Marine Couret

Damien Michez

Juliette Letellier

Anne Castelan

Julien Pernot

Nicolas C Rouger

Field-Plated D3MOSFET design for breakdown voltage improvement

Keywords: Diamond, D3MOSFET, Power device, Breakdown voltage, Field plate, Fabrication process

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

Introduction

Thanks to its ultrawide bandgap of 5.5 eV and related high critical electrical field above 7 MV/cm, together with a high free hole mobility of more than 2000 cm²•V -1 •s -1 , diamond offers theoretically the best performances for unipolar power devices [START_REF] Masante | Recent progress in deep-depletion diamond metal-oxide-semiconductor field-effect transistors[END_REF]. Moreover, the high thermal conductivity of 22 W/(m.K) for diamond material makes it an excellent asset for power applications to reduce the size of the thermal packaging [START_REF] Donato | Diamond power devices state of the art, modelling, figures of merit and future perspective[END_REF]. For those multiple reasons, the interest within diamond-based transistors has grown over the past years with the development of several architectures for diamond devices as summarized in [START_REF] Donato | Diamond power devices state of the art, modelling, figures of merit and future perspective[END_REF][START_REF] Umezawa | Recent advances in diamond power semiconductor devices[END_REF]. As such, field effect transistors (FET) have been demonstrated, including inversion MOSFET [START_REF] Matsumoto | Inversion channel diamond metal-oxide-semiconductor field-effect transistor with normally off characteristics[END_REF], deep depletion MOSFET [START_REF] Pham | mv/cm lateral diamond mosfet[END_REF][START_REF] Masante | MV/cm, 50 mΩ.cm 2 at 250 • C diamond MOSFET and its reverse conduction[END_REF] or Metal Semiconductor FET [START_REF] Umezawa | Diamond metal-semiconductor field-effect transistor with breakdown voltage over 1.5 kv[END_REF]. A survey of the different architecture performances is presented in figure 1 showing the specific on-state resistance according to the breakdown voltage. While lateral diamond transistors such as MESFETs or MOSFET with NO 2 doping have shown breakdown voltages above or larger than 2kV [START_REF] Kitabayashi | Normally-off c-h diamond mosfets with partial c-o channel achieving 2-kv breakdown voltage[END_REF][START_REF] Kawarada | Diamond mosfets using 2d hole gas with 1700v breakdown voltage[END_REF][START_REF] Umezawa | Diamond metal-semiconductor field-effect transistor with breakdown voltage over 1.5 kv[END_REF][START_REF] Saha | 345-mw/cm² 2608-v no p-type doped diamond mosfets with an alo passivation overlayer on heteroepitaxial diamond[END_REF][START_REF] Saha | 875-mw/cm² low-resistance no p-type doped chemical mechanical planarized diamond mosfets[END_REF][START_REF] Saha | 3326-v modulation-doped diamond mosfets[END_REF], diamond MOSFETs using the deep depletion regime have only shown a breakdown voltage around 200 V [START_REF] Pham | mv/cm lateral diamond mosfet[END_REF][START_REF] Masante | MV/cm, 50 mΩ.cm 2 at 250 • C diamond MOSFET and its reverse conduction[END_REF]. However, it was reported for this architecture, the highest peak electric field in off-state (above 5 MV/cm), with stable and efficient transistor characteristics above 250°C [START_REF] Masante | MV/cm, 50 mΩ.cm 2 at 250 • C diamond MOSFET and its reverse conduction[END_REF]. The premature breakdown has been attributed to an electric field crowding effect at the gate-edge, as typically reported on lateral power devices, inducing an electrical breakdown of the Al 2 O 3 gate-oxide layer around 200 V [START_REF] Masante | Recent progress in deep-depletion diamond metal-oxide-semiconductor field-effect transistors[END_REF]. This outcome, also observed in Ga 2 O 3 power devices, can be overcome with the employment of field-plate as demonstrated in [START_REF] Wong | Field-plated ga2o3 mosfets with a breakdown voltage of over 750 v[END_REF][START_REF] Zeng | 1.85 kv breakdown voltage in lateral field-plated ga2o3 mosfets[END_REF][START_REF] Joishi | Breakdown characteristics of β -(al0.22ga0.78)2o3/ga2o3 field-plated modulation-doped field-effect transistors[END_REF]. For the field-plate design, it is necessary to introduce a passivation layer commonly made of SiO 2 , Al 2 O 3 or Si 3 N 4 . As the dielectric material must sustain very high electric fields, its critical electric field must be as high as possible. Therefore, the main objective of this article is to present a field-plate design adapted to a lateral deep-depletion-diamond MOS-FET architecture with a requested breakdown voltage of 1 kV. Based on 2D TCAD numerical simulations, the device architecture will be presented together with its figures of merit. Then, a focus on the field-plate will be made with a design methodology for its geometrical parameters. The impact of the field-plate on the device performances will be assessed regarding both the breakdown voltage and the dynamic properties.

Concerning the device fabrication process, a particular interest will be dedicated to the passivation layer development and the characterization of its critical electric field. Finally, the article will be concluded. Figure 1: Specific on-state resistance vs. breakdown voltage performances of power diamond FETs reported in the literature. Experimental points are taken from [START_REF] Pham | mv/cm lateral diamond mosfet[END_REF][START_REF] Masante | MV/cm, 50 mΩ.cm 2 at 250 • C diamond MOSFET and its reverse conduction[END_REF][START_REF] Umezawa | Diamond metal-semiconductor field-effect transistor with breakdown voltage over 1.5 kv[END_REF][START_REF] Kitabayashi | Normally-off c-h diamond mosfets with partial c-o channel achieving 2-kv breakdown voltage[END_REF][START_REF] Kawarada | Diamond mosfets using 2d hole gas with 1700v breakdown voltage[END_REF][START_REF] Saha | 345-mw/cm² 2608-v no p-type doped diamond mosfets with an alo passivation overlayer on heteroepitaxial diamond[END_REF][START_REF] Saha | 875-mw/cm² low-resistance no p-type doped chemical mechanical planarized diamond mosfets[END_REF][START_REF] Saha | 3326-v modulation-doped diamond mosfets[END_REF][START_REF] Saha | 145-mw/cm2 heteroepitaxial diamond mosfets with no2 p-type doping and an al2o3 passivation layer[END_REF][START_REF] Iwasaki | 600 v diamond junction field-effect transistors operated at 200 • c[END_REF][START_REF] Kawarada | C-h surface diamond field effect transistors for high temperature (400°c) and high voltage (500v) operation[END_REF][START_REF] Syamsul | High voltage stress induced in transparent polycrystalline diamond field-effect transistor and enhanced endurance using thick al2o3 passivation layer[END_REF][START_REF] Oi | Vertical-type two-dimensional hole gas diamond metal oxide semiconductor field-effect transistors[END_REF][START_REF] Choi | High breakdown voltage of boron-doped diamond metal semiconductor field effect transistor grown on freestanding heteroepitaxial diamond substrate[END_REF] 2. D3MOSFET design

Description

The presented device is a lateral deep-depletion-diamond MOSFET which is directly based on the architecture proposed in [START_REF] Masante | Recent progress in deep-depletion diamond metal-oxide-semiconductor field-effect transistors[END_REF] as displayed in figure 2a. The principle of the D3MOSFET is a lateral conduction between the drain and the source ohmic contacts made through a p-bulk diamond layer (boron doped) grown on a Ib diamond semi-insulating substrate. Using the 1D ionization integral [START_REF] Rouger | Design of diamond power devices: Application to Schottky barrier diodes[END_REF] with the coefficients from [START_REF] Hiraiwa | Blocking characteristics of diamond junctions with a punch-through design[END_REF] for diamond material, the doping for the channel, N a , is decreased down to 8•10 16 atom.cm -3 considering a 1D breakdown voltage of 1400 V in non-punch through (NPT) condition implying a drift length, L gd , larger than 3.35 µm. This decreased doping density will thereby reduce the constraints on the breakdown voltage. From this doping density value, it is rather simple to calculate the corresponding resistivity of the channel layer using equation [START_REF] Masante | Recent progress in deep-depletion diamond metal-oxide-semiconductor field-effect transistors[END_REF]. The models for the free holes density [START_REF] Seok | Effects of incomplete ionization on forward current-voltage characteristics of p-type diamond schottky barrier diodes based on numerical simulation[END_REF][START_REF] Sze | Physics of semiconductor devices[END_REF], and the hole mobility [START_REF] Volpe | High hole mobility in boron doped diamond for power device applications[END_REF] are dependent of the temperature, the boron-doping density and the compensation level in donor atoms. Note that, in this study, the compensation level, N d /N a , is set to 4 %. From figure 2b, it can be observed a large reduction of the resistivity with increasing temperatures, demonstrating the benefits of using diamond devices at elevated junction temperatures. At 450 K, the targeted temperature for the device operation, the resistivity reaches a value of 2 Ω.cm.

ρ(T, N a , N d ) = 1 q • µ p (T, N a , N d ) • p(T, N a , N d ) (1)
where µ p is the hole mobility and p is the free holes density. Moreover, in order to keep an equivalent threshold voltage of 30 V compared to the device reported in [START_REF] Masante | Recent progress in deep-depletion diamond metal-oxide-semiconductor field-effect transistors[END_REF], the thickness of the channel, t channel , is increased up to 500 nm. This value does not take into account the depletion caused by the PN junction from the substrate but only the depletion induced by the gate bias, meaning that the thickness of the channel will be larger than 500 nm. Regarding the sizing of the D3MOSFET, the gate-drain length is fixed to 6 µm as the device must (i) sustain the 1D breakdown voltage of 1.4 kV, which implies a drift length larger than 3.35 µm and (ii) account for the future field-plate architecture which requires an increase of the gate-drain distance. Finally, the gate length, L g , and the gate-source length, L gs , are respectively fixed to 5 µm and 3 µm due to process limitations and technology constraints. The resulting device presents a specific on-state resistance, R on,s , calculated from equation ( 2), of 75 mΩ.cm² at 450 K. This value can be further optimized, while reducing safety fabrication margins (lower L g and L gs ) but it is out of scope of this contribution. Nonetheless, as presented in figure 1, this would be a strong improvement for these lateral diamond MOSFETs.

