C Attaccalite

M Bockstedte

A Marini

A Rubio

L Wirtz

Coupling of excitons and defect states in boron-nitride nanostructures

Keywords: numbers: 71, 35, -y 71, 55, -I

The signature of defects in the optical spectra of hexagonal boron nitride (BN) is investigated using many body perturbation theory. A single BN-sheet serves as a model for different layered BNnanostructures and crystals. In the sheet we embed prototypical defects such as a substitutional impurity, isolated Boron and Nitrogen vacancies, and the di-vacancy. Transitions between the deep defect levels and extended states produce characteristic excitation bands that should be responsible for the emission band around 4 eV, observed in luminescence experiments. In addition, defect bound excitons occur that are consistently treated in our ab initio approach along with the "free" exciton. For defects in strong concentration, the co-existence of both bound and free excitons adds sub-structure to the main exciton peak and provides an explanation for the corresponding feature in cathodo and photo-luminescence spectra.

I. INTRODUCTION

Boron nitride crystals and nano structures are excellent wide band gap materials for the realization of UVlasers. [START_REF] Taniguchi | Production of a hexagonal boron nitride crystal body capable of emitting out ultraviolet radiation[END_REF] Defects formed during the BN-synthesis or introduced as impurities can bind excitons and act as centers for their recombination. This leads to (desired or undesired) luminescence bands and losses in the photon frequency range of interest. In fact, recent experiments [START_REF] Watanabe | [END_REF][3][4][5][6][7][8][9] have given evidence that defects strongly effect the luminescence of BN-based materials. In order to achieve optimal optical properties, the relevant defects have to be identified such that methods can be devised to tune their concentration in the material.

Boron nitride (BN) is known to exist in three different crystals structures, namely hexagonal BN (h-BN), cubic BN (c-BN), and wurtzite BN (w-BN). Among these structures, h-BN is the stable one at room temperature and ambient pressure. Like graphite, it consists of stacked BN-layers with a honeycomb structure. BN nanostructures 10,[START_REF] Wirtz | Optical and vibrational properties of boron nitride nanotubes in B-C-N Nanotubes and Related Nanostructures[END_REF] ranging from single and multi-wall nanotubes to BN-fullerens and nanocones can be envisaged as having been tailored from a single BN sheet. Unlike in graphite, the partially ionic character of the BN bond results in a wide band-gap of about 6.5 eV for bulk h-BN 12 . The combination of a large gap with a strong electron-hole attraction makes the optical properties of BN nanostructures largely independent of the layer arrangement. [START_REF] Wirtz | Optical and vibrational properties of boron nitride nanotubes in B-C-N Nanotubes and Related Nanostructures[END_REF]13 A single layer of h-BN is thus a generic system for studying the optical properties of crystalline and nanostructured BN including the exciton-defect interaction.

Detailed experiments on the cathodo and photo-luminescence of h-BN crystals and BN nanotubes have been performed in the recent years [START_REF] Watanabe | [END_REF][3][4][5][6][7][8][9]14 . The origin of some of the features in the BN luminescence spectra remains unsettled. For instance, the occurrence of a defect-related emission band at 4 eV, composed of regularly spaced peaks compatible with phonon replicas 5,7,15 , can be associated with a deep level impurity. However, the defect that is responsible for this deep level has not yet been identified. Furthermore, the emission band at 5.77 eV shows different sub-peaks on the low energy side.

The origin of its sub-structure is unclear, since the main peak can be understood as being due to the recombination of a Frenkel-type exciton 12 with a degenerate recombination line. 13,16 Interpretations of the experimental findings are based on different mechanisms: quasidonor-acceptor pairs 6 , excitons coupled with structural defects 9,15 , breaking of the exact hexagonal symmetry (e.g., by the presence of defects) 16 , or a dynamical Jahn-Teller effect 17 (symmetry breaking due to geometry relaxation in the excited state).

Theoretical works on defects in BN-structures have so far focused on the electronic and structural properties of point defects [18][19][20][21][22][23][24] . The abundance (related to the formation energy) and stability of different defects has been addressed with detailed density-functional theory (DFT) calculations. 20,[25][26][27][28][29] . The position of deep levels within the fundamental band gap has been calculated on the DFT level. However, DFT calculation suffer from the well-known underestimation of the band-gap. Therefore, besides the neglect of the strong excitonic effects in BN, (de)excitation spectra based on DFT energy levels may not be reliable enough for comparison with experimental spectra.

In the present paper we address the optical properties of point defects in BN going beyond the approximation of DFT. We focus on a substitutional carbon impurity on the nitrogen site, the isolated vacancies and the BN divacancy that are common defects in BN. We will demonstrate that defects are responsible for the emission lines around 4 eV. Their presence can also explain the experimentally observed splitting of the main exciton peak. The paper is organized as follows: in section II we summarize the computation methods employed in this work; in section III we present the ground state results and the quasi-particle(QP) band structures; in section IV we investigate the role of defects on the optical properties and finally in section V we compare our results with experiments.

II. COMPUTATIONAL METHODS

Density functional theory (DFT) within the local spin density (LSDA) or gradient corrected approximations (GGA) for the exchange-correlation energy has been widely used to study defects in semiconductors and insulators. Although DFT is in principle exact, it is limited to the ground state and as a closed expression for the exchange-correlation energy functional is unknown, approximations like the LSDA or GGA have to be employed. Albeit theoretically well-founded, these approximate functionals bear inherent limitations like the wellknown band gap problem which concomitantly affect the prediction of defect levels, in particular in wide band gap insulators. In the last years, many-body perturbation theory within the GW approximation 30 has become a valid and accurate alternative to predict defects properties in semiconductors and insulators [31][32][33] starting from the DFT ground state. Moreover, in going beyond the ground state, the optical properties of defects and the host insulator can be addressed by including the electron-hole interaction in the excited state via the Bethe-Salpeter equation (BSE) [34][35][36] .

Here we use this approach to study the role of prototypical defects in the optical properties of BN nanostructures: a substitional carbon impurity on the nitrogen sublattice (C N ), the boron vacancy (V B ), the nitrogen vacancy (V N ) and the boron-nitrogen di-vacancy (V BN ). Calculations proceed in three steps: first we obtain the DFT ground state employing the LSDA approximation and a plane wave pseudopotential method, 37 (see 38 for details). Second we evaluate the quasi-particle band structure within the G 0 W 0 [START_REF] Aulbur | Solid State Physics[END_REF]39,40 approach and third we address the optical spectra via the BSE. 41 A very large supercell [START_REF]was 20 a.u. A single k-point was used. We used the exchange-correlation functional of Ref. 49[END_REF] is employed to accommodate the defect and to enable at the same time the description of defect bound and free excitons. This becomes possible due to the strong localization of the bulk exciton state in h-BN, 12 in the layer, and tubes 13,42 .

As it has been shown that a large class of impurities, when introduced into h-BN, is in a spin-polarized state and acquires a local magnetic moment 24 , we have performed spin-polarized calculations for the C N , V B , and V N cases while this was not necessary for the di-vacancy V BN that does not display a net spin polarization. The quasi-particle properties have been calculated starting from the Kohn-Sham Hamiltonian and orbitals within the G 0 W 0 approximation 43 :

E nk = ǫ nk + Z nk Re∆Σ nk (ǫ nk ), (1)
where ∆Σ = Σ -V xc and V xc is the LSDA exchangecorrelation functional, ǫ nk are the Kohn-Sham eigenvalues, and Z nk = [1-∂Re∆Σ/∂w] -1 is the renormalization factor [START_REF] Aulbur | Solid State Physics[END_REF] . The screened electron-electron interaction W has been evaluated within the random-phase approximation (RPA) in terms of the dielectric function ǫ G,G ′ (q, ω) using a plasmon-pole model. 44 Neutral excitations are investigated on the basis of the quasiparticle spectrum by solving the Bethe-Salpeter equation (see 45 for details). Excitonic effects are analysed through a comparison of the absorption spectra for independent particles with absorption spectra where electron-hole attraction is included on the level of the BSE.

As we are studying an isolated monolayer employing three-dimensional periodic boundary conditions, the size of the quasi particle energies and the energy of the electron-hole attraction will depend on the inter-sheet distance. In Ref. 13 we had shown that the value of the gap of the pure h-BN sheet is 0.6 eV higher for an intersheet distance is 80 a.u. than for an inter-sheet distance of 20 a.u. (value which we are using in the calculations presented here). At the same time we had observed an increase of the exciton binding energy with increasing inter-sheet distance that almost exactly cancels the variation of the band-gap. We expect a similar cancellation effect in the presence of defects but cannot exclude a minor effect of the inter-sheet distance on the absolute position of defect peaks.

III. DFT-GROUND STATE AND QUASI-PARTICLE STATES

In this section we address the electronic and geometric structure of the four defects and then compare the DFT-electronic structure with the quasiparticle states obtained from G 0 W 0 calculations.

