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Theory of statistical dislocation dynamics in photomechanical effects
Volume
108A, number 5,6
PHYSICS LETTERS
THEORY OF STATISTICAL DISLOCATION IN PHOTOMECHANICAL EFFECTS
8 April 1985
DYNAMICS
T. HAGIHARA a, Y. HAYASHIUCHI b and T. OKADA b a Department of Physics, Osaka Kyoiku University, Tennoji, Osaka 543, Japan b The Institute of Scientific and Industrial Research, 8 - 1, Mihogaoka, Ibaraki, Osaka 567, Japan Received 20 December
1984
A simple theory is proposed for the photoplastic effect in alkali halides containing F-centers based on statistical dislocation dynamics. The present theory depicts well several dependences of flow stress increment on, e.g., defect concentration, strain rate and light intensity.
Numerous investigations concerning photomechanical effects in various materials, such as semiconductors [ 1,2] , organic crystals [3] and alkali halides containing the F-center [4-81 have been reported so far. In many alkali halides containing the F-center Nadeau [4] has initiated the observation of the photoplastic effect, PPE, and pointed out that there exist two types of PPE, i.e., a flow stress increment during optical excitation of the F-center (positive PPE) and vice versa. He proposed that the positive PPE may give rise to the electrostatic interaction between the F--center [9] and the charged dislocation [lo] . Cabrera and Agull& L6pez [5] have performed active as well as detailed experiments on the various factors affecting the development of the PPE and have proposed an empirical relation regarding the flow stress increment, Au, during light illumination, as Au=
‘1
1 + c$u/I ’ where v is the crosshead speed, I the intensity of illuminating light and C,,C’, are constants. But, decisive experiments as well as comprehensive theories based on dislocation dynamics [ 1 l-131 giving systematic understanding as to PPE have scarcely been done, so far. In order to understand the mechanism of PPE, the interaction of moving dislocations with relevant point defects as well as of the point defects with photons should be clarified. In this paper we propose a simple theory regarding 0.375-9601/85/$ 03.300 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
PPE in terms of statistical dislocation dynamics. The motion of dislocations interacting with the optical sensitive defects is described by a master equation [ 141. Introducing several characteristic interaction times, we extend the theory to the photomechanical effects in alkali halides containing F-centers, of which photochemical reactions, such as, 2F * F- + F+, are expressed by a simple rate equation. The characteristic behavior of PPE is discussed on the basis of the newly proposed theoretical relations. When dislocations move in the lattice field, they encounter the point defects and experience a resistive force, which brings about the flow stress increment in the constant strain-rate system. Since the Interaction between the dislocations and the point defects takes place at random, the motion of dislocations becomes random. We are now interested in the average motion of many dislocations. This may be described by the motion of a small segment on a dislocation line, provided that the motions of dislocations are independent of each other and segments distribute uniformly on the dislocation line. A useful equation to describe such a random motion of the segment is a master equation [ 141 of the form dpildt= C(PjW, -PiWii>o
i
(2)
Here Piis the probability of the segment to be in state Wiidenotes the transition rate of the segment
i and
263
Volume 108A, number 5,6
PHYSICS LETTERS
from state i to statej, which depends on the segment positions x i, x i. We calculate the average velocity v of the segment. Differentiating the average position Y, ixiei with respect to time, we have the statistical expression of o of the form ~,#(x i - x/) W#Pi. In particular, if the segment can move a distance ;~ in one jump, this reduces to
o = G/;K(WI+ - Wj-)Pj = ~j oyPI,
(3)
where Wj+, Wi_ are the forward and backward displacement rates from the position x/, respectively, and we put vj = ~(Wj+ - Wj_ ). When the dislocation segment moves in the lattice, the segment encounters potential barrier due to either pure Peierls or point defects. The sum of eq. (3) can then be divided into two terms, i.e., j = rr for xTr in the Peierls potential and j = a for x a in the point defect potential. For a system containing several kinds of point defects, eq. (3) becomes
8 April 1985
only the change in Pa due to illumination. Since the dislocation segment moves a restricted distance k in one jump, the motion of Pa can be described by a rate equation of the form
We here p u t
=
1/W, and
