Marko V Jankovic

A Proof of Polignac's Conjecture

In this paper a proof of the Polignac's Conjecture is going to be presented. The proof represents an extension of the proof of the twin prime conjecture. It will be shown that primes with a gap of size g could be obtained through two stage sieve process, and that will be used to prove the conjecture.

Here it is going to be proved that m in (2) is represented by threads defined by odd prime numbers. Now, the form of (2) for some values of i will be checked. Case i = 1: m = 3j + 1, Case i = 2: m = 5j + 2, Case i = 3: m = 7j + 3, Case i = 4: m = 9j + 4 = 3(3j + 1) + 1, Case i = 5: m = 11j + 5, Case i = 6: m = 13j + 6 , Case i = 7: m = 15y + 7 = 5(3j + 1) + 2, Case i = 8: m = 17j + 8,

Introduction

In number theory, Polignac's conjecture states: For any positive even number g, there are infinitely many prime gaps of size g. In other words: there are infinitely many cases of two consecutive prime numbers with the difference g [START_REF] De Polignac | Recherches nouvelles sur les nombres premiers[END_REF].

In literature it has been proved that exists infinitely many primes with gaps not bigger than 246 [START_REF] Polymath | Variants of the Selberg sieve and bounded intervals containing many primes[END_REF].

Recently, the conjecture was proved for gaps of the size 2 and 4 [START_REF] Jankovic | A Proof of the Twin Prime Conjecture (together with the Proof of Polignac's Conjecture for Cousin Primes)[END_REF], 6 [START_REF] Jankovic | Proof of Polignac's Conjecture for Gap Equal to Six (Proof that exists infinitely many sexy primes)[END_REF] and 8 [START_REF] Jankovic | Proof of Polignac's Conjecture for Gap Equal to Eight[END_REF]. The problem was addressed in generative space, which means that prime numbers were not analyzed directly, but rather their representatives that can be used to produce them. This paper represents an extension of the previous work [START_REF] Jankovic | A Proof of the Twin Prime Conjecture (together with the Proof of Polignac's Conjecture for Cousin Primes)[END_REF][START_REF] Jankovic | Proof of Polignac's Conjecture for Gap Equal to Six (Proof that exists infinitely many sexy primes)[END_REF][START_REF] Jankovic | Proof of Polignac's Conjecture for Gap Equal to Eight[END_REF].

Basically, three groups of gaps bigger than 4 exist: the gaps of the size 6k, the gaps of the size 6k+2 and gaps of the size 6k+4, k ϵ N. In the text that follows we mark the prime numbers in the form 6k -1 as mps primes and prime numbers in the form 6k +1 as mpl primes, k ϵ N. The gaps of the size 6k could be related to the prime pairs in both (mps, mps) and (mpl, mpl) form. The gaps in the form 6k + 2 can only be related to the pair of primes in (mps, mpl) form, while gaps in the form 6k + 4 can only be related to the pair of primes in (mpl, mps) form. In other words there is not a single prime in mpl form that has consecutive prime that is 6k + 2 apart, and there is not a single prime in mps form that has consecutive prime 6k + 4 apart. It is trivial to show that by simple calculation.

Although last two groups are different, they can be treated analogously. Here, three different cases are going to be analyzed in order to explain how conjecture can be proved in general case.

First, the analysis from [START_REF] Jankovic | Proof of Polignac's Conjecture for Gap Equal to Eight[END_REF] for the gaps of the size 8 is going to be recapitulated. After that, cases in which gaps are equal to 10 and 12 are going to be analyzed briefly. Then, the proof can easily be generalized for the gaps of any size g. It will be shown that primes with gap of a size g (Gg-primes) could be generated by two stage sieve process. This process will be compared to other two stage recursion sieve process that leaves infinitely many numbers. Fact that sieve process that generate Gg-primes leaves more numbers than the other sieve process will be used to prove that infinitely many Gg-primes exist.

Remark 1: In this paper any infinite series in the form c 1 •l ± c 2 is going to be called a thread defined by number c 1 (in literature these forms are known as linear factors -however, it seems that the term thread is probably better choice in this context). Here c 1 and c 2 are numbers that belong to the set of natural numbers (c 2 can also be zero and usually is smaller than c 1 ) and l represents an infinite series of consecutive natural numbers in the form (1, 2, 3, …).

Remark 2:

In the text that follows we will use fact that number of even/odd numbers is equal to the one half of the number of natural numbers (also the 1/3, 4/5 and other fractions of the number of natural numbers are used). This is not an usual way of discussing the number of numbers that are infinitely big, but it is quite suitable in this context. It is important to notice that there is no context in which is correct to state that number of natural numbers is equal to the number of odd or even numbers (or that number of natural numbers is equal to the number of numbers divisible by 3, or 5, and so on). What can be said is that it is possible to generate the same number of unique labels for odd or even numbers using the same number of unique labels for natural numbers. However, if we want to produce all even and odd labels at the same time, obviously, it is necessary to have two sets of natural numbers at the same time (it is necessary to clone the set of natural numbers), which means that the number of the natural numbers in that moment is two times bigger than the number of even or odd numbers. This analysis can also be made in quantum probabilistic context, but it is beyond the scope of this paper.

2 Recapitulation of the proof of Polignac's conjecture for gap equal to eight

As it was already explained in the introduction part, if two consecutive prime numbers have the gap of the size 8 it is clear that smaller of those numbers has to be in mps form while the bigger one has to be in mpl form. Here, we are going to recapitulate a two stage process, that was presented in [START_REF] Jankovic | Proof of Polignac's Conjecture for Gap Equal to Eight[END_REF], that can be used for generation of the smaller primes of the G8-prime pairs that are in mps form.

In the first stage prime numbers are going to be produced by removal of all composite numbers from the set of natural numbers.

In the second stage, the following numbers are going to be removed:

-Number 2 (since it cannot generate G8-prime pair).

-All numbers in mps form that represent a smaller numbers in twin prime pairs.

-A quarter of sexy primes (or more precisely one half of mps sexy primes). In order to understand why is it so, we are going to consider 5 consecutive numbers 6k-1, 6k+1, 6k+3, 6(k+1)-1, 6(k+1)+1. We can see that 6k-1 and 6(k+1)+1 can create a G8-primes only in the case when they are both primes and when 6k+1 and 6(k+1)-1 are not primes (6k+3 is obviously composite number divisible by 3). Here, we are going to analyze three situations. When 6k+1 and 6(k+1)-1 are both primes, and in the case when only 6k+1 is prime, removal of twin primes will remove mps primes that cannot have G8-prime pair. However, when 6k+1 is not prime and 6(k+1)-1 is prime, it is also necessary to remove smaller number of sexy prime pair (6k-1, 6(k+1)-1) since it also cannot create a G8-prime pair.

-A half of the number of twin primes and quarter of the number of sexy primes from mpl primes is going to be removed in such a way that Dirichlet's theorem for arithmetic progressions [START_REF] Dirichlet | Bewies des Satzes, dass jede unbregenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftluchen Factor sind, unendlich viele Primzahlen enthält[END_REF] still holds after removal of all those numbers that are previously mentioned.

-All prime numbers in mpl form that are left.

-All prime numbers in mps form that have a bigger odd G8-neighbor (odd number that is by 8 bigger than the prime of interest) that is a composite number.

At the end, only the prime numbers in the mps form, that represent the smaller numbers of the G8prime pairs, are going to stay. Their number is a half of the number of G8-primes. It is going to be shown that that number is infinite.

STAGE 1

Prime numbers can be obtained in the following way:

First, we remove all even numbers (except 2) from the set of natural numbers. Then, it is necessary to remove the composite odd numbers from the rest of the numbers. In order to do that, the formula for the composite odd numbers is going to be analyzed. It is well known that odd numbers bigger than 1, here denoted by a, can be represented by the following formula

a = 2n +1,
where n ϵ N. It is not difficult to prove that all composite odd numbers a c can be represented by the following formula

a c =2 (2 i j+i+ j)+1=2((2 j+1)i+ j )+1. (1)
where i, j ϵ N. It is simple to conclude that all odd composite numbers could be represented by product (2i + 1)(2j + 1), where i, j ϵ N. If it is checked how that formula looks like for the odd numbers, after simple calculation, equation ( 1) is obtained. This calculation is presented here. The form 2m + 1, m ϵ N will represent odd numbers that are composite. Then the following equation holds 2 m+1=(2 i+1)(2 j+1) .

From that, it follows that m must be in the form

m = 2ij + i + j = (2i + 1) j + i. (2)
When all numbers represented by m are removed from the set of odd natural numbers bigger than 1, only the numbers that represent odd prime numbers are going to stay. In other words, only odd numbers that cannot be represented by [START_REF] De Polignac | Recherches nouvelles sur les nombres premiers[END_REF] will stay. This process is equivalent to the sieve of Sundaram [START_REF] Aiyar | Sundaram's Sieve for prime numbers[END_REF].

Let us denote the numbers used for the generation of odd prime numbers with m2 (here we ignore number 2). Those are the numbers that are left after the implementation of Sundaram sieve. The number of those numbers that are smaller than some natural number n, is equivalent to the number of prime numbers smaller than n. If we denote with π(n) number of primes smaller than n, than the following equation holds [START_REF] Rosser | Approximate formulas for some functions of prime numbers[END_REF] π( n)≈ n ln (n)

.

