Microscopic mechanism responsible for radiation-enhanced diffusion of impurity atoms
MICROSCOPIC MECHANISM RESPONSIBLE FOR RADIATION-ENHANCED DIFFUSION OF IMPURITY ATOMS
O. I. Velichko
E-mail address (Oleg Velichko): firstname.lastname@example.org
Abstract. Modeling of radiation-enhanced diffusion of boron and phosphorus atoms during irradiation of silicon substrates respectively with high- and low-energy protons was carried out. The results obtained confirm the previously arrived conclusion that impurity diffusion occurs by means of the “impurity atom – intrinsic point defect” pairs and that the condition of the local thermodynamic equilibrium between substitutional impurity atoms, nonequilibrium point defects created by irradiation, and the pairs is valid. It is shown that using radiation-enhanced diffusion, one can form a special impurity distribution in the semiconductor substrate including retrograde profiles with increasing impurity concentration into the bulk of a semiconductor. The calculations performed give clear evidence in favor of further investigation of various doping processes based on radiation-enhanced diffusion, especially the processes of plasma doping, to develop a cheap method for formation of specific impurity distributions in the near surface region.
PACS: 88.40.jj, 81.40.Wx, 66.30.Dn, 61.72.U-, 61.72.sh
Keywords: silicon; boron; phosphorus; radiation-enhanced diffusion; modeling
The increase in the cost of traditional energy resources has been a very important tendency during the past years. Therefore, investigation and implementation of alternative renewable energy sources are particularly urgent now. One of the promising alternative energy sources are solar cells. It is worth noting that the best crystalline silicon photovoltaic modules are 5 % more efficient than the best modules based on polysilicon films [1, 2]. Production of solar elements includes doping of silicon with impurity of p- and n-types of conductivity [3, 4, 5, 6]. During doping, simultaneously with the high concentration diffusion of dopant atoms, the gettering of undesirable impurities such as iron occurs [3, 4] that results in improvement of the electrophysical parameters of semiconductor devices. It is evident that the production of solar elements could be successful when a high value of energy conversion was achieved at a low cost of technological processes. Therefore, the introduction of impurity atoms into the silicon lattice by means of the low temperature doping from a gas discharge plasma (plasma immersion ion implantation or PIII) [7, 8, 9, 10] is rather promising. it is worth noting that thermal [7, 8] or laser annealing  is applied for electrical activation of the induced impurity atoms. During PIII, the silicon substrate can be heated up. On the other hand, a strictly assigned elevated temperature of it can be provided by a special heater. In such a case, the distribution of impurity atoms will be determined by both the low energy impurity implantation and low temperature radiation-enhanced diffusion (RED) that occurs due to the generation of a great amount of nonequilibrium point defects during plasma treatment.
One of the first experimental investigations of plasma doping has been carried out in the paper of Strack et al. . As follows from the numerous experimental data concerning radiation-enhanced diffusion that the redistribution of the impurity atoms previously introduced also occurs during the implantation of hydrogen ions [13, 14, 15, 16, 17, 18, 19], silicon ions , or heavy ions of inert gases [21, 22] in silicon substrates having an elevated temperature. The experimental results obtained have shown that the radiation-enhanced diffusion takes place at very low temperatures insufficient for the ordinary thermal diffusion. For example, it was reported in  that the RED of boron atoms is observed below 100 C. It means that for local ion implantation the radiation-enhanced diffusion occurs only in the local region with an increased concentration of point defects and there is no redistribution of impurity atoms in the other regions of the semiconductor. This feature of radiation-enhanced diffusion can be very useful for the production of various semiconductor devices including solar elements. Let us consider the characteristic features of impurity transport processes under conditions of radiation-enhanced diffusion.
2 The equation describing the radiation-enhanced diffusion of impurity atoms
The equation for impurity diffusion due to the formation, migration, and dissociation of the “impurity atom – vacancy” and “impurity atom – silicon self-interstitial” pairs has been obtained in . This equation takes into account different charge states of all mobile and immobile species and the drift of the pairs in the built-in electric field, although only the concentrations of neutral point defects are involved in the explicit form. It is supposed that nonuniform distributions of nonequilibrium point defects including defects in the neutral charge state can be formed and the mass action law is valid for the pairs, substitutionally dissolved impurity atoms, and self-interstitials or vacancies.
