Optimization of annealing to decrease quantity of radiation defects in a bipolar transistor

It has recently been shown that manufacturing of diffusive- and implanted-junction bipolar transistors in semiconductor heterostructure and optimization of annealing give us possibility to increase compactness of dopant distributions. In this paper, we analyze the possibility of decreasing the quantity of radiation defects by choosing regimes of annealing.


Introduction
One of the intensive solving problems of solid-state electronics is decreasing the dimensions of elements of integrated circuits (IC) (Gusev and Gusev 1991;Grebene 1983;Lachin and Savelov 2001;Gotra 1991). Another intensive solving problem is increasing the performance of such elements of IC as p-n-junctions and their systems (bipolar transistors and thyristors).
To fabricate p-n-junctions and their systems, some approaches can be used. One of the approaches is implantation of ions of dopant in a semiconductor sample or epitaxial layer (EL) of a heterostructure (H). It has been recently shown (see Pankratov 2008a, b) that implantation of dopant in a two-layer (TwL) H or in a three-layer (ThL) H at appropriately chosen energy of ions, thickness and materials of layers of H and optimal value of annealing time gives us the possibility to increase homogeneity of dopant distribution in doped area and sharpness of the p-n-junctions. Increasing the homogeneity gives us the possibility to decrease local overheats in the doped area during operation of the p-njunction or to decrease the depth of the p-n-junction at fixed tolerance on the value of the local overheats. A disadvantage of ion implantation is radiation damage of H. The main aim of the present paper is determination of the conditions to decrease the radiation damages or to decrease the influence of the damages on the characteristics of devices.

Statement of the problem
In this paper we consider a ThLH, which consists of a substrate (S) and two epitaxial layers EL 1 and EL 2 (see Fig. 1). The type of conductivity of S and other parameters (thickness, dopant diffusion coefficient, etc.) are known. Thicknesses and other parameters of EL 1 and EL 2 are known. A dopant has been implanted in the EL 1 to produce the type of conductivity, which is reverse to the type of conductivity of S. Another dopant has been also implanted in EL 2 to produce the type of conductivity, which coincides with the type of conductivity of S. Further annealing of radiation defects was done. After appropriate choosing of the thicknesses of the epitaxial layers and energies of ions, the dopants achieve the interfaces between EL 1 and S and also EL 1 and EL 2 . If the interfaces were not achieved by the dopants, additional annealing to shift the p-n-junctions to the interfaces attracted interest. If the dopants achieved the interface, maximal compromise between increasing the homogeneity of dopant distribution and sharpness of the p-n-junctions was obtained. Let us consider an alternative approach to annealing. In the framework of the alternative approach, the first dopant was implanted in EL 1 . Further radiation defects were annealed. After that, another dopant was implanted in EL 2 and new defects were annealed. The main aim of the present paper is comparison of restoring abilities of ThLH in the situations.