From [START_REF] Rosser | Approximate formulas for some functions of prime numbers[END_REF] we also know that following holds

π( n)> n ln( n) , n⩾ 17. ( 3
)

STAGE 2

What was left after the first stage are prime numbers. With the exception of number 2, all other prime numbers are odd numbers. All odd primes can be expressed in the form 2n + 1, n ϵ N. It is simple to understand that their bigger odd G8-neighbor must be in the form 2n + 9, n ϵ N. Now, we should implement a second stage in which we are going to remove:

A. Number 2 (since 2 cannot make a G8-prime pair);

B. All twin primes in mps form, a quarter of sexy primes (smaller sexy numbers in mps form). Also, a number of primes in mpl form that is equal to the number that represents the sum of the half of the numbers of twin primes and quarter of the numbers of sexy primes, in such a way that Dirichlet's theorem for arithmetic progressions [START_REF] Dirichlet | Bewies des Satzes, dass jede unbregenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftluchen Factor sind, unendlich viele Primzahlen enthält[END_REF] still holds after removal of all those numbers that are mentioned-so number of numbers that is going to be left is the number of primes minus the number of twin primes minus one half of the number of sexy primes (number 2 is ignored, and that has no impact on the analysis that follows). It is not difficult to prove that number of numbers left, is infinite. It is not difficult to understand it having in mind that relative density of twin and sexy primes to the density of all primes is zero.

C. The rest of mpl primes -it is trivial to see that it can be done by one thread that is defined by 3so in this step it is going to be removed, one half of the numbers that are left after step B; D. All odd primes in the form 2m + 1 such that 2m + 9, m ϵ N represents a composite number (all primes whose bigger odd G8-neighbor is composite number). If we make the same analysis, like in the Stage 1, it is simple to understand that m must be in the form

m = 2ij + i + j -4 = (2i + 1) j + i -4. (4)
All numbers (in observational space) that are going to stay must be numbers in mps form and they represent a smaller primes of the G8-prime pairs that is in mps form. What has to be noticed is that thread in (4) that is defined by prime number 3 (for i = 1) is not going to remove any additional number from the numbers left.

Let us mark the number of G8-primes with π G8 . Also, let us define the number of numbers that is left after two consecutive implementations of Sundaram sieve as pd8. The prime numbers that are obtained after removal of number 2, twin primes in mps form and a quarter of sexy primes (a half of sexy primes in mps form) and the corresponding number of mpl primes (equal to the sum of a half of the number of twin primes and a quarter of the number of the sexy primes) are going to be called G8-fs primes. The numbers obtained after recursive implementation of two Sundaram sieves (where the second Sundaram sieve is implemented on G8-fs primes) are going to be called G8double primes. The second stage sieve that is identical to the first stage sieve can be obtained if the G8-fs primes are lined up next to each other and then the numbers are removed from the exactly same positions like in the first stage -the second stage sieve is applied on the indexes of G8-fs primes. In that case it is easy to understand that the following equation would holds (n ϵ N)

pd8 ( n)≈ π (n)-π G2 (n)-0.5 π G6 (n) ln ( π( n)-π G2 ( n)-0.5 π G6 (n) ) , (5)
where pd8(n) represents the number of G8-double primes smaller than some natural number n (this number can be easily found out (by simple counting), once it is known which mpl numbers are removed in the process of obtaining the G8-fs primes). Here, it will be shown that number of G8double primes, marked as pd8, is smaller than the number π G8 /2. In order to understand why it is so, we are going to analyze ( 2) and ( 4) in more detail.

It is not difficult to understand that m in ( 2) and ( 4) is represented by the threads that are defined by odd prime numbers. For details see Appendix A. Now we are going to compare stages 2 for generation of G8-double primes and mps G8-primes, step by step, for a few initial steps (analysis can be easily extended to any number of steps). Starting point for the second stage is the moment when the primes G8-fs are generated. Remove numbers defined by thread defined by 11 (obtained for i = 5) amount of numbers left is 9/10 of the numbers that are left after previous step It should be kept in mind that in the case of G8-double primes sieve is implemented on indexes of G8-fs prime numbers (indexes are consecutive numbers 1, 2, 3, … ).

Values of the fractions presented in the Table 1 are asymptotically correct (in the finite case those values are only approximately correct -for details see [START_REF] Jankovic | A Proof of the Twin Prime Conjecture (together with the Proof of Polignac's Conjecture for Cousin Primes)[END_REF] ). Here is important to understand that it is assumed, without a proof, that Dirichlet's theorem on arithmetic progressions [START_REF] Dirichlet | Bewies des Satzes, dass jede unbregenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftluchen Factor sind, unendlich viele Primzahlen enthält[END_REF] holds on the subset of primes marked as G8-fs primes (in other words, after removal of twin primes in mps form and after removal of the quarter of the sexy primes (half of the sexy primes in mps form) from prime numbers, that it is possible to select the same number of mpl primes that are going to be removed in such a way that Dirichlet's theorem on arithmetic progressions [START_REF] Dirichlet | Bewies des Satzes, dass jede unbregenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftluchen Factor sind, unendlich viele Primzahlen enthält[END_REF] still holds). For the arithmetic progressions that are defined by finite numbers this is a simple task, but it requires additional attention when we are speaking about progressions defined by infinite numbers.

From the table, it can be noticed that threads defined by the same number in first and second column will not remove the same percentage of numbers. The reason is obvious -consider for instance the thread defined by 3: in the first column it will remove 1/3 of the numbers left, but in the second column it will remove ½ of the numbers left, since the thread defined by 3 in stage 1 has already removed one third of the numbers (odd numbers divisible by 3 in observation space). So, only odd numbers (in observational space) that give residual 1 and -1 when they are divided by 3 are left, and there is asymptotically the same number of numbers that give residual -1 and numbers that give residual 1, when the number is divided by 3 (see [START_REF] Jankovic | A Proof of the Twin Prime Conjecture (together with the Proof of Polignac's Conjecture for Cousin Primes)[END_REF]Appendix C]). Same way of reasoning can be applied for all other threads defined by the same prime in different columns.

From Table 1 can be seen that in every step, except step 1, threads in the second column will leave bigger percentage of numbers than the corresponding threads in the first column. This could be easily understood from the analysis that follows:

suppose that we have two natural numbers j, k such that j -1 ≥ k (j, k ϵ N ), then the following set of equations is trivially true

j+k-1⩾ 2k -j-k +1⩽ -2k jk-j-k +1⩽ jk-2k ( j -1)(k -1)⩽ ( j-2) k k-1 k ⩽ j-2 j-1
The equality sign holds only in the case j = k + 1. In the set of prime numbers there is only one case when j = k + 1 and that is in the case of primes of 2 and 3. In all other cases p(i) -p(i -1) > 1 , (i > 1, i ϵ N, p(i) is i-th prime number). So, in all cases i > 2

p(i-1)-1 p (i-1) < p (i)-2 p (i )-1 .
From Table 1 (or last equation) we can see that bigger number of numbers is left in every step of stage 2 then in the stage 1 (except 1 st step). From that, we can conclude that after every step bigger than 1, part of the numbers that is left in stage 2 is bigger than number of numbers left in the stage 1

(that is also noticeable if we consider amount of numbers left after removal of all numbers generated by threads that are defined by all prime numbers smaller than some natural number).

From previous analysis we can safely conclude that the following equation holds

π G8 > π G8 2 > pd8= lim n → ∞ pd8(n) .
Having in mind (3) and ( 5), we can say that for some n big enough the following inequality holds

pd8 ( n)> π( n)-π G2 ( n)-0.5 π G6 (n) ln ( π (n)-π G2 (n)-0.5 π G6 ( n) ) . ( 6
)
(The n for which equations holds can be specified once the numbers in mpl form that are going to be eliminated are known -for instance it can be said that n ≥ 317 can be chosen, since 317 is the mpl twin primes, and a quarter of the sexy primes are eliminated from the prime numbers set, and if we assume that all corresponding mpl numbers (that "balance" a quarter of sexy primes that are numbers in mps form) that are going to be removed are all bigger than 317). It is not difficult to prove that the number of primes that are left when 2 and twin primes in mps form and quarter of sexy primes, and corresponding number of mpl primes are eliminated, is infinite. One way to understand it, is the fact that the relative density of twin primes and all primes is zero as well as the relative density of sexy primes and all primes is zero.

Having that in mind, it it easy to show that following holds

lim n → ∞ π(n)-π G2 ( n)-0.5 π G6 ( n) ln ( π( n)-π G2 (n)-0.5 π G6 ( n) ) =∞ .
Then, the following equation holds

pd8= lim n → ∞ pd8 ( n)=∞ .
Now, we can safely conclude that the number of G8-primes is infinite. That concludes the proof.

