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Simulation of dislocation source dynamics in silicon for a 60° dislocation
Pergamon PII: S1359-6454(96)00310-2
SIMULATION
Actu ,mzrer. Vol. 45. No. 6. pp. 2639-2653. 1997 c’ 1997 Acta Metallurgica Inc. Published by Elsev~er Science Ltd Printed in Great Britain. All rights reserved 1359-6454 97 $17.00 + 0.00
OF DISLOCATION SOURCE DYNAMICS SILICON FOR A 60” DISLOCATION
IN
T. GEIPEL Institut
fiir Anorganische
Chemie
der UniversitCt
Bonn,
(Received 28 March
Riimerstr.
164, 53117 Bonn.
Germany
1996)
Abstract-A new approach for the simulation of a Frank-Read source in directionally bonded, e.g. covalent, cubic materials based on an f.c.c. lattice is presented [I] (Geipel, Ph.D. Thesis, Case Western Reserve University, 1993). In this algorithm, a segment of a perfect dislocation, initially lying along a (li0) Peierls valley, is pinned at its two ends on a { Ill} glide plane. Under the application of a shear stress. this dislocation first develops into a semi-hexagonal half-loop and then swings around the pinning centers to form a full dislocation loop surrounding a reformed dislocation source. In the present study the temperature has explicitly been included through the thermal activation of kink pairs and their influence on dislocation velocity. One of the conclusions drawn from this work is that the critical stress for a 60 source dislocation in silicon appears to be highly temperature sensitive. f,cs 1997 Acta Mefallurgicu Inc,. Zusammenfassung-Es wird ein neuer Weg vorgestellt zur Simulation der Frank-Read Quelle in Materialien mit gerichteter Bindung, z.B. kovalenten, kubischen Materialien basierend auf einem f.c.c. Gitter. In diesem Algorithmus liegt zunachst ein Segment einer perfekten Versetzung in einem (I i0) Peierls Tal und wird an ihren zwei Enden festgehalten, wobei sie auf einer { 1I I} Gleitebene beweglich ist. Bei Einwirkung einer Scherspannung entwickelt die Versetzung eine (sechseckige) Schleife und wickelt sich dann urn die Pinning-Zentren bis sie eine volle Schleife bildet. Die wichtige Neuerung liegt in der expliziten Beriicksichtigung der Temperatur in Form der thermischen Aktivierung der Kink-Paare und deren Einflul3 auf die Versetzungsgeschwindigkeit. Einer der Schliisse. die gezogen werden kijnnen. besteht darin, daI3 die kritische Scherspannung fiir eine 60” Quellenversetzung stark temperaturabhiingig erscheint.
1. INTRODUCTION
a Frank-Read source in Si ($5). Finally, a few conclusions drawn from the work will be presented
The mechanism proposed by Frank and Read [2] for the continuous multiplication of dislocation loops solved one of the outstanding problems in the application of dislocation theory by explaining the plastic properties of crystalline solids. Ever since the conception of this mechanism, and the direct experimental evidence of its existence by Dash [3], this mechanism has been used to explain numerous phenomena. The first simulations of a Frank-Read (F-R) source appear to have been carried out by Dorn and coworkers [4]. Later, Frost and Ashby [5] considered the effect of drag forces on the operation of the F-R source in metals at 0 K. In this paper, temperature is first incorporated in the formulation of Frost and Ashby [5] and the resulting equations are used to simulate a Frank-Read source in a metal at different stresses at 1000 K (52). Following this, it is argued that this algorithm cannot be applied to covalent materials. In order to develop a new algorithm applicable to directionally bonded materials, details of dislocation dynamics will have to be employed and thus, in the following section (§3), a brief overview of the necessary basic equations will be presented. The details of the new algorithm will then be described (94) and applied to the operation of 2639
(YJ). 2. INCORPORATION OF TEMPERATURE THE FROST-ASHBY APPROACH
IN
The method of Frost and Ashby [5] for the operation of a Frank-Read source is based on the balance of forces in bowing out a dislocation segment under an applied stress in a crystal with metallic bonding. In this treatment, the following force balance was used to describe the expansion of the source dislocation: R6 = (76 - I-;)“‘.
(1)
Incorporating the notations of Ref. [5], the variables have the following meanings: T is the resolved shear stress acting on a dislocation with Burgers vector 8 leading to its bow-out to a local radius of curvature ;I; thus ~6 describes the force (per unit length) perpendicular to the dislocation line. r = p/?/2 is the line tension of the dislocation, and ~1 is the shear modulus of the crystal. The force (per unit length). I-c, describes the line tension of the dislocation arising from an increase in its length by the curvature <; it acts against the force applied to the dislocation. BP is the drag force in the linear-viscous case on a
2640
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"
DISLOCATION
SOURCE DYNAMICS IN SILICON
Simulated
8
II
*
g
8
p 1 * -5000
*
3
Frank-Read u
II
1
source 8
s
8
f
I
*
8
'
8
a
2
8
*
5000 -
-5ooo-
-10000 8 -10000
*
0
13 8 0 x [nm] (L=lOOOnm)
1 5000
10000
Fig. 1. Simulation (50 sampling points) of a Frank-Read dislocation source forming a loop. The dashed line shows the first loop that forms just after the “jump” occurs: r = 22,,,, z 52 MPa, At = 6.25 s, floop= 26.6 s. dislocation moving with velocity fi, and B is the (temperature-dependent) drag coefficient. The empirical equation describing the dislocation velocity at an absolute temperature T is u = u~(z/~~)~ exp( - E/kT)
(2)
where E is the activation energy for dislocation motion, k is the Boltzmann constant, and the exponent m varies between 1 and 2.t This is in compliance with equation (1) if it is assumed that B = y
exp(E/kT).
In the case where an external applied stress Z,,~ is applied which is greater than a critical value r Crlt- pb/L, the dislocation segment bows out and forms a semicircle with a curvature K,,, = l/R0 = 2/ L, where R. is the radius of the semicircle. Thus, the stress required for the source dislocation to form a single closed dislocation loop is rext > rent. Typical numerical values for Si are F = pb’/ 2 = 2 eV/nm B = 5 x 10m5eV s/nm3 (at and T = 1000 K). The bow-out of a pinned dislocation segment forming a loop is shown in Fig. 1 for r = 2r,,,1 (Z 52 MPa). For this value of B, the time between two plotted positions of the loop is At = 6.25 s. The dashed line indicates the shape of the to a time first closed loop which corresponds t = 26.6 s after the activation of the source dislocation. Figure 2 shows a simulated Frank-Read source for an applied stress larger by an order of tstrictly speaking, equation (1)is valid only for rn = 1.For m # 1, the arrow on the vectors should be omitted.
magnitude, i.e. r = 20~,,,~ (~520 MPa) and for the same value of B (5 x 1O-5 eV s/nm3). In this case, the loop closes after a time t = 0.59 s. Note the change of scale in comparing Figs 1 and 2. A quantitative simulation of the Frank-Read source in silicon cannot be developed along these lines because of the specifics of dislocation dynamics in directionally bonded materials, for the following reasons: ??There is no easy algorithm to include the so-called Peierls stress. This modifies the round shape of a dislocation to a hexagonal shape, with the edges of the hexagon parallel to the (110) Peierls valleys on the { 111) slip plane. The latter is the actual shape of a dislocation loop at lower temperatures where the effect of the Peierls stress is more pronounced. A previous experimental observation from the work of Dash [3] is shown in Fig. 3; this figure shows dislocation loop multiplication by a symmetrical Frank-Read source on a { 11 l} plane in Si. ??For a given strain in a creep experiment, the temperature dependence of the dislocation bow-out is included only in the parameter B. However, changing B only changes the time required for the formation of a loop without affecting its geometrical shape, i.e. the geometrical configuration and the resulting dislocation density are independent of temperature. This does not conform with experimental expectations that higher temperatures should correlate with higher dislocation densities. It should be mentioned, however, that to the author’s knowledge, there are no reported experimental creep data to verify this.
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Simulated Frank-Read i’l~“‘l~~l~“‘~““l’~‘~“‘~‘l’~“~~‘~~,~~~”~”~
2000
263 I
IN SILICON
source
0
-20001~~~~~~~~~1~~~~~~~~‘i~‘~~~~~~~~~~~~~~~~~I~‘~~~~~~~~‘~’~~~’~~l -1000 0 -3000 -2000 1000 x [nm] (L=lOOOnm)
2000
3000
Fig. 2. Simulation of a Frank-Read dislocation source with T = 20r,,,, z 520 MPa. Ar = 0,125 s. t,,,,,,= 0.59 s. Note that the loop size is much smaller than in Fig. I.
??As seen in Fig. 2, there are numerical problems at higher stresses, e.g. the dislocation bows out too much at the pinning points just before closure of a loop.
