].

Order 64

Two matrices were constructed from two complex Hadamard matrices of order 8 obtained from the database [14] upon using our Theorem 1. Another matrix was obtained from a real Hadamard matrix of order 8, using the proposition 3.

Theorem 2

 2 If there is a Hadamard matrix H of order 4t 2 with structureH = R S -S R then the matrix E given by 2E = (R + S) -i(R -S) is a complex Hadamard matrix. If, furthermore, X Y is a self-dual bent sequence for H = S -R R S then U + iV is a self-dual bent sequence for E with U = X + Y and V = X -Y.

4 . 5 . 6 .

 456 For all Z ∈ Ω k 4 solve the system in Y given by Z = Y B k Compute the remaining v -k entries of Y B. If these entries are in Ω 4 declare Y B a self-dual bent sequence attached to C

Proof.

The first assertion is Lemma 4 in [START_REF] Kharaghani | Regular complex Hadamard matrices[END_REF]. The second assertion is a simple calculation starting from

and replacing E by its value. Separating real and imaginary parts we get the system

Remark: By [START_REF] Rj Turyn | An infinite class of Williamson matrices[END_REF], we know that the matrix H in this theorem can be constructed in relation with Williamson matrices.

Coding theoretic interpretation

Let C be a quaternary code of length n over the alphabet Ω 4 . Let Z be the Z 4 code determined by i Z = C. The distance properties of C for the squared Euclidean distance d E are equivalent to the distance properties of Z for the Lee distance d L because of the following identity, easily verified by induction on n :

A Hadamard code H is a code of length n over Ω 4 with |H| = n codewords that are pairwise orthogonal for the standard hermitian inner product <, > in dimension n. Thus we can think of its codewords as the rows of a complex Hadamard matrix of size n. The deviation θ(C, x) of an arbitrary vector x ∈ Ω n 4 from C is defined as

It can be seen by expanding < x -y, x -y > that (< x, y >) = n -d L (u, v), for all x, y ∈ Ω n 4 . The total deviation of the code C is then

Proposition 6 If there is a bent sequence for a complex Hadamard matrix H of order n, then its corresponding Hadamard code C has deviation θ(C) = √ n. Proof. See [START_REF] Solé | Bent Sequences over Hadamard Codes for Physically Unclonable Functions[END_REF]Th. 1] for a euclidean inner product version.

Recall that the covering radius of a code Z ⊆ Z n 4 is given by

The simple inequality (< x, y >)

Combining this fact with the above Proposition yields the following bound.

Corollary 1 If there is a bent sequence for a complex Hadamard matrix H of order n, then the covering radius of its attached

This is less satisfying that [11, Lemma 1].

5 Search methods

Brute force

This method is only applicable for small v's.

1. Construct C a Hadamard matrix of order v like in [START_REF] Solé | Bent Sequences over Hadamard Codes for Physically Unclonable Functions[END_REF] by using Magma database.


Diana O Nekrasova

Alexander A Tsirlin

Marie Colmont

Oleg I Siidra

Ángel M Arévalo-López

Olivier Mentré
email: olivier.mentre@univ-lille.fr

From (S = 1) spin-hexamer to spin-tetradecamer by CuO interstitials in A2Cu3O(CuO)x(SO4)3, A = alkali

Na, K)2Cu3O(SO4)3 compounds form structural chains of Cu6 hexameric units with nominal S=1 spins due to the interplay between inner strong antiferromagnetic and ferromagnetic exchanges. We show here that the lattice relaxation after the replacement of alkali by bigger Rb and Cs ones is accompanied by the insertion of neutral CuO species into (Rb,Cs)2Cu3O(CuO)x(SO4)3 phases. Structurally, interstitial CuO link two next Cu6 units in longer Cu14 tetradecameric ones. For A=Cs (x=0.5), the cationic ordering is perfect inside a doubled-cell superstructure. Magnetically, the original Cu14 units consist of frustrated fragments of S=1/2 spin-ladder, with ferromagnetic rung-like but antiferromagnetic leg-like and next-nearest neighbors (NNN) couplings. It returns S=1 Cu14 spin-clusters, effective around 100K. Our DFT calculations and susceptibility fits also show at low temperature, that they interact in 2D-lattices, despite the existence of short inter Cu-Cu distances between two next clusters along pseudo 1D-chains.

