Cyril Grunspan
email: cyril.grunspan@devinci.fr

Ricardo Pérez-Marco
email: ricardo.perez.marco@gmail.com

Ricardo P Érez-Marco

* Balzac

Père Le

Goriot

Profit lag and alternate network mining

Keywords: Mathematics Subject Classification. 68M01, 60G40, 91A60 Bitcoin, proof-of-work, selfish mining, intermittent selfish mining, smart mining

For a mining strategy we define the notion of "profit lag" as the minimum time it takes to be profitable after that moment. We compute closed forms for the profit lag and the revenue ratio for the strategies "selfish mining" and "intermittent selfish mining". This confirms some earlier numerical simulations and clarifies misunderstandings on profitability in the literature. We also study mining pairs of PoW cryptocurrencies, often coming from a fork, with the same mining algorithm. This represents a vector of attack that can be exploited using the "alternate network mining" strategy that we define. We compute closed forms for the profit lag and the revenue ratiofor this strategy that is more profitable than selfish mining and intermittent selfish mining. It is also harder to counter since it does not rely on a flaw in the difficulty adjustment formula that is the reason for profitability of the other strategies.

1. Introduction 1.1. Nakamoto consensus. The founding paper "Bitcoin, a peer-to-peer electronic cash system" was announced at the end of October 2008 on a cryptography mailing list [START_REF] Nakamoto | Bitcoin P2P e-cash paper[END_REF][START_REF] Nakamoto | Bitcoin: a peer-to-peer electronic cash system[END_REF]. A first version of the software implementing the new protocol developed in the article was then released on January 2009 [START_REF] Nakamoto | [END_REF].

Bitcoin gradually enjoyed worldwide success and is today the cornerstone of the new crypto-economy valuated at several hundreds of billions of dollars.

Two main reasons have contributed to its success. The decentralization of the network allows to transfer value on the internet from user to user without the assistance of a third party. Therefore, no central server, nor bank, nor jurisdiction can block payments [START_REF] Nakamoto | Bitcoin: a peer-to-peer electronic cash system[END_REF]. A second reason is the invention of "smart-contracts" which made bitcoin the first programmable currency [START_REF] Antonopoulos | Mastering Bitcoin: Unlocking Digital Cryptocurrencies[END_REF][START_REF] Song | Programming Bitcoin: Learn How to Program Bitcoin from Scratch[END_REF]. This allows to construct the "Lightning Network", on top of the Bitcoin network that offers instant and secure money transfer [START_REF] Dryja | The bitcoin Lightning Network: scalable off-chain instant payment[END_REF].

Bitcoin consensus, sometimes referred today as Nakamoto Consensus, is based on the use of proof of work (PoW). PoW was originally invented to fight e-mail spam [START_REF] Dwork | Pricing via processing or combatting junk mail[END_REF]. It was then used by H. Finney in the design of RPOW, the first cryptocurrency based on PoW, but was not decentralized. PoW plays a crucial role to secure Bitcoin protocol and in the minting process of new bitcoins. Academically speaking it combines computer security and classical probability theory. The main problem solved by Bitcoin and its PoW design is the prevention of double spend attacks without relying on a central server. Instead of searching for a flawless deterministic distributed consensus, Nakamoto's point is that, under reasonable conditions, the probability of success of such attacks is negligible [START_REF] Nakamoto | Bitcoin: a peer-to-peer electronic cash system[END_REF][START_REF] Rosenfeld | Analysis of hashrate-based double spending[END_REF][START_REF] Grunspan | Double spend races[END_REF][START_REF] Georgiadis | A combinatorial-probabilistic analysis of bitcoin attacks[END_REF] and economically non-profitable [START_REF] Grunspan | On profitability of Nakamoto double spend[END_REF].

1.2. Mining Process. At any time, miners are working to build a new block from new transactions. This is achieved solving a laborious cryptographic puzzle which involves heavy computation and use of energy. Miners iterate the calculation of images of a cryptographic hash function of data from a new block by varying a "nonce" (and an "extra-nonce") until finding a result fulfilling certain criterion.

If successful, the miner broadcasts his discovery to the rest of the network which checks that the solution is legit. Then the new block is added to the previous known chain of blocks. The miner is then rewarded in newly minted bitcoins determined by the protocol and by the transaction fees of transactions included in the validated block. This sequence of blocks forms the "Bitcoin blockchain". This is a secured distributed ledger recording all past validated transactions [START_REF] Wattenhofer | Blockchain Science: Distributed Ledger Technology[END_REF].

1.3. Selfish Mining. On the early days, the community of bitcoiners met in different internet forums, among them bitcointalk.org, a forum originally created by Nakamoto in November 2009. In particular, they tried to understand in depth how Bitcoin works. Some people had doubts of the specific point in the protocol that requires miners to broadcast their blocks as soon as they are validated through the proof-of-work [START_REF] Rhorning | Mining cartel attack. bitcointalk[END_REF]. This fact allows the blockchain to record several thousand transactions every ten minutes in average. If a miner witholds secretly a block, he risks to loose the reward awarded to this block (coinbase and transaction fees) to a faster miner.

It is implicit in the founding paper that Nakamoto believed this to be the optimal strategy for miners [START_REF] Nakamoto | Bitcoin: a peer-to-peer electronic cash system[END_REF]. Moreover, it is required in a decentralized protocol that the private economic interests of participants are in line with the protocol rules. However, several deviant strategies were proposed in the bitcointalk.org forum and a seminal paper by Rosenfeld examined the problem in 2011 [START_REF] Rosenfeld | Analysis of Bitcoin Pooled Mining Reward Systems[END_REF]. Then by the end of 2013, two articles showed that other alternative strategies than the honest one can be more profitable under suitable conditions [START_REF] Eyal | Majority is not enough: bitcoin mining is vulnerable[END_REF][START_REF] Bahack | Theoretical bitcoin attacks with less than half of the computational power[END_REF].

By modeling the progression of the blockchain with a Markov chain, the authors showed that a certain deviant strategy, the so-called " selfish mining " is more profitable in the long run than honest mining [START_REF] Eyal | Majority is not enough: bitcoin mining is vulnerable[END_REF]. This is in particular the case when a mining pool detains more than 33 % of the whole network hashpower.

The assumptions underlying the result, such as the cohesion of mining pools, were challenged. In particular, the fact that the model considers only two groups of miners and only one selfish mining pool in the network [START_REF] Felten | Bitcoin isn't so broken after all[END_REF][START_REF] Bahack | On Subversive Miner Strategies and Block Withholding Attack in Bitcoin Digital Currency[END_REF].

These objections are founded. For example, one could imagine that some miners participating in a mining pool decide to selfish mine not only against honest miners but also against their own mining pool.

Using new martingale techniques and a rigorous profitability model, it was proved in [START_REF] Grunspan | On profitability of selfish mining[END_REF] that the true reason for the profitability of selfish mining resides in a flaw in the difficulty adjustment formula. An important result obtained by these new techniques is that without difficulty adjustment, the honest mining strategy is the most profitable one, hence vindicating Nakamoto's original belief. The flaw in the difficulty adjustment formula can be easily corrected. In the presence of a blockwithholding attacker, the network, that does not track the production of orphan blocks, underestimates the real hash power deployed. Ultimately this makes the production of blocks easier than normal, which boosts miners income.

In particular, the selfish mining strategy only becomes profitable after a first difficulty adjustment, and after recouping all the cost employed reducing the difficulty (at the expense of regular honest mining income). At first, blocks are generated at a slower pace and all miners have their profitability reduced. Then after a first difficulty adjustment, blocks arrive faster and the selfish miner profits. This theoretically explains why the strategy has never been implemented. No miner risks to mine at a loss for several weeks while assuming that mining conditions remain the same i.e., one selfish miner and same overall hash power over time. Moreover, if a miner has substantial hashpower, then probably double spend attacks based on 51% hashrate dominance are easier to achieve by increasing the hashpower.

The original selfish mining strategy assumes also the non-realistic hypothesis that there is no arrival of new miners to the network attracted by the lower mining difficulty.

1.4. Smart mining. As explained, the advantage of selfish mining is based on reducing the difficulty but continue to profit from block rewards, even before the first difficulty adjustment. Another idea presented in [START_REF] Goren | Mind the mining[END_REF] consists in withdrawing temporarily from the network to lower the difficulty. Then come back after a difficulty adjustment to take advantage of the lower hash rate mining. The authors call such a strategy "smart mining" (sic). By considering the cost of mining (fixed cost and variable cost per time unit), it has been shown that this strategy can be more profitable than honest mining, even for low hashrates. In some sense, this attack is more serious than the selfish mining strategy because it does not exploit a particular flaw of the difficulty adjustment formula. 1.5. Intermittent selfish mining. Another possible strategy is the strategy of "Intermittent selfish mining" (ISM) that consists in alternating selfish and honest mining during consecutive difficulty periods. This was early discussed in social media and then numerically studied in [START_REF] Negy | Selfish Mining Re-Examined[END_REF].