In Fig. 1 the equilibrium geometry of the C N and the vacancies V N ,V B and V BN is shown along with the BN bond distances in the vicinity of the defect center. In the case of subsitutional impurity all the first neighbor B atoms of C N undergo a small symmetry conserving outward relaxation of 0.113 a.u. (4% of the BN-bond distance) with respect to the ideal lattice positions. The defect retains its threefold symmetry. In the case of V B and V N the the outward displacement of the first-neighbor atoms is larger. More importantly, in agreement with Ref. 24, we find that a (pseudo) Jahn-Teller effect breaks the three-fold symmetry slightly. In the di-vacancy case, the three-fold symmetry is already reduced to a mirror symmetry by the defect configuration itself. We observed that the first N and B neighbors decrease their mutual distance upon relaxation, as shown in Fig. 1d. The defect geometries found here are in good agreement with those reported in Ref. 20 and 18.

V BN B B B N B B N N N N N N N C 2 .
As established already in earlier work, the C N and the vacancies possess localized defect levels in the band gap. Here we briefly review the electronic structure of the defects as obtained in DFT-LSDA and then compare it with the quasiparticle states obtained on the level of the G 0 W 0 approximation where the DFT band gap-error is corrected. Since the prediction of defect emission and absorption spectra hinges on the defect level positions, a correct prediction of these quantities is a crucial point for the investigation of optical properties. The defect levels as obtained within the DFT-L(S)DA are reported in the left panel of Fig. 2.

In the case of C N the defect levels arise from the three bonds of the substitutional carbon impurity with its boron neighbors. These bonds combine into two fully occupied states located within the valence band and a third half-occupied localized state within the band gap. As a result of the occupation with only one electron there is a spin-splitting of ∼ 0.5 eV between the occupied (spin-up) and unoccupied (spin-down) state.

The situation is more complex for the boron vacancy. In this case a two-fold degenerate and a non-degenerate defect state are formed from the unpaired orbitals at the three nitrogen neighbors. In agreement with Ref. 20, 26, and 29, these defect levels are located within the band gap. In the neutral vacancy the two-fold degenerate level is occupied by three electrons and the non-degenerate level remains empty. Due to the Jahn-Teller distortion a small splitting of the degenerate level arises in addition to the exchange-splitting of the half-and unoccupied states. V B behaves as a triple acceptor.

For V N we obtain a half-filled level in the band gap, and two un-occupied levels almost degenerate in energy with the conduction bands. Due to spin-splitting their counter parts of opposite spin are found as resonances in the conduction band. Finally, in the case of the di-vacancy V BN , there are four orbitals originating from the two boron and nitrogen neighbors. Only three are located in the band gap, one being occupied by two electrons and the upper two remaining empty.

The above picture obtained from the Kohn-Sham eigenvalues gives a qualitative description of the defect levels in the band gap. The position of the states is, however, quantitatively affected by the LSDA-band gap error and the approximate treatment of exchange with this XC-functional as discussed above. Furthermore, the ordering of occupied and unoccupied states on the Kohn-Sham level may be different from that of the "true" quasiparticle energy levels (see e.g. 30). The G 0 W 0 method yields true quasiparticle energies and an improved description of exchange and screening effects. Due to the reduced dimensionality of the BN mono-layer there is an incomplete screening of the Coulomb interaction, such that the quasi-particle corrections are expected to be quite large. We calculated the quasi-particle band structure for all the cases considered above. The calculated band gap is in agreement with the results of Ref. 13. In the right panel of Fig. 2 the position of the defect quasi particle energies within the band gap is shown schematically. The G 0 W 0 corrections to the LSDA levels depend on the orbital occupation and on the wave-function character [START_REF] Aulbur | Solid State Physics[END_REF] . The exchange splitting between occupied and unoccupied defect levels is strongly enhanced and the ordering of defect levels within the band gap is altered in some cases. In all cases we find that occupied levels are pushed closer to the valence bands and unoccupied ones closer to conduction bands. For C N the spin-splitting is significantly increased from 0.5 eV to 3.1 eV. A large exchange splitting is also present in the case of V N . For V B all un-occupied levels are pushed up by more then 2 eV, depending on the character of the wave-function, and the level order is modified as well. In the case of V BN , un-occupied levels are found close to the conduction band edge, while the occupied ones remain in their position close to the valence band edge. This underlines the fact that the di-vacancy contains both donor and acceptor levels. The general picture obtained from G 0 W 0 shows the clear deficiencies of the DFT-LSDA in the de- scription of defect level positions in the case of BN, in particular, for unoccupied states. Vested with the defect quasiparticle spectrum we now turn to the optical properties.

IV. OPTICAL PROPERTIES

In order to discern excitonic effects on the optical properties, we present first in Fig. 3 the optical spectra in the independent particle picture (IP) 46 (i.e. without the inclusion of the electron-hole coupling). This yields an overview over relevant excitation mechanisms and later allows one to trace down the effect of the electron-hole coupling.

We start the discussion with the case of C N . Comparing the corresponding panel of figure 3 with the schematic representation of the defect levels in the band gap (Fig. 2) one can identify the different absorption peaks. The first band at 3.4 eV is due to transitions form the extended states at the valence edge to the defect levels within the band gap; the other band at 4.4 eV involves extended states below the valence band edge and defect levels. Above 6 eV we find the bulk-like spectrum that is only slightly perturbed by the presence of the defect. For the boron vacancy V B there are two bands at 3.5 eV and 4.7 eV that are associated with transitions from defect resonances in the valence band to the localized levels in the band gap and a series of small peaks around 5.9 eV originating from valence band to defect transitions. The defect-related excitation bands of V N at 4.1 eV, in contrast to the previous cases, involves extended states near the conduction band edge as final states. Only the faint defect-related excitation band around 6.6 eV, not visible in the figure, originates from valance band states. Another faint peak around 5 eV is due to transitions among the defect levels in the band gap. Finally, for the V BN di-vacancy, we find peaks between 5.6 eV and 6.0 eV due to defect-defect transitions and valence-band to defect transitions, respectively.

We now turn to the spectra 46 including the electronhole interaction via the Bethe-Salpeter equation. These are shown in Fig. 5. The inclusion of the electron-hole interaction has essentially two effects on the defect-related spectra: (i) a general red-shift of the defect excitation energies due to electron-hole attraction and (ii) a coupling of the independent particle excitations. This coupling can give rise to bound electron-hole states, the excitons. In our case the BSE leads to a red-shift of defect-defect transitions in the range from 0.6 to 1.3 eV depending on the character and the localization of the wavefunction. A common feature of all spectra is the large peak around 5.5 eV which corresponds to the strongly localized exciton of pure h-BN 12,13,42 which is composed of a large number of independent electron-hole pairs. In the presence of a defect, electron-hole coupling leads to defect bound excitons besides the "free" excitons.

FIG. 3. Independent-particle absorption spectra (arbitrary units) based on the G0W0 quasi-particle band structure. In the left panel, the absorption cross section is increased by a factor 15 with respect to the right panel in order to make the defect-related peaks visible. .

In the case of C N and V B , as one can see from Fig. 5, an additional small peak appears just below the main exci-tons. In the case of V BN the main exciton band has three sub-peaks due to the strong defect field with reduced symmetry. In order to disentangle the nature of the different peaks, we analyze the amplitude of the electronhole pairs that compose each of the exciton eigenstates λ, as a function of the energy difference of the quasiparticle states of the corresponding electron and hole states:

A λ (ω) = η={eh} | η|λ | 2 δ(ω -E η ), (2)
For brevity, we report A λ (ω) only for V B and V BN (see Fig. 4). The plots for the other defects show generally a similar behavior. Comparing for V B the electron-hole amplitudes of the four excitation energies with the schematic diagram of Fig. 2, we can distinguish three kinds of excitations: (i) transitions from valence bands or defect resonances to the defect levels, that generate the peaks at 4.2 eV, (ii) the main exciton at 5.6 eV that is entirely composed of extended states, and (iii) a peak at 5.4 eV generated by the coupling of bulk excitations with transitions from and to defect levels in the band gap. The C N and V N cases present a structure similar to the one discussed above. From these results it is possible to infer three important consequences: (i) transition from defect levels to valence/conduction bands with an energy close to the band gap energy mixes with the bulk excitation giving rise to an additional peak close to the main exciton; (ii) transition between defect levels are renormalized by the electron-hole interaction but do not mix with the bulk excitations; (iii) the main exciton peak maintains a structure similar to the bulk system.

The excitonic band of the di-vacancy V BN is distinct from the other spectra as the di-vacancy possess both acceptor and donor states of low symmetry in the vicinity of the band edges. The direct transitions between acceptor and donor states give rise to the absorption peaks around 4.8 eV, while the resonant coupling of transitions between the extended conduction and valence band states with donor and acceptor states splits the main exciton in three sub-peaks respectively at 5.5 eV, 5.43 eV and 5.37 eV (see Fig. 4).

The observed side peaks to the main excitonic peak are compatible with the fine structure observed in different luminescence experiments 5,6,15,17 .