Introduction

Generally speaking, 'intercalation' in inorganic chemistry refers to a reversible ion insertion in host materials, the typical working principle of rechargeable batteries at both the positive and negative electrodes. However, a plethora of synthetic routes have been developed for tailoring, for instance, mixed anion compounds keeping a common structural framework, at low temperature by rational anionic exchange. [START_REF] Kageyama | Expanding Frontiers in Materials Chemistry and Physics with Multiple Anions[END_REF] The cationic part of the crystalline network itself can be modified by a exsolution that enables the creation of cation-depleted lattices and their reversible refilling upon oxidation, for instance after the exsolution of nanometric nickel clusters from perovskites phases 2 with enhanced catalytic performances, but also more rarely in oxidizing media as observed after exsolution of nano-scaled hematite in LiFe2-x(PO4)2 [START_REF] Hamelet | The Effects of Moderate Thermal Treatments under Air on LiFePO4-Based Nano Powders[END_REF] or BaFe2-x(PO4)2. [START_REF] David | Reversible Topochemical Exsolution of Iron in BaFe2 2+ (PO4)2[END_REF] The latter process gives rise to a variety of original ordered Fe-vacancy depleted 2D triangular lattices. [START_REF] Alcover | Reversible Exsolution of Nanometric Fe2O3 Particles in BaFe2-x(PO4)2 (0 ≤ x ≤ 2/3): The Logic of Vacancy Ordering in Novel Metal-Depleted Two-Dimensional Lattices[END_REF] In all of the aforementioned examples the insertion of cations involves a redox-change of the parent or pristine compound.

