Romain Michon
email: michon@grame.fr

Daniel Overholt

Stéphane Letz

Yann Orlarey

Dominique Fober

Catinca Dumitrascu

A FAUST ARCHITECTURE FOR THE ESP32 MICROCONTROLLER

This paper introduces faust2esp32, a tool to generate digital signal processing engines for the ESP32 microcontroller family. It can target both the C++ and the Arduino ESP32 programming environment and it supports a wide range of audio codecs, making it compatible with most ESP32-based prototyping boards. After demonstrating how to use faust2esp32 and providing technical details about its implementation, we evaluate its performances and we present the FAUST Gramophone which is a programmable instrument taking advantage of this technology. Finally, future plans around embedded systems for real-time audio processing and FAUST are presented.

INTRODUCTION

For a long time, embedded Linux systems such as the Raspberry Pi 1 (RPI) or the Beagle Bone 2 were the only embedded platforms for real-time audio Digital Signal Processing (DSP) accessible to members of the music technology community without a programming background [START_REF] Berdahl | How to make embedded acoustic instruments[END_REF]. The Operating System (OS) running on this kind of platform makes the on-board implementation of audio processing algorithms through high-level graphical patching environments such as PureData [START_REF] Puckette | Pure data: another integrated computer music environment[END_REF] possible. Specialized Linux distributions targeting this type of application such as Satellite CCRMA [START_REF] Berdahl | Satellite CCRMA: A musical interaction and sound synthesis platform[END_REF] were developed in that context. However, the ease of use provided by the OS -Linux, for instance -comes at the cost of simplicity and efficiency, both in terms of hardware and software. Additionally, since embedded Linux systems are typically used by the music technology community to create musical instruments, art installations, etc., [START_REF] Jordà | Digital lutherie crafting musical computers for new musics' performance and improvisation[END_REF] the lack of a proper audio input/output and of Analog to Digital Converters (ADCs) for sensors are often compensated by the use of external hardware (e.g., USB audio interface, USB microcontroller like the Arduino, etc.), significantly increasing the cost and complexity of such setups.

These issues were addressed by the BELA3 [START_REF] Moro | Making high-performance embedded instruments with bela and pure data[END_REF][START_REF] Mcpherson | Bela: An embedded platform for lowlatency feedback control of sound[END_REF] for a subset of the Beagle Bone boards, or the Elk 4 for the RPI. These systems take the form of sister boards/hats providing high-quality audio inputs and outputs (audio codec) as well as sensor analog inputs from a single standpoint. They also run audio DSP tasks outside of the OS, which translates into a faster access to the audio inputs and outputs for a reduced audio latency. The overall design of such systems remains quite complex though for what they're actually used for and their price is relatively high (>$150). On the other hand, microcontrollers offer a promising lightweight, efficient, and cheap platform for real-time audio DSP applications. While microcontrollers were primarily designed for data/sensor acquisition and basic processing, new generations such as the ARM Cortex-M5 provide extended computational power and memory. This combines with the fact that an increasing number of microcontrollers host a Floating Point Unit (FPU), greatly simplifying the implementation of DSP algorithms. The Teensy microcontroller family 6 played a pioneering role in this field by distributing an Audio Shield7 (which is essentially a breakout board for an audio codec) and providing an "audio library 8 " for high-level audio DSP programming. The Teensy 3.6 and 4.0 (180MHz and 600MHz, respectively), when combined with the Audio Shield provide a powerful platform for real-time audio applications (70 sine waves can be ran in real-time on the 3.6 and up to 600 on the 4.0) [START_REF] Michon | Real time audio digital signal processing with Faust and the Teensy[END_REF].

While tools such as the Teensy Audio Library or libpd [START_REF] Brinkmann | Embedding PureData with libpd[END_REF] allow for the programming of DSP algorithms at a highlevel, they are limited to their built-in/pre-implemented DSP objects (e.g., filters, oscillators, etc.) and custom elements must be written in C++. In 2019, we introduced faust2teensy [START_REF] Michon | Real time audio digital signal processing with Faust and the Teensy[END_REF] which is a command-line tool that can be used to generate custom objects for the Teensy audio library using the FAUST programming language [START_REF] Orlarey | New Computational Paradigms for Computer Music[END_REF]. FAUST 9 is a functional Domain Specific Language (DSL) for real-time audio DSP that can target a wide range of platforms (e.g., Linux, macOS, Windows, Android, iOS, embedded systems, etc.) and standards (e.g., audio plugins, standalone applications, externals, etc.). While FAUST can easily add various features to generated objects such as polyphony for synthesizers, MIDI and OSC (Open Sound Control) support, etc., faust2teensy only provides basic functionalities because of the limited amount of memory available on the Teensy -i.e.,:

• control of the parameters of a FAUST DSP object (e.g., declared using UI elements such as hslider, nentry, etc.),

• stereo audio input and output.

FAUST-generated objects for the Teensy Audio Library can be combined with built-in objects, playing to the strengths of both approaches. Since its release, faust2teensy has been used for teaching (e.g., Music 250a -Physical Interaction Design for Music,10 the Embedded DSP With Faust Workshop11 at Stanford University, Low Latency Embedded DSP for New Musical Instruments at Aalborg University Copenhagen, etc.) and as a basis for many projects.

While the Teensy gained a special status in the makers community in recent years because of its power, reliability, and affordability, the ESP32 microcontroller12 has been increasingly used as well, offering a solid alternative to the Teensy for real-time audio processing applications.

In this paper, we introduce faust2esp32, a tool to generate DSP engines for the ESP32 microcontroller with FAUST. After giving an overview of this type of board, we demonstrate how faust2esp32 can be used to generate DSP objects and we provide technical details about the implementation of this system. We then evaluate its performance and present the FAUST Gramophone which is a programmable instrument based on the platform. Finally, we give future directions for this type of work.

THE ESP32

The ESP32 is a series of low-cost, low-power embedded systems with integrated WiFi, Bluetooth, and microcontroller distributed by Espressif Systems. It is based on a 32 bits dual core Tensilica Xtensa LX6 microprocessor operating at 240MHz. It provides 520 KiB of internal RAM, but almost all ESP32-based boards host an external SRAM module expanding its memory. The main asset of the ESP32 is its cost since basic ESP32-based boards can be found for less than $3.

The ESP32 is used at the heart of a wide range of more specialized boards targeting specific applications such as real-time audio processing (see Figure 1). In particular, a series of development boards to prototype smart speakers (e.g., Google Home, Amazon Alexa, etc.) have been available for about a year now and are particularly well suited for embedded musical instrument design by providing audio inputs and outputs as well as additional RAM required to implement algorithms with a large memory footprints such as echos, reverbs, etc. Table 1 gives a few examples of such boards and of their corresponding features.

While ESP32 boards can be used bare-metal (without an OS), the most convenient and effective way to program them is FreeRTOS13 which is a low-level real-time OS system kernel for embedded devices. It is accessible through a C++/MakeFile environment built on top of a Python tool-chain to communicate with the board. The ESP32 can also be programmed through the Arduino environment/IDE. FreeRTOS' scheduler/task manager is very convenient in the context of real-time audio DSP applications.

Espressif also recently released a series of ESP32-based boards hosting a Digital Signal Processors (DSP) such as the ESP32-LyraTD-DSPG, the ESP32-LyraTD-SYNA, and the ESP32-LyraTD-MSC14 which are not yet supported by FAUST (see §6).