V. CONCLUSION

We have investigated the role of defects on the optical properties of BN nanostructures. We have treated four different plausible defects in a single sheet of hexagonal BN: a subsitutional carbon impurity, a boron vacancy, a nitrogen vacancy, and a BN di-vacancy. Quasiparticle effects strongly modify the defect-level positions within the band-gap and increase the exchange splitting between occupied and unoccupied levels. Electron-hole interaction (treated on the level of the the Bethe-Salpeter equation) not only leads to a strongly bound excitonic state in pure BN but also strongly renormalizes transitions from and to defect levels. This means that the peaks in the optical spectra which are related to "deep level impurities"are strongly affected by the e-e and eh interaction and can be reliably calculated only on the level of many-body perturbation theory (as opposed to the frequently used random-phase approximation using DFT wave-functions and energies). Experimental luminescence spectra (as presented, e.g., in Ref. 15) display a prominent peak close to 4 eV with phonon replica on the low energy side. Comparing the four theoretical absorption spectra (Fig. 5) with the experimental luminescence spectrum, the best agreement is obtained for the Boron vacancy where the main absorption peak of the low energy regime is at 4.2 eV. The Stokes shift due to possible relaxation of the excited state is not included in our calculation and limits the comparison of experimental luminescence with theoretical absorption spectra. In the case of a strong (0.7 eV) Stokes shift, also the BNdivacancy (where a defect related absorption peak is at 4.8 eV) could be responsible for the experimental observation. The nitrogen vacancy (absorption peak at 4.5 eV) might also explain the experiment. However, in this case a second strong luminescence peak should also be observable below 3 eV which seems not to be the case in experiments.

The 10×10 supercell employed in our calculations is considerably larger than the extension of the wavefunction of the "free" exciton in pure h-BN. Therefore, besides the defect-related absorption peaks, our calculations also show the prominent absorption peak due to "free" excitions at 5.6 eV. This peak is doubly degenerate in the pure BN sheet. Coupling of the free exciton with one or several transitions from or to defect levels gives rise to a splitting of this peak and additional side peaks. The details of the splitting depend on the nature of the impurity but also on its periodic arrangement 47 . The fine-structure 2 of the luminescence peak at 5.7 eV (with additional side peaks at 5.64 eV and 5.46 eV) can possibly be explained through our calculations which clearly display defect-related side peaks of the "free" excitonic peak. Furthermore, any kind of breaking of the exact hexagonal symmetry can lead to a splitting of the bright doubly degenerate "free" excitonic peak into two peaks and will also (for bulk h-BN) render the "dark" exciton (located 0.1 eV below the bright exciton) visible 16 . For reasons of computational feasibility, we have performed calculations only on point defects. Other kinds of defects such as dislocations and grain boundaries lead similarly to a breaking of symmetry and/or to the formation of localized states and could thus as well explain the fine structure of luminescence spectra one would expect a similar interaction of the "free" exciton with danglingbond states). Indeed, Jaffrenou et al. had observed that the luminescence peaks at 5.64 eV and 5.46 eV occur mostly at grain boundaries and dislocations 15 where defect states due to dangling bonds are supposed to occur. The aim of our present study is thus not to unambiguously assign a certain type of defect to the ob-served luminescence spectra, but rather to show the general mechanism leading to the deep impurity levels and the formation of defect bound excitons. We also note that we did not study all possible point defects. A recent study of the chemical composition of an isolated h-BN sheet suggests the incorporation of oxygen atoms in the honeycomb lattice 48 . The same study also demonstrated that substitutional carbon atoms occur mostly in pairs, occupying neighboring nitrogen and boron sites. Further experimental and theoretical studies are necessary to decide if the latter two types of defects can also occur during the synthesis of BN nanostructures or if they are due to the experimental conditions in the transmission electron microscope where vacancies in the h-BN sheet are induced by the electron bombardment and may be subsequently filled by a substitutional atom. An alternative route to symmetry breaking was proposed by Watanabe et al. 17 . They suggested that the dynamical Jahn-Teller effect (splitting of the degenerate exciton by polaronic effects) could explain the excitonic splitting in hexagonal BN. Further experimental and theoretical studies are necessary to disentangle the two ef- starting from DFT-LDA wave-functions, using a plasmon pole approximation for the dielectric constant 44 , with the code YAMBO 51 . We use a single k-point, 3000 bands for expanding the Green's function, 40.000 G-vectors for the exchange and 35.000 for the correlation part of the selfenergy. In contrast to Ref. 13 we did not add any scissor operator to correct the underestimation of the band gap in the G0W0 approximation. 44 The same dielectric constant has been employed in all the cases with and without defects, both for quasi-particle and Bethe-Salpeter equation. The dielectric constant has been calculated in the pure system, using 3000 bands, 3833 Gvectors for the dielectric matrix, and 35.000 G-vectors for the wave-functions entering in the calculation. 45 The optical spectra have been calculated using the Bethe-Salpeter equation (BSE) 41 , with bands 300 to 500 (all π and π * bands of the 10×10 supercell), a 10 eV cutoff on the electron-hole pair energy entering in the BSE and 20.000 G-vectors for the electronic wave-functions. 46 The independent-particle (IP) spectra are obtained for the 1×1 k-point set (corresponding to a 10×10 sampling of the elementary unit cell) and are not fully converged with respect to k-point sampling for transitions involving extended band states, despite of the large supercells em-ployed in the present calculations. Since we are only concerned with transitions involving localized defects levels, this is irrelevant for the analysis of excitation mechanism conducted here. Moreover we verified that peaks due to defects transition do not change increasing the number of k-points. In the BSE case, the main exciton collects most of the oscillator strengths. This speeds up the k-point convergence of the spectra and makes the presented spectra almost converged, contrary to the IP ones. In both cases IP and BSE we always considered the polarization parallel to the sheet. 47 In our simulation, we use a 60 degree unit cell which corresponds to a trigonal ordering of the defects. If a 90 degree super-cell was chosen, the ordering of the defects would be rectangular. Only in the limit of very large unit-cells, the splitting of the excitonic peak will be independent of the defect ordering and correspond to the coupling of a "free excitons" with an isolated defect. However, with the present computational ressources, we cannot reach the limit of isolated impurities by increasing the super-cell.

FIG. 1 .

 1 FIG. 1. (color online) Geometry of the defects in BN mono-layer with defects: (a) CN ; (b) VB; (c) VN ; (d) VBN . Defect induced changes of BN bond distances are indicated (in atomic units) and have to be compared with the BN distance of the perfect sheet (2.724a.u.).

FIG. 2 .

 2 FIG. 2. (color online) Quasiparticle energies of CN, VB, VN, and VBN as obtained from LDA and G0W0 calculations: schematic representation of energy position with in the band gap. The filled arrows indicate levels occupied with electrons with spin up or down. Dotted arrows indicate unoccupied states (holes) correspondingly.

FIG. 4 .

 4 FIG.4. Amplitude of the different electron-hole pairing entering in the BSE, for the boron vacancy and the BN divacancy, compared with the pure h-BN monolayer.

FIG. 5 .

 5 FIG. 5. (color online)Optical absorption spectra calculated by the Bethe-Salpeter equation . In the left panels the absorption cross section is increased by a factor of ten in order to make the defect-related peaks visible. Spectra are shown with a broadening of 0.1 eV (to simulate a typical experimental broadening) and 0.005 eV (in order to investigate the fine-structure of the spectrum).

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ACKNOWLEDGMENTS

We acknowledge funding by the European Community through e-I3 ETSF project (Contract Number 211956). AR acknowledges funding by the Spanish MEC (FIS2007-65702-C02-01), ACI-promciona project (ACI2009-1036), "Grupos Consolidados UPV/EHU del Gobierno Vasco" (IT-319-07). LW acknowledges funding by the French National Research Agency through project ANR-09-BLAN-0421-01. This work was performed using HPC resources from GENCI-IDRID (No. 100063 and No. 091827). CA thanks X. Andrade and J. Alberdi Rodriguez for the efficient computer-cluster installation in San Sebastian. This work has been supported by the project no. FIS2010-21282-C02-01 (MCINN) from Spain.


T Vidal

S Saint-Jean

P Lusley

M Leconte

S Ben Krima

A.-L Boixel

Wheatamix Consortium

C De Vallavieille-Pope

Main effects of disease resistance and yield potential of component cultivars in diverse mixtures are little affected by diversity in plant height and flowering date

Keywords: Cultivar mixture, wheat diseases, Puccinia striiformis f.sp. tritici, yield, component number, intrafield diversity

Cultivar mixtures can stabilize yield and reduce pathogen spread in plant populations. A field experiment was performed to determine whether (i) a large difference between the cultivars in the mixture (e.g. plant height or earliness) would have an impact on mixture performance and whether (ii) such differences would modify the classical rules for mixture design. Mixtures were constituted from cultivars with diversity for many traits, including plant height, flowering date, disease resistance and yield potential. The field experiment was conducted in three years testing each year 72 to 90 mixtures of two, four or eight cultivars, and their corresponding pure stands. Disease severity and yield of cultivar mixtures were strongly related to the mean values of the component cultivars in pure stands. Despite the considerable diversity of the mixtures tested, the classic rules (e.g. proportion of susceptible cultivars) already tested in mixtures with similar height and earliness were effective for decreasing disease severity. Agronomic heterogeneity for traits such as plant height, yield potential or earliness of the cultivars in mixtures did not have a negative impact on disease severity and yield relative to pure stands. Increasing the number of cultivars in the mixture from two to eight had no impact on the mean disease severity and yield of the mixtures, but reduced the variability of disease severity and yield in the mixture relative to pure stands. These results suggest that it may be possible to increase within-field wheat diversity by combining more contrasted cultivars in mixtures than was previously thought.

