- Email: [email protected]
Migration and trapping of defects in a cubic lattice
Diamond and Related Materials 7 (1998) 1257–1258
Letter
Migration and trapping of defects in a cubic lattice A.T. Collins * Wheatstone Physics Laboratory, King’s College London, Strand, London, WC2R 2LS, UK Received 22 April 1998; accepted 2 June 1998
Abstract A computer simulation has been carried out, using Monte Carlo methods, to determine the probability of trapping for a mobile defect in a simple cubic lattice. The analysis shows that the probability P(n) of a defect reaching a trap after a random walk of n steps is P(n)=C ln(2) exp[−nC ln(2)], where C is the concentration of traps. © 1998 Elsevier Science S.A. Keywords: Cubic lattice; Defects; Radiation damage; Computer simulation;
1. Introduction This investigation was carried out to support annealing studies of radiation damage in diamond [1]. The subject of that work was the migration of a mobile species to a random distribution of fixed trapping centres. Defects in a crystalline solid can experience thermally activated migration, with a probability, g(T ), at temperature T of g(T )=A exp(−E /k T), (1) a B where E is the activation energy, and k is Boltzmann’s a B constant [2]. The pre-exponential constant A is the attempt-to-escape frequency, and is normally equated with a typical phonon frequency (~4×1013 Hz for diamond). Here, we might be considering an interstitial jumping to an equivalent location at an adjacent lattice site [1], or a vacancy jumping to an adjacent lattice site [2]. The defect will continue making thermally activated jumps until it reaches a trapping site, and the question arises: ‘‘If the defect moves through the crystal by jumping randomly from one lattice site to an adjacent lattice site, how is the average number of jumps related to the concentration of traps?’’ The author was not aware of any analytical treatment of this problem, and therefore, a computer simulation was performed. To simplify the analysis, without losing * Fax: +44 171 8732160; e-mail: [email protected] 0925-9635/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S 09 2 5 -9 6 3 5 ( 9 8 ) 0 0 22 0 - 9
any insight into the mechanism, the simulation was carried out for a simple cubic lattice. We will consider the case where the concentration of traps is much larger than the concentration of migrating defects, and so the former may be regarded as remaining approximately constant.
2. Procedure The simulation was performed using a simple QuickBASIC program. A cubic array with x, y and z values ranging from 0 to 30 was set up in memory, and 149 (approximately one in 200) randomly-chosen array elements were set to unity; the remainder were set to zero. The elements set to unity are the ‘‘trapping centres’’. A ‘‘defect’’ was introduced at (15, 15, 15) and allowed to move randomly with equal probability to one of the six surrounding nearest-neighbour ‘‘lattice sites’’. This random walk was allowed to continue until the defect reached a trapping centre, and the total number of steps taken was then noted. Periodic boundary conditions were imposed so that, if the defect tried to move to an x, y or z coordinate of −1 or 31, that coordinate wrapped round to 30 or 0, respectively. After the defect reached the trap, a new distribution of trapping centres was set up and the process repeated. To obtain reasonable statistics, the program looped round 10 000 times, after which a histogram could be
1258
A.T. Collins / Diamond and Related Materials 7 (1998) 1257–1258
plotted showing the number of trapping events as a function of the number of steps taken. The program was then re-run with different concentrations of trapping centres (C=0.01, 0.002 and 0.001). Using a 166-MHz Pentium PC, the shortest and longest runs took 4.5 and 18 min, respectively.
3. Results and discussion Fig. 1 shows the number of trapping events, as a function of the number of steps taken to reach the trap, for C=0.005. To reduce noise, the n-values have been grouped together in blocks of 20. It is clear from
inspection of the figure that the distribution is welldescribed by a decaying exponential function, and that this drops to approximately half of its initial value at n=1/C. This same general behaviour was noted for all the values of C used. We may infer from these simulations that the probability P(n) of trapping after n steps is P(n)=C ln(2) exp[−nC ln(2)].
(2)
When there are a large number of defects migrating and being trapped, the average number N of steps taken is given by
P
N
P
exp[−nC ln(2)]dn=
2
exp[−nC ln(2)]dn,
(3)
0 N which leads to the simple result that
N=1/C.
(4)
The overall probability P(T,C ) that a defect will undergo thermally activated migration at temperature T until it reaches a trap is g(T )/N, and combining the expressions from Eqs. (1) and (4), we may write that P(T,C )=Cn exp(−E /k T ), (5) a B where n is a typical phonon frequency. The validity of this result has been demonstrated by Allers et al. [1] in their study of interstitials migrating to, and annihilating with, a larger concentration of vacancies.
Fig. 1. Number of trapping events T(n), as a function of the number of steps n, taken to reach the trap, for a migrating defect performing a random walk in a random distribution of traps with a concentration of one part in 200. Data are shown for 10 000 events, and the n-values are grouped in blocks of 20. The curve through the data is the leastsquares fit T(n) =705×exp(−0.00358n).
References [1] L. Allers, A.T. Collins, J. Hiscock, Diamond Relat. Mater. 7 (1998) 228–232. [2] G. Davies, S.C. Lawson, A.T. Collins, A. Mainwood, S.J. Sharp, Phys. Rev. B 46 (1992) 13157–13170.