R on,s (T ) = ρ(T ) (L gs + L g + L gd ) 2 t channel (2)

Simulation results

Using the 2D numerical simulations from Silvaco ATLAS, the working principle of the D3MOSFET is verified. Figure 3a presents the device transconductance for a V gs ranging from -10 V to 60 V at low electric field (V ds = -1 V). This large scale allows us to observe all the device working regimes: the accumulation regime for V gs < -5 V, the device on-state for V gs up to 30 V and the device off-state for V gs beyond 30

V. One can also note that the device is normally-on as a negative current around 1.2 mA/mm is flowing between the source and the drain at a V gs of 0 V. For the three operation modes, the hole concentration and the current flowlines are also displayed in figures 3b for the accumulation regime, 3c for the device on-state and 3d for the device off-state. As the bulk mobility is considered during the simulations, an accumulation of holes underneath the gate is observed during the accumulation mode at V gs = -10 V, which leads to current flowlines confined into a very thin layer. The device depletion mode is shown at a V gs of 10 V with a partial depletion of the channel layer during the device on-state. The channel is fully-depleted for V gs larger than the threshold voltage of 30 V and no current can flow from the source to the drain.

Limits associated with the elementary lateral D3MOSFET

Concerning the device breakdown voltage, figure 4a presents the 2D cartography of the device electric field for V gs = 40 V and V ds = -200 V, in off-state. As a reminder, the theoretical 1D breakdwon voltage with a similar vertical device in NPT is 1.4 kV. From this graph, we observe an electric field crowding effect under the gate, at the drain-side, in both diamond and gate-oxide layers. The corresponding electric field along a lateral cutline is reported in figure 4b where a maximum value of 6.6 MV/cm, respectively 5

MV/cm, is observed in the gate-oxide, resp. in the diamond, layer. It is important to note that the value of the peak is largely dependent of the mesh grid used during the simulations. Here, the mesh spacing in the vicinity of the gate-edge at the drain side is fixed at 50 nm. Therefore, the reported values for the peak electric field in both diamond and oxide layers are only an estimation, mainly as a trade-off between accuracy, convergence and simulation time, and the values could be larger than the one reported using a thinner mesh spacing. However, in any cases, the peak electric field in the gate-oxide leads to a premature breakdown of the device as the theoretical critical electric field of the 50 nm thick Al 2 O 3 is estimated around 6.8 MV/cm [START_REF] Masante | Recent progress in deep-depletion diamond metal-oxide-semiconductor field-effect transistors[END_REF]. Thus, while the active layer of the device can normally reach much higher critical electric fields (between 7.7 MV/cm and 10 MV/cm for diamond material [START_REF] Donato | Diamond power devices state of the art, modelling, figures of merit and future perspective[END_REF]), the transistor is limited here by the gate-oxide layer and will not be able to reach the 1D breakdown voltage of 1.4 kV. 3. Field-plated D3MOSFET design

Description

One common solution used to improve the breakdown voltage of power devices is the design of a fieldplate such as in Ga 2 O 3 [START_REF] Wong | Field-plated ga2o3 mosfets with a breakdown voltage of over 750 v[END_REF][START_REF] Zeng | 1.85 kv breakdown voltage in lateral field-plated ga2o3 mosfets[END_REF][START_REF] Joishi | Breakdown characteristics of β -(al0.22ga0.78)2o3/ga2o3 field-plated modulation-doped field-effect transistors[END_REF] or GaN [START_REF] Liao | Simulation of algan/gan hemts' breakdown voltage enhancement using gate field-plate, source field-plate and drain field plate[END_REF][START_REF] Sun | Gan power integration for high frequency and high efficiency power applications: A review[END_REF] based devices. The purpose of this architecture is to smooth the electric field over a greater distance and thus reduce the electric field peak at the gate edge. A typical architecture is displayed in figure 5 which is called the source field-plate. This topology requires the addition of a passivation layer underneath the field-plate and of a thick metal layer to realize the field-plate.

Regarding the state-of-the-art of dielectric material, the silicon nitride, Si 3 N 4 , presents the best figure-ofmerit with a ϵ R • F BR product of 75 MV/cm [START_REF] Lei | Field plate engineering for GaN-based schottky barrier diodes[END_REF]. Moreover, power devices have already demonstrated high breakdown voltages using Si 3 N 4 as a dielectric material [START_REF] Joishi | Breakdown characteristics of β -(al0.22ga0.78)2o3/ga2o3 field-plated modulation-doped field-effect transistors[END_REF][START_REF] Mao | Reverse blocking characteristics and mechanisms in schottky-drain AlGaN/GaN HEMT with a drain field plate and floating field plates[END_REF]. In order to improve as much as possible the device voltage capability, two parameters, displayed in figure 5, need to be optimized: the thickness of the passivation layer, h F P , and the extension of the field-plate at the drain side, L F P . Although the geometry in an actual device would be different, the simplified structure of figure 5 has been considered as a good trade-off between accuracy and complexity.

Drain

Design optimization

The optimal sizing of the field-plate is performed through 2D numerical simulations using Silvaco ATLAS by varying the two geometrical features, meaning L F P and h F P , and by tracking the device electric field at the hot spots. The sensitive spots are displayed in figure 6a within: (i) the diamond, (ii) the gate-oxide and (iii) at the field-plate edge with the corresponding cutlines along the x-axis. Indeed, the addition of the field-plate leads to a relocation of the hot spot at the field-plate edge with a larger peak electric field observed than under the gate in the diamond and gate-oxide material. Therefore, for each variation of the geometrical features, the maximum electric field is evaluated and reported in figures 6b, 6c and 6d for the cutlines along the x-axis within the diamond, the gate-oxide and at the field-plate edge. Here, the simulation results for the source field-plate are presented for V gs = 40 V and V ds = -200 V. From these simulation results, one can clearly see the benefits of the field-plate on the peak electric field in both diamond and gate-oxide layers. Indeed, the peak value in the diamond, respectively gate-oxide, has decreased from 5 MV/cm, resp. 6.6 MV/cm, without the field-plate to an average value of 3 MV/cm, resp. 3.4 MV/cm, with the presence of a field-plate.

Concerning the peak electric field variation for various geometrical features, the same trend can be observed within the diamond and gate-oxide layers with a decrease of the peak electric field with increasing L F P which is consistent with the fact that the electric field will be smoothed over a larger distance. Also, the peak electric field is increasing with increasing h F P which is also consistent as the field-plate will be more efficient close to the gate-edge. On the contrary, an opposite behavior is observed for the peak electric field evolution at the field-plate edge. Indeed, the peak value is increasing with increasing L F P since the border of the field-plate is closer to the drain metal, thereby increasing the electrical constraint. Moreover, for the same reason, the peak electric field is increasing with decreasing h F P . Those simulations results

show us that a trade-off must be found between the decrease of the electric field within the diamond and gate-oxide material and the peak electric field observed at the field-plate edge, within the silicon nitride.

Therefore, thanks to the simulation results and due to the limitations of the fabrication process, the fieldplate extension at the drain side, meaning L F P , will be comprised between 1 µm and 3 µm while the passivation layer thickness, meaning h F P , shall be between 400 nm and 600 nm. For ease of comparison with the non-field-plated device, it is fixed L F P = 2 µm and h F P = 500 nm for the rest of the section.

Expected performances

The evolution of the peak electric field along the V ds for an optimized structure is displayed in figure 7 for the source field-plate architecture. This figure shows the peak electric field value within the diamond (in black), the gate-oxide (in red) and in the silicon nitride (in blue) together with their corresponding critical electric fields in dashed lines. From this graph, one can clearly see that the peak electric field in both diamond and gate-oxide layers is nearly constant along the V ds which demonstrates the benefit of the field-plate. Meanwhile, the peak electric field at the field-plate edge is largely increasing with increasing V ds Despite the improved breakdown voltage, the passivation layer addition underneath the field-plate leads to an increase of the device total oxide capacitance, and, thereby, of the device output capacitance. This aspect is highlighted in figure 8a, where it is displayed the evolution of the output surface capacitance according to the drain-source voltage for the field-plated and non-field-plated diamond lateral device. The corresponding surface used for the calculations is S = (L gs + L g + L gd ) • W, W being the total gate width.

From this graph, it is clearly observed an increase of the output surface capacitance from 0.5 nF/cm² for the non-field-plated device to 5.5 nF/cm² for the field-plate one. Moreover, on this figure, both architectures are compared to a vertical diamond one as similarly described in [START_REF] Couret | Analytic modeling of a hybrid power module based on diamond and sic devices[END_REF]. In that case, the device specific onstate resistance and the output capacitance are calculated using equations ( 3) and ( 4). The doping density and the drift layer thickness are the ones calculated using the 1D impact ionization integral of section 2.

Concerning the active surface of the vertical device it is fixed in order that the on-state resistance of both architectures (vertical and lateral) is identical, thereby leading to similar conduction losses. Compared to the lateral devices, we can see that the output surface capacitance is largely decreasing with increasing drain-source voltage with a minimum value around 3 nF/cm² at |V ds | = 400 V, larger than the estimated output surface capacitance for the lateral device without the field-plate.

R on,s = ρ(T )L drif t (3)
C oss = ϵ R S L drif t BV V ds (4)
where ϵ R is the dielectric permittivity of diamond material, L drif t is the drift layer thickness, S is the active surface and BV is the device breakdown voltage of the vertical device. Based on the estimated output capacitances, the switching energy of the three architectures are calculated in figure 8b. On this graph, one can see that the non-field-plated lateral device exhibits the lowest switching energy with a value of 0.01 µJ at |V ds |=400 V compared to 0.02 µJ for the vertical architecture and 5 µJ for the field-plated one. Therefore, the improvement of the breakdown voltage leads to a degradation of the switching energy due to the increase of the device MOS capacitance. The increase of the MOS capacitance could be further limited by increasing the thickness of the Si 3 N 4 as the corresponding oxide capacitance is inversely proportional to the oxide thickness. However, it will lead to the increase of the peak electric field in the diamond and gate-oxide layers. Once more, a trade-off must be found between those two figure-of-merits.

Evaluation of the Si 3 N 4 electrical properties for field-plate processing

According to the previous simulations, the use of a Si 3 N 4 source field plate on the D3MOSFET should allow to increase the breakdown voltage up to 1kV. But this is true only if the real material properties are the same as the one used in the simulations. The goal in this section is to show that the material, that will be used for the real device, has a dielectric constant close to 7.5 and a breakdown field close to 10 MV/cm. For this purpose, metal insulating metal (MIM) capacitors have been manufactured and characterized. Each device has several sizes of square MIM capacitors and other test devices. Corners are rounded to avoid field peaks. In this paper we will only focus on MIM capacitors with areas between 400x400 µm 2 and 50x50 µm 2 .