In the case of low impurity concentration , the equation obtained can be presented in the form
Here is the concentration of substitutional impurity atoms; is the intrinsic carrier concentration; and are the diffusivities of impurity atoms in intrinsic silicon due to the “impurity atom – vacancy” and “impurity atom – silicon self-interstitial” pairs, respectively; and are the concentrations of vacancies and silicon interstitial atoms in the neutral charge state normalized to the thermally equilibrium concentrations of these species and , respectively.
In a number of doping processes only one kind of defect is involved in diffusion of the main fraction of impurity atoms. For example, during a low-temperature oxidation of the surface, silicon self-interstitials are the dominating defects in the silicon crystal  and diffusion of boron, arsenic, and phosphorus occurs due to the interaction with these interstitial atoms. On the other hand, it follows from experimental data that antimony atoms diffuse due to the interaction with vacancies . Then, Eq. (1) can be presented in a simplified form as
where is the concentration of the neutral defects responsible for the impurity diffusion normalized to the thermally equilibrium concentrations of this species .
The equation obtained retains the basic character of the origin equations from , namely, the ability to describe segregation of impurity atoms, including the “uphill” impurity diffusion. Indeed, we shall present Eq. (2) in the following form:
It is clearly seen from Eq. (3) that depending on the gradient of the concentration of neutral point defects, the drift term is added to the right hand of the equation that has the type of Fick’s second law. It means that an additional drift flux of impurity atoms proportional to the gradient of concentration of point defects in the neutral charge state is added to the flux caused by an impurity concentration gradient. If these fluxes are directed in opposition, a component of the “uphill” diffusion is included into the general impurity flux. It means that segregation of impurity atoms can be observed at great values of the gradient of point defects in the neutral charge state instead of the leveling effect for nonuniform impurity distribution. If a component of the “uphill” diffusion exceeds the ordinary diffusion flux described by Fick’s first law, an unusual form of the impurity profile will be observed.
As has been mentioned above, the term on the right-hand side of the diffusion equation that describes the “uphill” diffusion component has a form of a drift flux caused by a force field. As such fields, a built-in electric field or a field of elastic stresses can be considered. It follows from experimental data and theoretical calculations that at a corresponding direction of the built-in electric field the “uphill” impurity diffusion is really observed [26, 27]. The “uphill” diffusion is also observed under conditions of significant elastic stresses that influence the diffusion of mobile species (the Gorsky effect [28, 29]). However, in the case under consideration the “uphill” diffusion component is due to the so-called fictitious thermodynamic forces [30, 31, 32] that qualitatively differ from the real forces acting on mobile particles. As follows from [30, 31, 32], the fictitious thermodynamic forces arise due to the gradient of the concentrations of the species participating in transport processes. In the present investigation, such species are represented by vacancies and silicon self-interstitials being in the neutral charge state.
It is worth noting that the equation similar to the diffusion equation (4) has been obtained in  for the so-called “kick-out” mechanism of diffusion that is related in general to the substitutional-interstitial diffusion mechanism when the silicon self-interstitial displaces an immobile impurity atom from the substitutional position to the interstitial one. A migrating interstitial impurity atom in turn replaces the host atom becoming substitutional again. Such analogy is not surprising, because in later publications [33, 34] it has been shown that there is no difference in the mathematical description of the impurity transports processes occurring due to “impurity atom – silicon self-interstitial” equilibrium pairs and due to the kick-out mechanism if impurity interstitials are in local thermodynamic equilibrium with substitutional impurity atoms and nonequilibrium silicon self-interstitials. At the same time, there is a substantial difference between the mathematical descriptions of impurity transport processes due to the equilibrium impurity-vacancy pairs and to the simple vacancy mechanism of diffusion when the impurity atom and neighboring vacancy exchange places. Indeed, in the papers [35, 36] the equation of impurity diffusion due to such a simple vacancy mechanism has been obtained in [35, 36]:
where is the concentration of the vacancies normalized to the thermally equilibrium vacancy concentration .
