J Vialle

A Zampa

A Badel

P Tixador

Stabilizers for REBCO Conductors for High Performance SFCL

This equation is valid for a three-phase grid where Vr is the operating line-to-neutral RMS voltage. Of course, this value becomes fully theoretical when using a SFCL, which will limit the fault current at a much lower value.

For both low and high prospective current cases, the conductor will dissipate energy, either along its full length or along a short length, before a switchgear isolates it from the grid after a



Abstract-Resistive REBCO Fault Current Limiter (FCL) is a promising solution to the fault current issue in electric grids, which is not fully satisfactorily solved. But the cost of the Superconducting (SC) REBCO conductor should be lowered to facilitate the deployment of this innovative device. What we refer to as REBCO conductor is usually a bare REBCO tape bonded to a stabilizer. The design of the stabilizer is fundamental to optimize the conductor and lower the SFCL cost. The goal is to limit the temperature rise for any fault conditions, notably for any amplitude of the prospective fault current. The specificities of SC tapes, including critical current inhomogeneities along the length, must be considered in the design. Here we study various stabilizer configurations. First the case of "dielectric stabilizer", electrically insulating is presented. The theoretical performances are interesting in terms of electric field under limitation, but the low thermal diffusivity remains an issue, as well as the implementation. Some experimental investigations are shown. The results are compared to classical metallic stabilizer designs, which are simpler to implement. To enhance the electric field under limitation, a variant of metallic stabilizer is then introduced, using a corrugated structure. A design showing an electric field under limitation of 200 V/m (clearing time of 50 ms) is presented. Further optimizations and higher electric fields seem achievable even if practical implementation remains very challenging. Index Terms-Superconducting Fault Current Limiter, optimization, limitation, hot spot, stabilizer

I. INTRODUCTION

uperconducting Fault Current Limiter (SFCL) is a satisfying and unique response to today's demand for higher security and quality in power grids. This is why SFCL, along with SC cables, shows the highest TRL (Technology Readiness Level) among SC applications [START_REF] Marchionini | High-Temperature Superconductivity: A Roadmap for Electric Power Sector Applications, 2015-2030[END_REF]. Several SFCLs have already been put into operation in grids. A prominent recent example is the SuperOx 220 kV/460 MVA 3 phase device installed in Moscow, Russia, since 2019 [START_REF] Moyzykh | First Russian 220 kV superconducting fault current limiter (SFCL) for application in city grid[END_REF]. So even under very high voltages, solutions close to commercialization are available. However, the length of conductor required remains an obstacle in terms of costs. The SuperOx 220 kV SFCL uses 8 km per phase. The electric field under limitation is about 16 VRMS/m (clearing time: 400 ms). Brought back to a clearing time of 50 ms considered in this paper the electric field reaches 45 VRMS/m. This electric field has to increase significantly to lower the superconducting conductor length and make the cost of the SFCL acceptable. That was the main goal of the EC project FASTGRID [START_REF] Tixador | Status of the European Project FASTGRID[END_REF] and this article aims to go further. We will only study here conductors based on REBCO tape as they show the best performances for SFCL among commercially available HTS. A conductor refers to a bare REBCO tape bonded to a stabilizer. We will consider resistive FCL. In the FASTGRID project, it was shown that conductors sustaining electric fields up to 130 V/m during 50 ms are experimentally possible even when operating at 65 K, a significant enhancement compared to classical values (50 VRMS/m during 50 ms at 77 K in the Eccoflow project [START_REF] Noe | Conceptual Design of a 24 kV, 1 kA Resistive Superconducting Fault Current Limiter[END_REF] for example). We want to go further with an objective of 300 VRMS/m. The conductor developed in FASTGRID derives from previous works on conductor optimization for SFCL [START_REF] Tixador | Superconducting Fault Current Limiter optimized design[END_REF][START_REF] Badel | REBCO FCL Modelling: Influence of Local Critical Current Non-Uniformities on Overall Behavior for Various Tape Architectures[END_REF]. Within FASTGRID a solution using REBCO tape based on a sapphire tape substrate is also investigated. It sustains electric fields overstepping 1000 VRMS/m [START_REF] Deutscher | High voltage superconducting fault current limiters based on high diffusivity substrates[END_REF]. A lot of research and development are still required to enhance the Ic, to industrialize this breaking solution. Its cost remains an open question. So in this article we only consider conventional REBCO tapes with a metallic substrate. The SC conductor design must fulfill two contradictory requirements which are the safe operations for both low and high prospective currents experienced in power grids. Due to the variety of possible faults, fault currents in power grid may vary from just above the rated current to the highest prospective RMS current (𝐼 𝑃𝑟𝑜𝑠 𝑀𝑎𝑥 ), which is given by the short circuit power of the source (SSC): predefined clearing time (∆t). The design approach is basically a thermal study: the temperature of the SC conductor must be kept below a threshold value set by either the SC layer degradation or the melting of solder in the conductor. For high prospective currents, the power grid behaves like a voltage source (V) and the dissipative power is given by V 2 /R1 where R1 is the resistance of the SFCL. This case is called limitation regime, in which a large conductor resistance is favorable to lower the dissipation. But the prospective fault current may take any value up to 𝐼 𝑃𝑟𝑜𝑠 𝑀𝑎𝑥 (equation ( 1)) whereas the critical current of a REBCO tape intrinsically varies along its length. Reference [START_REF] Irie | Characteristics of Critical Current Distribution for Oxide Superconductor Estimated from V-I Characteristics Using Weibull Function[END_REF] shows that the Ic distribution follows a Weibull probability distribution characterized by a minimum critical current Ic* which decreases with the conductor length. Fig. 1 plots the measured Ic every 1.1 mm along 100 m for a THEVA tape at 77 K. The minimum, maximum and mean Ic are respectively 642.7 A, 785.6 A and 743.5 A. A SFCL does not require very long unit lengths. It consists of modules put in series or/and in parallel to meet the specifications. The voltage for one module will be in the range of tens of kV, corresponding to a length of the order of a hundred meter. Fig. 1 then represents right the typical length of a SFCL module. Fig. 1 shows that the minimum Ic is very localized in the length. In the case of a prospective current in the range of this minimum Ic only this short spot would initially dissipate.

The expansion of this dissipative spot is very limited since the Normal Zone Propagation Velocity (NZPV) is low for REBCO tape, up to about 0.1 m/s for a bare tape at 77 K at the critical current [START_REF] Giguère | High normal zone propagation velocity in copper-stabilized 2G HTS coated conductors[END_REF], except if using special architecture like Current Flow Diverter (CFD) [START_REF] Lacroix | Engineering of second generation HTS coated conductor architecture to enhance the normal zone propagation velocity in various operating conditions[END_REF]. The normal zone still remains very short at the time of clearing (50 ms: 5 mm): it is a "spot" which rapidly may become hot, hence the name of "hot spot" given to this regime. The hot spot shows a negligible resistance (proportional to its length) compared to the fault impedance. The source then behaves like a current source (I). The dissipation power is therefore R2 I 2 where R2 is the hot spot resistance. Contrary to the limitation regime and much like the case of a magnet, a small resistance per unit length is favorable. We can see that a compromise has to be found to optimize the conductor to cope with both low and high prospective currents. The stabilizer plays a key part in the conductor optimization and we present some solutions in this work. Fig. 2 summarizes the behaviour of a REBCO conductor with typical inhomogeneity for the full range of prospective RMS fault current related to its critical current. It shows the maximum temperature reached at the clearing time for any fault. The two strongest thermal constraints are for the hot spot regime (Ipros ≈ Ic) and the limitation one (here Ipros = 15 Ic). This curve is a calculated plot for the FASTGRID THEVA conductor, in agreement with experimental curves [START_REF] Zampa | Experimental observations of the hot-spot regime on 2G HTS coated conductors[END_REF]. The FASTGRID conductor uses a 12 mm wide THEVA tape [12] with 1.6 µm of silver above the 3 µm HTS layer and 1 µm silver on the edges and the bottom. The tape is soldered to a 500 µm thick Hastelloy ® tape (stabilizer) with a Ni coating (1 µm) to make the soldering possible. The solder is tin and 10 µm thick. This conductor has been designed to get the same peak temperature (400 K) for the hot spot regime and the limitation one. The quantitative values of the maximum temperature depend on the conductor. The peak temperature depends on its length for example [START_REF] Zampa | Effect of the Conductor Length on the Hot-Spot Regime for Resistive-Type Superconducting Fault Current Limiter[END_REF]. But we nearly have always the same behaviour with an increase of the maximum temperature with the prospective current then a decrease and an increase again with Limitation reg.

an asymptotic value. References [START_REF] Kudymow | 2G HTS material selection, insulation, optimization, and the resulting component design of the Eccofow resistive fault current limiter[END_REF][START_REF] Liang | The Impact of Critical Current Inhomogeneity in HTS Coated Conductors on the Quench Process for SFCL application[END_REF] show similar experimental curves for SuperPower tapes and reference [START_REF] Colangelo | Inhomogeneity effects in HTS coated conductors used as resistive FCLs in medium voltage grids[END_REF] gives a comparable simulation plot. This article presents first some analytical conductor design with a solution to reach 200 VRMS/m. Strong questionable hypothesis will be of course made. Nevertheless, analytical expressions have the great advantage to clearly show the relevant parameters to optimise the conductor. They show in particular that a dielectric and corrugated stabilizers could be attractive. The real designs are then carried out using numerical approach which limits the expected performances. A few experiences about dielectric stabilizer are presented. They confirm the limitations and the rather good agreement between measurements and simulations even if differences exist. Preliminary optimisations of the corrugated stabilizer are presented. They also highlight the difficulties to implement this solution.

II. ANALYTICAL CONDUCTOR DESIGN

A SC conductor is a SC tape bonded to a stabilizer. These two elements are electrically in parallel and thermally connected. That is the case when using a metallic stabilizer soldered to the tape, as for the FASTGRID conductor. It must be noted that the design approach presented below remains valid for studying a tape bonded to a dielectric material, by simply setting the electrical conductivity of the stabilizer to zero. This makes the use of bonded dielectric material simply a particular case of electric stabilizer, hence the name of dielectric "stabilizer" that we will use later on.

A very strong assumption for analytical expressions is the isothermal cross section of the conductor. This is questionable even in case of a metallic stabilizer but mandatory to get analytical expressions to identify the relevant parameters. For the real design numerical simulations will be used to avoid this strong assumption.