USING FAUST2ESP32

3.1 Generating DSP Engines faust2esp32 is a command line tool that can be used to generate ready-to-use C++ DSP engines from a FAUST program. Given a simple FAUST program such as: import("stdfaust.lib"); f = nentry("freq",440,50,2000,0.01); g = nentry("gain",1,0,1,0.01); t = button("gate"); process = os.sawtooth(f) * g * t;

which just implements a band-limited sawtooth oscillator whose frequency (freq) and gain (gain) can be controlled. Additionally, the gate parameter allows us here to turn it on and off.

Running:

faust2esp32 -lib FaustObject.dsp will produce a package containing two C++ files: FaustObject.cpp and FaustObject.h which can be imported into a C++ or an Arduino ESP32 project. In both cases, FaustObject.h should be included at the beginning of the main program:

#include "FaustObject.h"

The corresponding FAUST DSP object can be instantiated with:

FaustObject * faustObject = new FaustObject(48000,8);
where 48000 is the sampling rate and 8 the block size. Any value can be chosen for these parameters as the long as the hardware can handle it. Note that block size has very little impact on hardware performances (see Table 2). Audio computing can be started by calling the start method:

faustObject->start();

Audio computing is taking place in its own high-priority thread (FreeRTOS task). Audio samples are retrieved from and transmitted to the audio codec of the board using I2S. This implies that the audio codec must be configured before using a driver. faust2esp32 comes with a series of drivers supporting some audio codecs such as the Wolfson WM8978 or the X-Powers AC101. These can be integrated to the generated package using the -wm8978 and the -ac101 options (respectively) when calling faust2esp32. Since functionalities differ between audio codecs, they are not presented here and we advise you to refer to their corresponding documentation.

The parameters of the FAUST DSP object (i.e., freq, gain, and gate here) can be updated at any time by calling the setParamValue() method which takes the name of the parameter and its corresponding value as arguments. E.g.:

faustObject-> setParamValue("freq",1000);

will set the frequency of the generated sawtooth wave to 1000 Hz. This procedure works both in the Arduino and native ESP32 C++ environments.

MIDI Support

MIDI support can be easily added to the generated DSP engine by using the -midi option when calling faust2esp32. MIDI events are received directly from the corresponding UART (Universal Asynchronous Receiver-Transmitter) serial interface on the ESP32 (see §4). Standard FAUST MIDI metadata15 can be used as well to configure the MIDI behavior of the DSP engine.

Polyphony

FAUST synthesizers declaring the freq, gain, and gate parameters (such as the one presented at the beginning of §3.1) can be automatically turned into polyphonic synths by using the -nvoices option when calling faust2esp32. Hence, faust2esp32 -lib -midi -nvoices 12

FaustObject.dsp will produce a polyphonic synth with MIDI support with a maximum number of twelve voices of polyphony. Standard FAUST polyphony metadata16 can be used to further configure the behavior of this feature.

IMPLEMENTATION

Unlike the Teensy, the ESP32 doesn't provide a high-level audio library, so we had to implement almost everything from scratch here. Audio DSP is running in its own high-priority FreeRTOS task (see Figure 2) on the first core. Control-rate computations (like MIDI, sensor, etc.) happen in a separate task on the second core of the ESP32.

External events are handled in an asynchronous manner, hence their timestamp is not taken in account. The generated C++ code hosts the DSP compute method which samples the values of all controllers at the beginning of the code, and use the same values during the entire audio block. Thus, only the most recent values are used.

This simple model may have to be improved with more sophisticated queue-based event handling code, if we want to ensure all events are taken in account (which is mandatory in some cases like MIDI clock-based synchronization, for instance).

External PSRAM can be integrated to the heap to make it available to the audio DSP object. This can be configured from the ESP32 project settings (typically accessible by running make menuconfig). Audio samples are received and sent to the audio codec using I2S through the corresponding ESP32 API. 17 match that of the audio codec. The audio codec is configured through I2C using the corresponding driver provided as part of the package generated by faust2esp32. Only a few audio codecs are currently supported (see §3). MIDI events are received through the corresponding UART interface and are parsed using jdksmidi, 18 then transmitted to the Faust MIDI handling code with a esp32_midi glue class. The general implementation of faust2esp32 is similar to most other FAUST architectures and is based on a script (i.e., BASH) taking care of assembling the various constituent elements of the generated DSP engine. faust2esp32 is now fully integrated to the main FAUST distribution. 19

EVALUATION

Running DSP code on microcontrollers presents two main advantages compared to using a more general board and operating system:

• very small buffer size (i.e., 8 samples) can be used, minimizing latency;

• the OS is minimal and is usually more real-time robust, diminishing the risk for audio glitches to happen.

Performance

The ESP32 offers plenty of computational power to run most "standard" DSP algorithms.

, phasor(tablesize,freq)); process = par(i,N,osc(50+i)); Each oscillator is using a 2 14 samples-long table read directly from memory (the sin function is only called during initialization). The table is read using a phasor which corresponds to one multiply, one add, one subtract, and one call of the math floor function.

As expected, the ESP32 clearly outperforms the Teensy Figure 3. The FAUST Gramophone.

3.6, but it is much less powerful than the Teensy 4.0 (which runs at 600MHz). Since most ESP32-based boards host an external PSRAM module, running DSP algorithms with a large memory footprints is not an issue in most cases on the ESP32 when it can be problematic on most Teensy boards. For example, while dm.zita_light20 (which is a feedback delay network reverb that can be found in the FAUST libraries) can't be ran on any Teensy board without making some modifications to the algorithm, the LyliGO TTGO-TAudio can "digest" it without any issue.

Use Case: the Gramophone

The Gramophone (see Figure 3) is a programmable musical instrument designed as part of the Amstramgrame project, 21 which is currently in its pilot phase. Amstramgrame aims at teaching physics and math through programming exercises with the Gramophone. The Amstramgrame website hosts pedagogical content as well as code examples that can be run directly in the FAUST Web IDE. FAUST programs prototyped in the web can then be exported to the Gramophone directly from the Amstramgrame website.

The Gramophone is based on a LilyGO TTGO-TAudio board which hosts a single 9-DoF sensor (comprising an accelerometer, a gyroscope, and a magnetometer), a lithium battery manager, an audio codec, etc. Additional sensors (e.g., photoresistor, potentiometers, buttons, etc.) are mounted on the body of the Gramophone. The Gramophone can be powered by a lithium battery directly connected to the board. The speaker of the instrument is directly connected to the built-in amplifier of the TTGO-TAudio's audio codec.

The Gramophone can be entirely programmed in FAUST without writing a single line of C++ or Arduino code. The -gramophone option can be used for that when calling faust2esp32. The built-in motion sensors (a full 9-axis IMU, Inertial Measurement Unit) can be mapped using the standard FAUST metadata system,22 and trans-mitted to the FAUST DSP using the Esp32SensorUI class. Gramophone-specific FAUST metadata were created as well to access the instrument's built-in sensors (i.e., potentiometers, buttons, etc.) and transmitted to the FAUST DSP using the Esp32ControlUI class.

Multiple FAUST programs can be installed in the Gramophone using the -multi option when calling faust2esp32, by following a specific coding convention in the DSP source code to describe them all in a single file. FAUST programs can then be selected using an encoder mounted on the body of the instrument. To save memory, only a single DSP is active at one point, so the old one is deleted before a new one is instantiated.