Fig. 3. Experimental
observation
For these and related reasons, it was felt useful to develop a new algorithm for quantitative simulation of a Frank-Read source in directionally bonded materials such as semiconductors. In order to do this.
of a Frank-Read
source
in Si( I I I) after
Dash [3]
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we first give a brief description of the required dislocation theory in the following section based on the work of Hirth and Lothe [6]. 3. THE BASIC
EQUATIONS
A dislocation has a low energy when it lies in a Peierls valley which, in materials based on the f.c.c. lattice, is parallel to a (110) direction. The Peierls energy is the height of the energy barrier that a dislocation has to overcome in order to move from one Peierls valley to a neighboring one under zero applied stress. The Peierls stress is the stress required to move a dislocation from one Peierls valley to a neighboring one at 0 K, i.e. without the help of thermal activation in forming kink pairs and migrating them. A high Peierls potential is related to a large activation energy, E, for dislocation motion. The net force on a dislocation line is the result of the following forces: external, e.g. the applied stress, ~~~~~ internal, e.g. the Peierls stress, r,,, ??the line tension, rlt, which describes the self energy of a dislocation. ?? ??
3. I. Dislocation dynamics If there is a shear stress on a dislocation segment at a finite temperature, it moves initially by the migration of the existing (geometrical) kinks. However, the dislocation movement is soon controlled by the generation and migration of kink pairs, each kink of which is thermally activated. Under the action of the shear stress, the two individual kinks in a kink pair migrate in opposite directions. As the two kinks traverse the dislocation segment, they move it from one Peierls valley to a neighboring one. Therefore, to a first approximation, the activation energy, E, is the sum of the kink formation energy, Fk, and the kink migration energy, W,. If the self energy of a dislocation is larger than the Peierls potential, the dislocation line can readily cross over from one Peierls valley to another; the net result is that the dislocation line will have a smooth curvature. This is usually the case in metals. On the other hand, if the converse situation is true, i.e. the Peierls energy is larger than the dislocation self energy, abrupt dislocation segments are energetically favorable. In this case, long segments of a dislocation lie along the Peierls valleys which are connected by short kinks if they are in parallel valleys, or by geometrical kinks if they are in non-parallel valleys. This is particularly the case for dislocations in covalent, or ionic-covalent, materials (e.g. semiconductors of the diamond cubic, zinc blende, or wurtzite structure) because the bonding in these materials is very directional. If the core of a dislocation is reconstructed, then the formation and migration of kinks entails the breaking of covalent bonds which implies a high kink formation and migration energy.
DYNAMICS
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Thus, as a consequence of the high Peierls energy in Si (and relatively low temperature in this case), the dislocations in Fig. 3 have a hexagonal rather than a curved (round) shape (as in Figs 1 and 2). At high temperatures and low shear stresses it can be assumed that kinks migrate by a diffusive (Brownian) motion, i.e. a dislocation motion can be considered as a diffusive glide. Then the velocity of a straight dislocation is given by
x exp(
-jC$+&)exp($)
where h is the distance between two Peierls valleys, vn is the Debye frequency (Z lOI Hz) and a is the periodicity along the dislocation line. S = Sk + S, is the entropy of dislocation motion consisting of kink S,, and formation, Sk and kink migration, F’ = Jm is the Seeger-Schiller correction energy [7]. The latter factor takes into account the experimental observation that the activation energy, E, in equation (1) is not a true constant but is a weak function of stress; the Seeger-Schiller correction approximately describes the stress dependence of the activation energy, i.e. the function E(z). Thus, the energy for dislocation motion is activation E= W,+Fk-if’. For a short dislocation segment of length L, the dislocation velocity is given by
(4) This equation is valid when the segment length L is less than the critical separation of thermal kinks, X, given by
X=2aexp(g-&)exp(-$).
(5)
In this regime (i.e. for L < X), E = W,,, + 2Fk - F’. The generation of a kink pair may be thermally activated, but very narrow kink pairs are not stable and they soon collapse. A stable kink pair has a minimum width, x* + x’, where:
x*+x’=~+~(~)‘;‘.
(6)
It is important to note that the velocity of a dislocation segment with length L > X is length independent [see equation (3)] while it is linearly
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proportional to the length for L < X [see equation (4)]. On the other hand, the critical length decreases with increasing temperature. Therefore the dislocation velocity depends predominantly on the segment length for short segments and high temperatures. 3.2. Dislocutior~ The energy
DYNAMlCS
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(a)
energy
of a piece-wise
straight
dislocation
bl
is
x’dxx2 x /
w = 1 w, + 1 w,,
(7)
‘
W, =
2
(i;.z)2+ (
f+$)L( ln(“>P
-
lb) qq
1) (8) Xl
51 where < is the line direction of the dislocation and v is Poisson’s ratio; p z h is the so-called innrr cut-qfj rmdius and indicates the breakdown of linear elasticity theory inside the dislocation core. The interaction energy between two non-parallel coplanar dislocations? is
X
X2
Fig. 4. Coordinates used for calculating interaction force between dislocations (1) and (2): (a) non-parallel dislocations. (b) parallel dislocations (after Hirth and Lothe [6]).
Thus, for .Y = 0 or J = 0, equation
(10) reduces
to
I(0. J) = J’In( I - cos 0) I(s,
0) = .Y In(
1 - cos 0)
I(0. 0) = 0.
X
WA: _I),,) O[.u, In t(.u,, J,,$)+ yii In s(.x,, J,P)]
i -cos
(9) where$
1(x. 1.) = .Y In
+
R(s.
1‘
R(x, J,) + J’ - .Y cos 0 .Y
In
>
R(x, 4‘) + s - Y cos o J‘ >
y) = \/x2 + y2 -
Equation (9) is based on a non-orthogonal coordinate system as shown in Fig. 4. One of the difficulties in applying equation (9) is that its validity range is limited to x, J’ > 0. although this is not mentioned in Ref. [6]. For x < 0 (J, < 0), the coordinate system should be transformed %, + -;,($+ -&), .Y- -.Y(J -+ -y), as cos 0 + -cos 0. Considering that this limited validity range is particularly important for intersecting dislocation lines, it can be seen that separate calculations for intersecting segments are required. For cos 0 = i 1. i.e. z,//& (two parallel dislocations). the interaction energy is given by
(10)
2.x~ cos 0
t(s. _v) = _Ycos 0 - J + R(s,
y)
(6,.&)(1;&) +[(A,
x(x, y) = J’ cos 0 - s + R(x, y). iThe coordinate
system chosen is defined by the unit vectors
&. 2:. and PI where ;, = ps,; @ (SI x 21,. j/(-V*. y,,) = /(.\_I.1’1) + f(.Y?,),I?)-.f(.\Y, .l?) - flxz. JI ).
x I(&.
x
~,)+?][&~(& x
.l‘,;) + ~~ 4n(;_
I’) (~,,P,)(I;?.P;)R(s,~..l.,i) (I 1)
with
1(.x,y) = R(s, J,) ~ (J, ~ s)ln[R(.u. J) + J’- x]
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or
m, Y) = w,
Y) - (x - Y)ww,
W, Y> =
Y) + x - yl
dm
yeis the distance between the two parallel dislocations. If q approaches zero, one must choose q - 2p so that 1WI21= 2LW, for the annihilation case. 4. DESCRIPTION OF THE ALGORITHM
4.1. Movement of dislocations In $2 it was stated that, for a material that has a high Peierls energy, dislocation configurations are made up of straight segments lying in the Peierls valleys with each segment moving as a straight line. For the diamond cubic and zinc blende structures, the glide plane is { 11 l} and the Peierls valleys are parallel to the (110) directions. Hence, on any of the { 11 l} glide planes, there are three possible Peierls valleys making an angle of 120” with respect to each other. In the present simulations, the dislocations are forced to lie in these three Peierls valleys. The velocity of each dislocation segment is given by equations (3) and (4). Of course this discretization of the dislocation loop imposes difficulties. However, as Devincre and Condat [8] have recently shown, discretization of dislocations leads to the same results as the continuous case.