Measurements and results

All measurements are made under vacuum (P ≈ 10 -4 mbar) and at room temperature. To obtain the dielectric constant and the breakdown field of Si 3 N 4 two types of measurements are realized. The capacitance is measured to calculate the dielectric constant using equation [START_REF] Pham | mv/cm lateral diamond mosfet[END_REF]. Measurements are done with a Modulab XM MTS at 10 kHz frequency. The capacitance limit detection of this equipment is 0.1 pF.

ϵ = eC ϵ 0 S (5)
In equation ( 5), ϵ is the dielectric constant, e is the thickness of Si 3 N 4 , C is the capacitance, ϵ 0 is the vacuum permittivity, and S is the surface area of the capacitor. Here we make the approximation that the surface area of the capacitor is equal to the top contact surface. Then, the breakdown field is calculated by plotting the current-voltage characteristic of the component until reaching the breakdown voltage and using the equation ( 6). An example of this characteristic is shown on figure 10. Measurements are done with a 2410 Keithley which has a current detection limit of 10 pA.

E c = V BD e (6)
where E c is the breakdown field and V BD is the breakdown voltage. Results presented are the average of what is obtained for the 3 processed samples. According to the measurements made, the dielectric constant is equal to 6.6. Table 1 shows the values obtained for the breakdown electric field. The larger the electrical contact, the lower the breakdown field. This is due to a higher probability to have defects when contacts are larger. According to the results, and the graph of figure 7, the improved D3MOSFET transistor should therefore have a breakdown voltage between 700 V and 900 V which agrees with the targeted values.

Validation of results on diamond substrates

To support the obtained results, MIM capacitors were then performed on two diamond samples with the same procedure as in section 4.1. The aim is to show that the properties of Si With the breakdown electric fields obtained on these samples, it would be theoretically possible to have a device with a breakdown voltage of more than 1000 V. The difference of results between silicon and diamond samples has not yet been explained, but some hypotheses can be done. During the PECVD deposition, constraints appear in the deposited layer and the smaller the sample, the more the layer is constrained.

This could be an explanation. It is also possible that the deposition process is not perfectly stable and that the properties of Si 3 N 4 vary a little from one time to another. Nonetheless, the obtained values for the two different substrate materials validate the suitability of Si 3 N 4 as a source field plate dielectrics.

  et al. (2021 & 2022) N.C. Saha et al. (2020,2021 & 2022)

Figure 2 :

 2 Figure 2: (a) Schematic cross-section of a deep-depletion diamond MOSFET (D3MOSFET) and (b) Resistivity according to the temperature for a D3MOSFET with a boron-doped channel at Na = 8 • 10 16 atom.cm -3 and a compensation level in donor atoms of N d /Na = 4 %.

Figure 3 :

 3 Figure 3: (a) Drain current vs. Gate voltage at T = 450K. Hole concentration and current flowlines in (b) the accumulation mode at Vgs = -10 V, (c) in the on-state mode at Vgs = 10 V and (d) in the off-state mode at Vgs = 40 V.

Figure 4 :

 4 Figure 4: (a) Finite element simulation of the electric field distribution in the off-state with a drain bias of -200 V and a gate bias of 40 V. (b) The lateral electric field profiles on the oxide surface (red) and the top surface of the diamond (black) are extracted from the simulation.

Figure 5 :

 5 Figure 5: Schematic of a source-field-plate D3MOSFET with a passivation layer made of Si 3 N 4 .

FieldFigure 6 :

 6 Figure 6: (a) Finite element simulation of the electric field distribution in the off-state with a drain bias of -200 V and a gate bias of 40 V with highlighted lateral cutlines. Electric field evolution according to the drain extension of the field-plate for different field-plate heights in (b) the diamond cutline, (c) the gate-oxide cutline and (d) the field-plate edge cutline.

4 Figure 7 :

 47 Figure7: Maximum electric field evolution along the V ds within the diamond (black curve), the gate-oxide (red curve) and the passivation layer (blue curve) for a source-field-plate. Is also displayed in dashed lines, the theoretical critical electric field of each material.

2 ]Figure 8 :

 28 Figure 8: (a) Output surface capacitance and (b) the corresponding switching energy according to the drain-source voltage for various device architectures : one vertical architecture and two lateral devices with and without the presence of a field-plate.

4. 1 .

 1 Design and manufacturing process of MIM capacitorMIM capacitors are initially fabricated on three 1 cm 2 silicon subtrates. The first step is the deposition of the bottom metal contacts by evaporation.The metal stack consists of 30 nm Ti, 40 nm Pt and 50 nm Au. Then, 400 nm of Si 3 N 4 are deposited by PECVD (Plasma-Enhanced Chemical Vapor Deposition), this corresponds to one of the values used in the simulations. Finally, the top metal contacts are deposited with a standard lift off process. Same metals have been used than the bottom contact. Figure9shows a schematic cross-section of the MIM capacitors (a) and an optical micro-graph of one of the samples (b).

Figure 9 :

 9 Figure 9: Design of MIM capacitors : (a) Schematic cross-section of MIM capacitors, (b) MIM capacitor device optical micrograph.

3 N 4

 34 stay the same on diamond substrate Ib and IIb 4x4 mm². They only have 50x50µm² MIM capacitors. Current versus voltage characteristics of a MIM capacitor is shown on the figure 10.

Figure 10 :

 10 Figure 10: Current versus electric field characteristics for 50 x 50 µm² MIM capacitor on diamond and silicon substrates.

Table 1 :

 1 The results obtained are added to table 1. Summary table of the results obtained on the electric fields of breakdown of Si 3 N 4 on silicon and diamond samples.

		Breakdown electric field [MV/cm]
	Metal contact sizes [µm x µm] Si samples Ib Diamond samples
	400 x 400	7.9 ± 0.2
	200 x 200	8.3 ± 0.2
	100 x 100	8.8 ± 0.1
	50 x 50	9.1 ± 0.3	10.7 ± 0.3

Preprint submitted to Diamond and Related MaterialsDecember 1, 2022

Acknowledgments

This project has received funding from the Clean Sky 2 Joint Undertaking (JU) under grant agreement No 101007868. The JU receives support from the European Union's Horizon 2020 research and innovation program and the Clean Sky 2 JU members other than the Union.

Data availability

The figures raw data presented in this manuscript are available on Zenodo [START_REF] Couret | Field-Plated D3MOSFET design for breakdown voltage improvement[END_REF].

Conclusion

The feasibility of a lateral field-plated deep depletion diamond MOSFET has been investigated. The study was based on 2D TCAD simulations in order to determine the geometrical features of the field-plate.

The simulations results show a potential breakdown voltage between 700 V to 900 V, close to the targeted value of 1 kV. Investigations were also performed regarding the passivation layer made of Si The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


N Castin
email: ncastin@sckcen.be

G Bonny

M J Konstantinović

A Bakaev

F Bergner

C Courilleau

C Domain

B Gómez-Ferrer

J M Hyde

L Messina

G Monnet

M I Pascuet

B Radiguet

M Serrano

L Malerba

Radiation-induced embrittlement of nuclear steels is one of the main limiting factors for safe long-term operation of nuclear power plants. In support of accurate and safe reactor pressure vessel (RPV) lifetime assessments, we developed a physics-based model that predicts RPV steel hardening and subsequent embrittlement as a consequence of the formation of nano-sized clusters of minor alloying elements. This model is shown to provide reliable assessments of embrittlement for a very wide range of materials, with higher accuracy than industrial correlations. The core of our model is a multiscale modelling tool that predicts the kinetics of solute clustering, given the steel chemical composition and its irradiation conditions. It is based on the observation that the formation of solute clusters ensues from atomic transport driven by radiation-induced mechanisms, differently from classical nucleation-and-growth theories. We then show that the predicted information about solute clustering can be translated into a reliable estimate for radiation-induced embrittlement, via standard hardening laws based on the dispersed barrier model. We demonstrate the validity of our approach by applying it to hundreds of nuclear reactors vessels from all over the world.

Multiscale modelling in nuclear ferritic steels: from nano-sized defects to embrittlement.

Introduction

Nuclear power plant lifetime extension and new builds are assets to fight climate change [START_REF]IEA report, Nuclear Power in a Clean Energy System[END_REF]. The main lifetime limiting factor for nuclear power plants is the integrity of the steels composing the pressure vessel, as their mechanical properties degrade under neutron irradiation. Reactor pressure vessel (RPV) steels in "western" nuclear power plants, as well as their Russian counterpart, "VVER" steels, become increasingly brittle with increasing irradiation dose at operating temperature [START_REF] Odette | On the history and status of reactor pressure vessel steel ductile to brittle transition temperature shift prediction models[END_REF]. Ferritic steels are also foreseen in new generations of nuclear reactors, where they will be subjected to harsh environments never experienced at large industrial scale before, noticeably in terms of temperature and neutronic dose. Ferritic/martensitic (F/M) steels, candidate materials for fourth generation fission reactors and fusion applications [START_REF] Zinkle | structural materials for fission & fusion energy[END_REF][START_REF] Fazio | European cross-cutting research on structural materials for Generation IV and transmutation systems[END_REF][START_REF] Cabet | Ferritic-martensitic steels for fission and fusion applications[END_REF], are also subjected to radiation-induced embrittlement. Systematic microstructural investigations on all these steels to determine the features of potentially embrittling defects are clearly not viable. By understanding the fundamental processes that take place in materials under irradiation [START_REF] Samaras | Modelling in nuclear energy environments[END_REF], however, it is possible to develop models that provide this information and enable high quality predictions, which can be used in support of degradation monitoring procedures and safe plant lifetime extension. A model of this type is proposed in this work.

