Thibault Formal

Benjamin Piwowarski
email: benjamin.piwowarski@lip6.fr

Stéphane Clinchant

A White Box Analysis of ColBERT

Keywords: Information Retrieval, Term Matching, Transformer, BERT

Transformer-based models are nowadays state-of-the-art in adhoc Information Retrieval, but their behavior is far from being understood. Recent work has claimed that BERT does not satisfy the classical IR axioms. However, we propose to dissect the matching process of Col-BERT, through the analysis of term importance and exact/soft matching patterns. Even if the traditional axioms are not formally verified, our analysis reveals that ColBERT (i) is able to capture a notion of term importance; (ii) relies on exact matches for important terms.

Introduction

Over the last two years, Natural Language Processing has been shaken by the release of large pre-trained language models based on self-attention, like BERT [START_REF] Devlin | BERT: pre-training of deep bidirectional transformers for language understanding[END_REF]. Ranking models based on BERT are currently state-of-the-art for adhoc IR, and rank first on leaderboards 3 of the MSMARCO passage and document (re-)ranking tasks by a large margin [START_REF] Nogueira | Passage re-ranking with bert[END_REF], as well as on more standard IR datasets such as Robust04 [START_REF] Dai | Deeper text understanding for IR with contextual neural language modeling[END_REF][START_REF] Macavaney | Cedr: Contextualized embeddings for document ranking[END_REF][START_REF] Nogueira | Document ranking with a pretrained sequence-tosequence model[END_REF]. They have excelled where previous neural models had been struggling so far [START_REF] Yang | Critically examining the "neural hype[END_REF]. It is thus interesting to understand better what is happening inside those models during ranking, and what phenomena are captured. Some works have been conducted in this direction [START_REF] Câmara | Diagnosing BERT with retrieval heuristics[END_REF][START_REF] Rennings | An Axiomatic Approach to Diagnosing Neural IR Models[END_REF], but focused on whether IR axioms are respected -or not -by neural and transformer-based models. In [START_REF] Câmara | Diagnosing BERT with retrieval heuristics[END_REF], BERT has been shown to not fully respect axioms that have proved to be important for standard IR models, such as the axiom stating that words occurring in more documents are less important (IDF effect). Instead of investigating whether these models behave like standard ones, in this paper, we make a step towards understanding how they manage to improve over traditional models through their specific matching process.

There exists a wide variety of BERT-based ranking models, as summarized in the recent overview [START_REF] Lin | Pretrained Transformers for Text Ranking: BERT and Beyond[END_REF]. Canonical BERT models are difficult to analyse because they require a thorough analysis of attention mechanisms, which is a complex task [START_REF] Brunner | On identifiability in transformers[END_REF]. We rather choose to focus on contextual interaction models [START_REF] Hofstätter | Interpretable & time-budgetconstrained contextualization for re-ranking[END_REF][START_REF] Khattab | Colbert: Efficient and effective passage search via contextualized late interaction over bert[END_REF][START_REF] Macavaney | Cedr: Contextualized embeddings for document ranking[END_REF],

where query and document are encoded independently -contrary to the usual case [START_REF] Nogueira | Passage re-ranking with bert[END_REF]. Among such models, ColBERT [START_REF] Khattab | Colbert: Efficient and effective passage search via contextualized late interaction over bert[END_REF] exhibits the best trade-off between effectiveness and efficiency, with performance on par with standard BERT, suggesting that the power of these models comes from learning rich contextual representations, rather than modeling complex matching patterns. Moreover, the structure of ColBERT (sum over query terms of some similarity scores) is similar to standard IR models like BM25, and makes the analysis easier, as the contribution for each term is explicit.

In this paper, we hence focus on ColBERT, and look at two research questions. In Section 3, we investigate the link between term importance as computed by standard IR models, and the one computed by (Col)BERT. In Section 4, we look at how (Col)BERT is dealing with exact and soft matches as this is known to be critical for IR systems.

Experimental setting

Dataset For our analysis, we use the passage retrieval tasks from TREC-DL 2019 and 2020 [START_REF]Proceedings of the Twenty-Eighth Text REtrieval Conference, TREC 2019[END_REF] (400 queries in total). We consider a re-ranking setting, where for a given query q, the model needs to re-rank a set of documents S q selected by a first stage ranker. Following the MSMARCO setting, we consider candidates from BM25, and |S q | ≤ 1000. In order to study the model properties, we are interested in how it attributes scores to each query token, for documents in S q .

ColBERT We now introduce the variant of ColBERT [START_REF] Khattab | Colbert: Efficient and effective passage search via contextualized late interaction over bert[END_REF] we used to simplify the analysis -we checked each time that the drop in performance was minor. In particular, we did not include query/document specific tokens ([Q] and [D]), since these tokens could bias the representation of query/document terms. Second, while query augmentation has been shown to be beneficial in [START_REF] Hofstätter | Improving efficient neural ranking models with cross-architecture knowledge distillation[END_REF][START_REF] Khattab | Colbert: Efficient and effective passage search via contextualized late interaction over bert[END_REF], we omit this component to avoid analysis of the induced implicit query expansion mechanism. We however keep the compression layer, that projects token representations from the BERT representation space (d = 768) to the ColBERT representation space (d = 128). By fine-tuning our model in a similar fashion to [START_REF] Khattab | Colbert: Efficient and effective passage search via contextualized late interaction over bert[END_REF], we obtain a MRR@10 of 0.343 on MSMARCO dev set (versus 0.349). This shows that the above simplifications are negligible performance-wise, and would not invalidate our analysis. In order to understand what is learned during training, we also consider a non fine-tuned version of the model (without projection layer), that relies on the output of a pre-trained BERT model.

The formal definition of ColBERT, given the BERT embeddings E q = (E qi ) i for the query q (after WordPiece tokenization) and E d = (E dj ) j for the document d, is given by the following relevance score:

s(q, d) = i∈q max j∈d cos(E qi , E dj ) = i∈q max j∈d C ij = i∈q C id (1)
In the following, we say that a query token i matches the document token j * if C ij * = C id . We denote this token j * by d i .

ColBERT term importance

Our first research question focuses on comparing the term importance of standard IR models (e.g. BM25) with the term importance as determined by Col-BERT. With respect to the former, given that documents are (small) passages, term frequency is close to 1 for most terms. Moreover, passage length does not vary much, and is caped at 512 tokens. Hence, we can reasonably assume that a term BM25 score roughly corresponds to its IDF -this might not be true for terms with very low IDF values, but it is a good enough approximation for other terms.

For ColBERT, it is difficult to measure the importance of a term because it depends on both document and query contexts. We hence resort to an indirect mean, by measuring the correlation between the original ColBERT ranking and the ranking obtained when the corresponding word is masked, i.e. when we remove from the sum in Equation (1) all the contributions of subwords that compose the word. Finally, to compare rankings, we use AP-correlation4 τ AP [START_REF] Yilmaz | A new rank correlation coefficient for information retrieval[END_REF], which is akin to Kendall rank correlation, but gives more importance to the top of the ranking. Values close to 1 indicate a strong correlation, meaning that the two rankings are similar, implying a low contribution of the term in the ranking process. Note that such measure of importance is query dependent: when the term appears in several queries, we consider the average as a final measure of importance. In Figure 1, we show how IDF and τ AP are connected. There is a linear negative correlation between both metrics (Pearson correlation coefficient r = -0.4), showing that (Col)BERT implicitly captures IDF. Note that words with higher IDF tend to be longer, and hence to be split into multiple subwords more often -increasing the importance of such terms.

We also observe that the link between IDF and term importance is not so direct for high IDF values (> 8). We believe that there are three reasons explaining this behavior: (i) ColBERT has correctly learnt that this term was not so important; (ii) as most of the documents contain the term, the effect on τ AP might not be high; (iii) another query term (with no semantics) is bearing the same semantics as the target one.

The first hypothesis is probably true since ColBERT improves over BM25. As for the second one, this is a more general observation regarding the re-ranking setting, where IR axioms might not fully apply. Finally, to investigate the hypothesis (iii), we looked, for each query token, at the frequency of exact matching (i.e. the max similarity is obtained with the same token in a document) and at the frequency with which it matches in documents other terms of the query. We observed that stopwords (the, of, etc.) did indeed match terms in the documents that were other query terms. For instance, in the query (and associated τ AP ) "the (0.94) symptoms (0.87) of (0.93) shingles (0.88)", the word "of" actually mostly matches with "shingles" in documents from S q .

Analysis of Exact and Soft matches

After having looked at term importance, we now turn our attention into the issue of exact matches, i.e. how exact string matching is processed by ColBERT.

Because it has been trained to re-order a standard term-based IR model, it is interesting to check whether it might be less sensitive to such signals.

To look into this, we need to define a measure indicating when ColBERT asserts whether a term should be an exact match or not (i.e., soft match). To do so, we compute, for each query term i, the difference between the average ColBERT scores when i matches the same term within a document (i.e, when d i → t) or not (i.e., when d i → t). We then average at the query level, to obtain one measure per term (for terms appearing in several queries). This measure is formally defined as:

∆ ES (t) = mean i,q/i→t mean d∈Sq/d i →t {C id } -mean d∈Sq/d i →t {C id } (2)
where j → t means that the j th token corresponds to token t. For a word w composed of several WordPiece components t 1 , . . . , t n , we use t∈w ∆ ES (t), which corresponds to the way ColBERT works (summing over subwords). Then, for all query words w, we plot ∆ ES (w) with respect to IDF (w) (Figure 2). We can observe that there is a moderate positive correlation between terms focusing more on exact matching by ColBERT -larger ∆ ES -and IDF Fig. 2. ∆ES with respect to IDF: we observe a moderate correlation (0.667) between ∆ES and IDF, showing that the less frequent a term is, the more it is likely to be matched exactly.

(r = 0.667). Interestingly, this effect is already observable for BERT, but finetuning has an important impact for words with an IDF above 8: ColBERT thus learns to favor exact matches for such words. For instance, in the query (and associated ∆ ES ) "causes (0.35) of (0.11) left (0.64) ventricular (1.14) hypertrophy (1.62)", we can see that the model relies a lot on exact match for the last two terms.

To explain this behavior, our hypothesis is that exact matches correspond to contextual embeddings that do not vary much, while terms that carry less "information" are more heavily influenced by their context (they act as some sort of reservoirs to encode concepts of the sequence), and thus their embeddings vary a lot. To check this hypothesis, we conducted a spectral analysis of contextual term embeddings. More specifically, we use an SVD decomposition of the matrix composed of all the contextual representations for a given term t, on the test documents, and look at the relative magnitude of the singular values λ 1 ≥ ... ≥ λ d where d is the dimension of the embedding space. If the magnitude of λ 1 is much larger than the others, it means that all the contextual representations point to the same direction in the embedding space. In Figure 3, we report the ratio of the first eigenvalue λ 1 with respect to k λ k for terms that appear in the test queries. This figure confirms the above hypothesis, as the ratio increases with the subword IDF (correlation r = 0.77). Moreover, this effect is much stronger when fine-tuning ColBERT, indicating that training on relevance indeed Fig. 3. Ratio of the first eigenvalue to sum of the eigenvalues with respect to IDF (subword level). The less frequent the term is, the higher the ratio is, showing that all contextualized embedding for a rare term are concentrated in the same direction.

promotes exact matches. By looking at the distribution of singular values (not shown here), we can confirm this trend. In particular, words with a low IDF tend to point each time in a different direction, showing that what they capture is more about their context. For instance, in the query "when did family feud come out ? " (a TV show), the term "come", for all the documents in S q , matches 97% of the time to document terms that are not in the query, but are synonyms (in a broad sense) e.g. {july, happen, item, landing, released, name, en, going, it, rodgers}.

