Molecular dynamics analysis on diffusion of point defects

Journal of Crystal Growth 210 (2000) 54}59

Molecular dynamics analysis on di!usion of point defects K. Kakimoto*, T. Umehara, H. Ozoe Institute of Advanced Materials Study, Kyushu University, 6-1, Kasuga-Koen, Kasuga 816-8580, Japan

Abstract Molecular dynamics simulation was carried out to estimate di!usion constants and mechanism of point defects such as a single vacancy and a self-interstitial atom under hydrostatic pressure. The Stillinger}Weber potential [1] was used as a model potential, which is widely accepted for modeling of silicon crystals and melts. We obtained the following results on a self-interstitial atom from the calculation. (1) Di!usion constants of self-interstitial are almost independent of pressure in the range from !50 to #50 kbar. (2) A self-interstitial atom di!uses with the formation of dumbbell structure, which is aligned in [1 1 0] direction. For single vacancy, the following clari"ed. (1) Di!usion constants of vacancy are also independent of pressure in the range from !40 to #40 kbar. (2) A vacancy di!uses with a switching mechanism to the nearest-neighbor atoms in lattice site. ( 2000 Elsevier Science BV. All rights reserved. PACS: 68.35.!p; 68.60.!p Keywords: Molecular dynamics; Point defects; Pressure

1. Introduction Control of point defect distribution in growing crystals of silicon is a key to overcome the breakdown problems in oxide layers of "eld e!ect transistors [FETs]. Although requirement of large diameter crystals increases, it is some times hard to control distributions of point defects such as vacancies and interstitial atoms. Furthermore, it is hard to estimate how point defects of vacancies and interstitial atoms distribute in the crystals during growth. This is mainly attributed to unknown factors such as transport mechanism and thermo-

* Corresponding author. Tel.: #81-92-583-7836; fax: #8192-583-7838. E-mail address: [email protected] (K. Kakimoto)

dynamical data of di!usion constants of the defects, although scattered di!usion constants of the defects have been reported so far [2}16]. Brown et al. reported di!usion constants of defects using molecular dynamics simulation based on constant volume scheme [2]. Furthermore, Unger et al. also reported di!usion constants of an interstitial atom and a vacancy by calculation of Gibbs free energy of silicon system using Monte Carlo and molecular dynamics techniques under a condition of constant volume [3]. Since the diameter of the crystal increases, large stress might be accumulated in the crystals during growth due to temperature distribution in growing crystals [17]; however, there are no reports as to how the pressure a!ects the di!usion mechanism and/or equilibrium concentrations of the defects.

0022-0248/00/$ - see front matter ( 2000 Elsevier Science BV. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 9 9 ) 0 0 6 4 6 - 6

K. Kakimoto et al. / Journal of Crystal Growth 210 (2000) 54}59

This paper aims to clarify the e!ects of pressure on mechanism of di!usion and di!usion constants of defects by using molecular dynamics simulation based on constant pressure algorithm. This paper especially focus on the e!ects of positive and negative pressures on di!usion constants of the defects, since both positive and negative pressures are always accumulating in crystals during growth.

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step; therefore, constant pressure conditions can be obtained during the calculation. Di!usion constants of vacancy and interstitial atom were calculated by

T

U

1 1 N D"lim + DR (t)!R (0)D2 , i i 6t N t?= i/1

(2)

where N is set to one, since only one atom is migrating during the calculation.

2. Calculation 3. Di4usion constants and di4usion mechanism Molecular dynamics simulation was carried out in the following scheme. Atom positions, volume and velocity were revised in each time-step in the calculation using Gear's "fth-order algorithm [18]. Predictor}collector and book keeping methods were adopted to reduce iterative error and enhance calculation speed, respectively. Total number of silicon atoms used in the calculation was 64 and 215 for the case of a single vacancy, while it was 65 and 217 for the case of an interstitial atom. Duration time of the present calculation was set to 0.3]10~9 s, which corresponds to 1]106 calculation-steps to obtain di!usion constants, statically [2]. Temperature of the system was controlled by momentum scaling method. Since hydrostatic pressure was applied to the system in the study, isotropic deformation was obtained in the calculation. Virial numbers of the system were calculated in each time-step from summation of two-and threebody forces. Subsequently, the numbers were converted to pressure of the system in each time-step using the Virial theory expressed by

T T

U UB

1 1 P" Nk ¹# + r )F i i < B 3< i N 1 " k ¹# + r )F , (1) B i i < 3N i where <, r , k , ¹ and F are the volume of the i B i system, the positions vector of atom i, the Boltzmann constant, the temperature and force of the ith atom, respectively. The present calculation employed a constant pressure algorithm based on Anderson's method [19]. Volume of the system was revised using the calculated pressure in each time-

A

Figs. 1(a) and (b) show mean-square displacements of the atoms in the calculating system as a function of duration time, which correspond to a term of summation in Eq. (2) with an interstitial atom and the a vacancy, respectively. Temperature and pressure were "xed at 1600 K and 0 kbar for Figs. 1(a) and (b), respectively. Di!usion constants of an interstitial atom and a vacancy in the system could be calculated from a gradient of mean square

Fig. 1. Mean-square displacements of silicon atoms as a function of time. These systems include an interstitial atom (a) and a vacancy (b).