The idea is to fully profit from the decrease of the difficulty but the downside is that the difficulty does not stabilize and recovers after each phase of honest mining. 1.6. Alternate network mining. In this article we introduce and study a strategy similar to "smart mining" that we name "alternate network mining" (ANM). The difference is that when the miner withdraws from the network he goes on to mine in another network with similar mining algorithm. Clearly, if honest mining is profitable, "alternate network mining" is more profitable than "smart mining", and we prove that it is also more profitable than "selfish mining" and "intermittent selfish mining". It is also effective for low hashrates. Therefore, the existence of cryptocurrencies operating with identical PoW algorithms represent a vector of attack for both networks (this was already observed in [START_REF] Grunspan | On profitability of selfish mining[END_REF]). Our study focus mainly on Bitcoin and Bitcoin Cash, but can be adapted to other pairs such as BCH / BSV or ETH / ETC. 1.7. Organization of this article. We start briefly recalling the mathematics of Bitcoin mining and the profitability model for comparing mining strategies [START_REF] Grunspan | On profitability of selfish mining[END_REF][START_REF] Grunspan | The Mathematics of Bitcoin[END_REF]. Then, we review the selfish mining strategy and show the equivalence between the Markov chain approach and the martingale approach for the computation of profitabilities [START_REF] Eyal | Majority is not enough: bitcoin mining is vulnerable[END_REF][START_REF] Grunspan | On profitability of selfish mining[END_REF]. Also we review how to compute time before profit with both approaches.

Then, we turn to ISM strategy and compute in closed form the apparent hashrate of the strategy as well as the time before profit. These results are new since before only numerical simulations were available. In passing we correct false claims and misunderstanding of the authors of [START_REF] Negy | Selfish Mining Re-Examined[END_REF]. Then, we compute the profitability of the ANM strategy. The present article is self contained.

Modelization

2.1. Mining and difficulty adjustment formula. From new transactions collected in the local database (the "mempool"), a miner builds a block B containing a trace of a previous block and a variable parameter (called a "nonce") until he finds a solution to the inequality h(B) < 1 δ where δ is the difficulty of the hashing problem and h is a cryptographic hash function (h = SHA 256 • SHA 256 for Bitcoin). This solution is the "proof-of-work" because it proves, under the usual assumptions on the pseudorandom properties of the hash function, that the miner has spend energy. For Bitcoin, the difficulty is adjusted every 2016 blocks, which corresponds on average to 2 weeks. For Bitcoin Cash and for for Bitcoin SV, the difficulty is adjusted at each block using timestamps of the previous 144 blocks (about one day). For Ethereum, the difficulty is adjusted at each block using the lapse for the discovery of the previous block. Ethereum difficulty adjustment algorithm was modified in June 2017 [START_REF] Buterin | Change difficulty adjustment to target mean block time including uncles[END_REF]. Since then, the adjustment takes into account the production of special orphan blocks also called "uncles". These are orphan blocks that are not too far from the "parent block". Ethereum Classic did not implement this new difficulty adjustment formula, and is more vulnerable to selfish mining [START_REF] Feng | Selfish Mining in Ethereum[END_REF][START_REF] Grunspan | Selfish Mining in Ethereum[END_REF] 2.2. Notations. In the article, we use Satoshi's notations from his founding paper. Thus, we denote by p (resp. q = 1 -p) the relative hash power of the honest miners (resp. the attacker). So, at any time, q is the probability that the attacker discovers a new block before the honest miners. We also denote by N (t) (resp. N (t)) the counting process of blocks validated by the honest miners (resp. the attacker) during a period of t seconds from an origin date 0. The hash function SHA 256 is assumed to be perfect, the time it takes to find a block follows an exponential distribution and hence N (t) (resp. N (t)) is a Poisson process. Initially (before a difficulty adjustment) the parameter of the Poisson process N (t) (resp. N (t)) is p τ 0 (resp. q τ 0 ) with τ 0 = 600 seconds. We also denote by S n (resp. S n ) the time it takes for the honest miners (resp. the attacker) to mine n blocks, and Sn the time it takes for the whole network to add n blocks to the official blockchain. The random variables S n and S n follow Gamma distributions. The same occurs for Sn when the attaker mines honestly (see [START_REF] Grunspan | Double spend races[END_REF]). The multiplicative difficulty adjustment parameter is denoted by δ. It is updated every n 0 = 2016 blocks. Thus, Sn 0 represents a complete mining period of n 0 official blocks and at this date, the protocol proceeds with a new difficulty.

We use the notation γ for the connectivity of the attacker. In case of a competition between two blocks, one of which has been mined and published by the attacker, γ represents the fraction of honest miners mining on top of the attacker's block.

2.3.

Profitability of a mining strategy. We consider a miner which is active over a long period compared to the average mining time of a block. What counts for his business is its Profit and Loss (PnL) per unit of time. We denote by R(t) the total income of the miner between time 0 and time t > 0. Similarly, we denote by C(t) the total cost he incurs during this period. Note that C(t) is not restricted to direct mining costs but also includes all expenditures. So, his PnL is P nL(t) = R(t) -C(t) and seeks to maximize

P nL(t) t = R(t) t -C(t) t
for t → ∞. We assume that the mining cost is independent of the mining strategy. Indeed, whether the miner broadcasts his blocks or keeps them secret, it has no impact on its mining costs. Also, whether he mines a certain cryptocurrency or another with the same hash algorithm, this does not change the mining cost per unit of time. This last quantity essentially depends on electricity costs, price of his machines, salaries of employees, etc. In particular, when the miner mines at full regime, then the cost of mining per unit of time does not depend on the strategy. In this situation, the relevant quantity is the revenue ratio Γ = lim t→+∞ R(t) t . A strategy ξ is more profitable than a strategy ξ if and only if its revenue ratio is greater: Γ ξ ≥ Γ ξ (see Corollary 2.4 below).

Note 2.1. The literature is often obscure about profitability model and sometimes disregards mining costs without proper justification. For example, in [START_REF] Negy | Selfish Mining Re-Examined[END_REF] one can read: "We omit transaction fees and mining costs from our analysis". But, as proved in [START_REF] Grunspan | Double spend races[END_REF], the only necessary assumption in order to compare strategies is to have the mining cost per unit of time to be the same for both strategies. Otherwise, the profitability analysis without considering costs does not make sense. Thus, in the smart mining strategy considered in [START_REF] Goren | Mind the mining[END_REF], the authors are naturally led to consider mining costs which depend on the miner being active or not. In the alternate network mining strategy considered in Section 5, the miner never remains inactive assuming that the mining costs are equal in both networks.

Attack cycles.

A strategy consists in attack cycles. The end of a cycle is determined by a stopping time. At the start of a cycle, the attacker and the honest miners have the same view of the official blockchain and mine on top of the same block. In general, thought this is not mandatory, during the attack cycle, the attacker mines on a fork that he keeps secret.

Example 2.2. The sequence SSSHSHH corresponds to a particular attack cycle for the selfish mining strategy: the attacker first mines three blocks in a row that are kept secret (blocks "S"); then the honest miners mine one (block "H"); then the attacker mines another one (still secret); then the honest miners mine two blocks in a row and so the attacker decides to publish his entire fork because he only has a lead of 1 on the official blockchain. The attack cycle then ends. In this case, the attacker is victorious: all the blocks he has mined end up in the official blockchain.

The attacker iterates attack cycles. It follows that by noting R i the miner's income after a i-th cycle and by T i the duration time of this cycle, the revenue ratio of the strategy is equal to

n i=1 R i n i=1 T i for n → +∞. This quantity converges to E[R] E[T ]
by the strong law of large numbers, provided that the duration time of a cycle attack T is integrable and non-zero. Likewise, assuming that costs are integrable, the cost of mining by time unit converges to

E[C] E[T ]
. Thus we can state the following corollary. Proposition 2.3. Let ξ and ξ be two mining strategies. Let R and R (resp. C and C ) be the revenue (resp. cost) of the miner by attack cycle. We denote also by T and T the duration times of attack cycles for ξ and ξ . Then, ξ is more profitable than ξ if and only if

E[R ξ ] E[T ] - E[C ξ ] E[T ] > E[R ξ ] E[T ] - E[C ξ ] E[T ]
Corollary 2.4. If moreover we assume that the cost of mining per unit of time of ξ and ξ are equally distributed, then ξ is more profitable than ξ if and only if

E[R ξ ] E[T ] > E[R ξ ] E[T ] .
Example 2.5. The attack cycles for the honest strategy are simply the time taken by the whole network to discover a block. Consequently, when the miner mines honestly, his revenue ratio is qb τ 0 where b is the mean value of a block (coinbase and transaction fees).