Conclusion

While the axiomatic approach is appropriate to analyze traditional IR models, its application to BERT-based models remains limited and somehow inadequate.

To the best of our knowledge, our study is one of the first to shed light on some matching behaviors of BERT, through the analysis of a simpler counterpart, ColBERT. We showed that (i) even if the IDF effect from the axiomatic theory is not enforced, (Col)BERT does have a notion of term importance; (ii) exact matching remains an important component of the model, and is amplified after fine-tuning on relevance; (iii) our analysis gave some hints on the properties of frequent words which tend to capture the contexts in which they appear.

Although this work is a first step towards understanding matching properties of BERT in IR, we believe there is much more to uncover by either analyzing a wider range of models, or by extending our analysis of ColBERT to first stage ranking, where retrieval axioms might be more critical.

Fig. 1 .

 1 Fig. 1. ColBERT term importance (as computed using τAP ) with respect to IDF (standard term importance).

using the Python implementation provided by[START_REF] Urbano | The treatment of ties in ap correlation[END_REF].


Keywords: ultra-wide bandgap, gallium oxide, packaging, Schottky barrier diodes, thermal resistance

The low thermal conductivity of Ga2O3 has arguably been the most serious concern for Ga2O3 power and RF devices. Despite many simulation studies, there is no experimental report on the thermal resistance of a large-area, packaged Ga2O3 device. This work fills this gap by demonstrating a 15-A double-side packaged Ga2O3 Schottky barrier diode (SBD) and measuring its junction-to-case thermal resistance (RθJC) in the bottom-side-and junction-side-cooling configurations. The RθJC characterization is based on the transient dual interface method, i.e., JEDEC 51-14 standard. The RθJC of the junction-and bottom-cooled Ga2O3 SBD was measured to be 0.5 K/W and 1.43 K/W, respectively, with the former RθJC lower than that of similarly-rated commercial SiC SBDs. This low RθJC is attributable to the heat extraction directly from the Schottky junction instead of through the Ga2O3 chip. The RθJC lower than that of commercial SiC devices proves the viability of Ga2O3 devices for high-power applications and manifest the significance of proper packaging for their thermal management.

I. INTRODUCTION

ltra-wide-bandgap semiconductor gallium oxide (Ga2O3) has been promoted for years as a promising candidate for power electronics and RF applications, due to its high critical electrical field, controllable n-type doping, and the availability of large-diameter wafers by the melt growth [START_REF] Pearton | A review of Ga2O3 materials, processing, and devices[END_REF]- [START_REF] Li | Single and multi-fin normally-off Ga2O3 vertical transistors with a breakdown voltage over 2.6 kV[END_REF]. Whereas a fundamental limitation of Ga2O3 is its low thermal conductivity (kT = 0.1-0.3 Wcm -1 K -1 [START_REF] Pearton | A review of Ga2O3 materials, processing, and devices[END_REF]), which is about 1/6 of the kT of Si, 1/10 of GaN, and 1/20 of SiC. The resulting high thermal resistance of Ga2O3 chip has brought serious concerns regarding the current and power scalability of Ga2O3 devices and their competitiveness in industrial power and RF applications.

The thermal resistance is an essential metric in the datasheet of any power device. Despite some simulation and modeling works [START_REF] Paret | Thermal and Thermomechanical Modeling to Design a Gallium Oxide Power Electronics Package[END_REF]- [START_REF] Buttay | Surge current capability of ultra-wide-bandgap Ga2O3 Schottky diodes[END_REF], there has been no experimental reports of the thermal resistance of a large-area, packaged Ga2O3 device. The lack of this data makes it difficult to compare Ga2O3 with commercial device technologies (e.g., Si, SiC, GaN) and evaluate the application space of Ga2O3 devices. Some recent works characterized the channel (or junction) temperatures in Ga2O3 devices [START_REF] Blumenschein | Self-Heating Characterization of β -Ga2O3 Thin-Channel MOSFETs by Pulsed I -V and Raman Nanothermography[END_REF]- [START_REF] Wong | Characterization of channel temperature in Ga2O3 metal-oxide-semiconductor field-effect transistors by electrical measurements and thermal modeling[END_REF] and studied different approaches to lower device temperatures, e.g., heterogenous integration [START_REF] Xu | First Demonstration of Waferscale Heterogeneous Integration of Ga2O3 MOSFETs on SiC and Si Substrates by Ion-Cutting Process[END_REF]- [START_REF] Wang | β-Ga2O3 MOSFETs on the Si substrate fabricated by the ion-cutting process[END_REF] and substrate thinning [START_REF] Yu | Device Topological Thermal Management of β-Ga2O3 Schottky Barrier Diodes[END_REF]. However, all of these devices have small areas with a current much lower than 1 Amp, and none of these devices are packaged.

This work fills this critical knowledge gap by demonstrating a 15-A double-side-packaged vertical Ga2O3 Schottky barrier diode (SBD) and characterizing its junction-to-case thermal resistance (RθJC) following the JEDEC 51-14 standard [START_REF]TRANSIENT DUAL INTERFACE TEST METHOD FOR THE MEASUREMENT OF THE THERMAL RESISTANCE JUNCTION[END_REF]. The RθJC of the same device were measured in the bottom-side-and junction-side-cooling schemes, where the bottom-side cooling is dominant in the packages of commercial devices. The RθJC of the junction-cooled Ga2O3 SBD was found to be smaller than that of similarly-rated commercial SiC SBDs. These results remove some of the key thermal concerns for Ga2O3 devices.

II. DEVICE FABRICATION AND PACKAGING

Fig. 1(a) shows the schematic of the packaged Ga2O3 SBD. The Ga2O3 wafer consists of a 10-µm n-Ga2O3 drift layer (Si: ~10 16 cm -3 ) grown on a 2-inch n + -Ga2O3 substrate. The substrate was thinned down to 500 µm [START_REF] Allen | Vertical Ga2O3 Schottky Barrier Diodes With Small-Angle Beveled Field Plates: A Baliga's Figure-of-Merit of 0.6 GW/cm 2[END_REF]. The device fabrication is similar to the ones in [START_REF] Allen | Vertical Ga2O3 Schottky Barrier Diodes With Small-Angle Beveled Field Plates: A Baliga's Figure-of-Merit of 0.6 GW/cm 2[END_REF], [START_REF] Wang | High-voltage vertical Ga2O3 power rectifiers operational at high temperatures up to 600 K[END_REF]. The cathode ohmic contact was formed by Ti/Au, and the anode Schottky contact by Ni/Au. A planar field plate was made by 1-µm SiO2. A Ti/Ag (100/200 nm) stack was deposited on both contacts as adhesion layers to the sintered nanosilver bond-line. Ti also serves as a barrier layer to prevent the metal diffusion in the sintering process.

The device packaging process was similar to that in [START_REF] Xiao | Packaged Ga2O3 Schottky Rectifiers With Over 60-A Surge Current Capability[END_REF]. For die attach, a 50-µm-thick nanosilver paste was sintered without pressure at 250 o C [START_REF] Calata | Sintered nanosilver paste for high-temperature power semiconductor device attachment[END_REF]. Each side of the chip was bonded to a 1-mm-thick silver (Ag) plate. Some low-kT silicone gel was applied to encapsulate the chip. The Schottky contact area was 3×3 mm 2 , the total Ga2O3 chip size was 4.6×4.6 mm 2 , and the Ag plate size was 7.3×7.3 mm 2 .

Fig. 1(b)-(d) show the forward I-V, reverse C-V, and reverse I-V characteristics of the packaged Ga2O3 SBD, revealing a turn-on voltage (VON) of 0.83 V extracted at 1 A/cm 2 , a forward current of 15 A at 2.15 V, an on/off ratio of ~10 10 extracted at 2 V/-50 V, and a breakdown voltage (BV) over 600 V. Note that a simple planar field plate was used in this work, hence the BV and reverse leakage current have much room for improvement. By adding a mesa, a BV up to 1100 V was demonstrated on a similar wafer in small-area devices [START_REF] Allen | Vertical Ga2O3 Schottky Barrier Diodes With Small-Angle Beveled Field Plates: A Baliga's Figure-of-Merit of 0.6 GW/cm 2[END_REF]. The ND extracted from the C-V characteristics is ~2×10 16 cm -3 , and it shows small temperature dependence.

III. THERMAL RESISTANCE MEASUREMENTS

The RθJC measurement was based on the transient dual interface method (TDIM) (i.e., JEDEC 51-14 standard [START_REF]TRANSIENT DUAL INTERFACE TEST METHOD FOR THE MEASUREMENT OF THE THERMAL RESISTANCE JUNCTION[END_REF]). This TDIM method relies on two transient thermal impedance curves (Z~t) measured with different contact thermal resistances between the package case surface and the ambient. The Z value at the separation point of the two curves is close to the device steady-state RθJC [START_REF]TRANSIENT DUAL INTERFACE TEST METHOD FOR THE MEASUREMENT OF THE THERMAL RESISTANCE JUNCTION[END_REF]. This method avoids the errors caused by traditional thermocouple methods [START_REF] Deng | Temperature Influence on the Accuracy of the Transient Dual Interface Method for the Junctionto-Case Thermal Resistance Measurement[END_REF], and has been widely used for Si [START_REF] Székely | Fine structure of heat flow path in semiconductor devices: A measurement and identification method[END_REF], [START_REF] Rencz | Measuring partial thermal resistances in a heat-flow path[END_REF], SiC [START_REF] Kim | Thermal simulation and measurement of SiC MOSETs[END_REF], and GaN [START_REF] Zhai | Transient dual interface measurement of junction-to-case thermal resistance in AlGaN/GaN HEMT utilizing an improved infrared microscope[END_REF] devices. In the TDIM method, the junction temperature (Tj) is usually monitored by continuously measuring a thermo-sensitive electrical parameter (TSEP) [START_REF] Avenas | Temperature Measurement of Power Semiconductor Devices by Thermo-Sensitive Electrical Parameters-A Review[END_REF]. The forward voltage at 10 mA was selected as the TSEP for our Ga2O3 SBDs, which shows an excellent linearity with the temperature (Fig. 2(d)). This test was performed in an oven with the thermocouple and electrical wires placed into the oven via small holes on the oven surface to allow the TSEP and temperature measurements.

The RθJC measurements started by applying a forward dc bias to the SBD for self-heating, producing a power (PH), until the steady state was reached with a constant Tj (Tj0). Subsequently, the dc power was cut off, and the TSEP was monitored to obtain the evolution of Tj(t) in the cooling phase. The Z~t curve was calculated by 𝑍 𝜃𝐽𝐶 (𝑡) = (𝑇 𝑗0 -𝑇 𝑗 (𝑡)) 𝑃 𝐻 ⁄ [START_REF]TRANSIENT DUAL INTERFACE TEST METHOD FOR THE MEASUREMENT OF THE THERMAL RESISTANCE JUNCTION[END_REF], [START_REF] Deng | Temperature Influence on the Accuracy of the Transient Dual Interface Method for the Junctionto-Case Thermal Resistance Measurement[END_REF]. As our TSEP is in the SBD subthreshold region, minimal heating is produced in its testing (< 7.6 mW), and a high signal-to-noise ratio are ensured in the Z measurement.