It can be seen from Eq. (6) that the second term in the right-hand side of this equation has a “minus” sign, whereas the second term in the right hand side of Eq. (5) has a “plus” sign. It means that the flux caused by the vacancy gradient in the simple vacancy mechanism has an opposite direction to that for the case of impurity diffusion due to the impurity-vacancy pairs. It follows from the analysis of Eqs. (5) and (6) that investigation of impurity redistribution under conditions of nonuniform distribution of the point defects responsible for the impurity diffusion allows one to make a conclusion about the character of the diffusion mechanism.
3 Modeling of the radiation-enhanced diffusion
In the paper of Baruch et al. , an analysis of the experimental data obtained for the boron redistribution under proton bombardment of homogeneously doped silicon revealed that the impurity diffusion occurs due to the kick-out mechanism. A similar conclusion in favor of the kick-out mechanism has been drawn in [17, 18] on the basis of boron redistribution modeling in nonuniform doped silicon layers with different maximal concentrations of impurity atoms. Modeling of boron redistribution due to silicon bombardment by protons with an energy of 140 keV for 3 hours at a temperature of 695 C has been carried out in . The mechanism of impurity diffusion due to equilibrium pairs and due to the simple vacancy mechanism was investigated. For comparison, the experimental data obtained in  were used. In the experiments of Akutagawa et al.  the initial boron profile in the (111) oriented silicon substrates was formed due to ion implantation with an energy of 300 keV and a dose of 1.510 ions/cm. The choice of a very low dose was necessary for boron profiling by the differential C-V technique to avoid the effects caused by avalanche breakdown. Simultaneously, the effects caused by the concentration-dependent diffusion were avoided. After the implantation and before the proton-enhanced diffusion (140 keV protons at a beam current of 1.1 A/cm), the wafers were annealed at a temperature of 750 C for 30 min in a purified argon ambient for electrical activation of the implanted boron. After the enhanced diffusion, the samples are left in the target chamber at 700 C for times greater than about 30 min for postannealing treatment, which can be expected to produce full electrical activity of the impurity and remove the residual radiation damage. Both the initial and the final experimental profiles of boron concentration are presented in Figs. 1 and 2.
The results of modeling carried out in  for a simple vacancy mechanism of boron diffusion and impurity diffusion due to the pairs made its possible to draw a conclusion that the radiation-enhanced diffusion under proton bombardment occurs due to the formation, migration, and dissolution of the “boron atom - intrinsic point defect” pairs that are in a local thermodynamic equilibrium with substitutional impurity atoms and nonequilibrium point defects. In the present, work we repeat the calculations carried out in  for the case of an improved approximation of initial impurity distribution and of the assumption on the additional generation of point defects on the surface of a semiconductor. Calculated boron concentration profiles for the pair diffusion mechanism and diffusion mechanism due to the exchange of the places between an impurity atom and a neighboring vacancy (the simple vacancy mechanism) are presented in Figs. 1 and 2. The following values of the parameters that describe implantation of hydrogen ions have been used for modeling: = 1.235 m; = 0.1124 m; = -5.46; = 1.27 m . Here and are the average projective range of implanted ions and straggling of the projective range, respectively; and are the skewness and the position of the maximal value of the implanted ion profile, respectively. It was supposed that the generation rate of the nonequilibrium point defects responsible for the diffusion of impurity atoms is proportional to the distribution of implanted protons. The value of boron diffusivity = 3.61410 m/s has been calculated from the temperature dependence presented in . The stationary distribution of nonequilibrium defects that takes into account its additional generation on the surface of a semiconductor was obtained from the solution of the diffusion equation for point defects and is presented in Fig. 1 by the dotted curve. The average migration length of neutral point defects in intrinsic silicon , derived from the fitting of the calculated boron concentration profile after diffusion to the experimental one, is equal to 0.32 m. This value is greater than the value = 0.2 m, which was used in .