Introduction

Cultivar mixtures have long been used to manage plant diseases, particularly in cereals (Finckh et al 2000;[START_REF] Borg | Unfolding the potential of wheat cultivar mixtures: A meta-analysis perspective and identification of knowledge gaps[END_REF], with some major successes. Mixtures have sometimes been chosen as a default solution in the past, when other disease management tools were ineffective, unavailable or not desirable (e.g. no fungicide or resistance genes available). Long-term success was limited in such cases. For example, barley

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This article is protected by copyright. All rights reserved mixtures were abandoned in former East Germany (RDA) after German reunification, because West European maltsters preferred to buy grain from single-cultivar crops, even though this required fungicide use to manage barley powdery mildew (Wolfe, 1992). In Denmark, mixture use declined because the mlo gene conferring resistance to barley powdery mildew was durably effective (Jørgensen, 1992). However, agriculture is currently facing new challenges that have revived interest in the use of cultivar mixtures [START_REF] Barot | Designing mixtures of cultivars for multifunctional agriculture with the help of ecology. A review[END_REF]. In addition to the continuous need to increase food production, there is now a demand to decrease agricultural inputs in general and fungicides in particular, as illustrated by the Ecophyto plans in France (https://agriculture.gouv.fr/ecophyto). Agricultural practices favoring diversity within cropping systems are becoming an attractive option for enhancing natural interactions and increasing agroecosystem resilience, particularly in a context of lower inputs and unpredictable climate variability.

Cultivar mixtures have been shown to have effects on diseases, such as barley powdery mildew (caused by Blumeria graminis f.sp. hordei), wheat yellow rust (caused by Puccinia striiformis f.sp. tritici, Wolfe, 1985;[START_REF] Mundt | Use of multiline cultivars and cultivar mixtures for disease management[END_REF] and Septoria tritici blotch (caused by Zymoseptoria tritici: [START_REF] Cowger | Effects of wheat cultivar mixtures on epidemic progression of Septoria tritici blotch and pathogenicity of Mycosphaerella graminicola[END_REF]Gigot et al., 2012;[START_REF] Vidal | Reduction of fungal disease spread in cultivar mixtures: Impact of canopy architecture on rain-splash dispersal and on crop[END_REF]. This cropping practice has also been shown to increase yield stability [START_REF] Kiaer | Grain yield increase in cereal variety mixtures: A meta-analysis of field trials[END_REF][START_REF] Creissen | Impact of disease on diversity and productivity of plant populations[END_REF]. Cultivar mixtures have the potential to reduce disease progression, but results can be highly variable and an appropriate design is crucial to reach the expected disease level and yield performances. Extensive studies of a range of design criteria for mixtures have been performed, as reviewed by [START_REF] Newton | Plant disease control through the use of variety mixtures[END_REF] and [START_REF] Barot | Designing mixtures of cultivars for multifunctional agriculture with the help of ecology. A review[END_REF].

A first wave of studies highlighted the need for diversity within mixtures. For example, it was shown that the effect of the mixture can be enhanced by growing more than two cultivars together [START_REF] Mille | Assessing four-way mixtures of winter wheat cultivars from the performances of their two-way and individual components[END_REF][START_REF] Newton | Analysing the contribution of component cultivars and cultivar combinations to malting quality, yield and disease in complex mixtures[END_REF]. In situations in which disease control is the main objective of mixture use, contrasting levels of disease resistance are often considered a key criterion (Gigot et al., 2014). Functional diversity can be modulated by choosing cultivars with complementary characteristics (e.g. one susceptible cultivar mixed

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This article is protected by copyright. All rights reserved with three resistant cultivars) or by choosing different proportions of the various cultivars in the mixture (e.g. 25% of a susceptible cultivar and 75% of a resistant cultivar). In general, higher proportions of resistant cultivars are expected to provide higher levels of disease protection [START_REF] Cox | Cultivar mixtures for the simultaneous management of multiple diseases: tan spot and leaf rust of wheat[END_REF][START_REF] Dai | Performance of hard red spring wheat cultivar mixtures[END_REF]. The susceptible cultivar can be replaced by a moderately susceptible or resistant cultivar carrying resistance QTLs. This makes it possible to reduce the proportion of the highly resistant cultivar required for effective protection in the mixture [START_REF] Sapoukhina | Quantitative plant resistance in cultivar mixtures: wheat yellow rust as a modeling case study[END_REF].

However, diversity in cultivar mixtures is limited by practical constraints, and agronomic homogeneity is often recommended for criteria such as plant height and flowering date. For example, in Denmark, barley mixtures are allowed only if they satisfy the following criteria: (i) less than five days difference in ripening date, (ii) less than 15 cm difference in height, (iii) the yield of each cultivar must be at least 95% the yield of the mixture and (iv) the mean disease susceptibility of component cultivars must be below a threshold established by experts [START_REF] Labarthe | Diffusion des mélanges variétaux pour la production de blé : une comparaison entre France et Danemark[END_REF]. Reviews on mixtures classically recommend limiting the agronomic differences between cultivars (Wolfe, 1985). Similarly, wheat multiline development has aimed to maximize agronomic homogeneity while providing diverse mechanisms of resistance to yellow and brown rusts [START_REF] Mundt | Use of multiline cultivars and cultivar mixtures for disease management[END_REF].

Ecological principles contrast with agronomic concerns, in that they suggest that the stability of an ecosystem increases with its diversity [START_REF] Barot | Designing mixtures of cultivars for multifunctional agriculture with the help of ecology. A review[END_REF]. Indeed, increasing diversity levels could enhance positive interactions, ecological services and yield [START_REF] Reiss | Cultivar mixtures: a meta-analysis of the effect of intraspecific diversity on crop yield[END_REF]. In wheat, plant height and flowering date affect disease progression by determining the likelihood of a plant escaping the disease by spatial or temporal displacement. For example, growing taller plants can affect vertically spreading diseases such as Septoria leaf blotch. Plants with different flowering times can reduce the effect of occurrence of rain in the case of epidemics caused by Fusarium spp.. Finally, enhanced disease control has been observed in

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This article is protected by copyright. All rights reserved experiments involving mixtures with contrasting plant heights and/or flowering dates (Zhu et al., 2005;[START_REF] Li | The effect of wheat mixtures on the powdery mildew disease and some yield components[END_REF][START_REF] Vidal | Reduction of fungal disease spread in cultivar mixtures: Impact of canopy architecture on rain-splash dispersal and on crop[END_REF].

Despite the relevance of these results, too few experimental data are currently available to draw firm conclusions on the balance between the positive (possible enhancement of disease control) and negative (agronomic management of heterogeneous mixtures) impacts of such phenotypic contrasts. The objective of this field experimentation was to assess the possible impact of increasing diversity on disease severity and yield in cultivar mixtures.

Materials and methods

We designed a range of field experiments for assessing the impact of cultivar diversity within mixtures on yield and disease severity, over three cropping seasons. Full details can be found elsewhere [START_REF] Dubs | An experimental design to test the effect of wheat variety mixtures on biodiversity and ecosystem services[END_REF]. Two experiments were performed in 2014. We first phenotyped 58 cultivars, for plant height, flowering date and disease, in particular (E1-2014). In parallel, 20 cultivars were chosen to constitute 90 cultivar mixtures, for which yield and disease severity were recorded (E2-2014). Finally, in 2015, 16 cultivars were chosen based on the traits measured in E1 to constitute 72 cultivar mixtures (E3-2015). This E3-2015 experiment was repeated in 2016 (E3-2016).

Field experiments

The experiments were conducted over three consecutive wheat-growing seasons: 2013-2014 (hereafter called 2014), 2014-2015 (hereafter called 2015) and 2015-2016 (hereafter called 2016) .Fields were located at the INRAE Versailles Research Station in France (20 km west of Paris, 48°48'26"N 2°05'13"E, elevation 114 m). The specific features of each experiment are listed in Table 1.

Table 1 around here

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This article is protected by copyright. All rights reserved In the phenotyping experiment (E1-2014), each cultivar was grown in a pure stand on a single plot. The three experiments including mixtures (E2-2014, E3-2015 and E3-2016) followed a two-block randomized design with two replicates for pure stands (one in each block or subfield, Figure S7) and one replicate for each cultivar mixture (randomly distributed in one of the subfields). Plot size depended on the experiment (Table 1).

The cultivar mixtures were sown with equal proportions of each cultivar (in terms of seed number), at the same density as pure stands. Cross-contamination between plots was limited by sowing triticale borders (×Triticosecale Wittm. ex A. Camus) of at least 3 m in width (250 seeds/m²). In all experiments, wheat crops were sown at a density of 180 seeds per m². We chose to use a sowing rate lower than that generally used in agronomic practice, to limit the airborne progression of disease. The microplots were sown after a previous crop of maize, rather late in the season, to limit autumn disease contamination (November 26 th 2013, November 5 th 2014 and October 21 st 2015, respectively). The field trial was managed as with an integrated crop protection system, with relatively low input levels and a target yield of 6 t/ha in E2-2014 and 7 t/ha in E3-2015 and E3-2016 (slightly lower than the mean national wheat yield of 7.5 t/ha in France in 2015). We ensured that nitrogen was not a limiting factor for plant growth, by determining the appropriate total dose of ammonium nitrate fertilizer to be applied by the balance-sheet method (as in Gigot et al., 2012 and[START_REF] Vidal | Reduction of fungal disease spread in cultivar mixtures: Impact of canopy architecture on rain-splash dispersal and on crop[END_REF]. The total dose of nitrogen was 120 kgN/ha in 2014, 160 kgN/ha in 2015 and 170 kgN/ha in 2016.