Here we will state the following conjecture: for n big enough, number of g8-primes is given by the following equation

π G8 (n) ∼ 4C 2 ⋅ ( ( π (n) -π G2 (n)-0.5 π G6 ( n) ) ln ( π( n) -π G2 ( n)-0.5 π G6 (n) ) ) .
Constant C 2 is known as twin prime constant [START_REF] Hardy | Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes[END_REF]. The equation can be expressed by using only n, but it would be cumbersome. Why it is reasonable to make such conjecture can be understood from the procedure that is similar to one used in [3, Appendix B]. If we mark the number of primes smaller than some natural number n with π(n) ≈ f (n), where function f (n) gives good estimation of the number of primes smaller than n, than π G8 (n), for n big enough, is given by the following equation

π G8 ( n) ∼ 4C 2 ⋅ ( f ( f (n) -π G2 ( n)-0.5 π G6 ( n) )) .
Here we can see that constant C 2 has a misleading name. It is connected with repeated implementation of a sieve that produces prime numbers which is also, but not exclusively, connected to the twin primes.

3 Analysis of the Polignac's conjecture for gaps equal to ten

As it was already explained in the introduction part, if two consecutive prime numbers have the gap of the size 10 it is clear that smaller those numbers has to be in mpl form while the bigger one has to be in mps form. Proof that number of the G10-primes is infinite can be done by trivial extension of the proof that infinitely many G8-primes exist. Here we are only going to analyze a two stage process that can be used for generation of the smaller primes of G10-prime pairs (mpl G10-primes), and how it differs from two stage process used for generation of smaller primes of G8-prime pairs.

The first stage in both processes are equivalent and they produce prime numbers by removing all composite numbers from the set of natural numbers. Here we are going to analyze the second stage of the process that produces mpl G10-primes.

In order to do that we will consider 6 consecutive numbers 6k+1, 6k+3, 6(k+1)-1, 6(k+1)+1, 6(k+1)+3 and 6(k+2)-1.

We can see that 6(k+1) and 6(k+2)-1 can create a G10-primes only in the case when neither 6(k+1)-1 nor 6(k+1)+1 are primes, and at the same time both 6k+1 and 6(k+2) -1 are primes (numbers 6k+3 and 6(k+1)+3 are obviously composite). Here we are going to analyze three cases:

1. Both 6(k+1)-1 and 6(k+1)+1 are primes. We can see that removal of all smaller primes of cousin prime pairs, removes primes in mpl form that potentially cannot have a G10-prime pair.

2. 6(k+1)-1 is a prime number and 6(k+1)+1 is a composite number. Again, we can see that removal of all smaller primes of cousin prime pairs removes primes in mpl form that potentially cannot have a G10-prime pair.

3. 6(k+1)-1 is composite number and 6(k+1)+1 is prime number. Now, we can see that beside the removal of all smaller primes in cousin prime pairs, it is necessary to remove smaller primes in mpl sexy prime pairs (which is one half of mpl sexy primes and a quarter of all sexy primes), in order to remove all primes in mpl form that potentially cannot have a G10-prime pair.

4.

Remove appropriate number of primes in mps form that match the number of mpl numbers that are removed in steps 1-3 of the second stage in such a way that Dirichlet's theorem on arithmetic progressions still holds.

From previous analysis we can conclude that in this part of the second stage it is necessary to remove all number of prime numbers that is equal to the number of cousin primes and one half of all sexy primes. After this has been done, the procedure can follow the line of reasoning that is used in proof that infinitely many G8-primes exist. The only small difference would be to prove that exist infinitely many primes that are not cousin primes or one half of sexy primes (and that follows from the fact that the ratio of density of cousin and sexy primes to all primes is zero). The other difference comes from the fact that 10/2=5 and that means that the thread that is defined by 5, in the second stage, will not remove any additional numbers. That will be analyzed in Appendix C.

This analysis can easily be extended to gaps g, where g=6k+2, k ϵ N.

Analysis of the Polignac's conjecture for gaps equal to twelve

The G12-primes are successive prime numbers with the gap 12. It was already mentioned that in the case when gap g=6k, where k is natural number, we can have G12-prime pairs in (mps, mps) as well as (mpl, mpl) forms.

It is clear that we can generate G12-primes using the two stage sieve process that is similar to the process used for generation of G6-primes [START_REF] Jankovic | Proof of Polignac's Conjecture for Gap Equal to Six (Proof that exists infinitely many sexy primes)[END_REF]. The only difference will be in the part of the second stage that is going to be analyzed here. The first stages are identical. The difference in the second stage is related to the removal of the primes that potentially cannot produce a G12-prime pairs.

In order to see what kind of primes should be removed from the set of prime numbers. Here, we are going to analyze mps G12-primes (mpl G12-primes can be analyzed analogously). We are going to analyze the following 7 consecutive odd numbers 6k-1, 6k+1, 6k+3, 6(k+1)-1, 6(k+1)+1, 6(k+1)+3 and 6(k+2)-1.

So, if want to have G12-prime pair (6k-1, 6(k+2)-1) it is clear that numbers 6k+1, 6(k+1)-1 and 6(k+1)+1 cannot be prime numbers. Here we are going to analyze all possible cases.

1. All three numbers 6k+1, 6(k+1)-1 and 6(k+1)+1, are primes. In that cases it is clear that removal of all smaller primes in the twin prime pairs remove mps primes that potentially cannot have G12-prime pair.

2. 6k+1, 6(k+1)-1 are prime numbers and 6(k+1)+1 is composite number. In this case beside removal of twin primes it is necessary to remove a smaller number in all mps sexy prime pairs (that represent half of mps sexy primes, or a quarter of all sexy primes).

3. 6k+1, 6(k+1)+1 are prime numbers and 6(k+1)-1 is composite number. Removal of the smaller primes in twin prime pairs is enough, 4. 6(k+1)+1, 6(k+1)-1 are prime numbers and 6k+1 is composite number. In this case beside removal of twin primes it is necessary to remove a smaller number in all mps sexy prime pairs (that represent half of mps sexy primes, or a quarter of all sexy primes).

5. 6k+1 is prime number and 6(k+1)+1, 6(k+1)-1 are composite numbers. What is required is removal of twin primes.

6. 6(k+1)-1 is prime number and 6k+1, 6(k+1)+1 are composite numbers. In this case removal of smaller number in the mps sexy prime pairs is required.

7. 6(k+1)+1 is prime number and 6k+1, 6(k+1)-1 are composite numbers. In this case it is necessary to remove smaller prime from all G8-prime pairs. Now it is necessary to remove appropriate number of mpl primes that is equal to the number of mps primes removed in all cases (1-7) that were analyzed. That should be done in such a way that Dirichlet's theorem on arithmetic progressions holds (this is assumed as possible without proof). Now, the rest of the procedure that is applied in the G6-primes case has to be done in G12-primes case.

Case of mpl G12-primes can be analyzed analogously. Of course in that case it would be necessary to remove all cousin and G10-primes, as well as mpl sexy primes, instead of twin primes, G8primes and mps sexy primes.

This analysis can easily be extended to gaps g, where g=6n, n ϵ N.

Proof that the number of Gg-primes is infinite

Here we are going to analyze the general case of the gap of the size g > 6. From previous analysis, it is very simple to understand that Gg-primes could be obtained by two stage process.

STAGE 1

Using the same methodology as previously, generate all prime numbers. In order to do that, from the set of all natural numbers bigger than 1, remove all even numbers (except 2) and all odd numbers generated by equation (2).

STAGE 2

What was left after the first stage are prime numbers. With the exception of number 2, all other prime numbers are odd numbers. All odd primes can be expressed in the form 2n + 1, n ϵ N. It is simple to understand that their bigger Gg-odd neighbor must be in the form 2n + 1 + g, n, g ϵ N, g > 6. Now, we should implement a second step in which we are going to remove:

A. Number 2 (since 2 cannot make Gg-prime pair), but this has no impact on the analysis, B. If mod (g, 6) = 2: all smaller primes of twin prime pairs, all smaller primes in s-prime pairs, where s is even natural number in the form 6x + 2 and s < g and a quarter of all k-primes (half of the number of k-primes in mps form), where k represents number in the form 6y and k < g (x, y ϵ N).

The adequate number of primes in mpl form that is equal to the number of mps primes that is removed in this step. The number of numbers that is left after this is number of primes minus the number of twin primes minus all s-primes minus one half of k-primes. It is not difficult to prove that number of primes left after this step is infinite.

If mod(g, 6) = 4: all smaller primes in cousin prime pairs and all smaller primes in l-prime pairs, where l is even natural number in the form 6x + 4 and l < g, and a quarter primes of all k-prime pairs (a half of the k-primes in mpl form), where k represents number in the form 6y and k < g (x, y ϵ N). The adequate number of primes in mps form that is equal to the number of mpl primes that is removed in this step. The number of numbers that is left after this is number of primes minus the number of cousin primes, minus all l-primes, minus one half of all k-primes. It is not difficult to prove that number of primes left after this step is infinite.

If mod(g, 6) = 0: Assume that we are working with numbers in mps form, we remove smaller primes from k-primes pairs, k is even natural number in the form 6x, and k < g; smaller prime of all twin prime pairs and s-prime pairs, where s is even natural number in the form 6x + 2 and s < g. The adequate number of primes in mpl form that is equal to the number of mps primes that is removed in this step.

If we work with primes in mpl form, we remove smaller primes from k-primes pairs, k is even natural number in the form 6x, and k <g, smaller prime of all cousin prime pairs and all l-prime pairs, where l is even natural number in the form 6x + 4 and l < g, The adequate number of primes in mps form that is equal to the number of mpl primes that is removed in this step.