For less than $40 (all parts included), the ESP32 through the TTGO-TAudio board allowed us to make a fully embedded programmable instrument with more than seven hours of battery life that can run most existing FAUST programs. We believe that this would not have been possible with any other board/type of system.

FUTURE DIRECTIONS

Around the ESP32

The ESP32 FAUST support could be extended in many ways. In particular, since the ESP32 provides built-in WiFi support, we hope to be able to improve the existing architecture by enabling OSC support.

Finally, we'd like to develop architectures for the ESP32 DSP boards that have been recently introduced by Espressif such as the LyraTD-DSPG, the LyraTD-SYNA, and the LyraTD-MSC. Since they are based on floating-point DSPs this should be pretty straight forward.

Towards More Advanced FAUST Architectures for Embedded Processing

We recently started investigating the idea to use FAUST to program more advanced systems such as GPUs and FPGAs. The latter is particularly challenging because it is usually programmed at an extremely low-level using hardware description languages such as VHDL or Verilog. While we managed to program Xilinx FPGAs from FAUST through C++ using HLS (High Level Synthesis),23 this solution remains relatively heavy and slow and we hope to be able to generated VHDL code directly from FAUST for real-time DSP applications.

In parallel to that, we've been working on a RPI baremetal FAUST architecture for low-latency applications. While we have a working prototype, many issues remain to be solved so we'd like to keep working on that in the future.

CONCLUSIONS

We believe that FAUST greatly simplifies the programming of embedded systems for real-time audio DSP applications by providing a high-level environment with hundreds of pre-implemented functions/algorithms. Systems such as the EP32 are currently revolutionizing the field of music technology by providing a comprehensive platform for musical instrument design/audio processing at a very low cost. Moreover, their simplified architecture (bare-metal or close to bare-metal) make them more stable and reliable than OS-based systems. We hope that in a near future, Domain-Specific Languages such as FAUST will make all types of embedded/specialized systems such as FPGAs accessible to makers, hobbyist, and members of the music technology community.

Figure 1 .

 1 Figure 1. ESP32-based audio Processing boards (the TTGO TAudio on the left and the AI-Thinker Audio Dev Kit on the right).

Figure 2 .

 2 Figure 2. Overview of the implementation of faust2esp32. Thick continuous frames on the left represent objects generated by faust2esp32 on the user's computer. Empty arrows on the left link the various steps in generating the DSP engine. Full arrows on the right represent streams computed in real-time.

Table 1

 1

	Board	Features
	LilyGO	4MB PSRAM, motion sensors,
	TTGO-TAudio audio in/out (codec WM8978),
		battery management,
		speaker amplifier
	AI-Thinker	8MB PSRAM, audio in/out
	ESP32-A1S	(codec AC101), speaker
		amplifier
	Espressif	4MB PSRAM, audio in/out
	ESP32-LyraT	(codec ES8388), battery
		management, speaker amplifier
	Espressif	8MB PSRAM, audio in
	ESP32-LyraT	(codec ES8311), battery
	Mini	management, speaker amplifier

. Examples of ESP32-based boards for real-time audio signal processing.

Table 2

 2

	presents a com-
	parison of the performance in terms of processing power
	of the ESP32 with other microcontrollers supported by
	FAUST. The ESP32 used for these tests was a WROVER-B

18 https://github.com/jdkoftinoff/jdksmidi 19 https://github.com/grame-cncm/faust

Table 2 .

 2 Maximum number of FAUST sine waves ran in parallel on various boards for different block sizes.

	on a LyliGO TTGO-TAudio board. All tests were carried
	out at a sampling rate of 48KHz with the following FAUST
	program implementing a series of sine wave oscillators in
	parallel:
	import("stdfaust.lib");
	N = 600;
	tablesize = 16384;
	time = (+(1)˜_) -1;
	sinwaveform(tablesize) =
	time * (2.0 * ma.PI)/tablesize : sin;
	decimal(n) = n -floor(n);
	phasor(tablesize,freq) =
	freq/ma.SR : (+ : decimal)˜_ :
	* (tablesize) : int;
	osc(freq) =
	rdtable(tablesize,
	sinwaveform

https://bela.io/

https://elk.audio/

https://developer.arm.com/ip-products/ processors/cortex-m

https://www.pjrc.com/teensy/

https://www.pjrc.com/store/teensy3_audio.html

https://www.pjrc.com/teensy/td_libs_Audio. html

https://faust.grame.fr

https://ccrma.stanford.edu/courses/ 250a-winter-2020/

https://ccrma.stanford.edu/workshops/ faust-embedded-19/

https://www.espressif.com/en/products/ hardware/esp32/overview

https://www.freertos.org/

https://www.espressif.com/en/products/ hardware/development-boards

https://faust.grame.fr/doc/manual/index.html# midi-and-polyphony-support

https://faust.grame.fr/doc/manual/index.html# midi-polyphony-support

https://docs.espressif.com/projects/esp-idf/ en/latest/api-reference/peripherals/i2s.html

https://faust.grame.fr/doc/libraries#dm.zita_ light

https://www.amstramgrame.fr

https://faust.grame.fr/doc/manual# sensors-control-metadatas

https://faust.grame.fr/syfala


Gaëtan Perez

Aurélien Maréchal

Gauthier Chicot

Pierre Lefranc

Pierre-Olivier Jeannin

David Eon

Nicolas C Rouger

Aurélien Maréchal

Diamond semiconductor performances in power electronics applications

Keywords: Diamond, Wide Bandgap, semiconductors, self-heating effect, semiconductor losses, heatsink volume, performance comparison

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

I. INTRODUCTION AND REVIEW OF DIAMOND POWER DEVICES

Wide bandgap (WBG) semiconductors have been used since several years to improve performances of power electronics converters. Materials such as silicon carbide (SiC) or gallium nitride (GaN) allow power converters to operate at higher voltages, higher temperatures, and higher switching frequencies than silicon (Si) [START_REF] Millan | A Survey of Wide Bandgap Power Semiconductor Devices[END_REF]. These benefits are offered by the superiorities of the WBG materials [START_REF] Chow | Smart power devices and ICs using GaAs and wide and extreme bandgap semiconductors[END_REF], [START_REF] Fujita | Wide-bandgap semiconductor materials: For their full bloom[END_REF]. Among WBG semiconductors, diamond is the most promising material for power electronics applications. Whereas experimental benefits of SiC and GaN devices in actual power converters have been widely demonstrated, the system level benefits of diamond power devices have not been yet clearly described. This article will offer a system level benchmark of diamond devices in power converters, based on the characteristics of fabricated diamond Schottky diodes, their implementation in power converters and a theoretical benchmark versus SiC power devices. The physical properties of diamond such as a high electric field (10 MV/cm), a high hole mobility (1300 cm 2 /(V.s)) [START_REF]Model implementation towards the prediction of J(V) characteristics in diamond bipolar device simulations[END_REF] make it the most suitable semiconductor for high power converters. These values are calculated at room temperature for a 1 kV breakdown voltage with a 1.6x10 17 cm -3 doping level for the required drift region. These properties combined with a high thermal conductivity (22 W/(cm.K)) and a very low intrinsic carrier concentration at PRE-PRINT (initial submitted article). Accepted version (Elsevier, DRM2020): https://doi.org/10.1016/j.diamond.2020.108154 room temperature (2.2x10 -27 cm -3 ) lead to the highest figures of merit (FOM) [START_REF] Chow | Smart power devices and ICs using GaAs and wide and extreme bandgap semiconductors[END_REF], higher than SiC or GaN materials.