In this article, we show an original A2Cu3O(CuO)x(SO4)3 series of topologically related crystalstructures obtained by a charge-neutral insertion of sizeable amount of CuO in lacunar Cu 2+ sulfates. The crystallographic principles engaged here imply a steric driving force mediated by the ionic size of the alkali. The resulting compounds feature 0D magnetic clusters acting as single magnetic units with distinct inner and inter energy scales. In such systems, an interesting physics may emerge as, for example, in Cu2OSeO3 where the Cu4 tetrahedra act as building blocks for several skyrmionic phases. [START_REF] Janson | The Quantum Nature of Skyrmions and Half-Skyrmions in Cu2OSeO3[END_REF] Besides large 3D-shaped large molecule-like units such as found in polyoxometalate, for instance giving rise to S = 1/2 effective spin in the V15 compound [START_REF] Chiorescu | Butterfly Hysteresis Loop and Dissipative Spin Reversal in the S = 1/2 , V15 Molecular Complex[END_REF] , most common examples and theoretical efforts [START_REF] Khatua | Effective Theories for Quantum Spin Clusters: Geometric Phases and State Selection by Singularity[END_REF] rely on the trimerization, or tetramerization of local spins, most often within trigonal layered topologies in real compounds, e.g. LiVO2, [START_REF] Pen | Orbital Ordering in a Two-Dimensional Triangular Lattice[END_REF] LiZn2Mo3O8. [START_REF] Sheckelton | Possible Valence-Bond Condensation in the Frustrated Cluster Magnet LiZn2Mo3O8[END_REF] It leads to non-magnetic, spin singlet or low collective pseudospins with exotic behavior, see the so-called hybrid electrons in S = 1/2 in V3 triangles of the recent-GaV4O8 phase. [START_REF] Aguilar-Maldonado | Hybrid Electrons in the Trimerized GaV4O8[END_REF] In oxides, only rare examples show spin condensation into more larger units, as observed in La3Cu 2+ 2V 5+ O9 with 8-spin and 9-spin clusters of S = 1/2, both sub-units of the parent 2D-Kagomé lattice. [START_REF] Robert | Formation of Collective Spins in Frustrated Clusters[END_REF] To the best of our knowledge, only few examples concern the spin condensation in 1D fragments of S = 1/2 spins, as observed in the title A2Cu3O(CuO)x(SO4)3 series. Here, the A = Na, K members (x = 0) consist of condensed individual S = 1/2 Cu 2+ ions into S = 1 Cu6 hexamers. [START_REF] Fujihala | Cluster-Based Haldane State in an Edge-Shared Tetrahedral Spin-Cluster Chain: Fedotovite K2Cu3O(SO4)3[END_REF][START_REF] Nekrasova | Magnetic Hexamers Interacting in Layers in the (Na,K)2Cu3O(SO4)3 Minerals[END_REF][START_REF] Furrer | Spin Triplet Ground-State in the Copper Hexamer Compounds A2Cu3O(SO4)3 (A = Na,K)[END_REF] The structural characterization of puninite (A = Na) [START_REF] Siidra | Copper Oxosulphates from Fumaroles of Tolbachik Volcano: Puninite, Na2Cu3O(SO4)3 a New Mineral Species and Structure Refinements of Kamchatkite and Alumoklyuchevskite[END_REF] euchlorine (mixed A = Na,K) [START_REF] Scordari | The Crystal Structure of Euchlorine, NaKCu3O(SO4)3, Locality: Vesuvius, Italy[END_REF] , and fedotovite (A = K) [START_REF] Starova | The Crystal Structure of Fedotovite, K2Cu3O(SO4)3[END_REF] and their synthesis and crystallization by the solid-state route [START_REF] Nekrasova | Magnetic Hexamers Interacting in Layers in the (Na,K)2Cu3O(SO4)3 Minerals[END_REF] have been followed by an intensive research on their intriguing magnetic properties. [START_REF] Fujihala | Cluster-Based Haldane State in an Edge-Shared Tetrahedral Spin-Cluster Chain: Fedotovite K2Cu3O(SO4)3[END_REF][START_REF] Nekrasova | Magnetic Hexamers Interacting in Layers in the (Na,K)2Cu3O(SO4)3 Minerals[END_REF][START_REF] Furrer | Spin Triplet Ground-State in the Copper Hexamer Compounds A2Cu3O(SO4)3 (A = Na,K)[END_REF][START_REF] Furrer | Spin-Coupling Topology in the Copper Hexamer Compounds A2Cu3O(SO4)3 (A = Na,K)[END_REF][START_REF] Hase | A Possible Magnetic Structure of the Cluster-Based Haldane Compound Fedotovite K2Cu3O(SO4)3[END_REF] In agreement with the crystal structure, there is a consensus on the occurrence of the magnetic Cu6 hexamers with the nominal S = 1 macrospins below 100 K. These hexamers were proposed to form an exotic Haldane state due to the dominant one-dimensional interactions along the depleted chains (b direction). [START_REF] Fujihala | Cluster-Based Haldane State in an Edge-Shared Tetrahedral Spin-Cluster Chain: Fedotovite K2Cu3O(SO4)3[END_REF] However, later studies argued that the true interaction geometry should be 2D involving layers of the Cu6 hexamers (bc plane), and quantum magnetic order sets in at low temperatures despite the vanishingly small interlayer coupling (a direction). [START_REF] Nekrasova | Magnetic Hexamers Interacting in Layers in the (Na,K)2Cu3O(SO4)3 Minerals[END_REF] Figure 1a shows the chains of edgesharing OCu4 anti-tetrahedra as described in ref. [16] holding the Cu6 magnetic hexamers, arranged in 1D-chains along b-and separated by the sulfate groups into 2D-layers in-plane.

The interleaves along the a-axis are occupied by apical sulfate ligands and alkali species, see Fig S1 . In the following, we extend this series to larger A = Rb and Cs for which the lattice expansion allows the incorporation of sizeable stoichiometric-or-not amount of CuO up to the formation of a well-ordered double superstructure. It follows that not only magnetic interactions but also the size of the magnetic cluster can be chemically tuned giving rise to a unique magnetic tetradecamer built by the spin-1/2 Cu 2+ ions associated with a S = 1 collective spin. The magnetic response and the underlying interactions of these exotic tetradecamers are probed by magnetic susceptibility measurements along with the ab-initio (DFT) calculations and exact diagonalization.

Experimental Section

Synthesis. The various materials were prepared by a solid-state reaction from a stoichiometric mixture of anhydrous precursors A2(SO4) (A = Rb, Cs), CuSO4 and CuO in 1:2:1 ratio. The mixtures were ground in an agate mortar and loaded into gold plates and kept at 560 °C for 12 hours in the air followed by a cooling back to room temperature in 10 hours. The resulting solid products consist of single crystals (typical crystal size 0.15×0.15×0.10 mm) and were ground in an agate mortar to prepare the polycrystalline phase.

