LOCAL EXISTENCE RESULT OF THE SINGLE DOPANT DIFFUSION INCLUDING CLUSTER REACTIONS OF HIGH ORDER

We consider the pair diffusion process which includes cluster reactions of high order. We are able to prove a local (in time) existence result in arbitrary space dimensions. The model includes a nonlinear system of reaction-drift-diffusion equations, a nonlinear system of ordinary differential equations in Banach spaces, and a nonlinear elliptic equation for the electrochemical potential. The local existence result is based on the fixed point theorem of Schauder.


Introduction
During the doping process impurity atoms of higher or lower chemical valence as silicon are introduced into a silicon layer to influence its electrical properties.Such dopants penetrate under high temperatures, usually around 1000 • C, with the so-called pair diffusion mechanism into the (homogeneous) layer.A precise description of the process can be found in [2,3,4] and in the literature cited therein.
Usually, dopant atoms (A) occupy substitutional sites in the silicon crystal lattice, loosing (donors, such as Arsenic and Phosphorus) or gaining (acceptors, such as Boron) by this an electron.The dopants move by interacting with native point defects called interstitials (I) and vacancies (V).Interstitials are silicon atoms which are not placed on a lattice site and move through the crystal unconstrained, and vacancies are empty lattice sites.Both can form mobile pairs with dopant atoms (AI, AV), while the unpaired dopants are immobile.The formation and decay of such pairs as well as the recombination of defects cause a movement of the dopants.We additionally include cluster formations, where a certain number of dopant atoms accumulate to immobile clusters (A cl ) in the silicon lattice.
These interactions can be modelled in terms of chemical reactions of arbitrary order.The resulting nonlinear model contains a set of reaction-drift-diffusion equations for the point defects and pairs, reaction equations for the immobile dopants and clusters as well as a Poisson equation for the electrochemical potential.

The model
For i ∈ {I,V ,AI,AV ,A,A cl } we consider the species X i and denote their concentrations by C i .We distinguish between mobile and immobile species defining respectively.We denote by C = (C I , C V , C AI , C AV , C A , C A cl ) the corresponding concentration vector.Each of the X i , i ∈ J ∪ J , is considered as the union of charged species X (j ) i , with the charged states j ∈ S i , where each S i ⊂ Z.Thus, if C (j ) i denotes the concentration of X (j ) i , the total concentrations C i are defined as ( The immobile species X i , i ∈ J , usually obey one fixed charged state.The chemical potential of the electrons is denoted by ψ.The charge density of the electrons n and holes p are assumed to obey the Boltzmann statistics, meaning that Moreover, are reference concentrations with positive constants K (j ) i .Set which represents the electrochemical activity of the ith component.

R. Bader and W. Merz 15
The mobile species for i ∈ J obey reaction-drift-diffusion equations (2.6) The immobile dopant concentration C A obeys the reaction equation (2.7) The immobile cluster concentration C A cl also obeys a reaction equation (2.8) The equation for the chemical potential of the electrons reads where , e are physical quantities.For i ∈ J , the drift-diffusion term is given by with the diffusivity represents the total charge of the ith species for i ∈ J ∪ J .Next, we put the reactions in concrete form.The source terms R i (C, ψ) result from the reactions occurring during the redistribution of the dopants.All relevant reactions (including cluster formations of high order) occurring during the (single) dopant diffusion are due to (2.5) of the form as well as the cluster reaction where l, m, n, s, r ∈ N, cl := l + m + n (the size of the cluster) and for i, h, k ∈ J ∪ J , the reaction rate coefficients are i,h,k > 0 are constants and S i,h,k ⊂ Z are special sets of indices.Thus, the source terms R i (C, ψ) are for i ∈ J of the form (2.16) and for i ∈ J we have (2.17) For the detailed description and physical meaning of the coefficients mentioned above, (see for instance [3].)Moreover, we set the constants , e, U T , 2n i equal to one for the analytical investigations.

Problem (P)
Now we summarize the basic properties of the coefficients appearing in the equations.The notation of the function spaces corresponds to that in [5,6].If we consider some function space Y , we denote by Y + the cone of its nonnegative elements.Operations on vectors have to be understood componentwise.Throughout the paper, > 0 denotes a generic constant, which we supply with indices if the occasion arises.