The most challenging aspect of the problem is that it all starts at the nanometer [START_REF] Marquis | Nuclear reactor materials at the atomic scale[END_REF], or even at the atomic [START_REF] Nordlund | Improving atomic displacement and replacement calculations with physically realistic damage models[END_REF], scales, with consequences on the integrity of massive components that weigh several tons. It is indeed well-established that the degradation of most structural materials under irradiation unfolds from the formation of nano-sized defects (diameter D = 1-4 nm) that are finely dispersed in the bulk (number density N = 10 22 -10 24 m -3 ) and cause hardening and ensuing embrittlement, at temperatures relevant to operating light water reactors (LWRs) [START_REF] Zinkle | structural materials for fission & fusion energy[END_REF][START_REF] Bergner | Application of a three-feature dispersed-barrier hardening model to neutron-irradiated Fe-Cr model alloys[END_REF]. Such defects result from the diffusion and interaction of radiationgenerated point defects between themselves, with the pre-existing microstructure and with alloying elements and impurities [START_REF] Nordlund | Improving atomic displacement and replacement calculations with physically realistic damage models[END_REF][START_REF] Fu | Multiscale modelling of defect kinetics in irradiated iron[END_REF]. In the specific case of ferritic steels, atomic transport mechanisms [START_REF] Messina | Systematic electronic-structure investigation of substitutional impurity diffusion and flux coupling in bcc iron[END_REF][START_REF] Messina | Solute diffusion by selfinterstitial defects and radiation-induced segregation in ferritic Fe-X (X=Cr, Cu, Mn, Ni, P, Si) dilute alloys[END_REF] enrich point-defect clusters with Ni, Mn, Si, P and Cu [START_REF] Castin | The dominant mechanisms for the formation of solute-rich clusters in low-Cu steels under irradiation[END_REF]. Except in the case of the last element, enrichment is largely proportional to the abundance of each element in the matrix. This is revealed by atom probe tomography (APT) techniques [START_REF] Marquis | Nuclear reactor materials at the atomic scale[END_REF][START_REF] Castin | The dominant mechanisms for the formation of solute-rich clusters in low-Cu steels under irradiation[END_REF][START_REF] Almirall | Precipitation and hardening in irradiated low alloy steels with a wide range of Ni and Mn compositions[END_REF][START_REF] Soneda | Embrittlement Correlation Method for the Japanese Reactor Pressure Vessel Materials[END_REF][START_REF] Miller | Evolution of the nanostructure of VVER-1000 RPV materials under neutron irradiation and post irradiation annealing[END_REF][START_REF] Miller | Atom probe tomography characterizations of high nickel, low copper surveillance RPV welds irradiated to high fluences[END_REF], as illustrated in Fig. 1, and by small angle neutron scattering (SANS) techniques [START_REF] Mühlbauer | Magnetic small-angle neutron scattering[END_REF]. These nano-sized clusters of solute elements obstruct the motion of dislocations, thereby hindering the blunting of crack tips. The dispersed-barrier hardening model [START_REF] Seeger | On the theory of radiation damage and radiation hardening[END_REF] relates the increase y of the yield strength (hardening) with √𝑁𝐷 of the nano-clusters. In turn, the shift 𝛥𝑇 in the ductile-to-brittle transition temperature (DBTT) is correlated to y [START_REF] Nanstad | Some Useful Mechanical Property Correlations for Nuclear Reactor Pressure Vessel Steels[END_REF]. (The DBTT is the main parameter that is used at industrial level to define the lifetime of the vessel.) Thus, if N and D are known for a given steel, irradiated under known conditions, the corresponding 𝛥𝑇 can be estimated. [START_REF] Miller | APT characterization of high nickel RPV steels[END_REF][START_REF] Miller | APT characterization of irradiated high nickel RPV steels[END_REF][START_REF] Kuleshova | Degradation of RPV steel structure after 45 years of operation in the VVER-440 reactor[END_REF], while closed symbols denote NiMnSiPCu-rich precipitates in 74 different materials [START_REF] Castin | The dominant mechanisms for the formation of solute-rich clusters in low-Cu steels under irradiation[END_REF][START_REF] Almirall | Precipitation and hardening in irradiated low alloy steels with a wide range of Ni and Mn compositions[END_REF][START_REF] Miller | Atom probe tomography characterizations of high nickel, low copper surveillance RPV welds irradiated to high fluences[END_REF][START_REF] Almirall | On the use of charged particles to characterize precipitation in irradiated reactor pressure vessel steels with a wide range of compositions[END_REF][START_REF] Auger | Synthesis of atom probe experiments on irradiation-induced solute segregation in French ferritic pressure vessel steels[END_REF][START_REF] Dohi | Effect of Neutron Flux at High Fluence on Microstructural and Hardness Changes of RPV Steels[END_REF][START_REF] Edmondson | Using complimentary microscopy methods to examine Ni-Mn-Si precipitates in highly-irradiated reactor pressure vessel steels[END_REF][START_REF] Styman | Postirradiation annealing of Ni-Mn-Si-enriched clusters in a neutron-irradiated RPV steel weld using Atom Probe Tomography[END_REF][START_REF] Wells | The Character, Stability and Consequences of Mn-Ni-Si Precipitates in Irradiated Reactor Pressure Vessel Steels[END_REF][START_REF] Jenkins | The effect of composition variations on the response of steels subjected to high fluence neutron irradiation[END_REF][START_REF] Takeuchi | Effects of chemical composition and dose on microstructure evolution and hardening of neutron-irradiated reactor pressure vessel steels[END_REF][START_REF] Lambrecht | On the correlation between irradiation-induced microstructural features and the hardening of reactor pressure vessel steels[END_REF][START_REF] Kuleshova | Mechanisms of radiation embrittlement of VVER-1000 RPV steel at irradiation temperatures of (50-400)°C[END_REF][START_REF] Odette | Precipitation in reactor pressure vessel steels under ion and neutron irradiation: On the role of segregated network dislocations[END_REF]. Note that Cu is the only element that strongly deviates from the linear correlation with the element abundance in the alloy. Mn also slightly deviates because it is a carbide former, so part of the content of this element is found in carbides.

Thermodynamic-driven precipitation of Cu and Cr occurs under irradiation below 300°C in materials containing more than ~0.1at.% Cu or ~9at.% Cr [START_REF] Cabet | Ferritic-martensitic steels for fission and fusion applications[END_REF][START_REF] Buswell | Irradiation-induced microstructural changes, and hardening mechanisms, in model PWR reactor pressure vessel steels[END_REF]. In this class of materials, Cu-rich or Cr-rich precipitates are a clear cause of embrittlement [START_REF] Fisher | Identification of the Precipitate Accompanying 885°F Embrittlement in Chromium Steels[END_REF] and thermodynamic models can be largely used to predict their formation. We address here the formation of clusters rich in Ni, Mn, Si and P in materials where no element exceeds the solubility limit of the binary alloy. The mechanism of solute cluster formation in this case is still debated. Is it an irradiation-enhanced process, dictated by thermodynamics [START_REF] Wells | Evolution of Manganese-Nickel-Silicon-Dominated Phases in Highly Irradiated Reactor Pressure Vessel Steels[END_REF][START_REF] Ke | Thermodynamic and kinetic modeling of Mn-Ni-Si precipitates in low-Cu reactor vessel steels[END_REF], or an irradiationinduced process that would not occur without irradiation [START_REF] Meslin | Characterization of neutron-irradiated ferritic model alloys and a RPV steel from combined APT, SANS, TEM and PAS analyses[END_REF][START_REF] Huang | Atom Probe Tomography characterization of the microstructural evolution of a low copper reactor pressure vessel steel under neutron irradiation[END_REF]? Recently, Ke et al. [START_REF] Ke | Thermodynamic and kinetic modeling of Mn-Ni-Si precipitates in low-Cu reactor vessel steels[END_REF] developed a model based on the assumption that irradiation mainly accelerates the thermodynamics-driven kinetics of phase separation, due to the increased vacancy concentration. This model also included a corrective factor to account for possible "radiation-induced heterogeneous nucleation" of precipitates on point defect clusters produced in atomic displacement cascades. The application of the model revealed that the latter term is dominant by several orders of magnitude over the thermodynamic component, in some cases. This suggests that thermodynamics-driven processes are not dominant under irradiation.

APT provides several notable clues for the dominant radiation-induced driving force for solute clustering (Fig. 1): a) Segregation is ubiquitous: in addition to finely dispersed clusters in the bulk, high concentrations of solutes (especially P [START_REF] Marquis | Atomic Scale Analysis of Solute Behaviours by Atom-Probe Tomography[END_REF]) occur along dislocation segments, at carbides and at grain boundaries. b) Two classes of solute clusters can be distinguished by their chemical compositions. The composition of many solute clusters strongly correlates with the overall composition of the material (closed symbols in Fig. 1b). In contrast, in materials with high enough Cu or Cr content, Cu-rich [START_REF] Miller | APT characterization of high nickel RPV steels[END_REF][START_REF] Buswell | Irradiation-induced microstructural changes, and hardening mechanisms, in model PWR reactor pressure vessel steels[END_REF] (open symbols in Fig. 1b) or Cr-rich [START_REF] Cabet | Ferritic-martensitic steels for fission and fusion applications[END_REF] clusters are found. c) Clusters are dilute, as they contain a significant fraction of Fe [START_REF] Almirall | Precipitation and hardening in irradiated low alloy steels with a wide range of Ni and Mn compositions[END_REF][START_REF] Miller | Evolution of the nanostructure of VVER-1000 RPV materials under neutron irradiation and post irradiation annealing[END_REF][START_REF] Miller | Atom probe tomography characterizations of high nickel, low copper surveillance RPV welds irradiated to high fluences[END_REF]. APT often finds iron contents ≥50% [START_REF] Edmondson | Using complimentary microscopy methods to examine Ni-Mn-Si precipitates in highly-irradiated reactor pressure vessel steels[END_REF], while other techniques (electron microscopy, SANS) [START_REF] Mühlbauer | Magnetic small-angle neutron scattering[END_REF][START_REF] Edmondson | Using complimentary microscopy methods to examine Ni-Mn-Si precipitates in highly-irradiated reactor pressure vessel steels[END_REF] suggest instead a content 20% Fe. These clues advocate for a continuous process of solute deposition on a large density of nucleation sites, triggered and sustained by the continuous fluxes of vacancies and selfinterstitials due to irradiation. This mechanism is different from classical precipitation by nucleation and growth. It is the theoretical foundation that we rely on to present in this work a model describing the formation of NiMnSiPCu-rich clusters under irradiation.