A. Limitation regime

The power source is in that case a voltage source. The high prospective current quenches the full length. With the isothermal assumption the thermal equation is then:

𝑣(𝑡) 2 𝑅 1 𝑑𝑡 = (𝑉𝑜𝑙 𝑡𝑎𝑝𝑒 𝑐 𝑝 𝑡𝑎𝑝𝑒 + 𝑉𝑜𝑙 𝑠𝑡𝑎𝑏 𝑐 𝑝 𝑠𝑡𝑎𝑏 ) 𝑑𝑇 (2)
where v(t) is the instantaneous voltage, R1 the resistance of the conductor, Voltape Volstab the volumes of the tape and the stabilizer respectively and cp the specific heat per unit volume (J/m 3 /K). The regime is supposed adiabatic since the dissipation is huge with heat fluxes much higher than what can be absorbed by the cooling fluid. Temperature sensors show a very good agreement (∆T/T < 5 %) of the measured temperature with the calculations assuming adiabatic conditions at least for 30 ms [START_REF] Tixador | Electrothermal phenomena about current limitation with Coated Conductor[END_REF]. The calculations are also very close to the temperature extracted from the R(T) of the tape or conductor. The tape and the stabilizer being electrically in parallel the equation becomes:

( 1 𝑅 𝑡𝑎𝑝𝑒 + 1 𝑅 𝑠𝑡𝑎𝑏 ) 𝑣(𝑡) 2 𝑑𝑡 = (𝐿 𝑡𝑎𝑝𝑒 𝑤 𝑡𝑎𝑝𝑒 𝑒 𝑡𝑎𝑝𝑒 𝑐 𝑝 𝑡𝑎𝑝𝑒 + 𝐿 𝑠𝑡𝑎𝑏 𝑤 𝑠𝑡𝑎𝑏 𝑒 𝑠𝑡𝑎𝑏 𝑐 𝑝 𝑠𝑡𝑎𝑏 ) 𝑑𝑇 (3)
The tape and the stabilizer do not have necessarily the same lengths (Ltape and Lstab) nor the same widths (wtape and wstab). The thicknesses of the tape and of the stabilizer are respectively etape and estab. tape and stab are the equivalent resistivity of the tape and of the stabilizer respectively. One only has to integrate to get the maximum temperature (Tmax) when the SFCL is electrically isolated after ∆t (clearing time). Let us fix the starting temperature not to the initial one (To) but to the critical one (Tc) so that the regime under Tc where the resistivity of the tape depends on the current is neglected. This has a very limited impact on the results while simplifying greatly the calculations. The only consequence is that the maximum temperature is slightly overestimated by 10 K as order of magnitude. With this hypothesis the integration of equation ( 4) gives:

𝐸 𝑙𝑖𝑚 𝑡𝑎𝑝𝑒 2 ∆𝑡 = ∫ 𝑒 𝑡𝑎𝑝𝑒 𝑐 𝑝 𝑡𝑎𝑝𝑒 (𝑇) + 𝐿 𝑠𝑡𝑎𝑏 𝐿 𝑡𝑎𝑝𝑒 𝑤 𝑠𝑡𝑎𝑏 𝑤 𝑡𝑎𝑝𝑒 𝑒 𝑠𝑡𝑎𝑏 𝑐 𝑝 𝑠𝑡𝑎𝑏 (𝑇) 𝑒 𝑡𝑎𝑝𝑒 𝜌 𝑡𝑎𝑝𝑒 (𝑇) + 𝑒 𝑠𝑡𝑎𝑏 𝜌 𝑠𝑡𝑎𝑏 (𝑇) 𝑤 𝑠𝑡𝑎𝑏 𝑤 𝑡𝑎𝑝𝑒 𝐿 𝑡𝑎𝑝𝑒 𝐿 𝑠𝑡𝑎𝑏 𝑇 𝑚𝑎𝑥 𝑇 𝑐 𝑑𝑇 (5)

𝐸 𝑙𝑖𝑚

𝑡𝑎𝑝𝑒 is the RMS value of the tape electrical field.

In case of a thick (estab > 5 etape) stabilizer design, we can even neglect the tape itself in the thermal balance and the equation of the tape electrical field simplifies to:

𝐸 𝑙𝑖𝑚 𝑡𝑎𝑝𝑒 ≈ 𝐿 𝑠𝑡𝑎𝑏 𝐿 𝑡𝑎𝑝𝑒 √ ∫ 𝜌 𝑠𝑡𝑎𝑏 (𝑇) 𝑐 𝑝 𝑠𝑡𝑎𝑏 (𝑇) 𝑑𝑇 𝑇 𝑚𝑎𝑥 𝑇 𝑐 ∆𝑡 (6)
The objective is of course to maximize the tape electric field 𝐸 𝑙𝑖𝑚 𝑡𝑎𝑝𝑒 to use the shortest length of the SC tape. Equations ( 5) and ( 6) show that we have to:  Increase the ratio Lstab/Ltape.  Foster high electric resistivities for the tape and the stabilizer.  Use tape and stabilizer with high specific heats.  Lower the clearing time. On the other hand, the thicknesses and the widths play little part, but we will see that they are important for the hot-spot regime. Commercial REBCO tape structures are more or less similar. The tape is made of a substrate, electrical insulated thin buffer layers, the REBCO layer and an Ag metallization layer above. REBCO tape based on a Hastelloy ® substrate are preferable due to the Hastelloy ® high resistivity. Silver is not ideal for a high resistance but tests carried out to pollute it to increase its resistivity were unsuccessful within FASTGRID. Likewise, its thickness cannot be lowered too much, as a continuous and regular protection layer is required. A total Ag thickness of about 2 µm is reasonable. Nevertheless, we will see that it would be useful to increase the resistance of the metallization layer by decreasing its thickness and/or its electrical conductivity. The substrate thickness may be adapted: it can vary typically from 30 µm to 100 µm. The clearing time ∆t is fixed by the speed of the switch gear but also by the risk that it fails to open. In that event the opening of a second switch clears the fault. Considering this back-up the clearing time reaches about 50 ms [START_REF] Creusot | SuperGrid Institute[END_REF]. This value will be considered in the rest of the article knowing that most of the time (normal operation without failure of the first switch gear) the clearing time is about 20 ms, decreasing the maximum temperature a lot. This low clearing time is not common for today switch gears but fully possible and demonstrated The specific heat per unit volume does not significantly vary between materials above 60 K. The use of a stabilizer materials with zero electric conductivity, so called dielectric stabilizers is an option and some experiments will be shown in part II A. Longer and wider stabilizer compared to the tape can be imagined with corrugations for example (Fig. 3). We will see that the corrugated stabilizer will be very different compared to Fig. 3 due to thermal considerations. But before we study more in detail these stabilizers, we have to consider the conductor design for the hot spot regime.

B. Hot spot regime

The power source is in that case a current source. The thermal equation is then:

𝑅 2 𝑖(𝑡) 2 𝑑𝑡 = 𝑉𝑜𝑙𝑢𝑚𝑒 𝑐 𝑝 𝑑𝑇 (7)
The regime is also supposed to be adiabatic even if this hypothesis is less valid compared to the limitation regime. This effect of the thermal exchanges has to be studied especially in subcooled nitrogen at 65 K since the exchanges may be better compared to 77 K saturated nitrogen [START_REF] Jeong | Transient Heat transfer Experiment in Subcooled Liquid Nitrogen[END_REF].

The tape and the stabilizer being in parallel electrically equation [START_REF] Deutscher | High voltage superconducting fault current limiters based on high diffusivity substrates[END_REF]

The RMS prospective fault current which leads to the hot spot regime is close above the critical current (Fig. 2) so we fix it at Ic in the calculations. As for the limitation regime the integration starts at the critical temperature: The hot spot regime mainly defines the stabilizer thickness. To reduce it (thinner and more flexible stabilizer) equations ( 10) and [START_REF] Zampa | Experimental observations of the hot-spot regime on 2G HTS coated conductors[END_REF] show that we have to:  Increase the ratio wstab/wtape.  Foster low resistivities for the tape and the stabilizer.

𝐼
 Use tapes and stabilizers with high specific heats.  Lower the clearing time.

The relative lengths play little part contrary to the limitation regime.

Of course, for the hot spot regime high resistivities are logically not favorable but we have however to maximize the resistivities to lower the tape length. That is the objective of this work. The stabilizer has to be adapted to meet the hot spot regime as we will see. It will be thick (< 0.5 mm). Increasing the width of the stabilizer is also interesting to lower the recovery time of the SFCL since it increases the heat transfer surfaces with the cooling fluid and the area over volume ratio.

Unfortunately, this increase of the stabilizer width compared to the tape will be limited by the thermal diffusion in the total width of the stabilizer. In addition, we will see that a wider stabilizer introduces a harmful thermal gradient within the tape.

C. Intermediate conclusion

The main conclusions deduced from the previous expressions to get an "optimized" conductor for the SFCL are:  A stabilizer longer compared to the tape (corrugation e.g.).

 A stabilizer wider compared to the tape.  A highly resistive stabilizer, even possibly an electric insulator (dielectric material). We have nevertheless to emphasize that these conclusions are based on a questionable isothermal section, as thermal simulations will show. These conclusions will be applied now to two types of stabilizers:  Dielectric stabilizer.  Metallic corrugated stabilizer.

III. STABILIZER

A. Dielectric stabilizer Such a stabilizer opens the way of high electric fields (equation ( 5) and ( 10) with infinite resistivity for the stabilizer): Neglecting the temperature dependence of the parameters, the expression of the electric field under limitation becomes very simple:

𝐸 𝑙𝑖𝑚 𝑡𝑎𝑝𝑒 = 𝜌 𝑡𝑎𝑝𝑒 𝑒 𝑡𝑎𝑝𝑒 𝐼 𝑐-𝑤 (14)
The electrical conductivity and the thickness of the tape should be as low as possible: tape with only its thin metallization layer and thin Hastelloy ® substrate (50 µm, even 30 µm). Using equation ( 14), the electric field may be very high: tape = 0.2 µΩm, etape = 55 µm, Ic-w = 120 kA/m (65 K):

𝐸 𝑙𝑖𝑚 𝑡𝑎𝑝𝑒 = 440 𝑉/𝑚
This is extremely attractive but the hypothesis behind this extreme value must be fulfilled, in particular the isothermal section of the tape and above all the stabilizer. Using equation ( 13) the stabilizer thickness is about 2 mm (Lstab/Ltape = 1; wstab/wtape = 1; ∆T = 350 K; ∆t = 50 ms). This is a lot taking into account the low diffusivity of dielectric materials during the clearing time ∆t. We carried out some tests on SuperOx 4 mm wide tapes encapsulated in bulk epoxy and on SuperPower 2 mm wide tapes with Stycast ® coatings on both sides.

1) SuperOx epoxy encapsulated tape

The 4 mm wide tape is encapsulated in a bulk epoxy block 10 mm thick and 10 mm wide. Fig. 4 gives its longitudinal cross section with two external connections and three temperature sensors (Pt100 resistances, S1, S2 and S3). S1, S2 and S3 are respectively 1 mm, 2 mm and 5 mm above the tape (S3 is near the external surface). The SuperOx tape has a Hastelloy ® substrate 100 µm thick. In addition to the micronic Ag layer, 3 µm of Cu and 10 µm of solder (Sn62Pb32Ag2) are deposited on both sides of the tape. Outside the 90 mm central part, a second Cu layer 10 µm thick protects the tape in the parts without encapsulation (200 mm long) (Fig. 4). Fig. 6 and Fig. 7 show the temperatures of the 3 sensors (S1, S2 and S3, see Fig. 4) for this short circuit for different durations (0-0.2 s for Fig. 6 and 0-10 s for Fig. 7). In addition, for Fig. 6, we also plot the calculated hot spot temperature (Tmax) of the conductor including the epoxy encapsulation in adiabatic condition [START_REF] Nguyen | Influence of Architecture coated conductor YBCO to fault current limiter-visualisation analyse[END_REF]:

∫ 𝑖(𝑥) 2 𝑑𝑥 𝑡 0 = 𝐴 ∫ 𝑐 𝑝 (𝑇) 𝑅 𝑙 (𝑇) 𝑑𝑇 𝑇 𝑚𝑎𝑥 𝑇 𝑐 => 𝑇 𝑚𝑎𝑥 (𝑡) ( 15
)
where A is the conductor cross section of the tape (including epoxy) and Rl its resistance per unit length. The integration of the square of the current starts at t = 0 and the temperature is supposed to be Tc (Tmax(t = 0) = Tc) to avoid the regime under Tc where the resistance depends of the current. This equation assumes again an isothermal conductor cross section, the real temperature of the tape will be much higher as some simulations will show. However, the encapsulation definitely plays a part and protects the tape. Indeed, the theoretical hot spot temperature calculated considering only the tape without epoxy exceeds several thousands of K. This is clearly not possible since the tape survived several tests: the tape shows a non-linear behaviour. However, no Ic measurement at 100 µV/m was carried out to observe possible slight degradations. The temperature curves show the low thermal diffusion in the thickness of the epoxy stabilizer: the temperature variation quickly decreases as soon as we move away from the tape. Even close to the tape the temperature only rises after the clearing time. The external temperature (TS3) increases only a little and after a very long delay. Due to the diffusion in the stabilizer thickness the maximum temperature within the epoxy occurs after the end of the fault and it takes about 10 s to go back to the initial temperature (77 K, Fig. 7).