This dose was applied in two installments in 2014: 70% at tillering and 30% when the second node became detectable. In 2015 and 2016, there were three installments: one quarter of the total dose at tillering, half at the beginning of stem elongation and one quarter when the second node became detectable. No fungicides or growth regulators were applied. One herbicide treatment was applied each year, in early spring, at GS31 (50 g/ha Harmony extra®, 250 g/ha Archipel® and 1 l/ha adjuvant Actirob 842 g/l esterified rapeseed oil base).

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The cultivars included in the mixtures (E2 and E3) were chosen from a panel of 58 cultivars of four types:

(i) elite cultivars (ELT: modern commercial cultivars registered for use in conventional agriculture), (ii) organic cultivars (ORG: modern commercial cultivars registered for use in organic farming), (iii) MAGIC lines (MGC: inbred lines developed from multiparental and highly recombinant INRA populations, each line was hereafter considered as a cultivar) and (iv) landraces (LDR: old traditional cultivars that evolved over decades and adapted locally under the unconscious selection of farmers, commonly cultivated until 1930).

The whole panel of cultivars was screened for morphological, phenological and physiological traits (see summary of diversity assessment for root, architecture, yield and disease resistance traits in [START_REF] Dubs | An experimental design to test the effect of wheat variety mixtures on biodiversity and ecosystem services[END_REF]. Groups of cultivars had different numbers of cultivars and more or less heterogeneous characteristics, so that individual cultivars and their specific properties were considered in analyses rather than groups of cultivars and their mean properties. Disease resistance was assessed in 2014 (E1-2014). Plant height and flowering date were measured in a separate experiment in the Paris region (see [START_REF] Dubs | An experimental design to test the effect of wheat variety mixtures on biodiversity and ecosystem services[END_REF] for details).

Cultivar characteristics are summarized in Table S1. In E2-2014, cultivars were chosen so as to ensure an equal representation of the four types of cultivar (ELT, ORG, MGC, LDR), whereas, in E3-2015/2016, the choice was based on functional groups established on the basis of the phenotyping performed in 2014 [START_REF] Dubs | An experimental design to test the effect of wheat variety mixtures on biodiversity and ecosystem services[END_REF]. This second design criterion resulted in the inclusion of a smaller number of MAGIC lines in mixtures, with six such lines included in E2-2014 but only two namely a highly resistant and a highly susceptible line in E3-2015/2016 (Table 1).

Many combinations of the considered cultivars were possible. For example, with 16 pure lines (E3-2015/2016), there were 120 possible two-cultivar mixtures (2CMs), 1820 possible four-cultivar mixtures (4CMs) and 12870 possible eight-cultivar mixtures (8CMs). It was therefore necessary to select particular mixtures for implementation in the field. The mixtures had two, four or eight component cultivars, selected so as to ensure that all cultivars were included in the same number of mixtures and to maximize the range of

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This article is protected by copyright. All rights reserved functional diversity from very homogeneous to highly heterogeneous mixtures (Table 1), as described by [START_REF] Dubs | An experimental design to test the effect of wheat variety mixtures on biodiversity and ecosystem services[END_REF]. We chose to increase the number of mixtures to cover a wider diversity, rather than repeating a smaller number of mixtures. Each mixture was therefore considered as a statistical individual, rather than an experimental treatment.

Disease and yield assessment

In all field experiments, disease levels were assessed three times during the cropping season. Only assessments in which scoring was considered adequate (well-developed symptoms and limited senescence) were taken into account for the analyses. In E1-2014, severity measurements were carried out at plot scale (rather than leaf scale) for the 58 pure stands. In the three years of experimentation on cultivar mixtures (E2 and E3), yellow rust and Septoria tritici blotch disease severity were visually assessed, on the three upper leaves of single stems, at GS59 (heading stage), based on the percentage of the total leaf area covered by sporulating lesions. The number of main stems scored per plot was proportional to the number of cultivars included in the mixture, to generate data with similar sampling rates for a similar number of plants of each cultivar, whether in pure stands or mixtures. Thus, eight stems were scored for PS (pure stands), four groups of four adjacent main stems for 2CM (i.e. 16 main stems), four groups of eight adjacent main stems for 4CM (i.e. 32 main stems), and four groups of 16 adjacent main stems for 8CM (i.e. 64 main stems). Main stems were chosen in the plots so as to avoid bias, with the same number of plants assessed in each of the two dedicated sowing strips for disease assessment.

In E3-2016, the trial was also scored for two other diseases that were present in the field: Fusarium head blight (FHB) and barley yellow dwarf virus (BYD). FHB severity was recorded at GS59 (heading stage), as the percentage of diseased spikelets per spike for each type of mixture. As for yellow rust scoring, the number of spikes collected per plot was proportional to the number of cultivars present in the mixture: 8 spikes for PS, 16 spikes for 2CM, 32 spikes for 4CM and 64 spikes for 8CM. The severity of barley yellow dwarf virus

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This article is protected by copyright. All rights reserved infections was assessed by eye, for each plot, at GS37 (stem elongation stage), as the percentage of the plot area displaying discoloration (stunting chlorosis and reddening of leaves).

All plots were harvested in early August, with a MB Hege 140 combine harvester with a cut-width of 1.75 m (Hege Maschinen GmbH, Waldenburg, Germany). Grain yield estimates are expressed as mean grain weight measured at 15% humidity over the central section of the plot, 1.75 m x plot length (10 m in E2, 8 m in E3) per plot, in tons per hectare. Mixture effects were assessed by comparing disease severity or yield in the mixture with the mean value of the corresponding pure stands:

𝑀𝐸 𝑌 = 𝑌 𝑚 -𝑌 𝑝 𝑌 𝑝 (1)
with 𝑀𝐸 𝑌 the mixture effect for variable 𝑌, 𝑌 𝑚 the variable measured in the mixture, 𝑌 𝑝 the mean value of the variable of each component cultivar grown in pure stands. Mixture effects were defined as the decrease in disease severity (a positive ME value indicating a decrease in disease severity) or increase in yield (a positive ME indicating a yield increase) in mixtures relative to the mean value for the corresponding pure stands.

The three years of experimentation differed in terms of disease pressure and yield potential, as detailed in Table 2. Yellow rust pressure was exceptionally high in 2014. The Magic lines were chosen as a special genetic group from recombinant populations but their high susceptibility to YR was not known before the first year of experimentation. The very high yellow rust severity observed in 2014 accounts for the replacement of some cultivars (in particular, susceptible MAGIC lines) in the following two years of the experiment. The 2015 cropping season was considered a typical epidemiological year in terms of yellow rust disease pressure, with a relatively dry summer. Finally, 2016 was an atypically unfavorable year for the crop, resulting in very low yields in France [START_REF] Ben-Ari | Causes and implications of the unforeseen 2016 extreme yield loss in the breadbasket of France[END_REF]. In 2016, there was a very high and early general disease pressure, with the presence of uncommon diseases, such as barley yellow dwarf virus, in our experiment. Figure S8 compares yield and disease severity for each mixture of E3, in 2015 and 2016.

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Attribution of disease resistance classes

Yellow (stripe) rust caused by Puccinia striiformis f.sp. tritici was the main disease during the three years of the experiment, particularly in 2014 and 2015, when other diseases were absent or of very low severity. Each cultivar was assigned a yellow rust severity class based on expert judgement and qualitative analyses of experimental measurements of disease severity in pure stands. The severity scores obtained for each cultivar in pure stands in the different experiments, and the associated resistance class, are detailed in Table S1. We obtained a large range of severity scores over the various years of the experiment. It was therefore possible to provide detailed classes for cultivar resistance, ranging from highly resistant to highly susceptible, and to consider moderately resistant and susceptible cultivars. systematically had a disease severity below the mean value. Resistant cultivars were subdivided into two classes that had small but significant differences in terms of mean disease severity: moderately resistant (MR, mean severity of 16.3% in 2014, 5.3% in 2015 and 4.5% in 2016) and highly resistant (HR, mean severity of 5.4% in 2014, 0.2% in 2015 and 1.6% in 2016). In E2-2014 there were as many susceptible cultivars (including 4 HS cultivars) as resistant cultivars, whereas, in E3, there were nine resistant cultivars and seven susceptible cultivars (including 3 HS cultivars).

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This article is protected by copyright. All rights reserved In 2016, disease resistance classes were assigned for the four diseases observed in the E3 experiment, as detailed in Table S2. The other diseases were of lower severity than yellow rust, with a much narrower range of severity scores, such that only resistant and susceptible classes could be established. Classes were defined according to whether the disease severity of the pure stands was higher (susceptible) or lower (resistant) than the mean disease severity for pure stands.