4.2. Growth of a single dislocation source In the simulations, when the source is activated, a loop expands (or “grows”) by a step-by-step occupation of the Peierls valleys. Thus the three Peierls valleys [Fig. 5(a)] are occupied in six separate steps as shown in Fig. S(b)-(d). In Fig. S(b) the initial configuration is shown where a dislocation segment is pinned between points A and F. In the first step, 3
i
\/
DYNAMICS
IN SILICON
the dislocation segments CD and DF “grow” longer in Peierls valleys # 2 and # 3, respectively, until their lengths are L = x* + x’ leading to the configuration shown in Fig. 5(c). In the next step, the dislocation segments AB, BC, DE, and EF expand and occupy Peierls valleys # 3 (for AB), #2 (for BC), # 1 (for CD), #3 (for DE), and #2 (for EF) leading to the configuration shown in Fig. 5(d). In this way, the dislocation keeps expanding until two segments meet below the initial dislocation line between the obstacles A and F. Then a jump occurs, and a complete loop is formed, which begins to expand radially outwards. Figure 6 illustrates how a complete dislocation loop forms and keeps expanding. The different loops show the loop position after each step. The six innermost loops correspond to the first six steps of a complete loop formation. Loop 7 shows the shape of the dislocation loop just before it closes. After this, a complete loop, denoted by l’, has formed which keeps expanding to positions shown by 2’, 3’, . . The formation of the second loop restarts from step 1 and the process repeats itself. Once a complete loop is formed, it keeps expanding with a constant velocity (assuming that the net stress is approximately constant) because all of its line segments are usually longer than the critical length. The outer seven loops, l’-7’, correspond to steps 7 + l’, 7 + 2’, etc., and show how the first complete loop expands. It should be mentioned that the time durations between the different steps are not equal. From equations (3) and (4), the velocity of dislocation segments for which L < X, is smaller than those for which L > X. As the initial length of the dislocation, i.e. the distance AF between the two obstacles, is assumed to be much larger than X (e.g. 1 pm compared to 50 nm), the first step is completed very rapidly. The dislocation travels the distance between
2 L ”
n
C
D
(4 Fig. 5. First steps of algorithm: initial dislocation configuration;
E
04 (a) the three different Peierls valleys that are occupied in six steps; (b) (c) dislocation configuration after step 1, Ad = (x* + x’)sin(60”); (d) dislocation configuration after step 2.
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,
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DISLOCATION
,
(
s
*
SOURCE
Dislocation II
60’
(
DYNAMICS
2645
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in Si 1 “I
1
I
I
*
a
*
II
5000 -
A IiN
O-
V
-7 c x
-5000
-
-100001
-
t
10000
1
-5000
3
8
8
c
1
0
t
0 x [rim]
’
*
5000
I 10000
Fig. 6. Simulation of Frank-Read source in Si with b = i( IiO), T = 60 MPa. T= 1000 K, L = I mn. The small numbers indicate the different steps in the formation of the first complete closed loop. The large numbers indicate the shapes of the expanding first loop while the second loop forms step-wise. It should be noted that the time required for each step is different. Steps l-3 require an increasing time until the maximum back stress due to the line tension is reached, while steps 47 require a decreasing time each: d indicates the size of the first closed loop and b is the direction of the Burgers vector of the dislocation.
the outer loops (i - 1)’ and i’ in the time interval At(i) which is required for the formation of step i. It can be seen that the distance between the loop in step 7 and the loop in step 1’ is almost zero, leading to At(1’) z 0. Therefore step 1’ is completed very rapidly. The distances between positions 4’-6’ of the loop are almost constant, indicating that the time intervals required for steps 5 and 6 are similar. The relatively large distances between 2’4’ indicate long durations, At, for steps 3 and 4. The line tension is calculated as ~1,= ( Wu - Wo)/(bL,otJd)
(12)
where WO is the total energy of the initial dislocation configuration including the self energies and the interaction energies of all dislocation segments [as calculated from equations (7-9)], and it is assumed that all dislocations (screw and segments of mixed character) have the same velocity. W,, is the total energy of the dislocation configuration which arises from isotropic expansion of the initial dislocation by a small length Ad. L,,,, is the arithmetic average of the total length of the initial and final dislocation configurations. The line tension is predominantly determined by the changes in self energy, A W,, of the dislocation segments. AW, describes the increase in self energy by an increase in the dislocation length, AL. The interaction energy. AW,,, between the segments i and j is much smaller than A W,.
4.3. Simultaneous
grobvth of’man_y dislocations
At the same time that the first complete dislocation loop is expanding, the source dislocation is in the process of generating the second loop whose progress follows the same sequence as the formation of the first loop. However, when more than one loop is present, the effective stress on the inner (nascent) half-loop is T = z,,~ - sit - rbilck:r,, is the line tension of the nascent half-loop, and th,lci is the back stress which arises from the repelling force of the already existing loops on the nascent loop. In general, interactions between different dislocation loops are weak and, to a first approximation, can be neglected. Only in the case of a nascent dislocation half-loop, and when r = rrr, - r,, is small, does thail have a non-negligible influence on the simulation. In such a case, the nascent dislocation half-loop can still expand but the previously generated complete loops may temporarily st.op the operation of the source. 4.4. General remarks In the case of metals, the line tension is a maximum for a semicircular configuration because this is the configuration that has the highest curvature [see equation (l)]. In the case of semiconductors, on the other hand, the configuration with the maximum line tension occurs at the critical stress and this does not correspond to a semicircular configuration. The line
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tension is high if the increase in the line length from one configuration to the next is large; this favors higher steps when the dislocation has developed out. On the other hand, the line tension is also high if the differences in interaction energies are large, i.e. the dislocation segments are close to each other. Thus, depending on the size of the dislocation configuration, the maximum line tension often occurs at steps 3 or 4. The size of the first closed loop decreases with increasing applied stress in both metals and semiconductors. For semiconductors, this can be understood from the decrease of the stable kink-pair width with increasing applied stress [see equation (6)] which favors smaller loops, and a decrease in the critical length with increasing stress [see equation (5)], which increases the dislocation velocity of smaller dislocation segments and in effect favors a faster bowing of the dislocation around the obstacle. For metals, this effect can be explained geometrically as follows: for all possible stresses, the maximum dislocation velocity occurs at the mid point of the dislocation between the obstacles, the velocity at the obstacles being zero. The greater the maximum velocity between the obstacles, the more the dislocation tends to bend around the obstacles. Therefore the size of the first closed loop for a higher stress is smaller than for a lower stress; this can be seen from a comparison of Figs 1 and 2. As was already stated in $2, the simulations of Ref. [5] for metals are temperature independent. An increase in the temperature may lead to an increase in dislocation velocity, say, by a factor of n. But the closing time for the first loop also increases by a factor n. Therefore the dislocation configuration, say, after closing of the mth loop, is temperature independent. In the present simulations for directionally bonded materials, on the other hand, the sizes of the dislocation loops are temperature dependent. This is due to the fact that a length dependence (and therefore, indirectly, a temperature dependence) for the velocity has been assumed for dislocation segments for which L < X. The closure time for the first loop is the sum of the time required for steps 1-6 and the time required for step 7. If, by changing the temperature, the required time for step 7 is changed by a factor of n, the expansion time of an already existing loop also changes by n because the lengths, L,, of the segments of the corresponding dislocation configuration are all larger than the critical length, i.e. L, > X. The required times for steps l-6 do not tThe factor sin(60”) arises from geometry. Ad for step 7 is not as critical and is selected similar to Ad of step 6. $In the present algorithm, the kink-pair width is calculated from equation (6) based on t = z..~. If z were to be replaced by the effective stress, i.e. by r = z.., - tit - rhck, the kink-pair width would change to x* + x’. However, this change would not strongly affect the simulation, as the dominant parameter for the simulation is the critical length X which does not depend on the applied stress.
DYNAMICS
IN SILICON
scale with the same factor because for these steps there are segments with lengths L, for which L, < X. In the case where X is assumed to be zero, steps l-6 take place almost instantaneously. Hence, basically, only step 7 determines the time-to-closure of the loop. In this case the simulation is temperature independent. The present algorithm takes the Peierls stress into account because the dislocations are forced to lie in the Peierls valleys. Kink formation and kink migration are statistical processes. This is included in the algorithm: for steps l-6 it is assumed that Ad = (x* + x’)sin(60°).~ Therefore, only if the length of a dislocation segment exceeds the stable kink pair widthi, can the dislocation expand perpendicular to its corresponding Peierls valley. On the other hand, the other dislocation segments move simultaneously into adjoining Peierls valleys at a distance Ad. Therefore, the algorithm is not continuous and a minimum step size for occupying a neighboring Peierls valley is required. Additionally, as the details of the dislocation theory presented in $3 are still subject to controversy, the present simulations can just show the tendencies of temperature- and stress dependence of dislocation formation and cannot be very accurate. Despite the reservations mentioned above, the algorithm overcomes the numerical present difficulties associated with smooth dislocations in Ref. [5] because only a small number of interaction energies need to be considered. In addition, the simulations include the temperature dependence of dislocation loop formation, as well as the effect of the Peierls stress on the dislocation configurations.
5. RESULTS AND DISCUSSION 5.1. Test of the algorithm Simulations were performed for an undissociated 60” dislocation in Si with its Burgers vector lying in the (111) glide plane. The simulation parameters were taken to be a0 = 0.543 nm (lattice constant), a = dOI1= aO/2, h = dIIz = aa/6, vD = 1.34 x 10” Hz, p = 68.1 GPa, and v = 0.218. The entropy factor was taken to be S = 5k [9]. Since no accurate experimental values are available, it was assumed that Fk = l.OeV and W,,, = 1.2eV [lo]. Including the Seeger-Schiller factor, for an applied stress of r = 30-80 MPa and temperatures of T = 9001100 K, this results in an activation energy of E N 2.15 eV and a dislocation velocity [see the pre-factor of equation (2)] of v0 - 2 mm/s. The distance between the obstacles, corresponding to the dislocation source length, was assumed to be L=lpm. As a verification test, equations (7k(9) were first applied to calculate and compose the energies of different loop configurations for various orientations. This test was satisfactory.