Fig. 8 and Fig. 9 plot the temperatures of the tape at the locations of the 3 sensors calculated by finite elements for two different durations (0-0.5 s for Fig. 8 and 0-10 s for Fig. 9). The tape temperature increases very rapidly and is strongly limited after the clearing time. The simulated and measured temperatures are in rather good agreement especially in terms of maximum value (Fig. 10) but there is a time lag of about 1 s between them. This difference may be due to thermal contacts between the layers not considered in the simulation, the thermal conductivity is also not accurately known. In addition, the locations of the sensors are not very precise. We put indeed the sensors after we received the encapsulated tapes. Simulations show that due to the thermal diffusivity of epoxy the thickness of the epoxy stabilizer should be lower than 0.5 mm. This is what we tried in the next step.

2) SuperPower 2 mm wide tapes with Stycast ® coatings

We have coated a 2 mm wide SuperPower tape (Ic = 50 A (100 µV/m, 77 K sf)) with 0.5 mm Stycast ® 2850FT with catalyst 24LV on both sides. The tape has an Hastelloy substrate of 50 µm thick and its total thickness is about 52 µm, so 1 µm of silver. Fig. 11 shows a limitation curve. The electric field nearly reachs 180 V/m during 33 ms and the tape survived this test but the Stycast ® layers were fully delaminated after. The calculated hot spot temperatures reached about 400 K and 1000 K at the clearing time considering the total section or the section of the bare tape respectively. Since the tape still was superconducting after the tests, the Stycast ® definitely plays a part. However, the Ic was not measured after the test so that slight tape degradations could occur. By lowering the electric field, the delamination of the stabilizer occurred less quickly. For 170 V/m the delamination occurred during the second limitation test. In conclusion the dielectric stabilizer concept shows interesting perspectives. Nevertheless, the poor diffusivity of dielectric material limits the performance. Indeed, contrary to the case of metallic stabilizer where the stabilizer itself dissipates, here the stabilizer acts only as a thermal drain. All the power is still dissipated in the tape and diffuses to the stabilizer by thermal conduction. Temperature differences should be limited to avoid differential contraction mechanical issues. The stabilizer thermal expansion should be adapted to that of the tape for the same reason. The good thermal contact between the tape and the stabilizer is another challenge. References [START_REF] Pekarčíková | Composite Heat Sink Material for Superconducting Tape in Fault Current Limiter Applications[END_REF] and [START_REF] Pekarčíková | Analysis of thermo-physical properties of materials suitable for thermal stabilization of superconducting tapes for high-voltage superconducting fault current limiters[END_REF] give some studies about insulated or high cp stabilizer. The recovery time may be an issue but it depends on the SFCL use and location. For some locations the recovery time can be short but for other locations, a grid with cables for example the recovery time is unproblematic. In opposition to an overhead line where a fault may be fugitive for a cable, the fault cannot be fugitive with no automatic reclosing. Due to these difficult implementation issues, we moved from this solution toward metallic corrugated stabilizer.

B. Metallic corrugated stabilizer

Equation ( 5) clearly shows the advantage to increase the length of the stabilizer compared to that of the tape to enhance the tape electric field. Without increasing the stabilizer length compared to the tape length, the ultimate electric field (metallization layer neglected) using an Hastelloy ® stabilizer is:

𝐸 𝑙𝑖𝑚 𝑢𝑙𝑡 ≈ √ 𝜌 𝐻𝑎𝑠𝑡 (𝐻 ℎ𝑎𝑠𝑡 (𝑇 𝑚𝑎𝑥 )-𝐻 ℎ𝑎𝑠𝑡 (𝑇 𝑐 )) ∆𝑡 ( 16
)
where H is the conductor enthalpy per unit volume. For Tmax = 415 K and ∆t = 50 ms the ultimate electric field reaches 160 V/m. The stabilizer is Hastelloy ® for two reasons: this superalloy shows a high electrical resistivity (1.25 µΩm) and matches the expansion of the tape, mainly Hastelloy ® . Fig. 12 gives the increase of the tape electric field with the ratio Lstabilizer/ Ltape. The tape is a 55 µm thick tape with a Hastelloy ® substrate and 2 µm Ag layer. The geometrical ratio estabilizer wstabilizer/wtape is calculated using equation ( 10) and considering a maximum temperature of 415 K (50 ms). The critical current per unit width is 1200 A/cm-w since we operate at 65 K to lower the quantity of tape. This is the lowest temperature possible with liquid nitrogen, the only cheap and industrially available cryogenic fluid. Fig. 12 shows that 300 V/m is reached for a stabilizer 3.2 times longer than the tape.

In addition to the increase of the stabilizer length compared to the tape length, the corrugated configuration increases the exchange surfaces with the cooling fluid. This decreases the recovery time after a limitation and increase the NZPV for the tape parts between the ribs. Indeed, the NZPV is larger for a bare tape compared to that of an electrically-stabilizered tape [START_REF] Giguère | High normal zone propagation velocity in copper-stabilized 2G HTS coated conductors[END_REF] since the dissipation per unit length is higher. If equation [START_REF] Tixador | Superconducting Fault Current Limiter optimized design[END_REF] shows the large interest to increase the length and width of the stabilizer compared to the tape, it is based on a strong hypothesis, that the temperature remains uniform in the section. This hypothesis is nearly valid when the stabilizer is fully bonded with the tape despite the low thermal diffusivity of Hastelloy ® since the stabilizer itself is heated by Joule effect during the limitation phase. Nevertheless, the power dissipation per unit volume in the stabilizer remains in general much lower than in the tape: the stabilizer is often heated mainly by conduction. The thermal diffusivity of Hastelloy ® is about 3 10 -6 m 2 /s (it varies with temperature). This means that temperature diffuses over about 200 to 300 µm in the 50 ms of the clearing time (Fig. 13) without internal heating. Since the stabilizer length is higher than the tape, its resistance is also higher and less current flows in the stabilizer still lowering its heating by Joule effect. This is why it would be all the more interesting to increase the resistance of the tape by lowering the metallization thickness and its electrical conductivity to divert more current in the stabilizer to heat it more. This poor diffusivity of Hastelloy ® stabilizer hinders from increasing the corrugation height above typically 700 µm. This low Hastelloy ® diffusivity also prevents from increasing too much the stabilizer width compared to the tape one.

To take into account the thermal diffusion within the section, 2D finite element simulations have been carried out using COMSOL ® software. They are based on following hypothesis:  Perfect thermal contact between the different areas.

 Adiabatic conditions.  The superconducting tape under Tc is modeled by two branches in parallel; one is the entire tape with the superconducting layer in the normal state. The other branch represents the superconducting layer considering the power load for the electric field versus current to deduce an equivalent resistance. Both edges of the conductor are an electrical equipotential. The conductor is inserted in an electrical circuit operating in voltage mode (limitation regime) or current mode (hot spot regime). For the hot spot regime, the Ic is lower (40 % reduction) in the center (spot 100 µm long). The conductor is made by 5 layers:  Hastelloy ® (substrate, 100 µm).  Superconductor (3 µm).  Silver (2 µm).  Tin (solder) (5 µm).  Hastelloy ® corrugated stabilizer. The data (cp(T)…) comes from handbook and from [START_REF] Lu | Physical properties of Hastelloy ® C+276TM at cryogenic temperatures[END_REF] for Hastelloy. Fig. 14 shows the simulated temperature distributions 50 ms after having overstepped Tc in limitation conditions for different widths of the stabilizer (12 mm, 14 mm and 16 mm (tape width: 12 mm)) and for Lstabilizer/Ltape = 2 (it influences the current then dissipation distributions). The stabilizer section has been kept constant. The temperature distributions are similar in hot spot condition. If the temperature remains very homogeneous across the whole conductor when the widths are the same, gradients appear when the stabilizer is wider than the tape. It even introduces an unfavorable temperature gradient in the tape width itself. We can conclude that the stabilizer cannot be larger than the tape by 1 mm (12 mm wide tape) and that a same width is better.

The second term depends little (within 30 %) on the silver/metallization thicknesses. So the only free parameter to limit the maximum temperature is L, distance between two ribs. If we want a limitation field of 300 V/m the distance L should be lower than 200 µm to keep the temperature difference under 50 K. A lower L favorably increases the ratio Lstabilizer/Ltape :

𝐿 𝑠𝑡𝑎𝑏 𝐿 𝑡𝑎𝑝𝑒 = 𝐿 + ℎ 𝑡𝑜𝑡 𝐿+ 𝑒 𝑠𝑡𝑎𝑏 (18)
Thermal considerations then give the limits for the dimensions of the stabilizer:  height (htot) lower than 700 µm  distance between two ribs (L) lower than 200 µm. The increase of the tape thickness (substrate) or electrical conductivity decreases the stabilizer thickness but lowers the electric field under limitation. It favorably forces more current in the stabilizer in normal state which then contributes more to the temperature rise limitation. But the increase of the stabilizer length compared to the tape one reduces the amount of current flowing in the stabilizer. Fig. 16 shows a possible conductor with two corrugated Hastelloy ® stabilizers on both sides of the tape. The limitation of the stabilizer height leads to this configuration. In addition, two stabilizers protect mechanically the tape and makes it possible to have the SC layer on the neutral fiber to avoid mechanical stresses when wound. The positions of the two stabilizers may be shifted like in Fig. 16. The stabilizer has the same width compared to the tape to reduce the temperature gradients in the conductor section. The tape has a Hastelloy ® substrate of 100 µm, a standard silver metallization layer of 2 µm. The total height of the conductor is about 1.5 mm (htot = 700 µm). The distance between two ribs (L) is 100 µm and the stabilizer thickness 300 µm (estab). Such a conductor should withstand an electric field of 200 V/m (Fig. 17) and be safe in hot spot condition (Fig. 18). The temperature difference between the tape and the extremity of the stabilizer remains a little high (about 100 K) and we have nearly the same thermal conditions for limitation and hot spot regimes as designed. Enhanced performances are possible using a metallization layer with reduced thickness and electrical conductivity. Optimizations should be carried out and above all experimental validations. Nevertheless, further work is needed to develop low cost and long length practical realization of such corrugated stabilizer, which is made very challenging by the small dimensions. The distance between the corrugations is tiny. Its filling by the solder must be avoided for example. The industrial implementation will require a lot of developments.

IV. STABILIZER COMPARISON

The previous studies enable to make a very preliminary comparison between the corrugated stabilizer and dielectric stabilizer. Still a lot of works are required for both solutions. Stabilizer design is a key element for optimizing REBCO conductors for SFCL while keeping bare standard REBCO tape as a basis. We show that a dielectric stabilizer may be an attractive solution but a lot of issues have to be solved. This is due in particular to the thermal diffusivity within the stabilizer. Implementation is also a concern. We then concentrated on corrugated metallic (Hastelloy ® ) stabilizer on both sides of the tape. First investigations have showed that electric fields of 200 V/m are possible. Still higher values are certainly possible using further optimizations. In addition to these theoretical works corrugated stabilizer conductors must be realized and tested to verify and qualify this solution, as well as assess its feasibility as a cost-effective industrialized solution.