2.5.

Performant strategy and profit lag. Definition 2.6. A mining strategy is a performant strategy if its revenue ratio is greater than the revenue ratio of the honest strategy, i.e. E[R(T )] E[T ] > qb τ 0 , where R(T ) is the revenue per attack cycle and T is the duration time of an attack cycle.

For a performant strategy ξ, we have

E[R(T )] ≥ qb E[T ]
τ 0 . So, at T -time, the miner is better off, on average, following strategy ξ that mining honestly. However, by continuing the strategy ξ, it is possible that the miner will incur in looses for a longer period of time. Nothing prevents having E[R(τ )] < qb E[τ ] τ 0 for a certain stopping time τ > T . This, indeed, happens for the ISM strategy. Below we define the notion of profit lag of a mining strategy. Definition 2.7. Let ξ be a mining strategy and τ a stopping time.

• ξ is profitable at date τ if E[R(τ )] ≥ qb E[τ ] τ 0 . • ξ is definitely a performant strategy at date τ if E[R(τ )] ≥ qb E[τ ]
τ 0 for any stopping time τ > τ a.s.

• The profit lag is the smallest stopping time τ with this property.

This definition is sound since the infimum of stopping times is a stopping time. Note that if ξ is a performant strategy with duration time T for an attack cycle, then ξ is profitable at a date T . But in general, ξ is not definitely profitable at date T : the profit lag is in general longer than T . However we can prove that if ξ is a performant strategy then the profit lag is finite.

Proposition 2.8. Let ξ be a performant mining strategy. Then, there exists τ a stopping time such that ξ is definitely a performant strategy at date τ .

Proof. We keep the same notations as above. Let

X(t) = E[R(t) -qbt τ 0 ] for t ∈ R + .
Since ξ is a performant repetitive strategy, we have X(T ) > 0 and

Inf {X(τ ) ; τ > 0 stopping time} = Inf {X(τ ) ; τ stopping time ∈ [0, T ]} > -∞
Let us denote by m ∈ R -this quantity and let n be an integer with n > |m| X(T ) . Then, if τ > nT is a stopping time, we have:

X(τ ) = X(nT ) + X(τ -nT ) ≥ nX(T ) + m > 0
Hence, we get the result.

Selfish mining revisited

Selfish mining strategy can be described by the stopping time which defines the end of an attack cycle.

Definition 3.1. The end of an attack cycle for the selfish mining strategy is given by the stopping time

T = Inf t > S 1 / N (t) = N (t) -1 + 2 • 1 S 1 <S 1 + 2 • 1 S 1 <S 1 <S 2 <S 2 .
Example 3.2. In the case considered above, S 1 < S 2 < S 3 < S 1 < S 4 < S 2 < S 3 , we have

1 S 1 <S 1 = 1 S 1 <S 1 <S 2 <S 2 = 0 and T = Inf t > S 1 / N (t) = N (t) -1 = S 3 .
We note also by L the number of blocks added to the official blockchain after an attack cycle. It is clear that |N (T ) -N (T )| = 1 so L = N (T )+N (T )+1 2 . In the previous example, we have N (T ) = 3, N (T ) = 4 and L = 4. Proposition 3.3. We have:

E[R] = (1 + pq)(p -q) + pq p -q qb -(1 -γ)p 2 q b , E[T ] = (1 + pq)(p -q) + pq p -q τ 0 E[L] = 1 + p 2 q + p -q p -q b
Proof. The counting process N (t) (resp. N (t)) is a Poisson process. So, T (resp. T ) is an integrable stopping time. Then the result follows from Doob stopping time theorem applied to the compensated martingales N (t) -αt (resp. N (t) -α t) with α = p τ 0 (resp. α = q τ 0 ) and to the finite stopping time T ∧ t for a fixed t > 0. Then, we take the limit when t → ∞ using the monotone convergence theorem. We get as limits

E[T ], E[N (T )], E[N (T )] and E[L]. Finally, we also get E[R] by observing that E[R] = E[N (T )]b -(1 -γ)p 2 qb
since the only attack cycle when R = N (T ) happens when the attacker mines first a block, then the honest miners mine one block and the honest miners mine another block on top of the honest block.

We set the date t = 0 when the selfish miner starts his attack, at the beginning of a new period of 2016 official blocks, just after a difficulty adjustment.

Corollary 3.4. Let δ be the first difficulty parameter. We have E[δ] = p-q+pq(p-q)+pq p 2 q+p-q and E[

Sn 0 ] = E[δ] • n 0 • τ 0
Proof. Let ν be the number of cycles before we get a difficulty adjustment. By Wald theorem, we get

E[L] • E[ν] = n 0 and E[ Sn 0 ] = E[ν] • E[T ]. So, E[δ] = E[ Sn 0 ] n 0 τ 0 = n 0 E[T ]
E[L]τ 0 . Hence, we get the result using Proposition 3.3

We can now calculate the apparent hashrate of the strategy which is defined as the fraction of blocks of the attacker in the official blockchain on the long term.

Corollary 3.5. The apparent hashrate of the selfish mining strategy is q = ((1 + pq)(p -q) + pq)q -(1 -γ)p 2 q(p -q) p 2 q + p -q Proof. After a difficulty adjustment, the mining difficulty is divided by δ. So the parameters of the Poisson processes N (t) and N (t) are each multiplied by δ and also the duration times of attack cycles are divided by δ. On the other hand, the revenue of the selfish miner remains constant after each attack cycle since the probability to be first to discover a new block remains the same. Therefore, his new revenue ratio after difficulty adjustment is

q b τ 0 with q = E[R] E[T ] • δ • τ 0 b .
Hence, we get the result using Proposition 3.3.

By rearranging terms, we observe that q is nothing but the quantity R pool from [START_REF] Eyal | Majority is not enough: bitcoin mining is vulnerable[END_REF]. But, more importantly, we have proven the following proposition. Proposition 3.6. Before a difficulty adjustment, official blocks are discovered on average every period of δ • τ 0 . The revenue ratio of the selfish miner is q δ b τ 0 (lower than q b τ 0 ). After a difficulty adjustment, officials blocks are discovered on average every period of τ 0 . The revenue ratio of the selfish miner equal q b τ 0 (which is now greater than q b τ 0 ).

3.1.

Previous state-machine approach revisited. We can model the different states taken by the network with the help of a Markov chain (X n ) as was done in [START_REF] Eyal | Majority is not enough: bitcoin mining is vulnerable[END_REF]. The set of states is N ∪ {0 }. The state {0 } corresponds to the state when there is a competition between two blocks in the network. The state {n} corresponds to the state when the attacker has a lead of n blocks over the honest miners. In this framework, the duration time of an attack cycle is proportional to the first return time ν = Inf {n ∈ N * / X n = 0}. It turns out that the Markov chain is irreducible and all states are positive recurrent. It has naturally a stationary probability (π n ) and the computation shows that π 0 = 1-2q 1-4q 2 +2q 3 [START_REF] Eyal | Majority is not enough: bitcoin mining is vulnerable[END_REF]. Therefore, a classical probability result on Markov chains shows that E[ν] = 1 π 0 [START_REF] Douc | Markov Chains[END_REF]. Then it is easy to check that

E[ν] = (1 + pq)(p -q) + pq p -q
We recover the formula obtained for E[T ] in Proposition 3.3. Also, for r pool and r others defined in [START_REF] Eyal | Majority is not enough: bitcoin mining is vulnerable[END_REF] we have

1 r pool + r others = 1 -4q 2 + 2q 3 1 -q -2q 2 + q 3 = E[δ]
So we get the difficulty adjustment parameter given in Proposition 3.4. Finally, the revenue ratio before difficulty adjustment is

E[R] E[T ] = r pool • b τ 0 .
The formula given in [START_REF] Eyal | Majority is not enough: bitcoin mining is vulnerable[END_REF] for the apparent hashrate q = r pool r pool +r others is the same that we find: r pool is the revenue ratio before difficulty adjustment and 1 r pool +r others > 1 is the (mean) difficulty adjustment parameter δ. For the purposes of computation of the profitability there are two equivalent models: one with Poisson processes (as considered first by Satoshi Nakamoto in his founding paper and continued in [START_REF] Grunspan | On profitability of selfish mining[END_REF]) and another with Markov chains. They are not equivalent for other purposes, as for example to show the optimality of honest mining when there is no difficulty adjustment. Note also that there exists a third very effective combinatorical model that captures directly the long term behavior of a deviant strategy. For the selfish mining strategy, stubborn strategy and trailing strategy, it uses the combinatorics of Dyck words and Catalan distribution (see [START_REF] Nayak | Stubborn mining: generalizing selfish mining and combining with an eclipse attack[END_REF][START_REF] Grunspan | On profitability of stubborn mining[END_REF][START_REF] Grunspan | On profitability of trailing mining[END_REF]).