It can be seen from Fig. 1 that the better agreement of the calculated boron profile after RED with experimental data is obtained than in . Mainly, it is due to taking into account the additional generation of point defects on the surface of a semiconductor. Similar calculations for the simple vacancy mechanism of impurity diffusion are presented in Fig. 2. As follows from Fig. 2, a qualitative disagreement between the calculated boron profile and the experimental one is observed. Thus, the present calculations confirm the conclusion made in  that the radiation-enhanced diffusion of impurity atoms during proton bombardment of silicon substrates at elevated temperature occurs due to the “impurity atom – intrinsic point defect” equilibrium pairs. It is worth noting that due to the exchange of its place with the neighboring vacancy the impurity atom moves over one interatomic distance, whereas the pair makes a great number of jumps before the dissolution and it transfers the impurity atom to a much larger distance. In any case, we can claim a greater efficiency of diffusion due to the formation, migration, and dissolution of the pairs.
Determination of the mechanism of boron diffusion during proton bombardment allows us to investigate the form of impurity distributions that can be obtained using the radiation-enhanced diffusion. Let us consider, for example, the doping of silicon from hydrogen-containing plasma with addition of some amount of a diffusant. To provide the radiation-enhanced diffusion, it is supposed that the substrate temperature is equal to 620 C. Such a low temperature of the substrate has been chosen to completely exclude the thermal diffusion of impurity atoms. Indeed, boron diffusivity calculated for this temperature according  = 9.06410 m/s is too small for diffusion. On the other hand, during the treatment of silicon substrates in hydrogen-containing plasma with addition of a diffusant a great number of point defects are generated on the surface of a semiconductor due to the bombardment with low-energy ions. Migration of these nonequilibrium defects into the bulk of the semiconductor provides the radiation-enhanced diffusion of impurity atoms that enter into the vicinity of the surface from the gas discharge plasma.
The solution of the diffusion equation for nonequilibrium neutral point defects in the case of their generation on the surface of a semiconductor has the form
where is the average migration length of nonequilibrium point defects. Here and are the diffusivity and average lifetime of point defects in intrinsic silicon, respectively; is the concentration of the nonequilibrium neutral defects on the surface of a semiconductor normalized to the thermally equilibrium concentration of this species .
The results of modeling of silicon doping with boron from the gas discharge plasma obtained for the distribution of nonequilibrium point defects described by expression (7) are presented in Fig. 3. Equation (4) obtained for the mechanism of impurity atom migration by the “impurity atom – intrinsic point defect” pairs has been used for the description of impurity diffusion. Taking into account the need the formation of shallow p-n junction, the value of the average migration length of point defects = 0.1 m has been chosen.
It can be seen from Fig. 3 that the impurity profiles formed by the RED differ dramatically from the distribution described by the erfc-function which characterizes doping from a constant source. This difference results from the spatial dependence of the impurity diffusivity and from the presence of the additional flux due to the concentration gradient of point defects in the neutral charge state. Indeed, whereas at the beginning of the treatment, the impurity distribution is similar to the profiles formed by diffusion with a high concentration dependence of diffusivity, the continuation of treatment leads to the formation of a retrograde impurity profile characterized by an increasing concentration of impurity atoms in the bulk of a semiconductor.
Unfortunately, there is lack of experimental data on doping of silicon from a gas discharge plasma due to the radiation-enhanced diffusion. For example, the experimental data of  were obtained for the case of a long-term treatment when the sputtering of the surface of a semiconductor plays a significant role and, therefore, distribution of impurity atoms becomes stationary. For the calculation of impurity distribution measured in , it is reasonable to introduce a new coordinate system bound to the moving surface of the semiconductor and solve the diffusion equation in this mobile coordinate system. Here is the coordinate measured from the initial position of the surface of the semiconductor; is the projection of the surface velocity on the immobile axis ( in the case of sputtering of the surface of the semiconductor).
Then, the equation of impurity diffusion (4) in the moving coordinate system takes the following form:
When the impurity atoms entering from the plasma are compensated by those removed from the surface of the semiconductor due to sputtering, the distribution of impurity atoms becomes stationary in the moving system of coordinates. Then, Eq. (8) will be transformed into a stationary diffusion equation:
The ordinary differential equation (9) can be solved analytically. Let us obtain such an analytical solution for the following Dirichlet boundary conditions:
where is the impurity concentration on the surface of the semiconductor.