Fig. 1 .

 1 Fig. 1. Ic (77 K sf) along 100 m of a REBCO tape (THEVA data), zoom on the right (z in mm) for the lowest Ic zone.

Fig. 2 .

 2 Fig. 2. Typical maximum temperature of the SC conductor at the clearing time (50 ms) versus the ratio of the prospective RMS current to the critical one.

Fig. 3 .

 3 Fig. 3. REBCO tape with corrugated stabilizer (same width).

  𝑝 𝑠𝑡𝑎𝑏 (𝑇)) 𝑑𝑇[START_REF] Lacroix | Engineering of second generation HTS coated conductor architecture to enhance the normal zone propagation velocity in various operating conditions[END_REF] Ic-w is the critical current per unit width of the tape (Ic/wtape). This expression makes possible to design the stabilizer (thickness and width) so that the conductor survives the hot spot regime. Once again for thick stabilizer we may neglect the tape influence to get an ultra-simplified expression:

Fig. 4 .

 4 Fig. 4. SuperOx tape encapsulated in epoxy with current leads and thermal sensors (not to scale).

Fig. 5

 5 Fig. 5 shows the time evolutions of the current, voltage for a 70 ms short circuit in DC conditions at 77 K. The short-circuit power of the source is insufficient to keep the voltage constant. The critical current is about 150 A (100 µV/m, 77 K sf). The peak current is high due to the low resistance even in the middle (solder and Cu) and the high electric field. Limitation is strong due to the reduction of the voltage and the high sample heating.

Fig. 5 .

 5 Fig. 5. DC short-circuit test, current and voltage for the SuperOx tape encapsuled in epoxy at 77 K.

Fig. 6 .

 6 Fig. 6. DC short-circuit test, measured temperatures (TS1, TS2 and TS3) and calculated hot spot temperature (Tmax) including epoxy at the beginning.

Fig. 7 .

 7 Fig. 7. DC short-circuit test, measured temperatures evolution up to the recovery.

Fig. 8 .

 8 Fig. 8. Simulated temperatures (tape and at the 3 sensors) during the limitation and just after.

Fig. 9 .

 9 Fig. 9. Simulated temperatures (tape and at the 3 sensors) up to the recovery.

Fig. 10 .

 10 Fig. 10. Simulated and measured temperatures (sensors 1 and 2).

Fig. 11 .

 11 Fig. 11. Current and electric field during a limitation experiment on a SuperPower tape with 0.5 mm epoxy on both sides.

Fig. 12 .

 12 Fig. 12. Electric field versus stabilizer to tape length ratio.

Fig. 13 .

 13 Fig. 13. Temperature distribution in a semi infinite Hastelloy ® plan submitted to a temperature step at t = 0.

Fig. 14 .

 14 Fig. 14. Simulations of temperature distribution for different widths of stabilizer 50 ms after a quench.But temperature diffuses also very hardly along the tape. Between two ribs the maximum temperature difference (Fig.15) reaches (steady state, full conduction):

Fig. 15 .

 15 Fig. 15. Tape (in red) bonded to a corrugated stabilizer (in grey), location of the maximum temperature between two ribs.

Fig. 16 .

 16 Fig. 16. Possible corrugated stabilizer conductor with top and bottom stabilizer offset.

Fig. 17 .

 17 Fig. 17. Temperature distribution at the clearing time in limitation regime (200 V/m) (htot = 700 µm, L = 100 µm, estab = 300 µm).

Fig. 18 .

 18 Fig. 18. Temperature distribution at the clearing time in hot spot regime (I = Ic) (htot = 700 µm, L = 100 µm, estab = 300 µm).

TABLE CORRUGATED AND

 CORRUGATED DIELECTRIC STABILIZER COMPARISON

	V. CONCLUSION
		Corrugated	Dielectric
		stabilizer	stabilizer
	Implementation	Challenging	Difficult
	Theorical electric	High	Ultimate higher
	field under limitation	(200 V/m)	(440 V/m)
	Issues	Stabilizer realization at low cost	Dielectric material with good thermal conductivity and suitable elongation

This project has received funding from the European Union's Horizon 2020 research and innovative programme under grant agreement No 721019. (Corresponding author: Pascal Tixador.) P. Tixador is with Univ. Grenoble Alpes, CNRS, Grenoble INP, G2Elab-Institut. Neel, 38000 Grenoble, France (e-mail: Pascal.Tixador@grenoble-inp.fr).


Edwin Kermarrec

A Zorko

Fabrice Bert

R H Colman

B Koteswararao

Frédéric Bouquet

P Bonville

A Hillier

A Amato

J Van Tol

A Ozarowski

A S Wills

P Mendels

Spin dynamics and disorder effects in the S = 1 2 kagome Heisenberg spin-liquid phase of kapellasite

Keywords: numbers: 75.10.Kt, 76.60.-k, 76.30.-v, 76.75.+i 1

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

I. INTRODUCTION

The S = 1 2 Heisenberg antiferromagnet model on a kagomé lattice has proven to be a fascinating and enduring problem for more than 20 years. Currently, there seems to be a broad consensus on the nature of the ground state, with recent DMRG studies pointing to a resonating valence bond state that features a singlettriplet gap, though the proposed value of ∆ = 0.13J 1,2 is still uncertain. 3 Contrastingly, fermionic approaches which suggest a gapless ground state remain debated. [4][5][6][7] Meanwhile, on the experimental side, new spin liquid candidates have been discovered in recent years with S = 1 2 spins. [8][9][10][11] One of the most puzzling results, in view of the DMRG results, is the absence of any gap in the excitations of these experimental realizations. The reasons for such a discrepancy, and whether it is related to deviations from the ideal model, remain key open issues in the field. Several research directions have been proposed based on both theoretical developments and experiments such as Dzyaloshinskii-Moriya anisotropic interaction and next nearest-neighbours (nnn) exchange, [12][13][14][15] which are believed to drive the system towards a magnetically ordered phase, [16][17][18] or lattice distortions.

In this landscape of new spin liquid candidates and drive to understand the role of perturbations to the ideal nearest-neighbour (nn) Heisenberg antiferromagnetic model, kapellasite, 9,19 a polymorph of herbertsmithite γ-Cu 3 Zn(OH) 6 Cl 2 , has come as a surprise. Instead of being a mere variant of herbertsmithite, it has been demonstrated to be a kagomé spin liquid candidate of a new type. µSR and inelastic neutron scattering (INS) experiments evidenced a gapless spin liquid state down to 20 mK, with unusual dynamical short-range spin correlations. It is an experimental representative of a previously uninvestigated class of models on the kagomé lattice with frustrated competing interactions, namely ferromagnetic nn-exchange J 1 and second nn (along the edges) J 2 , and antiferromagnetic nnn exchange (across the diagonal of the hexagons) J d , with J 2 ≪ J 1 ∼ J d . 20,21 The values of interactions extracted from a high-T series analysis of macroscopic susceptibility and specific heat lead to an estimate of J 1 = -12 K, J 2 = -4 K and J d = 15.6 K, 20,22 which, for classical spins, would stabilize a non-coplanar chiral state of the cuboc2 type. In kapellasite, quantum fluctuations are argued to destabilize the latter state and to lead to related dynamical correlations, as detected by INS. 20 The magnetic structure of kapellasite, α-Cu 3 Zn(OH) 6 Cl 2 , exhibits well decoupled magnetic planes with a kagomé geometry formed from the depletion of a Cu 2+ (S = 1 2 ) triangular lattice by diamagnetic Zn 2+ [Fig. 1(a)].

These planes are only weakly coupled along the third dimension (the hexagonal c-axis) via interlayer Cl• • • OH hydrogen bonding, which guarantees highly bidimensional characteristics. This essential feature causes the flat shape of micron scale grains constituting our powder sample [Fig. 1(b)]. Neutron diffraction revealed a non-distorted kagomé lattice, but showed a sizeable shortage of Cu compared to the ideal stoichiometry. 9 The observed formula determined by ICP-AES chemical analysis is (Cu 0.73 Zn 0.27 ) 3 (Zn 0.88 Cu 0.12 )(OH) 6 Cl 2 .

Importantly, the local octahedral environment of nominally Cu site is more distorted than the Zn one, and has an extension along the Cl-Cl direction [Fig. 1(c)]. The more symmetric bonding environment of the Zn site, an octahedral site coordinated by six oxygens, leads to strong preferential occupation by Zn 2+ , with 95 % of the Jahn-Teller active Cu 2+ residing on the more distorted kagomé sites coordinated by four oxygens and two chlorines. The major consequence of this non-stoichiometry is then the depletion of the kagomé lattice by non-magnetic Zn 2+ ions. The kagomé site occupancy, p = 0.73, still remains above the percolation threshold of the kagomé lattice p c ≃ 0.65. 23,24 In this article, we investigate the magnetic properties and the effects of structural disorder in kapellasite by means of NMR, µSR, ESR and specific heat measurements. This paper is organized as follows. In section II, we evaluate by 35 Cl-NMR the effect of disorder within the kagomé planes and report evidence for local homogeneity of the magnetic properties, proving the robustness of the underlying physics against dilution. In section III, we use ESR to estimate the symmetric exchange anisotropy term in the Hamiltonian and confirm that the choice of an isotropic Heisenberg Hamiltonian to describe the magnetic properties of kapellasite is reasonable in a first approach. While the main exchange interactions J 1 and J d have already been derived from magnetic susceptibility and specific heat fits, we detail in section IV an analysis of the specific heat under magnetic fields which confirms that its low-T part is magnetic and intrinsic, and thus provides a pertinent quantity for extracting the exchange integrals. Finally, in the last section V, we investigate with NMR and µSR the spin fluctuations at low energy in the low-T frustrated antiferromagnetic phase induced by these competing interactions on the S = 1 2 kagomé lattice.

II. NMR DETERMINATION OF LOCAL STRUCTURE AND SPIN SUSCEPTIBILITY

NMR experiments were performed on the most abundant Cl isotope, 35 Cl (abundance of 75.8 %) using an oriented powder aligned along the hexagonal c-axis of the kapellasite's rhombohedral structure. The high quality orientation leads to accurate refinements of the local structure and unambiguous determination of the local spin susceptibility and the spin dynamics. 35 Cl is a quadrupolar nucleus (I = 3/2), hence one has to consider two sets of interactions resulting from the coupling of the nucleus to its magnetic and charge environments. The nuclear Hamiltonian can be written [START_REF] Slichter | Principles of Magnetic Resonance[END_REF][START_REF] Cohen | [END_REF]

H = H mag + H quad (1)
with

H quad ≫ H mag
As detailed in Appendix A, this generally leads for a given environment to a spectrum consisting of three lines.

The central line, corresponding to the -1 2 ↔ 1 2 transition, and the satellite lines, corresponding to - would expect a unique Cl environment, where each Cl nucleus couples to the three adjacent Cu sites located at the corners of a given triangle. In disagreement with this simple scheme, we instead observe multiple lines, which are best resolved for the central -1 2 ↔ 1 2 transition line at T = 30 K taken for θ = 0 • , where θ is the angle between the applied field and the c-axis (see Fig. 2 and section II B). The multiple lines can be associated with different Cl environments induced by the Cu/Zn mixing disorder, namely four different Cu/Zn configurations on a probed triangle: Cu 3 (1), Cu 2 Zn (2), CuZn 2 (3) and Zn 3 (4). This site assignment also agrees with the high-T shifts which scale with the number of Cu to which each Cl is coupled, see Fig. 4(b). Since the spin-spin (T 2 ) and spinlattice (T 1 ) corrections were found to be negligible, the integrated intensity of each line is directly proportional to the number of triangles of each configuration which is found to perfectly match with a random occupation of the Cu sites (Table I), with a filling ratio p = 0.75(1). This agrees with the value of 0.73 extracted from neutron diffraction refinements 9 .