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2647
stress
Fig. 7. Critical stress vs initial length of dislocation between obstacles, i.e. the dislocation source length It may be seen that, approximately, r:,,, c L ~2
In a second test, simulations for different f( 110) dislocations were compared; apart from small numerical differences, identical results were obtained implying that the line tension is the same for, say, a screw and a 60” dislocation. In a third test, the discretization of the dislocation loop was checked from a comparison of the critical stresses for different values of L. From the linear
0
2
elasticity theory for materials with metallic bonding it follows that s,,,~ = pb/L. In this case it is expected that r’,,,, cc l/L. For Si, Fig. 7 shows that, approximately, r:,,, ,X L-‘. The difference between the two cases may be due to the geometric configuration for the maximum line tension which, in some cases for semiconductors, strongly deviates from a semicircular shape (see also Ref. [S]).
4
8 x*
IO
+ x’ [“ml”
Fig. 8. Line tension for varying x* + x’ and parameters as in Fig. 6. The * indicates the reference value of x* + x’ (see also Table 2).
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line tension
50 -
E z r c
_ _
40 -
Fig. 9. Line tension
for varying
X and parameters as in Fig. 6. The * indicates x* + x’ (see also Table 2).
Figure 8 shows the influence of the stable kink-pair width on the line tension stress, i.e. on the maximum stress. It can be seen that the stable kink-pair width, which in the present simulation determines the minimum length of a dislocation segment before it grows into an adjoining Peierls valley, does not significantly affect the line tension stress, or the size of the closing loops. Thus, increasing x* + x’ by an order of magnitude leads to an increase in rlt of about 30%. The influence of the critical length X upon the line tension is shown in Fig. 9. An increase in the critical length by a factor 4, leads to a decrease in the line tension by -5O%, indicating that, in the present simulations, the critical length is the dominant quantity determining the critical stress. The origin of the temperatureand stress dependence of the simulations (compared to those in Ref. [5]) can be understood as follows: equation (6) shows that the stable kink-pair width weakly depends on temperature and stress. Further, variations in x* + x’ do not strongly affect the line
the reference
tension stress, as can be seen from Fig. 8. It may be concluded that this quantity does not explain the temperature dependence of the simulations. On the other hand, X is a strong function of temperature [see equation (4)] leading to the variation in ra shown in Fig. 9. It should be noted that only quantities that enter equations (4) and (6) affect rlt and d; thus variations in vd, W,, and S only affect the closure time of the loop, in a similar manner as in Ref. [5], but do not affect the stress- and temperature dependence of the configurations. If the critical length is set to zero, then short dislocation segments have the same velocity as longer ones. The simulations then become temperature independent and similar results as in Ref. [5] are obtained. A remark should be made about the self energy approximation, i.e. the fact that no interaction energies between different segments have been considered. In Table 1 the self- and interaction energies of a regular hexagonal loop before and after a 1% expansion are shown.
Table 1. Self and interaction energies of all dislocation segments in a regular hexagonal loop before and after a 1% expansion W,/Cq
Before (%) After (%) (wa - W) (%)
91.181 98.908 1.121
w,,+,/rl+q
w,,+z/rlq
10.935 11.044
-4.714 -4.761
wu+l/r, WJ -4.008 - 4.048
KPU;” 100.000 101.143 1.143
W,,+3= -3.970 The initial total segments), interaction of opposite
value of
loop energy Zq is 100%. W1 is the (sum of the) self energy (of the six different W,,+ I is the appropriate interaction energy of neighboring elements, W,,+ 2 is the energy of next-but-one neighboring elements and W,,+ 3is the interaction energy elements. W” - wb gives the differences of the self and total loop energies after and before expansion. WC,+ 3is the interaction energy of opposite elements where the distance is increased by 1% while the lengths of the segments are kept constant.
GEIPEL:
DISLOCATION
SOURCE
DYNAMICS
2649
IN SILICON
I pm and b = A
Table 2. Simulated dislocation sources for different temperatures and external stresser with L = T(K)
71cxl (MPa)
.x* + Y’; X (nm)
900
21 30 40 60 80 39 40 60 80 41 50 60 80 55 60 80 63 70
6.9; 1200 6.4; 1165 5.4; 1067 4.1: 920 3.5; 812 5.5; 565 5.4: 560 4.2; 481 3.5; 433 5.0; 299 4.8; 293 4.3; 215 3.6; 246 4.6; 169 4.3; 164 3 6: 147 42: IO1 4.0: 97 3.6: 92
950
1000
I050
1100
80
I: 81,(MPa); d (nm):
f (s)
21; -; ~ 27; 4553; 523 26: 4451; I31 2X;4236; 44.4 30; 4086: 22.9 39: -: ~ 39: 3596: 79.6 39; 3385; 7.60 40;3118;370 47; -; ~ 46; 2386; 5.31 46; 2282; I 95 47; 2171: 0.829 55: --; 55; 1725, 0.800 56: 1633; 0.235 63; -: ~~ 64: 1382; 0 173 64: 1409: 0.088
34; 4541: 1795(*) 32; 4446: 30 I 32; 4239: 94.1 34; 4091: 47.6 p: em: 43; 3591: 307(*) 45; 3399: 16.7 46; 31 IO: 7.X5 54: 2378::19.6(*) 54; 2281: 4.95 55; 2153: 1.82 -; -: ~~ 61; 1706: 2.58(*) 65; 1641: 0.551 6X; 1359:0.564 73: 1357: 0.225
~~~~ is the applied external stress. 71,is the maximum line tensmn during formation of the first loop, and ihril 1s the back stress of the first loop upon the nascent second half-loop: d mdicates the size of the first and second closed loops which are shown in Fig. 6; f is the closure time for the first and second loop; (*) mdlcates that there 1s no synchronous expansion of the first and second loop, i.e. the first loop expands until its back stress upon the second loop IS belo\+ a threshold level ~~~~~ - T,,- T~.,‘~) < 0.5 MPa.
appreciate formation
For parallel segments, the interaction energies are positive, while for anti-parallel segments the interaction energies are negative. It can be seen that the difference in the self energies before and after expansion is comparable to the difference in the total loop energies. Therefore the self-energy approximation leads to comparable results to those where all interactions are considered. However, when the lengths of the dislocation segments are kept constant, the interaction energies of opposite elements decrease, as can be seen from the table. In the present simulations, all interactions have to be considered, to
-
4400
-
4200
-
I
II
I
.
outer
loops
upon
the
5.2. Results Table 2 shows the stressand temperature dependence of the critical stress (defined as the maximum line tension stress) and other corresponding parameters (such as the kink-pair width, .Y* + s’. the critical length, X, the time taken for a complete loop to form, t, the loop size, (1, the temperature, T. and the applied stress. Go,.
size
loop 4600
the impact of of new loops.
’
I
I”
,
,
I
40
8
,
,
,
I
f
,
,
80
(1
100
Text ?JPol Fig. IO. Size of first closed loop for varying zexlwith T = 900 K. It can be seen that the loop size decreases with increasing stress.
2650
DISLOCATION
GEIPEL: ,:_I
,
,
,
I
,
SOURCE
DYNAMICS
FO", Di,sloyti,on,in
IN SILICON
Si<,lll,> ,
,
,
,
, 1
I
A Il.-
-5000
-
b’_ \_ I
-10000
I
I
- 10000
Fig. 11. Simulation
I
I -5000
I
I
I
in Table 2 can be
??The geometric shape of a closed dislocation loop is close to a regular hexagon (see, e.g., Fig. 6). This is in agreement with experimental observations (Fig. 3).
800
900
I
b
0 x [nm]
as in Fig: 6 for r = 80 MPa, T= 6, it can be seen that d decreases
The results of the simulations summarized as follows:
I
s
3
I
I 5000
I
I
I
1000 K, L = 1 ,um. From a comparison slightly with increasing stress.
1
_ 10000
with Fig.
??The size of the first closed loop decreases with increasing stress. This can be seen from Fig. 10, or from a comparison of Figs 6 and 11 where the applied stress has been increased from 60 MPa to 80 MPa. This also happens in the simulations for metals (compare Figs 1 and 2).
1000
1100
1200
T WI Fig. 12. Size of first closed loop for varying T with 7 = 80 MPa. It can be seen that the loop size decreases with increasing temperature. However, for T = 1175 K, it is rent = 80 MPa, and if ~~~~ z ~~~~~~ the loop size increases because of the increasing closing time.