It was noticed that the synthesis did not give a pure phase. The maximal purity was reached by the preparation of Cs2Cu3+xO1+x(SO4)3 (x = 0, 0.25, 0.5, 0.75, 1, 1.25) samples of syntheses for the cesium phase, increasing the CuO amount, as discussed latter.

One impurity of formula Cs2Cu(SO4)2 was synthesized by a solid-state reaction in vacuum. A mixture of Cs2SO4, CuSO4, and CuO was pelletized (ca. 5 × 3 mm pellet size) and loaded into a quartz ampoule. The ampoule was heated up to 700 °C for 3 hours and kept for 10 h. Then it was cooled to 500 °C for 96 hours and further cooled to room temperature in 12 hours. The products contained dark blue crystals of Cs2Cu(SO4)2, which was not obtained as a single phase material yet.

Single-crystal X-ray diffraction.

Single crystals were visually isolated under microscope and mounted on thin glass fibers for X-ray diffraction analysis using Bruker APEX II DUO X-ray diffractometer with a microfocus X-ray tube operated with MoKα radiation at 50 kV and 0.6 mA. The data were integrated and corrected from absorption using a multiscan type model implemented in the Bruker programs APEX3 21 . More than a hemisphere of X-ray diffraction data was collected for each crystal. Main crystallographic information, atom coordinates, thermal displacement parameters and selected interatomic distances are provided in Supporting Information (Table S1 including crystal and refinement data for the full series, and Tables S2,S3, S4 for A= Rb and Tables S5, S6, S7 for A= Cs. CCDC numbers 2098142 and 2098187 contain the supplementary crystallographic data concerning new structures of this paper. These data can be download free of charge from the Cambridge Crystallographic Data Centre through the link: www.ccdc.cam.ac.uk/structures. Powder X-ray diffraction. Typical powder XRD analyses of the compounds were collected at room temperature in the 2θ range of 10-60 ° with a scan step width of 0.02 ° and a scan rate of 0.5 s/step using a D8 steps of 1.0s/0.02°. The Profile-matching refinements were carried out using JANA2006 [START_REF] Petrícek | Crystallographic Computing System JANA2006: General Features[END_REF] . The background was fitted using Chebyshev polynomial function, and the peak shapes were described by a Pseudo-Voigt function. The firs are shown Fig1.a-c and validate the preparation of single phase materials.

EDS study.

It was performed on a FEI Technai G2-20 twin TEM microscope. The powder was crushed and dropped in the form of alcohol suspension on carbon supported nickel grids followed by evaporation under ambient condition. The EDS study performed on 20 crystals are given in the concerned sections. However, electron-diffraction before and after EDS analysis proved a collapsing of the crystal structures in the electron beam.

Physical properties measurements. Magnetic properties and heat capacity were measured out using a PPMS Dynacool (9T) from Quantum Design using zero field cooling (ZFC) and field cooling (FC) procedures under fields 1000 Oe. Magnetization versus H was measured at 2K and 300K. Specific heat was measured from 1.9K to 300K using materials pressed into pellets.

First-principle calculations. All calculations were performed in the VASP code [START_REF] Kresse | Efficient Iterative Schemes for Ab Initio Total-Energy Calculations Using a Plane-Wave Basis Set[END_REF] with the Perdew-Burke-Ernzerhof exchange-correlation potential. [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] Experimental atomic positions were relaxed, and exchange couplings for the fully relaxed crystal structure were subsequently obtained by a mapping procedure. [START_REF] Xiang | Predicting the Spin-Lattice Order of Frustrated Systems from First Principles[END_REF] The exchange couplings Jij are defined similar to Ref. 14.

All calculations were performed on the 2x2x2 k-mesh within DFT+U using the on-site Coulomb repulsion Ud = 9.5 eV, Hund's coupling Jd = 1 eV, and double-counting correction in the atomic limit. [START_REF] Nath | Quasi-Two-Dimensional S = 1/2 Magnetism of Cu[C6H2(COO)4][C2H5NH3]2[END_REF][START_REF] Nath | Magnetization and Spin Dynamics of the Spin S = 1/2 Hourglass Nanomagnet Cu5(OH)2(NIPA)4⋅10H2O[END_REF] For structural relaxation, the ferromagnetic spin configuration was used.