One of the most specific property of bulk diamond semiconductors is the incomplete ionization of boron (p-type layers) and phosphorus dopants (n-type layers) at room temperature, these dopants are then activated by increasing the junction temperature. Other devices based on a two dimensional hole gas (2DHG) in Hydrogen-terminated diamond have been also presented [START_REF] Kawarada | Durability-enhanced two-dimensional hole gas of C-H diamond surface for complementary power inverter applications[END_REF], where a large sheet carrier density is available even at low temperatures. In our analysis, we will further consider only bulk Chemical Vapor Deposition (CVD) grown diamond devices, since 2DHG-based diamond devices are still subject to instabilities and reliability issues [START_REF] Daicho | High-reliability passivation of hydrogen-terminated diamond surface by atomic layer deposition of Al2O3[END_REF], albeit with already attractive performances for power electronics [START_REF] Kitabayashi | Normally-Off C-H Diamond MOSFETs with Partial C-O Channel Achieving 2-kV Breakdown Voltage[END_REF]. Concerning the carrier mobility in CVD diamond, both electron and hole mobilities are reduced at higher temperatures [START_REF] Pernot | Hall electron mobility in diamond[END_REF], [START_REF] Pernot | Hall hole mobility in boron-doped homoepitaxial diamond[END_REF]. As the device on-state resistance is impacted by the carrier concentration and their mobility, the two opposite evolutions of incomplete ionization and mobility lead to, first a negative temperature coefficient (NTC) and then a positive temperature coefficient (PTC) of the diamond device onstate resistance. The on-state resistance is then minimized at a high temperature [START_REF] Tarelkin | Power diamond vertical Schottky barrier diode with 10 A forward current[END_REF], [START_REF] Umezawa | 1ohm On-Resistance Diamond Vertical-Schottky Barrier Diode Operated at 250 C[END_REF], this temperature being doping (and breakdown voltage) dependent. Such an evolution is shown experimentally in Figure 1, where the measured on-state resistance of a 200 µm x 200 µm pseudovertical diamond Schottky Barrier Diode (SBD), fabricated at Néel Institute (Grenoble, France) is presented. It highlights the NTC of the diode resistance between 300 K (RT) and 450 K and a PTC for temperatures higher than 450 K. A picture and a 3D schematic of the device are shown in Figure 2. A 200 nm p + layer (10 20 cm -3 ) is grown on a Ib HPHT diamond substrate (Electrically insulating, nitrogen doped, monocrystalline High Pressure High Temperature diamond substrate), a 2 µm p -layer (5.10 15 cm -3 ) is then grown on the p + layer. A ohmic contact (Ti (20 nm) / Pt (20nm) / Au (10 nm)) is deposited on the p + layer, several Schottky contacts (Zr (30 nm) / Pt (20 nm) / Au (10 nm)) are deposited at the center of the device on the p -layer. The realization process is described in [START_REF] Traoré | Zr/oxidized diamond interface for high power Schottky diodes[END_REF]. The diamond die is then composed of several SBD connected with a common anode (Ohmic contact) and electrically isolated cathodes (Schottky contacts). 3. The diode breakdown voltage has not been reached; the reverse characterization has been stopped at -100 V due to the high leakage current seen on the reverse characteristic (10 -4 A/cm²). Such a high leakage current can be caused by defects in the diamond substrate and the p-type layers [START_REF] Umezawa | Leakage current analysis of diamond Schottky barrier diode by defect imaging[END_REF]. Higher breakdown voltages for similar diamond SBD have been experimentally obtained, such as 1 kV [START_REF] Traoré | Zr/oxidized diamond interface for high power Schottky diodes[END_REF] and 10 kV with thicker drift regions and a planar SBD structure [START_REF] Volpe | Extreme dielectric strength in boron doped homoepitaxial diamond[END_REF]. Even if diamond physical properties are very promising for power electronics applications, performances of diamond devices are currently lower than the theoretical expected values. Several phenomena can be responsible for these limitations. One of them is a limitation of the breakdown voltage (BV) caused by the electric field crowding at the PRE-PRINT (initial submitted article). Accepted version (Elsevier, DRM2020): https://doi.org/10.1016/j.diamond.2020.108154 edge of the electrodes [START_REF] Driche | Characterization of breakdown behavior of diamond Schottky barrier diodes using impact ionization coefficients Characterization of breakdown behavior of diamond Schottky barrier diodes using impact ionization coe ffi cients[END_REF] and the lack of efficient junction termination extension solutions. The avalanche breakdown mechanism appears at the edge of the electrode for lower voltages than designed, the device structure is then responsible for this limitation. Several edge termination techniques are investigated in the literature in order to solve this issue. Benefits of field plates have been shown [START_REF] Umezawa | High temperature application of diamond power device[END_REF], [START_REF] Ikeda | Fabrication of a field plate structure for diamond Schottky barrier diodes[END_REF], the use of floating metal guard rings (FMGR) is also demonstrated on diamond Schottky diodes [START_REF] Driche | Electric field distribution using floating metal guard rings edge-termination for Schottky diodes[END_REF]. Even if these improvements are required to reach diamond physical limits in terms of critical electric field, these limitations are not a particularity of diamond semiconductors. Indeed, edge termination techniques have been used for SiC and GaN semiconductors [START_REF] Jiang | Electrical Characteristics of 10-PRE-PRINT[END_REF], [START_REF] Khadar | 820-V GaN-on-Si Quasi-Vertical p-i-n Diodes[END_REF]. Apart from off-state improvements of diamond devices, other studies are conducted to optimize the on-state characteristics. An optimization of diamond devices drift layers has been investigated in the way to minimize the drift layer resistance for given breakdown voltages. A trade-off between the doping concentration, the layer thickness and the punch through (PT) factor have been used to optimize the drift layers [START_REF] Chicot | Optimal drift region for diamond power devices[END_REF].

PRE-PRINT (initial submitted article).