For each RθJC test, two Z~t curves were acquired by using two different thermal interface materials (TIMs) between the indium foil and the cold plate, i.e., some silicone oil (lower kT) and some thermal grease (higher kT). The separation point of the two heating Z~t curves was extracted as RθJC by the Analyzer software following the JEDEC standard [START_REF]TRANSIENT DUAL INTERFACE TEST METHOD FOR THE MEASUREMENT OF THE THERMAL RESISTANCE JUNCTION[END_REF]. The RθJC of a commercial SiC SBD (SCS220KGHR) was first measured. The measured value (0.6 K/W) agreed with the datasheet value (0.62 K/W), validating our test setup and procedure.

Fig. 3(a) and (b) show the measured Z~t curves of our packaged Ga2O3 SBD in the bottom-side-and junction-sidecooling schemes, respectively, revealing a much lower RθJC (0.5 K/W) under the junction-side cooling as compared to the RθJC (1.43 K/W) under the bottom-side cooling.

A cumulative structure function can be calculated from each Z~t curve [START_REF]TRANSIENT DUAL INTERFACE TEST METHOD FOR THE MEASUREMENT OF THE THERMAL RESISTANCE JUNCTION[END_REF], which gives the sum of thermal capacitances (C∑) with respect to the sum of thermal resistances (R∑) in the packaged device structure, measured from the point of heating excitation (i.e., junction) toward the ambient. Each slope in this function represents either a new material or an increase in the cross sectional area of the heat flow or both [START_REF] Rencz | Measuring partial thermal resistances in a heat-flow path[END_REF].

Fig. 3(c) and (d) show the calculated structure function of the packaged Ga2O3 SBD in the bottom-and junction-side-cooling schemes, respectively, each scheme with two TIMs. The separation points show good consistence with those extracted from Z~t curves. The function before the separation point provides the structure information within the package case. The key difference between the junction-and bottom-cooling functions is that the former shows an almost constant slope before the separation point while the latter shows two regions with different slopes. The slope in the junction-cooling function corresponds to the nano-Ag attach and Ag plate (Fig. 3(c)). The almost constant slope suggests little heat up into the Ga2O3 chip. The first slope in the bottom-cooling function corresponds to the Ga2O3 chip (Fig. 3(d)), as its span (~0.8 K/W) is close to the calculated Rθ using the Ga2O3 kT and chip geometries (Rθ = 0.5mm/0.25-Wcm -1 K -1 /22-mm 2 = 0.9-K/W). The second slope corresponds to the bottom attach and Ag plate. Its R and C spans (~0.5 K/W and ~0.1 Ws/K) before the separation point are similar to the counterparts in the junction-cooling function.

Based on the measured RθJC, 3-D TCAD simulations were performed in Silvaco Atlas to evaluate the device Rθ in various external cooling conditions. The electrothermal models were similar to [START_REF] Zhang | Electrothermal Simulation and Thermal Performance Study of GaN Vertical and Lateral Power Transistors[END_REF], and the material models (e.g., kT of Ga2O3 and nanosilver, interface Rθ) were based on [START_REF] Xiao | Packaged Ga2O3 Schottky Rectifiers With Over 60-A Surge Current Capability[END_REF]. A copper plate with a geometry similar to the experimental setup was added to the package surface where the external cooling is applied. The simulated RθJC and I-V characteristics were calibrated with the experimental data. The calibration revealed that the kT of sintered nanosilver is ~1 Wcm -1 K -1 [START_REF] Xiao | Packaged Ga2O3 Schottky Rectifiers With Over 60-A Surge Current Capability[END_REF], suggesting the room for further improvement of the sintering process. Table I benchmarks the RθJC of our Ga2O3 SBDs against commercial 600-V SiC SBDs with a similar current rating and different TO-series packages (the dominant packages for commercial power devices), as well as a small-area unpackaged Ga2O3 SBD reported previously [START_REF] Chatterjee | Thermal characterization of gallium oxide Schottky barrier diodes[END_REF]. The RθJC of our junctionside cooled Ga2O3 SBD is lower than that of commercial SiC SBDs with a similar package size and current rating. As a more direct comparison, if TO-263-2 package is used for our Ga2O3 SBDs, RθJC is estimated to be 0.61 and 1.54 K/W for the anode facing up and down, respectively. This estimation assumes the use of solder alloy as die attach with a thickness of 150-µm and kT of 0.23 Wcm -1 K -1 instead of Ag sintering.

  Fig. 1 (a) Schematic of the double-side packaged Ga2O3 SBD encapsulated in silicone gel. Temperature-dependent (b) forward I-V, (b) C-V and (c) reverse I-V (up to 600 V) characteristics of the packaged device.

Fig. 2 (

 2 a) shows our RθJC measurement set-up using an Analysis Tech Phase 12 Semiconductor Thermal Analyzer. The Ga2O3 SBD was placed on a water-cooling cold plate with a 26 o C constant temperature. An indium foil was attached to each Ag plate to conduct electric signals. A top plastic clamp applied a ~15-psi pressure to ensure good and consistent contacts. As this clamp has very low thermal conductivity, this setup allows heat extraction dominantly towards the bottom water-cooling plate. Fig. 2(b) and (c) show the bottom-and junction-cooling measurements of the same double-side packaged Ga2O3 SBD.

Fig. 2 .

 2 Fig. 2. (a) Photo of the test setup. Schematic of Rθ measurements under (b) bottom-side cooling and (c) junction-side cooling. (d) The forward voltage at 10 mA current as a function of temperature of the packaged Ga2O3 SBD.

Fig. 3 .

 3 Fig. 3. Transient thermal impedance curves of the Ga2O3 SBD measured with two TIMs under the (a) junction-and (b) bottom-side cooling. The insets show the zoom-in plot of the separation point. Calculated structure function with two TIMs in the (c) junction-and (d) bottom-side cooling.

Fig. 4 (

 4 a) shows the simulated heat flux contours in a doubleside-cooled SBD with 26 o C fixed on both package surfaces [i.e., infinite heat transfer coefficient (HTC)]. Most heat flows via the junction side of the package, agreeing with the expectation from the much lower junction-cooled RθJC. Fig. 4(b) shows the simulated junction-to-coolant (-ambient) Rθ as a function of HTC (representing different cooling methods) for our Ga2O3 SBD in the bottom-, junction-and double-side cooling schemes. The results suggest that the junction-cooling is essential for Ga2O3 devices and the double-side-cooling can further reduce Rθ by 30~40%. An HTC over 10 3 W/m 2 K (e.g., forced water cooling) is preferable for external cooling; a lower HTC may lead to a fast increase in Rθ for Ga2O3 devices.

Fig. 4 . 5 *

 45 Fig. 4. (a) Simulated heat flux contours in the Ga2O3 SBD cross-section under the double-side cooling. The positive and negative values represent the upwards and downwards flux, respectively. The bottom and top case temperatures are set as 26 o C (i.e., HTC = ∞). (b) Simulated junction-tocoolant thermal resistance of the Ga2O3 SBD as a function of HTC at the case surface, under the bottom-, junction-and double-side cooling. The HTC ranges for different external cooling techniques are marked in green.

  Sasaki is with Novel Crystal Technology, Inc., Sayama 350-1328, Japan. This work was supported in part by the National Science Foundation under Grant ECCS-2100504 and in part by the Center for Power Electronics Systems High Density Integration Industry Consortium at Virginia Polytechnic Institute and State University.

K.

IV. SUMMARY

This work presents the first RθJC data of large-area, packaged Ga2O3 devices measured following the JEDEC standard. The packaged Ga2O3 SBD shows over 15 A current and 600 V BV. The RθJC under the junction-side-and bottom-side-cooling is 0.5 W/K and 1.43 W/K, respectively. The difference is primarily attributable to the low-kT Ga2O3 chip. The RθJC of our Ga2O3 SBD under the junction cooling is lower than RθJC of similarlyrated commercial SiC SBDs, suggesting the feasibility of using the proper packaging to overcome the low kT of Ga2O3 and thereby enable Ga2O3 devices for high-power applications.


Y Zhang
email: yhzhang@vt.edu

B Wang

M Xiao

J Spencer

R Zhang

J Knoll

C Dimarino

G Q Lu

K Sasaki

C Buttay

(Invited) How to Achieve Low Thermal Resistance and High Electrothermal Ruggedness in Ga2O3 Devices?

Ultra-wide bandgap gallium oxide (Ga2O3) devices have recently emerged as promising candidates for power and RF electronics. The low thermal conductivity of Ga2O3 has arguably been the most serious concern for these devices. Despite many simulation studies, there still lacks an experimental report on the thermal resistance and electrothermal ruggedness of a large-area, packaged Ga2O3 device. Recently, our team for the first time demonstrated largearea Ga2O3 devices with different packaging configurations and measured the thermal resistance and surge current capabilities of these packaged Ga2O3 devices. This paper reviews the key results in our efforts. It is shown that, contrary to some popular belief, Ga2O3 devices with proper packaging can achieve high thermal performance in both short transients and the steady state. The double-side-packaged Ga2O3 Schottky rectifiers show a junctionto-case thermal resistance lower than that of the similarly-rated commercial SiC Schottky rectifiers. In addition, these Ga2O3 rectifiers can survive a higher peak surge current as compared to SiC rectifiers. The critical enabler for these excellent performances is the direct junction cooling with minimal heat going through the Ga2O3 chip. Our work proves the viability of Ga2O3 devices for high power applications and manifests the significance of packaging for their die-level thermal management.

Introduction

Ultra-wide-bandgap (UWBG) semiconductor gallium oxide (Ga2O3) has been promoted for years as a promising candidate for power electronics and RF applications, due to its high critical electrical field (E-field), controllable n-type doping, and the availability of large-diameter wafers by the melt growth [START_REF] Pearton | A review of Ga2O3 materials, processing, and devices[END_REF][START_REF] Higashiwaki | Guest Editorial: The dawn of gallium oxide microelectronics[END_REF]. Recently, kilovolt-class Ga2O3 Schottky barrier diodes (SBDs) have been demonstrated with a peak junction E-field exceeding the critical E-field of GaN and SiC [START_REF] Allen | Vertical Ga 2 O 3 Schottky Barrier Diodes With Small-Angle Beveled Field Plates: A Baliga's Figure-of-Merit of 0.6 GW/cm 2[END_REF][START_REF] Zhang | Ultra)Wide-Bandgap Vertical Power FinFETs[END_REF][START_REF] Li | Single and multi-fin normally-off Ga2O3 vertical transistors with a breakdown voltage over 2.6 kV[END_REF]. However, a fundamental limitation of Ga2O3 is its low thermal conductivity (kT = 0.1-0.3 Wcm -1 K -1 ), which is about 1/6 of the kT of Si, 1/10 of GaN, and 1/20 of SiC. The resulting high thermal resistance of Ga2O3 chip has brought serious concerns regarding the current and power scalability of Ga2O3 devices and their electrothermal ruggedness. As a result, questions have long persisted on the true viability of Ga2O3 devices for industrial power and RF applications. Despite many simulation and modeling works on the thermal management of Ga2O3 devices [START_REF] Paret | Thermal and Thermomechanical Modeling to Design a Gallium Oxide Power Electronics Package[END_REF][START_REF] Mahajan | Electrothermal performance limit of β-Ga2O3 field-effect transistors[END_REF][START_REF] Yuan | Modeling and analysis for thermal management in gallium oxide field-effect transistors[END_REF][START_REF] Chatterjee | Device-Level Thermal Management of Gallium Oxide Field-Effect Transistors[END_REF], there has been no experimental report on the packaging and thermal management of large-area Ga2O3 devices. The lack of these data makes it difficult to compare Ga2O3 with commercial device technologies (e.g., Si, SiC, GaN) and evaluate the true application space of Ga2O3 devices. Some recent works characterized the channel (or junction) temperature in Ga2O3 devices [START_REF] Blumenschein | Self-Heating Characterization of β -Ga2O3 Thin-Channel MOSFETs by Pulsed I -V and Raman Nanothermography[END_REF][START_REF] Pomeroy | Raman Thermography of Peak Channel Temperature in β-Ga2O3 MOSFETs[END_REF][START_REF] Chatterjee | Thermal characterization of gallium oxide Schottky barrier diodes[END_REF], and developed various methods to reduce the temperature [START_REF] Xu | First Demonstration of Waferscale Heterogeneous Integration of Ga2O3 MOSFETs on SiC and Si Substrates by Ion-Cutting Process[END_REF][START_REF] Zhang | Overcoming the limitations of gallium oxide through heterogeneous integration[END_REF], but all of these devices have small areas with a current much lower than 1 Amp, and none of these devices are packaged.