o~r = X(W~r+ - Wrr_) ~ o c exp [-(U~r -
V~ro)/kT],
(5)
where o c is constant and V~r is the activation volume for the Peierls potential. Inserting eq. (5) into eq. (4), we obtain a simple relationship between o and Pa:
V~r°= U~ + kT[ln(°/°c)-in( 1 - ~ P~)I ~ UTr + kT[in(o/oc) + ~ Pc~],
(6)
provided the probability Pa of the segment to be trapped by point defects is small. Light illumination on the system makes the states of optical sensitive point defects change, which brings about a change in o through the dislocation-point-defect interactions. As the increment of mobile dislocations density, p, will be small during the illumination, the dislocation velocity u in eq. (6) is expected to be constant on the basis of the relation ~ = boo [11], and thus we may consider 264
=
and G
have the meanings of inter-trapping and pinning times of dislocations moving on a glide plane, respectively. T t is inversely proportional to the surface concentration ×a of a defect a on the glide plane. If we introduce an effective interaction length regarding the defect a and denote it by ~a, 1/~ay,a is the mean path of the segment moving between the successive trapping at the defect a. T t is then given by 1/v~aXa, where b is the Burgers length. Assuming that ~a is independent of the kind of a, we write ~ for ~a for simplicity. The rate equation for Pa is then
dP(~/dt ~ lIT t -PJT~p ~ v~Xa -
where we assume that the forward velocity in the point defect potential, o~, is much smaller than the velocity in the Peiefls potential, on. These velocities depend on the applied flow stress a and temperature T. Taking account of the thermal activation process [12,13] in particular, one gets
(7)
dP ~ l d t = P,,W,,~ - P~ W~,, .
p cJTp.
(8)
We distinguish the states of point defects by o~, i.e., we consider that the defects in different states belong to different kinds. The optical excitation then makes Xa change through the photon-defect interaction. Therefore combining eqs~ (6) and (8), we can calculate the flow stress change Ao due to optical excitation. In order to obtain more realistic information on/Xo we will apply the theory to the system of photomechanical effects in alkali halides containing the F-centers as below. By optical excitation at around RT, the Fcenter converts into F - - and F+-centers, such as T" 2F ~ F - + F +, T'
(9)
where T" and T' are time constants for respectively excitation and lifetime of F - - as well as F÷-centers. For simpficity, we label the F-, F - - and F+-centers by t~ = F, - , +, respectively. For the photochemical reaction (7), ×_ for the F--center approximately satisfies the rate equation
d×_ Idt = - x _ / T ' + XFIT" - × _ / T ' + [×F(0) -- 2×_1 IT".
(10)
Solving eqs. (8), (10) and inserting the solution of Pa into eq. (6), we obtain the Ao during/after the optical excitation with sufficient long duration At as follows,
Volume 108A, number 5,6
PHYSICS LETTERS
8 April 1985
Table 1 for
ao _ 1 - exp(-t/Tp), Aomax =
exp [- (t - At)lr
The strain rate dependence on Aamax in colored KCI at 166 K.
0 ~
l
exp [ - (t - At)/T'] - exp [ - (t - At)/Tp ] +__ for
t1>At,
(11)
~(s-1 )
AOmax(MPa)
3.7 × 10-s 9.3 × 10-s
0.066 0.14
(12) and (13), the C-dependence of AOmax, i.e., Ao/Aomax = C2/3,
where Aomax is defined as VTrA Omax = 2k TO~XF(0) Tp
=C2/3/(1 + A C ) ,
(12)
2 + r"/r' It is clear from eq. (11) that Ao increases with time constant T~ and the decrease of Ao after ceasing the optical excitation is governed by the decay constants T' and T - , which is the very case for the PPE. We h P e derived the fundamental relations for Ao on PPE. In particular, the formulae (12) can well explain the several empirical relations on maximum flow stress increment in PPE. These comparisons are now given. (1) Dependence on light intensity. In eq. (12) T" depends on the photon flux or intensity I for the excitation of F-centers. If we denote the cross-sectional area of the F-center for the optical excitation by s, the photon flux at depth x in the specimen reduces to Ie - s e x , C being the initial concentration of the Fcenter. The inverse of T" for the specimen with thickness D is then given by 1IT" = (I/CD) (1 - e-sCD),
(13)
which is proportional to 1. Hence we have, from eqs. (12) and (13),
A°max
=
A1 1 + A 2/1 '
(14)
where A 1, A 2 are constants. This formula, as a function of I, is consistent with the empirical relation (1). (2) Concentration dependence. When the concentration of the F-center, C, is varied under the condition of constant temperature and light intensity I, Aomax will change with C, since in eq. (12) the F-center concentration XF(0) on the glide plane is approximately C 2/3 and T" depends on C as given by eq. (13). If T' is independent of C, we have, with the help of eqs.