Method of solution
Let us determine the spatiotemporal distribution of dopant concentration to solve our aim. We determine the distribution by solving the second Fick's law (Gusev and Gusev 1991;Grebene 1983;Lachin and Savelov 2001;Gotra 1991) with boundary and initial conditions Here, C k (x, t) is the spatiotemporal distribution of the kth type dopant concentration; J Ck (x, t) is the spatiotemporal distribution of the kth type dopant flow; D Ck is the dopant diffusion coefficient. The value of dopant diffusion coefficient depends on the properties of materials of layers in H, on the rate of heating and cooling of SH and on spatiotemporal distributions of dopant and defect concentrations. The concentration dependence of the diffusion coefficient could be approximated by the following functions (Gotra 1991; Zorin et al. 1975) Here, P k (x, T) is the limit of solubility of the kth type dopant in H; D Lk (x, T) is the diffusion coefficient of the kth type dopant for low level of doping; parameter c k , which depends on the properties of materials of H, can be the integer usually in the interval c k 2 ½1; 3 (Gotra 1991); T is the temperature of annealing; V(x, t) is the spatiotemporal distribution of concentration of vacancies; V * is the equilibrium distribution of concentration of vacancies. Spatiotemporal distributions of concentration of defects were determined by the following system of equations (Zorin et al. 1975, Fahey et al. 1989 with boundary and initial conditions J I ð0; tÞ ¼ 0; J I ðL; tÞ ¼ 0; Iðx; 0Þ ¼ f I ðxÞ; Here I(x, t) is the spatiotemporal distribution of concentration of interstitials; D I (x, T) and D V (x, T) are diffusion coefficients of vacancies and interstitials; quadratic terms in Eq. 4 correspond to generation of divacancies and diinterstitials (see, for example Pankratov 2008c and appropriate references in the paper); k I,V (x, T) is the recombination parameter; k I,I (x, T) and k V,V (x, T) are parameters of generation of diinterstitials and divacancies, respectively. In this paper, in comparison with Pankratov (2008a, b), generation complexes of radiation defects have been accounted for. Let us transform Eqs. 1-4 to the following integral form Here, Wðx; tÞ Let us determine the solution of Eqs. 6 and 7 using the method of averaging of function corrections (Sokolov 1955) with a decreased number of iteration steps (Pankratov 2005). In the framework of the approach, we consider the solutions of Eqs. 1 and 4 with averaged value D 0l of diffusion coefficients D l (x, T) (l = L, I, V) and zero value of parameter n k ; f j ; K I;V ðx; TÞ; K I;I ðx; TÞ; K V;V ðx; TÞ instead of the averaged value of concentrations of dopants and radiation defects a 1C , a 1I and a 1V as a more exact initial-order approximation of solution of Eqs. 6 and 7. The solutions of Eqs. 1 and 4 could be written in the form Here, replaced on C). The first-order approximations of dopants and radiation defects concentrations could be obtained by substitution of the relation in Eq. 7 into the right side of Eqs. 6 and 7 instead of the functions q(x, t). After the substitution, the first-order approximations take the form Appl Nanosci (2011) 1:123-129 125 Let us to determine the approximations of dopants and radiation defects concentrations with the higher order by using the standard procedure of the method of averaging of function corrections (Pankratov 2008c;Sokolov 1955), i.e., by substitution of the functions q (x, t) in the right side of Eqs. 6 and 7 in the following sum a iq ? q i-1 (x, t), where i is the order of approximation. The substitution gives us the possibility of obtaining the second-order approximations of concentrations in the following form The parameter a 2q could be calculated by the standard relation (Pankratov 2008c;Sokolov 1955) One can obtain the following equation to determine the parameter a 2C for a fixed value of parameter c k by substituting Eqs. 10 and 11 in the relation in Eq. 13 Values of the parameter a 2Ck for several values of parameter c k are presented in the Appendix. By substitution of Eqs. 9 and 12 in the relation in Eq. 13, one can obtain the relations for parameters a 2I and a 2V in the following form 5 a 2I S I;V1100 þ S I;V1110 À Á À1 : tÞdxdt; a 1 ¼ S I;I0100 ðS V;V0100 S I;I0100 À S 2 I;V1100 Þ; a 2 ¼ ½ð2S I;I0110 þ S I;V1101 þ H L 2 2 ÞÂ 2S V;V0100 S I;I0100 À S I;I0100 ðS I;V1110 þ 2S V;V1101 þ S I;V1100 S I;V1110 þ H L 2 2 Þ þ S I;V1101 S 2 I;V1100 À ð L 2 2 Â H þ S I;V1110 þ 2S V;V1101 þ S I;V1100 S I;V1110 Þ S 2 I;V1100 ; a 3 ¼ fS 2 I;V1100 ½S I;V1111 þ S I;V1001 þS I;V0102 þQ V þ R L 0 F I ðxÞdx À ðS I;V1110 þ H L 2 2 þ 2S V;V1101 þ S I;V1100 S I;V1110 Þ 2 À S 2 I;V1100 ½S I;I0120 þ Q I þ N I þ S I;V1111 þ R L 0 F I ðxÞdxþ 2S I;V1101 S I;V1100 S I;V1110 À S I;V1110 S I; Let us analyze the redistribution of dopants in H. The obtained analytical results usually gives us the possibility to analyze demonstrably the redistribution of dopants during annealing of radiation defects as a functions of parameters. The second advantage of analytical solution is decreasing the calculation time of dopant and defect distribution, because calculation of the sums of elementary functions needs lesser time than numerical solution of differential or integral equation. On the other hand, using numerical approaches gives us the possibility to obtain more exact results. Therefore, we used both analytical and numerical approaches to analyze redistributions of dopants (we used numerical solution to check the analytical solution).

Discussion
Let us analyze the redistribution of dopant in the H (see Fig. 1). During the analysis, it is worthwhile to take into account matching energy of ions and thicknesses of epitaxial layers to achieve or almost achieve the appropriate interfaces of H by dopants during annealing of radiation defects. In this situation, the first dopant should achieve the interface between EL 1 and S and also between EL 1 and Increasing the number of curves corresponds to increasing the difference between diffusion coefficients. In this case, the position of the layer interface is a = L/2 EL 2 ; the second dopant should achieve the interface between EL 1 and EL 2 . If the dopants almost achieved the interfaces, additional annealing of dopants attracted interest to the shift of the p-n-junctions to the appropriate interfaces. The achievement attracted an interest to increase the sharpness of the p-n-junctions and at one time to increase the homogeneity of dopant distributions in the doped area (see, for example, Figs. 2, 3 and (Pankratov 2008a, b). Both the effects (increasing the sharpness of p-n-junction, and increasing the homogeneity of dopant concentrations) will be obtained in both situations: after annealing of all radiation defects, after implantation of both dopants and after annealing of individual groups of radiation defects, particularly after each ion implantation. However, annealing of each individual groups of defects gives us the possibility to accelerate recrystallization of H in the area of collision of both distributions of radiation defects. At one time during this type of annealing, one can obtain decreasing generation of complexes of defects.

Conclusion
In this paper we compare two types of annealing of radiation defects, which have been generated in a semiconductor heterostructure during ion implantation to produce a bipolar transistor. The first of them is one-stage implantation (implantation of all dopant into the heterostructure). The second of them is two-stage implantation (at the first stage only one half of the dopant is implanted, then radiation defects are annealed; at the second stage, the second half of the dopant is implanted, then radiation defects are annealed again). The comparison gives us the possibility to determine that the second type of implantation corresponds to larger decrease in radiation damage.