Letter
Migration and trapping of defects in a cubic lattice A.T. Collins * Wheatstone Physics Laboratory, King’s College London, Strand, London, WC2R 2LS, UK Received 22 April 1998; accepted 2 June 1998
Abstract A computer simulation has been carried out, using Monte Carlo methods, to determine the probability of trapping for a mobile defect in a simple cubic lattice. The analysis shows that the probability P(n) of a defect reaching a trap after a random walk of n steps is P(n)=C ln(2) exp[−nC ln(2)], where C is the concentration of traps. © 1998 Elsevier Science S.A. Keywords: Cubic lattice; Defects; Radiation damage; Computer simulation;
1. Introduction This investigation was carried out to support annealing studies of radiation damage in diamond [1]. The subject of that work was the migration of a mobile species to a random distribution of fixed trapping centres. Defects in a crystalline solid can experience thermally activated migration, with a probability, g(T ), at temperature T of g(T )=A exp(−E /k T), (1) a B where E is the activation energy, and k is Boltzmann’s a B constant [2]. The pre-exponential constant A is the attempt-to-escape frequency, and is normally equated with a typical phonon frequency (~4×1013 Hz for diamond). Here, we might be considering an interstitial jumping to an equivalent location at an adjacent lattice site [1], or a vacancy jumping to an adjacent lattice site [2]. The defect will continue making thermally activated jumps until it reaches a trapping site, and the question arises: ‘‘If the defect moves through the crystal by jumping randomly from one lattice site to an adjacent lattice site, how is the average number of jumps related to the concentration of traps?’’ The author was not aware of any analytical treatment of this problem, and therefore, a computer simulation was performed. To simplify the analysis, without losing * Fax: +44 171 8732160; e-mail: [email protected] 0925-9635/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S 09 2 5 -9 6 3 5 ( 9 8 ) 0 0 22 0 - 9
any insight into the mechanism, the simulation was carried out for a simple cubic lattice. We will consider the case where the concentration of traps is much larger than the concentration of migrating defects, and so the former may be regarded as remaining approximately constant.
2. Procedure The simulation was performed using a simple QuickBASIC program. A cubic array with x, y and z values ranging from 0 to 30 was set up in memory, and 149 (approximately one in 200) randomly-chosen array elements were set to unity; the remainder were set to zero. The elements set to unity are the ‘‘trapping centres’’. A ‘‘defect’’ was introduced at (15, 15, 15) and allowed to move randomly with equal probability to one of the six surrounding nearest-neighbour ‘‘lattice sites’’. This random walk was allowed to continue until the defect reached a trapping centre, and the total number of steps taken was then noted. Periodic boundary conditions were imposed so that, if the defect tried to move to an x, y or z coordinate of −1 or 31, that coordinate wrapped round to 30 or 0, respectively. After the defect reached the trap, a new distribution of trapping centres was set up and the process repeated. To obtain reasonable statistics, the program looped round 10 000 times, after which a histogram could be
1258
A.T. Collins / Diamond and Related Materials 7 (1998) 1257–1258
plotted showing the number of trapping events as a function of the number of steps taken. The program was then re-run with different concentrations of trapping centres (C=0.01, 0.002 and 0.001). Using a 166-MHz Pentium PC, the shortest and longest runs took 4.5 and 18 min, respectively.
3. Results and discussion Fig. 1 shows the number of trapping events, as a function of the number of steps taken to reach the trap, for C=0.005. To reduce noise, the n-values have been grouped together in blocks of 20. It is clear from
inspection of the figure that the distribution is welldescribed by a decaying exponential function, and that this drops to approximately half of its initial value at n=1/C. This same general behaviour was noted for all the values of C used. We may infer from these simulations that the probability P(n) of trapping after n steps is P(n)=C ln(2) exp[−nC ln(2)].
(2)
When there are a large number of defects migrating and being trapped, the average number N of steps taken is given by
P
N
P
exp[−nC ln(2)]dn=
2
exp[−nC ln(2)]dn,
(3)
0 N which leads to the simple result that
N=1/C.
(4)
The overall probability P(T,C ) that a defect will undergo thermally activated migration at temperature T until it reaches a trap is g(T )/N, and combining the expressions from Eqs. (1) and (4), we may write that P(T,C )=Cn exp(−E /k T ), (5) a B where n is a typical phonon frequency. The validity of this result has been demonstrated by Allers et al. [1] in their study of interstitials migrating to, and annihilating with, a larger concentration of vacancies.
Fig. 1. Number of trapping events T(n), as a function of the number of steps n, taken to reach the trap, for a migrating defect performing a random walk in a random distribution of traps with a concentration of one part in 200. Data are shown for 10 000 events, and the n-values are grouped in blocks of 20. The curve through the data is the leastsquares fit T(n) =705×exp(−0.00358n).
References [1] L. Allers, A.T. Collins, J. Hiscock, Diamond Relat. Mater. 7 (1998) 228–232. [2] G. Davies, S.C. Lawson, A.T. Collins, A. Mainwood, S.J. Sharp, Phys. Rev. B 46 (1992) 13157–13170.