Exact diagonalization. Magnetic susceptibility of the Cu14 tetradecamer was calculated by a full diagonalization performed in ALPS. [START_REF] Albuquerque | The ALPS Project Release 1.3: Open-Source Software for Strongly Correlated Systems[END_REF]

Results and Discussion

The synthesis for the A = Rb and Cs analogs was first attempted from a 1:2:1 stoichiometric mixture. After isolating a single crystal out of the Rb sample, the structure refinement using the parent model (A = Na, K; monoclinic, C2/c) returns R1,obs = 4.11%. However, the Cu and O over-stoichiometry was detected by significant residual electron-density peak corresponding to 7% of interstitial CuO in the empty sites separating the original Cu6 hexamers, see Fig. 1b. The final Rb2Cu3.07O1.07(SO4)3 stoichiometry was systematically refined on several crystals, and their consideration lowered Robs by 0.9%, which is significant at this stage. In the cesium compound, the inter-layer a-parameter shows a drastic expansion, whereas only modest or even negative expansion is seen along the other directions, as reported in Fig. 1. The crystal structure refinement shows same features as in the Rb case, but with a larger amount of the interstitials, resulting in the refined composition Cs2Cu3.5O1.5(SO4)3. Concerning the average unit cell, the consideration of CuO interstitials (ref. occ. 47%) lowered the Robs value from 12.7% to 6.6%. It perfectly matches our EDS analyzis. Most of the tested crystals demonstrated a perfect ordering of the interstitial ions, resulting in a doubled unit cell. Figure 2 shows a comparison between the hk1 reconstructed XRD precession layers for A = K and A = Cs samples with evidences of i) the breaking of the C-centering of the parent subcell. ii) satellites with k = n + ½ indices indicating the (a, 2×b, c) supercell. Group-theory analysis using Isosubgroup [START_REF] Stokes | An Internet Tool for Generating Isotropy Subgroups of Crystallographic Space Groups[END_REF] with the (0, ½, 0) super structure over the C2/c subgroup (which describes nicely the disordered average structure), leads to P2/a, P2/c and P2 as the possible space groups. Prior to the magnetic characterization, polycrystalline material was prepared for Rb2Cu3.07O1.07(SO4)3 using the stoichiometric mixture of the reactants listed above. The Le Bail fit shown in Fig. S2 confirms that the sample is a single phase. The estimated composition by EDS (to consider with caution due to the structure collapsing under electron beam and the exposition in air) gives atomic ratio Rb:2.00, Cu:3.02, S:3.12. For A = Cs, the preparation turned out to be more complicated due to the increased stability of secondary phases in our experimental conditions. As we mentioned before we synthesized the samples corresponding to the Cs2Cu3+xO1+x(SO4)3 (x = 0, 0.25, 0.5, 0.75, 1, 1.25) stoichiometries. For x=0, we found by EDS a major composition with atomic ratio: Cs:2.00, Cu:3.45, S:2.97 with same comments as for the previous compound.

The XRD patterns, refined parameters, and magnetic susceptibility data are given in Fig. S3 and Fig. S4. The refined lattice parameters show no significant evolution and prove that there is no other composition that the one refined from single crystal data, i.e. x=0.5 : However, for the low Cu 2+ experimental excess (x = 0 and 0.25), the Cs2Cu(SO4)2 impurity was observed (the space group is P21/n, unit cell parameters: a = 9.685 is responsible for a significant χ(T) upturn at low-T, such that its absence in the sample is mandatory for a decent magnetic study.

Finally, the x = 0 mixture shows the lowest amount of impurity and, according to the following chemical balance additionally verified by XRD (see its Lebail fit on Fig. S5).: Cs2Cu3.0O1.5(SO4)3 → 0.8 Cs2Cu3.5O1.5(SO4)3 + 0.2 Cs2Cu(SO4)2 + 0.2 SO3 This sample was selected for the study of magnetic properties in agreement with our EDS results. In that aim, the magnetic data discussed below have been corrected for the corresponding 6.6% of Cu 2+ ions of the solid Cs2Cu(SO4)2 impurity, assumed to be paramagnetic in the absence of experimental magnetic data. Indeed, in this impurity (to be published elsewhere) the refined crystal structure reveals only Cu-O-O-Cu exchange inside isolated dimers, with d(Cu-Cu)=3.9Å, but parallel CuO4 plaquettes hosting the magnetic dx 2 -y 2 orbital and Cu-O-O-Cu dihedral angles unfavorable for significant J exchanges.