3.2. Profit lag. We recall that according to notations in Section 2.2, Sn 0 denotes the time used by the network to reach the first difficulty adjustment. Proposition 3.7. Set t 0 = qδ-q q -q n 0 τ 0 . On average, the selfish mining strategy is not profitable before Sn 0 + t 0 and is profitable after this date.

Proof. We denote by ∆(t) the difference between the revenue of the selfish miner at date t and the revenue of honest mining. As we have seen, the revenue ratio of the selfish miner before the first difficulty adjustment is lower than the revenue ratio of the honest strategy. So, the selfish mining strategy cannot be profitable before this date. Moreover, using Proposition 3.6, we get: E[∆( Sn 0 )] = ( q δ -q)n 0 δb = -(qδ-q )n 0 b < 0 and for t > 0, we have

E[∆( Sn 0 + t)] = (q -q) t τ 0 b + E[∆( Sn 0 )] = (q -q) t τ 0 b -(qδ -q )n 0 b = (q -q)(t -t 0 ) b τ 0 Figure 1.
Difference of average revenue between the selfish miner and honest miner for q = 0.1 and γ = 0.9. X-axis: progression of the official blockchain, in difficulty adjustment units. Y -axis: revenue of the miner in coinbase units. The strategy definitely not profitable before 5 difficulty adjustments.

Example 3.8. When q = 0.1 and γ = 0.9, the computation shows that we have exactly qδ-q q -q = 4. Therefore, the selfish mining strategy becomes definitely profitable at the date of the fifth difficulty adjustment which is roughly 10 weeks of mining (it is exactly 10.1839 weeks) as asserted in [START_REF] Grunspan | On profitability of selfish mining[END_REF]. Before this date, the strategy is not profitable. Note 3.9. In the introduction of [START_REF] Negy | Selfish Mining Re-Examined[END_REF] the authors claim (with no justification) that the previous computation from [START_REF] Grunspan | On profitability of selfish mining[END_REF] is erroneous. However, in their simulations, they obtain E[ Sn 0 ] = 15.29 days which agrees, up to rounding, with the exact theoretical value from Corollary 3.4 which is 15.2872. And from here, it is not hard to get Proposition 3.7 as we did above. Obviously, our theoretical exact results agree with selfish mining simulators that one can find online [START_REF] Khosravi | A Full-fledged Simulation for Selfish Mining in Bitcoin[END_REF]. So, apparently, the authors of [START_REF] Negy | Selfish Mining Re-Examined[END_REF] are confused, not only about this, but also about the profitability of the Intermittent Selfish Mining (ISM) that they analyze only numerically. In the next section we give rigorous analysis of ISM with closed form formulas.

Intermittent selfish mining strategy

This strategy is composed by attack cycles with two distinct phases. Each phase corresponds to a complete period of mining of n 0 official blocks until the next difficulty adjustment. In the first phase, the attacker follows the selfish mining strategy. In the second phase, he mines honestly.

Proposition 4.1. The mean duration time of the first (resp. second) phase is n 0 τ 0 δ (resp. n 0 τ 0 δ ). During the first (resp. second) phase, the revenue ratio of the miner is q δ • b τ 0 (resp. q δ • b τ 0 ) where q and δ are given in Corollary 3.4 and Coreollary 3.5 (denoting by δ its expected value). Moreover, denoting by R (resp. T ) the revenue (resp. duration time) at the end of an attack cycle, we have:

E[T ] = δ + 1 δ n 0 τ 0 E[R] = (q + q)n 0 b
Proof. The duration time and the revenue ratio of the attacker during the selfish mining phase has been computed in the previous section. After a first difficulty adjustment, the mining difficulty is divided by δ. So, blocks arrive on an average period τ 0 δ and the duration time of the second phase is n 0 τ 0 δ . Each time, there is a probability q that the attacker is the first one to discover the next block. Hence, during the second phase, his revenue ratio is q b τ 0 δ = q δb τ 0 . Finally, the revenue of the ISM miner at the end of an attack cyle is

E[R] = q δ b τ 0 • n 0 τ 0 δ + q δ b τ 0 • n 0 τ 0 δ
and the result follows.

Corollary 4.2. The apparent hashrate of the Intermittent selfish strategy is q" = q+q δ+ 1 δ , and we have the closed form formula

q" = q (1 -4q 2 + 2q 3 ) (1 + γ + (3 -4γ)q + (5γ -11)q 2 + (5 -2γ)q 3 ) 2 -2q -11q 2 + 10q 3 + 18q 4 -20q 5 + 5q 6
Example 4.3. This formula checks well with simulations from [START_REF] Negy | Selfish Mining Re-Examined[END_REF]. The threshold ISM/HM is obtained when q" = q. When γ = 0, we get q" = 0.365078. The authors from [START_REF] Negy | Selfish Mining Re-Examined[END_REF] obtain 0.37 with their simulations.

Corollary 4.4. When ISM is more profitable than honest mining, then SM is more profitable than ISM.

Proof. By hypothesis, we have q+q δ+ 1 δ > q, which implies

q q > δ + 1 δ -1 > 1 > 1 δ + 1 δ -1 since δ + 1 δ > 2 for δ > 0. So, q+q δ+ 1 δ
< q and ISM is less profitable than SM.

In Figure 2 regions in (q, γ) ∈ [0, 0.5]×[0, 1] are colored according to which strategy is more profitable (HM is the honest mining strategy).

Figure 2. Dominance regions in parameter space (q, γ). The threshold SM/HM (resp. ISM/HM, resp. SM/ISM) in black (resp. blue, red). When ISM is more profitable than HM then SM is always more profitable than ISM.

4.1. Profit lag. As before, we denote by ∆ the difference between the average revenue of selfish and honest mining. According to Proposition 4.1, after n attack cycles i.e., alternatively n phases of selfish mining and n phases of honest mining, the at-

tacker earns q" • δ + 1 δ n • n 0 b during δ + 1 δ n • n 0 τ 0 . So, at this date, we have ∆ = (q" -q) • δ + 1 δ • n n 0 b
After this date, if the attacker selfish mines again, then at the end of this phase of duration δn 0 τ 0 , his revenue is q"

• δ + 1 δ n • n 0 b + q δ • n 0 δ τ 0 . At this date, we have ∆ = (q" -q) • δ + 1 δ • n n 0 b + q δ -q • n 0 δ b = (q" -q) • δ + 1 δ • n n 0 b -(qδ -q ) • n 0 b
By the end of an attack cycle, we have on average ∆ = (q" -q) n 0 δ + 1 δ b which is positive when ISM is more profitable than honest mining. Thus when q" > q, ISM is more profitable than HM before a second difficulty adjustment as noticed in [START_REF] Negy | Selfish Mining Re-Examined[END_REF]. This comes as no surprise since the duration time of an attack cycle of the ISM strategy corresponds to a period of 2 × 2016 official blocks, and any performant mining strategy is always profitable at the end of an attack cycle. See the discussion at the end of Subsection 2.5.

The authors [START_REF] Negy | Selfish Mining Re-Examined[END_REF] fail to understand that after another difficulty adjustment the revenue falls because the difficulty increases and ISM becomes less profitable than HM. It is only after several difficulty adjustment periods of 2016 blocks that ISM can become definitely more profitable than HM. For instance, when q = 0.1 and γ = 0.9 this only happens approximatively after the 13-th difficulty adjustment. Therefore, on average, it takes more than 6 months for the strategy to be definitely profitable, and this is much longer than for classical SM. Obviously the authors of [START_REF] Negy | Selfish Mining Re-Examined[END_REF] are overall confused with a naive notion of profitability and don't follow a rigorous profitability model.

It is important to understand what happens. First, the selfish miner invests in lowering the difficulty, and, at any moment, he can get reap immediate profits, even before the profit lag, by just mining honestly. But if the miner wants to repeat the attack cycle, he will need to burn again these profits for the purpose of lowering the difficulty. This is what happens in the first cycles of the ISM strategy. In Figure 3 we have the plot of the progression of ∆.

