Optimization of Pulse Laser Annealing to Increase Sharpness of Implanted-junction Rectifier in Semi- conductor Heterostructure

It has been recently shown that inhomogeneity of a semiconductor heterostructure leads to increasing of sharpness of diffusion-junction and implanted-junction rectifiers, which are formed in the semiconductor heterostructure. It has been also shown that together with increasing of the sharpness, homogeneity of impurity distribution in doped area increases. The both effect could be increased by formation of an inhomogeneous distribution of temperature (for example, by laser annealing). Some conditions on correlation between inhomogeneities of the semiconductor heterostructure and temperature distribution have been considered. Annealing time has been optimized for pulse laser annealing.

Increasing performance and reliability of microelectronic devices and integrated circuits has attracted great interest recently. One way to increase performance of semiconductor devices is decreasing capacitance of p-n-junctions [1,2]. The increase of homogeneity of dopant distribution in doped areas of a semiconductor structure allows to operate with higher current densities and to decrease local overheats or to decrease depth of p-n-junction [1][2][3]. Another actual problem is the increase of exactness of theoretical description of dynamics of technological process. The increase leads to higher predictability of dopant dynamics and, as following, higher reproducibility of parameters of solid state electronic devices.
Different types of technological processes could be used for production p-n-junctions (see, for example, [1][2][3][4][5]). One of them is dopant diffusion into a semiconductor sample or in an epitaxial layer (EL). Another one is ion implantation in the same cases. In this paper we consider a semiconductor heterostructure (SH), which is presented in Fig. 1. The SH consists of two layers. First of them is a substrate (a x  L) with diffusion coefficient D 2 , thickness L-a and known type of conductivity (n or p). The second layer of the SH is an EL (0 x  a) with diffusion coefficient D 1 and thickness a. Let us consider a dopant, which is implanted across the boundary x=0 into the EL to produce the opposite type of conductivity (p or n). At the time t=0 annealing of radiation defects is started with continuance . The annealing of radiation defects after production of the implanted-junction rectifier leads to a decrease of quantity of the defects and an increase of depth of the p-n-junction. The increasing is unwanted, because the process leads to deviation of characteristics implanted-junction rectifier from scheduled values. It has been recently shown, that inhomogeneity of a SH leads to increasing of sharpness of diffusion-junction (see, for example, [6,7]) and implanted-junction (see, for example, [8]) rectifiers, which are

Method of Solution
Spatiotemporal distribution of dopant concentration in the considered SH (see Fig. 1) has been described by the second Here V(x,t) and V * are spatiotemporal and equilibrium distributions of concentrations of vacancies. P(x,T) is the limit of solubility of dopant in SH. The fitting parameters ,  and  depend on properties of layers of SH. Parameter  characterizes degree of radiation damage of SH. Parameter  characterizes doping degree of SH. Parameter  usually is equal to an integer value in the interval   [1,3] (see [2]). In the following let us consider the limiting case, when the number of different complexes (for example, complexes of defects) is negligible in comparison with the number of point defects. Spatiotemporal distribution of vacancies concentration is described by the following system of equation [3] where I(x,t) and I * are the spatiotemporal and the equilibrium distributions of interstitials, respectively.   [12]), T d is Debye temperature [12],  and  are fitting parameters.
where T r is the equilibrium distribution of temperature, which coincides with room temperature.
First of all let us estimate spatiotemporal distribution of temperature. The parabolic equation has been transformed as x v r L d ass x v L d ass ass Let us determine the solution of the system Eqs. (6) by averaging functional corrections (see, for example, [14].
Substitution of the average value of the functions  (x,t) ( = ,T;  =I,V,L) and their partial derivatives in the right side of the Eqs.
(6) instead of the considered functions gives us possibility to obtained the first-order approximations  1 (x,t) of the functions  (x,t). To decrease steps of the iterative process, let us consider more accurate initial-order approximation (see, for example, [8]).
As such approximation we consider the solutions of the equations of the system (6), which correspond to average values of diffusion coefficients D 0L , D 0I and D 0V , thermal diffusivity  0ass and zero parameter of recombination. The solutions can be written in the form Substitution of the Eqs. (7) into the right side of the equations of the system (6) instead of the functions  (x,t) gives us possibility to obtain the first-order approximations (in the modified method of averaging of function corrections) of the appropriate functions. The algorithm is presented in details in [8] and will not be considered in this paper. The second-order approximations of the functions  (x,t), by using the method of averaging of function corrections can be determined by using the standard procedure (see, for example, [8]), i.e. one shall substitute the sums  2 + 1 (x,t) instead of the functions  (x,t) in the right side of the equations of the system (6). The substitution gives us possibility to obtained the second-order approximation of the functions  2 (x,t). The algorithm is presented in details in [8,14] and will not be considered in this paper.
The parameter  2 is determined by the following relation [8,14],

Discussion
Let us analyze the dynamics of redistribution of dopant in the SH (Fig. 1) for step-wise approximations of spatial distribution of diffusion coefficients of radiations defects and dopant and thermal diffusivity. In the case the approximations can be written as   should be noted, that annealing by laser pulse with optimal continuance could be substituted by some laser pulses with smaller continuance, but with high frequency. It should be noted, that optimal annealing time for the laser annealing case is smaller than optimal annealing time for the volumetric annealing case (see Fig. 5). It has been obtained that annealing times for Fig. 2 almost equal to optimal values of annealing times in Fig. 5. The difference could be explained by two reasons. The first of them is insufficient continuance of annealing in Fig. 2. The second one is finite exactness of mathematical approach.

Conclusion
In this paper we consider pulse laser annealing of radiation