The impact of dilution on the local structure can be traced by the satellite lines which positions depend to first order upon the parameters which govern H quad , namely the quadrupolar frequency ν q and the asymmetry η, and the orientation of the principal axis of the elec- 2) are displayed in Fig. 3(b) and (c). For θ = 0 • , if one assumes that the principal axis of the EFG remains along the symmetry axis Oz for all the Cl sites, the distance between the two satellites should be 2ν q (eqs.A9,A10), independently of the η value. Under the latter assumption, we find two different values for sites (1) and (2), respectively ν q,1 = 3.4(1) MHz and ν q,2 = 4.0(2) MHz [Fig. 3(b)], which reflect two different local electrostatic environments around Cl. While for site (1) the spectrum is consistent with a η = 0 value, for site (2) the asymmetry of the spectrum obtained in the perpendicular direction (θ = 90 • ) can only be accounted for by assuming a surprisingly large value of the asymmetry parameter η = 0.3 (1). Finding such a large value might rather invalidate our initial assumption and indicate that for this site, the EFG and shift tensor principal axis do not coincide anymore. The local distortions deduced through this extensive analysis of the quadrupolar parameters can easily be attributed to the different nature of the two Cu 2+ and Zn 2+ ions. Whereas they possess the same charge and a similar radius, Cu 2+ is known to be Jahn-Teller active and to elongate the octahedron built on the nearby anions. Such local effects can then be expected when the ratio Zn/Cu is varied on a given triangle but do not affect the average threefold axis structural symmetry.

B. Spin susceptibility

Below we extend the concise presentation of Ref.20 and detail the analysis of the 35 Cl NMR data taken on the central line, and further compare the susceptibilities extracted for sites (1), ( 2) and (3) from the lineshifts taken for an applied field along the c-axis, K z,1 , K z,2 from 287 K down to 1.2 K and K z,3 down to 80 K (Fig. 4). The broadening of the NMR lines observed at low temperature is discussed in section VI A.

Following a typical two-step procedure, we derive the NMR spin shifts which are induced by the local susceptibilities denoted χ i : from a linear high-T fit of K z,i plotted versus the macroscopic susceptibility χ macro measured by SQUID magnetometry, K z,i = A hf,i •χ macro +σ i , we extract the hyperfine coupling constants A hf,i and the chemical shifts σ i [Fig. 4(b)]. The local susceptibilities are then given by χ i = (K z,i -σ i )/A hf,i . The ratios of hyperfine couplings A hf,1 /A hf,2 = 1.53(2) and A hf,3 /A hf,2 = 0.58 (2), with A hf,1 = -4.98(3) kOe/µ B , are found to be in good agreement with the ratios of the number of Cu 2+ probed, respectively 1.5 and 0.5. This further strengthens the previous analysis of the local structure, and again indicates that at high T the S = 1/2 spins are rather uncorrelated, depicting a fluctuating paramagnetic state. This validates the overall consistency of our analysis.

The local susceptibility derived for site (1) is certainly the least perturbed by defects and is the closest to that of the ideal system. It is found to scale with χ macro over the whole temperature range, down to 1.2 K, which invalidates the possibility of a dominant contribution of defects at low-T in χ macro [Fig. 4(b)]. One should note that this finding strongly contrasts with the case of many correlated oxides where a Curie-like tail in macroscopic susceptibility is commonly attributed to defects, either spin vacancies in the magnetic lattice or other species of weakly coupled spins. In these systems, the disorder contributes to the NMR linewidth only, and not to the lineshift. 15,[27][28][29] This uncommon divergence as T → 0 of χ spin in comparison with the reported χ spin of nn-Heisenberg kagomé antiferromagnets 15,27,30 could only be explained within the framework of competing interactions on the kagomé lattice, J 1 = -12 K, J 2 = -4 K and J d = 15.6 K, evaluated by high temperature series expansion as emphasized in Ref.20 and 22. The Curie-Weiss temperature θ CW ∼ 9 K is consequently the result of an averaging of these competing interactions.

A closer comparison of the low T local susceptibilities χ i at sites ( 1) and ( 2) brings an additional insight into the nature of the interactions. While all shifts display a low-T upturn, a minor effect is detected below T = 50 K, where the ratio of the local susceptibilities of site (1) and site (2), χ 1 /χ 2 , is not T -independent anymore and shows instead a clear increase as T is lowered (Fig. 5). In order to infer a physical understanding to this deviation, χ 1 /χ 2 is fitted to the ratio of two Curie-Weiss susceptibilities C(T -θ 2 )/C(T -θ 1 ). The parameters θ 1 and θ 2 account for the interactions respectively probed at sites (1) and ( 2) and are expected to be relevant in the mean field regime, i.e. for T 2θ CW ≃ 20 K. The main result of this simple analysis is that the upturn at low T could only be reproduced when θ 1 > θ 2 . In a mean field approach and ignoring the weakest J 2 coupling, the Curie-Weiss temperature of each site (i) of the J 1 -J d Heisenberg model on a diluted kagomé lattice can be simply expressed as

θ i = -α i J 1 -β i J d /2.
For a fully occupied kagomé lattice, one has α 1,2 = β 1,2 = 1. Taking into account the random depletion of the lattice by 27 %, a straightforward calculation of probabilities leads to α 1 = 0.865, β 1 = 0.73 and α 2 = 0.615, β 2 = 0.73. Therefore, one can directly infer from the relation θ 1 > θ 2 that J 1 < 0 (ferromagnetic), in agreement with Ref. 20 and 22.

The related scheme of interactions is summarized in the diagram of Fig. 5.

III. ELECTRON SPIN RESONANCE

ESR measurements were performed at NHMFL, Tallahassee, Florida, on a custom-made ESR spectrometer with homodyne detection working in transmission mode at the fixed frequency of ν 0 = 326.4 GHz. The same ori- ented sample was used as in NMR. A single ESR line was observed in the temperature range between 5 and 100 K. We notice first that this is consistent with the absence of any magnetic parasitic phase in the sample. Moreover, since the copper ions are present on both the nominally Cu and Zn sites, the observation of a single line indicates that there is a sizable exchange coupling between sites, yielding a Lorentzian line shape typical for an exchange narrowing limit.

There are three distinct T -regimes as revealed in (Fig. 6). Above 100 K, the ESR line could not be followed reliably due to its decreasing intensity with increasing temperature. The reduction of the signal amplitude was additionally enhanced by line broadening. Between 10 and 100 K (intermediate T ), the linewidth increases regularly with increasing T . Below 10 K, an opposite behaviour is observed. In the two latter regimes, the resonance field exhibits a monotonic variation, similar to the macroscopic susceptibility.

Although the ESR line in kapellasite is similarly broad to that seen in herbertsmithite which points to a sizable anisotropy, 12 the temperature dependence of the ESR parameters; i.e., the line position B 0 (g-factor; g = ν0 µB B0 , where µ B is the Bohr magneton) and the linewidth ∆B, are quite different. In herbertsmithite, a maximum in ∆B and small temperature changes of the g-factor of less than 0.02 were observed. Contrastingly, in kapellasite, a minimum is observed in the linewidth at 10 K. This behaviour rather resembles the one found in another Cu-based kagomé compound, vesignieite, where it was assigned to a phononic contribution. 13 Lastly, we note that the measured line-shifts with decreasing temperature in kapellasite are much larger than in herbertsmithite and vesignieite. This indicates that a different type of magnetic anisotropy is responsible for these shifts than the Dzyaloshinsky-Moriya (DM) interaction that had been found to be dominant in the other two cases. (Fig. 6). Below, we discuss successively the information which can be extracted from the ESR linewidth and from the g-factor.

A. Linewidth

In the paramagnetic regime of herbertsmithite (T ≫ J), the ESR linewidth was constant so this asymptotic value could be used to extract the dominant magnetic anisotropy term. 12 It is notable that, in kapellasite, one cannot reach this regime even at 100 K, a temperature already as high as ∼ 6J d [Fig. 6(b)]. This indicates that the linewidth in the intermediate-T regime cannot be accounted for solely by the spin system and an additional line-broadening mechanism must originate from other degrees of freedom. A spin-phonon coupling, responsible for a similar increase in vesignieite, 13 is quite commonly encountered in copper-based systems and is perfectly in line with the increase of the linewidth between 10 and 100 K. [31][32][33][34] . This mechanism is dominant at high-T so that the magnetic anisotropy cannot be extracted from the linewidth. The low-T regime (below 10 K ∼ J) resembles more the behavior found in herbertsmithite. At such low temperatures the phonon population vanishes and the ESR broadening becomes dominantly spininduced, reflecting the development of spin correlations.

B. g-factor

Large changes of the g-factor with decreasing temperature in kapellasite [Fig. 6(c)] could, in principle, be due to either a locally inhomogeneous susceptibility or a significant magnetic anisotropy of the system. The first possibility can be easily discarded. Indeed, in inhomogeneous spin systems (e.g., spin glasses, impure systems and systems with several non-equivalent sites) nonuniform local fields develop with increasing susceptibility (decreasing T ) leading exclusively to negative B 0 shifts, irrespective of the orientation of the applied field. 35,36 In addition line-broadening that scales with the susceptibility is generally encountered. Such shifts and broadening result from staggered local fields. 36 In kapellasite this is clearly not the case since both positive and negative line shifts are observed depending on the orientation of the applied field and the ESR width is non-monotonic when the susceptibility is increasing steeply at low temperature. We are then led to assign these shifts to magnetic anisotropy.

When the magnetic anisotropy H ′ is small compared to the exchange and/or Zeeman energy Nagata's theory for g-factor shifts can be applied. 37 Then, the shift is determined by the correlation function 37,38 ∆g =

[S -, [S + , H ′ ]] 2µ B B 0 S z , (2)
where [..., ...] denotes a commutator, ... canonical averaging and S α = i S α i the α-component of the total spin operator. Here, we refer to the general form of the magnetic anisotropy in the spin Hamiltonian of a spin-1/2 system,

H ′ = 1 2 i,j =i S i • K ij • S j , (3)
where the coupling tensor

K ij = d ij + D ij (4)
includes both the long-range dipolar coupling d ij and the short-range symmetric anisotropic exchange D ij on bonds J 1 and J d . We note that the antisymmetric Dzyaloshinsky-Moriya interaction does not contribute to the ESR line shift at the level of first order [eq.( 2)] 38 and so we omit it. We proceed to the evaluation of both anisotropy terms from eq.( 4) in kapellasite, by noting that in the paramagnetic regime the Nagata's theory predicts a linear scaling of the line shift with susceptibility, 37,39

∆g z (T ) = S z 2µ B B 0 j =i 2K zz ij -K xx ij -K yy ij , (5)
where S z = i S z i = χ mol (T )B0 NAgµ0µB , z denotes the orientation of the applied field and x, y the two perpendicular directions, N A is the Avogadro number and µ 0 the vacuum permeability. A convincing scaling is observed in kapellasite for both directions in the whole investigated temperature range [Fig. 6(d)], an additional proof that the increase of the bulk susceptibility is intrinsic even at low temperatures. The scaling yields the limiting infinite-temperature values of the g-factor within the kagomé planes, g 90 • ∞ = 2.140, and along the c axis, g 0 • ∞ = 2.115. These values correspond to the values averaged over the structural triangular unit.