Concerning the presence of the Cu14 tetradecamers and their ordered arrangement in the bulk polycrystalline sample employed in the magnetic measurements, it is worth noting that the lattice parameters refined for all members of the x = 0.0 -1.25 series show only minor deviations from those of the Cu-ordered x = 0.5 single crystalline specimen. This similarity validates the nominal x 0.5 composition of the target phase. Structurally, the alternation of the tetradecamers in the 1D chains is also mandatory due to their specific coordination by the SO4 groups. Low angle-XRD powder patterns systematically reveal broad and weak supercell peaks (1 ½ 0) and (0 ½ 1) with their intensities growing with x, see Fig. S7. Similar to the single crystal, one cannot exclude minor deviations from the Cs2Cu3.5O1.5(SO4)3 over-stoichiometry and, therefore, the broadness of 2xb diffracting satellites suggest the plausible loss of the ideal long-range Cu 2+ ordering in-plane as well as between the layers.

All compounds of the series show a characteristic bend of the inverse susceptibility around 100 K (Fig. 4a) that marks the transition from the paramagnetic regime with individual Cu 2+ ions to the collective regime, where each cluster acts as a single spin. [START_REF] Nekrasova | Magnetic Hexamers Interacting in Layers in the (Na,K)2Cu3O(SO4)3 Minerals[END_REF] This behavior indicates the separation of energy scales, with strong couplings within the clusters and weak couplings between them. Between 400 and 150 K, Curie-Weiss fits return eff = 1.77 and 1.97 B/Cu for A = Rb and Cs, respectively, in agreement with the expected value for S = 1/2. However, in the low-temperature part, the effective moment is reduced to 0.84 B/Cu for A = Cs, significantly lower compared to A = Na and K (1.11 and 1.16 B/Cu respectively), but intermediate for A = Rb (1.01 B/Cu). Altogether, this suggests a weaker contribution of individual spins in the Cu14 tetradecamers than in the Cu6 hexamers. At first sight, the evolution of the heat capacity divided by temperature (normalized per one Cu unit) between the various members of the series, shows the vanishing of the long-range anomaly increasing the A cationic radius, which was for A = K correlated to a possible crystallinity loss [START_REF] Nekrasova | Magnetic Hexamers Interacting in Layers in the (Na,K)2Cu3O(SO4)3 Minerals[END_REF] though magnetic ordering was detected experimentally by PND [START_REF] Hase | A Possible Magnetic Structure of the Cluster-Based Haldane Compound Fedotovite K2Cu3O(SO4)3[END_REF] , see Fig. 4b. For A = Na and K, the magnetic couplings within the hexamers were analyzed in several previous studies, by means of the energies of local spin excitations probed by inelastic neutron scattering [START_REF] Furrer | Spin Triplet Ground-State in the Copper Hexamer Compounds A2Cu3O(SO4)3 (A = Na,K)[END_REF][START_REF] Hase | A Possible Magnetic Structure of the Cluster-Based Haldane Compound Fedotovite K2Cu3O(SO4)3[END_REF] , and also via systematic DFT calculations. [START_REF] Nekrasova | Magnetic Hexamers Interacting in Layers in the (Na,K)2Cu3O(SO4)3 Minerals[END_REF] According to a fragment of spinladder nomenclature, a strong ferromagnetic Cu-Cu exchange (rung-like) is found for edgesharing CuO4 plaquettes with the Cu-O-Cu angles close to 90°, while both the leg-like and diagonal ones are also strong but antiferromagnetic (AFM). Most of the intra-hexamer bringing frustration into the Cu2-3-4 and Cu4-5-6 triangles, exchange couplings within the tetradecamer tend to stabilize the S = 1 ground state, sketched in Fig. 3b and Fig. 3c.