The nuclear industry typically estimates the evolution of 𝛥𝑇 with the received neutron dose, for given steel and irradiation conditions, employing empirical regressions [START_REF] Soneda | 11 -Embrittlement correlation methods to identify trends in embrittlement in reactor pressure vessels (RPVs)[END_REF] that are fitted to experimental data. Even though these correlations are inspired by physics (e.g., explicit terms stand for the so-called "matrix damage", others for the effects of solute-rich clusters, etc.), their extrapolation capabilities depend entirely on the choice of the mathematical expression that is used. In this work, we aim at setting the bases for the definition of models that are more strongly rooted in physics. For this purpose, we develop a multiscale tool that describes the evolution of the microstructure under irradiation. We combine first principles calculations, molecular dynamics simulations and atomistic modelling within a single coarse-grained tool which links the atomic to the nanometer scales, allowing the simulated time to be extended up to decades of neutron irradiation. The tool is therefore driven by elementary, atomic-level reactions, which implement the abovediscussed evidence for radiation-induced solute clustering. This tool is a generalization of a previous model [START_REF] Castin | The dominant mechanisms for the formation of solute-rich clusters in low-Cu steels under irradiation[END_REF] which, given material composition and irradiation conditions, predicts the evolution of N and D, and thus of the related cluster volume fraction, f, with the received irradiation dose. We then employ classical hardening laws, also rooted in physics, to correlate this information with the corresponding 𝛥𝜎 and the ensuing 𝛥𝑇. This step can be regarded as a shortcut in the classical multiscale approach [START_REF] Samaras | Modelling in nuclear energy environments[END_REF] towards the desired macroscopic level, as we replace mesoscale methods such as dislocation dynamics, continuous field models or finite elements methods with simpler laws and the correlation between hardening and embrittlement. However, this route remains open for future work based on the knowledge of N and D.

Materials and Methods

Materials and experimental databases

This work is focused on the following families of materials:

 Base materials RPV steels: The pressure vessels of LWR, i.e. pressurized (P) and boiling (B) water reactors (P/BWR) are made of low carbon (<0.9-1.3 at%), lowalloyed (> 95 at% Fe) bainitic steels. The main alloying elements common to all RPV steels are Mo (~0.3 at%), Ni (~0.5-1.0 at%), Mn (~0.5-1.5 at%), and Si (~0.2-0.8 at%), chiefly added for strengthening purposes. We consider here experimental data from samples irradiated at different dose rates: low and medium, corresponding to surveillance samples (~2.5•10 -10 dpa/s and ~3•10 -9 dpa/s), and high, irradiated in a material test reactor, MTR (~10 -7 dpa/s).  Weld materials RPV steels: these materials are also bainitic steels, but their composition is different, typically they are characterized by higher Ni and Mn content and more impurities; in particular, this class includes steels with very highfor RPV -Ni and Mn content, i.e. up to 2 at%. The irradiation conditions considered here are similar to those of the RPV-BM.  Materials used for the RPV of Russian design nuclear reactors (VVER): these are bainitic steels that contain, in addition to the alloying elements added to western Europe RPV steels, also Cr (~2.0-2.5 at%) and V (0.1-0.35 at%), to provide higher thermal ageing resistance through formation of several types of Cr-and Vcontaining carbides and carbonitrides. The Ni content varies in a wider range for these materials, as it can be either relatively high (~1.5 at%) or very low (< 0.1 at%).

The samples considered here were irradiated in MTR at high dose rate (~10 -7 dpa/s) and in nuclear power plants at low dose rates (10 -10 to 10 -9 dpa/s).

Two databases of experimental data were collected from the literature for the calibration and validation of our model:  The microstructure database collects 104 experimental measurements with APT [13, 14, 16, 17, 21, 25-31, 32, 33, 39, 40, 43-49] from 55 irradiated materials (46 RPV steels, 6 VVER steels and 3 model alloys for RPV steels). For each case, there are 9 input data: the materials chemical composition (Ni, Mn, Si, P and Cu contents in at%), the irradiation conditions (temperature in °C and the dose rate in dpa/s), and the received neutron dose (in dpa, which is equivalent to the irradiation time given the dose rate). The output data are N and D, and therefore f, of the solute clusters found in bulk, with the associated error bars, measured with APT when available.  The embrittlement database collects 1707 experimental measurements from 359 irradiated materials (2 model alloys, 347 RPV steels, 10 VVER steels). Most of the data is taken from the PLOTTER database of international nuclear power surveillance [START_REF]Adjunct for E900-15 Technical Basis for the Equation Used to Predict Radiation-Induced Transition Temperature Shift in Reactor Vessel Materials[END_REF], which is complemented with data found in the literature: see the microstructure database and the references provided above, further complemented with VVER data [START_REF] Ulbricht | Small angle neutron scattering analysis of the radiation susceptibility of reactor pressure vessel steels[END_REF] and recent embrittlement data derived for high Ni model RPV steels [START_REF] Kolluri | Influence of Ni-Mn contents on the embrittlement of PWR RPV model steels irradiated to high fluences relevant for LTO beyond 60 years[END_REF]. Overall, we only retained the materials that correspond to a Cu content < 0.1 at%. The input data are identical as in the microstructure database, while the output data is the embrittlement ∆𝑇 in °C, and/or the hardening Y in MPa. The specific range of uncertainty associated with each of these measurements should ideally be provided case by case, from the experimental procedure that is applied. Since precise information is scarce, we assume that the uncertainty on ∆𝑇 is ± 25°C.

Both databases are provided as supplementary materials. In both cases, original neutron fluence and neutron flux data provided in n cm -2 or in n cm -2 s -1 , respectively, are converted into displacement-per-atom, dpa, or dpa s -1 using a constant conversion factor: 1 dpa = 6.67 10 20 n cm -2 [START_REF] Jones | Influence of manganese on mechanical properties, irradiation susceptibility and microstructure of ferritic steels, alloys and welds[END_REF]. In addition, this work aims at predicting the embrittlement ∆𝑇 induced by radiation, but experimental data found in the literature often concern the increase in the yield strength (hardening) y instead. As both are correlated [START_REF] Nanstad | Some Useful Mechanical Property Correlations for Nuclear Reactor Pressure Vessel Steels[END_REF], hardening data is here converted into an estimated embrittlement value with the relation:

∆𝑇(°C) = 0.5913 ∆𝜎 𝑦 (MPa) (1)
The coefficient of proportionality was derived from the experimental data in the microstructure database. It is clear that the use of a constant conversion factor both from fluence to dpa and from hardening to embrittlement will be a source of scatter; however, as will be seen, the overall uncertainty remains acceptable.

Multiscale model for microstructure evolution under irradiation

In this section, we describe our physical model aimed at predicting N, D and f of the solute clusters, as functions of the material chemical composition (Ni, Mn, Si, P and Cu contents in at%), the irradiation conditions (temperature and dose rate), and the received neutron dose (equivalent to the irradiation time, given the dose rate).

The novelty of our approach compared to standard theories (such as, e.g., in [START_REF] Ke | Thermodynamic and kinetic modeling of Mn-Ni-Si precipitates in low-Cu reactor vessel steels[END_REF]) is to explicitly account for the radiation-induced mechanisms discussed above, relying on elementary processes of diffusion, transport and binding at the atomic level. It is implemented through the coarse-grained object kinetic Monte Carlo (OKMC) computational method: see the schematic representation in Fig. 2. The fundaments of the model were defined in [START_REF] Castin | The dominant mechanisms for the formation of solute-rich clusters in low-Cu steels under irradiation[END_REF], where we demonstrated that the dominant mechanism for nano-sized solute cluster formation is the interaction of solute atoms with self-interstitial defects generated under irradiation. The OKMC simulation volume (in this work 270 a0 × 300 a0 × 420 a0, thus including 68 million atoms in a perfect bcc lattice) is initially filled with the number of Ni, Mn, Si, P and Cu atoms that correspond to the targeted chemical composition. It is noteworthy that Cr atoms, that are present in the VVER steels in significant proportion, are not explicitly included in the simulation box, but their effect on the diffusivity of SIA defects is taken into account via an established grey alloy approximation [START_REF] Dubinko | Effect of Si-Ni-P on the emergence of dislocations loops in Fe-9Cr matrix under neutron irradiation: TEM study and OKMC modelling[END_REF][START_REF] Chiapetto | Effect of Cr content on the nanostructural evolution of irradiated ferritic/martensitic alloys: An object kinetic Monte Carlo model[END_REF]. Next, primary point-defects (vacancies, self-interstitials and their clusters) are injected at the right frequency (according to the specified neutron dose rate) using libraries of atomic collision cascade debris previously derived from molecular dynamics. Key features of the model are solute transport phenomena induced by the injected point-defects, and their deposition at sinks. When single vacancies or selfinterstitials encounter solute atoms dissolved in the matrix, they form stable pairs, with given binding energies. These pairs migrate with given activation energies towards sinks.

The characteristic energy values were calculated using density functional theory (DFT) methods [START_REF] Messina | Systematic electronic-structure investigation of substitutional impurity diffusion and flux coupling in bcc iron[END_REF][START_REF] Messina | Solute diffusion by selfinterstitial defects and radiation-induced segregation in ferritic Fe-X (X=Cr, Cu, Mn, Ni, P, Si) dilute alloys[END_REF]. In addition to pre-existing microstructural features, such as dislocations and grain boundaries, point-defect clusters also act as sinks, especially self-interstitial atom clusters (denoted as C1 in Fig. 2). These are immobilized because of strong binding interactions with solute atoms accumulated round them, thereby acting as nuclei for further solute clustering. While C1 self-interstitial complexes can be safely considered as indissoluble, the dissolution of C2 vacancy-solute complexes (Fig. 2) needs an appropriate description. The corresponding reactions are depicted in Fig. 3. Dissolution is a thermally activated process that takes place via two possible paths: emission of a single vacancy (henceforth denoted as "Va"), or emission of a vacancy-solute pair (henceforth "VaSo", where So is either Ni, Mn, Si, P or Cu). We estimate the dissociation energy 𝐸 𝑑𝑖𝑠𝑠 (𝑉𝑎,𝑉𝑎𝑆𝑜) with the well-established approximate relation:

𝐸 𝑑𝑖𝑠𝑠 (𝑉𝑎,𝑉𝑎𝑆𝑜) = 𝐸 𝑚 (𝑉𝑎,𝑉𝑎𝑆𝑜) + 𝐸 𝑏 (𝑉𝑎,𝑉𝑎𝑆𝑜) (2)
Here, 𝐸 𝑚 (𝑉𝑎,𝑉𝑎𝑆𝑜) is the migration energy of a Va or VaSo pair in bulk Fe, which is known from DFT calculations [START_REF] Messina | Systematic electronic-structure investigation of substitutional impurity diffusion and flux coupling in bcc iron[END_REF]. In contrast, the binding energy 𝐸 𝑏 (𝑉𝑎,𝑉𝑎𝑆𝑜) of a Va or a VaSo pair to C2 complexes requires special attention. In the absence of a theoretical framework, the approach of Ref. [START_REF] Castin | The dominant mechanisms for the formation of solute-rich clusters in low-Cu steels under irradiation[END_REF] can be used: they are determined by individual fitting, taking, for each given material, the binding energy that allows the OKMC model predictions for N, D and f to best match the experimental values from APT. However, in this way the model can only be applied to materials for which detailed microstructural data are available, such as those in the microstructure database described in section 2.1. It cannot be applied to the nuclear power plant surveillance materials collected in the PLOTTER database [START_REF]Adjunct for E900-15 Technical Basis for the Equation Used to Predict Radiation-Induced Transition Temperature Shift in Reactor Vessel Materials[END_REF] because, to our knowledge, no APT analysis is available for these materials. The goal of the present work is to make such microstructural analysis unnecessary for the application of the model, given that in most cases it is not given: the model should be applicable with input variables that are always given, i.e. material composition and irradiation conditions. Therefore, as described in section 2.3, in this work we remove the limits of applicability of the OKMC method and propose, based on atomistic modelling and first principle calculations, a general recipe to estimate 𝐸 𝑏 𝑉𝑎 and 𝐸 𝑏 𝑉𝑎𝑆𝑜 a priori, as functions of the chemical composition and the size of the solute clusters, as schematically shown in Fig. 3 and described next.