It is not difficult to prove that number of primes left after this step is infinite.

C. If mod (g, 6) = 2, all mpl primes -it is trivial to see that it can be done by one thread that is defined by 3 -so in this step it is going to be removed, approximately, one half of the numbers that are left after first stage;

If mod (g, 6) = 4, all mps primes with same consequences as previously;

If mod (g, 6) = 0, if we are working with mps primes, remove rest of mpl primes. If we work with mpl primes remove rest of mps primes.

D. Assume now that we are working with numbers in mps form (everything should be done analogously for numbers in mpl form). Remove all odd primes in the form 2m + 1 such that 2m + g, m ϵ N, represents a composite number (those that have composite bigger g-odd neighbor). If we make the same analysis again, it is simple to understand that m must be in the form

m = 2ij + i + j -1 = (2i + 1) j + i -g/2. ( 7
)
All numbers (in observational space) that are going to stay must be numbers in mps form and they represent a smaller primes of the Gg-prime pairs.

It should be noticed that if g/2 is divisible by the prime (2i+1) that defines thread in ( 7), there will be no additional removals of the numbers defined by that thread in STAGE2. That will lead to an additional correction of the formula for the estimation of number of Gg-primes smaller than some natural number n (see Appendix C).

Following identical procedure like in the case of sexy or G8-primes it is possible to it possible to prove that the number of Gg-primes is infinite.

Here, we are going to conjecture several formulas for the number of Gg-primes. Why this is reasonable can be seen in Appendix B and C. If we mark the number of primes smaller than some natural number n with π(n) ≈ f (n), where function f (n) gives good estimation of the number of primes smaller than n, than π Gg (n), for n big enough, is given by the following equation

a. Case mod(g, 6) = 2 π G g (n) ∼ C G ( g)⋅ 4C 2 ⋅ ( f ( f ( n)-π G2 ( n)-∑ 2 <S <g ( π GS (n) ) -0.5 ∑ 0< K< g ( π GK ( n) ) )) ,
where S are even numbers in the form 6x+2 and K is an even number in the form 6y, x, y ϵ N, and C G (g) is correction constant for gap g (for details, see Appendix C). If we us f (n) = MoLi (n) [START_REF] Jankovic | A New Approximation of Prime Counting Function Based on Modified Logarithmic Integral[END_REF] and we define P g (n) as

P g ( n) =π(n) -π G2 (n)-∑ 2<S <g ( π GS (n) ) -0.5 ∑ 0< K <g ( π GK ( n) ) , then we have π G g (n) ∼ C G ( n)⋅ 4C 2 ⋅ ∫ 2 P g (n) dx ln( x+ √ P g (n)) . b. Case mod(g, 6) = 4 π G g (n) ∼ C G ( g)⋅ 4C 2 ⋅ ( f ( f (n)-π G4 (n) -∑ 4< L< g ( π GL ( n) ) -0.5 ∑ 0<K <g ( π GK (n) ) )) ,
where L are even numbers in the form 6x+4 and K is an even number in the form 6y, x,y ϵ N, and

C G( (g
) is correction constant for gap g (for details, see Appendix C).

If we us f (n) = MoLi (n) [START_REF] Jankovic | A New Approximation of Prime Counting Function Based on Modified Logarithmic Integral[END_REF] and we define P g (n) as

P g (n) =π( n) -π G4 (n)-∑ 4< L< g ( π GL ( n) ) -0.5 ∑ 0<K <g ( π GK (n) ) , then we have π G g (n) ∼ C G ( n)⋅ 4C 2 ⋅ ∫ 2 P g (n) dx ln( x+ √ P g (n)) . c. Case mod(g, 6) = 0 π G g (n) ∼ C G ( g)⋅ 4C 2 ⋅ ( f ( f ( n)-π G2 ( n) -∑ 1⩽ K < g ( 0.5 π GK (n) ) -∑ 2<S <g ( π GS (n) ) )) +C G ( g)⋅ 4C 2 ⋅ ( f ( f ( n) -π G4 ( n) -∑ 1⩽ K < g ( 0.5 π GK (n) ) -∑ 4<L< g ( π GL (n) ) )) ,
and S, L, K and C G (g) are defined as previously.

If we us f (n) = MoLi (n) [START_REF] Jankovic | A New Approximation of Prime Counting Function Based on Modified Logarithmic Integral[END_REF] and we define P gS (n) as

P gS ( n) =π( n) -π G2 (n) -∑ 1⩽ K< g ( 0.5 π GK (n) ) -∑ 2<S <g ( π GS ( n) ) ,
and P gL (n) as

P gL (n) =π( n)-π G4 (n) -∑ 1⩽ K< g ( 0.5 π GK (n) ) -∑ 4< L< g ( π GL ( n) ) , then we have π G g (n) ∼ C G ( g)⋅ 4C 2 ⋅ ∫ 2 P gS (n ) dx ln (x + √ P gS ( n)) +C G ( g )⋅ 4C 2 ⋅ ∫ 2 P gL (n) dx ln ( x+ √ P gL (n))
.

It can be seen that m is represented by the threads that are defined by odd prime numbers. From examples (cases i = 4, i = 7), it can be seen that if (2i + 1) represent a composite number, m that is represented by thread defined by that number also has a representation by the the thread defined by one of the prime factors of that composite number. That can be proved easily in the general case, by direct calculation, using representations similar to [START_REF] Polymath | Variants of the Selberg sieve and bounded intervals containing many primes[END_REF]. Here, that is going to be analyzed. Assume that 2i + 1 is a composite number, the following holds In the same way this can be proved for (4), ( 5) and [START_REF] Rosser | Approximate formulas for some functions of prime numbers[END_REF].

Note: It is not difficult to understand that after implementation of stage 1, the number of numbers in residual classes of some specific prime number are equal. In other words, after implementation of stage 1, for example, all numbers divisible by 3 (except 3, but it does not affect the analysis) are removed. However, the number of numbers in the forms 3k + 1 and 3k + 2 (alternatively, 3k -1) are equal. The reason is that the thread defined by any other prime number (bigger than 2) will remove the same number of numbers from the numbers in the form 3k + 1 and from the numbers in the form 3k + 2. It is simple to understand that, for instance, thread defined by number 5, is going to remove 1/5 of the numbers in form 3k + 1 and 1/5 of the numbers in form 3k + 2. This can be proved by elementary calculation. That will hold for all other primes and for all other residual classes.

APPENDIX B.

Here asymptotic density of numbers left, after implementation of the first and second Sundaram sieve is calculated.

After first k steps of the first Sundaram sieve, after removal of all composite even numbers, density of numbers left is given by the following equation

c k = 1 2 ∏ j= 2 k +1 (1- 1 p( j ) ),
where p(j) is j-th prime number.

In the case of second "Sundaram" sieve the density of numbers left after the first k-steps is given by the following equation

c2 k = ∏ j=2 k+1 ( 1-1 p ( j)-1 ) = ∏ j=2 k+1 ( p( j)-2 p ( j)-1 ) .
So, if implementation of first sieve will result in the number of prime numbers smaller than n which we denote as π(n), than implementation of the second sieve on some set of size π(n) should result in the number of numbers gp(n) that are defined by the following equation (for some big enough n)

gp( n)=r S2S1 ( n)⋅ π( n) ln (π (n))
, where r S2S1 (n) is defined by the following equation (k is the number of primes smaller or equal to n1 = sqrt(n), where sqrt marks square root function)

r S2S1 ( n)= c2 k c k = ∏ p>2, p≤ n1 ( p -2 p -1 ) ∏ p≤ n1 ( p -1 p ) =2 ∏ p>2, p≤ n1 ( p -2 p -1 )( p p -1 ) ≈ 2C 2 .
For n that is not big, gp(n) should be defined as

gp( n)= f COR ( n)⋅ 2C 2 ⋅ π(n) ln(π( n))
,

4

 4

2i + 1 =

 1 (2l + 1)(2s + 1) where (l, s ϵ N). That leads to i = 2ls + l + s. The simple calculation leads to m = (2l +1) (2s + 1) j + 2ls + l + s = (2l + 1)(2s+1)j + s(2l + 1) + l or m = (2l+1)((2s+1)j + s) + l which means m = (2l + 1)f + l, and that represents the already exiting form of the representation of m for the factor (2l + 1), where f = (2s + 1)j + s.

Table 1

 1 Comparison of the stage 2 for generation of G8-double primes and mps G8-primes -

	threads defined by a few smallest primes
	Step	Stage 2 -G8-double primes	Step	Stage 2 -mps G8-primes
	1	Remove even numbers (except 2)	1	Remove numbers defined by thread
				defined by 3 (obtained for i = 1)
		amount of numbers left is 1/2		amount of numbers left is 1/2
	2	Remove numbers defined by thread	2	Remove numbers defined by thread
		defined by 3 (obtained for i = 1)		defined by 5 (obtained for i = 2)
		amount of numbers left is 2/3 of the		amount of numbers left is 3/4 of the
		numbers that are left after previous step		numbers that are left after previous step
	3	Remove numbers defined by thread	3	Remove numbers defined by thread
		defined by 5 (obtained for i = 2)		defined by 7 (obtained for i = 3)
		amount of numbers left is 4/5 of the		amount of numbers left is 5/6 of the
		numbers that are left after previous step		numbers that are left after previous step
	4	Remove numbers defined by thread
		defined by 7 (obtained for i = 3)
		amount of numbers left is 6/7 of the
		numbers that are left after previous step

APPENDIX C.