GEIPEL:
DISLOCATION
SOURCE
60’ 10000~
'
r
c
'
0
0
8
a
DYNAMICS
Dislocation
1
n
'
r
'
1
1
a
*
c
8
1
265 I
IN SILICON
in Si I
a
r
'
8
3
1
I
r
’
s
(
n
1
5000 -
A IkI_ hl
O-
V
7
24 x
-5000
-
-100001
-10000
Fig. 13. Simulation
-5000
0 x [nm]
5000
as in Fig. 6 for 5 = 60 MPa, T = 950 K, L = 1 pm. A comparison that d increases with decreasing time.
??The size of the first closed loop increases with decreasing temperature. This is shown in Fig. 12, and can also be seen from a comparison of Figs 6 and 13 where the temperature has been decreased by 50 K. However, in Fig. 12 it can also be seen that in the case size increases with where zeX1- L,,~, the loop increasing temperature. In this case, the effective stress, ~~~~ - tit, is small leading to a large closure time which finally leads to a bigger loop size also. ??With increasing stress and increasing temperature, there is a substantial increase in dislocation velocity which is indicated by the decreasing closure time for the first loop. ??In general, the size of the second loop is smaller than the size of the first loop due to the repulsion of the outer loop. However, the repelling force of the outer loop increases the closure time for the inner loop also. Therefore, in some cases, the second loop is larger than the first loop which stems from numerical errors in the computations. ??The dislocation density increases with increasing stress and increasing temperature; the second loop stress and acquires a larger area with decreasing temperature. This indicates a lower decreasing density with decreasing stress dislocation and temperature. This is in accord with decreasing experimental expectations that high stresses should correlate with high dislocation densities and high temperatures. ??For an applied stress slightly greater than the critical stress, the back stress of the already existing
8
j
1 10000
with Fig. 6 shobs
loops determines when the operation of a FrankRead source stops. ??The critical stress increases roughly linearly with increasing temperature, which can be seen in Fig. 14 (solid line). A comment should be made about the last result. It is very important to distinguish between the so-called yield stress in a constant strain-rate experiment and the microscopic critical stress. The macroscopic yield stress is a global property arising from a number of active sources; for an overview on yield stress properties refer to Refs [l1. 121. It is difficult to relate the yield stress to the critical stress for the operation of a single source. Therefore the above result does not mean that it is more difficult for a dislocation to move at higher temperatures. something which would be contrary to experimental observations. For low temperatures the mobility of dislocations is low and there are no time constraints for the solid line, i.e. the closing time for the loop is very high for low temperatures. Figure 14 also shows the case where the closing time of the loop is kept below a line; threshold closing time (tLhr= 0. I s dashed tihr = 0.01 s dotted line). It can be seen that for the low-temperature range the critical stress is decreasing with increasing temperature. Because yield stress experiments are performed for finite (small) times the dashed and dotted curves reflect the behaviour of the materials in common constant strain rate
2652
GEIPEL:
DISLOCATION critical
b r i 0 a n i I ’ I a n n”I
800
SOURCE DYNAMICS IN SILICON stress
I ( ’ I * 8 I #“I
(L= 1 OOOnm) G( I I ( “I 8 0 n ’ 1 ’ 8 1 0 mn c ”
600 -j
Fig. 14. Critical stress vs T with L = 1 lrn. It can be seen that, approximately, 7cnloc T. The dashed (dotted) line shows the case where the closing time of the loop is kept below a threshold value fthr= 1 s (&h,= 0.1 s).
experiments better for the low-T range rather than the solid line. 6.
CONCLUSIONS
The aim of this paper was to simulate the operation of a Frank-Read source for dislocation multiplication in directionally bonded materials, e.g. tetrahedrally bonded semiconductors. As an example, the source dislocation was taken to be a perfect 60” dislocation in Si. However, the present simulations can be extended to other dislocations including partials by changing the appropriate Burgers vector and, for the latter case, introducing an additional term in the back stress, y/b, where y is the (intrinsic) stacking fault energy and b is the magnitude of the Burgers vector of the leading partial dislocation. In Si, for which y - 60 mJ/m*, this leads to an additional back stress of - 200 MPa. For the 30” and 90” partial dislocations, the numerical values for critical stress, etc., are similar to those for a 60” perfect dislocation (Table 2). Thus, at T = 1000 K, the minimum stress required for the operation of a l/6(112) partial dislocation can be estimated to be rmln- 250 MPa. Assuming that deformation twinning occurs by the operation of a partial Frank-Read source [13], this would be the minimum stress required for twin formation. However, this estimate assumes a stationary trailing partial and the repelling force between the leading and trailing partial has not been taken into account. In the case where the distance between the leading and trailing partial is small (large enough, however, for a faulted loop to form), this
repelling force may reduce the effective critical stress considerably. Therefore the above-estimated value for the operation of a partial dislocation (the critical stress for twinning) is probably too high, and the “correct” value may be much smaller. Twinning occurs predominantly at lower temperatures below the brittle-ductile transition temperature. According to the model of Ref. [13], for twin formation the leading and trailing partials must have different mobilities, something that is favored at lower temperatures. On the other hand, dislocation loops have to close, i.e. a critical stress has to be overcome. The present simulations indicate that the critical stress decreases with decreasing temperature, making closed loops easier to form at lower temperatures. This supports the proposed twinning model. The twinning model of Ref. [13] has also been extended to explain polytypic transformations in certain semiconductors with very low stacking-fault energies, e.g. SIC [ 141.The present algorithm can also be used for this purpose. If the applied stress is slightly higher than the critical stress, the back stress due to (a small number of) already existing dislocation loops upon a nascent loop (which is much smaller than the line tension stress) brings the operation of the Frank-Read source to a temporal stop. Hence, the number of closed loops in this model is highly temperature- and stress sensitive. The present algorithm can be used to make quantitative estimates of the stress and temperature required for polytypic transformations. In fact, the present results for a perfect dislocation source already indicate that
GEIPEL:
DISLOCATION
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such phase transformations should be highly sensitive to temperature and stress. The extension of the algorithm to partial dislocations, including the consideration of the stacking-fault energy and the repelling force between the leading and trailing partials, is currently in progress. Acknowledgemenrs-A large part of this work was performed during my PhD research at the Department of Materials Science and Engineering, Case Western Reserve University, Cleveland OH, which was supported by grant #FG02-93ER45496 from the Department of Energy. I would like to acknowledge fruitful discussions with Pirouz Pirouz (DMSE) and Ladislas Kubin (CNRSONERA, Chitillon. France); many improvements to the manuscript were initiated by them.
REFERENCES I.Geipel,
T., Applications of HRTEM in Materials Science Problems and Dislocation Simulations. Ph.D. thesis, Case Western Reserve University, 1993. 2. Frank, F. C. and Read, W. T., f’hys. Rec., 1950, 79, 722-123.
DYNAMICS
IN SILICON
3. Dash, W. C. in Dislocations
2653
and Mechanical Propertie., qf Crystals. ed. J. C. Fisher, W. G. Johnston. R. Thomson and T. Vreeland Jr. John Wiley & Sons, Inc.. New York, 1956, pp. 57-68. 4. Campbell. J. D., Simmons, J. A. and Dorn, J. E.. J. appl. Mech.. 1961, 28, 441-453. 5. Frost, H. J. and Ashby. M. F.. J. uppl. Plz~~.c..1971. 42, 5273-m5279. 6. Hirth, J. P. and Lothe. J.. T/7ror~ of Disloc~utions. Wiley, New York. 1982. 7. Seeger, A. and Schiller. P., Acta mrrall.. 1962. 10, 3488357. 8. Devincre. B. and C‘ondat, M., Acra meta//. n7ufer.. 1992, 40, 262992637. 9. Marklund, S.. Solid State Commun. 1985. 54, 5555558. 10.Hirsch, P. B., Ourmazd, A. and Pirouz. P.. in Math&. Vol. 60, ed. A. Microscopy of Semiconducting G. Cullis and D. C. Joy. Inst. Phys. Conf. Ser. No.. Bristol, 1981, pp. 29934. 11. Alexander. H. and Haasen, P.. Solid S/u/e Ph~,s.. 1968. 22, 27-l 58. 12.George. A. and Rabier. J., Rewe Phxv. .4ppl. 1987. 22, 941-966. 13.Pirouz, P., Scripra Metall., 1987. 21, 1463-1468. 14.Pirouz. P.. in Strwfurr and Properties of’Dislocafions in Semiconductors, Vol. 104, ed. S. G. Roberts, P. Wilshaw and D. B. Holt. Institute of Physics Conference Series No.. Bristol, 1989. pp. 49956.
SIMULATION
Actu ,mzrer. Vol. 45. No. 6. pp. 2639-2653. 1997 c’ 1997 Acta Metallurgica Inc. Published by Elsev~er Science Ltd Printed in Great Britain. All rights reserved 1359-6454 97 $17.00 + 0.00
OF DISLOCATION SOURCE DYNAMICS SILICON FOR A 60” DISLOCATION
IN
T. GEIPEL Institut
fiir Anorganische
Chemie
der UniversitCt
Bonn,
(Received 28 March
Riimerstr.