Statistical analysis

Statistical analyses were carried out with R statistical software (R Core Team, 2018). The Shapiro-Wilk normality test ('shapiro.test' function) was used to test the normality of the variables. The Bartlett test ('bartlett.test' function from the 'stats' package) was used to test for the equality of variance across groups (homoscedacity). Means were compared by Student's t-tests ('TukeyHSD' function) for normally distributed variables that passed the homoscedacity test, and by Kruskal-Wallis tests ('kruskal' function from the 'agricolae' library) for non-normally distributed variables or those that did not pass the homoscedacity test.

Linear models were fitted with the 'lm' function. Correlation tests were performed with the 'cor.test' function, using the Pearson method for normally distributed variables and the Kendall method for nonnormally distributed variables.

Results

Disease severity and yield in mixtures and pure stands

Cultivar mixture disease severity and yield were strongly related to the mean performances of the component cultivars grown in pure stands (Figure 1). The three experimental years differed in yield potential, with mean yields of 3.11 t/ha in E2-2014, 5.95 t/ha in E3-2015 and 1.80 t/ha in E3-2016.

The difference in yield between a cultivar mixture and the mean of its components in pure stands ranged from -1.7 t/ha to +1.3 t/ha in E2-2014 (55/90 mixtures had higher yields than the corresponding pure stands),

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This article is protected by copyright. All rights reserved from -1.1 t/ha to +1.2 t/ha in E3-2015 (28/72 mixtures had higher yields) and from -0.4 to t/ha +0.6 t/ha in E3-2016 (40/72 mixtures had higher yields). The difference in yellow rust severity between a cultivar mixture and the mean value for its components in pure stands ranged from -12.5% to +29.2% in E2-2014 (31/90 mixtures had lower disease severity than the pure stands), from -30.7% to +9.4% in E3-2015 (39/72 mixtures had lower disease severity) and from -17.1% to 12.5% in E3-2016 (37/72 mixtures had lower disease severity). The mean difference in yield and disease severity between cultivar mixtures and the mean of pure stands was small and not significantly different from zero according to Student's t-tests (and Wilcoxon tests for disease in E3-2016), for all the years considered. It was therefore possible to anticipate the performances of cultivar mixtures (disease severity and yield) from the characteristics of the component cultivars in pure stands, at least approximately. However, some mixtures displayed both positive (higher yield and/or lower disease severity) and negative (lower yield and/or higher disease severity) mixture effects on performance. Thus, cultivar mixtures did not automatically increase yield or decrease disease, but some of the mixtures tested had very interesting properties (e.g. yield increase of more than 1 t/ha). The characteristics of the three best and three worst mixtures in terms of yield increase (Table S3) and disease reduction (Table S4), for each year, are detailed in the supplementary tables. The relationships between mixture characteristics, yield and disease are explored further below.

Figure 1 around here

Tables 3 and4 summarize the correlations of mixture characteristics with yield and disease severity, respectively. The principal correlations observed were those with the mean yield and disease severity of the component cultivars in pure stands. Yield was strongly correlated with the severity of yellow rust, the main disease in these experiments. This correlation was particularly strong in E2-2014 (-0.45, p=7.0 x 10 -6 ), in which disease pressure was particularly high, and was weaker in E3-2015 (-0.29, p=4.1 x 10 -4 ) and E3-2016 (-0.35, p=2.8 x 10 -3 ). Significant correlations were observed between yellow rust severity in mixtures and

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This article is protected by copyright. All rights reserved the contrast in resistance levels of the mixture components in all three experiments. A significant correlation between mixture yellow rust severity and the proportion of susceptible cultivars was identified in E3-2015 and E3-2016 but not in E2-2014.

The difference in height between the cultivars in the mixture had no significant impact on yield (Table 3) or disease severity (Table 4). However, mixture yield was negatively correlated with the mean plant height of the mixtures in E3 experiments (2015: -0. 42, p-value=2.6 x 10 -7 ; 2016: -0.36 p-value=1.8 x 10 -3 ) but not in E2-2014. This is illustrated in Figure S1 and Figure S2. On the other hand, the difference in flowering date among the cultivars of the mixture had an impact on two variables: (i) yield in E3-2015 (negative correlation of 0.17, p-value = 0.04) and (ii) yellow rust severity in E2-2014 (positive correlation of 0.23, p-value = 0.03).

This correlation was much smaller and not significant in E2-2014 and E3-2016.

Table 3 around here Table 4 around here

Number of cultivars in the mixture and variability of mixture disease severity and yield

The number of cultivars in the mixture (from 2 to 8) had no impact on the mean disease severity and yield of the mixtures (Table 3 and4). However, increasing the number of cultivars within a mixture reduced the variability of yield and disease severity (Figure 2). Mean disease severity and mean yield varied considerably between years (Table 2), but similar patterns were observed for E2-2014, E3-2015 and E3-2016.

Figure 2 around here

The data for experiment E3 (mean of 2015 and 2016) are presented in Figure 3. The decrease in variability was particularly striking for yellow rust severity, which was the main disease during the experiment.

Considering the means of the two years, an inverse correlation between yellow rust severity and yield was observed (-0.72 for all plots considered together, including pure stands), suggesting that disease interfered with yield formation in this experiment. In 2015, the correlation between yellow rust severity and yield was

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This article is protected by copyright. All rights reserved -0.50 for the eight-cultivar mixtures compared and -0.78 for pure stands. Pure stands performed both the best and worst in terms of yield and disease resistance. Overall, cultivar mixtures did not increase mean yield or decrease disease severity, but they did greatly decrease the variation in yield and disease severity.

Figure 3 around here

In E3-2016, a number of diseases were observed in the field. The mean characteristics of pure stands and mixtures with different numbers of component cultivars were similar to those for the previous two years (Figure S3). However, variability among plots decreased with increasing number of cultivars in the mixture. Some pure stands were quite susceptible to one or two diseases in particular. Overall, cultivar mixtures were less susceptible to disease, and this was particularly true for eight-cultivar mixtures. Each mixture seemed to have a different disease profile and this effect was amplified by decreasing the number of cultivars in the mixtures. The severities of FHB and STB severity were positively correlated (0.25), whereas inverse correlations were observed between yellow rust and STB severities (0.25) and between BYD and FHB severities (0.25).

Cultivar mixture yield and disease severity were modulated by the proportions and resistance levels of the component cultivars included in the mixture

Yield was strongly related to yellow rust severity in this experiment, in all three years considered (Figure 4, Table 3). The composition of the mixtures, in terms of the disease resistance of the component cultivars, had various impacts on the disease severity and yield of the mixtures. The proportion of susceptible cultivars affected yellow rust severity and yield. The proportion of susceptible cultivars in the mixture had a visible impact for all the diseases considered. Disease severity increased with the proportion of susceptible cultivars in the mixture (Figure 5, Table 3).

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The difference in yellow rust severity between the cultivars included in a mixture had a significant impact on the mixture effect on disease severity (Figure S5, E2: cor=-0.41, p-value =6.9 x 10 -5 ; E3: cor = 0.48, p-value=2.5 x 10 -5 ). However, this mixture effect decreased with increasing contrast in E2-2014, whereas it increased with increasing contrast in E3 (2015-2016). In E2-2014, positive mixture effects were observed in situations in which resistance differences between cultivars were small, mostly with a high proportion of resistant cultivars. Mixtures including similar proportions of resistant and susceptible cultivars had a high resistance contrast and null to negative mixture effects. The high disease pressure probably increased the severity of disease on moderately resistant cultivars grown with highly susceptible cultivars. By contrast, in experiment E3 (2015-2016), a greater difference in disease resistance between the cultivars in a mixture increased the mixture effect on disease severity. This was the case for all proportions of susceptible cultivars, and the effect increased with the proportion of susceptible cultivars. This suggests that the protective effect sought in mixture design was greater in this experiment, due to a lower disease pressure and a better balance in the range of resistance levels of the cultivars used in the mixtures.

Mixture effect increased with mean yellow rust severity of the cultivars in the mixture in E3 (2015-2016), but not in E2-2014 (Figure S6). This relationship was particularly marked for E3 mixtures in which at least 50% of the cultivars were susceptible to yellow rust. For mixtures with high proportions of resistant cultivars, which had the lowest mean disease severity, mixture effects were mostly negative, meaning that disease severity in the mixture was slightly higher than the mean value for the individual cultivars in in pure stands, although the overall level of resistance remained high. The best mixture effects for yellow rust severity were observed with 50 to 75% susceptible cultivars in the mixture, but such mixtures had a high level of disease severity, as illustrated in Figure 4. Few mixtures with high proportions of susceptible cultivars were included in the experiments but those tested gave highly variable results concerning disease severity in the E2 experiment.

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In the case of yellow rust, the level of resistance to disease in pure stands varied considerably, making it possible to define more detailed resistance classes, as shown in Table S1. Table 5 summarizes the relationships between the proportion of cultivars of different resistance classes and disease severity in mixtures. Detailed data were provided in Figure 3. The proportion of highly susceptible cultivars was strongly related to disease severity. For example, increasing the proportion of highly susceptible cultivars by 10% increased disease severity by absolute values of 5.2% in E2-2014, 1.9% in E3-2015 and 1.3% in E3-2016. The proportion of moderately susceptible cultivars had a small but significant impact only in E3-2015.