Here we are going to repeat equation [START_REF] Rosser | Approximate formulas for some functions of prime numbers[END_REF] m = 2ij + i + j -1 = (2i + 1) j + i -g/2.

From equation is quite clear that thread defined by the prime (2i + 1) in STAGE 2 will not produce new removals of numbers in addition to removals produced by the thread defined by the same prime in the STAGE 1, in the case when (2i+1) divides g/2. That means that r S2S1 defined in Appendix B has to be changed. From Appendix B we know that r S2S1 is defined by the following equation

while c2 k is defined by the following equation

In the case when prime p(j) divides g/2 corresponding term in c2 k is actually 1 instead of p( j)-2 p ( j)-1 .

That means that c2 k has to be multiplied by the following term

in order to obtain the correct equation.

If we mark with Z set of all primes 3 < p ≤ n1 (n1 = sqrt(n)) that divide g/2, a constant C G (g) is defined by the following equation

Now, we can write the proper values for c2 k and r S2S1 as


domain: info shs

text:
The cost of data production

• OCR and HTR = fast and powerful tools for transcription • collect text data; accelerate indexation of collections or build digital editions but!

• we need for training examples (ground truth) • it's expensive to produce and project don't always have the resources so!

• it's helpful to rely of pre-existing models or pre-existing data • especially to fine-tune models

Findability of data

• it is difficult to find data and models • we need greater habits of publishing them Why it is hard to find?

• lack of clear keywords and locations to publish • uncertainty on the formats • uncertainty on the (copy)right status Combining them is also difficult

• variation in annotations format • variation in annotation rules

This is why: HTR-United ! A catalog, a schema and some imperatives HTR-United's core: a catalog

• provide access to the datasets! • foster findability (filtering)

• shed light on the importance of citation

Human Readable

Machine readable

• no pre-existing shared way to describe transcription ground truth • writing and controling a entry in the catalog had to stay simple/easy 3 formats options :

• XML • JSON • YAML
The schema: technical choices Catalog which format? which schema?

• no pre-existing shared way to describe transcription ground truth • writing and controling a entry in the catalog had to stay simple/easy 3 formats options :

• XML • JSON • YAML
The schema: technical choices Catalog which format? which schema?

+ JsonSchema

• no pre-existing shared way to describe transcription ground truth • writing and controling a entry in the catalog had to stay simple/easy 3 formats options :

• XML • JSON • YAML
The schema: technical choices Catalog which format? which schema?

+ JsonSchema

Why a schema?

• a machine actionable way to check conformity of a submission to the catalog • clearer documentation on the evolutions of the schema

The schema: what does it include

• dataset title, link and short description

• license of use and format used for data (ALTO vs PAGE)

• software used to produce the data

• period covered by the documents

• languages and the scripts

• collaborators of the project

• script type (e.g. print vs. manuscript or mixed)

• number of hands or of fonts

• metrics (mainly lines, characters and files)

Controlled vocabularies:

• format • language • script • script type
• number of hands/fonts

• collaborators' roles

• currently: 1 field of free text for "transcription guidelines"

The longer term goals:

• automatic parsing of the ground truth (character sets, and normalization systems) • build a controlled vocabulary for transcription guidelines

The schema: about transcriptions examples:

• a collaborative enterprise for the community

• friendly to consumer and producers of data

• as low tech as possible (because $$)

Imperatives

• we are editors of the catalog

• but we have our own domains of expertise and interest (mainly French from the medieval period to today)

• we don't know every use case

• therefore the initiative benefits from users' contributions to the ecosystem

Imperatives: Welcoming to users

We ad de d a qu ali fie r op tio n fo r ea ch ite m in to the ISO lis t of sc rip ts

• we created a form to help generate the catalog entry • updating the form along with the schema takes time but it's worth it • we want a form as simple as possible, even though the schema gains complexity

We use Continuous Integration via Github Actions. For each update in the catalog:

• automatic validity control of the metadata (.yml)

• automatic validity control of the files (.xml)

• automatic generation of metrics (.yml and badges) CI requires skills, for we made a form for that too!

Imperatives: User friendly

Ecosystem

• we started with cataloging and normalizing but didn't (couldn't ?) stop there

• four common actions between different projects:

○ an htr-united.yml catalog entry ○ people want metrics on their dataset ○ people want cohesion within their dataset

○ people want compatibility between their datasets

We developed or integrated tools to address theses actions

Ecosystem: Consolidating workflows

Why?

• users provide catalog entries as YAML files, which content is added to the core catalog • we need to control this entry before adding it to the catalog What is it? Ecosystem: Control the catalog with HTRuc

• control schema validity in XML files

• can control conformity to ontologies like SegmOnto (correctness of the tagging)

• can control occurrences of empty elements (lines or regions)

Ecosystem: Control the data with HTRVX

Why?

• we like metrics, you like metrics • but computing metrics takes time and is boring What is it?

• a command line actionable with Github Actions • creates badges • update the YML file with the metrics Ecosystem: Update information with HUMG(enerator)

• originally developed for the CREMMA medieval corpus (Pinche & Clérice 2021) • originally tracked variations of encoding for medieval characters Why?

• consistency character sets is sometimes hard to maintain What does it do?

• control character or sequences of characters in a dataset • convert characters according to a conversion table

Ecosystem: Control the transcription with ChocoMufin

Perspectives by way of concluding

• browsing and reading the description requires a human for the free text sections

• we want to add more filters to add automation

• we could have a lab bench to combine and compose new datasets

Perspectives: A dataset builder

• necessary to add automation over fields like "transcription guidelines"

Perspectives: Control vocabularies in the schema?

• offer server versions of the tools

• alternatives to running them locally or via Github Actions

• but requires having servers

Perspectives: Tools as web services?

• can we make more than a catalog (and a ecosystem)? Yes !

• foster publication of data papers for HTR ground truth

• catalog entries having a data paper would be considered to be better described


Ali Ammar
email: ali.ammar@insa-lyon.fr

Viet Luong

Phung

Dominique Planson
email: cdominique.planson@insa-lyon.fr

Herve Morel
email: herve.morel@insa-lyon.fr

Camille Sonneville
email: ecamille.sonneville@insa-lyon.fr

Design and Methodology of Silicon Carbide High Voltage Termination Extension for Small Area BJTs

Keywords: Bipolar Power Devices, Silicon Carbide, Junction Termination Extension, High Electric Field, TCAD Modeling

An optimized termination extension tutorial is reported for a small area (≈ 1mm²) 4H-SiC NPN BJT. The extension system is based on a parametric JTE with isolation rings. Additional aspects such as (1) MESA angle, (2) relieving electric field on the base-drift junction, and (3) blocking the electric field from reaching border structure are investigated. A breakdown voltage greater than 10 kV is recorded, when drift region is n-doped to 8x10 14 cm-3.

Introduction

4H-SiC power bipolar devices are considered one of the most reasonable candidates for low-loss and high power semiconductor devices when compared to Silicon (Si) due to [START_REF] Locatelli | Silicon carbide against silicon: a comparison in terms of physical properties, technology and electrical performance of power devices[END_REF] a high bandgap allowing it to reach a low intrinsic carrier density even when operating over high temperatures (greater than 500 °C) [START_REF] Locatelli | Silicon carbide against silicon: a comparison in terms of physical properties, technology and electrical performance of power devices[END_REF], (2) a high saturation drift velocity enabling the operation over high frequencies, and (3) a huge breakdown electric field resulting in the reduction of ON resistance (RON) plus avoiding a second breakdown avalanche effect [START_REF] Cooper | SiC Power-Switching Devices-The Second Electronics Revolution[END_REF]. Similarly, 4H-SiC power BJTs, are free from gate oxide problems giving it the ability to sustain high currents with low-voltage drop when compared to SiC MOSFETs [START_REF] Krishnaswami | 1000-V, 30-A 4H-SiC BJTs With High Current Gain[END_REF]. Since BJTs are current controlled devices, making it a disadvantage over MOSFETs, an input drive current is needed on the base terminal in the on-state operation mode leading to an increase in the total power dissipation of the device, thus it is always useful to maximize the current gain in BJTs. The recent reported high voltage 4H-SiC BJTs include a 10 kV open-base breakdown voltage (BVCEO) and specific on-resistance of 130 mΩ.cm 2 at (RT) with a 120 µm, 6x10 14 cm -3 n-doped drift layer [START_REF] Zhang | 10 kV, 10 A Bipolar Junction Transistors and Darlington Transistors on 4H-SiC[END_REF]; a 15 kV BVCEO at a leakage current of 0.1 mA/cm 2 with 125 µm drift layer n-doped to 1.1x10 14 cm -3 [START_REF] Salemi | 15 kV-Class Implantation-Free 4H-SiC BJTs 15 kV-Class Implantation-Free 4H-SiC BJTs[END_REF], and a 9.2 kV BVCEO with a 50 µm, 7x10 14 cm -3 n-doped drift layer and RSP_ON = 78 mΩ.cm 2 [START_REF] Zhang | Demonstration of first 9.2 kV 4H-SiC bipolar junction transistor[END_REF]. For high electric field bipolar applications, it is mandatory to implant a highly sophisticated base-drift junction termination extension (JTE) system, which serves in minimizing the electric field plate crowd generated at the MESA edge of the power device. Several approaches were applied to optimize the extension performance; (1) by applying variations over the length and etching depth of the JTE [START_REF] Salemi | Area-and Efficiency-Optimized Junction Termination for a 5.6 kV SiC BJT Process with Low ON-Resistance[END_REF], (2) by implanting different doses over the total length of the JTE [START_REF] Miyake | 21-kV SiC BJTs With Space-Modulated Junction Termination Extension[END_REF], and (3) by the addition of extra implanted rings [START_REF] Planson | Edge termination design improvements for 10 kV 4H-SiC bipolar diodes[END_REF]. This paper presents different design methodologies for high-voltage SiC BJT termination for obtaining both high-open-emitter (BVCBO) and high open-base (BVCEO) breakdown voltage greater than 10 kV based on simulation measurements done using Synopses TCAD [START_REF] Sentaurus | TCAD Simulation Tool by Synopses Inc[END_REF]. The realized results are relative to several technical models associated to the SiC simulation parameters given by TCAD, which were used as follows: Arora model for Doping Dependence., Caughey-Thomas model for High Field Dependence, Scharfetter and VanOverstraetendeMan used for Recombination and Impact Ionization models, and finally Bandgap Narrowing for the band gap model.