The contribution to the above sum related to the dipolar field arising from the finite spin S z in a polarized paramagnet can be calculated exactly,

B dd = S z µB j =i 2d zz ij -d xx ij -d yy ij .
Following Lorentz, we write 40 B dd = B ′ dd + B Lor + B dem , where the first term is a sum of dipolar fields within a (Lorentz) sphere with a radius large enough to provide convergence, the second term B Lor = 1 3 χB 0 is the Lorentz field that originates from the polarization of the sphere's surface and the third term B dem = -N χB 0 is the demagnetization field depending on the shape of the sample. By distributing Cu 2+ spins randomly on both Cu and Zn sites according to the sample stoichiometry, we find B ′ 0 • dd /χ mol = -1.80 T(emu/mol Cu) -1 and B ′ 90 • dd /χ mol = 0.90 T(emu/mol Cu) -1 . Further, based on the oblate shape of the crystalline grains (2r/d ∼ 10) we estimate N 0 • = 0.80 (5) and N 90 • = 0.10(3), 41 which yields

(B Lor + B dem ) 0 • /χ mol = -1.61 T(emu/mol Cu) -1 and (B Lor + B dem ) 90 • /χ mol = 0.81 T(emu/mol Cu) -1 .
The effect of the dipolar fields (including demagnetization) is too small to explain the large experimental shifts [see Fig. 6(a) for a comparison of corrected and noncorrected data]. We subtract this contribution and derive "pure" g-shifts [Fig. 6(c)] arising from the symmetric anisotropic exchange. The linear scaling of the shift with macroscopic susceptibility [Fig. 6(d)] can be used to evaluate the magnitude of the exchange anisotropy. Since the Cu-Cl bonds defining the principal axis of the uniaxial local g-tensor on each site are tilted from the crystallographic c direction towards the center of the triangle by 43 • , the direction of the principal axis of D ij of each ij bond is not trivial. Therefore, we only inspect the shift data for θ = 0 • , for which employing the three-fold rotational symmetry of the crystal structure, we derive

∆g 0 • (T ) = χ mol (T ) N A gµ 0 µ 2 B 4D ′cc , (6)
where D ′cc = D cc 1 + 1 2 D cc d corresponds to an effective component of the magnetic anisotropy arising from both dominant bonds J 1 and J d .

From the slope ∆g 0 • (T )/χ mol (T ) = -2.0 (emu/mol Cu) -1 we obtain D ′cc = -0.4 K, yielding the ratio |D ′cc /J 1 | = 3% with respect to the dominant exchange. This is somewhat larger than the conventional Moriya estimate |D/J| ∼ (∆g/g) 2 ∼ 2%. However, D ′cc sums two contributions and, moreover, similarly enhanced values of the symmetric anisotropic exchange interaction were reported before in copper-based systems. 32,34 Our results thus suggest that kapellasite is quite close to the isotropic Heisenberg limit. We stress that the DM interaction, which is generally larger than the symmetric exchange anisotropy in copper-based systems (of the order ∆g/g ∼ 15%) could not be assessed in this study, although it may well be present also in kapellasite. Last, we note that the large line shifts in kapellasite observed at low temperatures are due to rather high values of the intrinsic magnetic susceptibility. In contrast, in herbertsmithite and vesignieite, a suppressed susceptibility did not allow an unambiguous assignment of the symmetric exchange anisotropy based on the ESR line shift.

IV. SPECIFIC HEAT

The specific heat C(T ) was measured with a Quantum Design Physical Property Measurement System (PPMS) from 300 K to 0.4 K on thin pressed pellets of kapellasite with a typical mass of 8 mg, in magnetic fields up to 9 T (Fig. 7). The analysis of C(T ) under zero magnetic field has been the subject of two recent publications, where its T dependence was used to refine the model Hamiltonian of kapellasite, based on high temperature series calculations 20,22 . In the following we present its field dependence and confirm that its low-T part is magnetic and intrinsic, and thus provides a pertinent quantity for extracting the exchange integrals.

In the absence of a non-magnetic compound isostructural to kapellasite, the phonon contribution is estimated at high temperatures (50 ≤ T ≤ 160 K) with a phenomenological fit made of a linear combination of Debye and Einstein functions. The lattice contribution is found to be negligible below T = 10 K (see Fig. 7). Therefore, the broad peak singled out in C(T )/T around 2 K has clearly a magnetic origin.

The specific heat is found to be insensitive to fields smaller than 3 T, however larger fields cause a shift of the peak towards higher T , up to 4 K for µ 0 H = 9 T, as well as a decrease of intensity. This field dependence definitely shows the magnetic character of C(T ) at low T . The peak that is observed cannot be attributed to the presence of orphan spins which would contribute as a Schottky anomaly in the specific heat as observed for example in herbertsmithite 42 . As well as displaying a peak, such a contribution would have a linear dependence in field, contrary to our observations (Fig. 7, inset). This is completely consistent with the fact that the upturn of the susceptibility is not driven by weakly coupled spins but is intrinsic as clearly shown by the perfect match of the measured NMR and ESR shifts with the macroscopic susceptibility. Therefore, in contrast with other frustrated systems with larger magnetic couplings, the field dependence of C(T ) observed here is due to the comparable energy scale between the applied magnetic field and the rather low magnetic exchanges in kapellasite. As a main consequence, the specific heat for µ 0 H = 0 T is characteristic of the model Hamiltonian of kapellasite and is relevant for the evaluation of the exchange interactions as performed by Bernu et al. 22 .

The T dependence of the magnetic part of C(T ) could be reproduced in the range 0.4-10 K within a J 1 -J 2 -J d model, provided one assumes a ∼ 14 % missing entropy, in this T -range. In fact, the total magnetic entropy expected for a spin S = 1 2 , ∆S = R ln 2, is fully recovered only by T = 40 K, a quite extended regime when compared to the former model. This release of the missing entropy at higher T appears in good agreement with the extended T -regime where inelastic neutron scattering studies reveal the presence of magnetic correlations. 20 Two different hypothesis were proposed in Ref. 22 to account for this missing entropy based on the presence of spin vacancies: the existence of weakly coupled spins or the formation of spin singlets. The former scenario can be ruled out, as already discussed, since it would give rise to a linear field dependence of the maximum of the specific heat below 10 K. The second scenario, namely spin singlet freezing, remains, on the contrary, plausible since the exchange along the diagonal J d ∼ 15.6 K is antiferromagnetic in nature. The identification of the involved spins is still unclear, although the singlet could result from the coupling of a a spin sitting on a kagomé site and a spin defect on the center of an hexagon. Their contribution would have to show up at higher temperatures (T ∼ 10 K), and would therefore remain almost unaffected by fields µ 0 H < 9 T (∼ 12 K). While we are not able to single out the exact behavior of these defects from these measurements, this last scenario is consistent with our experimental results. It is not clear, however, if a spin singlet can be stabilized in the presence of other large competing interactions.

V. SPIN DYNAMICS

A. 1/T1 NMR spin lattice relaxation rate

The 35 Cl spin-lattice relaxation rate 1/T 1 was determined through the saturation recovery method, with a standard π/2 -∆t -π/2 -τ -π pulse sequence (70 < τ < 100 µs), from 40 to 1.3 K. The measurements were performed at the center of lines ( 1), ( 2) and (3), which gave similar average values and distributions (see below) of T 1 at low temperature. The evolution of T 1 with temperature was then extracted from measurements performed on the line with the highest intensity (1).

Within this T -range, the spin dynamics can be divided into two regimes described below: (i) a high-T paramagnetic one where T 1 remains constant and (ii) a low-T fluctuating regime where a broad distribution of relaxation times is observed.

(i) High T paramagnetic relaxation: Above 10 K, the magnetization-recovery curve M (t) is consistent with that expected for a relaxation process of magnetic origin for a nuclear spin I = 3 2 with a single spin-lattice relaxation time T 1 43 ,

M (t, T 1 ) = M sat 1 -0.1e -t/T1 -0.9e -6t/T1 . ( 7
)
Fitting our recovery curves to eq.( 7) yields the T 1 values plotted in [Fig. 8(c)] with a constant T 1 = 12(3) ms value for 10 < T < 40 K.

In conventional insulating paramagnets, for T ≫ J, the spin-lattice relaxation 1/T 1 is indeed temperature independent. The latter was explicitly calculated by Moriya 44 to be where z 1 = 3 is the number of Cu 2+ probed and ω e = J 2zS(S + 1)/3/ is the exchange frequency with z = 4 being the number of Cu 2+ neighbours on the kagomé lattice. Using experimental values A hf = -4.98 kOe/µ B for the 3-Cu triangle and the average exchange J = θ CW = 9.5 K, the formula leads to T 1 = 11 ms, which is in excellent agreement with the measured T 1 . 46 (ii) Low T relaxation: At lower T , in the regime where correlations become noticeable, one has to take into account the q-dependence of the excitations. Indeed, T 1 is given by

1/T 1 = √ 2πγ 2 N g 2 A 2 hf S(S + 1)/3z 1 ω e , (8)
1 T 1 = γ 2 µ 2 B k B T q |A(q)| 2 χ ′′ ⊥ (q, ω 0 ) ω 0 (9)
where A(q) acts as a filtering factor for some specific values of q, associated here with the triangular geometry. The functions |A(q)| 2 are plotted in Fig. 9 for the various Cl sites. It is notable that at the peak of S(q) detected in neutron experiments 20 , the filtering effect varies by only ∼ 30 % from one site to the other, with a minimal transmission of ∼ 60 %. In the following we give our results for the Cl site (1) situated at the center of a fully occupied triangle.

If the conventional exponential shape of relaxation is indeed observed above 10 K, the situation is remarkably different at T = 1.3 K. The saturation recovery curve there had to be fitted using the phenomenological stretched exponential variant of eq.( 7) [Fig. 8(b)],

M (t) = M sat 1 -0.1e -(t/τ0) β -0.9e -(6t/τ0) β . (10)
This points to a broad distribution of relaxation times τ , with a probability density function ρ(τ ), so that

M (t) = +∞ 0 M (t, τ )ρ(τ )dτ (11)
Physically, τ 0 is related to the mean of the probability density function while β characterizes its dispersion. The probability density function itself can be derived from an inverse Laplace transform of eq.( 11), whose explicit expression can be found elsewhere 45 , and which, following Ref. 47, can be numerically evaluated once β and τ 0 are known. In order to follow the evolution of the distribution of relaxation times with temperature, we fitted the magnetization recovery curve with eq.( 10) for each measured temperature. The pair of parameters (τ 0 , β) extracted from these fits allows then to compute the ρ(τ ) profiles for 1. It then asymmetrically broadens at lower T , exhibiting a τ → 0 divergence when T → 0, while β approaches smoothly its lowest value of ∼ 0.5. The same qualitative behavior was observed for all Cl sites.

-2 -1 0 1 2 -2 -1 0 1 2 q x (Å -1 ) q y (Å -1 ) -2 -1 0 1 2 -2 -1 0 1 2 2 Cu / 1 Zn q x (Å -1

B. LF µSR relaxation

Complementary information on the spin dynamics can be obtained from µSR measurements in longitudinal field, since it directly probes the spectral fluctuation density. As discussed in Ref.20, the µ + polarisation is described by P (t) = f e -t/T1 + (1 -f ), where 1 -f = 0.14 is a slowly relaxing muons fraction completely decoupled for µ 0 H > 100 G and thus responsible for the non-zero value observed at long time 20 (Fig. 10). The application of a small longitudinal field µ 0 H = 100 G also enables one to negate the static nuclear contribution and allows the electronic fluctuations to be probed separately.