164, 53117 Bonn.
Germany
1996)
Abstract-A new approach for the simulation of a Frank-Read source in directionally bonded, e.g. covalent, cubic materials based on an f.c.c. lattice is presented [I] (Geipel, Ph.D. Thesis, Case Western Reserve University, 1993). In this algorithm, a segment of a perfect dislocation, initially lying along a (li0) Peierls valley, is pinned at its two ends on a { Ill} glide plane. Under the application of a shear stress. this dislocation first develops into a semi-hexagonal half-loop and then swings around the pinning centers to form a full dislocation loop surrounding a reformed dislocation source. In the present study the temperature has explicitly been included through the thermal activation of kink pairs and their influence on dislocation velocity. One of the conclusions drawn from this work is that the critical stress for a 60 source dislocation in silicon appears to be highly temperature sensitive. f,cs 1997 Acta Mefallurgicu Inc,. Zusammenfassung-Es wird ein neuer Weg vorgestellt zur Simulation der Frank-Read Quelle in Materialien mit gerichteter Bindung, z.B. kovalenten, kubischen Materialien basierend auf einem f.c.c. Gitter. In diesem Algorithmus liegt zunachst ein Segment einer perfekten Versetzung in einem (I i0) Peierls Tal und wird an ihren zwei Enden festgehalten, wobei sie auf einer { 1I I} Gleitebene beweglich ist. Bei Einwirkung einer Scherspannung entwickelt die Versetzung eine (sechseckige) Schleife und wickelt sich dann urn die Pinning-Zentren bis sie eine volle Schleife bildet. Die wichtige Neuerung liegt in der expliziten Beriicksichtigung der Temperatur in Form der thermischen Aktivierung der Kink-Paare und deren Einflul3 auf die Versetzungsgeschwindigkeit. Einer der Schliisse. die gezogen werden kijnnen. besteht darin, daI3 die kritische Scherspannung fiir eine 60” Quellenversetzung stark temperaturabhiingig erscheint.
1. INTRODUCTION
a Frank-Read source in Si ($5). Finally, a few conclusions drawn from the work will be presented
The mechanism proposed by Frank and Read [2] for the continuous multiplication of dislocation loops solved one of the outstanding problems in the application of dislocation theory by explaining the plastic properties of crystalline solids. Ever since the conception of this mechanism, and the direct experimental evidence of its existence by Dash [3], this mechanism has been used to explain numerous phenomena. The first simulations of a Frank-Read (F-R) source appear to have been carried out by Dorn and coworkers [4]. Later, Frost and Ashby [5] considered the effect of drag forces on the operation of the F-R source in metals at 0 K. In this paper, temperature is first incorporated in the formulation of Frost and Ashby [5] and the resulting equations are used to simulate a Frank-Read source in a metal at different stresses at 1000 K (52). Following this, it is argued that this algorithm cannot be applied to covalent materials. In order to develop a new algorithm applicable to directionally bonded materials, details of dislocation dynamics will have to be employed and thus, in the following section (§3), a brief overview of the necessary basic equations will be presented. The details of the new algorithm will then be described (94) and applied to the operation of 2639
(YJ). 2. INCORPORATION OF TEMPERATURE THE FROST-ASHBY APPROACH
IN
The method of Frost and Ashby [5] for the operation of a Frank-Read source is based on the balance of forces in bowing out a dislocation segment under an applied stress in a crystal with metallic bonding. In this treatment, the following force balance was used to describe the expansion of the source dislocation: R6 = (76 - I-;)“‘.
(1)
Incorporating the notations of Ref. [5], the variables have the following meanings: T is the resolved shear stress acting on a dislocation with Burgers vector 8 leading to its bow-out to a local radius of curvature ;I; thus ~6 describes the force (per unit length) perpendicular to the dislocation line. r = p/?/2 is the line tension of the dislocation, and ~1 is the shear modulus of the crystal. The force (per unit length). I-c, describes the line tension of the dislocation arising from an increase in its length by the curvature <; it acts against the force applied to the dislocation. BP is the drag force in the linear-viscous case on a
2640
GEIPEL: 10000
"
DISLOCATION
SOURCE DYNAMICS IN SILICON
Simulated
8
II
*
g
8
p 1 * -5000
*
3
Frank-Read u
II
1
source 8
s
8
f
I
*
8
'
8
a
2
8
*
5000 -
-5ooo-
-10000 8 -10000
*
0
13 8 0 x [nm] (L=lOOOnm)
1 5000
10000
Fig. 1. Simulation (50 sampling points) of a Frank-Read dislocation source forming a loop. The dashed line shows the first loop that forms just after the “jump” occurs: r = 22,,,, z 52 MPa, At = 6.25 s, floop= 26.6 s. dislocation moving with velocity fi, and B is the (temperature-dependent) drag coefficient. The empirical equation describing the dislocation velocity at an absolute temperature T is u = u~(z/~~)~ exp( - E/kT)
(2)
where E is the activation energy for dislocation motion, k is the Boltzmann constant, and the exponent m varies between 1 and 2.t This is in compliance with equation (1) if it is assumed that B = y
exp(E/kT).
In the case where an external applied stress Z,,~ is applied which is greater than a critical value r Crlt- pb/L, the dislocation segment bows out and forms a semicircle with a curvature K,,, = l/R0 = 2/ L, where R. is the radius of the semicircle. Thus, the stress required for the source dislocation to form a single closed dislocation loop is rext > rent. Typical numerical values for Si are F = pb’/ 2 = 2 eV/nm B = 5 x 10m5eV s/nm3 (at and T = 1000 K). The bow-out of a pinned dislocation segment forming a loop is shown in Fig. 1 for r = 2r,,,1 (Z 52 MPa). For this value of B, the time between two plotted positions of the loop is At = 6.25 s. The dashed line indicates the shape of the to a time first closed loop which corresponds t = 26.6 s after the activation of the source dislocation. Figure 2 shows a simulated Frank-Read source for an applied stress larger by an order of tstrictly speaking, equation (1)is valid only for rn = 1.For m # 1, the arrow on the vectors should be omitted.
magnitude, i.e. r = 20~,,,~ (~520 MPa) and for the same value of B (5 x 1O-5 eV s/nm3). In this case, the loop closes after a time t = 0.59 s. Note the change of scale in comparing Figs 1 and 2. A quantitative simulation of the Frank-Read source in silicon cannot be developed along these lines because of the specifics of dislocation dynamics in directionally bonded materials, for the following reasons: ??There is no easy algorithm to include the so-called Peierls stress. This modifies the round shape of a dislocation to a hexagonal shape, with the edges of the hexagon parallel to the (110) Peierls valleys on the { 111) slip plane. The latter is the actual shape of a dislocation loop at lower temperatures where the effect of the Peierls stress is more pronounced. A previous experimental observation from the work of Dash [3] is shown in Fig. 3; this figure shows dislocation loop multiplication by a symmetrical Frank-Read source on a { 11 l} plane in Si. ??For a given strain in a creep experiment, the temperature dependence of the dislocation bow-out is included only in the parameter B. However, changing B only changes the time required for the formation of a loop without affecting its geometrical shape, i.e. the geometrical configuration and the resulting dislocation density are independent of temperature. This does not conform with experimental expectations that higher temperatures should correlate with higher dislocation densities. It should be mentioned, however, that to the author’s knowledge, there are no reported experimental creep data to verify this.
GEIPEL:
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DYNAMICS
Simulated Frank-Read i’l~“‘l~~l~“‘~““l’~‘~“‘~‘l’~“~~‘~~,~~~”~”~
2000
263 I
IN SILICON
source
0
-20001~~~~~~~~~1~~~~~~~~‘i~‘~~~~~~~~~~~~~~~~~I~‘~~~~~~~~‘~’~~~’~~l -1000 0 -3000 -2000 1000 x [nm] (L=lOOOnm)
2000
3000
Fig. 2. Simulation of a Frank-Read dislocation source with T = 20r,,,, z 520 MPa. Ar = 0,125 s. t,,,,,,= 0.59 s. Note that the loop size is much smaller than in Fig. I.
??As seen in Fig. 2, there are numerical problems at higher stresses, e.g. the dislocation bows out too much at the pinning points just before closure of a loop.
Fig. 3. Experimental
observation
For these and related reasons, it was felt useful to develop a new algorithm for quantitative simulation of a Frank-Read source in directionally bonded materials such as semiconductors. In order to do this.
of a Frank-Read
source
in Si( I I I) after
Dash [3]
2642
GEIPEL:
DISLOCATION
SOURCE
we first give a brief description of the required dislocation theory in the following section based on the work of Hirth and Lothe [6]. 3. THE BASIC
EQUATIONS
A dislocation has a low energy when it lies in a Peierls valley which, in materials based on the f.c.c. lattice, is parallel to a (110) direction. The Peierls energy is the height of the energy barrier that a dislocation has to overcome in order to move from one Peierls valley to a neighboring one under zero applied stress. The Peierls stress is the stress required to move a dislocation from one Peierls valley to a neighboring one at 0 K, i.e. without the help of thermal activation in forming kink pairs and migrating them. A high Peierls potential is related to a large activation energy, E, for dislocation motion. The net force on a dislocation line is the result of the following forces: external, e.g. the applied stress, ~~~~~ internal, e.g. the Peierls stress, r,,, ??the line tension, rlt, which describes the self energy of a dislocation. ?? ??