Accepted version (Elsevier, DRM2020): https://doi.org/10.1016/j.diamond.2020.108154 Even if the actual performances of diamond devices are lower than expected, several diamond device structures have been experimentally demonstrated [START_REF] Umezawa | Recent advances in power semiconductor devices[END_REF]. Diamond Schottky diodes with high electric field (7.7 MV/cm) [START_REF] Volpe | Extreme dielectric strength in boron doped homoepitaxial diamond[END_REF], with a high power figure of merit (244 MW/cm²) [START_REF] Traoré | Zr/oxidized diamond interface for high power Schottky diodes[END_REF] and with 20 A current rating [START_REF] Bormashov | Thin large area vertical Schottky barrier diamond diodes with low on-resistance made by ion-beam assisted lift-off technique[END_REF] have been successfully fabricated. Even if n-type layers are challenging for diamond, bipolar diodes can be seen in the literature [START_REF] Suzuki | Electrical characterization of diamond PiN diodes for high voltage applications[END_REF] where a 1 kV breakdown voltage has been reached and diamond Schottky PN diodes with better compromises in on-state losses and off-state leakage currents [START_REF] Makino | Diamond Schottky-pn diode without trade-off relationship between onresistance and blocking voltage[END_REF]. Concerning transistors, the diamond MESFET (BV = 1.5 kV) [START_REF] Umezawa | Diamond Metal-Semiconductor Field-Effect Transistor With Breakdown Voltage Over 1.5 kV[END_REF] and the deep depletion MOSFET (BV = 200V, breakdown field = 4 MV/cm) [START_REF] Pham | Deep-depletion mode boron doped monocrystalline diamond metal oxide semiconductor field effect transistor[END_REF] use the deep depletion regime which is stable thanks to the diamond properties [START_REF] Pham | Deep depletion concept for diamond MOSFET[END_REF]. Diamond transistors using a two dimensional hole gas (2DHG) have also been demonstrated with a 1700 V breakdown voltage [START_REF] Kawarada | Durability-enhanced two-dimensional hole gas of C-H diamond surface for complementary power inverter applications[END_REF]. This emergence of diamond based semiconductor devices shows the necessity to PRE-PRINT (initial submitted article). Accepted version (Elsevier, DRM2020): https://doi.org/10.1016/j.diamond.2020.108154 work in advance on the integration of these devices in power converters. This preliminary study is required to develop power converters adapted to diamond semiconductors physical properties in the way to use their full potentials. The predicted performances of diamond semiconductors in a power converter are then studied in this paper. The potential of diamond devices in power electronics applications is highlighted and used to define the requirements of power converters based on diamond devices.

II. METHOD FOR A SYSTEM-LEVEL BENCHMARK

In order to analyze the benefits allowed by diamond semiconductors in power converters, the performances of diamond semiconductors are compared to those of SiC in the same converter. For this comparison, diamond components characteristics are based on analytical state of the art models, fitted from up to date experimental data. This article is then presenting the expected performances of diamond unipolar devices and it highlights the rules to design power converters due to the particularities of diamond material. The diamond analytical models for both on and off-state parameters are described in [START_REF]Model implementation towards the prediction of J(V) characteristics in diamond bipolar device simulations[END_REF], [START_REF] Chicot | Optimal drift region for diamond power devices[END_REF], where an optimized diamond device drift layer for a given breakdown voltage is considered. These models use the hypothesis of a one dimensional avalanche breakdown based on the impact ionization coefficients proposed in [START_REF] Hiraiwa | Blocking characteristics of diamond junctions with a punchthrough design[END_REF]. The drift layer resistance is calculated by using the temperature dependence of the incomplete ionization model [START_REF]Model implementation towards the prediction of J(V) characteristics in diamond bipolar device simulations[END_REF] and the empirical hole mobility model [START_REF] Volpe | High hole mobility in boron doped diamond for power device applications[END_REF] in CVD diamond. The diamond device characteristics are compared to a SiC commercial off-the-shelf (COTS) power device. As both diamond and SiC semiconductors are dedicated to high voltage, high temperature and fast switching applications [START_REF] She | Review of Silicon Carbide Power Devices and Their Applications[END_REF], it is natural to compare the benefits allowed by diamond PRE-PRINT (initial submitted article). Accepted version (Elsevier, DRM2020): https://doi.org/10.1016/j.diamond.2020.108154 devices to other WBG devices for the same application. In this article, all the comparisons are done by using Schottky diodes. However, this study can be applied to all unipolar semiconductor devices based on a bulk conduction (albeit unipolar devices where the conduction is based on a two dimensional gas are not included in the analysis). Some differences need to be taken into account to accurately specify the performances of each diamond component (Schottky, MOSFET, JFET…). The differences can be either the barrier height for the Schottky diode (conduction losses) or the impact of the gate driver circuit for the diamond MOSFET and JFET (switching losses). All these differences are not considered in this article.

Two parameters are compared in the first analysis, the semiconductor losses and the volume of semiconductors heatsink. These comparisons are performed for different junction temperatures in a 900 V / 10 A buck converter where a transistor is associated to a Schottky diode; a 25 kHz switching frequency (fsw) is used (unless otherwise specified), the duty cycle (α) is considered as 50% and the ambient temperature is 300 K. A second analysis is then used to include the power converter's switching frequency in the comparison. We assume that the associated transistor and diode are based on the same material and are well designed, their voltage and current ratings are the same. The breakdown voltage of each semiconductor is considered as 1.2 kV. The SiC Schottky diode CPW4-1200-S010B (1.2 kV / 10 A) from Wolfspeed corporation is selected for the comparison. For a 1.2 kV breakdown voltage and a Non Punch Through (NPT) design, the diamond Schottky diode drift layer is optimized with a 2.5 µm thickness and a 1.2 × 10 17 𝑐𝑚 -3 Boron doping level [START_REF] Chicot | Optimal drift region for diamond power devices[END_REF]. The on-state resistance as a function of the junction temperature of the modeled diamond SBD and the selected SiC Schottky diode are presented in Figure 4. The surface of the diamond SBD has been set to allow the same PRE-PRINT (initial submitted article). Accepted version (Elsevier, DRM2020): https://doi.org/10.1016/j.diamond.2020.108154 conduction losses between the SiC and the diamond diodes at the temperature, which minimizes the on-state resistance. For an area of 0.37 mm², the diamond conduction losses at 640 K are equal to those of the SiC diode at 300 K (3.92 mm² area). The negative temperature coefficient (from 300 K to 640 K) and the positive temperature coefficient (higher than 640K) of the diamond SBD on-state resistance is shown. The SiC Schottky diode on-state resistance is extracted from the datasheet at four temperatures, the evolution of its resistance is then estimated in the whole temperature range. It shows the PTC of the SiC diode on-state resistance from 300 K to higher temperatures. For these two diodes, the thermal resistance of the required heatsink is estimated as a function of the desired operating junction temperature. This estimation is done by using equation [START_REF] Millan | A Survey of Wide Bandgap Power Semiconductor Devices[END_REF], where Tj, Tamb and Iload are respectively the desired device junction temperature in steady state operation, the ambient temperature (300 K) and the power converter load current (10 A). Only the conduction losses are taken into account in a first step, an estimation of the switching losses will be further presented. A difference can be seen by comparing the design of the thermal resistance for the two semiconductor materials. By using the diamond semiconductor, the design of the heatsink thermal resistance is of great interest to reduce the semiconductor losses. Indeed, the thermal resistance has to be high enough to allow the diamond device to increase its temperature thanks to a self-heating effect.

This method has to be used to reduce the device on-state resistance. This assessment is not the same for SiC materials where the losses are minimized at room temperature. In that case, the thermal resistance has to be minimized if the objective is the limitation of the semiconductor losses.

𝑅 𝑡ℎ = 𝑇 𝑗 -𝑇 𝑎𝑚𝑏 𝑅 𝑂𝑁 × 𝐼 𝑙𝑜𝑎𝑑 ² × (1 -𝛼) (1)
For the following comparisons, several conditions are considered. Concerning the temperature, high temperature power modules have been already demonstrated with a maximum operating temperature between 475 K and 525 K [START_REF] Axelrod | A Novel High-Temperature Planar Package for SiC[END_REF]- [START_REF] Chen | A 1200-V, 60-A SiC MOSFET multichip phase-leg module for high-temperature, high-frequency applications[END_REF]. We then assume in this study that the maximum operating junction temperature allowed by the power modules for diamond and SiC diodes is 450 K. Moreover, the following hypotheses have been used:



The surface of the diamond Schottky diode is sized by three methods, the performances of these designs are then compared. The selected areas are compared in Figure 5.