for
C ~ 1]sD,
for
C>> I/sD,
(15)
A being constant. Aomax then increases as C 2/3 for small C but decreases as C -1/3 for large C. (3) Strain rate dependence. For the system under constant strain rate ~, the dislocation velocity o is related to ~ as ~ = bpo [11]. Hence from eq. (12), one obtains the proportionality relation, AOmax ~ ~.
(16)
In deriving this relation, we have not considered the o-dependence of T~-, although physically T~ will decrease with increas'mg o. Table 1 shows a tyf~ical example ofAomax observed in KC1 under different b, which is in good agreement with the theoretical prediction from relation (16). This result is, however, not consistent with the data obtained by Cabrera and Agull6L6pez [5]. As pointed out in a previous paper [7], AOmax depends on the amount of plastic strain. Hence, the evaluation of the AOmax should be done at the same strain level, but this condition is rather difficult to arrange in the course of various strain-rate measurements. To explain present discrepancy more sophisticated experiments are required. We do not give further comments about this point. (4) Temperature dependence. It is generally accepted that the temperature plays a significant role in the dislocation-point-defect interaction through the thermal activation process [12,13] as well as in the optical conversion of the F-center [8,15]. Eq. (12) shows that AOmax is determined by the parameters T', Tp and VTr,which depend heavily on temperature. Tlierefore, the temperature dependence of these parameters should be clarified in detail. Although the present theory was concerned with PPE and PPAE where the photochemical conversion of the F-centers into F - - and F +-centers occurs only above a characteristic transition temperature as reported 265
Volume 108A, number 5,6
PHYSICS LETTERS
previously [ 8 ] , this theoretical approach and results are generally applicable to p h o t o m e c h a n i c a l effects where light illumination causes conversion o f optical sensitive defects that play a main role to result in the flow stress i n c r e m e n t or d e c r e m e n t .
References [ 1] Yu.A. Osip'yan and I.A. Savchenko, JETP Lett. 7 (1968) 100. [2] K. Nakagawa, K. Maeda and S. Tekeuchi, J. Phys. Soc. Japan 48 (1980) 2173. [3] K. Kojima, Appl. Phys. Lett. 38 (1981) 530. [4] J.S. Nadeau, J. Appl. Phys. 35 (1964) 669. [51 J.M. Cahrera and F. AguU6-L6pez, J. Appl. Phys. 45 (1974) 1013. [6] E.M. Korovkin and Ya.M. Soifer, Soy. Phys. Solid State 18 (1976) 1435.
266
8 April 1985
[7] Y. Inoue, T. Okada and T. Hagihara, Tech. Rep. Osaka Univ. 29 (1979) 33. [8] T. Hagihara, Y. Inoue and T. Okada, Phys. Lett. 88A (1983) 191. [9] W.B. Fowler, Physics of color centers (Academic Press, New York, 1968). [10] R.W. Whitworth, Adv. Phys. 24 (1975) 203. [11] J.J. Gilman and W.G. Johnston, in: Solid state physics Vol. 13 (Academic Press, New York, 1962) p. 147. [ 12 ] J.C.M. Li, in: Dislocation dynamics, eds. A.R. Rosenfield, G.T. Hahn, A.L. Bement and R.I. Jaffe (McGraw-Hill, New York, 1968). [ 13 ] G. Schoek, in: Dislocations in solids, ed. F.R.N. Nabarro, Vol. 3 (North-Holland, Amsterdam, 1980) p. 61. [14] A.T. Bharueha-Reid, Elements of the theory of Markov processes and their application (McGraw-Hill, New York, 1960). [ 15 ] J.J. Markham, F-centers in alkali halides, Solid state physics, Suppl. 8 (Academic Press, New York, 1966).