3. I. Dislocation dynamics If there is a shear stress on a dislocation segment at a finite temperature, it moves initially by the migration of the existing (geometrical) kinks. However, the dislocation movement is soon controlled by the generation and migration of kink pairs, each kink of which is thermally activated. Under the action of the shear stress, the two individual kinks in a kink pair migrate in opposite directions. As the two kinks traverse the dislocation segment, they move it from one Peierls valley to a neighboring one. Therefore, to a first approximation, the activation energy, E, is the sum of the kink formation energy, Fk, and the kink migration energy, W,. If the self energy of a dislocation is larger than the Peierls potential, the dislocation line can readily cross over from one Peierls valley to another; the net result is that the dislocation line will have a smooth curvature. This is usually the case in metals. On the other hand, if the converse situation is true, i.e. the Peierls energy is larger than the dislocation self energy, abrupt dislocation segments are energetically favorable. In this case, long segments of a dislocation lie along the Peierls valleys which are connected by short kinks if they are in parallel valleys, or by geometrical kinks if they are in non-parallel valleys. This is particularly the case for dislocations in covalent, or ionic-covalent, materials (e.g. semiconductors of the diamond cubic, zinc blende, or wurtzite structure) because the bonding in these materials is very directional. If the core of a dislocation is reconstructed, then the formation and migration of kinks entails the breaking of covalent bonds which implies a high kink formation and migration energy.
DYNAMICS
IN SILICON
Thus, as a consequence of the high Peierls energy in Si (and relatively low temperature in this case), the dislocations in Fig. 3 have a hexagonal rather than a curved (round) shape (as in Figs 1 and 2). At high temperatures and low shear stresses it can be assumed that kinks migrate by a diffusive (Brownian) motion, i.e. a dislocation motion can be considered as a diffusive glide. Then the velocity of a straight dislocation is given by
x exp(
-jC$+&)exp($)
where h is the distance between two Peierls valleys, vn is the Debye frequency (Z lOI Hz) and a is the periodicity along the dislocation line. S = Sk + S, is the entropy of dislocation motion consisting of kink S,, and formation, Sk and kink migration, F’ = Jm is the Seeger-Schiller correction energy [7]. The latter factor takes into account the experimental observation that the activation energy, E, in equation (1) is not a true constant but is a weak function of stress; the Seeger-Schiller correction approximately describes the stress dependence of the activation energy, i.e. the function E(z). Thus, the energy for dislocation motion is activation E= W,+Fk-if’. For a short dislocation segment of length L, the dislocation velocity is given by
(4) This equation is valid when the segment length L is less than the critical separation of thermal kinks, X, given by
X=2aexp(g-&)exp(-$).
(5)
In this regime (i.e. for L < X), E = W,,, + 2Fk - F’. The generation of a kink pair may be thermally activated, but very narrow kink pairs are not stable and they soon collapse. A stable kink pair has a minimum width, x* + x’, where:
x*+x’=~+~(~)‘;‘.
(6)
It is important to note that the velocity of a dislocation segment with length L > X is length independent [see equation (3)] while it is linearly
GEIPEL:
DISLOCATION
SOURCE
proportional to the length for L < X [see equation (4)]. On the other hand, the critical length decreases with increasing temperature. Therefore the dislocation velocity depends predominantly on the segment length for short segments and high temperatures. 3.2. Dislocutior~ The energy
DYNAMlCS
2643
IN SILICON
(a)
energy
of a piece-wise
straight
dislocation
bl
is
x’dxx2 x /
w = 1 w, + 1 w,,
(7)
‘
W, =
2
(i;.z)2+ (
f+$)L( ln(“>P
-
lb) qq
1) (8) Xl
51 where < is the line direction of the dislocation and v is Poisson’s ratio; p z h is the so-called innrr cut-qfj rmdius and indicates the breakdown of linear elasticity theory inside the dislocation core. The interaction energy between two non-parallel coplanar dislocations? is
X
X2
Fig. 4. Coordinates used for calculating interaction force between dislocations (1) and (2): (a) non-parallel dislocations. (b) parallel dislocations (after Hirth and Lothe [6]).
Thus, for .Y = 0 or J = 0, equation
(10) reduces
to
I(0. J) = J’In( I - cos 0) I(s,
0) = .Y In(
1 - cos 0)
I(0. 0) = 0.
X
WA: _I),,) O[.u, In t(.u,, J,,$)+ yii In s(.x,, J,P)]
i -cos
(9) where$
1(x. 1.) = .Y In
+
R(s.
1‘
R(x, J,) + J’ - .Y cos 0 .Y
In
>
R(x, 4‘) + s - Y cos o J‘ >
y) = \/x2 + y2 -
Equation (9) is based on a non-orthogonal coordinate system as shown in Fig. 4. One of the difficulties in applying equation (9) is that its validity range is limited to x, J’ > 0. although this is not mentioned in Ref. [6]. For x < 0 (J, < 0), the coordinate system should be transformed %, + -;,($+ -&), .Y- -.Y(J -+ -y), as cos 0 + -cos 0. Considering that this limited validity range is particularly important for intersecting dislocation lines, it can be seen that separate calculations for intersecting segments are required. For cos 0 = i 1. i.e. z,//& (two parallel dislocations). the interaction energy is given by
(10)
2.x~ cos 0
t(s. _v) = _Ycos 0 - J + R(s,
y)
(6,.&)(1;&) +[(A,
x(x, y) = J’ cos 0 - s + R(x, y). iThe coordinate
system chosen is defined by the unit vectors
&. 2:. and PI where ;, = ps,; @ (SI x 21,. j/(-V*. y,,) = /(.\_I.1’1) + f(.Y?,),I?)-.f(.\Y, .l?) - flxz. JI ).
x I(&.
x
~,)+?][&~(& x
.l‘,;) + ~~ 4n(;_
I’) (~,,P,)(I;?.P;)R(s,~..l.,i) (I 1)
with
1(.x,y) = R(s, J,) ~ (J, ~ s)ln[R(.u. J) + J’- x]
2644
GEIPEL:
DISLOCATION
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or
m, Y) = w,
Y) - (x - Y)ww,
W, Y> =
Y) + x - yl
dm
yeis the distance between the two parallel dislocations. If q approaches zero, one must choose q - 2p so that 1WI21= 2LW, for the annihilation case. 4. DESCRIPTION OF THE ALGORITHM
4.1. Movement of dislocations In $2 it was stated that, for a material that has a high Peierls energy, dislocation configurations are made up of straight segments lying in the Peierls valleys with each segment moving as a straight line. For the diamond cubic and zinc blende structures, the glide plane is { 11 l} and the Peierls valleys are parallel to the (110) directions. Hence, on any of the { 11 l} glide planes, there are three possible Peierls valleys making an angle of 120” with respect to each other. In the present simulations, the dislocations are forced to lie in these three Peierls valleys. The velocity of each dislocation segment is given by equations (3) and (4). Of course this discretization of the dislocation loop imposes difficulties. However, as Devincre and Condat [8] have recently shown, discretization of dislocations leads to the same results as the continuous case.
4.2. Growth of a single dislocation source In the simulations, when the source is activated, a loop expands (or “grows”) by a step-by-step occupation of the Peierls valleys. Thus the three Peierls valleys [Fig. 5(a)] are occupied in six separate steps as shown in Fig. S(b)-(d). In Fig. S(b) the initial configuration is shown where a dislocation segment is pinned between points A and F. In the first step, 3
i
\/
DYNAMICS
IN SILICON
the dislocation segments CD and DF “grow” longer in Peierls valleys # 2 and # 3, respectively, until their lengths are L = x* + x’ leading to the configuration shown in Fig. 5(c). In the next step, the dislocation segments AB, BC, DE, and EF expand and occupy Peierls valleys # 3 (for AB), #2 (for BC), # 1 (for CD), #3 (for DE), and #2 (for EF) leading to the configuration shown in Fig. 5(d). In this way, the dislocation keeps expanding until two segments meet below the initial dislocation line between the obstacles A and F. Then a jump occurs, and a complete loop is formed, which begins to expand radially outwards. Figure 6 illustrates how a complete dislocation loop forms and keeps expanding. The different loops show the loop position after each step. The six innermost loops correspond to the first six steps of a complete loop formation. Loop 7 shows the shape of the dislocation loop just before it closes. After this, a complete loop, denoted by l’, has formed which keeps expanding to positions shown by 2’, 3’, . . The formation of the second loop restarts from step 1 and the process repeats itself. Once a complete loop is formed, it keeps expanding with a constant velocity (assuming that the net stress is approximately constant) because all of its line segments are usually longer than the critical length. The outer seven loops, l’-7’, correspond to steps 7 + l’, 7 + 2’, etc., and show how the first complete loop expands. It should be mentioned that the time durations between the different steps are not equal. From equations (3) and (4), the velocity of dislocation segments for which L < X, is smaller than those for which L > X. As the initial length of the dislocation, i.e. the distance AF between the two obstacles, is assumed to be much larger than X (e.g. 1 pm compared to 50 nm), the first step is completed very rapidly. The dislocation travels the distance between
2 L ”
n
C
D
(4 Fig. 5. First steps of algorithm: initial dislocation configuration;
E
04 (a) the three different Peierls valleys that are occupied in six steps; (b) (c) dislocation configuration after step 1, Ad = (x* + x’)sin(60”); (d) dislocation configuration after step 2.