To fill these gaps, our team for the first time demonstrated large-area, packaged Ga2O3 devices and characterized their thermal resistance and transient electrothermal ruggedness. This paper reviews the key results, with the details published in [START_REF] Buttay | Surge current capability of ultra-wide-bandgap Ga2O3 Schottky diodes[END_REF][START_REF] Xiao | Packaged Ga2O3 Schottky Rectifiers With Over 60-A Surge Current Capability[END_REF][START_REF] Wang | Low Thermal Resistance (0.5 K/W) Ga2O3 Schottky Rectifiers with Double-Side Packaging[END_REF][START_REF] Wang | High-voltage vertical Ga2O3 power rectifiers operational at high temperatures up to 600 K[END_REF]. Vertical Ga2O3 SBDs with a 3×3 mm 2 Schottky contact area were fabricated, showing a forward current over 20 A and a breakdown voltage (BV) over 600 V. Small-area Ga2O3 SBDs fabricated on the same wafer exhibited capabilities to operate at high temperatures up to 600 K [START_REF] Wang | High-voltage vertical Ga2O3 power rectifiers operational at high temperatures up to 600 K[END_REF]. The fabricated large-area SBDs were then packaged in the bottom-and double-side-cooling configurations using the nanosilver sintering as the die attach.

To evaluate the device's steady-state thermal performance, the junction-to-case thermal resistance (RθJC) of a double-side-packaged Ga2O3 SBD was measured in the bottom-side-and junction-side-cooling configurations. The RθJC is an essential metric in the datasheet of any commercial power device. The RθJC characterization was based on the transient dual interface method, i.e., JEDEC 51-14 standard [START_REF]TRANSIENT DUAL INTERFACE TEST METHOD FOR THE MEASUREMENT OF THE THERMAL RESISTANCE JUNCTION[END_REF]. The RθJC of the junction-and bottom-cooled Ga2O3 SBDs was measured to be 0.5 K/W and 1.43 K/W, respectively. The former RθJC was found to be lower than that of similarly-rated commercial SiC SBDs. This low RθJC is attributable to the heat extraction directly from the Schottky junction instead of through the Ga2O3 chip.

Surge current is an essential ruggedness metric listed in any power diode's datasheet and the most important indicator of a device's transient electrothermal ruggedness [START_REF] Buttay | Surge current capability of ultra-wide-bandgap Ga2O3 Schottky diodes[END_REF][START_REF] Liu | Surge Current and Avalanche Ruggedness of 1.2-kV Vertical GaN p-n Diodes[END_REF]. It measures the device's capability of temporarily sustaining a current much higher than the rated current. A surge-current test circuit was prototyped to produce a 10-mswide half-sinusoidal current waveform based on the JEDEC standard. The surge-current tests revealed a critical surge current of 37.5 A for the single-side-packaged Ga2O3 SBD and 68 A for the double-side-packaged Ga2O3 SBD. The latter Ga2O3 SBD shows a ratio between the peak surge current and the rated current higher than that of the similarlyrated commercial SBDs. This superior capability is attributable to the small temperature dependence of on-resistance (RON) of Ga2O3 devices, which strongly reduces the thermal runaway, and the double-side-cooled packaging, which allows direct junction cooling.

Our results have removed some critical concerns regarding the thermal performance and ruggedness of Ga2O3 devices and suggested the strong need for the device-packaging co-design for Ga2O3 devices. The co-optimization should also be performed in the context of circuit operations, e.g., transient dynamics in a surge (short) current profile.

Bare-Die Device: Fabrication and High-Temperature Characteristics

High-temperature characterizations of bare-die devices are the pre-requisites for the thermal study of packaged Ga2O3 devices. Hence, we first fabricated small-area vertical Ga2O3 SBDs [START_REF] Allen | Vertical Ga 2 O 3 Schottky Barrier Diodes With Small-Angle Beveled Field Plates: A Baliga's Figure-of-Merit of 0.6 GW/cm 2[END_REF][START_REF] Wang | High-voltage vertical Ga2O3 power rectifiers operational at high temperatures up to 600 K[END_REF]. A critical knowledge gap of Ga2O3 devices in the literature was the lack of reports on the high-voltage blocking capability at high temperatures. This gap makes the applicability of Ga2O3 devices in harsh-environment power applications questionable. To this end, we focused on evaluating the high-voltage blocking capabilities of our bare-die Ga2O3 SBDs at high temperatures.

The commercially available Ga2O3 wafer from Novel Crystal Technologies consists of a 10-µm Si-doped n-Ga2O3 epitaxial drift layer (net donor concentration ~2×10 16 cm -3 ) grown on a 2-inch n + -Ga2O3 (001) substrate (Sn: 1.3×10 19 cm -3 ). The substrate was thinned down to a thickness of 500 µm. The device fabrication is detailed in [START_REF] Allen | Vertical Ga 2 O 3 Schottky Barrier Diodes With Small-Angle Beveled Field Plates: A Baliga's Figure-of-Merit of 0.6 GW/cm 2[END_REF][START_REF] Wang | High-voltage vertical Ga2O3 power rectifiers operational at high temperatures up to 600 K[END_REF][START_REF] Wong | TCAD-Machine Learning Framework for Device Variation and Operating Temperature Analysis With Experimental Demonstration[END_REF][START_REF] Dhillon | TCAD-Augmented Machine Learning With and Without Domain Expertise[END_REF]. A layer of SiO2 was deposited, followed by patterned wet etch, which functions as the hard mask for Ga2O3 mesa etch. Then a 1-µm-thick spin-on-glass (SOG) was deposited and selectively wet etched to produce a controllable bevel angle (3), which functions as the field plate (FP) dielectrics. A Ti/Au Ohmic contact is formed on the backside of the substrate, and a Ni/Au stack is deposited as the Schottky and FP metals. The device structure is illustrated in Fig. 1(a). Fig. 1(b) shows scanning electron microscopy (SEM) images of the edge termination region, revealing a bevel angle of 45 o in the FP dielectrics and Ga2O3 mesa. Fig. 1(c) and (d) show the forward I-V and highbias reverse I-V characteristics of the fabricated Ga2O3 SBDs, respectively, demonstrating the capability of blocking at least 500 V up to 600 K. The high-bias leakage current can be explained by a combination of the thermionic-field emission (TFE) across the Schottky barrier and the electron hopping via the defect states in the depletion region [START_REF] Wang | High-voltage vertical Ga2O3 power rectifiers operational at high temperatures up to 600 K[END_REF]. The latter mechanism was widely reported in other wide-bandgap high-voltage power devices, e.g., GaN [START_REF] Zhang | Gallium nitride vertical power devices on foreign substrates: a review and outlook[END_REF][START_REF] Zhang | GaN FinFETs and trigate devices for power and RF applications: review and perspective[END_REF][START_REF] Xiao | Leakage and breakdown mechanisms of GaN vertical power FinFETs[END_REF][START_REF] Zhang | Design space and origin of off-state leakage in GaN vertical power diodes[END_REF][START_REF] Liu | Trap-Mediated Avalanche in Large-Area 1.2 kV Vertical GaN p-n Diodes[END_REF] and SiC [START_REF] Kozak | Degradation of SiC MOSFETs under High-Bias Switching Events[END_REF][START_REF] Kozak | Physics of Degradation in SiC MOSFETs Stressed by Overvoltage and Overcurrent Switching[END_REF][START_REF] Kozak | Robustness Evaluation and Degradation Mechanisms of SiC MOSFETs Overstressed by Switched Stimuli[END_REF]. These results verify the high-temperature stability of high-voltage Ga2O3 SBDs. Subsequently, large-area vertical Ga2O3 SBDs with a Schottky contact area of 3×3 mm 2 were fabricated on the same wafer. To simplify the device structure to maximize the device fabrication yield (and leave margin for packaging yield), our first-generation large-area Ga2O3 SBDs employ only a planar FP without the mesa structure. According to the simulation (3), a small FP bevel angle can reduce the E-field crowding. Hence, a ~15 o bevel angle in the FP was fabricated. Fig. 2(a) and (b) show the forward and reverse I-V characteristics of the fabricated large-area Ga2O3 SBDs, respectively, revealing a 0.83 V turn-on voltage (VON) extracted at 1 A/cm 2 , a forward current of 15 A at 2.15 V, an on/off ratio of ~10 10 , and a capability to block at least 600 V.

Device Packaging

The packaging started with depositing 100-nm-thick Ti and 200-nm-thick Ag on both sides of contacts as adhesion layers to the sintered nanosilver bond-line. Ti also serves as a barrier layer to prevent metal diffusion during the sintering process. A nanosilver paste from NBE Technologies was then used for the die attach by a pressureless sintering process in air [START_REF] Wang | Low-Temperature Sintering with Nano-Silver Paste in Die-Attached Interconnection[END_REF][START_REF] Calata | Sintered nanosilver paste for hightemperature power semiconductor device attachment[END_REF]. More details on the sintering process were reported in [START_REF] Xiao | Packaged Ga2O3 Schottky Rectifiers With Over 60-A Surge Current Capability[END_REF]. Devices with single-side-and double-side-cooled packages were then prototyped, as shown in Fig. 3(a) and (b). For the single-side-cooled package, the cathode of the Ga2O3 chip was sintered on a 1-mm thick Ag plate, and wire-bonds were attached on the top anode. For the double-side-cooled package, each terminal of the Ga2O3 chip was sintered on a 1-mm thick Ag plate. Fig. 3(c) shows a double-side-cooled package mounted on a ceramic substrate with wire-bond connections, ready for surge tests. Substrate and wire-bonds were not used for the thermal resistance tests.