SiC BJT Termination Model and Technology

Figure 1: Schematic cross-sectional view of a BJT structure with optimized termination Figure 1 represents an illustration of a 4H-SiC NPN optimized BJT structure. The structure is built with a 0.5 µm, 1x10 18 n-doped collector layer. The lowly n -doped drift layer has a thickness of 120 µm and a doping concentration 8x10 14 cm -3 . The 2 µm base layer is p-doped to 2x10 17 cm 3 , with 400 nm highly p + implants under the base contacts to ensure a low resistive ohmic contact. The 0.5 µm emitter layer is n-doped to 1x10 19 cm -3 meeting another 0.5 µm highly n-doped emitter contact layer. Theoretically, the highest breakdown voltage of a PN junction with such drift parameters can reach to 14 kV according to Equation 1, where BV represents the blocking voltage in volts, EFmax is maximum electrical field that SiC can sustain in V/cm, 𝜀 𝑠 is the dielectric permittivity of the semiconductor material, and ND is the doping concentration of the drift region in cm -3 . The thickness of the epitaxial layer also influences the breakdown voltage, since the electrical field 𝐸 𝐵 can be expressed as follows:

𝐵𝑉 = 𝐸𝐹

𝐸𝐹 𝑚𝑎𝑥 2 = 𝑞 𝑁 𝐷 𝑊 𝐷 𝜀 𝑠 …………………………………….….…………………… (Equation 2, [11])
where WD is the drift thickness in meters, giving another form of the blocking voltage:

𝐵𝑉 = 𝑞 𝑁 𝐷 𝑊 𝐷 ² 2𝜀 𝑠 ……………………………………………...…………………. (Equation 3, [11])
From Equations 1 and 3, we realize that increasing WD, and decreasing ND, both serve in increasing the blocking voltage. This feature can be observed in [START_REF] Miyake | 21-kV SiC BJTs With Space-Modulated Junction Termination Extension[END_REF], where the highest breakdown voltage for a BJT to-date is recorded 21 kV. This lab in Kyoto University/Japan, in addition to an optimized termination extension and a proper fabrication process, was able to reach this value with a very low ND = 2.3x10 14 cm -3 and a WD greater than 180 µm. Putting these values in Equation 1, a maximum theoretical breakdown voltage up to 50 kV can be obtained, which is considered far from the recorded results. Normally, it is impossible to reach the theoretical breakdown due to electric field crowding close to junction edges, but research is always working to get near this value by optimizing the termination extension. Figure 1 describes several approaches when designing an optimized extension.

From geometrical point of view, fabrication of MESA structures with negative inclination aid in avoiding problems such as edge breakdown due to a significant reduction of electric field when compared to the bulk one [START_REF] Lebedeva | Edge-Termination Technique for High-Voltage Mesa-Structure 4H-SiC Devices: Negative Beveling[END_REF]. Figure 2, shows the electric field behavior of same BJT structure but with different MESA angles. An increase in the angle reduces the electric field crowding generated on the junction. Relieving this electrical field crowding effect is also done by implanting P + isolation rings. These rings can be capped along the JTE as in Figure 1 and integrated at the edge of structure with an ohmic contact performing a PN isolation diode as described in [START_REF] Liang | Optimization Design of Isolation Rings in Monolithically Integrated 1200V SiC Transistor and Anti-Parallel Diode for Improved Blocking Voltage[END_REF]. The efficiency of the JTE is influenced by parameters such as the length (LJTE), and the implanted dose (DJTE). The idea is to provide a smooth flow for the electric field by choosing the suitable dose given for the implanted JTE along its length. The chosen dose should be relative to the base doping concentration, for permitting a continuous flow all over the structure. The thickness of the JTE is limited and fixed to 1µm, as we vary the dose.

Moreover, is it possible to complete the extension by adjoining rings with the same dose and thickness associated to the JTE (Figure 1). The performance of the rings is relative to their number (NRings) and the space given between them (SRings.), which may vary between each ring. The rings length (LR) which is not considered a critical parameter is fixed to 5 µm. The number of rings with their associated spaces plus the LJTE determines the overall length of the termination system. This length is important when dealing with small sized structures.

EF Crowd θ EF Crowd θ EF @ border = 3x10 5 V/cm SC No EF @ border

In normal cases, no electric field crowd should be present at the edge of the structure, and hence an isolation length should be given after the given extension system. For small structures, where no to little space is available, a highly n+-doped stop channel can be added after the termination extension (Figure 1) in order to prevent the lateral electric field from reaching the chip edge which is a phenomenon where capability of breakdown voltage decreases as shown in Figure 3.

Results and Discussion

Table 1 describes three different general small BJT structures with different designs. Structure 1 as mentioned before refers to an unoptimized device and will be used as a reference. The BV recorded is nowhere from reaching the 10 kV, with a 0% of enhancement due to the absence of JTE and Rings. Structure 2 is built just with a JTE. Several lengths between 250 µm and 500 µm were tested with different doses in order to detect the optimum length and dose that match the base doping we have.

No rings are integrated in this structure and the recorded BV is increased by 300% since the breakdown voltage obtained is almost triple the one we got from structure 1, translating the ability of peripheral protection to get closer to the theoretical 1D target. After dealing with the JTE parameters, an isolation ring system is added to structure 3 with constant spacing between them. This structure burdened more the electrical field and hence increased significantly the efficiency of the system to reach up to 430%. Finally, referring to the electrical field profile shown in Figure 4 we can clearly notice the importance and potential of adding a JTE and ring system on the electric field. The continuous curve in Figure 4 refers to the electric field associated to structure 1. We realize the peak of the curve is located immediately at the base-drift junction, and then the field sharply vanishes, which explains the low breakdown voltage with such a structure. Paradoxically, the dashed curve represents an optimized structure. We notice how the electric field is distributed all over the structure. The first two peaks of this field determine the starting and ending point of the JTE, while the following ones represent the electric field on each ring border to reach the last one. It is highly recommended to conserve or keep the same electric field along the border of the JTE and rings, in other words; keeping the same level for each peak. This phenomenon also affects the ability of increasing the breakdown voltage and it is controlled by the number and space given between each ring.

Conclusion

A high power, with small area 4H-SiC NPN bipolar transistor structure with optimized termination extension was investigated in this paper. The termination is highly effective and efficient. A targeted breakdown voltage was reached with a proper electric profile all over the structure. A final structure was ended up with specific parameters where the breakdown voltage recorded is 12.6 kV which forms 90% from the theoretical value (14 kV) with 120 µm drift thickness n-doped to 8x10 14 cm -3 . Additional points were discussed concerning high P + doped implanted rings to decrease the electric field peak at the base-drift junction, and highly N + doped stopping channel for the aim of blocking high electric field on the edge of the structure.

  .... (Equation 1,[START_REF] Buono | Simulation and Characterization of Silicon Carbide Power Bipolar Junction Transistors[END_REF])

Figure 2 :Figure 3 :

 23 Figure 2: Electrical field (EF) distribution at MESA with various angle θ: (a) θ = 0°, (b) θ = 10°

Figure 4 :

 4 Figure 4: Variation of electrical field as a function of JTE and Rings presence

  Horizontal Electric Field [V/cm]Half Length of BJT Structure (0-600)[µm]

Table 1 :

 1 Variation of breakdown voltage as function of structure design

	Structure	JTE	LJTE [µm] DJTE [x10 13 cm -2 ]	NRings	BVCBO [kV]
	1	Absence 	Absence  Absence 	Absence  Absence 	3
	2	Presence ✓	LJTE Є [250, 500] DJTE Є [0.8, 1.5]	Absence  Absence 	9
	3	Presence ✓	LJTE Є [250, 500] DJTE Є [0.8, 1.5]	NRings Є [1, 8]	12.6

Acknowledgement

This work was supported by a grant overseen by the French National Research Agency (ANR), the project HV-PhotoSW, ANR-18-CE05-0020-01.