For a comparison with the experimental data, we performed exact diagonalization for the Cu14 cluster and fitted high-temperature part of the magnetic susceptibility, see Fig. 5. We used the simplified tetradecamer model with one ferromagnetic coupling Jrung and one antiferromagnetic coupling Jlegs = Jdiag. Additionally, the frustrating NNN coupling JNNN was included. The fitted values ( Jlegs = Jdiag = 250K, and Jrung =-250K, JNNN=350K, g=2.23) have the same order of magnitude as the DFT values. Moreover, we find a better fit when including the coupling JNNN that appears to be integral to the Cu14 tetradecamers in this system. Nevertheless, with or without JNNN, the Cu14 units develop the same S = 1 state below about 100 K. Field-dependent magnetization displays a unsaturated plateau around 1.2 B/Cu14, which is also compatible with the S = 1 state. For A = Na and K, weak interactions between the hexamers trigger the long-range magnetic order below 3.0-3.5 K (Fig. 4b). [START_REF] Nekrasova | Magnetic Hexamers Interacting in Layers in the (Na,K)2Cu3O(SO4)3 Minerals[END_REF] Similar weak AFM interactions should be present in the Cs case according to our DFT calculations. In details, the inter-cluster exchanges calculated by DFT are listed in Table S9. Our results suggest that no significant coupling takes place along the b direction. Indeed between the terminal Cu 2+ of the Cu14 clusters along b, the exchanges are calculated very weak and ferromagnetic (J/Kb =-0.8 K) along SSE's due to unfavorable, nearly perpendicular Cu-O bonds with the Cu-O-O-Cu dihedral angles  80°. On the opposite, the in-plane SSEs with the -type overlap between the two Cu-O bonds (45°<Cu-O-O-Cu <2°) and short O-O distances promote two significant J's of 53 and 74 K between two Cu14 units, see Fig. 3d, despite the longer Cu-Cu distances above 6.13 Å. These values refer to the interactions between individual S=1/2 centers and should be renormalized when calculating an effective interaction between the S=1 Cu14 at lower temperature. [START_REF] Nekrasova | Magnetic Hexamers Interacting in Layers in the (Na,K)2Cu3O(SO4)3 Minerals[END_REF] Experimentally, these interactions manifest themselves in the departure of the magnetic susceptibility from the Curie law for S = 1. We fitted the low-temperature susceptibility with

Concluding Remarks:

The structural principles engaged in the A2Cu3O(CuO)x(SO4)3 series allow for the passage from

.

  The over-stoichiometric CuO creates local junctions between two next Cu6 hexamers and transforms them into a single Cu14 tetradecamer. Such tetradecamers are random in the A = Rb case and involve severe local re-orientation of the sulfate groups involved in the Cu 2+ coordination. It may become ordered for A = Cs when the full occupation of the Cu 2+ interstitials is reached.

Figure 1 :

 1 Figure 1: Main crystallographic parameters for A2Cu3O(CuO)x(SO4)3 vs. A and projected "oxo-centered" crystal representation of a) A = Na, K and (Na,K), x = 0. b) A = Rb (x = 0.07). c) A = Cs (x = 0.5).

Figure 2 :Figure 3 :

 23 Figure 2: Reconstructed hk1 precession image from single-crystal XRD data for a) A = K, b) A = Cs

  (3) Å, b = 7.920(3) Å, c = 12.141(4) Å, β = 91.416(8)º, V = 931.0(5) Å 3 , R1 = 4.38%) (see Lebail fit on Fig. S5 and Fig.S6). Above x = 0.25, this latter Cs2Cu(SO4)2 is absent, but an unidentified phase labelled X occurs along with minor amounts of CuO, which is responsible for the χ(T) anomaly around 250 K. The X phase

Figure 4 :

 4 Figure 4: a) inverse susceptibility and b) Cp/T versus temperature normalized per one Cu 2+ ion for pertinent members of the series.