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K. Kakimoto et al. / Journal of Crystal Growth 210 (2000) 54}59

Fig. 3. Activation energy of di!usion constants of vacancy as a function of hydrostatic pressure.

Fig. 2. Arrhenius plots of an interstitial atom (a) and a vacancy (b) under various hydrostatic pressures.

displacements shown in Figs. 1(a) and (b) by using Eq. (2). Figs. 2(a) and (b) show the Arrhenius plots of calculated-di!usion constants of an interstitial atom and the vacancy as a function of external pressure. The pressure ranged from !40 to #40 kbar for interstitial atom, and from !50 to #50 kbar for vacancy, respectively. N in the "gures shows total number of atoms in the system. The results indicate that calculated di!usion constants of interstitial atom and vacancy are almost independent of external hydrostatic pressure. Consequently, activation energy of a vacancy can be evaluated from Fig. 2(b) as 0.4$0.04 eV in the range from !40 to #40 kbar which is shown in Fig. 3. Furthermore, activation energy of an interstitial atom can be obtained from Fig. 2(a) as 0.78$0.08 eV, which is also almost constant in the pressure range from !50 to #50 kbar.

Figs. 4(a) and (b) show sequences of atom con"gurations including one migrating interstitial atom at a temperature of 1600 K which are observed from both [1 1 0] and [1 1 0] directions, respectively. We can recognize that the dumbbell structure migrates downward. Fig. 5 shows schematic sequence of migration of an interstitial atom. When one additional atom labeled A was inserted in a perfect diamond lattice of silicon, the added atom formed a dumbbell pair with the nearest-neighbor atom B, which is parallel to [1 1 0] in the "rst stage of migration shown in Fig. 5(a). Subsequently, one of the atoms B in the dumbbell pair migrates to [1 1 0] direction, while the other atom A was inserted in lattice site shown in Fig. 5(b). Finally, the migrating atom B pushed out the other two atoms of C and D. The two atoms formed the new dumbbell pair. Thus, the result indicates that the migration of interstitial atom is estimated to be based on interstitialcy mechanism. This phenomenon could be observed under the pressure range from !50 to #50 kbar, although the phenomenon has been reported elsewhere under the condition of constant volume [2]. Figs. 6(a) and (b) show the present results and the published data of di!usion constants of interstitial atoms and vacancies, respectively. Open circles and dashed lines represent the present results, while published data are listed at the bottom of the "gures. Reported di!usion constants of vacancy rather converged into the present result except for several experimental data, while the values of interstitial atoms diverged in the range from 2]10~6 to

K. Kakimoto et al. / Journal of Crystal Growth 210 (2000) 54}59

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Fig. 4. Di!usion sequence of an interstitial silicon atom. Time duration from (a) to (b) is 6.72 ps.

2]10~3 cm/s. While quantitative discussion is required concerning di!usion constant of an interstitial atom, interstitialcy mechanism, switching mechanism, might exist in silicon system at high temperature even under hydrostatic pressure (Fig. 7).

4. Conclusion We obtained the following results on self-interstitial atoms from the calculation. (1) Di!usion constants of self-interstitial are almost independent of pressure in the range from !40 to #40 kbar.

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K. Kakimoto et al. / Journal of Crystal Growth 210 (2000) 54}59

Fig. 5. Schematic sequence of an interstitial atom.

Fig. 7. Di!usion constants of vacancy with several data reported elsewhere [1}16].

Acknowledgements This work was conducted as JSPS Research for the Future Program in the Area of Atomic-Scale Surface and Interface Dynamics. The New Energy and Industrial Technology Development Organization (NEDO) through the Japan Space Utilization Promotion Center (JSUP) supported part of this work. This work was partially carried out under the support of Grant-in Aid to the Science Research by the Ministry of Education, Science and Culture. Fig. 6. Di!usion constants of an interstitial atom with several data reported elsewhere [1}16].

(2) Self-interstitial atoms di!use with the formation of S1 1 0T dumbbells in diamond lattice structure. Therefore, switching of atoms between an additionally inserted interstitial-atom and host crystal might be the main mechanism of interstitial di!usion even under hydrostatic pressure. For the vacancy case, di!usion constants are almost independent of the pressure.

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