Hossein Rabiee Golgir

Yang Gao

Yun Shen Zhou

Lisha Fan

Premkumar Thirugnanam

Kamran Keramatnejad

Lan Jiang

Jean-François Silvain

Yong Feng Lu

Hossein Rabiee Golgir

Low-Temperature Growth of Crystalline Gallium Nitride Films Using Vibrational Excitation of Ammonia Molecules in Laser-Assisted Metalorganic Chemical Vapor Deposition

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Introduction

Gallium nitride (GaN), a III-V compound semiconductor with a wide, direct band gap of ∼3.4 eV, has been utilized in widespread applications, such as optoelectronics and high-power electronic devices. (1)(2)(3) High-quality crystalline GaN films are in demand for their ability to enhance the performance and reliability of GaN-based devices. (4)(5)(6)(7) However, growth of high-quality crystalline GaN films requires growth techniques using high temperatures, such as metalorganic chemical vapor deposition (MOCVD, ∼950-1100 °C), molecular beam epitaxy (MBE, ∼800 °C), and hydride vapor phase epitaxy (HVPE, ∼750 °C). (6)(7)(8) A sufficiently high temperature is necessary to overcome the activation barriers to precursor chemisorption and adatom surface diffusion. However, high substrate temperatures can also cause adverse effects, such as biaxial stress within GaN films, nitrogen loss, and GaN decomposition, which degrade the efficiency of GaN-based devices. (9)(10) The biaxial stress is caused by the difference between thermal expansion coefficients of the GaN epitaxial layer and substrates (e.g., sapphire and silicon (Si)) and results in a poor light extraction in GaN-based light-emitting diodes (LEDs). (10) To reduce the thermal stress in GaN films, substrates with a matching lattice, including lithium aluminate (LiAlO2), lithium gallate (LiGaO2), and silicon carbide (SiC), are preferred. (11)(12)(13) However, such lattice-matching substrates are too expensive to be commercialized.

A high growth temperature will also lead to nitrogen reevaporation and GaN decomposition, which limits the growth rate of GaN films. (14) So far, the growth rates of GaN films synthesized by MOCVD and MBE were reported to be 4 and 1 μm/h, respectively. (15)(16) Therefore, a low-temperature synthetic technique is highly desired in growing crystalline GaN films.

P a g e | 2

In conventional GaN synthetic methods, energy is added into the gas precursors in the form of thermal heating. Energy is first deposited to kinetic energy of the reactant molecules and eventually makes its way into the internal modes (electronic, vibrational, and rotational) via collisions. (6)(7)(8) However, universal thermal heating is short of selectivity in chemical reactions.

Attempts at promotion of material synthesis have been investigated by exploring infrared (IR)-laser-assisted vibrational excitations of precursor molecules, in which energy is directly coupled into specific molecules toward selective reaction pathways. (17)(18) Highly efficient energy coupling through vibrational excitation provides reactant molecules with sufficient energy to surmount reaction barriers and influence reaction pathways. (19)(20) In our previous studies, substantial enhancement in diamond growth was observed with resonant vibrational excitation of ethylene molecules in an IR-laserassisted combustion CVD process. (21)(22) In this study, low-temperature growth of highly c-oriented GaN films was achieved at a growth rate of up to 12 μm/h using laser-assisted metalorganic chemical vapor deposition (LMOCVD). Resonant excitation of the NH-wagging mode (v2) in NH3 molecules was realized using a wavelength-tunable carbon dioxide (CO2) laser at a matching wavelength of 9.219 μm. GaN films were successfully deposited on sapphire at a substrate temperature as low as 250 °C. A GaN growth rate of up to 12 μm/h was achieved at 600 °C using the LMOCVD method, which is 4.6 times faster than that of conventional MOCVD (2.6 μm/h). The influence of laser resonant vibrational excitation of NH3 on the GaN film growth was investigated.

Experimental Section

Measurement of Laser Power Absorption by NH3 Molecules

The absorption spectra of CO2 laser power by gaseous NH3 were measured in a vacuum chamber with an absorption path length of 40.64 cm (Figure 1). Absorption measurement of laser power at different laser wavelengths (9.2-10.9 μm) was performed at three different gas pressures of 1, 10, and 100 Torr, respectively. In the experiments, the chamber was evacuated to a base pressure of 1 × 10 -2 Torr. NH3 gas was subsequently introduced into the chamber. The incident laser power was kept at 80 W. A power meter was used to measure the laser power before and after passing through the chamber.

The drop in laser power was calculated as the absorption percentage.

Growth of GaN Films

Figure 1 shows the experimental setup of the LMOCVD system for the growth of crystalline GaN films at low temperatures.

GaN films were grown on c-plane sapphire substrates at different temperatures (250-600 °C). The sapphire substrates, with a dimension of 10 × 10 mm 2 , were ultrasonically cleaned with organic solvents (acetone and methanol) and deionized water, dried, and loaded into the LMOCVD chamber, sequentially. Then, the chamber was evacuated to a base pressure of 1 × 10 - 2 Torr using a mechanical pump. Trimethylgallium (TMGa) and ammonia (NH3) precursors were used as gallium (Ga) and nitrogen (N) precursors, respectively. The gas flow rate of NH3 was 1200 standard cubic centimeters per minute (sccm), and TMGa was carried into the reaction chamber using nitrogen as the carrying gas at a flow rate of 16 sccm. The growth pressure was maintained at 100 Torr during the growth process. A wavelength-tunable CO2 laser (PRC, wavelength range from 9.2 to 10.9 μm) was used to achieve resonant vibrational excitation of the NH3 molecules. The laser was tuned at a wavelength of 9.219 μm with a power of 80 W to resonantly excite the rotational-vibrational transition of the NH-wagging mode (ν2, 1084.63 cm -1 ) of NH3 molecules and couple the laser energy into the molecules. The laser beam, with a diameter of around 6-9 mm, was irradiated in parallel to the substrate surface inside the chamber through a zinc selenide (ZnSe) window, as shown in Figure 1. The distance between the laser beam and substrate surface was maintained at about 20 mm.

The substrate temperature was maintained at a constant temperature of 150, 250, 350, 450, and 600 °C, respectively. The deposition time was kept at 1 h. To understand the effects of laser-induced energy coupling, GaN films were also synthesized by the conventional MOCVD technique under the same growth conditions (deposition temperature, deposition time, gas flow rate, and growth pressure) without laser irradiation.

Characterization of GaN Films

The crystallinity of the GaN films was examined using a powder X-ray diffractometer (Rigaku D/Max B diffractometer, Co Kα1 λ = 1.788 Å). Surface morphologies and dimensions of the GaN films were studied using a field emission scanning electron microscope (FESEM, S4700). An energy-dispersive X-ray spectrometer (EDX, Oxford X-max 20 mm 2 ) was applied to analyze the composition of the GaN films. The optical properties of GaN films were studied using a photoluminescence spectrometer (iHR320 photoluminescence spectrometer with Si detector and indium gallium arsenide (InGaAs) detector).

Results and Discussion

Roles of Resonant Vibrational Excitation of NH3 in GaN Growth

NH3 is a dominant nitrogen source in GaN synthesis. However, NH3 has low decomposition efficiency due to a high NH bond energy (93-105 kcal/mol). (23) A sufficient amount of active N species are required for GaN growth. The growth rate of GaN depends on the molecular flux of active N and Ga species that transport to substrates. (24) However, it is difficult to grow GaN films at temperatures lower than 500 °C using a conventional MOCVD method due to the low dissociation efficiency of NH3. (23) To increase the reactivity of NH3 is, therefore, critical in low-temperature growth of GaN. Through resonant vibrational excitation of the NH-wagging mode in NH3 molecules, the reactivity of NH3 is selectively enhanced, and the dissociation efficiency of NH3 at low temperature is promoted. (25)(26) The absorption spectra of the CO2 laser power by NH3 gas at different gas pressures of 1, 10, and 100 Torr are shown in Figure 2. Three strong absorption peaks were observed at 9.219, 10.35, and 10.719 μm, respectively, at the NH3 pressures of 10 and 100 Torr. At a pressure of 100 Torr, the laser energy at these three wavelengths was completely absorbed by NH3 gas, whereas only one absorption peak at 9.219 μm was observed at a pressure of 1 Torr. The absorptions at these wavelengths were attributed to the resonant vibrational excitation of the NH-wagging mode (ν2) of the NH3 molecules. (25- 26) NH3 has a pyramidal shape with three hydrogen atoms forming the base and a nitrogen atom at the top. The NH-wagging mode vibrates in an umbrella inversion way. There is a barrier to umbrella inversion that the nitrogen atom faces on its travels through the hydrogen plane. (25) The existence of the barrier results in a splitting of a fundamental vibrational level of the NH-wagging mode into two components at 932.51 (ν2+) and 968.32 (ν2-) cm -1 , giving rise to the observed absorption peaks at laser wavelengths of 10.719 (932.92 cm -1 ) and 10.35 μm (966.18 cm -1 ), respectively. (25)(26) The strongest absorption peak at 9.219 μm is attributed to a rotational-vibrational transition (J = 5 → J′ = 6, K = 0) of the v2 mode at 1084.63 cm -1 .