Determination of key binding energies from atomistic modelling.

Small vacancy-solute complexes located on the left-hand side of Fig. 3 have been studied using DFT [START_REF] Castin | The dominant mechanisms for the formation of solute-rich clusters in low-Cu steels under irradiation[END_REF]. On the right-hand side of the figure, the other limiting case is a large complex, where the effects of the interface with the Fe matrix do not evolve with size any longer. The corresponding "saturated" binding energy values are denoted as 𝐸 𝑏 * 𝑉𝑎 and 𝐸 𝑏 * 𝑉𝑎𝑆𝑜 .

Intermediate cases can be interpolated using the 𝑉 𝐶2 -1/3 capillary law, where VC2 is the volume of the complex C2. The OKMC model is therefore fully parameterized once 𝐸 𝑏 * 𝑉𝑎 and 𝐸 𝑏 * 𝑉𝑎𝑆𝑜 are known, which solely depend on the chemical composition of the solute clusters.

To calculate 𝐸 𝑏 * 𝑉𝑎 and 𝐸 𝑏 * 𝑉𝑎𝑆𝑜 , we need a Hamiltonian for an atomic system that contains the Fe, Ni, Mn, Si, Cu and P species. This is offered by the DFT-rooted cluster expansion (CE) model derived in [START_REF] Bonny | The impact of alloying elements on the precipitation stability and kinetics in iron based alloys: An atomistic study[END_REF]. Given a chemical composition of the C2 complex, we first evaluated 𝐸 𝑏 * 𝑉𝑎 and 𝐸 𝑏 * 𝑉𝑎𝑆𝑜 using the method developed in [START_REF] Bonny | Exact mean field concept to compute defect energetics in random alloys on rigid lattices[END_REF], which enables the calculation of defect energetics in random alloys on a rigid lattice based on an exact mean field approach. This corresponds to assuming that the atoms are randomly distributed inside the complexes, which is a fair assumption, given their mechanism of formation by segregation on pointdefect clusters rather than via precipitation as phases. We applied this method for the materials listed in the microstructure database. Regarding the evidence shown in Fig. 1b, we assume that the relative chemical contents in Ni, Mn, Si, P and Cu are identical to those derived from the overall chemical content of the material. Finally, the Fe content in the solute clusters can be varied between 20 at% and 50 at% to cover the limiting cases between experimental evidence from SANS and APT, respectively.

Determination of embrittlement from solute clustering

According to the dispersed barrier hardening model [START_REF] Seeger | On the theory of radiation damage and radiation hardening[END_REF], the radiation hardening ∆𝜎 𝑦 is a function of the density and the size of the array of obstacles to the motion of dislocations. In this work, we consider that the obstacles are the nano-sized solute clusters and apply the model originally proposed by Bacon, Kocks and Scattergood (BKS) [START_REF] Bacon | The effect of dislocation self-interaction on the orowan stress[END_REF], as formulated in [START_REF] Monnet | Multiscale modeling of irradiation hardening: Application to important nuclear materials[END_REF], which considers the elastic interactions between segments of bowed-out dislocations by a population of randomly distributed obstacles. Accounting for Eq (1), we write:

∆𝑇 (𝐵𝐾𝑆) = 0.5913 𝑀 𝛼 𝐵𝐾𝑆 3/2 ( 𝑙𝑛(𝐷 𝑏 ⁄ ) 𝑙𝑛(𝑙 𝑏 ⁄ ) ) 3/2 𝑙𝑛(𝑙 𝑏 ⁄ ) 2𝜋 𝜇𝑏√𝑁𝐷 (3)
Here, M is Taylor's factor (taken to be 3), l is defined as the free distance between solute clusters (𝑙 = 1/√𝑁𝐷 -𝐷), 𝐷 is a harmonic sum (𝐷 = 𝑙𝐷 / (𝑙 + 𝐷)),  is the shear modulus and b is the dislocation's Burgers vector module. The factor 𝛼 𝐵𝐾𝑆 accounts for the strength of solute clusters impeding the flow of dislocations under the applied strain. Following Friedel's interpretation [START_REF] Monnet | Multiscale modeling of irradiation hardening: Application to important nuclear materials[END_REF], 𝛼 𝐵𝐾𝑆 3/2 can be directly related to the cosine of the bowing angle of the dislocations.

The solute clusters diameter D is typically observed to vary within narrow ranges of values in databases of irradiated ferritic steels, at least in the range of temperatures and neutron doses that were studied the most. Considering that the volume fraction f ≈ ND 3 and therefore √𝑁𝐷 ≈ √𝑓/D, D being almost constant, for simplicity and seeking for more accurate regressions, some authors [START_REF] Almirall | Precipitation and hardening in irradiated low alloy steels with a wide range of Ni and Mn compositions[END_REF][START_REF] Soneda | Embrittlement Correlation Method for the Japanese Reactor Pressure Vessel Materials[END_REF][START_REF] Wagner | Influence of the copper impurity level on the irradiation response of reactor pressure vessel steels investigated by SANS[END_REF] preferred to rely on this measurement of solute clustering directly. We hence write:

∆𝑇 (𝑉𝐹) = 𝛼 𝑉𝐹 √𝑓 (4)
Here, the superscript (VF) stands for "volume-fraction-based model". Given our embrittlement database described in section 2.1 which provides values for ∆𝑇, and having √𝑁𝐷 and √𝑓 either provided by our microstructure database or estimated by our theoretical model described in section 2.3, numerical values for 𝛼 𝐵𝐾𝑆 and 𝛼 𝑉𝐹 can be estimated from linear regressions. We should emphasize that, for the sake of practicality, we adopt here a specific convention to define the volume fraction f, due to the fact that the latter is not systematically reported in APT examination work. We thus define 𝑓 = 𝜋 6 𝑁𝐷 3 , irrespective of whether N and D are calculated using the OKMC model, or obtained from APT measurement.

3 Results

Calibration and validation of the microstructure evolution model

Following the methodology described in section 2.3, we performed atomistic calculations of 𝐸 𝑏 * 𝑉𝑎 and 𝐸 𝑏 * 𝑉𝑎𝑆𝑜 for the materials listed in the microstructure database described in section 2.1. We found that the Fe content can be varied between 20 at% and 50 at% without qualitatively affecting the results. We also found that 𝐸 𝑏 * 𝑉𝑎 and 𝐸 𝑏 * 𝑉𝑎𝑆𝑜 are strongly correlated with the relative Ni content in the materials, while the content in the other chemical elements play only a secondary role. Looked from that perspective, the values fall within a narrow range of variability. This is a crucial result of the physical atomistic model, as it greatly simplifies the form of the correlation we are looking for. As shown in Fig. 4, it is accordingly possible to write the following estimate functions, which hold for all chemical species (S = Ni, Mn, Si, P or Cu):

𝐸 𝑏 * 𝑉𝑎 | 𝐶𝐸 ≅ 1.2 -0.4 𝑥 𝑁𝑖 ± 0.01 (eV) (5)
𝐸 𝑏 * 𝑉𝑎𝑆𝑜 | 𝐶𝐸 ≅ 1.56 -0.85 𝑥 𝑁𝑖 ± 0.15 (1 -𝑥 𝑁𝑖 ) (eV) (6)
Here, 𝑥 𝑁𝑖 is the relative Ni content defined as CNi / (CNi + CMn + CSi + CP + CCu) and CNi,…,CCu are the chemical contents in the materials. The difference between the 𝐸 𝑏 * 𝑉𝑎 | 𝐶𝐸 curve and the 𝐸 𝑏 * 𝑉𝑎𝑆𝑜 | 𝐶𝐸 curve for a given Ni content is consistent with estimates deduced from PAS experiments [START_REF] Konstantinović | Radiation induced solute clustering in high-Ni reactor pressure vessel steel[END_REF]. The estimates in Eq. (5-6) could in principle be used directly to parameterize OKMC simulations. However, the superposition of independent migration and binding energies in Eq. ( 2) does not account for dynamic effects that may affect the true value of 𝐸 𝑑𝑖𝑠𝑠 (𝑉𝑎,𝑉𝑎𝑆𝑜) . To improve accuracy without undue complications, corrections are introduced only on 𝐸 𝑏 * 𝑉𝑎𝑆𝑜 . In the OKMC model, we therefore assume that:

𝐸 𝑏 * 𝑉𝑎 | 𝑂𝐾𝑀𝐶 = 𝐸 𝑏 * 𝑉𝑎 | 𝐶𝐸 (7)
In contrast, a better estimate for 𝐸 𝑏 * 𝑉𝑎𝑆𝑜 is derived from another function of the material chemical composition, determined as follows. OKMC simulations were performed for the materials listed in the microstructure database, in the corresponding irradiation conditions. In each case, 𝐸 𝑏 * 𝑉𝑎 was taken from Eq. ( 7) and 𝐸 𝑏 * 𝑉𝑎𝑆𝑜 was varied between 𝐸 𝑏 * 𝑉𝑎 | 𝐶𝐸 and 𝐸 𝑏 * 𝑉𝑎𝑆𝑜 | 𝐶𝐸 , used as reference physical values, to determine the one that leads to the closest predictions of the solute clusters' volume fraction f, as compared to the APT data. The sofitted values for individual materials are shown by dots in Fig. 4. The error bars indicate the typical range of tolerance for 𝐸 𝑏 * 𝑉𝑎𝑆𝑜 to achieve a good match with APT. Clearly, the optimal 𝐸 𝑏 * 𝑉𝑎𝑆𝑜 values can be approximated by a regression of 𝑥 𝑁𝑖 , similarly to the cluster expansion finding in Eq. ( 5) and Eq. ( 6). We therefore derived an overall single regression that relates 𝐸 𝑏 * 𝑉𝑎𝑆𝑜 with 𝑥 𝑁𝑖 , which minimizes the average prediction error by the OKMC model for √𝑓 within the whole microstructure database, as follows: With Eqs [START_REF] Marquis | Nuclear reactor materials at the atomic scale[END_REF][START_REF] Nordlund | Improving atomic displacement and replacement calculations with physically realistic damage models[END_REF][START_REF] Bergner | Application of a three-feature dispersed-barrier hardening model to neutron-irradiated Fe-Cr model alloys[END_REF][START_REF] Fu | Multiscale modelling of defect kinetics in irradiated iron[END_REF][START_REF] Messina | Systematic electronic-structure investigation of substitutional impurity diffusion and flux coupling in bcc iron[END_REF], the OKMC model is fully parameterized, and can therefore be applied to assess solute clustering in any RPV material (i.e. of given composition in terms of minor solutes), under any irradiation condition. Fig. 5 shows a comparison between the predictions achieved with the model and APT data for the materials listed in the microstructure database. In Fig. 5a, we see that a reasonable correlation exists between the predicted and experimental volume fraction f of solute clusters, despite large scatter with the largest values. In Fig. 5b and Fig. 5c, we see that the OKMC model achieves this reasonable prediction of f by slightly overestimating the experimental N and, in general, underestimating the average D. The latter is predicted to vary within a much narrower range than observed experimentally. We interpret this discrepancy as the result of an expectable divergence in resolution between, on the one hand, an experimental technique where the smallest defects are invisible (APT), and, on the other, a theoretical model where all defects are accounted for.

𝐸 𝑏 * 𝑉𝑆 | 𝑂𝐾𝑀𝐶 ≅ -

Predictions of radiation-induced embrittlement

Via the relations in Eq (2) and Eq (3), the coefficients 𝛼 𝐵𝐾𝑆 and 𝛼 𝑉𝐹 can be determined from linear regressions using a given set of data about solute clusters (√𝑁𝐷), and the corresponding T.

We start by performing this exercise using the experimental data from APT collected in the microstructure database. The obtained coefficients are listed in Tab. 1, and the achieved predictions of T are compared with the desired values in the top panels in Fig. 6. Both embrittlement laws appear to correlate the embrittlement T with the amount of solute clustering on the average. Next, we use the fitted 𝛼 𝐵𝐾𝑆 and 𝛼 𝑉𝐹 coefficients of Tab. 1, to correlate the OKMC model predictions of N, D and f with the embrittlement. This can be now done without further calibration to all materials and their respective irradiation conditions collected in the embrittlement database. The comparison between OKMC predicted and target values is shown in the bottom panels of Fig. 6. Remarkably, both embrittlement laws in Eq (3) and Eq (4) work well for nearly all T data-points, with high accuracy.

Discussion

Our results show that the linear between √𝑁𝐷 or √𝑓 and T as defined in Eq (3)

and Eq (4) are general and can be applied to any RPV material via our physical model, without the need of APT data from time-consuming and costly microstructural examination. Importantly, these relations work well also for the high dose data points of the database, suggesting that the model enables accurate extrapolation to higher irradiation doses than those of surveillance. These simple embrittlement laws, coupled with √𝑁𝐷 or √𝑓 computed using the OKMC model, can therefore be used to evaluate the integrity of the RPV in any of the world's operating nuclear power plants. Examples are shown in Fig. 7. It is worth noting that the rightmost irradiation dose of 0.4 dpa correspond to about 80 years of operation for pressurized-water reactors. To verify the robustness of our model, in Fig. 8 and Fig. 9, three industrial embrittlementdose correlations are applied to the same embrittlement database as in Fig. 6 and compared with the current model's results. The generated predictions are fully listed in the supplementary materials (Part 2 in the "Embrittlement database" tab). The study included: (a) The EONY correlation [START_REF] Eason | A Physically Based Correlation of Irradiation-Induced Transition Temperature Shifts for RPV Steels[END_REF], although thought for industrial use, is conceived as a physicsinspired model, by including embrittlement contributions from the so-called "stable matrix damage" and from Cu-rich precipitates; (b) The Regulatory Guide 1.99 Rev. 2 [START_REF]Radiation Embrittlement of Reactor Vessel Materials[END_REF], in contrast, is a fully industrial trend curve with embrittlement contributions from a chemistry factor and a fluence function, which is currently the reference correlation in the US and not only for integrity assessment; (c) The ASTM E900-15 correlation [START_REF]Standard Guide for Predicting Radiation-Induced Transition Temperature Shift in Reactor Vessel Materials[END_REF], which can be regarded as another trend curve fitted to more recent surveillance data, including some for which the Regulatory Guide 1.99 Rev. 2 noticeably failed. These three embrittlement-dose correlations were numerically fitted to surveillance data. It is clear from Fig. 8 and Fig. 9 that all correlations are accurate in most cases. However, in some cases, supposedly outside of the range of fitting, the industrial trend curves clearly fail and the error of prediction is large (up to 300°C). In Fig. 9 the evolution of the prediction error is also plotted as a function of the three most influencing input variables of the models, i.e., the Cu content, the Ni content, the temperature, and the neutron fluence. We observe that:

 The industrial correlations fail to make accurate predictions when the Ni content is high (above ~1.5%). This is of course expected for very high Ni contents (> 3 w%) that are atypical compared to industrially-relevant steels: these materials have been chemically tailored for specific scientific purposes [START_REF] Almirall | Precipitation and hardening in irradiated low alloy steels with a wide range of Ni and Mn compositions[END_REF]. Ni contents between 1.5 wt% and 2 wt%, however, are industrially relevant. The model proposed in this work achieves overall more accurate predictions compared to the industrial trend curves for these materials.  The industrial correlations commit the largest error of predictions when the neutron fluence is larger than 3 ⋅ 10 19 n cm 2 (equivalent to 0.05 dpa). On the contrary, the prediction error committed by the model proposed in this work is contained in a noticeably narrower range.  The model proposed in this work performs with equal accuracy in the whole range of irradiation temperatures (from 255°C to 300°C), even though a slight bias towards underestimation is detectable in the lowest temperature range. More low temperature (<275ºC) experimental data would be required to perform a detailed study, but the model proposed in this works seems to be at least as accurate as the Regulatory Guide trend curve for temperatures < 275°C, while the EONY model is perhaps slightly more accurate in this range. The success of our model strongly suggests that the effect of solute clusters dissolved in the bulk of grains is indeed the dominant factor to cause radiation-induced hardening and embrittlement in RPV steels. It also suggests that the fundamental physical mechanisms introduced in the microstructure evolution model are correct, or at least sufficient, to provide fair predictions of the density, size and volume fraction of these clusters. It is however important to discuss the limits of the assumptions made: a) The ensuing model is here applied up to 0.1 at% Cu. As discussed above, higher Cu contents imply an increasingly dominant effect of precipitation as dictated by thermodynamics. Consequently, the embrittlement evolution with dose should rapidly saturate, a trend that the model in its present form cannot predict. b) Irradiation doses well beyond those of relevance to RPV steel operation are known to induce the development of other microstructural features that influence their mechanical properties. For example, near or above 1 dpa, ferritic-martensitic steels develop dislocation loops and nanovoids, which also act as obstacles to the motion of dislocations [START_REF] Gaganidze | Assessment of neutron irradiation effects on RAFM steels[END_REF]. While the dispersed barrier hardening theory still holds, these additional obstacles are not explicitly accounted for in the present model: embrittlement laws that superpose different defect populations should be applied. c) If the P concentration is high enough, its segregation at grain boundaries is known to favour intergranular fracture [START_REF] Lu | Radiation-and Thermally-Induced Phosphorus Inter-Granular Segregation in Pressure Vessel Steels[END_REF], leading to embrittlement without hardening. This cannot be described by the dispersed barrier hardening model adopted here. Such effects should be assessed separately from the effect of bulk precipitation.

We should emphasize that a similar microstructure evolution model as described here was previously applied with success to predict the evolution of solute cluster volume fraction in model alloys for F/M steels [START_REF] Dubinko | Effect of Si-Ni-P on the emergence of dislocations loops in Fe-9Cr matrix under neutron irradiation: TEM study and OKMC modelling[END_REF], as a function of the received neutron dose. This results further confirms the robustness of the assumptions made here in terms of microstructure evolution mechanisms, which extend from RPV to F/M steels. That is, the mechanism of solute cluster formation can be considered essentially identical in RPV, VVER, and F/M materials and is driven by single point-defect that drag solutes towards sinks. The sinks that dominantly provide nucleation sites for solute clusters are irradiation-generated pointdefect clusters. Thus, the radiation-induced character of the process is dominant over the radiation-enhanced one. A quantitative prediction of T in F/M steels based on this sole information like for RPV steels is most likely not possible, for at least two reasons: (1) the significant difference in composition and microstructure between RPV and F/M steels, and between different types of F/M steels, which produces features such as secondary phases and makes it difficult to define the actual grain sizes that need to be introduced in the underlying hardening law (see, e.g., Ref. [START_REF] Monnet | Multiscale modeling of irradiation hardening: Application to important nuclear materials[END_REF]); (2) unlike the majority of RPV and VVER steels, nanovoids and dislocation loops are not negligible in the case of F/M materials, especially considering that (much) higher neutron doses are generally reached there (> 1dpa in contrast to < 0.3 dpa for RPV and VVER materials). Discussing these aspects in detail goes beyond the scope of this paper. Nevertheless, our OKMC model is clearly applicable to predict the kinetics of solute clustering in any ferritic material relevant for nuclear application, i.e., RPV, VVER, and F/M steels, regardless of their intrinsic differences. This is an important point in support of the generality of the fundamental physical mechanisms that drive the microstructure evolution under irradiation.