Alternate network mining strategy

The strategy is described for alternate mining between BTC and BCH networks, but obviously it applies to any pair of networks with the same mining algorithm (with frictionless switching mining operations). This time we consider three distinct types of miners: the first ones mine on Bitcoin only, the second ones on Bcash (BCH), and the third ones alternate between Bitcoin and Bcash. The ones mining alternatively between BTC and BCH are labelled "attackers" (although his strategy is legit and Figure 3. Difference of average revenue between intermittent selfish and honest mining for q = 0.1 and γ = 0.9. X-axis: progression of the official blockchain, in difficulty adjustment units. Y -axis: revenue of the miner in coinbase units. The strategy is not definitely profitable before 13 difficulty adjustments. By comparison, SM is definitely profitable after 5 difficulty adjustments (compare with Figure 1). respects both network protocols). We assume that the revenue ratios of mining honestly on Bitcoin and Bcash are the same. This is approximativelly what is observed in practice since any divergence justifies a migration of hashrate from one network to the other. We denote by ρ this common value for the attacker. We assume also that there are no other miners and all miners mine with full power. In particular, the total hashrate remains constant. The attacker starts mining on Bitcoin at the beginning of a difficulty adjustment on BTC. An attack cycle is made of two phases. During Phase 1, the attacker withdraws from BTC and mines on BCH until n 0 blocks have been mined on Bitcoin. During Phase 2, the attacker comes back to mine on the BTC network until a new difficulty adjustment. We call this strategy: Alternate Mining Strategy. This is a variation of smart mining. The only difference being that the miner does not remain idle during Phase 1 but goes on to mine on BCH. Note that this mining strategy does not provoke periodic reorganizations of the blockchain. The main annoyance for users of both networks is that difficulty does not stabilize and for Bitcoins users blocks arrive regularly at a slower pace than in the steady regime with the miner fully dedicated to the BTC network.

We denote by δ the difficulty adjustment parameter after Phase 1. Since the miner comes back on BTC in the second phase, the second difficulty adjustment at the end of the second phase is 1 δ .

Proposition 5.1. The duration phase of Phase 1, resp. Phase 2, is n 0 τ 0 δ, resp. n 0 τ 0 1 δ . During Phase 1, resp. Phase 2, the revenue ratio of the attacker is ρ, resp. ρδ. The revenue ratio of the alternate mining strategy is 1+δ

δ+ 1 δ ρ.
Proof. By definition of the difficulty adjustment parameter, the duration time of Phase 1, resp. Phase 2, is δn 0 τ 0 , resp. 1 δ n 0 τ 0 . So the duration time of an attack cycle is δ + 1 δ n 0 τ 0 . During Phase 1, the attacker's revenue ratio is ρ because we assume that the attacker mines honestly during this phase with the assumption that the revenue ratio is the same for BCH and BTC. During Phase 2, the mining difficulty is divided by δ. So, the revenue ratio of the attacker during this period is δ•ρ. Therefore, the revenue of the attacker after an attack cycle is ρ • n 0 τ 0 δ + (ρ δ) • n 0 τ 0 δ . Corollary 5.2. The alternate mining strategy is always more profitable than honest mining and selfish mining for all values of q.

Proof. The first statement results from δ > 1 that implies 1+δ δ+ 1 δ > 1. To prove, the second statement, we remark that in Phase 1, blocks are only validated by honest miners. So this phase lasts on average n 0 τ 0 p and the difficulty parameter is updated accordingly: δ = 1 p with p = 1 -q. So, the revenue ratio of the attacker is 2-q 2-2q+q 2 b τ 0 as follows by replacing δ with 1 p in the formula from Proposition 5.1. On the other hand, in the most favorable case (when γ = 1), the revenue ratio of the selfish miner is q(2q 3 -4q 2 +1) q 3 -2q 2 -q+1 b τ 0 . Then, the result comes from 2-q 2-2q+q 2 > q(2q 3 -4q 2 +1) q 3 -2q 2 -q+1 for 0 < q < 0.5 that we prove studying the polynomial (2 -q) • (q 3 -2q 2 -q + 1) -(2 -2q + q 2 ) • (q 2q 3 -4q 2 + 1 ) This polynomial is non-increasing on [0, 1 2 ] and remains positive on this interval.

5.1. Profit lag. As before, we denote by ∆ the difference of the average revenue between selfish mining and mining honestly from the beginning.

Proposition 5.3. After n attack cycles, we have ∆ = ρ • 1 -1 δ n n 0 τ 0 = q 2 n n 0 b and ∆ stays constant during the BCH mining phase.

Proof. After a first phase of mining, we have ∆ = 0 since the revenue ratio on BCH and BTC are assumed to be equal. At the end of the second phase, we have ∆ = 1 -1 δ ρ • n 0 τ 0 . Indeed, the second phase lasts n 0 τ 0 δ and the revenue ratio of the attacker during this phase is ρδ. Now, we use that ρ = q b τ 0 and δ = 1 p because only honest miners are mining during Phase 1. The strategy is then a repetion of alterning phases 1 and 2 and the result follows.

We plot the graph of ∆ in Figure 4. for q = 0.1. X-axis: progression of the official blockchain, in difficulty adjustment units. Y -axis: revenue of the miner in coinbase units. The strategy is definitely profitable after one difficulty adjustment.

Figure 4 .

 4 Figure 4. Difference of average revenue between alternate and honest mining

Conclusion

We revisit the selfish and intermitent mining strategies, and define alternate mining strategy for pairs of networks with the same PoW. We define the notion of "profit lag". We compute exact formulas for different strategies of the profit lag and the revenue ratio. We correct misunderstandings and unfounded claims in [25] by clarifying the profitability setup using the new notion of "profit lag". We show that, under natural hypothesis, the alternate mining strategy is the best one: its revenue ratio is the largest and the profit lag is the least. So, the existence of two networks sharing the same PoW algorithm can lead to instabilities of the difficulty and with blocks validated slower than normal.


Alauzet Aline
email: aline.alauzet@ifsttar.frgwenaelle.raton@ifsttar.fr

RESIDENTIAL CHOICE AND ACCESS TO TRANSPORT FOR PEOPLE WITH DISABILITIES

Extended Abstract

This study aims to identify the constraints people with disabilities have to face when making residential choices, to understand how they make such choices and ascertain the role transport issues play in them.

Qualitative information was collected in the "Grand Lyon" area (France). The targeted population was people with all types of disabilities: physical, sensorial, psychological, mental or cognitive impairments; and several profiles: disability from birth or acquired later in life, people in work or the unemployed, several different family situations. A total of 57 disabled people were interviewed. Other interviews, aimed at collecting the experiences and opinions of people likely to be involved in disabled people's housing choice, were conducted, with members of 20 associations that provide assistance to disabled people and defend their rights and 5 associations or services responsible for housing issues.

Our results show that the residential choice process is difficult for people with disabilities, for all the steps individuals have to cross during this process: formulate and express their wishes, estimate their possibilities and compare them to the housing market, and then make a trade-off between wishes, the constraints that weigh on the household or individual, and the reality of the housing market.

Assessing their housing needs and choosing a dwelling is indeed a complex process for many disabled people. It begins right from the expression of preferences, notably for people with mental or psychological disabilities: due to decision-making difficulties they often need human support at this stage. Another possible difficulty is assessing one's own ability to live autonomously, which requires management skills, the ability to keep accounts and to cope with administrative procedures. To make a residential choice it is also necessary to have some knowledge about the housing market. People with disabilities do not always have the right contacts to enable them to find the information they need. In addition, they sometimes need to find adapted housing. But our interviewees found no place or service where information about available adapted housing was clearly recorded and accessible. Due to the constraints they have to face, some disabled people also need assistance to make trade-offs and decide on actions during the entire residential choice process. The nature and intensity of assistance vary according to the person and their ability to carry out the different stages of the choice process autonomously. The interviews revealed some general tendencies concerning the role of the caregiver according to the type of impairment. The need to have a caregiver to transcribe the person's needs can also lead to a higher level of dissatisfaction about the final choice. The greater the involvement of the caregiver, the higher this risk. Some disabled people have internal adaptation needs for their dwellings. Our interviews showed that each type of disability generates distinct housing needs, not restricted to the dimensions for wheelchairs users or some specific tools like videophones for the hearing impaired or warning lights for the visually impaired, for example: open spaces for the visually or hearing impaired; to be able to move easily inside the dwelling, to be able to enter and leave the dwelling easily, and to be able to add fixtures and fittings that increase their independence, for people with a motor impairment; and reassuring housing for people with a mental or psychological impairment. Specific needs in terms of location in the city were also expressed by our interviewees. They underlined the importance of access to transport, which play a crucial role when they have to choose between urban or rural location for their home. There are also shared preferences for the proximity of various elementssuch as shops and family caregivers -with the closeness of public transport a priority. But the proximity of public transport is not in itself enough, because it also has to be accessible. We observed differences between transport proximity preferences according to the person's impairment.