Thermal Resistance Measurement

The RθJC measurement was based on the transient dual interface method (TDIM) (i.e., JEDEC 51-14 standard [START_REF]TRANSIENT DUAL INTERFACE TEST METHOD FOR THE MEASUREMENT OF THE THERMAL RESISTANCE JUNCTION[END_REF]). This TDIM method relies on two transient thermal impedance curves (Z~t) measured with different contact thermal resistances between the package case surface and the ambient. The Z value at the separation point of the two curves is extracted as the device steady-state RθJC [START_REF]TRANSIENT DUAL INTERFACE TEST METHOD FOR THE MEASUREMENT OF THE THERMAL RESISTANCE JUNCTION[END_REF]. The RθJC measurements started by applying a dc bias to the SBD for self-heating until the junction reaches the steady state (junction temperature = Tj). The power was then cut off, and the Tj evolution was measured by monitoring a thermo-sensitive electrical parameter (TSEP) [START_REF] Avenas | Temperature Measurement of Power Semiconductor Devices by Thermo-Sensitive Electrical Parameters-A Review[END_REF]. For our Ga2O3 SBDs, the TSEP was selected as the forward voltage at 10 mA, which showed good linearity with Tj. With Tj, the Z~t curve was calculated. For each RθJC test, two Z~t curves were acquired by using two different thermal interface materials (TIMs) between the indium foil and the cold plate, i.e., some silicone oil (lower kT) and some thermal grease (higher kT). The separation point of the two heating Z~t curves was extracted as RθJC [START_REF]TRANSIENT DUAL INTERFACE TEST METHOD FOR THE MEASUREMENT OF THE THERMAL RESISTANCE JUNCTION[END_REF] Table I compares the RθJC of our Ga2O3 SBDs against commercial 600-V SiC SBDs with a similar current rating and different TO-series packages. The RθJC of our junctionside cooled Ga2O3 SBD is lower than that of commercial SiC SBDs with a similar package size and current rating. This suggests the feasibility of employing the proper packaging to overcome the low kT of Ga2O3.

Surge Current Test

Fig. 5(a) and (b) show the surge-current test circuit and the prototype, respectively. The working principle and implementation of the test circuit are detailed in [START_REF] Xiao | Packaged Ga2O3 Schottky Rectifiers With Over 60-A Surge Current Capability[END_REF][START_REF] Liu | Surge Current and Avalanche Ruggedness of 1.2-kV Vertical GaN p-n Diodes[END_REF]. A 10-ms-wide half-sinusoidal current waveform was produced by a resonance circuit comprising a 2.2-mH inductor and a 4.7-mF capacitor). SiC MOSFETs were used as the control switches. The peak surge current (Ipeak) was stepped up by increasing the power supply voltage (VDC). After each single-pulse surge-current test, the device was measured on the curve tracer to identify any possible degradation. It should be noted that the device cooling in this section is in the context of a 10-ms transient instead of the steady state. The thickness of Ag plate (1 mm) is designed to ensure that the heat diffusion is confined in the plate during the 10-ms transient [START_REF] Buttay | Surge current capability of ultra-wide-bandgap Ga2O3 Schottky diodes[END_REF][START_REF] Xiao | Packaged Ga2O3 Schottky Rectifiers With Over 60-A Surge Current Capability[END_REF], while the outer solder, DBC, and wire bond do not contribute to the heat dissipation in this transient. Fig. 5(c) and (d) show the current/voltage waveforms in the surge current tests with increased Ipeak for the Ga2O3 SBDs with the bottom-side and double-side packages. The Ga2O3 SBD with the bottom-side-cooling package was found to fail in the surge test with an Ipeak of 39 A. The failure Ipeak is much higher (70 A) in the double-side-cooled SBD. Fig. 5(e) and (f) show the surge I-V loops of the SBDs with both types of packages. Both I-V loops are clockwise, due to the increased Ron at higher Tj, and the loop area is correlated to the Ron (and Tj) increase in the surge test. With a similar Ipeak (e.g., 30 A), the loop area of the double-side-cooled SBD is smaller than that of the bottom-sidecooled SBD, suggesting a smaller Tj increase. Transfer characteristics of the double-sidecooled SBD were measured after each surge test with increased Ipeak. Almost no device degradation is shown with Ipeak up to 60 A. At 68 A Ipeak, higher leakage current is present, suggesting that degradation emerges in the Schottky contact. An important device ruggedness metric for practical power applications is the ratio between the maximum Ipeak in 10-ms surge current tests and the rated current. The rated currents of the bottom-side-cooled and double-side-cooled Ga2O3 SBDs were determined by the calibrated static electrothermal simulations [START_REF] Xiao | Packaged Ga2O3 Schottky Rectifiers With Over 60-A Surge Current Capability[END_REF] when the Tj reaches 150 o C, being 6.2 A for the bottom-side-cooled device and 9.2 A for the double-side-cooled device. For comparison, several commercial SiC SBDs with similar ratings (600-V voltage rating and 4~11 A current ratings) were tested in the same surge current test setup to identify their maximum surge currents. As shown in Table II, despite the low kT of Ga2O3 (1/20 of SiC), the fabricated Ga2O3 SBDs, particularly the ones with double-side-cooling package, show comparable, or even superior surge current capabilities as compared to the similarly-rated commercial SiC SBDs.

Mixed-Mode Electrothermal Simulations

To understand the electrothermal dynamics within the device structure, mixed-mode electrothermal TCAD simulations were performed in Silvaco Atlas. The mixed-mode simulation combines physics-based device models and circuit arrangements, enabling to reveal the device internal dynamics in at any switching transient. Some exemplar mixedmode TCAD simulations for power devices are available in [START_REF] Wang | Switching Performance Analysis of Vertical GaN FinFETs: Impact of Interfin Designs[END_REF][START_REF] Xiao | Design and Simulation of GaN Superjunction Transistors With 2-DEG Channels and Fin Channels[END_REF][START_REF] Zhang | Surge-Energy and Overvoltage Ruggedness of P-Gate GaN HEMTs[END_REF]. In this work, selfconsistent electrothermal device models are solved in a circuit arrangement consistent with that shown in Fig. 5(a). The electrothermal model settings and boundary conditions are similar to [START_REF] Zhang | Electrothermal Simulation and Thermal Performance Study of GaN Vertical and Lateral Power Transistors[END_REF]. The temperature-dependent kT, heat capacity, and electron mobility models for Ga2O3 and nanosilver die attach are detailed in [START_REF] Xiao | Packaged Ga2O3 Schottky Rectifiers With Over 60-A Surge Current Capability[END_REF]. The heat flux distribution in the double-sidecooled SBD reveals that most heat is dissipated directly from the Schottky junction instead of through the Ga2O3 die. This explains the lower Tj in the double-side-cooled SBD as compared to that in the bottom-side-cooled SBD at a similar Ipeak. As shown in Fig. 6(d), the simulated peak temperature is located within the Ga2O3 drift layer in the double-side-cooled SBD. By contrast, in the usual static operations of a SBD, the peak temperature is located at the Schottky junction. The double-side package moves the peak temperature from the Schottky contact region into the robust bulk Ga2O3, which allows the device to sustain a higher Tj before degradation of the Schottky contact.

Strategies for Further Improvement

The well-calibrated electrothermal simulations allow an exploration of the further improvement in the steady-state and transient thermal performance of Ga2O3 devices. Fig. 7 shows the simulated junction-to-coolant (-ambient) thermal resistance as a function of heat transfer coefficient (HTC), in which the HTC represents different cooling methods, for our vertical Ga2O3 SBDs in the bottom-, junction-and double-side cooling schemes. The results suggest that the junction-cooling is essential for Ga2O3 devices, as it allows over 60-70% reduction in the junction-to-ambient thermal resistance. The double-sidecooling can further reduce the junction-to-ambient thermal resistance by 30~40%. An HTC over 10 3 W/m 2 K (e.g., forced water cooling) is preferable for external cooling; a lower HTC may lead to a fast increase in the junction-to-ambient thermal resistance for Ga2O3 devices. The HTC ranges for different external cooling techniques are marked in green.

For the transient electrothermal ruggedness, the simulation in the previous section has shown that most of the heat does not reach the external surface of the 1-mm Ag plate. Hence, if this Ag plate continues to apply, the external cooling with different HTC is not expected to significantly impact the device's surge current capability. Instead, the device and package structure within the Ag plate are more critical.

To further understand the design space of the surge current capabilities of Ga2O3 devices, two additional thermal management approaches were considered: thinning of the Ga2O3 substrate, and bonding Ga2O3 device layers to a SiC substrate [START_REF] Xu | First Demonstration of Waferscale Heterogeneous Integration of Ga2O3 MOSFETs on SiC and Si Substrates by Ion-Cutting Process[END_REF][START_REF] Zhang | Overcoming the limitations of gallium oxide through heterogeneous integration[END_REF]. Using the calibrated simulation models, Fig. 8 shows the simulated peak Tj as a function of surge Ipeak for the different Ga2O3 device structures [START_REF] Xiao | Packaged Ga2O3 Schottky Rectifiers With Over 60-A Surge Current Capability[END_REF]. A similarly-rated SiC SBD with identical substrate thickness was also simulated as a reference. In Ga2O3 devices, the substrate thinning provides little improvement in the surge current capabilities when compared to the use of junction cooling, since most of the heat is directly extracted from the junction. Whereas, if low-kT SiC substrate is used in Ga2O3 devices, the heat extraction through the bulk chip can be improved significantly. Hence, the surge current capabilities can be further improved in the double-side-cooled Ga2O3-on-SiC device as compared to the bottom-side-cooled one. Finally, the simulation predicts that the Ga2O3 SBDs on SiC substrate can provide significantly superior surge current capability when compared to the similarly-rated SiC SBDs, as the Ga2O3-on-SiC SBD combines the inherent thermal stability of Ga2O3 devices and the high kT of SiC substrate. Caughey-Thomas model was used for the SiC electron mobility.

Summary

The low kT of Ga2O3 is a key roadblock to remove for the industrial applications of any Ga2O3 power and RF devices. To ensure the relevance to future applications, we believe that the thermal management of Ga2O3 devices has to be studied in the context of large-area, packaged device or, in the future, the power modules. This is because many thermal parameters revealed in small-area devices cannot be scaled to predict the performance of large-area devices by simply considering the device area enlargement, not to mention the electro-thermal coupled parameters such as the surge current capability.

To this end, we for the first time demonstrated the large-area Ga2O3 devices with different packaging configurations and measured the thermal resistance and surge current capabilities of these packaged Ga2O3 devices. We found that, contrary to some popular belief, Ga2O3 devices with proper packaging can achieve high thermal performance in both short transients and the steady state. The double-side-packaged Ga2O3 Schottky rectifiers show a junction-to-case thermal resistance lower than that of the similarly-rated commercial SiC Schottky rectifiers. In addition, these double-side-packaged Ga2O3 rectifiers can survive a higher peak surge current as compared to SiC rectifiers. The critical enabler for these excellent performances is the direct junction cooling with minimal heat going through the Ga2O3 chip. Strategies for further improving the transient and steady-state thermal performance of Ga2O3 devices have been identified, with the former being the transfer of Ga2O3 device layers to low-kT substrate, and the latter being the addition of external cooling technologies. Our work proves the viability of Ga2O3 devices for high power applications, manifests the significance of packaging for their die-level thermal management, and suggests some new capabilities that Ga2O3 devices can enable for power electronics.

Figure 1 .

 1 Figure 1. (a) Schematic of the fabricated small-area Ga2O3 SBD, and (b) the SEM image of the edge termination region. (c) Forward and (d) reverse I-V characteristics of the Ga2O3 SBD at various temperatures from 300 K to 600 K.

Figure 2 .