PRE-PRINT (initial submitted article).

Accepted version (Elsevier, DRM2020): https://doi.org/10.1016/j.diamond.2020.108154 Case 2. The diamond diode conduction loss density at 450 K is the same as the one of the SiC diode at 300 K.

Case 3. The diamond diode surface is equal to the SiC diode surface (3.92 mm²).



The semiconductor losses (Ptot) are the sum of the conduction losses (Pcond) and the switching losses (Psw). Pcond is calculated using equation [START_REF] Chow | Smart power devices and ICs using GaAs and wide and extreme bandgap semiconductors[END_REF], where RON is the on-state resistance of the diode at a given temperature (T) and Iload is the load current of the buck converter equal to 10 A.

𝑃 𝑐𝑜𝑛𝑑 (𝑇) = 𝑅 𝑂𝑁 (𝑇) × 𝐼 𝑙𝑜𝑎𝑑 2 × (1 -𝛼) (2)
PRE-PRINT (initial submitted article). Accepted version (Elsevier, DRM2020): https://doi.org/10.1016/j.diamond.2020.108154



The transistor associated to the Schottky diode determines the switching losses. We assume that for a diode and a well-designed transistor, the output capacitance of the transistor (COSS) is equal to the transition capacitance of the Schottky diode (CT). To include the impact of the Schottky diode in the switching losses, the hypothesis that CT is added to COSS is used to calculate Psw; the sum of these two capacitances is named Csw. We also consider that the charge and discharge of the input capacitance of the transistor (CISS) is fast compared to the hard switching of the power voltage and current. Hence the gate driver circuit is neglected and has no impact on the switching losses. These hypotheses lead to the lowest possible switching losses values, which would be further increased considering circuit parasitics and EMI-induced limitations [START_REF] Meade | Parasitic inductance effect on switching losses for a high frequency Dc-Dc converter[END_REF], [START_REF] Reusch | Understanding the Effect of PCB Layout on Circuit Performance in a High-Frequency Gallium-Nitride-Based Point of Load Converter[END_REF]. The switching losses Psw are determined by the switching frequency and the energy dissipated during the transistor turn-on (Pt.on) and turn-off (Pt.off). They are calculated as follows:

1.

For Pt.on, we consider that the charges stored in Csw are dissipated in the transistor's channel. By considering both transistor and diode with a NPT design, the non-linearity of the transistor's output capacitance and the diode transition capacitance can be estimated by equation (3) [START_REF] Kolar | Extreme efficiency power electronics[END_REF]. COSS(V), CT(V), CT(1000V) and V are respectively the output capacitance of the transistor at a given voltage, the diode transition capacitance at a given voltage, the transition capacitance of the diode at 900 V and the voltage across the semiconductor device. For the SiC Schottky diode, CT(900V) is determined by its datasheet, whereas for the diamond SBD it is calculated from equation ( 4), where ε0, εr, Sdiode and epp-are respectively the vacuum permittivity, the diamond relative permittivity (5.7), the diamond diode PRE-PRINT (initial submitted article). Accepted version (Elsevier, DRM2020): https://doi.org/10.1016/j.diamond.2020.108154 surface and the diamond drift layer thickness. Based on these equations, Pt.on is calculated by using equation ( 5), Vbus being the buck converter nominal voltage.

𝐶 𝑂𝑆𝑆(𝑉) = 𝐶 𝑇(𝑉) = 𝐶 𝑇(900𝑉) √ 900 𝑉

(3)

𝐶 𝑇(900𝑉) = 𝜖 0 × 𝜖 𝑟 × 𝑆 𝑑𝑖𝑜𝑑𝑒 𝑒𝑝 𝑝 - (4)
𝑃 𝑡.𝑜𝑛 = 2 3 × 𝐶 𝑇(900𝑉) × 𝑉 𝑏𝑢𝑠 2 × 𝑓 𝑠𝑤 (5) 2.
For Pt.off by assuming a fast discharge of CISS, we can consider that the transistor channel is opened rapidly. Once the channel is opened, the crossover between the power voltage and current during the turn-off is due to a charge of Csw. In this special case, the energy is stored in Csw and is not to be dissipated. This hypothesis has been considered in this study, the switching losses are then only due to the transistor's turn-on.



The volume of the heatsink is estimated for a given operating (steady state) junction temperature and the calculated total semiconductor losses. A natural convection is assumed in this study and we consider that the thermal resistance is only caused by the heatsink. The heatsink volume is finally estimated by equation ( 6), with the hypothesis that a typical volume thermal resistance is between 500 cm 3 .K/W and 800 cm 3 .K/W [START_REF] Tong | Advanced Materials for Thermal Management of Electronic Packaging[END_REF]. This estimated volume is used as a guide to compare volumes required for diamond and SiC semiconductors. Even if the volume of heatsinks can vary with a lot of parameters like the shape of the heatsink, its material, the altitude… this estimated value allows a comparison between diamond and SiC semiconductors for the same application, with similar hypotheses and realistic data.

III. BENCHMARK RESULTS

Evolutions of the heatsink volume and total losses of both semiconductors as a function of the junction temperature are presented in Figure 6 and Figure 7. For readability reasons, only the mean value of the range of volume values calculated from equation ( 6) are shown. For all diodes, the increase of the operating temperature allows a decrease of the heatsink volume.

However, as regards to the total losses evolution, the analysis is different. By increasing the junction temperature from 300 K to 450 K, the total losses of the SiC Schottky diode are almost doubled (multiplied by 1.9). This assessment is different for the diamond SBD where the total losses are decreased with the increase of temperature.

PRE-PRINT (initial submitted article).

Accepted version (Elsevier, DRM2020): https://doi.org/10.1016/j.diamond.2020.108154 For a diamond surface designed by the conduction losses density at 450 K (black squares), an operation at 450 K allows to divide by a factor of two the total losses of the diode in comparison to an operation at 300 K. The main difference between SiC and diamond components in power electronics applications is that by using SiC devices there is a tradeoff between the decrease of the converter volume (heatsink volume) and the increase of the semiconductor losses for an increase of the operating temperature. By using diamond semiconductors, increasing the operating temperature allows both the decrease of the converter volume and the semiconductor losses. Increasing the junction temperature is then of high interest for diamond power devices.

PRE-PRINT (initial submitted article).

Accepted version (Elsevier, DRM2020): https://doi.org/10.1016/j.diamond.2020.108154

This is particularly interesting relatively to the targeted junction temperatures which are high but realistic for the realization of high temperature power modules [START_REF] Axelrod | A Novel High-Temperature Planar Package for SiC[END_REF]- [START_REF] Chen | A 1200-V, 60-A SiC MOSFET multichip phase-leg module for high-temperature, high-frequency applications[END_REF]. However, increasing the junction temperature higher than 450 K for diamond devices has not so much interest, even for low breakdown voltages in the kV range. A decrease of the heatsink volume is noticed but the total losses are only lightly improved. Moreover, the realization of power modules for temperatures higher than 525 K is very challenging, even if the most suitable materials for high temperature packaging have been identified [START_REF] Khazaka | Survey of high-temperature reliability of power electronics packaging components[END_REF], [START_REF] Coppola | Survey on high-temperature packaging materials for SiC-based power electronics modules[END_REF]. From these analyses, diamond semiconductors can be very interesting for the realization of high performance power converters.