Another important consideration stemming from our model is that the direct use of microstructural information in embrittlement correlations, and damage models in general, seems to significantly simplify their mathematical form: this can be appreciated by comparing Eq (3) and Eq (4) with the mathematical expression of, e.g., the EONY model or other correlations [START_REF] Soneda | 11 -Embrittlement correlation methods to identify trends in embrittlement in reactor pressure vessels (RPVs)[END_REF]. The whole dependence of radiation embrittlement on chemical composition, temperature, dose and dose-rate is fully accounted for by N and D and f, which are computed applying a microstructure evolution theory. The rest of the mathematical expression is just a coefficient. The present work thus suggests that, even though the longterm goal of multiscale approaches [START_REF] Marquis | Nuclear reactor materials at the atomic scale[END_REF] is to predict materials behavior with models that describe all fundamental processes along increasing time and space scales, it is also possible to improve the reliability of engineering correlations by directly coupling them with microstructure evolution theories. Specifically, in this work we produced a tool that provides reasonable estimates of microstructural variables, starting from information that is always available (steel composition, temperature, dose and dose-rate), without the need to perform long and costly microstructural examinations (e.g., APT), and enabling the direct assessment of mechanical properties. We argue that, with the help of modern techniques, such as machine learning, this approach can be directly used for engineering purposes, by training an artificial intelligence model to provide similar, but faster, predictions of N and D or f , given the steel and the irradiation conditions [START_REF] Castin | Advanced atomistic models for radiation damage in Fe-based alloys: Contributions and future perspectives from artificial neural networks[END_REF]. Alternatively, this microstructural information can be used to feed more refined correlations with embrittlement, perhaps better rooted in physics, leading to even further reduced uncertainty in the prediction of T.

Conclusion

We have developed a robust microstructure evolution theory and model that quantitatively predicts the formation of solute clusters under irradiation in ferritic steels given their composition and the irradiation conditions. This piece of microstructural information proves to be sufficient to assess in a very simple way embrittlement as a function of the nuclear power plant operation time. The embrittlement of ferritic steels under irradiation is therefore dominated by the formation of nanometric solute clusters. The simple embrittlement law provided here in Eq. ( 3) and Eq (4), coupled with our microstructure evolution model for the assessment of solute clustering, proves to be even more reliable than existing correlations that are currently used to assess the integrity of reactor pressure vessel steels beyond the planned surveillance programs.

Figure 1 -

 1 Figure 1 -APT analysis of irradiated RPV steels. (a) 3-D atomic map reconstructed from a Cu-rich irradiated RPV steel. Several radiation-caused features are seen. Typical analyzed volumes contain 2-10 million atoms; (b) correlation between the chemical content in solute clusters in bulk and nominal concentration. Chemical contents are calculated in relative terms, excluding the Fe matrix atoms. Open symbols denote Cu-rich precipitates in 3 different materials[START_REF] Miller | APT characterization of high nickel RPV steels[END_REF][START_REF] Miller | APT characterization of irradiated high nickel RPV steels[END_REF][START_REF] Kuleshova | Degradation of RPV steel structure after 45 years of operation in the VVER-440 reactor[END_REF], while closed symbols denote NiMnSiPCu-rich precipitates in 74 different materials[START_REF] Castin | The dominant mechanisms for the formation of solute-rich clusters in low-Cu steels under irradiation[END_REF][START_REF] Almirall | Precipitation and hardening in irradiated low alloy steels with a wide range of Ni and Mn compositions[END_REF][START_REF] Miller | Atom probe tomography characterizations of high nickel, low copper surveillance RPV welds irradiated to high fluences[END_REF][START_REF] Almirall | On the use of charged particles to characterize precipitation in irradiated reactor pressure vessel steels with a wide range of compositions[END_REF][START_REF] Auger | Synthesis of atom probe experiments on irradiation-induced solute segregation in French ferritic pressure vessel steels[END_REF][START_REF] Dohi | Effect of Neutron Flux at High Fluence on Microstructural and Hardness Changes of RPV Steels[END_REF][START_REF] Edmondson | Using complimentary microscopy methods to examine Ni-Mn-Si precipitates in highly-irradiated reactor pressure vessel steels[END_REF][START_REF] Styman | Postirradiation annealing of Ni-Mn-Si-enriched clusters in a neutron-irradiated RPV steel weld using Atom Probe Tomography[END_REF][START_REF] Wells | The Character, Stability and Consequences of Mn-Ni-Si Precipitates in Irradiated Reactor Pressure Vessel Steels[END_REF][START_REF] Jenkins | The effect of composition variations on the response of steels subjected to high fluence neutron irradiation[END_REF][START_REF] Takeuchi | Effects of chemical composition and dose on microstructure evolution and hardening of neutron-irradiated reactor pressure vessel steels[END_REF][START_REF] Lambrecht | On the correlation between irradiation-induced microstructural features and the hardening of reactor pressure vessel steels[END_REF][START_REF] Kuleshova | Mechanisms of radiation embrittlement of VVER-1000 RPV steel at irradiation temperatures of (50-400)°C[END_REF][START_REF] Odette | Precipitation in reactor pressure vessel steels under ion and neutron irradiation: On the role of segregated network dislocations[END_REF]. Note that Cu is the only element that strongly deviates from the linear correlation with the element abundance in the alloy. Mn also slightly deviates because it is a carbide former, so part of the content of this element is found in carbides.

Figure 2 -

 2 Figure2-Schematic description of the object kinetic Monte Carlo method. From left to right, after the production of damage in collision cascades, elementary events take place, which are described as thermally-activated processes of diffusion, dissociation and transport by point-defects, all parameterized here based on first principle calculations. They spontaneously lead to the formation of solute rich clusters: complexes C1 and C2 contain a sufficient number of atoms to be resolvable with APT. The dominant mechanism[START_REF] Castin | The dominant mechanisms for the formation of solute-rich clusters in low-Cu steels under irradiation[END_REF] of formation is indicated by solid line arrows, while dashed arrows indicate secondary mechanisms which are, formally, not excluded from the model.

Figure 3 -

 3 Figure 3 -Schematic description of how the binding energy Eb of vacancies (Va) and vacancy-solute pairs (VaSo) to complexes C2 depend on cluster size. Values are normalized to the saturated value Eb*.

Figure 4 -

 4 Figure 4 -Saturated binding energy values estimated in different ways, as functions of the relative Ni content in the material. The acronyms PAS stands for positron annihilation spectroscopy (experimental measurement taken from [60]), and CE for cluster expansion (theoretical model based on density functional theory).

Figure 5 -

 5 Figure 5 -Comparison between the OKMC predictions and APT experimental data for the materials and irradiation conditions listed in the microstructure database.

Figure 6 -

 6 Figure 6-Application of the embrittlement laws from Eq (2) (right panels) and Eq (3) (left panels) using the microstructure database (top panels) and the embrittlement database (bottom panels) described in section 2.1. Black lines are the ideal match, and the dashed lines show the 95% interval of confidence (2 ). The horizontal error bars on the dots is set to 25°C, while the vertical error bars are derived from the experimental uncertainty of the APT measurements or of the OKMC predictions.

Figure 7 -

 7 Figure 7 -Physics-based law (volume-fraction model in Eq (3)) for radiation embrittlement applied to individual steels. Evolution with the received dose of the predicted (solid lines) and measured (dots) embrittlement ∆T for four individual steels (different colors). The 95% interval of confidence (±2 = 27.5°C) is shown as well (dashed lines). Chemical compositions are given in at%. The irradiation dose of 0.4 displacement per atoms corresponds to about 80 years of operation for pressurized water reactors.

Figure 8 -

 8 Figure 8 -Accuracy of prediction of T achieved by the EONY embrittlement correlation method[START_REF] Eason | A Physically Based Correlation of Irradiation-Induced Transition Temperature Shifts for RPV Steels[END_REF], The Regulatory Guide 1.99 Rev. 2[START_REF]Radiation Embrittlement of Reactor Vessel Materials[END_REF], and the ASTM E900-15 correlation[START_REF]Standard Guide for Predicting Radiation-Induced Transition Temperature Shift in Reactor Vessel Materials[END_REF], compared with the predictions achieved in this work. The black line is the ideal match, and the dashed lines show the 95% interval of confidence (2= 37 °C ).

Figure 9 -

 9 Figure9-Accuracy of prediction of T achieved by the EONY embrittlement correlation method[START_REF] Eason | A Physically Based Correlation of Irradiation-Induced Transition Temperature Shifts for RPV Steels[END_REF], The Regulatory Guide 1.99 Rev. 2[START_REF]Radiation Embrittlement of Reactor Vessel Materials[END_REF], and the ASTM E900-15 correlation[START_REF]Standard Guide for Predicting Radiation-Induced Transition Temperature Shift in Reactor Vessel Materials[END_REF], compared with the predictions achieved in this work. The panels plot the evolution of the prediction error with four key variables of the model.

  𝐶𝑢 is an additional binding energy term which accounts for the presence of Cu.We fitted a linear ramp step function that is effective from a moderate Cu content CCu:

	5.30𝑥 𝑁𝑖 2 + 0.109 𝑥 𝑁𝑖 + 1.178 + 𝐸 𝑏 * 𝑉𝑆 | 𝐶𝑢 ± 0.025 (eV)	(8)
	Here, 𝐸 𝑏 * 𝑉𝑆 | 𝐸 𝑏 * 𝑉𝑆 | 𝐶𝑢 = 0	if (CCu < 0.05 at%)	(9)
	= (CCu -0.05) * 0.035 (eV) if (0.05 at% ≤ CCu ≤ 0.07 at%)	(10)
	= 0.035 (eV)	if (CCu > 0.07 at%)	(11)

Table 1 -

 1 Coefficients derived for embrittlement laws using the BKS model or the volume fraction based model. The coefficients are either derived from APT experimental data, or from OKMC estimates, for N and D of the solute clusters. In each case, the 95% interval of confidence (2 ) and the correlation coefficients R2 are reported.

		BKS model Eq. (3)		Volume fraction model Eq. (4)
		𝜶 𝑩𝑲𝑺 (-)	2(°C)	R2	𝜶 𝒗𝒇 (°C)	2(°C)	R2
	N and D From APT	0.478 ± 0.0923	41.9	0.80	1383 ± 129.8	42	0.80
	N and D From OKMC	0.466 ± 0.0277	27.9	0.84	1550 ± 22.7	28.3	0.91

Acknowledgements

This work received partial financial support in the framework the 2019-2020 Euratom research and training programme (ENTENTE project, grant No 900018), as well as from the 2014-2018 Euratom research and training programme (SOTERIA project, grant No. 661913, and M4F project, grant No. 755039). It is also partly indebted to earlier 7 th framework programme projects (LONGLIFE project, grant No. 249360, and MATISSE project, grant No. 604862). Finally, this work contributes to the Joint Programme on Nuclear Materials of the European Energy Research Alliance (EERA JPNM). The views and opinions expressed herein do not necessarily reflect those of the European Commission. N. Castin acknowledges Prof. M. Miralles from the Buenos Aires Catholic university (UCA) for a critical review of this work.

Author contributions