Though the differences between moderately and highly resistant cultivars were small, they performed differently in mixture, justifying the distinction between these two classes. Increasing the proportion of moderately resistant cultivars by 10% decreased disease severity by 2.0% in E2-2014 and had no significant impact in E3-2015 and E3-2016. Finally, the proportion of highly resistant cultivars had a large impact on disease severity, although not as significant as that of the highly susceptible cultivars. Thus, increasing the proportion of highly resistant cultivars by 10% decreased disease severity in cultivar mixtures by 3.6% in E2-2014, 1.1% in E3-2015 and 0.6% in E3-2016.

Table 5 around here

Discussion

We tested diverse cultivar mixtures, some of which included highly contrasted phenotypes (Table 1). Despite this diversity, we observed that mixture disease severity and yields were strongly related to the mean performances of the component cultivars grown in pure stands (Figure 1). This suggests that, whatever the rules used to design mixtures, the mean disease severity and yield of pure stands can be used as a first indicator, to anticipate the behavior of the mixture, and that the mixture can then be refined on the basis of more precise knowledge of the mechanisms affecting disease severity and yield in mixtures. However, the predictability of mean disease severity and yield may conceal other effects on one of the component cultivars not considered in this study. For example, large decreases in disease severity have been reported for

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This article is protected by copyright. All rights reserved susceptible plants grown in mixtures (Gigot et al., 2012;[START_REF] Vidal | Reduction of fungal disease spread in cultivar mixtures: Impact of canopy architecture on rain-splash dispersal and on crop[END_REF], but these decreases might be concealed by calculations of the mean disease severity for a mixture of several cultivars including resistant ones, as in our study. Some of the properties of mixtures may reflect changes in cultivar proportions relative to those at sowing. Indeed, the proportion of cultivars within mixtures may change considerably between sowing and harvest (Finckh and Mundt, 1992Mundt, , 1993Mundt, , 1996;;Finckh et al., 2000;[START_REF] Fraj | The use of microsatellite markers to determine the relative proportions of grain produced by cultivars and the frequency of hybridisation in bread wheat mixtures[END_REF][START_REF] Vidal | Reduction of fungal disease spread in cultivar mixtures: Impact of canopy architecture on rain-splash dispersal and on crop[END_REF]. This effect may be enhanced if one cultivar is particularly competitive due to its greater height or earlier stem elongation. Such mechanisms may have occurred in the mixtures studied here, but they had no significant impact on mixture disease severity and yield. The mean effect of mixtures was relatively small, but this effect was also highly variable, ranging from -30.7% to +29.2% for yellow rust severity and from -1.7 t/ha to +1.3 t/ha for yield. In E2-2014, many mixtures had more disease than the mean of pure stands (Figure 1d). This could be explained by the presence of many susceptible cultivars in this experiment (Table 1 andS2), which was not the case in 2015 and 2016 when other cultivars were included. In a meta-analysis comparing 386 mixtures, Borg et al. (2018) reported mixture effects on wheat yield ranging from -40% to +60%, with larger mixture effects under conditions of high disease pressure. We observed mixture effects ranging from -33% to +27% for yield, the largest effects (either positive or negative) being observed under high disease pressure (E2-2014). In another meta-analysis comparing 161 bread wheat cultivar mixtures, Huang et al. (2012) reported decreases in yellow rust severity of 22.5% to 33.4%, with a mean decrease of 28.0% and larger mixture effects under high disease pressure. The variability of cultivar mixture effects could potentially be reduced by an appropriate design, including a large number of cultivars. The effects of some key mixture characteristics are described below.

Agronomic homogeneity is a classic recommendation for the design of cultivar mixtures. It simplifies crop practices and it is often claimed that a contrast in competitive ability would have a major impact on the least competitive cultivar, resulting in a poor performance of the mixture. Agronomic homogeneity

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This article is protected by copyright. All rights reserved has direct practical benefits, particularly for harvest scheduling, to our knowledge, no systematic negative impact of heterogeneity has ever been demonstrated experimentally. On the other hand, several studies have reported positive effects of mixing cultivars of cereal species with different heights or flowering dates. These effects concerned airborne diseases, yield, grain quality, lodging, water use efficiency and weed suppression, as reviewed in [START_REF] Vidal | Reduction of fungal disease spread in cultivar mixtures: Impact of canopy architecture on rain-splash dispersal and on crop[END_REF]. For example, mixtures of a tall traditional susceptible glutinous rice cultivar with a shorter modern resistant cultivar have been reported to display good control of panicle blast disease, which is caused by Magnaporthe grisea (wind-dispersed), due to a decrease in canopy humidity levels (Zhu et al., 2005). A certain degree of difference between components may be acceptable in mixtures if it confers properties of interest on the mixture. In our experiment, we mixed different types of cultivars (elite cultivars, organic agriculture cultivars, MAGIC lines from a multiparent population and landraces), thereby considerably extending the concept of a "mixture" beyond the more widely accepted mixtures of several elite cultivars with different resistance genes or a susceptible cultivar of superior agronomic characteristics with several resistant cultivars of lower yield potential. Agronomic contrast in height or earliness had little or no deleterious effect on mixture yield (Table 3) and disease severity (Table 4).

We investigated the impact of the number of cultivars in the mixture. In ecology, the number of species, or species richness, is recognized to be a key criterion with potential impacts on ecosystem performance in assessments of ecosystem diversity (Gaba et al., 2015;Yang et al., 2019). [START_REF] Sapoukhina | Quantitative plant resistance in cultivar mixtures: wheat yellow rust as a modeling case study[END_REF] reported that the use of three-component mixtures made it possible to design a broader range of effective control strategies than the use of two-component mixtures. The number of cultivars in mixtures has been shown to be positively correlated with overyielding (Mundt et al., 1995;[START_REF] Newton | Analysing the contribution of component cultivars and cultivar combinations to malting quality, yield and disease in complex mixtures[END_REF][START_REF] Kiaer | Grain yield increase in cereal variety mixtures: A meta-analysis of field trials[END_REF][START_REF] Reiss | Cultivar mixtures: a meta-analysis of the effect of intraspecific diversity on crop yield[END_REF]. We observed no significant impact of the number of component cultivars on mean mixture disease severity and yield. However, the variability of mixture performances strongly decreased with

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This article is protected by copyright. All rights reserved increasing number of cultivars (Figure 2). This pattern was reported in previous experimental studies and clearly emerged from meta-analysis results (Huang et al., 2012;[START_REF] Borg | Unfolding the potential of wheat cultivar mixtures: A meta-analysis perspective and identification of knowledge gaps[END_REF][START_REF] Reiss | Cultivar mixtures: a meta-analysis of the effect of intraspecific diversity on crop yield[END_REF]. The main advantage of increasing the number of cultivars seems to be a greater flexibility in mixture design, with the possibility of including cultivars with a wider range of desirable properties, such as resistance to various different diseases, resistance to abiotic stresses and good grain quality.

The proportion of susceptible cultivars in the mixture was strongly related to disease severity on the mixture (Figure 5). This finding confirmed that the proportion of susceptible cultivars has a crucial impact on disease protection, as already demonstrated in previous studies [START_REF] Cox | Cultivar mixtures for the simultaneous management of multiple diseases: tan spot and leaf rust of wheat[END_REF][START_REF] Dai | Performance of hard red spring wheat cultivar mixtures[END_REF]. This protection of a susceptible cultivar by resistant ones can be optimized in mixtures with a proportion of one third to one quarter susceptible plants (Gigot et al., 2012;Gigot et al., 2014). The proportion of highly susceptible cultivars had the largest and most significant impact on yellow rust severity (Table 5). The proportions of moderately susceptible cultivars had little impact on disease severity in the mixture, consistent with the results of the modeling study by [START_REF] Sapoukhina | Quantitative plant resistance in cultivar mixtures: wheat yellow rust as a modeling case study[END_REF].

When cultivar mixtures are used for disease management, it is generally recommended to associate cultivars with contrasting levels of disease resistance (Jeger et al., 1981;Mundt et al., 1995;Gigot et al., 2014). In this experiment, the contrast in disease resistance between the cultivars in a mixture had a significant impact on the effect of the mixture on yellow rust severity (Table 4, Figure S5). However, this impact was negative for E2-2014, under conditions of exceptionally high yellow rust pressure, but positive in E3-2015/2016.

The extent and direction of this mixture effect varied with disease pressure, consistent with previous observations [START_REF] Cowger | Effects of wheat cultivar mixtures on epidemic progression of Septoria tritici blotch and pathogenicity of Mycosphaerella graminicola[END_REF]Gigot et al., 2012). In E2-2014, yellow rust pressure was high and mixture efficacy was low. However, the highly susceptible elite cultivar included in some mixtures was protected in those mixtures. In E3-2015/2016, the mixture effect increased with the mean severity of disease on the component cultivars in pure stands (Table3, Figure S6), consistent with previous findings (Jeger et al., 1981;

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This article is protected by copyright. All rights reserved Mundt et al., 1995). When disease pressure was very high, with high levels of disease on both cultivar mixtures and component cultivars (up to 40-50% disease severity in E2), even positive mixture effects were insufficient for adequate disease protection. Similarly, when disease pressure was very low or if the mixture components were all resistant or moderately resistant, no benefit was expected from the cultivar mixtures, in terms of a decrease in disease levels in a particular season. Benefits were observed mostly in cases of moderate disease pressure, with a susceptible cultivar included in the mixture.