For a 450 K operation, they allow to divide by three both heatsink volume and total semiconductor losses in comparison to SiC semiconductors for the same application. Both the efficiency and the converter power density can be increased by using diamond semiconductors.

Concerning the impact of the surface on the diamond Schottky diode performance, Erreur ! Source du renvoi introuvable. shows the evolution of the conduction and switching losses as a function of the diode surface at a 450 K junction temperature. For a given application (Vbus, Iload, fsw), one can see that the total losses are minimized for the diode surface allowing an equality between the conduction and switching losses. For small diode surfaces, total losses are dominated by conduction due to the higher diode on-state resistance. For large diode surfaces, the transition capacitance of the diode is increased, increasing the switching losses. The three diamond diode surfaces previously designed are compared in this graph, which explains the differences between the volumes and the total losses previously estimated. The diamond diode surface sized at 1.77 mm 2 by comparing the power density at 450 K for diamond and at 300 K for PRE-PRINT (initial submitted article). Accepted version (Elsevier, DRM2020): https://doi.org/10.1016/j.diamond.2020.108154

SiC (Case 2), has lower losses and heatsink volume because of its closest surface to the optimized one for this temperature (2 mm 2 ). losses at ambient temperature show that even for an optimized diamond SBD at 300 K (Case 1), the diamond losses are higher than those of SiC diode. Finally, by comparing losses and volume of both SiC and diamond diodes at their optimized temperature, losses of SiC diode at 300 K (3.9 W) are higher than those of diamond diode at 450 K (2.2 W for Case 2). It highlights the benefits that diamond semiconductors could bring in power electronics applications.

Another advantage allowed by diamond and wide bandgap semiconductors is the increase of the switching frequency, leading to smaller and lighter passive elements [START_REF] Burkart | Comparative η-ρ-σ Pareto Optimization of Si and SiC Multilevel Dual-Active-Bridge TopologiesWith Wide Input Voltage Range[END_REF]. Figure 9 shows the estimated heatsink volume and semiconductors total losses of SiC and Diamond as a function of the temperature. The best diamond design at 450 K (Case 2) has been selected. For this comparison, the diamond switching frequency has been increased at 125 kHz, whereas reduced five times with SiC devices (25 kHz). It is shown that for a 450 K operation, diamond semiconductor losses (6.1 W) are still lower than the SiC losses (7.6 W) even with the increase of the switching frequency. This gain in semiconductor losses and heatsink volume (26.4 cm 3 for diamond and 32.8 cm 3 for SiC) are associated to a decrease of the power converter input filter due to the increase of the switching frequency. Indeed, as shown in [START_REF] Safari | The Impact of Switching Frequency on Input Filter Design for High Power Density Matrix Converter[END_REF], the volume of power converter's input filter is decreased with the increase of the switching frequency. The power converter efficiency and its power density are also improved by using diamond converter at higher switching frequency, showing another advantage allowed by diamond semiconductors.

PRE-PRINT (initial submitted article).

Accepted version (Elsevier, DRM2020): https://doi.org/10.1016/j.diamond.2020.108154 To highlight the improvement of diamond performances at high temperatures and to confirm the hypotheses of the benchmark presented in section II, practical test results with diamond devices are mandatory. For this purpose, the diamond diode device shown previously was integrated in a buck converter as presented in Figure 10. The diamond device is assembled on an alumina substrate (Al2O3) by using 25 µm wire bondings. This package is then plugged on the power converter, and this solution allows to apply a heating source on the back side of the alumina substrate. The diamond device temperature can then be controlled by keeping the entire converter at Room Temperature. A double pulse test is considered in this study, allowing lower semiconductor self-heating effect than a steady state operation. The diamond junction temperature is then controlled by the heating source, where the maximum temperature is limited to 450 K. The diamond diode is associated with a Si MOSFET (BSS87 from Infineon corporation), PRE-PRINT (initial submitted article). Accepted version (Elsevier, DRM2020): https://doi.org/10.1016/j.diamond.2020.108154 which is selected to match the diamond diode characteristics (breakdown voltage and parasitic capacitance). Due to a combination of a low current level and fast switching, the semiconductors switched currents were not measured. The comparisons have then been compared with voltage measurements. Figure 11 shows the diode voltage variation (dVD/dt) during the diode turn-on and turnoff for 50 V / 200 mA at three temperatures, highlighting that the temperature has no impact on the diode switching behavior. This is mainly due to the parasitic capacitor of the diamond SBD, which is not affected by the temperature. Figure 13 represents the experimental C-V measurement of the diamond Schottky diode, which is not influenced by the temperature. Figures 12 and13 clearly show that the switching losses of diamond SBD are temperature independent.

PRE-PRINT (initial submitted article).

Accepted version (Elsevier, DRM2020): https://doi.org/10.1016/j.diamond.2020.108154

The theoretical analysis demonstrates that the switching losses are not impacted by the temperature. Figure 13a) presents the diode voltage and the load current for the double pulse test at room temperature. A zoom on VD during the diode conduction is also presented in Figure 13b),

showing the modification of the diode voltage drop for the same load current at different temperatures. The diode voltage drop is decreased with the increase of temperature, leading to a decrease of the diamond diode conduction losses at high temperature. The constant switching losses combined with the decrease of conduction losses for an increase of temperature lead to an improvement of the diamond diode total losses at high temperature. This highlights the theoretical estimations previously realized. Finally, the impact of the power converter design on the diode losses is analyzed. Only the modeled diamond diode with a 1.77 mm² surface (best design at 450 K) is compared to the SiC diode. The thermal resistance associated to each diode heatsink (65 K/W for diamond and 19.2 K/W for SiC) is determined to allow a 450 K operation at 10 A and is kept constant. The evolution of diodes performances as a function of the load current is presented in Figure 14. It simulates the integration of semiconductors devices in a power converter where the operating conditions are modified. It can also present a case where both diodes and power modules are oversized regarding to the power converter nominal current. Results show that advantages allowed by diamond at 10 A are decreased for lower load currents. Moreover, losses of the diamond diode are even higher than SiC diode for very low currents (1 A for this example, 10% of the rated current). This evolution is explained by the modification of the self-heating effect on due to the modification of the load current, and the higher on-state resistance of diamond at lower temperatures. For diamond, the low currents do not sufficiently increase the device junction temperature to reduce its on-state resistance; the interest of diamond is then decreased. It shows that to be interesting compared to SiC power devices, diamond semiconductors have to operate at higher temperatures (relatively to room temperature). Even if the targeted temperatures are reasonable for power electronics applications, the self-heating phenomenon of diamond devices needs to be sufficient. However, this temperature increase has to be controlled to avoid thermal runaway of the devices temperature and to avoid power converter and semiconductor power modules damages. This is also particularly the case during overcurrent events, where a specific attention must be paid to Schottky diode operations. The temperature management of diamond devices is then of great interest in power converters.