Similarly to NMR, a constant relaxation rate 1/T µ 1 is observed for T > 5 K [Fig. 8(c)]. Assuming local fluctuations, the spin-lattice relaxation times are linked through the relation

T 1 /T µ 1 = (γ µ A µ /γ n A hf ) 2 .
At T = 20 K, the experimental scaling factor is found to be T 1 /T µ 1 ∼ 150, leading to the muon coupling constant A µ = 1.88 kOe/µ B , close to the dipolar coupling esti- mated to be A dip µ ∼ 1.3 kOe/µ B at the principal OH - muon site. Thus, the spin fluctuations probed in µSR above 10 K also originate from a paramagnetic regime.

A good agreement is observed for the 3 K onset-T where the relaxation rate departs from the paramagnetic limit while the evolution at lower T follows qualitatively that of the probability density function profiles extracted from 35 Cl NMR, as underlined by Fig. 8(c).

However, contrary to NMR, a broad distribution of relaxation times does not appear clearly in µSR experiments. Should the same distribution be observed, the µ + polarisation for µ 0 H = 100 G would vary as P (t) = e -(t/T1) β with β ∼ 0.5 at low T . Allowing a stretched exponential relaxation does not appreciably improve the quality of the fit. The β exponent is ill-defined but its value is found to remain above 0.7 in the whole T -range below 4 K which would suggest a moderate distribution of relaxation times as probed by the muon.

Lastly, we comment on the field-dependence of the relaxation rate. In disordered spin systems with a single relaxation time and exponentially decaying correlations, the spin-lattice relaxation time T 1 exhibits a field square dependence T 1 ∝ H 2 , characteristic of a Lorentzian spectral density. 48 Such a spectral-weight function, which is centered at zero energy, is commonly used to describe conventional fluctuations in the paramagnetic regime. Here, a power law dependence T 1 ∝ H α is observed at 60 mK, with α ≃ 1 (Fig. 10,inset). This observation is inconsistent with the existence of a single time scale, as underlined by our NMR measurements, and suggests a more exotic spectral density, such as the one at play in the spin-liquid phase of the S = 1 2 antiferromagnetic chain. 49 The contrast between the low-T relaxation extracted from the two NMR and µSR techniques is discussed in the following section.

VI. DISCUSSION

A. Effects of disorder

In this section, we discuss the information that can be extracted from the NMR spectra and the related impact of defects, that is deviations from the ideal kagomé lattice, on the ground-state of kapellasite. Let us first recall that the structural refinements yield the chemical formula (Cu 0.73 Zn 0.27 ) 3 (Zn 0.88 Cu 0.12 )(OH) 6 Cl 2 which points to both the dilution of the kagomé lattice (27 %) and a substantial occupation of the Zn site by Cu (12 %).

Spin vacancies are rather common in experimental materials, e.g. in the kagomé antiferromagnet where, for a nn-antiferromagnetic Hamiltonian, they are known to create a spin singlet on the sites first neighbours of the impurity and induce a staggered susceptibility at larger [50][51][52] In the case of kapellasite, where the scheme of interactions is different, little is known about such an impact of the spin vacancy. From a general view point, magnetic defects are expected to generate a distribution of susceptibilities. In addition, the rapid crystallisation of kapellasite during the synthesis process 9 may affect the homogeneity of the defect distribution, and could further influence the disorder. The latter can yield a distribution of hyperfine constants. Although we expect the distribution of susceptibilities to dominate the linewidth of our spectra, ∆H ∼ ∆(A hf χ), especially in the low T range, attributing all the width to a structural disorderinduced distribution of hyperfine constants would lead to a scaling ∆H ∼ χ. An upper bound of ∆A hf /A hf ∼ 33 % can then be extracted from the linear slope of the plot of ∆H versus χ in Fig. 11. In the low-T range, we note that, for site (2), the linear scaling is severely disrupted below 10 K which rather argues in favour of a distribution of susceptibilities, as expected.

In the context of a large dilution, one could wonder whether the absence of any spin freezing really reflects the spin liquid physics of the ideal J 1 -J 2 -J d Heisenberg Hamiltonian. We first recall that the level of dilution of d = 27 % is still above the percolation threshold of the kagomé lattice p c ≃ 0.652, 23,24 i.e. that d < d c = 1 -p c ≃ 0.35. On the contrary, delafossite LaCuO 2.66 , with a 33 % ( d c ) diluted kagomé lattice, displays an inhomogeneous spin frozen ground-state. 53 While the extreme two-dimensional character of kapellasite could also be invoked to support the absence of freezing at finite temperature, the Mg-analogue of kapellasite, haydeite, has a clear transition at T ∼ 4 K. 9,54 This gives support to the intrinsic character of the spin liquid behavior which is found in kapellasite.

To date, the existence of such a spin liquid phase is supported within two theoretical approaches: observations, 20 including the evolution of the µSR relaxation rate with temperature, but inherently fails to reproduce a gapless behavior as observed both in neutron and our experiments; and (ii) a pseudofermion functional renormalization group approach by R. Suttner et al.. 55 There, kapellasite would locate in a small non-frozen region between the cuboc and the ferromagnetic phase, in a slightly different J 1 -J 2 Heisenberg model.

B. Inhomogeneous dynamics

We now turn to the spin dynamics in the low temperature phase and discuss the intriguing difference observed between the relaxation probed by NMR and µSR. The form factor |A(q)| 2 in eq.( 9) could be a potential source of disagreement between the two measured relaxations. The muon most likely sits in a non-symmetric site and is dominantly coupled to Cu 2+ via dipolar interaction while the 35 Cl nucleus, located at a symmetric site above a Cu 2+ triangle probes each Cu at the corners of this triangle through hyperfine couplings.

While the distribution thought to be of hyperfine couplings mentioned previously could be responsible for the distribution of T 1 seen only in NMR, this can be ruled out as the broadening of the distribution function is clearly temperature dependent [Fig. 8(a)]; a distribution of hyperfine constants remains T -independent.

The q-dependence of the form factor may also filter out some modes in NMR as compared to µSR. Let us recall that the different sites are clearly resolved in the NMR spectra, and that the same stretching of the relaxation is observed at each site at low temperature. Since the intensity of the form factors differs from one site to another (see Fig. 9), it is unlikely that the q-dependence of |A(q)| 2 is responsible for the T 1 distribution. We are thus left with the firm conclusion that the distribution of T 1 observed in NMR mainly comes from the fluctuation spectrum. The difference between NMR and µSR relaxations could be ascribed to the difference of these couplings rather than being purely explained by the different time-scale probed with each technique. Indeed, for both techniques, T -1 1 scales with A 2 -A is the coupling constant -so that the long time tail of the T 1 distribution detected through NMR would yield unobservable relaxation times in µSR. Cutting off this tail by restricting the integration range of the probability density function extracted from NMR over [0, α] with α < +∞, yields a distribution of 1/T µ 1 = λ [see eq.( 11)] still too large to account for the smaller breadth of relaxation times (β ≥ 0.7) observed in µSR. It is therefore necessary to invoke very local modes that give distinct relaxations characteristics to the various sites probed by NMR and which have distribution that is smoothed out through the long range character of the dipolar coupling between the muon probe and the fluctuating Cu 2+ spins.

One of the characteristic features of the low T phase of kapellasite is the well localized intensity present in the inelastic channel measured by inelastic neutron scattering (INS) at the M point of the Brillouin zone. Both INS and NMR relaxation probe the dynamical susceptibility χ ′′ (q, ω), yet NMR has no q-resolution (except via the form factor) and probes the ω → 0 region. In order to extract some information from their comparison, we convert χ ′′ (q, ω) measured by INS in an equivalent 1/T 1 by integrating χ ′′ in the ranges q = [0.2, 0.8] Å and ω = [0.4, 0.8] meV. The values obtained are scaled so as to match the high T NMR and µSR ones. The result is plotted on Fig. 8(c). We note that no dramatic filtering effect is expected when applying the NMR form factor |A(q)| 2 of different sites, in agreement with the nonzero value of the form factors at the cuboc2 q-positions (Fig. 9). The T evolution of T 1 generated from the INS data is in good agreement with the flat dependence observed by local probes above 10 K, but fails to reproduce any distribution at lower T . Therefore, we can conclude that at low T the NMR relaxation is dominated by extra low-energy modes that are inaccessible to INS. The low energy fluctuation spectrum would then have spectral weight relative not only to the cuboc2 mode but also at many different q positions, not filtered out by the form factors.

It is interesting to note that the cuboc2 phase is chiral and consequently could allow for the observation of a chiral phase transition at finite temperature, studied theoretically in details in Refs.56 and 57. This transition is combined with a proliferation of topological point defects (Z 2 vortices), which preferentially nucleate around the walls separating domains with opposite chirality. Such excitations were previously suspected to be at the origin of the unconventional spin dynamics observed in the Heisenberg triangular antiferromagnet NaCrO 2 58 . Further work would be required to confirm that the peculiar low-energy excitations observed in kapellasite are indeed a signature of such an exotic transition.

VII. CONCLUSION

We reported the experimental investigation of the S = 1/2 Heisenberg kagomé system kapellasite, where the two main competing interactions J 1 and J d conspire to stabilize a spin liquid state at low T with spin correlations of the cuboc2 type. 35 Cl NMR data have confirmed the previously established level of dilution of 27% of the kagomé lattice and further evidenced its random character. Surprisingly, an identical local magnetic susceptibility is probed at high temperature for each Cu 2+ spin, irrespective of its neighbourhood. This is a strong evidence of the homogeneity of the physics in the system, despite the substantial level of disorder. Additional improvements in our understanding of the disorder effect would require a set of samples with various dilution, which remains at the moment a real chemical challenge, due to the metastability of the kapellasite phase. ESR showed that the Heisenberg model is relevant to kapellasite and established a moderate magnetic anisotropy of the symmetric anisotropic exchange type of |D ′cc /J 1 | = 3% while no conclusion can be given about the Dzialoshinskii-Moriya anisotropy. The intrinsic character of the specific heat and of the magnetic susceptibility are thoroughly assessed which gives a solid experimental credit to the theoretical procedure applied by Bernu et al. 22 to extract the exchange couplings of kapellasite from high temperature series fits. Unconventional spin dynamics is consistently revealed by NMR, µSR and inelastic neutron scattering 20 upon cooling below T = 5 K, where spin correlations effectively built up. The concomitant distribution of relaxation observed in NMR is then due to the emergence of extra low energy modes, inaccessible to the neutron time scale. It could either be a complex signature of the effect of disorder, or of the specific excitation spectrum of the low-T cuboc2 chiral phase.

In conclusion, while a ferromagnetic kagomé lattice is not frustrated, the existence of a further nearestneighbour exchange along the diagonal J d in kapellasite, comparable to |J 1 |, restores magnetic frustration via the competition of interactions. The effect of frustration through the geometry of interactions was only relatively recently explored on the kagomé lattice 55 . Up to now, most of the work on kagomé magnets has been dedicated to the pure antiferromagnet Heisenberg model and the consideration of the effects of additional energy scales is still in its infancy. Second neighbour interactions have also been argued to be necessary to explain the neutron data on herbertsmithite single crystals. 55 Kapellasite gives an order of magnitude for such interactions and more broadly opens the way towards the experimental study of the phase diagram of the J 1 -J 2 -J d Heisenberg model which displays non-coplanar chiral ordered phases and non-magnetic regions. The discovery of its spin liquid state certainly deserves further attention in order to reach a more comprehensive understanding, and provide new exciting perspectives to explore quantum spin liquids physics on the kagomé lattice.