To conclude, cultivar diversity for various traits, including plant height and flowering date, may have a positive impact (possible positive interactions such as compensation, facilitation or complementarity), or a negative impact (possible negative interactions, such as competition) on the performance of cultivar mixtures. We found that, despite contrasts between cultivars, mixture disease severity and yield were strongly related to the mean values for these variables obtained for pure stands of the component cultivars of the mixture. Some mixtures had very interesting properties, with yield increases of up to 1.3 t/ha relative to pure stands, and decreases in disease severity by up to 30%. For the large diversity of mixtures considered here, the variability of disease severity and yield decreased with increasing number of cultivars in the mixture. Adding more cultivars did not reduce disease or increase yield significantly, but gave more flexibility to mixture design, with the possibility of including cultivars with interesting characteristics and/or modifying the proportion of cultivars with an interesting trait (e.g. resistance to a particular disease). Despite the agronomic contrasts within the mixtures tested, we found that variations in yield and disease severity were related to known design criteria mostly established in experiments on agronomically homogeneous mixtures.

In particular, the proportion of susceptible cultivars affected disease severity, yield and mixture effects.

Disease pressure also had an impact on mixture effects. Our results confirm the potential of cultivar mixtures for use in disease management and the robustness of the established rules. Mixtures may also provide other This article is protected by copyright. All rights reserved

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This article is protected by copyright. All rights reserved Mixture yield (T/ha) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q (a)

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Yield difference between mixture and the mean of pure stands (T/ha) Mean yellow rust severity of component cultivars in pure stands (%) Yellow rust severity in mixture (%) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q (c) 2.0 q q q q q q q q q q q q q yellow rust Septoria tritici blotch draft yellow barley Fusarium head blight (b) Disease severity ted A

Mean yellow rust severity (%) Mean yield (qx/ha) 0 20 40 60 30 35 40 45 50 q q q q q q q q q q q q q q q q q q q q q q q q q q 1 cultivar 2 cultivars 4 cultivars 8 cultivars ed A Yield (T/ha) q q q q q q q q q q q q q q q q Aa Ab ABb Bc Yield (T/ha) q q q q q q q q q q q q q q q q q q q q Aa ABb Bbc Cc (b) 0 5 10 15 20 25 30 0.5 1.0 1.5 2.0 2.5 3.0 Yellow rust severity (%) Yield (T/ha) q q q q q q q q q q q q q q q q q q q q Aa Bb Cb Cb

  Yellow rust severity differed significantly between cultivars from different resistance classes (p<0.05) in E2-2014, E3-2015 and E3-2016. Highly susceptible cultivars (HS, mean severity of 74.6% in 2014, 52.8% in 2015 and 21.3% in 2016) systematically displayed disease of above-mean severity. Moderately susceptible cultivars (MS, mean severity of 43.6% in E2-2014, 19.4% in E3-2015 and 10.3% in E3-2016) displayed variable behavior in the years 2015 and 2016. Resistant cultivars (moderately or highly resistant)

Figure

  Figure 4 around here

Figure S3 :

 S3 Figure S3: Principal component analysis of the severity of four diseases in pure stands and cultivar mixtures

Figure S4 :Figure S5 :

 S4S5 Figure S4: Yellow rust severity in mixtures as a function of the proportion of cultivars with different levels

Figure S6 :

 S6 Figure S6: Mixture effect on wheat yellow rust severity as a function of the percentage of susceptible (HS

Figure S7 :

 S7 Figure S7: Yield and disease severity for two years of experiment. Each point corresponds to one mixture.

Figure S8 :

 S8 Figure S8: Yellow rust severity (%) in two repetitions of pure stands.

Figure legends Figure 1 :

 legends1 Figure legends

Figure 2 :

 2 Figure 2: Variability of yield (a) and disease severity (b) as a function of the number of

Figure 3 :

 3 Figure 3: Yield and yellow rust severity in a range of bread wheat pure stands and mixtures including 2, 4

Figure 4 :

 4 Figure 4: Relationship between yellow rust severity and yield in cultivar mixtures, for different

Figure 5 :

 5 Figure 5: Impact of the proportion of susceptible (moderately susceptible and highly susceptible)

(

  Table 2 around here.

Table 1 : Field experiment design

 1

	Experiment	E1-2014	E2-2014	E3-2015/2016
	Plot area	26 m²	52.5 m²	84 m²
	Plot size 5.25 x 5 m 5.25 x 10 m	10.5 x 8 m
	Number of plots	58	130	104
	Number of cultivars	58	20	16
	Susceptible cultivars*	34	10	7
	Highly susceptible cultivars*	15	5	3
	Elite cultivars	-	8	9
	Organic cultivars	-	4	4
	MAGIC lines	-	6	2
	Landraces	-	2	1
	Number of mixtures	0	90	72
	2-cultivar mixtures	-	30	24
	4-cultivar mixtures	-	30	28
	8-cultivar mixtures	-	30	20
	Low heterogeneity**	-	11	6
	High heterogeneity**	-	69	55

* Resistance classes were defined on the basis of yellow rust severity in pure stands for each cultivar, across different experiments. ** Heterogeneity was defined here according to the Danish rules, as reported by

[START_REF] Labarthe | Diffusion des mélanges variétaux pour la production de blé : une comparaison entre France et Danemark[END_REF]

. Low heterogeneity describes mixtures with flowering dates differing by less than five days and plant heights differing by less than 15 cm. High heterogeneity describes mixtures with flowering dates differing by at least five days and plant heights differing by at least 15 cm.

Table 2 : Summary statistics for the yield and disease severity of bread wheat cultivars grown in pure stands during three cropping seasons

 2 Yield os expressed in tonnes per hectare. Disease severity is expressed as a percentage (foliar area, spike, or plot area) for all diseases. YR: yellow rust, STB: Septoria tritici blotch; FHB: Fusarium head blight; BYD: barley yellow dwarf virus.

		E2-2014	E3-2015			E3-2016
		Yield	YR	Yield	YR	Yield	YRS	STB	FHB	BYD
	min	0.95	0.0	4.50	0.0	0.60	0.1	0.0	0	0.0
	max	5.57	85.6	7.86	75.5	2.57	38.4	34.3	98.6	80.0
	mean	3.11	35.5	5.95	17.2	1.80	9.9	7.2	60.0	23.2
	SD	0.75	20.3	0.95	21.8	0.51	10.7	10.2	31.3	20.5

Table 5 : Yellow rust severity in cultivar mixtures as a function of the proportion of cultivars with different levels of resistance to disease

 5

	Resistance class Experiment Int.	Slope R²	p-value **
	HS	E2-2014	23.6 52.0 0.59	< 10 -15 *
		E3-2015	11.0 19.2 0.19	6.6 x 10 -5 *
		E3-2016	7.1	13.0 0.15	3.1 x 10 -4 *
	MS	E2-2014	36.0 2.2	-0.01	0.77
		E3-2015	12.9 6.8	0.05	2.8 x 10 -2 *
		E3-2016	9.2	1.1	-0.01	0.64
	MR	E2-2014	40.6 -19.8 0.06	1.2 x 10 -2 *
		E3-2015	16.5 -6.2	0.03	7.9 x 10 -2
		E3-2016	10.1 -1.8	-0.007 0.49
	HR	E2-2014	47.3 -35.8 0.29	2.0 x 10 -8 *
		E3-2015	17.5 -11.5 0.14	6.3 x 10 -4 *
		E3-2016	10.9 -5.7	0.06	2.5 x 10 -2 *

HS: highly susceptible, MS: moderately susceptible, MR: moderately resistant, HR: highly resistant. Linear models were fitted to the data. Resistance classes -HS: highly susceptible; MS: moderately susceptible; MR: moderately resistant; HR: highly resistant. Stars indicate significant relations between yellow rust severity and the proportion of cultivars belonging to different resistance classes.

Acknowledgments

We thank our collaborators for their participation in field disease severity and yield assessments: Léa Boutard, Laurent Gérard and François Lecourt, and Christophe Montagnier and his team (INRAE Experimental Unit, Versailles) for crop management. We also thank Julie Sappa of Alex Edelman & Associates for improving our English usage. This work was supported by the ANR WHEATAMIX project (grant ANR-13-AGRO-0008, French National Research Agency).

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This article is protected by copyright. All rights reserved ecosystem services (e.g. weed reduction, resistance to water stress or lodging, grain quality) and such properties might require the mixing of cultivars with contrasting characteristics. Our results suggest that such contrasts would not modify the established rules and that the mean yield and disease severity values for cultivars in pure stands would remain a good indicator of the likely performance of mixtures.

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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This article is protected by copyright. All rights reserved Jørgensen, J.H., 1992. Discovery, characterization and exploitation of Mlo powdery mildew resistance in barley, in: Johnson, R., Jellis, G.J. (Eds.), Breeding for Disease Resistance, Developments in Plant

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This article is protected by copyright. All rights reserved Finckh M.R., Mundt C.C. 1992. Stripe rust, yield, and plant competition in wheat cultivar mixtures.

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