 2 Figure 2. (a) Forward and (b) reverse I-V characteristics of the large-area vertical Ga2O3 SBD at various temperatures from 30 o C to 90 o C.

Figure 3 .

 3 Figure 3. Schematic of the large-area Ga2O3 packaged in (a) single-side-cooled and (b) double-side-cooled configurations. (c) Photo of a double-side-packaged device.

Fig. 4 (

 4 Fig. 4(a) shows our RθJC measurement set-up using an Analysis Tech Phase 12B Semiconductor Thermal Analyzer. The Ga2O3 SBD was placed on a water-cooling cold plate with a 26 o C constant temperature. An indium foil was attached to each Ag plate to conduct electric signals. The top plastic clamp has very low thermal conductivity, ensuring the heat extraction dominantly towards the bottom water-cooling plate. Fig. 4(b) and (c) show the RθJC measurement of the double-side packaged Ga2O3 SBD under bottom-side-and junction-side-cooling.

Fig. 4 (

 4 d) and (e) show the measured Z~t curves of our packaged Ga2O3 SBD in the bottom-side-and junction-side-cooling schemes, respectively, revealing a much lower RθJC (0.5 K/W) under the junction-side cooling as compared to the RθJC (1.43 K/W) under the bottom-side cooling.

Figure 4 .

 4 Figure 4. (a) Photo of the thermal resistance test setup. Schematic of RθJC measurements under (b) bottom-side cooling and (c) junction-side cooling. Z~t curves of the Ga2O3 SBD measured with two TIMs under (d) junction-side-and (e) bottom-side-cooling, and the extracted RθJC under two cooling schemes.

Figure 5 .

 5 Figure 5. (a) Circuit diagram and (b) photo of the surge current test board. Current/voltage waveforms of the (c) bottom-side-and (d) double-side-cooled Ga2O3 SBDs in the surge current tests. I-V loops of the (e) double-side-and (f) bottom-sidecooled devices.

Fig. 6 (

 6 Fig. 6(a)-(d) show the simulated distributions of heat flux and temperatures in the double-side-cooled SBD at the peak Tj transient in the surge current test with Ipeak ~ 68 A. Fig. 6(a) and (c) show the simulated contours in the entire device structure, while Fig. 6(b) and (d) show the junction region.The heat flux distribution in the double-sidecooled SBD reveals that most heat is dissipated directly from the Schottky junction instead of through the Ga2O3 die. This explains the lower Tj in the double-side-cooled SBD as compared to that in the bottom-side-cooled SBD at a similar Ipeak. As shown in Fig.6(d), the simulated peak temperature is located within the Ga2O3 drift layer in the double-side-cooled SBD. By contrast, in the usual static operations of a SBD, the peak temperature is located at the Schottky junction. The double-side package moves the peak temperature from the Schottky contact region into the robust bulk Ga2O3, which allows the device to sustain a higher Tj before degradation of the Schottky contact.

Figure 6 .

 6 Figure 6. Simulated heat flux contour in the (a) entire packaged device and (b) device junction region, and simulated temperature distributions in the (c) entire device and (d) junction region, in a double-side-packaged Ga2O3 SBD at the peak Tj transient in the surge current test with 68 A Tpeak.

Figure 7 .

 7 Figure 7. Simulated junction-to-coolant thermal resistance of the Ga2O3 SBD as a function of HTC at the case surface, under the bottom-, junction-and double-side cooling.The HTC ranges for different external cooling techniques are marked in green.

Figure 8 .

 8 Figure 8. Simulated max peak Tj as a function of peak surge current in 10-ms surge tests for the double-side-cooled and bottom-side-cooled Ga2O3 SBDs on the 0.5-mmthick Ga2O3 substrate, 0.1-mm-thick Ga2O3 substrate, and 0.5-mm-thick SiC substrate. A thermal boundary resistance of 0.01 K/(Wꞏcm 2 ) was set at the Ga2O3/SiC bonding interface. Identical electrical conductivity was set for Ga2O3 and SiC substrates. The simulated SiC SBD has a 0.5-mm-thick substrate and a bottom-side-cooled package.Caughey-Thomas model was used for the SiC electron mobility.

TABLE I .

 I Thermal resistance comparison between Ga2O3 SBDs and commercial SiC SBDs with similar current ratings and package sizes.

	Device	Package Package Size * (mm 2 ) VON (V) IF (A) ** Cooling RθJC (K/W)
	Ga 2 O 3 SBD (this work)	Double-side	7.3×7.3	0.83	13	Junction Bottom	0.5 1.43
	SiC SBD (C3D10060G)	TO-263-2	6.5×7.9	0.85	18	Bottom	1.2
	SiC SBD (E3D08065G)	TO-263-2	6.5×7.9	0.85	14.5	Bottom	1.47
	SiC SBD (C6D04065E)	TO-252-2	5.2×4.3	0.85	12	Bottom	2.89

*Size of the die-attached thermal pad. **Forward current at 2 V.

TABLE II .

 II Comparison of the surge current capability of SiC and Ga2O3Schottky barrier diodes

	Device	Rated Current (A)	Max Surge Current (A)	Max surge current over rated current
	SiC SBD (CSD01060A)	4	20.3	5.1
	SiC SBD (CSD02060A)	8	26.9	3.36
	SiC SBD (CSD03060A)	11	31.8	2.89
	Bottom-side-packaged Ga 2 O 3 SBD	6.2	37.5	6.05
	Double-side-packaged Ga 2 O 3 SBD	9.2	68	7.4

Acknowledgments

The Virginia Tech authors acknowledge the partial support by the National Science Foundation under Grant ECCS-2100504 and the High Density Integration Industry Consortium of the Center for Power Electronics Systems, Virginia Tech.


Stefan Kaalen
email: kaalen@kth.se

Mattias Nyberg

Olle Mattsson
email: olle.mattsson@scania.se

Using State Transition Diagrams for Safety Quantification within the Automotive Industry 1 st

One method suggested in functional safety standards for quantitative analysis of systems is Markov processes of which Semi-Markov processes (SMPs) is an extension allowing more accurate models of the systems of interest. The paper suggests state transition diagrams modeled in Stateflow for modeling systems as SMPs since it will ease the adoption of the method in the automotive industry. Compared to the method of fault trees, which is frequently applied in industry today, SMPs have a significant advantage for some systems given their ability to explicitly model time-dependent aspects of the system. SMP-tool is suggested as a tool support for quantitative analysis of state transition diagrams modeled in Stateflow.

I. INTRODUCTION

This paper argues for a state-diagram approach based on stochastic processes for safety quantification of safety-critical systems containing timed aspects. This method is for certain situations preferable to Fault Tree Analysis (FTA), which is probably the most widely used approach in the industry today. Although the discussion is focused on safety, the results are applicable to quantification of any critical property, such as other dependability attributes.

In the automotive industry, systems are growing more and more safety-critical and complex, a transformation that increases even further with the development of autonomous vehicles. Therefore, performing quantitative safety analysis of safety-critical systems is more important than ever [START_REF]Functional Safety of Electrical/Electronic/Programmable Electronic Safety-related Systems (IEC61508)[END_REF], [START_REF]Road vehicles -functional safety[END_REF]. In these systems, proving the highest levels of safety through pure testing becomes unfeasible given the high amount of testing that would be needed. Furthermore, without an extensive preanalysis, unacceptable numbers of system failure can occur during the road testing endangering human lives in the process. The alternative to testing is to estimate the level of safety using some type of model of the system.

Several different methods for performing model-based quantitative analysis are suggested in the functional safety standard ISO26262 and its mother standard IEC61508 [START_REF]Functional Safety of Electrical/Electronic/Programmable Electronic Safety-related Systems (IEC61508)[END_REF], [START_REF]Road vehicles -functional safety[END_REF]: fault trees, Markov processes, reliability block diagrams, and Petri nets. Of the four methods, Markov processes and Petri nets are the most feasible to incorporate timed aspects [START_REF] Ruijters | Fault tree analysis: a survey of the state-ofthe-art in modeling, analysis and tools[END_REF] which

The authors acknowledge the following agencies and projects for general and financial support: the European H2020 -ECSEL PRYSTINE project and Vinnova FFI, through the AVerT project, and the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by Knut and Alice Wallenberg Foundation.

are often present in complex safety-critical systems and these will therefore be investigated further below. However, though standard fault trees lacks support for explicit modeling of important timed aspects [START_REF] Flammini | Using repairable fault trees for the evaluation of design choices for critical repaiable systems[END_REF], they are often used in the industry today.

The probably most well-known method for modeling timed probabilistic properties is Markov processes, which is also the only one mentioned in ISO26262 [START_REF]Road vehicles -functional safety[END_REF]. While Markov processes allow for easy computation, they can often not accurately describe real-world systems. The main reason for this is that they lack explicit modeling of non-exponential times until events occur. Semi-Markov Processes (SMPs) generalize Markov processes to allow non-exponential times, and consequently enable more accurate descriptions of realworld systems. The present paper therefore argues for the use of models of semi-Markov processes, in favor of models of Markov processes, in order to capture the underlying system more accurately.

Petri nets [START_REF] Trivedi | Reliability and availability engineering: modeling, analysis, and applications[END_REF] with probabilistic delays of transitions (here referred to as stochastic Petri nets) is a more general and powerful modeling formalism than Markov processes. Their quantitative analysis is performed on the underlying stochastic processes and can thereby be seen as a method of visualizing stochastic processes such as Markov processes or SMPs. However, while stochastic Petri nets have been around for a long time, they have still not received any major spread in the automotive industry, perhaps since they are experienced as too intricate. The present paper therefore argues for the use of state transition diagrams for the modeling of stochastic processes. The argument for this is that state transition diagrams, in tools such as Simulink Stateflow, are already widely used in the industry.

To summarize, the industry is in need of a powerful modeling formalism to support model-based safety analysis. It is also important that this formalism is easy to use and understand. A way of achieving this would be to base the modeling formalism on methods and tools already widely spread in the industry, such as state transition diagrams and Stateflow. However, an issue is that Stateflow lacks support for explicit analysis of models with probabilistic aspects. While the analysis phase is problematic, probability can be easily incorporated in the modeling phase in Stateflow as will be seen later. In order to aid the industry in analysing these models, the present paper proposes SMP-tool [START_REF] Kaalen | Tool-Supported Dependability Analysis of Semi-Markov Processes with Application to Autonomous Driving[END_REF], [START_REF] Kaalen | Branching Transitions for Semi-Markov Processes with Application to Safety-Critical Systems[END_REF] which can be downloaded for free from [9]. SMP-tool is a tool support for modeling state transition diagrams of SMPs in Stateflow and performing quantitative analysis, such as computing the reliability of the models. Through SMP-tools numerical engine the large computation time that can occur using Monte-Carlo simulation is avoided.

The outline of the paper is as follows: In Sec. 2, SMPtool is presented. Then, in Sec. 3, a simple redundant steering system is presented and modeled with state transition diagrams of SMPs. it is also analyzed using SMP-tool to highlight the advantages of SMPs over other techniques such as fault trees and Markov processes. Finally, in Sec. 4, it is discussed how state transition diagrams of SMPs can be used in practise.

