SEARCH

SEARCH BY CITATION

Keywords:

  • atomistic process simulation;
  • dopant clusters;
  • dopant diffusion;
  • kinetic Monte Carlo

Abstract

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Model
  5. 3 Simulation results
  6. Acknowledgements
  7. References

Arsenic diffusivity in heavily doped n-type silicon has been observed to increase strongly with donor concentration. This behaviour has been related to percolation effect, but other explanations, such as mobile As2V clusters (or, more generally, mobile AsDV clusters, with D being a donor), have also been postulated. In this work, we report the modelling and simulation of arsenic diffusion for high donor concentrations based on AsDV mobile clusters, within the framework of the atomistic non-lattice kinetic Monte Carlo model. Expressions for arsenic diffusivity in terms of microscopic parameters have been developed, and the parameter set have been calibrated from basic experiments and ab initio calculations. For donor concentrations above 2 × 1020 cm−3, the model predicts a concentration dependence of arsenic diffusivity with an exponent of 3.5, in close agreement to the experimental observations and quite similar to the percolation model. Long-hop distances of AsDV clusters have been also analysed. The model has been implemented in the atomistic simulator Sentaurus Process KMC. A good agreement with experimental arsenic profiles has been obtained for a wide variety of process conditions, including low ion energy, high dose and amorphizing implants, and annealing temperatures ranging from 750 to 1050 °C. The model has shown to allow efficient and accurate simulation, working together with all the other models accounting for the complex phenomenology of state-of-the-art processes.

1 Introduction

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Model
  5. 3 Simulation results
  6. Acknowledgements
  7. References

The ever-continuous shrinkage of electron device dimensions demands shallower dopant profiles and higher electrically active dopant concentrations [1]. Arsenic is one of the most commonly used donor dopants in silicon devices and, consequently, arsenic diffusion and activation has been a topic of interest during the last decades [2, 3]. Nevertheless, a wide variety of phenomena has been shown to play a role in arsenic-doped silicon, making modelling and simulation specially challenging. In particular, arsenic diffusion is driven by both vacancies (V) and self-interstitials (I), via mobile AsV or AsI pairs, respectively [2-4]. Moreover, immobile AsmVn clusters (and, notably, As4V) are known to be very stable and they could form at high concentrations even in equilibrium conditions [5-9]. AsmIn clusters can be also produced by ion-implant damage [5, 9, 10]. Both AsmVn and AsmIn clusters are responsible for the partial electrical deactivation of implanted arsenic. For arsenic concentrations below 2 × 1020 cm−3, the diffusion coefficient (DAs) was found to increase linearly with the electron concentration (which, in the extrinsic range, is approximately equal to the electrically active donor concentration) [11]. Above 2 × 1020 cm−3, a decrease of DAs with concentration was reported, and it has been related to immobile cluster formation [12]. In contrast, for much shorter anneals, a strong enhancement of DAs with concentration was found by Larsen Nylandsted et al. [13]. This last behaviour has been attributed to percolation [14] or, alternatively, to mobile As2V clusters [5].

The complex phenomenology sketched above poses a difficult challenge for predictive simulation of realistic processes. Recently, atomistic process simulators, based on the kinetic Monte Carlo method (kMC), have proved to be a choice in complex scenarios [15]. In this work, we report the modelling and simulation of arsenic diffusion for high donor concentrations, including mobile As2V clusters, within the framework of the non-lattice kMC approach.

2 Model

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Model
  5. 3 Simulation results
  6. Acknowledgements
  7. References

In our model we assume mobile As2V clusters. The contribution of these mobile As2V to arsenic diffusion coefficient is given by

  • display math(1)

inline image and [As2V] are the diffusivity and concentration of As2V clusters, and [Astot] is the total arsenic concentration. The diffusivity inline image is described by

  • display math(2)

where inline image is the migration energy of As2V and inline image is a prefactor. As2V can be formed by point-defect reactions and can break with a rate that depends on its binding energy. Thus, the ratio [As2V]/[Astot] can be expressed in terms of microscopic parameters as:

  • display math(3)

where [As] is the electrically active arsenic concentration, Ef,V is the formation energy of vacancies, inline image the binding energy of As2V, eAsV+ the positive charge level of AsV, ec conduction band edge, eF the Fermi level and Ω is a prefactor. For extrinsic conditions, the Fermi level is determined by the dopant concentration. Within the simple Maxwell–Boltzmann approximation (exp(eF/kT) ∝ [As]) and using Eqs. (1)-(3), the concentration dependence of arsenic diffusivity due to As2V could be approximated by