2 3 Na 2 3 Rb 2

 3232 Cu 3 O(SO 4 ) Cu 3 O(SO 4 ) 3 K 2 Cu 3 O(SO 4 ) 3 Rb 2 Cu 3.07 O 1.07 (SO 4 ) 3 Cs 2 Cu 3.5 O 1.5 (SO 4 ) Cu 3.07 O 1.07 (SO 4 ) 3 Cs 2 Cu 3.5 O 1.5 (SO 4 ) 3 interactions tend to align the outer four spins parallel to each other and antiparallel to the two central ones, thus giving rise to the S = 1 ground state of the hexamer, see Fig. 3a. Our DFT calculations reveal a similar coupling scheme for the Cu14 tetradecamers of Cs2Cu3.5O1.5(SO4)3 where the two independent crystallographic Cu14 sub-units a and b were considered equivalent for simplicity. The main calculated intra-exchanges are shown in Fig. 3c, and listed exhaustively in Table S8. All exchanges along legs, diagonal and next-nearest neighbor ones (NNN) are antiferromagnetic with calculated values up to J1-2= 359 K using the relaxed crystal structure and GGA+U (U = 9.5 eV, J = 1 eV). Similar to the A = Na, K cases, the edge-sharing rung-like exchanges are ferromagnetic with values up to -96 K for J3-3 because the Cu-O-Cu angles are close to 90°. Apart from the next-nearest neighbor (NNN) couplings J2-4 and J4-6

Figure 5 :

 5 Figure 5: (a) χ -1 (T) and (b) χ(T) QMC-fits using adapted models along the temperature range. The best low-T fit returns a square lattice of S = 1 clusters depicted in (b). The g values are 2.23, 2.21, 2.27 for the orange , blue and green calculated lines respectively.

  : J leg, diag = 250 K = -J rung , J NNN = 350 K 2D-coupled spins-1 (J = 3.3 K) c (emu/mol) of S = 1 spins interacting on the square lattice with the effective coupling of 3.3 K and g = 2.23. Interestingly, even this model does not describe the data below 12 K, and no signs of a magnetic transition are seen in the specific heat data. This may indicate that some structural disorder related to the intercalated CuO units (incomplete formation of the double-b superstructure) becomes important at very low temperatures and affects interactions between the Cu14 tetradecamers. One should also keep in mind that below 10 K, the experimental data are doubtful because spurred by the correction of secondary phase ideally considered as paramagnetic.

  1D-like Cu6 hexamers (x = 0, A = Na, K) arranged in depleted chains to longer Cu14 tetradecamers (0.07<x<0.5, A = Cs, Rb) after insertion of neutral CuO species. The compositional and topological changes are driven by the variation of the internal chemical pressure and cell volume dependent on the A alkali size. It is striking that the scheme of FM and AFM magnetic exchanges effective in the individual sub-units return S = 1 magnetic clusters in both cases, themselves interacting in 2D layers, rather than in 1D chains due to specific Cu-O-O-Cu magnetic paths. Ideally one could imagine for x = 1 the occurrence of infinite chains, not observed experimentally, as probably hampered by the mismatch between such 1D-units and the coordination of all Cu 2+ ions by the sulfate groups. The exact nature of the magnetic ground state in Cs2Cu3.5O1.5(SO4)3 remains an open question and requires additional studies using neutron scattering or local probes. While an AFM order in the bc-plane may be expected from the similarity to the Na and K compounds, the vanishingly small magnetic interactions between the planes and the incomplete structural order of the intercalated CuO units may cause deviations from this simple scenario. One should keep in mind the similar ambiguity in the A = K compound, where NPD results 20 revealed magnetic ordering, not detected by specific heat with a Schottky-like anomaly around TN, see Fig.4b. In this compound, it was previously concluded that the role of the magnetic anisotropy of the S = 1 clusters is predominant at TN in absence of inter-layer exchanges, Similarly the anisotropy of the Cu14 S = 1 clusters may also be significant for A = Cs.

S=1, hexamer S=1, tetradecamer +2 CuO

ACKNOWLEDGEMENTS.

This work was supported by the Embassy of France in Russia, the French Government, and managed by the Agency Campus France. The Fonds Européen de Développement Régional, CNRS, Région Nord Pas-de-Calais, and Ministère de l'Education Nationale de l'Enseignement Supérieur et de la Recherche are acknowledged for funding the X-ray diffractometers and PPMS system. This work was partially carried out under the framework of the LOVE-ME project supported by the ANR (Grant ANR ANR-16-CE08-0023). OIS was supported by the Russian Foundation for Basic Research, Grant No. 19-05-00413. Laurence Burylo (XRD) and Marielle Huvé (TEM and EDS) are thanked for their experimental help.

ASSOCIATED CONTENT.

The Supporting Information is available free of charge a thttps://pubs.acs.org/****** . It contains extra data on the crystallography, XRD of polycrystalline samples and details of the magnetic data and calculated exchanges.