R. Bader and W. Merz 17
As can easily be seen, the coefficients appearing in the equations for i ∈ J ∪J and k ∈ J have the following properties for all ψ ∈ R and derivatives (l = 0, 1, 2) of required order two.Furthermore, for i, h, k ∈ J ∪ J and where The source terms (2.16) and (2.17) obey the growth conditions respectively, where λ r ∈ C(R) for r = 1, . . ., 5, λ r (ψ) > 0 for all ψ ∈ R, and under the assumption of nonnegative concentrations C = (C k ) k∈J ∪J .For i ∈ J ∪ J , the source terms satisfy the property for all ψ ∈ R, C ∈ R 6 + and if C i = 0. Finally, we assume and satisfies (P) for some T f ∈ (0, ∞).

Ordinary differential equations
In this section, we consider the system of ordinary differential equations in Banach spaces (2.7) and (2.8).For given functions (C k ) k∈J and ψ, with the properties we state an existence result, which we need in the next section.
In accordance with the results and notation used in [7], we extend (2.7) and (2.8) to the whole interval [0, ∞) and write them in the form where l ∈ N, and we make the functions (the same with ψ) those functions, which are contained in the coefficients α i (i ∈ I A := {1, . . ., 5}) and β j (j ∈ I B := {1, 2}) due to (2.17).With (3.1) and (3.2) we conclude that Moreover, from (2.17) we conclude that R. Bader and W. Merz 19 where which is continuous and maps bounded sets into bounded sets.
Lemma 4.1.Let (4.4) and (4.5) be satisfied.Then system (4.2) has a unique, nonnegative solution which satisfies the estimate where , which depends on the coefficients α i , β j and the initial data.
Proof.We proceed in several steps.(I) We have to ensure that and for all t ∈ [0, ∞) and (u, w) (4.12) Similarly, we deduce that (4.15)We get a local Lipschitz condition, that is, where we set u we conclude the existence of a local, nonnegative solution to the right of (4.2) from the point (t 0 , (u t 0 , w t 0 )).
(III) Finally, we derive a priori estimates in order to extend the solution to the maximal right-open interval, which is in our case [0, ∞) as we will see.Now let (u, w) be a solution of (4.2) in some interval J ⊂ [0, ∞).Thus, for t ∈ J we have (4.17) Let x ∈ ¯ , then from (4.3), step (I), and (4.5), it follows that for all t ∈ J .From this we immediately get .20) which we summarize as , we conclude, with [7, Proposition 1.1, page 200], the existence of a global solution, that is, the solution exists for any t ∈ [0, ∞).
For later use, we state a compactness result concerning equation ( 4 then we define the operator where (u, w) is a solution of (4.2) in [0, T ].
We use Ascoli's theorem (see [7]) to state the following result.
] r be a sequence, satisfying with a constant ¯ > 0.
We consider the sequence We again proceed in several steps.
(I) We have to show that {u n , w n } n∈N is equicontinuous in the time variable.Let (4.26) For s < t we have the estimate, where the constant ˜ > 0 is independent of n.This proves the equicontinuity.
(II) Finally, we have to verify that, for any t ∈ [0, T ], the set {u n (t), w n (t)} n∈N ⊂ [C( ¯ )] 2 is relatively compact.We apply the theorem of Arcelá-Ascoli.
(1) From (4.9) we get the estimate for all x ∈ ¯ , which is independent of n ∈ N.
(2) It remains to prove the equicontinuity in ¯ .Let x = y ∈ ¯ .A short calculation and the application of Gronwall's lemma yield for all t ∈ [0, T ] and some constants 1 , 2 which are composed of the quantities ¯ , ˆ introduced in the present derivation.
Since each component of (α n i ) i∈I A , (β n j ) j ∈I B belongs to L 1 (0, T ; W 1 p ( )), it results, from the embedding theorems, (see [5]) that it also belongs to L 1 (0, T ; R. Bader and W. Merz 23 and similarly with the other coefficients.In summary, we conclude that which yields, combined with the continuity of the initial data, the desired equicontinuity and completes the proof of compactness.