IV. DISCUSSION

Such a precise temperature management by using diamond semiconductors is more sensitive than power converters made of "usual" devices such as Si, SiC or GaN. Indeed, PRE-PRINT (initial submitted article). Accepted version (Elsevier, DRM2020): https://doi.org/10.1016/j.diamond.2020.108154 considering these devices, the semiconductor losses will be decreased in the case of oversized heatsinks due to the positive temperature coefficient of their on-state resistance, unlike diamond semiconductors. Then, several solutions can be used to allow diamond devices to operate at their best performances. First, the power converter can operate in a confine enclosure where the high ambient temperature is regulated to use diamond semiconductors at the targeted temperature, considering that there is no temperature exchange between the inside and outside of the enclosure. This solution has not been analyzed in this article, where the entire power converter (gate drivers, passive components …) has to be able to operate at high temperature. The second solution is more traditional, where the environment of the power converter imposes the ambient temperature. This ambient temperature can be high or low depending on the application, but is not regulated. This is the solution analyzed in this article where the ambient temperature is imposed at 300 K, as an arbitrary example for this case study. With this solution, a cooling system has to be defined to control the junction temperature of diamond semiconductors. In this study the cooling system is a heatsink in natural convection which is designed to allow a targeted junction temperature for given operating conditions of the power converter. However, if the operating conditions are modified, this solution does not allow to regulate the junction temperature of the diamond device. To solve this lack of temperature management flexibility, a forced flow can be added to the heatsink where the flow velocity is regulated as a function of the instantaneous device junction temperature. A high flow velocity can be used when the power converter is operated at high currents, avoiding a thermal runaway of the diamond semiconductor devices. The flow velocity can then be decreased for lower currents to allow a sufficient device self-heating effect, allowing the decrease of diamond devices on-state PRE-PRINT (initial submitted article). Accepted version (Elsevier, DRM2020): https://doi.org/10.1016/j.diamond.2020.108154 resistance. Such a closed loop forced convection control can be possible by using a junction temperature estimation of the device in operation in the power converter [START_REF] Perez | Integrated temperature sensor with diamond Schottky diodes using a thermosensitive parameter[END_REF]. This solution can be analyzed with a particular care on the dynamic response of the temperature control loop to avoid the diamond device thermal runaway.

Considering the diamond devices, further works are still required to bring such attractive devices in power converters. The first main improvements are the developments of devices with larger areas, with the reduction of defect densities and perfect control of diamond surface quality over a wide area. The increase of diamond "wafer" size is desired and expected [START_REF] Shikata | Single crystal diamond wafers for high power electronics[END_REF] but even with 5 mm x 5 mm diamond substrates, there is a possibility to fabricate in a first development step diamond devices with current ratings higher than 100A. Moreover, some studies are done on heteroepitaxially grown diamond substrates [START_REF] Kawashima | Electronic properties of diamond Schottky barrier diodes fabricated on silicon-based heteroepitaxially grown diamond substrates[END_REF]. The parallelization of diamond devices, considering both the NTC and PTC coefficients and the use of higher thermal resistances must be investigated. A first step towards parallelization of diamond devices has been presented in [START_REF] Perez | Parallel and interleaved structures for diamond Schottky diodes . Keywords Diamond Schottky diode static characteristics[END_REF].

The second improvements on the diamond devices are the required solutions to increase the electric field during the off-state. Indeed, reaching higher electric field, closer to the theoretical 1D breakdown, will help to increase the doping level of the drift layer and therefore reducing even further the on-state losses. Innovative solutions have been proposed for diamond junction terminations [START_REF] Arbess | High termination efficiency using polyimide trench for high voltage diamond Schottky diode[END_REF]- [START_REF] Brezeanu | Termination Structures for Diamond Schottky Barrier Diodes[END_REF], which must be further investigated and fabricated.

The theoretical benchmark proposed here must be confirmed by a fair comparison between power converters with the same specifications. Since SiC power devices are well suited for 600 V -1 kV voltage ranges, current ratings above 10 A and high switching frequencies (>20 PRE-PRINT (initial submitted article). Accepted version (Elsevier, DRM2020): https://doi.org/10.1016/j.diamond.2020.108154 kHz), similar diamond devices must be fabricated. Unfortunately, many diamond devices available at the moment suffer from premature breakdown voltage and/or high defect density reducing the maximum total active area. Consequently, it is quite difficult to integrate diamond devices in a 600 V -1 kV, 10 A class power converter, to be further benchmarked with SiC-based power converters. However, we report here the fastest switching of a power converter with diamond devices (8 to 12 V/ns, Figure 12). We also demonstrate here experimentally that diamond devices will switch at the same speed when operating at high temperature (see Figure 12 and 14), and will have lower conduction losses at higher temperatures (see Figure 1 and13b).

V. CONCLUSION

This article presents a comparison between the performances of diamond and SiC power devices in the way to highlight the benefits that diamond could bring to the power electronics community.

The diamond device characteristics are based on analytical models and are compared to a commercial SiC Schottky diode. The comparisons are performed on the estimated total semiconductor losses in a 900 V / 10 A buck converter and the volume of the required heatsink as a function of the junction temperature in steady state operation of the converter. It is shown that by using diamond semiconductors both the heatsink volume and the power devices losses can be reduced or the switching frequency can be increased still with a smaller heatsink, allowing to increase the efficiency and the power density of converters. However, due to the incomplete ionization of diamond dopants at room temperature, the diamond devices have to operate at high temperature to overpass SiC components performances. Even if these results are based on several hypotheses, it shows that a precise design of converters and a temperature management PRE-PRINT (initial submitted article). Accepted version (Elsevier, DRM2020): https://doi.org/10.1016/j.diamond.2020.108154 are needed to use full benefits of diamond semiconductors in power converters. It shows that special integration methods and special temperature managements have to be developed for the realization of future high efficiency power converters based on diamond devices.

AKNOWLEDGMENT

The research leading to these results has been performed within the GreenDiamond project (http://www.greendiamond-project.eu/) and received funding from the European The LANEF framework (ANR-10-LABX-51-01) is acknowledged for its support with mutualized infrastructure.

Figure 1 :

 1 Figure 1: Measured on-state resistance of a 200 µm x 200 µm diamond Schottky Barrier Diode as

Figure 3 :

 3 Figure 3: Measured static characteristics of the 200 µm x 200 µm diamond SBD at room

Figure 4 :

 4 Figure 4: Evolution of the on-state resistance and the estimated thermal resistance of the

Figure 5 :

 5 Figure 5: Size of the selected dies for the performance comparison. The SiC die is extracted

Figure 6 :Figure 7 :

 67 Figure 6: Estimated heatsink volume for the SiC and diamond Schottky diodes as a function of the

Figure 8 :

 8 Figure 8: Detail of the conduction and switching losses of the diamond Schottky diode at 450 K as

Figure 8

 8 Figure 8 compares the heatsink volumes and total losses of each diode normalized to

Figure 8 :

 8 Figure 8: Total losses and heatsink volume of SiC Schottky diode and the three surfaces of

Figure 9 :

 9 Figure 9: a) Estimated heatsink volume and b) estimated total losses of the SiC diode and the best

Figure 10 :

 10 Figure 10: a) Picture of the realized buck converter with the integrated diamond Schottky diode

Figure 11 :Figure 12 :

 1112 Figure 11: Measured diode voltage (VD) derivative as a function of time, for different

Figure 13 :

 13 Figure 13: a) Diode voltage and load current for a double pulse test at 50 V / 200 mA (500 A/cm²)

Figure 14 :

 14 Figure 14: Comparison of the total losses of a diamond and SiC Schottky diode as a function of

Community' s

 s Horizon 2020 Program (H2020/2014-2020) under grant agreement n° 640947. This research is partially funded by French ANR Research Agency under grant ANR-16-CE05-0023 #Diamond-HVDC.