FIG. 1 .

 1 FIG. 1. (Color online) (a) The kagomé magnetic pattern arises from the regular arrangement of Cu 2+ (S = 1/2) within a Cu/Zn triangular lattice (Cu in blue, Zn in yellow). The main exchange couplings J1, J2 and J d are presented on the diagram. (b) Scanning electron microscope image of kapellasite. The width of the image is 20 µm. The flat shape of the micron scale grains evidences the bidimensional character of kapellasite. (c) Octahedral coordination environments of Cu and Zn. (d) The kagomé layers are only weakly coupled through Cl-OH hydrogen bonds. The 35 Cl NMR probe is dominantly coupled to three Cu. Zn atoms are not shown for clarity.

FIG. 2 . 1 2

 21 FIG. 2. (Color online)35 Cl NMR spectrum of the -1 2 ↔ 1 2 transition lines at T = 30 K, θ = 0 • (empty circles). The shift reference is indicated by the red vertical dotted line. The NMR spectrum is fitted using four gaussian lines (black line), corresponding to various35 Cl environments due to Cu/Zn (blue/yellow) mixing. The integrated intensity Ai of each line i reveals the site probability.

FIG. 3 .

 3 FIG. 3. (Color online) (a) 35 Cl NMR spectrum at ν0 = 47.950 MHz and T = 100 K. Three distinct groups of NMR lines are observed. (b) and (c): NMR satellites in two different angle configurations θ = 0 • (b) and θ = 90 • (c) at T = 100 K.The total fit (black line) is the sum of the two satellites pairs of the main sites (site 1 in blue, site 2 in red). For θ = 90 • , the simulation with η2 = 0 (green line) fails to represent the satellites of the site 2, contrary to the case where η2 = 0.3(1) (black line).

  FIG. 4. (Color online) (a) 35 Cl NMR spectra of the central lines plotted versus the lineshift from high (right) to low (left) temperature. The spectra are vertically shifted for clarity. The composition of the spectrum in terms of different Cl sites is shown at 80 K. The local susceptibilites can be directly extracted from the shift of the lines from the reference (dashed vertical red lines). As an example, the blue circles emphasize the shift of the NMR line corresponding to site (1) (left panel). (b) 35 Cl NMR shifts for the different Cl sites vs. χmacro at high T . The hyperfine couplings are extracted from the slope of the curves (A hf,1 = -4.98(3) kOe/µB). (c)Temperature dependence of the macroscopic susceptibility χmacro, the local NMR susceptibility χspin (left) and of 1/χmacro, 1/χspin (right). The Curie-Weiss parameter θCW = 9.5 ± 1 K evidences a weak ferromagnetic effective coupling at high T .

FIG. 6 .

 6 FIG. 6. Temperature dependence of (a) the ESR line position (b) linewidth and (c) g-factor measured at 326.4 GHz.(d) scaling of the g-factor with bulk susceptibility. Open symbols include all magnetic anisotropy contributions present in kapellasite, while solid symbols represent data corrected for the dipolar (including demagnetization) fields.

FIG. 7 .

 7 FIG. 7. (Color online) Molar specific heat divided by temperature C/T per formula unit vs. temperature and under magnetic field from 0 to 9 T. Below 10 K the phonon contribution C ph is negligible. Inset: The field dependence of the peak position (Tm) is non-linear, contrary to a Schottky behavior.

FIG. 8 .

 8 FIG. 8. (a) Probability density functions ρ(τ ) profiles for selected temperatures. (b) Magnetization-recovery curve at T = 30 and 1.3 K. The conventional relaxation form for I = 3/2 is successfully applied at high T (blue line) but non longer valid at 1.3 K where a stretched exponential curve is more appropriate (red line), due to a distribution of relaxation times τ . (c) Temperature dependence of the spin lattice relaxation rate measured in µSR (1/T1, right axis, reproduced from 20 ), in NMR (1/τ , left axis), and calculated from inelastic neutron scattering (INS), see text. The color map represents the intensity of the probability density function ρ(τ ) (left axis) extracted from NMR measurements (grey hatched area is for the absence of NMR data below 1.2 K). Below 10 K, the distribution of τ is too large to allow a single relaxation time to be defined.

FIG. 9 .

 9 FIG.9. Intensity of the normalized form factor |A(qx, qy)| 2 related to site 1 (left) and 2 (right), shown as a grey scale maps. White (black) corresponds to a full (no) transmission. The site 3 does not show any filtering effect. Blue (red) crosses indicate the wave-vectors of the cuboc2 (q = 0) state.

3 ≤

 3 T ≤ 40 K [Fig.8(a)]. Within the investigated T -range, β is found to change smoothly from 1.0(1) at T = 40 K to 0.54(4) at T = 1.3 K. The results of these fits are presented in the color map of Fig.8(c). Above 10 K the distribution ρ(τ ) is well peaked around τ 0 , with ρ(τ ) ∼ δ(τ -τ 0 ), which legitimately defines a single relaxation time T 1 = τ 0 ∼ 12 ms, with β → 1.

FIG. 10 .

 10 FIG. 10. (Color online) Muon polarization under longitudinal applied magnetic fields µ0H = 0 -2500 G, at T = 60 mK. Inset: H dependence of the spin-lattice relaxation time T1.

FIG. 11 .

 11 FIG. 11. (Color online) FWHM of the NMR lines corresponding to site(1) and (2), from the spectra displayed on Fig.4. Temperature is an implicit parameter.

TABLE I

 I

	satellite contribution from the two dominant main sites
	(1) and (2) can be reliably followed since the sites (3)
	and (4) represent a minor fraction (altogether 16 %) of
	the total spectrum intensity. Further details on satel-
	lites (1) and (
	. Cl sites occupancies evaluated by NMR lines inte-
	grals Ai and compared to binomial probabilities assuming a
	random disorder for a kagomé site filling ratio p = 0.75.
	Sites i / config.	Int. Ai	Bin. prob. (p = 0.75)
	1 Cu3	0.43(1)	0.42
	2 Cu2Zn	0.42(1)	0.42
	3 CuZn2	0.10(5)	0.14
	4 Zn3	0.05(4)	0.02
	tric field gradient (EFG) -see Appendix A. Hence they
	reflect the difference in electrostatic environments at the
	local scale. A complete NMR spectrum measured in the
	θ = 0 • orientation is shown in Fig.3(a). Three groups of
	NMR lines are clearly observed which correspond to the
	three expected transitions for a spin I = 3/2. Only the

Acknowledgements EK acknowledges very useful discussions with B. Bernu. This work was supported in part by the European Commission under the 6th Framework Programme Contract No. RII3-CT-2003-505925, by the french Agence Nationale de la Recherche under grants ANR-09-JCJC-0093-01, ANR-SPINLIQ-86998, ANR-HFM-86998, by the PHC Proteus program No. 24322SF and by Université Paris-Sud, grant MRM PMP. A.Z. acknowledges the financial support of the Slovenian Research Agency, projects No. BI-US/14-45-039. The NHMFL is supported by NSF Cooperative Agreement No. DMR-1157490 and by the state government of Florida.

A. Local structure

In this section we discuss the local structure of kapellasite as deduced from 35 Cl NMR experiments. For a perfect coverage of the kagomé lattice with Cu 2+ ions, one Appendix A: 35 Cl NMR measurements 1. Oriented sample and experiments NMR experiments were performed on an oriented powder of kapellasite, aligned along the c-axis. This sample was prepared by mixing together 200 mg of kapellasite powder, prepared following Ref. 19, with a stycast epoxy. The mixture was then placed in a cylindrical teflon mold and cured under a magnetic field of 7 T for 15 h. The level of orientation was estimated to be ∼ 90% from the comparison of intensities between the non-oriented and oriented NMR spectrum. Therefore, the oriented sample behaves like a single crystal in the c-axis direction, i.e. perpendicular to the kagomé planes. This orientation allows the observation of narrow NMR lines instead of conventional complex powder lineshape due to quadrupolar effects.

NMR experiments were performed on the most abundant Cl isotope, 35 Cl (abundance of 75.8 %), with a nuclear spin I = 3/2 and a gyromagnetic ratio γ n = 4.1716 MHz/T. There is only one Cl crystallographic site within the structure, which is coupled to three Cu 2+ ions of one triangle belonging to the kagomé plane [Fig. 1(d)]. NMR measurements were made both in fixed and sweep field configurations, under various magnetic fields from 3 to 13 T. A 4 He flow cryostat was used to reach temperatures from 1.2 to 300 K.

Nuclear Hamiltonian

35 Cl is a quadrupolar nucleus (I = 3/2), hence one has to consider two sets of interactions resulting from the coupling of the nucleus to its magnetic and charge environments. The nuclear Hamiltonian can be written [START_REF] Slichter | Principles of Magnetic Resonance[END_REF][START_REF] Cohen | [END_REF]

with

The first term of eq.(A2) corresponds to the nuclear Zeeman effect under a static applied field H 0 . The second term accounts for the magnetic interaction of the nuclear spin with its local environment through the shift tensor K. Its T -dependent part directly probes the spinsusceptibility of the surrounding magnetic moments. The quadrupolar Hamiltonian [eq.(A3)] is characterized by two parameters: ν q = 3e 2 qQ/2I(2I -1)h, which results from the coupling of the nuclear quadrupole moment Q with the electrostatic field gradient (EFG) characterized by its q value along its principal z ′ axis, and its asymmetry, η, in the (x ′ , y ′ ) plane (0 ≤ η ≤ 1).

For an "ideal" structure of kapellasite, i.e. with perfect stoichiometry and no local distortion which might result from the Zn/Cu substitution, a threefold axial symmetry exists at the Cl site. This simplifies considerably both the hyperfine tensor and the quadrupolar Hamiltonian which then have the same principal axis along z and a rotational invariance in the (x, y) plane, hence K x = K y and η = 0.

Under the applied fields of a few teslas, H quad is a small perturbation as compared to H mag for the experimental case considered in this work. In first order, the (2I + 1) energy levels E m write E m = m|H|m , where -I ≤ m ≤ I and |m is the eigenstate of the Zeeman Hamiltonian for an applied field H 0 . This energy splitting gives rise to the 2I resonance frequencies ν m-1,m = (E m-1 -E m )/h detected in NMR. Taking into account the previous symmetry arguments and expressing H 0 in spherical coordinates, H 0 = H 0 (sin θ cos φ, sin θ sin φ, cos θ) x,y,z , we obtain for the "ideal" case:

where ν 0 is the resonance frequency of the bare nucleus γ n H 0 corrected by the usual T -independent chemical shift. For 35 Cl-NMR (I = 3/2), one obtains three resonances: one with no quadrupolar shift to the first order approximation and is only governed by magnetic effects -the central line at ν -1/2,1/2 -, and two satellites ν 1/2,3/2 and ν -3/2,-1/2 . With the applied field along z axis (θ = 0 • ) and in the case of axial symmetry (η = 0), we simply derive:

Similar formula can be obtained for the applied field per-pendicular to the z axis.

Assuming as a first approximation that the susceptibility is isotropic, we can directly extract the variation of the shift when the field is applied along z direction, K z , from the variation of the susceptibility through δK z (T ) = A hf,z • δχ spin (T ) where A hf,z is the hyperfine coupling for H 0 z.

As a final note to this section, we underline that any local structural distortion will introduce deviations from these simple formula (η = 0 and different principal axis for the EFG and shift tensor not aligned along z).