The perfect match between the CO2 laser wavelength at 9.219 μm (1084.71 cm -1 ) and the rotational-vibrational transition line of the NH3 (1084.63 cm -1 ) makes a stronger absorption at 9.219 μm than that at 10.35 and 10.719 μm. The rotationalvibrational excitation of NH3 molecules at 9.219 μm contributes to dissociating of NH3 molecules at low temperatures.

Characterization of GaN Films

The X-ray diffraction (XRD) 2θ spectra of the GaN films grown at different temperatures (i.e., 150, 250, 350, 450, and 600 °C) are shown in Figure 3. Panels (a) and (b) in Figure 3 compare the XRD 2θ patterns of the GaN films grown by LMOCVD and MOCVD, respectively. A XRD peak attributed to the (0002) plane of GaN is observed in the GaN samples grown by LMOCVD at temperatures from 250 to 600 °C. This peak is indexed to wurtzite GaN with a hexagonal structure, (27) indicating the high c-plane orientation of the GaN films. Therefore, it can be confirmed that the synthesis of GaN films was achieved at a temperature as low as ∼250 °C by the LMOCVD technique with the laser resonant vibrational excitation of NH3. In contrast, the (0002) diffraction peak was only found for the sample grown by MOCVD at a temperature of 600 °C, suggesting that a much higher substrate temperature is required for the growth of GaN using conventional MOCVD (Figure 3b). As shown in Figure 3a, the intensity of the (0002) peak increases as the substrate temperature increases from 250 to 600 °C. The intensity increase can be attributed to the improved crystalline quality of the GaN films due to the substrate temperature increase. (28) Additionally, a 2θ signal at 86° was observed for the GaN film grown at 600 °C with LMOCVD (Figure 3a). This (0004) GaN peak also refers to c-axis oriented GaN films. were observed at intervals of 60° for all samples grown at temperatures from 250 to 600 °C, which confirmed the hexagonal structure of the epitaxial GaN films. It is also observed that there is a ∼30° rotation of the GaN unit cell with respect to the sapphire substrate due to the large lattice mismatch between GaN and the sapphire substrate (Figure 4). However, no diffraction peaks from the (200) plane of cubic GaN film were observed for all samples grown at different temperatures (250-600 °C) by LMOCVD (see the Supporting Information, Figure S1). The 2θ position and full width at half-maximum (FWHM) of ( 0002) diffraction peaks obtained from XRD are shown in Table 1. The FWHM of the (0002) plane reflects lattice distortion from screw dislocations and mixed dislocations. (28) As shown in Table 1, a small FWHM value of 0.18° was obtained for the GaN sample grown by LMOCVD at 600 °C. For the GaN film grown by MOCVD at 600 °C, the FWHM of (0002) was 0.20°. The distinctive reduction in the FWHM of GaN films deposited by LMOCVD indicates that the laser resonant vibrational excitation of NH3 leads to a decrease in lattice distortion and improvement in the quality of the GaN films. For the GaN films grown by LMOCVD, the FWHM of the (0002) plane decreased monotonically with the increase in the substrate temperature from 250 to 600 °C (Table 1). This means that the crystalline quality of GaN films improves continuously with the increase in the growth temperature, which is explained by the reduction in diffusion barriers and the increase in diffusion rates of Ga and N species as the substrate temperature increases. (28) The effectiveness of laser resonant vibrational excitation of NH3 on the uniformity and surface morphology of GaN films is clearly shown by the FESEM images shown in Figure 5. For the GaN film grown with LMOCVD at 250 °C (Figure 5a), small domains of about ∼30 ± 5 nm with hexagonal facets were obtained. The average domain sizes were ∼65 ± 10 and ∼100 ± 10 nm for the GaN films grown by LMOCVD at 350 and 450 °C, respectively (Figure 5b,c). With a further increase in the substrate temperature to 600 °C, the lateral size of the islands with hexagonal facets increased. With a coalescence of crystallite islands, GaN structures with flat facets were obtained by LMOCVD at 600 °C, as shown in Figure 5d. In contrast, the surface of the GaN sample grown at 600 °C by MOCVD was rough with obvious hexagonal hillocks (Figure 5f).

Additionally, the threshold temperature to grow GaN films by MOCVD is 600 °C. Below this temperature, no GaN deposition is observed on the substrate surfaces (Figure 5e), which agrees with the XRD results. μm/h; and MOCVD: ∼4 μm/h), (15)(16) laser resonant vibrational excitation contributes to a drastic enhancement in the GaN growth rate. Table 1 shows the GaN growth rates by LMOCVD at different temperatures from 250 to 600 °C. As the substrate temperature increased, the thickness of the GaN films increased. why the GaN growth rate was highly improved using LMOCVD, which is in agreement with other studies. (29) Furthermore, it is worthy to note that the impurity content (C, O, and Si) in the GaN film grown with LMOCVD decreased considerably (Figure 7). In MOCVD growth, Ga reacts with impurity elements in the absence of sufficient active N atoms or N-related intermediates. However, laser resonant vibrational excitation of NH3 in LMOCVD increased the production of active Nrelated intermediates, eliminating the impurity content incorporated into the film during GaN growth. With the reduction in impurities, the GaN crystal islands grow larger in size, leading to higher crystallinity, a full coalescence, and a smooth surface morphology of GaN films (Figure 5d). It is also worthy to note that EDX measurements here just express a semiquantitative analysis and the values do not reflect the actual composition of the GaN films. (30) The large concentration of oxygen and carbon in the films is just an artifact resulting from the very low detection efficiency of EDX for light elements. The optical quality of the GaN films was studied using a photoluminescence spectrometer at room temperature (see the Supporting Information, Figure S2). The GaN film grown at 600 °C by LMOCVD showed a near-band-edge (NBE) emission at about ∼3.398 eV, which was closer to the NBE emission of bulk GaN (3.47 eV), (31) compared to that of MOCVD with It is valuable to note that there are several studies of using the polycrystalline GaN films with columnar structures for photonic and electronic devices. (32)(33) In this study, the quality of the polycrystalline GaN films grown at low temperatures by LMOCVD are comparable to those reported values, (32)(33) suggesting a potential of polycrystalline GaN film grown by LMOCVD for electronic and photonic applications.

Conclusions

The LMOCVD technique was developed for low-temperature growth of GaN films on sapphire substrates through laser resonant vibrational excitation of NH3 molecules. The highly c-oriented GaN films were successfully grown at temperatures as low as 250 °C. Low-temperature growth of GaN films is ascribed to the enhanced decomposition efficiency of NH3 with resonant excitation of rotational-vibrational transition (1084.71 cm -1 ) of the NH-wagging mode at the laser wavelength of 9.219 μm. The FWHM of (0002) diffraction peaks obtained from XRD for GaN films grown at 600 °C decreased from 0.20° (MOCVD) to 0.18° (LMOCVD), indicating an improvement in crystalline quality by LMOCVD. SEM images showed that the laser resonant vibrational excitation of NH3 helped to grow GaN films with a smooth and uniform surface morphology. A high GaN growth rate of up to 12 μm/h was achieved at 600 °C by LMOCVD, which is ∼4.6 times faster than that of conventional MOCVD with 2.6 μm/h. This approach suggests that the laser resonant vibrational excitation of precursors is promising in increasing NH3 reactivity and promoting GaN synthesis, which could be extended to the synthesis of other nitride semiconductors.

Figure 1 .

 1 Figure 1. Illustration of the experimental setup for the CO2 laser-assisted MOCVD growth of GaN films at low temperatures.

Figure 2 .

 2 Figure 2. Absorption of CO2 laser energy by NH3 from 9.219 to 10.8 μm at pressures of 1, 10, and 100 Torr.

Figure 3 .

 3 Figure 3. X-ray diffraction spectra of the GaN films grown on sapphire substrates at different temperatures (250-600 °C) by (a) LMOCVD and (b) MOCVD.

Figure 4 .

 4 Figure 4. XRD φ-scan of (112̅ 0) plane of the sapphire substrate and (101̅ 1) plane of GaN films grown on sapphire at different temperatures (250-600 °C) by LMOCVD.

Figure 5 .

 5 Figure 5. SEM images of GaN films grown by LMOCVD at temperatures of (a) 250, (b) 350, (c) 450, and (d) 600 °C; and SEM images of GaN films grown by MOCVD at temperatures of (e) 450 and (f) 600 °C.

Figure 6 .

 6 Figure 6. Cross-sectional SEM images of GaN films grown at 600 °C by (a) LMOCVD and (b) MOCVD.

Figure 7 .

 7 Figure 7. EDX spectra of GaN films grown at 600 °C by (a) LMOCVD and (b) MOCVD.

∼3. 394

 394 eV. The NBE peaks of GaN films grown by LMOCVD at 250, 350, and 450 °C are ∼3.388, ∼3.391, and ∼3.392 eV, respectively.

Acknowledgment

This work was financially supported by the National Science Foundation (NSF CMMI 1068510 and 1129613). It was also performed in part in the Central Facilities of the Nebraska Center for Materials and Nanoscience, which is supported by the Nebraska Research Initiative.

Supporting Information

XRD φ-scan of (200) plane of GaN films grown by LMOCVD and PL spectra of GaN films acquired with a 325 nm laser. This material is available free of charge via the Internet at http://pubs.acs.org. The authors declare no competing financial interest.

Author Information