Poisson equation
Next, we collect results concerning the elliptic equation (2.9).We sketch the results and refer the reader for a detailed analysis to [1,8,9,10].The following results regarding the Poisson equation are valid in any space dimension.
Let n < p < ∞ and C ∈ [L p + ( )] 6 .Then we are able to show (with the help of Leray-Schauder's fixed point theorem, see [1]) that there exists a unique solution of (2.9).Moreover, there exists a constant p > 0 such that and the stability estimate 6 and the corresponding ψ, ψ satisfying the Poisson equation.Estimate (5.3) is also true, if only one of the concentrations is nonnegative.We will use this fact in (6.53).
If C i ∈ C([0, T ]; L p ( )) for i ∈ J ∪ J , then we immediately get that and that there exists a constant p > 0 such that and that there exists another constant p > 0 satisfying (5.7) Thus, we have summarized all results concerning ψ, which we need for further investigations.
The proof of this theorem consists of several steps, which we present in the next subsections.We start with a modification of our problem.
we denote the modified system by (P + ).
In the next subsection, we will show that, for any solution of (P + ), the concentrations are nonnegative.Then we will prove the existence of a strong solution of problem (P + ) with the help of Schauder's fixed point theorem in Sobolev spaces and use regularity results to get the desired smoothness.This (nonnegative) solution obviously solves (P), too.Finally, we have to show that there exists no other solution of (P), which concludes the proof of Theorem 6.1.

Problem (P
)) be a solution of (P + ), then C i ≥ 0 for i ∈ J .

R. Bader and W. Merz 25
Proof.For i ∈ J , we test the equation , where J i is defined in (2.10).We get with appropriate constants the estimate where we used Young's inequality and properties (3.1) and (3.2).Since C + i C − i = 0, we are able to apply property (3.7) to omit the reaction rates.We choose ε > 0 such that ε/2 = 1 , then we get We have ∇ψ ∈ L 2 (0, T ; C( ¯ )) and C − i ∈ C([0, T ]; L 2 ( )), so we can use Gronwall's lemma, saying that

Fixed point iteration for (P +
).Now we prove the existence of a local solution of (P + ) in Sobolev spaces by means of the fixed point theorem of Schauder.Let where the constants K i , k > 0 depend on known quantities only and will be specified below.
We define the set for some T ∈ (0, ∞).
We consider the vector-valued mapping by where C i , i ∈ J , is the solution of .11) and ψ is the solution of (6.12) Therefore, C i , i ∈ J , is the nonnegative solution of the ordinary differential equation in the Banach spaces, Now we check the properties of the mapping required in the fixed point theorem in the following steps (I), (II), and (III).
(Ia) The mapping Z is well defined, since system (6.11),(6.12) has a unique solution In order to see the solvability of (6.11), (6.12), we first note that, for i ∈ J each )) also belongs due to the embedding theorems to the space C([0, T ]; C( ¯ )) and so do the cuts.The function φ ∈ C([0, T ]; W 1 p ( )) is also continuous in both variables.Having this in mind, we can say that for given R. Bader and W. Merz 27 5 , the nonlinear system (6.13) has according to Lemma 4.1 a unique solution, that is, From (6.15) and embedding theorems, it follows that the coefficients as well as the right-hand side of (6.11) are continuous, and thus they also belong to the space L p (Q T ) for any p ≥ 1, especially for p ∈ (n + 2, ∞).In addition, the right-hand side of (6.12) belongs to the space W 1 p (0, T ; L p ( )).So with (3.8), the parabolic theory (see [5]) and the result (5.6) concerning the elliptic equation yield (6.14).
(Ib) For later use, we state an estimate.We get, by testing (6.13) with with the linear theory of ordinary differential equations in Banach spaces, and from the linear elliptic theory applied to (6.12) that there exists a constant > 0, such that the stability estimate holds for all φ 1 , φ 2 , Ꮿ 1+ i , Ꮿ 2+ i ∈ L p (0, T ; L p ( )) and the corresponding solutions C 1 i , C 2 i of (6.13) as well as ψ 1 , ψ 2 of (6.12).(II) We show, that there exists an instant of time At first, we state the constants K i , k > 0 defined in (6.7).In order to discuss K i > 0, we write (6.11) for i ∈ J in the form where Let T 0 ∈ (0, ∞) and set K i ≡ K i (T 0 ).Then the parabolic theory yields the estimate which is true for all T ∈ (0, T 0 ], and where K i > 0 remains bounded for any finite T 0 > 0 (see [5]).For (6.12) we get according to (4.9) and (5.5) the estimate with a constant k > 0.