108A, number 5,6
PHYSICS LETTERS
THEORY OF STATISTICAL DISLOCATION IN PHOTOMECHANICAL EFFECTS
8 April 1985
DYNAMICS
T. HAGIHARA a, Y. HAYASHIUCHI b and T. OKADA b a Department of Physics, Osaka Kyoiku University, Tennoji, Osaka 543, Japan b The Institute of Scientific and Industrial Research, 8 - 1, Mihogaoka, Ibaraki, Osaka 567, Japan Received 20 December
1984
A simple theory is proposed for the photoplastic effect in alkali halides containing F-centers based on statistical dislocation dynamics. The present theory depicts well several dependences of flow stress increment on, e.g., defect concentration, strain rate and light intensity.
Numerous investigations concerning photomechanical effects in various materials, such as semiconductors [ 1,2] , organic crystals [3] and alkali halides containing the F-center [4-81 have been reported so far. In many alkali halides containing the F-center Nadeau [4] has initiated the observation of the photoplastic effect, PPE, and pointed out that there exist two types of PPE, i.e., a flow stress increment during optical excitation of the F-center (positive PPE) and vice versa. He proposed that the positive PPE may give rise to the electrostatic interaction between the F--center [9] and the charged dislocation [lo] . Cabrera and Agull& L6pez [5] have performed active as well as detailed experiments on the various factors affecting the development of the PPE and have proposed an empirical relation regarding the flow stress increment, Au, during light illumination, as Au=
‘1
1 + c$u/I ’ where v is the crosshead speed, I the intensity of illuminating light and C,,C’, are constants. But, decisive experiments as well as comprehensive theories based on dislocation dynamics [ 1 l-131 giving systematic understanding as to PPE have scarcely been done, so far. In order to understand the mechanism of PPE, the interaction of moving dislocations with relevant point defects as well as of the point defects with photons should be clarified. In this paper we propose a simple theory regarding 0.375-9601/85/$ 03.300 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
PPE in terms of statistical dislocation dynamics. The motion of dislocations interacting with the optical sensitive defects is described by a master equation [ 141. Introducing several characteristic interaction times, we extend the theory to the photomechanical effects in alkali halides containing F-centers, of which photochemical reactions, such as, 2F * F- + F+, are expressed by a simple rate equation. The characteristic behavior of PPE is discussed on the basis of the newly proposed theoretical relations. When dislocations move in the lattice field, they encounter the point defects and experience a resistive force, which brings about the flow stress increment in the constant strain-rate system. Since the Interaction between the dislocations and the point defects takes place at random, the motion of dislocations becomes random. We are now interested in the average motion of many dislocations. This may be described by the motion of a small segment on a dislocation line, provided that the motions of dislocations are independent of each other and segments distribute uniformly on the dislocation line. A useful equation to describe such a random motion of the segment is a master equation [ 141 of the form dpildt= C(PjW, -PiWii>o
i
(2)
Here Piis the probability of the segment to be in state Wiidenotes the transition rate of the segment
i and
263
Volume 108A, number 5,6
PHYSICS LETTERS
from state i to statej, which depends on the segment positions x i, x i. We calculate the average velocity v of the segment. Differentiating the average position Y, ixiei with respect to time, we have the statistical expression of o of the form ~,#(x i - x/) W#Pi. In particular, if the segment can move a distance ;~ in one jump, this reduces to
o = G/;K(WI+ - Wj-)Pj = ~j oyPI,
(3)
where Wj+, Wi_ are the forward and backward displacement rates from the position x/, respectively, and we put vj = ~(Wj+ - Wj_ ). When the dislocation segment moves in the lattice, the segment encounters potential barrier due to either pure Peierls or point defects. The sum of eq. (3) can then be divided into two terms, i.e., j = rr for xTr in the Peierls potential and j = a for x a in the point defect potential. For a system containing several kinds of point defects, eq. (3) becomes
8 April 1985
only the change in Pa due to illumination. Since the dislocation segment moves a restricted distance k in one jump, the motion of Pa can be described by a rate equation of the form
We here p u t
=
1/W, and
o~r = X(W~r+ - Wrr_) ~ o c exp [-(U~r -
V~ro)/kT],
(5)