  • display math(4)

Migration and binding energies for As2V have been calculated by ab initio methods. In particular, a value of 2.0 eV has been found for inline image [6], whereas a value of 2.68 eV has been reported for inline image [9]. We choose the migration prefactor to fit the diffusion results at high arsenic concentration reported in Ref. [13]. These experiments, however, were performed using a doping background of phosphorous, instead of arsenic, in order to get a cleaner measurement. Therefore, Eqs. (1)-(3) were redone considering AsPV mobile pairs, obtaining an identical formalism substituting As2V by AsPV and [As] by [P], but without the dimer-related factor 2 in Eq. (1). A binding energy of 2.58 eV (instead of 2.68 eV) calculated for AsPV [9] was used. The other input parameters for mobile species (displayed in Table 1) were fitted to arsenic diffusion experiments at lower concentrations and, complementarily, to reproduce other experiments or calculations [3, 7, 9]. An interstitial fraction for arsenic diffusivity of about 0.3 is obtained from 800 to 1300 °C, well within the reported estimations [3]. A detailed explanation of the meaning of the parameters can be found in Ref. [4].

Table 1. Parameter set for mobile species used in this work
speciesEm (eV)Dm0 (cm2/s)Eb (eV)Db0 (cm2/s)eT (eV)
AsI+0.52 × 10−40.560.50
AsI01.31.7 × 10−2   
AsV+1.34.6 × 10−3 0.20.6
AsV01.687.11.34  
As2V2.04.02.68180 
AsPV2.04.02.58180 

Figure 1 displays the calculated arsenic diffusion coefficient as a function of donor concentration, showing the contribution of mobile clusters. In order to compare it to the experimental data of Ref. [13], also shown in Fig. 1, phosphorous donor doping and AsPV mobile clusters have been considered. Fermi–Dirac statistics has been used for Fermi level calculation, instead of Maxwell–Boltzmann approximation of Eq. (4). This yields an exponent close to 4 for the concentration dependence of mobile cluster contribution above 2 × 1020 cm−3, resulting in an exponent of about 3.5 for the total diffusion coefficient [3], in close agreement to the experimental observations and quite similar to the percolation model [14]. The calculated concentration dependence for arsenic doping is similar to that of Fig. 1 but with slightly higher mobile cluster contribution.

image

Figure 1. Calculated arsenic diffusivity as a function of donor (phosphorous) concentration at 1050 °C. Thin solid line: contribution of mobile AsPV clusters. Dashed line: diffusivity with no mobile clusters. Thick solid line: arsenic total diffusivity. Symbols: experimental measurements [11, 13]. White dashed line displays the 3.5 power dependence [3].

Download figure to PowerPoint

The long-hop distance, Λ, for AsDV clusters (As2V or AsPV) can be calculated as [4]

  • display math(5)

where Dm is the corresponding diffusivity (inline image or Dm,AsPV) and νbk is the cluster break-up rate. According to our parameter set, νbk is determined by the AsV emission, rather than by the V emission. The calculated long-hop distance for AsPV as a function of temperature for intrinsic silicon and as a function of donor concentration is shown in Figs. 2a and b. It can be seen in Fig. 2a, for intrinsic material at annealing temperatures, Λ is in the order of the interatomic distance range (0.2–0.3 nm) with activation energy of only −0.14 eV. The concentration dependence in Fig. 2b reflects that Λ ∝ exp(eF/2kT).

image

Figure 2. Long-hop distance for AsPV pairs (a) as a function of temperature for intrinsic conditions and (b) as a function of donor concentration at 1050 °C.

Download figure to PowerPoint

The model has been implemented in the atomistic simulator Sentaurus Process KMC [16], where it works together with all the other models related to the complex phenomenology taking place during silicon processing (ion damage, amorphization–recrystallization, charge effects, extended defects formation and dissolution, dopant clustering, etc.) [15]. Ion implant cascades are calculated within the binary collision approximation and loaded into the kMC simulator. The total binding energies used for arsenic clusters are listed in Table 2. These energies are based on ab initio calculations [9] but they have been fine-tuned to fit experimental arsenic composition profiles for a wide variety of experimental conditions, including ion implant doses from 5 × 1013 to 5 × 1015 cm−2, ion energies from 5 to 35 keV, annealing temperatures from 600 to 1050 °C, and both interstitial-rich and vacancy-rich scenarios [10, 17-19]. Formation energies used for V and I are 3.74 and 4.0 eV, respectively.