II. SMP-TOOL

SMP-tool has previously been presented in [START_REF] Kaalen | Tool-Supported Dependability Analysis of Semi-Markov Processes with Application to Autonomous Driving[END_REF] and [START_REF] Kaalen | Branching Transitions for Semi-Markov Processes with Application to Safety-Critical Systems[END_REF] as a tool for performing steady-state, transient, and sensitivity analyses of SMPs modeled as state transition diagrams. There is support for these analyses as long as all transitions have a time to occurrence belonging to the class of expolynomial distributions, containing the exponential, degenerate, and uniform distribution. Furthermore, in SMP-tool 2.0, support for analysis of hierarchical-SMPs has been added.

Stateflow, within Matlab and more specifically the Simulink product family, provides a graphical language for state transition diagrams and is widely used within the automotive industry. Based on this, the tool support given by SMP-tool is based on models specified in Stateflow. However, in order to include all stochastic behaviour of SMP-models, the models used by SMP-tool do, while they are specified in Stateflow, differ in some aspects from the models that can be analyzed directly in Stateflow. The main user interface for creating models for SMP-tool is the standard modeling Stateflow GUI shown in Fig. 3. The main user interface in SMP-tool for the analysis of models is illustrated in Fig. 1. By simply opening a model to be analyzed, or creating a new model through the "file"-menu, a transient analysis can be performed for any specified time by simply pressing the button "Run transient analysis".

III. CASE STUDY

In this section, a simple redundant steering system, which is based on a real system from the heavy vehicle manufacturer Scania, will be presented. This will be followed by an attempt to analyze the system utilizing both fault trees, Markov processes and SMPs.

A. Redundant steering system

In the redundant steering system considered, the steering can be performed by either a Primary System (PS) or a Secondary System (SS) that takes over in the event of a system failure in the primary system. The two systems both have an exponentially distributed time to failure with a failure rate of λ P S and λ SS respectively. Furthermore, each of the subsystems have diagnostic procedures in place to detect a system failure; a system failure in PS is detected with the probability (has a diagnostic coverage of) DC P S and a system failure in the SS is detected with the probability DC SS . It is assumed that the vehicle undergoes a maintenance every 30 days. If a failure in any of the subsystems has been detected, the subsystem will be repaired during the maintenance. The case study has been somewhat simplified with assumptions to fit the paper format. The assumptions does for example include that the driver will not actively take the vehicle to the workshop any sooner than planned if there is a system failure in either PS or SS. In order to build quantitative models, some assumptions on the system are made. Firstly, it is assumed that everything works initially in the lifetime of the system. Secondly, it is assumed that λ P S = λ SS = 10 -6 and that DC P S = DC SS = 0.99.

B. Fault tree

Fig. 2 visualizes one of several possible attempts to model the redundant steering system with a fault tree. In the fault tree, the events EV1 and EV10 have an event probability that is compensated by the repair of the corresponding subsystem. The model has been built in Isograph Reliability Workbench.

C. Semi-Markov Process

The same system will now be modeled as an SMP with a state transition diagram. The result is visualized in Fig. 3. The first stage in building a state transition diagram of an SMP through SMP-tool is to identify the states the system can be in. The states should be chosen so that the system is always in exactly one of the states. This concept of states is corresponding to the built-in standard states in Stateflow and the six states chosen for creating the model of the redundant steering system are visualized in Fig. 3.

The next step is to add the transitions that can occur between states. When a state, i, is entered, assume that for each event that will cause a transition from state i to some other state, a timer is started. Connected to each of these timers is a Cumulative Distribution Function (CDF) that, by drawing a sample, provides the time the timer starts counting down from. Each of these timers start counting down in the same pace when state i is entered and when the first one reaches zero, the corresponding state transition occurs. Furthermore, each transition t may have several possible target states, each possible target state j is assigned a probability that a transition t will lead to the next state being state j. For each transition, these probabilities should add up to 1. To model this, the transitions, modeled by the default transitions of Stateflow, may have a Stateflow junction as target from which several default transitions origins, one for each possible target state. Each default transition with a state as origin is labelled with the CDF of the corresponding timer, while each default transition with a junction as origin is labelled by the probability that the corresponding target state will be the outcome of a transition through the junction. For the redundant steering, junctions are utilized to model the diagnostic coverage.

The last things to add to the model is which state(s) should be considered as down state(s) (states corresponding to system failure), and what probability each state have of being the first state of the process. To make a state a down state, simply mark the state with "down state". To assign a probability, s, that a state i is the initial state of the process, simply mark the state with "p=s;". Fig. 3 illustrate a resulting state transition diagram after modeling the redundant steering system as an SMP.

D. Markov process

Trying to model the same system with continuous-time Markov processes, it can be seen in Fig. 3 that issues arise when trying to describe the transitions corresponding to re-pairing a failed subsystem since they are most realistically modeled as non-exponential. Markov processes only allow for exponential distributions of the timers for each transition in the model. While attempts can be made to approximate non-exponential distributions as exponential distributions, the accuracy of the results differ highly on the system being studied and it is a difficult task to estimate how much the approximation affects the overall accuracy of the results.

E. Comparison of SMPs and fault trees

The modeling of the redundant steering system will here be compared between the proposed method of SMP-based statetransition diagrams and fault trees. For the redundant steering system, a failure of the secondary system must occur while a failure in the primary system is active, or vice versa, in order to cause an overall system failure. Given the presence of the diagnosis and possible repair of each of the subsystems, no fault tree based on the standard fault tree formalism could be found that completely capture the behavior of the model. In order to approximately compensate the event probabilities for the repair time in the fault tree analysis, the repair time is assumed to be exponential [START_REF] Rausand | Reliability of safety-critical systems: theory and application[END_REF] (with the same mean as the original uniform distribution). In comparison, an SMP describing the model could easily be found modeled as a state transition diagram without approximations in Fig. 3. Estimating the effect of approximations, made in the modeling, on the analysis result is generally an intricate task. On the redundant steering case study with the parameters chosen as described earlier in this section, the unreliability was by the fault tree analysis found to be 5.1 • 10 -5 while it by SMP-tool was found to be 3.1 • 10 -5 . A Monte-Carlo simulation of the system found the real value to have a 95% confidence interval of [2.8 • 10 -5 , 3.6 • 10 -5 ]. This together with the result of analysing the fault tree result clearly indicates that the limited support of explicit modeling of timed aspects in fault trees leads to an analysis result that differs from the correct value. For this particular case, the result of the fault tree analysis still ended up in the same magnitude as the real value and could therefore be deemed acceptable. However, it is difficult to in advance estimate the impact that approximations in models will have on the result of analyses, and thereby difficult to verify models.

What is actually visualised in the model is also an important aspect. In the fault tree model, the dynamical course of events is not clear. The state transition diagram however needed only a few states and transitions to completely visualize the dynamics of the redundant steering system in a model that is still is easy to overlook and understand.

It should be noted that there are extensions of standard fault trees such as dynamic fault trees [START_REF] Ruijters | Fault tree analysis: a survey of the state-ofthe-art in modeling, analysis and tools[END_REF], which are better equipped for handling e.g. timed aspects of the systems. However, as the modeling possibilities increase the number of possible gates, the complexity of building a model of your system increases with it, which in the end increases the probability of creating an erroneous model. Furthermore, while the extensions are many, they are yet too see any wide spread in the automotive industry.

IV. USING STATE TRANSITION DIAGRAM MODELING IN SAFETY ENGINEERING

It is possible to, through the use of SMPs modeled as state transition diagrams, find different configurations of subsystems that assures that the system satisfies a needed level of reliability. This way, the configuration that shows most promise from e.g. an economical or technological standpoint can be chosen for the system. Finding possible configurations is done by assigning different values of the CDF for each timer and the probabilities from each junction in the model. This way, different configurations satisfying the same level of reliability can be found. As an example: consider that the maximal allowed probability of loosing steering within the lifetime of the vehicle is 3.2 • 10 -5 . One way of designing the systems and diagnosis procedure is as illustrated in Fig. 3: to have λ P S = λ SS = 10 -6 and DC P S = DC SS = 0.99. Another idea could be to instead make the secondary system have a higher failure rate, while the primary system has a lower failure rate or vice verse. It is also possible to look at how changing the proportion of failures that is caught by the diagnosis procedure, or the maintenance, affects the needed failure rate of each subsystem.

In order to assist in finding these configurations, SMP-tool has a built-in sensitivity analysis that can be utilized not only to find suitable configurations, but also to find possible aspects of the system that has low impact of the overall safety. This way the company can both save money and focus the workload on where it is most useful.

A question that often arises when looking at a state transition diagram of an SMP is: how can the probability distributions in the model be found? Four answers to this question will be presented here. Firstly, while the shape of some transitions may be given directly out of the context, such as the uniform transition in the redundant steering system, others that are found in the sensitivity analysis can be seen as requirements of the subsystems or components in order to satisfy a level of safety. For example, the failure rates of the primary and secondary steering systems in Fig. 3 can be considered to be requirements to be given to the suppliers delivering these subsystems. Secondly, the sensitivity analysis may show that the distribution of some timers barely affect the safety of the vehicle. The distributions of these timers can therefore be set almost arbitrarily. Thirdly, the probability distributions can be found through testing. By performing tests on components and subsystems, the number of test hours can often be made significantly lower than if tests are done on the whole system. Finally, in safety engineering, some assumptions are always made on the behaviour of the system. These assumptions are often conservative estimates, which can be fairly easy to find.

To avoid the problem of state-space explosion, models can be built on different abstraction levels of the vehicle and analyzed in isolation. An example from the case study could be to make two more detailed separate models describing the internal reliability of the primary and secondary steering systems respectively. This reliability can then be used as failure distributions in the model on the higher abstraction level.

V. CONCLUSION When modeling systems with timed aspects, as is often the case with real-world embedded systems for automotive vehicles, modeling and analysis are easily performed through SMPs modeled as state transition diagrams. Unlike fault trees, which are often used in the industry, state transition diagrams with underlying SMPs have general support for explicit modeling of timed aspects of the systems of interest. SMPs should therefore be preferable over fault trees for modeling complex systems where timed aspects often occurs, an importance that will only grow with the development of autonomous vehicles

To make the industry adapt the methods, it is suggested to model the state transition diagrams in Simulink Stateflow, which is already used widely in the automotive industry. SMP-tool was presented to compute the reliability for a SMP modeled as a state transition diagram in Stateflow.

The paper has focused on safety, yet the result are equally valid for other dependability attributes.

Fig. 1 .

 1 Fig. 1. Main user interface of SMP-tool.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Attempted fault tree breakdown of redundant steering


Charles Atwood Kofoid

John R Dolan
email: john.dolan@imev--mer.fr

Keywords: scientific illustration, history of protistology, dinoflagellates, tintinnid ciliates, Beulah Mitchell Clute Introduction

 Talbot 1936). While these comments and descriptions were not in scientific publications, but rather in publications concerned with book or bookplate collections, they nonetheless included

erroneous descriptions of the protists shown in Kofoid's bookplate. These range from descriptions of them as being bacteria (Talbot 1936), to correctly identifying dinoflagellates as such, but declaring the tintinnid ciliates to be radiolaria (Jackson 2003). To a large extent, the erroneous identifications are understandable since accurate identification requires not only familiarity with protists, but also more importantly a familiarity with the numerous publications of Kofoid, many of which are currently quite obscure. Here the focus will be on his bookplate and the protists shown. It will be dissected, so to speak, and all of the protists illustrated identified, as far possible. In addition, the apparent sources of the illustrations in Kofoid's publications will be given.