where o c is constant and V~r is the activation volume for the Peierls potential. Inserting eq. (5) into eq. (4), we obtain a simple relationship between o and Pa:
V~r°= U~ + kT[ln(°/°c)-in( 1 - ~ P~)I ~ UTr + kT[in(o/oc) + ~ Pc~],
(6)
provided the probability Pa of the segment to be trapped by point defects is small. Light illumination on the system makes the states of optical sensitive point defects change, which brings about a change in o through the dislocation-point-defect interactions. As the increment of mobile dislocations density, p, will be small during the illumination, the dislocation velocity u in eq. (6) is expected to be constant on the basis of the relation ~ = boo [11], and thus we may consider 264
=
and G
have the meanings of inter-trapping and pinning times of dislocations moving on a glide plane, respectively. T t is inversely proportional to the surface concentration ×a of a defect a on the glide plane. If we introduce an effective interaction length regarding the defect a and denote it by ~a, 1/~ay,a is the mean path of the segment moving between the successive trapping at the defect a. T t is then given by 1/v~aXa, where b is the Burgers length. Assuming that ~a is independent of the kind of a, we write ~ for ~a for simplicity. The rate equation for Pa is then
dP(~/dt ~ lIT t -PJT~p ~ v~Xa -
where we assume that the forward velocity in the point defect potential, o~, is much smaller than the velocity in the Peiefls potential, on. These velocities depend on the applied flow stress a and temperature T. Taking account of the thermal activation process [12,13] in particular, one gets
(7)
dP ~ l d t = P,,W,,~ - P~ W~,, .
p cJTp.
(8)
We distinguish the states of point defects by o~, i.e., we consider that the defects in different states belong to different kinds. The optical excitation then makes Xa change through the photon-defect interaction. Therefore combining eqs~ (6) and (8), we can calculate the flow stress change Ao due to optical excitation. In order to obtain more realistic information on/Xo we will apply the theory to the system of photomechanical effects in alkali halides containing the F-centers as below. By optical excitation at around RT, the Fcenter converts into F - - and F+-centers, such as T" 2F ~ F - + F +, T'
(9)
where T" and T' are time constants for respectively excitation and lifetime of F - - as well as F÷-centers. For simpficity, we label the F-, F - - and F+-centers by t~ = F, - , +, respectively. For the photochemical reaction (7), ×_ for the F--center approximately satisfies the rate equation
d×_ Idt = - x _ / T ' + XFIT" - × _ / T ' + [×F(0) -- 2×_1 IT".
(10)
Solving eqs. (8), (10) and inserting the solution of Pa into eq. (6), we obtain the Ao during/after the optical excitation with sufficient long duration At as follows,
Volume 108A, number 5,6
PHYSICS LETTERS
8 April 1985
Table 1 for
ao _ 1 - exp(-t/Tp), Aomax =
exp [- (t - At)lr
The strain rate dependence on Aamax in colored KCI at 166 K.
0 ~
l
exp [ - (t - At)/T'] - exp [ - (t - At)/Tp ] +__ for
t1>At,
(11)
~(s-1 )
AOmax(MPa)
3.7 × 10-s 9.3 × 10-s
0.066 0.14
(12) and (13), the C-dependence of AOmax, i.e., Ao/Aomax = C2/3,
where Aomax is defined as VTrA Omax = 2k TO~XF(0) Tp
=C2/3/(1 + A C ) ,
(12)
2 + r"/r' It is clear from eq. (11) that Ao increases with time constant T~ and the decrease of Ao after ceasing the optical excitation is governed by the decay constants T' and T - , which is the very case for the PPE. We h P e derived the fundamental relations for Ao on PPE. In particular, the formulae (12) can well explain the several empirical relations on maximum flow stress increment in PPE. These comparisons are now given. (1) Dependence on light intensity. In eq. (12) T" depends on the photon flux or intensity I for the excitation of F-centers. If we denote the cross-sectional area of the F-center for the optical excitation by s, the photon flux at depth x in the specimen reduces to Ie - s e x , C being the initial concentration of the Fcenter. The inverse of T" for the specimen with thickness D is then given by 1IT" = (I/CD) (1 - e-sCD),
(13)
which is proportional to 1. Hence we have, from eqs. (12) and (13),
A°max
=
A1 1 + A 2/1 '
(14)
where A 1, A 2 are constants. This formula, as a function of I, is consistent with the empirical relation (1). (2) Concentration dependence. When the concentration of the F-center, C, is varied under the condition of constant temperature and light intensity I, Aomax will change with C, since in eq. (12) the F-center concentration XF(0) on the glide plane is approximately C 2/3 and T" depends on C as given by eq. (13). If T' is independent of C, we have, with the help of eqs.