Table 2. Total binding energies (in eV) used for arsenic clusters
As2−0.3As2V2.68As2I2.40As2I24.09
As3−0.23As3V3.67As3I2.05As3I23.83
As40.0As4V4.83As4I2.40As4I24.50

3 Simulation results

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Model
  5. 3 Simulation results
  6. Acknowledgements
  7. References

Figures 3 and 4 display some representative examples of simulated and experimental [17-19] diffusion profiles with concentrations above 1020 cm−3. The role of As2V mobile clusters is illustrated in Fig. 4, in which simulations excluding mobile As2V clusters are displayed as a reference. In this case (35 keV, 5 × 1015 cm−2 As+ implant, followed by 1030 °C anneal) the electrically active arsenic concentration is high and the diffusion component related to As2V becomes dominant.

image

Figure 3. Arsenic concentration profiles for different process conditions: (a) 32 keV, 1 × 1015 cm−2 As+ implant followed by a 4 min 600 °C anneal and an additional 10 or 30 s 1050 °C anneal, (b) 12 or 25 keV, 2 × 1015 cm−2 As+ implant followed by a 1050 °C spike anneal, (c) 35 keV, 5 × 1015 cm−2 As+ implant followed by a 10 min or a 240 min 750 °C anneal. Experimental data are from [17-19] and they are represented with lines (thick solid: annealed profiles; thin dashed: as-implanted). Simulation results are plotted with symbols.

Download figure to PowerPoint

image

Figure 4. Arsenic concentration profiles for a 35 keV, 5 × 1015 cm−2 As+ implant and different annealing times at 1030 °C: (a) 5 s anneal, (b) 15 s anneal and (c) 30 s anneal. Solid lines are the experimental annealed profiles, taken from Ref. [19]. Circles are the simulated profiles including As2V mobile clusters whereas rhombuses are the simulated profiles excluding As2V-mediated diffusion. Dashed lines are the initial as-implanted experimental profiles [19] and triangles are the as-implanted simulations.

Download figure to PowerPoint

In contrast, we have found that the role of mobile As2V is not very relevant in the cases of Fig. 3, as inferred from the comparison with simulations with no mobile As2V clusters (not shown). This is explained by the effect of immobile clusters (notably As4V and As2) which make arsenic electrically inactive and, in the case of As2 clusters, compete with active arsenic in the capture of mobile AsV pairs. As2 competition plays a role in Fig. 3a whereas electrical deactivation dominates in Figs. 3b and c. In particular, Fig. 3c corresponds to the same implant conditions as in Fig. 4 but for lower annealing temperature (750 °C) and, hence, As4V dissolution is slow and the active arsenic concentration remains below 1020 cm−3.

As it can be seen in the examples above, the agreement to the experimental data obtained with the model is quite satisfactory. Although we cannot conclusively exclude the possibility of getting a good fitting without the use of mobile As2V clusters, we find that they help very much to fit high active concentration profiles (such as those of Fig. 4) keeping a good agreement to the other experimental results. Moreover, ab initio calculations [5, 6] support the physical base of the model used in this work. In addition, the analysis of Figs. 3 and 4 indicates the close interaction between the As2V mobile cluster model and the model for immobile AsmVn and AsmIn clusters formation and dissolution.

In summary, we have analysed and implemented a physically based arsenic diffusion model including mobile AsDV clusters. A relevant role of these mobile clusters is found in the simulation of arsenic diffusion profiles in n-type heavily doped silicon.

Acknowledgements

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Model
  5. 3 Simulation results
  6. Acknowledgements
  7. References

I. M.-B. acknowledges funding of the project MASTIC (PCIG-GA-2011-293783) by the Marie Curie Actions Grant FP7-PEOPLE-2011-CIG program. P. C. acknowledges funding from the Spanish Government (grant no. TEC2008-05301). Support from Synopsys Inc. is also acknowledged by I. M.-B. and P. C.

References

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Model
  5. 3 Simulation results
  6. Acknowledgements
  7. References