Kofoid's Bookplate

Kofoid the book collector, and his bookplate are shown in Fig. 1. The bookplate is of the type known as a '20th Century Pictorial '. Beulah Mitchell Clute (1873--1958) designed it. A full account of Beulah Clutes's life and work as a bookplate designer is given in Dolan (2021). At the time she undertook Kofoid's bookplate, she was the designer at the Three Redwoods Studio in Berkeley. The studio produced bookplates, greeting cards, and illustrations for publications. Beluah Clute was well--known for her designs of bookplates and over the years made bookplates for several notable Berkeley academics other than Kofoid. These included William Frederic Badè, the Dean of the Pacific School of Religion and Oliver Essig, the chair of the Department of Entomology at UCB as well as Robert Legge, a Professor of Hygiene and University Physician at UCB and also Armin Otto Leuschner, the distinguished astronomer and Dean of Graduate Studies at UCB. Kofoid's bookplate was likely created sometime between 1929 (as we will see based on the apparent age of the protists illustrations shown) and 1936, when the bookplate was first mentioned in a publication, Historic California Bookplates (Talbot 1936). Figure 1. Left panel, Kofoid at his desk in his office at the University of California Berkeley, courtesy of the Regents of the University of California. It appears to date from about 1936 when he retired after chairing the Department of Zoology for over 25 years. Note the bookshelves extending to the ceiling and the piles of pamphlets, books, and manuscripts on his desk. Right panel, Kofoid's bookplate as it appears, printed on off--white paper using a dark brown ink; actual size 7 x 10 cm (author's copy). In Kofoid's bookplate, the setting is his office in his Berkeley home. The view from the office window is dominated by the Campanile Tower of UCB. Irises, for which the university campus is well known, adorn the frame and the central crest resembles one of the Berkeley Family arms (Jackson 2003). Three medallions (on the left, the right and at the bottom), show protists clearly in some detail. The bottom third of the bookplate is a depiction of a ship pulling a plankton net through a sea populated by myriad, oddly shaped, tiny organisms. Identifying the protists shown in these minute sketches (actual size of the scene below the ship with the plankton the net and the protists is but 4.5 x 1.5 cm) proved to be challenging, but only a few could not be identified.

Dissecting Kofoid's Bookplate

To aid in the identification of the forms shown in Kofoid's bookplate, it has been enlarged (Fig. 2) and the image was enhanced to improve the definition as much as possible (e.g., sharpened, black added to greys, black reduced from whites, increased contrast, color data discarded). Then, Kofoid's protist publications were examined for illustrations used in the bookplate. These were the protist works (as indicated by the titles) from the comprehensive Kofoid bibliography given in Goldschmidt (1951). The majority of Kofoid's publications on protists were available through the Biodiversity Heritage Library and/or Open Archive. Two of the illustrations of tintinnid ciliates are somewhat problematic as they appeared in two publications; first in the 1929 Kofoid and Campbell monograph on tintinnids [START_REF] Kofoid | A conspectus of the marine and fresh--water ciliata belonging to the suborder Tintinnoinea, with descriptions of new species principally from the Agassiz expedition to the eastern tropical Pacific, 1904--1905[END_REF] and then reproduced, oddly with same figure numbering, in a 1930 article [START_REF] Kofoid | Factors in the evolution of the pelagic ciliata, the Tintinnoinea[END_REF]. It seems likely that Kofoid would have given or lent to Beulah Clute the 1930 article of 39 pages rather than the 1929 monograph of 403 pages. All in all, the illustrations of protists in the bookplate can be found in a pool of only 8 works published between 1899 and 1929 or 1930 (see Table 1). This represents a very small portion of Kofoid's considerable number of publications. The Kofoid bibliography given in Goldschmidt's biographical memoir on Kofoid (Goldschmidt 1951), lists 236 titles published between 1894 and 1948. Of the 236 titles, 156 dealt with protists. Figure 2. Kofoid's bookplate enlarged and enhanced to facilitate identification of the organisms shown. The signature of Beulah Mitchell Clute, BMC, is circled in white at the lower right. The bottom panel is a further enlargement of the plankton net scene below the waves and ship. The uppercase letters A to E mark the larger images of organisms, and the lowercase a to zz, the organisms of the plankton net scene. A key to the labeled organisms is given in Table 1. The article in which mitotic stages of Endamoeba dysteneriae (now known as Entamoeba histolytica), were shown [START_REF] Kofoid | Mitosis in Endamoeba dysenteriae in the bone marrow in arthritis deformans[END_REF], has only been cited twice perhaps in part because of the long--standing confusion of Endamoeba and Entamoeba and synonyms within each genus (e.g., [START_REF] Kirby | Entamoeba coli versus Endamoeba coli[END_REF]). It appears likely that the images of protists showcased in Kofoid's bookplate will remain obscure. However, their artistic value will endure. Kofoid's bookplate will likely endure as the only widely known protist bookplate.

  It seems likely that Beulah Clute would have sought Kofoid's wishes as to which illustrations he wanted featured, particularly with regard to the three 'medallions' of his bookplate, the circles at the left and right margins, and the bottom center. These prominent medallions show the colonial volvocine algae Platydorina caudata (Fig.2. A), the dinoflagellate Peridinium fatulipes (Fig.2. C), and a stage of mitosis of the amoeba Endamoeba dysenteriae (Fig.2, E). The presumed source publications for the illustrations are shown in Fig.3. The three illustrations correspond to three areas of his expertise. Platydorina caudata is representative of the part of Kofoid's career in which his studies concerned freshwater plankton. These were carried out when he was Superintendent of the Illinois Biological Station, and published from 1896 to 1908. Peridinium fatulipes was described from material Kofoid gathered during the cruise of the Albatross to the Eastern Tropical Pacific in 1904--1905. The samples he gathered provided much of the material used in his monographic studies of dinoflagellates and tintinnid ciliates of the marine plankton published from 1907 to 1939. The cruise of the Albatross is also represented in the plankton net scene filled with many distinctive species of dinoflagellates and tintinnid ciliates. However it should be noted that the Albatross was a two--masted steamer, different from the 3--masted sailing ship shown pulling the plankton net. The illustration of Endamoeba dysenteriae, is representative of the latter part of Kofoid's career in which he focused largely on parasitology(1915--1940).

Fig. 3 .

 3 Fig. 3. Examples of published illustrations, presumably used by Beulah Clute as sources for her illustrations of the protists shown in Kofoid's bookplate. The left panel shows the colonial flagellate Platydorina caudata (fig. 1, in bookplate labeled A) in Kofoid 1899. Clute omitted the flagella shown in the original. The middle panel shows the dinoflagellate Peridinium fatulipes from Kofoid 1906a. Note that in the bookplate Clute included some of the crosshatching used in the original (here fig. 30, in bookplate labeled C) to distinguish the different thecal plates. The right panel shows the illustrations from Kofoid & Swezy 1922. In the bookplate a stage of mitosis in Endamoeba dysenteriae (here fig. 6, in bookplate labeled E) is shown.

Table 1 . Key to the protists shown in Kofoid's bookplate.

 1 The organism names in the table are those as given in the presumed source publication. Most of the species shown in the bookplate were described as new species of protists by Kofoid as indicated by 'yes' in the column 'Kofoid sp?'; n/d indicates no data. Many are species are not currently recognized as valid first descriptions. The organisms shown cover nearly all of his scientific career beginning in Illinois in the late 1890's when he described Platydorina as a new colonial flagellate, the many species shown from his extensive work on dinoflagellates and tintinnids, and the later part of his career when his work focused on endosymbiotic and parasitic protists (i.e., Endamoeba). Of the 32 forms shown in the bookplate, 5 could not be matched with any certainty to an illustration in a Kofoid publication.

	Label Organism	Kofoid sp? Presumed Source
	A	Platydorina caudata	yes	Kofoid 1899, plate 38, fig.1
	B	Dictyocysta pacifica	no	Kofoid & Campbell 1929, Kofoid 1930,
				fig. 550
	C	Peridinium fatulipes	yes	Kofoid 1906a, plate 5, fig. 30
	D	Epiplocylis nervosa	no	Kofoid & Campbell 1929, Kofoid 1930,
				fig. 318
	E	Endamoeba dysenteriae	no	Kofoid & Swezy 1922, fig. 6
	a
		Gymnodinium puniceum	yes	Kofoid & Swezy 1921, plate 5, fig. 1
	b	Amphisolenia quinquecauda	yes	Kofoid 1906a, plate 13, fig. 75
	c	tintinnid ciliate	n/d	unknown
	d		yes	Kofoid & Swezy 1921, plate 12, fig. 133
		Erythopsis pavillardi
	e	Peridinium fatulipes	yes	Kofoid 1906a, plate 5, fig. 30
	f	Peridinium latissimum	yes	Kofoid 1906a, plate 5, fig. 32
	g	Peridinium tenissimum	yes	Kofoid 1906a , plate 5, fig. 34
	h	tintinnid ciliate	n/d	unknown
	i	Oxytoxum turbo	yes	Kofoid 1906a, plate 10, fig. 60
	j	dinoflagellate	n/d	unknown
	k	Acanthodinium spinosum	yes	Kofoid 1906a, plate 11, fig. 60
	l	tintinnid ciliate	n/d	unknown
	m	Histioneis garrettii	yes	Kofoid 1906a, plate 16, fig. 97
	n	dinoflagellate	n/d	unknown
	o	Peridinium rectum	yes	Kofoid 1907, plate 32, fig. 49
	p	Murrayella punctata	yes	Kofoid 1906a, plate 9, fig. 58
	q	Ceratium reticulum spirale	yes	Kofoid 1907, plate 27, fig. 27
		subsp. nov.
	r	Histioneis carinata	yes	Kofoid 1906a, plate 16, fig. 98
	s	Ceratium claviger	yes	Kofoid 1906a, plate 4, fig. 27
	t	Tintinnopsis dadayi divider	yes	Kofoid 1905, plate 26, fig. 4
	u	Heterodinium inaequale	yes	Kofoid 1906b, plate 18, fig. 9
	v	Ceratium ehrenbergii	yes	Kofoid 1906a, plate 2, fig. 16
	w	Tintinnopsis dadayi	yes	Kofoid 1905, plate 26, fig. 3
	x	Cyttarocylis fasciata	yes	Kofoid 1905, plate 26, Fig. 6
	y	Cyttarocylis torta	yes	Kofoid 1905, plate 28, fig.16
	z	Triplosolenia ambulatrix	yes	Kofoid 1906a, plate 4, fig 24
	zz	dinoflagellate	n/d	unknown

  ConclusionA protistological dissection of Kofoid's bookplate shows that it was mostly filled with drawings of protists that he himself had described as new species. Interestingly, the most prominent protist images in his bookplate, while well representing his career, were perhaps not well chosen as they are today fairly obscure. According to Google Scholar, the publication describing Platydorina caudata in 1899 has been cited only 8 times and the species has been termed 'rare' in recent years (Coleman 2014). The dinoflagellate Kofoid described as Peridinium fatulipes in 1906 was transferred to the genus Protoperidinium byBalech in 1974 (Balech 1974). As Peridinium fatulipes, it has been mentioned in only 9 publications and in another 9 publications as Protoperidinium fatulipes.

Acknowledgements

Graham Clarke was of great help in untangling Endamoeba/Entamoeba. The efforts of the reviewers and the Monitoring Editor led to significant improvements in the manuscript. However, I retain full responsibility for all errors of fact, omission, and interpretation in this work.

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