R. Bader and W. Merz 29
We estimate the first L p -norm in (6.24).Using the growth conditions (3.4), we see that there exists a constant 1 > 0, just depending on known quantities defined in (6.8), such that where s := l + m + n and λ i ∈ C(R), i ∈ J .We consider the second L p -norm in (6.24), which is We use (3.1) to get the estimate ≤ T 1/p ∇ψ C([0,T ];L p ( )) C i W 2,1 p (Q T ) , (6.27) where the last inequality is true for n + 2 < p < ∞, (see [5]) an explanation of our special choice of p.The other terms in (6.26) and the last L p -norm in (6.24) may be estimated similarly.Again we can say that there exists a constant 2 > 0, just depending on known quantities, such that (6.28) Thus, in summary, and so Z : X T f → X T f .(III) Compactness and continuity of the mapping.We proceed with the following lemma.
Lemma 6.4.The mapping is compact and continuous.
Proof.At first, we note that the embedding is compact.Thus, from (6.14), the mapping is compact in the second variable.
there exists a subsequence Ꮿ n i → Ꮿ i in Y for n → ∞.This is also true for the cuts, that is, (2) If we apply Lemma 4.2 to the system (6.13)we conclude that there exists a subsequence (3) From (6.33), (6.35), and (6.36) we get the convergence of a subsequence

.37)
Let C n i , C i , i ∈ J , be the solutions of (6.11).We set In Q T f , we consider the system for Cn i , i ∈ J , which is where and with the boundary conditions ∇ Cn i •n = 0 on T , and zero initial conditions.We do not discuss the coefficients A, B (supplied with indices) appearing in the linear equations in detail, but we mention that they belong at least to the space C([0, T f ]; L p ( )), which will be discussed in a moment.

.41)
In order to show the convergence of the left-hand side, we have to estimate the right-hand side of (6.41) with the help of (5.3) as well as (6.35) and (6.36).
For this, we use the mean value theorem to get (the same with the other coefficients which depend on ψ), where ψ(s) = sψ n + (1 − s)ψ.Representatively, we estimate the term , where a short calculation gives , whereas the other functions are continuous due to embedding results, so we get A 3 ∈ C([0, T f ]; L p ( )).Thus, The other terms in (6.41) may be estimated similarly.Thus, we are able to show that there exists a constant > 0 satisfying i∈J F n i L p (0,T f ;L p ( )) ≤ i∈J Ꮿn i L p (0,T f ;L p ( )) + i∈J Cn i L p (0,T f ;L p ( )) , (6.45) which implies convergence in the left-hand side of (6.41).This and (6.33) prove the compactness of the mapping Z.
The continuity can be obtained by similar arguments.More precisely, we take a sequence {(Ꮿ n i ) i∈J , φ n } n∈N ⊂ X T f .From this we get that in L p 0, T f ; W 1 p ( ) ∩ W 1 p 0, T f ; L p ( ) , (6.46) as well as for n → ∞.
From steps (I), (II), and (III), we conclude the existence of a local, nonnegative solution of problem (P + ).

Uniqueness and problem (P).
The solution of (P + ) obviously solves problem (P), too.In order to show that the (nonnegative) solution, which we denote by C 1 i , ψ 1 , i ∈ J ∪ J , is the only one, we assume the existence of another, not necessarily, nonnegative solution C 2 i , ψ 2 , i ∈ J ∪ J .We again consider the system for the respective differences Ci := C 1 i − C 2 i , i ∈ J ∪ J , which is for i ∈ J exactly the same as (6.39) if we replace the cuts (Ꮿ n+ k ) k∈J and (Ꮿ + k ) k∈J in the right-hand side Fi by the solution vectors (C 1 k ) k∈J and (C 2 k ) k∈J , respectively.We test the ith equation with Ci and if we set This proves the unique solvability of (P), which is Theorem 6.1.

Definition 6 . 2 .
If we replace in (P) the source terms by R i ((C + k ) k∈J , (C k ) k∈J , ψ) and the right-hand side in the Poisson equation by i∈J .39) R. Bader and W.Merz 31 .8) Since in our case, |J | = 4 and |J | = 2 are the numbers of mobile and immobile species, respectively, the formulation of the problem reads.