It appeared that the residential choice of people with disabilities is a cumulative disadvantage process: a number of specific criteria have to be met in order for people with disabilities to live and move autonomously. But their housing choice is made difficult by powerful constraints that make it difficult for them to meet these needs. Housing needs are responsible not only for considerable financial costs but they also require a considerable amount of time to look for and access information. Due to economic inequalities, social barriers and geographical issues, combined with the nature or the evolution of their impairment, people with disabilities don't often have the necessary financial and social resources to deal with these constraints.

Finally, to make their housing choice, disabled people have to make a trade-off between three main needs: welfare in the home, access to the local area and access to the town. Their independence is ensured only if all three of them are satisfied, but in reality it is rarely the case, and a priority has to be given to one of these needs, depending notably on professional or family status.

In conclusion, special attention should thus be paid to information about housing and to the way support is given to disabled people for their housing choice. Their housing adaptation needs, or needs in term of location in the city, should also be taken into consideration by housing suppliers. Finally, in view of the complexity of the housing choice process for disabled people, which sometimes makes it difficult for them to choose a dwelling that guarantees their full independence, we can conclude that housing choice constitutes the primary source of inequality for access to transport. But as transport is a key means of bringing about their social participation, in most cases it plays an important role in disabled people's housing choice.

Keywords: people with disabilities, transport accessibility, residential choice, housing


J Gié

M Lazar

D Planson

D Bergogne

P Bevilacqua

B Allard
email: bruno.allard@insa-lyon.fr

S M'rad

Towards an integrated inverter based on lateral JFET SiC

The feasibility of a hybrid inverter based on commercial silicon carbide vertical JFETs was tested. The Ronresistance, mainly due to the vertical channel, increases with the temperature, from 1.1 Ω measured at RT to 2.98 Ω at 300°C. These devices can drive current in both forward and reverse direction, eliminating the need for an external anti-parallel diode. A transient current-limited short circuit is observed with duration of 100 ns in turnon. The power losses estimated involve for the hybrid inverter a working frequency up to 300 kHz. Good results obtained for hybrid inverter predict a SiC monolithic inverter realization. For this propose, SiC lateral JFETs (LJFETs) were studied and designed by finite-element method. A double RESURF structure is retained for the SiC LJFET, in order to reduce the on-resistance and to improve the blocking voltage. The obtained Ronresistances are one order of magnitude lower than in vertical SiC JFETs. This result is obtained thanks to the high doping lateral channel and RESURF structure. Switching times of the LJFETs during turn-off and turn-on, are obtained in the 20 to 30 ns range. Eight levels of masks are designed to fabricate these devices. The masks structures were optimized in order to minimize both the Ron-resistance and the device size by utilizing interfingers structures.

INTRODUCTION

The development of power electronics applications involves the realization of specific devices which must work in harsh environment (high temperature >200°C, high voltage, 600 V for domestic applications). Hybrid approaches suffer from interconnection reliability. Monolithical integration seems to be a more effective method to assemble power devices and passive components. Due to their physical properties, semiconductors with large bandgap are very interesting in this case compared to the classical silicon, the material usually utilized in power device fabrication. Among the spectra of large bandgap semiconductors, silicon carbide (SiC) is the semiconductor material presenting the most advanced device technology [1][2].

SiC is known to exist in many different crystallographic structures (polytypes). To develop high temperature and/or high voltage devices, 4H-SiC and 6H-SiC are most frequently utilized. In the case of these polytypes, the breakdown critical electric field is an order of magnitude higher compared to silicon and the thermal conductivity is three times higher.

Thus the use of 4H or 6H-SiC allows to reduce the size (by a factor of 10 to 100), the number of devices and to increase their operating temperature.

Diminishing the size of elemental devices is an additional factor that commonly with the monolithic integration allows a significantly decrease of the switching power losses during the inverter operation.

Inverters are currently based on silicon IGBTs for medium and large power applications. Because it can not conduct current in reverse direction, an antiparallel power diode is needed. The most favorable SiC switch (technologically available) is the JFET [3,4].

Compared to the IGBT, this is a normally-on switch and presents a reversible behavior with a conducting current in forward and reverse bias. Thus, for the inverters based on SiC JFET, the anti-parallel power diodes are no more necessary. Furthermore, to turn the SiC JFET off, a negative gate voltage is required at values that may be superior than in IGBT based inverters with probably superior gate capacitances also.

Firstly this paper presents an experimental validation of a high temperature operating inverter based on commercial discrete Vertical-JFET (VJFET) in SiC.

The experimental results enable to consider the design of an integrated inverter in SiC. This integrated system based on the design of SiC lateral JFETs (LJFET) with a breakdown voltage of 900V is presented in the second part of this paper.

FEASIBILIY

The feasibility of an inverter based on silicon carbide JFETs was tested. Figure 1 In the off-state (V GS at -20 V), the I-V measurements of SiC VJFETs (Figure 1c) show a blocking voltage limited to about 1 kV with a leakage current increasing with temperature.

(b) V DS [V] Vgs= 0 V -4 V -8 V -1 V -5 V -9 V -2 V -6 V -10 V -3 V -7 V RT I DS [A] V DS [V] (a) T = 300°C Vgs=0 V -4 V -8 V -1 V -5 V -9 V -2 V -6 V -10 V -3 V -7 V
-

2 -1 0 1 2 -2 -1 0 1 2 3 RT 75°C 150°C 200°C 250°C 300°C

V DS [V] I DS [A]

V GS = 0 V Fig. 2 Reversible behavior of the SiC VJFET in forward and reverse bias for ambient temperatures varying from RT to 300°C.. These experimental investigations show also that these devices can conduct current in both forward and reverse direction (Figure 2), eliminating the need for an external anti-parallel diode. Waveform qualities are due to the used inductor in the current source and to a not so small gate resistor of 47Ω.

Fig. 5 Estimated total power losses per VJFET with junction temperature as a parameter

Losses from the inverter were estimated from electrical measurements for RT to 300°C operation. Total power losses as a function of frequency are presented in Figure 5. The performance of this circuit predicts the possibility to construct an integrated inverter with a working frequency up to 300 kHz.

INTEGRATED INVERTER DESIGN

Approach -design method

A monolithically integration of SiC JFETs allows to reduce considerably the switching time and to eliminate losses and parasitics related to the device inter-connections. Lateral devices are in need in this case because the silicon carbide technology is mainly developed only in the Si face. The substrate doping is still elevated to realize power devices by doping the substrate and the growth of epitaxial layers in the C face is also technologically difficult.

Reduced Surface Field (RESURF) technique is one of the most widely used method to realize the design of lateral high voltage devices in power integrated systems [START_REF] Ludikhuize | Proc. ISPSD[END_REF][START_REF] Krizaj | [END_REF]. This method allows to integrate different power devices such as bipolar and MOSFET transistors with high voltage diode.

The RESURF technology consists to introduce layers in the devices structures that are defined by their geometric dimensions (thickness and length), the type of semiconductor and the doping concentration. By dimensioning these parameters, the presence of these layers will generate a more homogenous distribution of the electric field in the device structures and thus will increase breakdown voltage. The RESURF method is applied here in order to reduce the on-resistance and to improve the blocking voltage of the SiC LJFETs. 947 cm 2 V -1 s -1 0 cm 2 V -1 s -1 -1,962 0 0.61 1.94×10 17 cm -3 124 cm 2 V -1 s -1 15.9 cm 2 V -1 s -1 -1.434 0 0.34 1.76 10 19 cm -3 500 cm 2 .V -1 .s -1 167 cm 2 .V -1 .s -1 50 ns 50 ns 10 30 cm -3 10 30 cm -3 4.08 10 5 cm -1 1.63 10 7 cm -1 1.67.10 7 V.cm -1 1.67.10 7 V.cm -1 SiC LJFETs, targeting inverter structure integration, were studied and designed by finite-element method simulation with MEDICI software tool. Semiconductor device behaviours are obtained by simulating of internal parameters in the device structure such as the distribution of potential, the density of carriers, the density of current, and the propagation of electric field. The device structure is composed of different areas (semiconductor, oxide, metal). A schematic SiC LJFET structure is illustrated in Figure 6. In each area, different parameters are defined such as doping profiles. A grid with a great number of nodes is realized on all the device structure. The equations of the physic of semiconductor such as equation of continuity and Poisson's equation are solved for each node. SiC physical parameters and analytical models are utilised for the mobility, incomplete ionization, life time of carriers and coefficients of ionization (Table 1). In the surface, for the different structure layers, realistic doping profiles were introduced. These profiles were obtained by Monte-Carlo ion implantation simulation or from SIMS preliminary analyses of implanted SiC samples. Drain, source, gate and the channel were designed as to be ion implanted in a low doped epitaxial p-type layer with a uniform doping concentration.

⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ n,p n,p n,p0 n,p 1 n,p0 P satn,p µ µ (x,y)= µ .E (x,y) 1+( ) v β β ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (a) 1 e x p D D FN D N N E E GCB kT + = - ⎛ ⎞ + ⎜ ⎟ ⎝ ⎠ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - + = - T FP A A A K E E GVB N N exp 1 (b) n,p0 n,p SRHn,p τ τ (x,y)= N(x,y) 1+ N ⎛ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎞ (

Fig. 6 Schematic presentation of the SiC LJFET with a double RESURF structure

A double RESURF structure is retained for the SiC LJFET. This is realized by the epitaxial layer forming a junction below the n-type channel (active layer) and by adding a top layer between the gate and the drain. The doping profile of the p-type top-layer is also controlled by ion implantation. In blocking state, the vertical electric field distribution is defined principally by the thickness and the doping of the p-type epitaxial and the ratio with the active layer doping. The low value of the epitaxial layer doping by its depletion allows a vertical distribution of the electric field inside the structure. But the ratio between the channel doping and the epitaxial layer doping must be kept low enough in order to deplete the channel active layer by its junction with the epitaxial layer. The channel depletion by this vertical SCR (space charge region) must occur before the lateral depletion of the channel by the lateral diode formed between the drain and the gate.

The presence of the p-type top layer constitutes the second RESURF structure. In blocking state this layer must be depleted vertically in order to extend the SCR towards the top junction before the lateral depletion of the n-type active layer and the p-top layer itself. Keeping a high breakdown voltage by improving electric field distribution, the RESURF technique provides also better lateral on-state conduction in JFETs. It allows to obtain a JFET with higher channel doping concentration, a small length and so a lower on-resistance.

Results and discussion

The lateral electric field distribution of the double RESURF JFET structure is shown in Fig. 7 for different doping concentration of the top layer. The doping concentration of the channel is fixed to 2.10 17 cm -3 . A low doping concentration of the top layer compared to channel doping supports the lateral depletion, and involves a high concentration of the electric field close to the gate. The potential lines are distributed mainly close to the device surface. The breakdown of junction is established between the top layer and the channel.

The presence of top layer with a high doping concentration (5.10 17 cm -3 ) will make difficult the depletion of this layer. The SCR extends mainly through the channel and the P-epi layer. The maximum value of electric field is obtained close to the drain where the potential lines are concentrated that involves the device to reach prematurely the breakdown voltage (V BR ). The lateral electric field distribution is optimised when the doping concentration of the top layer allows obtaining a trapezoidal electric field between the drain and the gate. The optimum potential distribution for this structure is obtained for a doping concentration of the top layer included between 1.10 17 cm -3 to 2.10 17 cm -3 . In this range, maximum values of the electric field are reached both at the drain and at the gate side of the structure, respectively. The SCR surface is wider, developed in the epi-layer, which increases the breakdown voltage of the JFET.

The figure 8 shows the breakdown voltage as a function of doping concentration of top RESURF layer. The maximum value of breakdown voltage reach 1kV with doping concentration of top layer equal to the doping concentration of channel. The doping concentration of the top layer is fixed to 2.10 17 cm-3. Fig. 13 shows the doping profile for the cut line indicated in Fig 6. The forward characteristics of SiC lateral JFETs are related to a technological limit (the minimum distance between two successive lateral layers). Figure 14 presents the spectra for a technological limit of 3 µm. The device surface area is 1.5 mm 2 . The high conduction level is mainly due to the low value of the on-resistance thanks to the high channel doping. Fig. 15 shows the evolution of Ron-resistance as a function of length of channel and the area of device. The length of channel is defined by the technological limit (L T ). The Ron-resistance is estimated to 0.3 Ω, by using an area device of 1.5 mm 2 with technology limit of 3 µm. The Ron-resistance can reach a value lower than 0.1Ω, by using an area of device of 2 mm 2 with technology limit of 0.5 µm. To obtain a value of 0.05Ω with a technological limit less advanced such as 1µm, the area of SiC LJFET must be larger than 5.5 mm 2 . These values are one order of magnitude lower than in vertical SiC JFETs. This result is obtained thanks to the high doping channel and the dissociation in the lateral JFET between the layers for the lateral conduction in on-state and the layers for the voltage sustain in off-state. For the moment, SiC devices size seems to be limited by the defects densities of the available wafers. Nevertheless a considerable effort was done these last years to decrease the micropipe and dislocation densities in SiC substrates. The optimisation of this resistance is important in order to increase the courant in on-state and to decrease the voltage over the LJFET in off-state (limiting the power losses in switching mode).

The SiC LJFETs are principally designed in order to be utilised in power converters, like inverter modules. The static performances show that the SiC RESURF JFET can be used as a lateral switching device with a breakdown voltage over 600V. A simple structure with a resistor and an inductive load (100nH) was realized for evaluating the switching performances of lateral JFETs. The dc-link voltage was 450 V and the load current was 2 A. A gate signal with amplitude of 450V was applied with a resistance of 220 Ω. Eight levels of masks are necessary to fabricate this device (Fig. 17). The masks structures were optimized in order to minimize both the Ron-resistance and the device size by utilizing inter-fingers structures. Three different channel lengths, 3, 6 and 9 µm have been considered. The distance between the gate and the drain areas was fixed to 10µm. The device fabrication process is currently under completion. Planar structures are privileged utilizing ion implantation to form box profiles. The number of epitaxial layers is reduced and the technology process excludes the use of plasma etching that generates significant surface defects.

Conclusion

The experimental results obtained from the realization of a hybrid inverter based on commercial SiC vertical JFETs enable to consider the design of a SiC monolithically integrated inverter. For this, SiC lateral JFETs structures were studied and designed by finiteelement method. The Ron-resistances and switching times were improved by one order of magnitude versus vertical SiC JFETs. These results are obtained thanks to the optimized double RESURF structures where high doping channel is present. In this structures also are dissociated the lateral conduction layer in on-state from the layers sustaining the high voltage in off-state. Eight levels of masks are designed to fabricate these devices by utilizing interfingers structures in order to minimize both the Ronresistance and the device size. Integration compatible planar structures are privileged by utilizing ion implantation to form box doping profiles. The device fabrication process is currently under completion.

Fig. 1 I

 1 Fig. 1 I DS -V DS of SiC commercial VJFET in the forward on and off-state: on-state at RT (a), on-state at 300°C (b) and off-state from 200°C to 300°C (c)

Fig. 3 Fig. 4

 34 Fig. 3 Schematic (a) and picture (b) of the hybrid inverter with two SiC commercial VJFET

Fig. 7 :

 7 Fig. 7 : Distribution of the lateral electric field of the JFET RESURF structure

Fig. 12 :

 12 Fig. 12: Breakdown voltage as a function of the top RESURF layer doping concentration with the active layer = 2.10 17 cm -3

Fig. 13 :

 13 Fig. 13: Vertical doping profile in the lateral device.

Fig. 14

 14 Fig. 14 : i-v characteristics of SiC lateral JFET

2 ]Fig. 15 :

 215 Fig. 15: R on resistance as a function of the area of SiC lateral JFET.

Fig. 16

 16 Fig. 16 Switching characteristic of the SiC lateral JFET with double RESURF structure, a) turn-off, b) turn-on

Fig. 17

 17 Fig. 17 Top view of the SiC lateral JFET mask layout with eight levels superposed.

  In a vertical configuration the on-resistance of JFETs are limited by the length of the channel which corresponds to the thickness of the low-doped SiC epitaxial layer. Its thickness value (a tenth micron) is directly related to the JFET blocking voltage in offstate (e.g.: a SiC tenth micron with 5×10 15 cm -3 doping corresponds to 1500V blocking voltage).

presents the forward onstate and blocking characteristics of commercial SiC VJFET purchased from SiCED

[5]

. The currentvoltage measurements were performed in air from room temperature (RT) to 300°C with a Keithley 2410 Source Measurement Unit and Tektronix 370, by a four point method in order to eliminate the resistances from the wire connections. The onresistance, mainly due to the vertical channel, increases with the temperature, from 1.1 Ω measured at RT to 2.98 Ω at 300°C. Usual mobility majority carrier behavior is involved.

Table 1 .

 1 SiC

	n,p	.IONIZA	⎛ ⎜ ⎜ ⎝	-	E	cn,p Para E (x,y)	⎞ ⎟ ⎟ ⎠	(d)

α (x,y)=n,p .e parameters and analytical models utilized in finite-element MEDICI simulations: (a) mobility, (b) incomplete ionization, (c) carrier life time and (d) ionization coefficients.