To obtain the analytical solution of the formulated boundary value problem, we present Eq. (9) in the following form:
and integrate (11) from up to to obtain
As soon as both the impurity concentration and the flux of impurity atoms are equal to zero on the right boundary , the ordinary differential equation can be obtained from Eq. (12):
This equation can be solved using the separation of variables:
Let us integrate Eq. (14) from 0 up to :
where is the normalized concentration of point defects in the neutral charge state on the surface of the semiconductor.
Using exponentiation of Eq. (16), one can obtain the expression for distribution of impurity concentration:
It follows from expression (17) that for the concentration of impurity atoms at . At the same time for the impurity concentration has a finite value at due to the faster decrease of the exponential function.
Let us consider the widespread case of defect generation on the surface of a semiconductor. Then, the distribution of neutral point defects can be described by expression (7). Substituting (7) into (17) and calculating the integral obtained, we find that the impurity distribution is determined by the expression
It follows from expression (18) that the concentration of impurity atoms increases with , attains a maximum, and then decreases to zero. The phosphorus concentration profile calculated by means of expression (18) for silicon doping from the gas discharge plasma is presented in Fig. 4. The experimental data are taken from . In the experiments the p-type silicon with a conductivity of 200 cm was used. The treatment temperature was equal to 820 C and the rate of surface sputtering was equal approximately to 5.2810 m/s. The impurity distribution profile was found by removing thin layers from the surface of the sample and measuring their sheet resistance. The thermal diffusivity of phosphorus for the above-mentioned temperature is equal to 1.91910 m/s , the intrinsic carrier concentration, to = 2.7610 m.
As can be seen from Fig. 4, the impurity concentration profile calculated from expression (18) agrees well with the experimental data obtained for phosphorus radiation-enhanced diffusion. The agreement is observed in the entire diffusion zone including the region near the surface of the semiconductor. To satisfy the experimental data , the average migration length of point defects was chosen to be equal to 0.34 m. This value is very close to the value of = 0.32 m, which has been derived from the experimental data of  on the boron radiation-enhanced diffusion during implantation of high energy hydrogen ions.
To investigate the microscopic mechanisms of impurity transport in semiconductors, modeling of boron radiation-enhanced diffusion during implantation of high energy protons into the silicon substrate being at an elevated temperature and modeling of phosphorus radiation-enhanced diffusion during the treatment of silicon substrate in a hydrogen-containing plasma with the addition of a diffusant have been carried out. It follows from the comparison of the calculated impurity profiles with experimental ones that the radiation-enhanced diffusion occurs by means of formation, migration, and dissolution of the “impurity atom – intrinsic point defect” pairs which are in a local thermodynamic equilibrium with the substitutionally dissolved impurity atoms and nonequilibrium point defects generated due to external irradiation. It is worth noting that a similar calculation for the simple vacancy mechanism of diffusion due to exchange of the places between impurity atom and neighboring vacancy qualitatively disagrees with the experimental data.
For the pair diffusion mechanism a theoretical investigation was carried out for the form of impurity profiles that can be created in the vicinity of the surface due to the radiation-enhanced diffusion during plasma treatment. It was shown that for diffusion from a dopant source that provides a constant impurity concentration on the surface the impurity profiles formed by the RED are similar to those formed in the case of the high concentration dependence of impurity diffusivity. Moreover, it is possible to form a retrograde impurity distribution characterized by an increasing concentration of impurity atoms in the bulk of the semiconductor. These characteristic features of the dopant profile are due to the nonuniform distribution of neutral point defects responsible for the impurity diffusion.
For simulation of the experimental phosphorus profile  formed during plasma treatment, the analytical solution of the equation that describes the radiation-enhanced diffusion under conditions of sputtering of silicon surface was obtained. From fitting to the experimental data the average migration length of point defects = 0.34 m was obtained. This value is very close to = 0.32 m which was derived from the experimental data of  on the boron radiation-enhanced diffusion during “hot” implantation of high energy protons.
The results of the calculations performed give a clear evidence in favor of further investigation of various doping processes based on the radiation-enhanced diffusion, especially the processes of plasma doping, to develop a cheap method for formation of special impurity distributions in the near-surface region.
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