for
C ~ 1]sD,
for
C>> I/sD,
(15)
A being constant. Aomax then increases as C 2/3 for small C but decreases as C -1/3 for large C. (3) Strain rate dependence. For the system under constant strain rate ~, the dislocation velocity o is related to ~ as ~ = bpo [11]. Hence from eq. (12), one obtains the proportionality relation, AOmax ~ ~.
(16)
In deriving this relation, we have not considered the o-dependence of T~-, although physically T~ will decrease with increas'mg o. Table 1 shows a tyf~ical example ofAomax observed in KC1 under different b, which is in good agreement with the theoretical prediction from relation (16). This result is, however, not consistent with the data obtained by Cabrera and Agull6L6pez [5]. As pointed out in a previous paper [7], AOmax depends on the amount of plastic strain. Hence, the evaluation of the AOmax should be done at the same strain level, but this condition is rather difficult to arrange in the course of various strain-rate measurements. To explain present discrepancy more sophisticated experiments are required. We do not give further comments about this point. (4) Temperature dependence. It is generally accepted that the temperature plays a significant role in the dislocation-point-defect interaction through the thermal activation process [12,13] as well as in the optical conversion of the F-center [8,15]. Eq. (12) shows that AOmax is determined by the parameters T', Tp and VTr,which depend heavily on temperature. Tlierefore, the temperature dependence of these parameters should be clarified in detail. Although the present theory was concerned with PPE and PPAE where the photochemical conversion of the F-centers into F - - and F +-centers occurs only above a characteristic transition temperature as reported 265
Volume 108A, number 5,6
PHYSICS LETTERS
previously [ 8 ] , this theoretical approach and results are generally applicable to p h o t o m e c h a n i c a l effects where light illumination causes conversion o f optical sensitive defects that play a main role to result in the flow stress i n c r e m e n t or d e c r e m e n t .
References [ 1] Yu.A. Osip'yan and I.A. Savchenko, JETP Lett. 7 (1968) 100. [2] K. Nakagawa, K. Maeda and S. Tekeuchi, J. Phys. Soc. Japan 48 (1980) 2173. [3] K. Kojima, Appl. Phys. Lett. 38 (1981) 530. [4] J.S. Nadeau, J. Appl. Phys. 35 (1964) 669. [51 J.M. Cahrera and F. AguU6-L6pez, J. Appl. Phys. 45 (1974) 1013. [6] E.M. Korovkin and Ya.M. Soifer, Soy. Phys. Solid State 18 (1976) 1435.
266
8 April 1985
[7] Y. Inoue, T. Okada and T. Hagihara, Tech. Rep. Osaka Univ. 29 (1979) 33. [8] T. Hagihara, Y. Inoue and T. Okada, Phys. Lett. 88A (1983) 191. [9] W.B. Fowler, Physics of color centers (Academic Press, New York, 1968). [10] R.W. Whitworth, Adv. Phys. 24 (1975) 203. [11] J.J. Gilman and W.G. Johnston, in: Solid state physics Vol. 13 (Academic Press, New York, 1962) p. 147. [ 12 ] J.C.M. Li, in: Dislocation dynamics, eds. A.R. Rosenfield, G.T. Hahn, A.L. Bement and R.I. Jaffe (McGraw-Hill, New York, 1968). [ 13 ] G. Schoek, in: Dislocations in solids, ed. F.R.N. Nabarro, Vol. 3 (North-Holland, Amsterdam, 1980) p. 61. [14] A.T. Bharueha-Reid, Elements of the theory of Markov processes and their application (McGraw-Hill, New York, 1960). [ 15 ] J.J. Markham, F-centers in alkali halides, Solid state physics, Suppl. 8 (Academic Press, New York, 1966).