Acknowledgements

The CEGELY would like to thank Dr Heinz Mitlehner and Dr Peter Friedrichs from SICED for his kind help in the supply of SiC VJFETs devices. 620 µm

GATE DRAIN

SOURCE POLP


Dominique Bergogne
email: dominique.bergogne@insalyon.fr

Pascal Bevilacqua

Dominique Planson

Hervé Morel

High temperature operating converter

Two sample SiC JFET are characterized and used to build a high operating temperature converter. Power losses are evaluated, the effect of temperature on ONState resistance and saturation current is measured and discussed. Temperature effects on dynamic behaviour is not a major issue. Total losses vary with temperature. A 600V/300V converter is build and tested. JFET and inductor are cooled at 300°C.

Introduction.

High temperature power electronics has become a possibility by the development of Silicon Carbide (SiC) components. Breakdown voltage remains high at high temperatures, over 250°C, where silicon devices reach an absolute limit. The perspective of high operating temperature power components permits to imagine heatsinkfree systems or converters embedded in the vicinity of internal combustion engines. Extreme environments are a new area for power electronics based on SiC. This is why we are studying converters with a cold cooling source at temperatures in the 300°C range. JFET components are becoming available outside laboratories, the devices used in this work comes from SiCED. A SiCJFET/SiMOSFET cascode assembly provided in a plastic flat package, rated 1200V2A. Cascode is used to get rid of the normallyON behaviour of JFETs or to use SiCJFET as an 'almost pin to pin' replacement of MOSFETs or IGBTs. Nevertheless, in this work cascode operation is not used, the MOSFET is disabled (G and S terminals are connected together) and the JFET is driven using a slightly modified standard integrated MOSFET IGBT driver. High junction temperature converter brings to the foreground leakage current limitations of diodes however the reverse characteristics of JFETs make possible diodeless converter. Free wheeling take place in the bulk structure of the JFET used. Another issue of high temperature is passive components. Capacitors are almost inexistent at 300°C while magnetic cores can be purchased on the web. Conductor designed for high temperature are also common. The converter presented in this paper is a 600V/350V buck converter with both inductor and switching cell cooled at 300°C.

Static characterization

SiC JFET power components are characterized to study temperature effects on majors parameters and to define the thermoelectrical limits. This knowledge is the basis for the design of the inverter. Temperature is set by a hot air furnace and verified by thermocouples placed on the metal case of the devices, ranging from 30°C to 300°C.

To start with, the JFET are characterized as power switches this is why the Drain to Source voltage is limited from 2V to 2V in the ON State, see Fig. 1.

It can be seen that the ONresistance varies with temperature from 1Ω at 30°C to 4Ω at 300°C. Saturation current is measured and drops by a factor of 4 as temperatures rises to 300°C. It should be noted here that the devices used in this paper are first generation JFET from SiCED and that newly released versions have a reduced temperature dependency. For testing reverse conduction properties of the JFET the same voltage range is used for comparison purposes. In the OFFstate the JFET exhibits a reverse conducting characteristic as a diode that is used for free wheeling, see

2 -1 0 1 2 -1 0 1 2 3 VDS [V] IDS [A] 300°C 250°C 200°C 150°C 75°C 30°C VGS = 0 V 300°C 250°C 200°C 150°C 75°C 30°C -2 -1 0 1 2 -0,5 -0,4 -0,3 -0,2 -0,1 0,0 300°C 200°C 250°C 150°C 75°C 30°C VDS [V] VGS = -15V IDS [A]
On Fig.

Switching losses measurement

In an inverter leg, deadtime, tm, and gate resistor, Rg, can have a dramatic effect on losses, Eon and Eoff, [START_REF] Morel | On the choice of blanking times at turnon and turnoff for the diode less SiCJFET inverter bridge[END_REF] An experimental test setup is used to electrically measure switching losses at turnON and turnOFF, see Fig. 5. with deadtime, tm, gate resistor, Rg and temperature, T, as parameters.

The inverter leg is built using sample SiC JFET connected to a ceramic capacitor on a double sided printed circuit board with a grounded back plane to reduce inductive parasitic effects . Q3 of Fig. 4 is used to connect a constant current source to the inverter leg during a switching sequence, thus allowing pulsed mode operation. Pulsed mode operation is chosen to limit self heating of the JFET under test, therefore case temperature can be assumed to be equal to internal junction temperature. Case temperature is set by placing the devices in a hot air furnace. When electrical measurements are stable in time it is assumed that thermal equilibrium is reached. Drain current is sensed using a coaxial wide bandpass shunt -1GHz -and voltage is measured with two paired probes placed in a differential arrangement, bandwidth is approximately 400 MHz.

Probes are compensated and deskewed. We have published [START_REF] Dominique | 300°C operating junction temperature inverter leg investigations[END_REF], this year at EPE, such work and results. The converted described here uses the same switching cell, so previous optimization work can be used. The optimal values of resistance and deadtimes are used in this paper. Fig. 5 summarizes it and presents the switching losses as a function of temperature.

Total power losses evaluation

A simple evaluation is made from static and dynamic data in order to set the maximum possible frequency range of the JFET switching cell built using the tested devices. Static measurements are used for conduction modelling. Dynamic measurements produced switching losses as functions of dead time and gate resistor. Switching losses, not much affected by temperature, are multiplied by frequency and addedup. While conduction losses are computed using the value of the current, conduction time and ONresistance varying with temperature Fig. 6 shows that 100kHz to 300kHz operation is possible.

Experimental converter

This time the converter is a buck converter aimed at accumulators powered power DCbus for traction. The accumulators bank voltage is 600V, the DCbus is 300V. A reversible converter is build around one inverter leg built with two JFET. For economical reasons and availability, very high temperature capacitors and interconnect is not used. Only the JFET and the inductor are cooled at 300°C using a hot air furnace. The rest of the circuit is at room temperature. The schematic is presented at Fig. 6, dashed rectangle indicates components cooled at 300°C.

A closeup picture of the converter is at Fig. 8. Top from left to right: Inductor L (the white wound component), Q1 and Q2 SiC JFETs from SiCED sitting just outside the hot air furnace outlet. Printed circuit (Insulated Metal Substrate) is visible on the bottom side of the picture, holding the ceramic tank capacitor Cin. Load and load capacitor C are not visible. Gate drivers are separate circuits connected to the power board by twisted pairs. Dead times, different at turnON and turnOFF, are generated by a programmable gate array circuit. Optical fibres transmit switching signals to the drivers.

Testing the converter

The converter is tested at a cooling temperature of 300°C with an input voltage of 600V (breakdown voltage/2). Switching frequency is 300KHz and duty cycle 0.5. The load draws an average 0.4A. Fig. 9 is an oscilloscope screen showing the converter's operation. Gate signal is a 0/20V square wave at 300KHz. Inductor peak current is 0.5A, a value quite close to the saturation current of the JFET at 300°C. Current ripple is 0.2A. JFET Drain Source voltage is a 600V square wave with 150ns rise/fall time. Variation of DrainSource voltage during conduction is visible, it is due to the relative high value of Drain current in respect with saturation of the JFET. The inductor is made of a special high temperature magnetic core from Magnetics [START_REF]Magnetics, inc. 111 Zeta Drive[END_REF], wound with a 1mm diameter copper wire insulated by two crossed layers of Teflon thread. Long term stability is not tested.

Conclusion

High temperature operation of a SiC JFET switching cell with inductor at 300°C is demonstrated. Operating frequency is 300kHz, input voltage is 600V, current is 0.5A limited by device size and technology version. The internal diode of JFET is used as a free wheeling diode.

The effect of temperature is mainly an increase of conduction losses as ONresistance is very temperature dependent (a 4.5 ratio on the 30to300°C range) Switching losses are slightly reduced by temperature increase due to the reduction of peak current during diode recovery (an average 0.75 ratio on the 30to300°C range). High temperature inductor is build using commercially available core and materials. The tank capacitor and solder interconnections still have to be operated at room temperatures.

Fig. 2 Fig. 1

 21 Fig. 2 Static measurement in the OFF state VGS=15

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Fig. 3 :Fig. 4 :Fig. 5 .

 345 Fig. 3: JFET ONresistance as a function of temperature measured at constant VDS=2V.

Fig. 8 :

 8 Fig. 8: A torus inductor and two JFET just outside the hot air outlet. Tank capacitor lies below, on the SMI substrate. A cold plate is used to maintain solder joints and capacitor at reasonable temperatures.

Fig. 6 :Fig. 9 :Fig. 7 :

 697 Fig. 6: Estimated total power losses of one JFET

Acknowledgements

The authors would like to thank Dr Heinz Mitlehner and Dr Peter Friedrichs from SiCED for their kind help in the supply of SiC JFET devices.