GEIPEL:
_^..^^.
,
1 vvvv
DISLOCATION
,
(
s
*
SOURCE
Dislocation II
60’
(
DYNAMICS
2645
IN SILICON
in Si
1
I
I
*
a
*
II
5000 -
A IiN
O-
V
-7 c x
-5000
-
-100001
-
t
10000
1
-5000
3
8
8
c
1
0
t
0 x [rim]
’
*
5000
I 10000
Fig. 6. Simulation of Frank-Read source in Si with b = i( IiO), T = 60 MPa. T= 1000 K, L = I mn. The small numbers indicate the different steps in the formation of the first complete closed loop. The large numbers indicate the shapes of the expanding first loop while the second loop forms step-wise. It should be noted that the time required for each step is different. Steps l-3 require an increasing time until the maximum back stress due to the line tension is reached, while steps 47 require a decreasing time each: d indicates the size of the first closed loop and b is the direction of the Burgers vector of the dislocation.
the outer loops (i - 1)’ and i’ in the time interval At(i) which is required for the formation of step i. It can be seen that the distance between the loop in step 7 and the loop in step 1’ is almost zero, leading to At(1’) z 0. Therefore step 1’ is completed very rapidly. The distances between positions 4’-6’ of the loop are almost constant, indicating that the time intervals required for steps 5 and 6 are similar. The relatively large distances between 2’4’ indicate long durations, At, for steps 3 and 4. The line tension is calculated as ~1,= ( Wu - Wo)/(bL,otJd)
(12)
where WO is the total energy of the initial dislocation configuration including the self energies and the interaction energies of all dislocation segments [as calculated from equations (7-9)], and it is assumed that all dislocations (screw and segments of mixed character) have the same velocity. W,, is the total energy of the dislocation configuration which arises from isotropic expansion of the initial dislocation by a small length Ad. L,,,, is the arithmetic average of the total length of the initial and final dislocation configurations. The line tension is predominantly determined by the changes in self energy, A W,, of the dislocation segments. AW, describes the increase in self energy by an increase in the dislocation length, AL. The interaction energy. AW,,, between the segments i and j is much smaller than A W,.
4.3. Simultaneous
grobvth of’man_y dislocations
At the same time that the first complete dislocation loop is expanding, the source dislocation is in the process of generating the second loop whose progress follows the same sequence as the formation of the first loop. However, when more than one loop is present, the effective stress on the inner (nascent) half-loop is T = z,,~ - sit - rbilck:r,, is the line tension of the nascent half-loop, and th,lci is the back stress which arises from the repelling force of the already existing loops on the nascent loop. In general, interactions between different dislocation loops are weak and, to a first approximation, can be neglected. Only in the case of a nascent dislocation half-loop, and when r = rrr, - r,, is small, does thail have a non-negligible influence on the simulation. In such a case, the nascent dislocation half-loop can still expand but the previously generated complete loops may temporarily st.op the operation of the source. 4.4. General remarks In the case of metals, the line tension is a maximum for a semicircular configuration because this is the configuration that has the highest curvature [see equation (l)]. In the case of semiconductors, on the other hand, the configuration with the maximum line tension occurs at the critical stress and this does not correspond to a semicircular configuration. The line
2646
GEIPEL:
DISLOCATION
SOURCE
tension is high if the increase in the line length from one configuration to the next is large; this favors higher steps when the dislocation has developed out. On the other hand, the line tension is also high if the differences in interaction energies are large, i.e. the dislocation segments are close to each other. Thus, depending on the size of the dislocation configuration, the maximum line tension often occurs at steps 3 or 4. The size of the first closed loop decreases with increasing applied stress in both metals and semiconductors. For semiconductors, this can be understood from the decrease of the stable kink-pair width with increasing applied stress [see equation (6)] which favors smaller loops, and a decrease in the critical length with increasing stress [see equation (5)], which increases the dislocation velocity of smaller dislocation segments and in effect favors a faster bowing of the dislocation around the obstacle. For metals, this effect can be explained geometrically as follows: for all possible stresses, the maximum dislocation velocity occurs at the mid point of the dislocation between the obstacles, the velocity at the obstacles being zero. The greater the maximum velocity between the obstacles, the more the dislocation tends to bend around the obstacles. Therefore the size of the first closed loop for a higher stress is smaller than for a lower stress; this can be seen from a comparison of Figs 1 and 2. As was already stated in $2, the simulations of Ref. [5] for metals are temperature independent. An increase in the temperature may lead to an increase in dislocation velocity, say, by a factor of n. But the closing time for the first loop also increases by a factor n. Therefore the dislocation configuration, say, after closing of the mth loop, is temperature independent. In the present simulations for directionally bonded materials, on the other hand, the sizes of the dislocation loops are temperature dependent. This is due to the fact that a length dependence (and therefore, indirectly, a temperature dependence) for the velocity has been assumed for dislocation segments for which L < X. The closure time for the first loop is the sum of the time required for steps 1-6 and the time required for step 7. If, by changing the temperature, the required time for step 7 is changed by a factor of n, the expansion time of an already existing loop also changes by n because the lengths, L,, of the segments of the corresponding dislocation configuration are all larger than the critical length, i.e. L, > X. The required times for steps l-6 do not tThe factor sin(60”) arises from geometry. Ad for step 7 is not as critical and is selected similar to Ad of step 6. $In the present algorithm, the kink-pair width is calculated from equation (6) based on t = z..~. If z were to be replaced by the effective stress, i.e. by r = z.., - tit - rhck, the kink-pair width would change to x* + x’. However, this change would not strongly affect the simulation, as the dominant parameter for the simulation is the critical length X which does not depend on the applied stress.
DYNAMICS
IN SILICON
scale with the same factor because for these steps there are segments with lengths L, for which L, < X. In the case where X is assumed to be zero, steps l-6 take place almost instantaneously. Hence, basically, only step 7 determines the time-to-closure of the loop. In this case the simulation is temperature independent. The present algorithm takes the Peierls stress into account because the dislocations are forced to lie in the Peierls valleys. Kink formation and kink migration are statistical processes. This is included in the algorithm: for steps l-6 it is assumed that Ad = (x* + x’)sin(60°).~ Therefore, only if the length of a dislocation segment exceeds the stable kink pair widthi, can the dislocation expand perpendicular to its corresponding Peierls valley. On the other hand, the other dislocation segments move simultaneously into adjoining Peierls valleys at a distance Ad. Therefore, the algorithm is not continuous and a minimum step size for occupying a neighboring Peierls valley is required. Additionally, as the details of the dislocation theory presented in $3 are still subject to controversy, the present simulations can just show the tendencies of temperature- and stress dependence of dislocation formation and cannot be very accurate. Despite the reservations mentioned above, the algorithm overcomes the numerical present difficulties associated with smooth dislocations in Ref. [5] because only a small number of interaction energies need to be considered. In addition, the simulations include the temperature dependence of dislocation loop formation, as well as the effect of the Peierls stress on the dislocation configurations.
5. RESULTS AND DISCUSSION 5.1. Test of the algorithm Simulations were performed for an undissociated 60” dislocation in Si with its Burgers vector lying in the (111) glide plane. The simulation parameters were taken to be a0 = 0.543 nm (lattice constant), a = dOI1= aO/2, h = dIIz = aa/6, vD = 1.34 x 10” Hz, p = 68.1 GPa, and v = 0.218. The entropy factor was taken to be S = 5k [9]. Since no accurate experimental values are available, it was assumed that Fk = l.OeV and W,,, = 1.2eV [lo]. Including the Seeger-Schiller factor, for an applied stress of r = 30-80 MPa and temperatures of T = 9001100 K, this results in an activation energy of E N 2.15 eV and a dislocation velocity [see the pre-factor of equation (2)] of v0 - 2 mm/s. The distance between the obstacles, corresponding to the dislocation source length, was assumed to be L=lpm. As a verification test, equations (7k(9) were first applied to calculate and compose the energies of different loop configurations for various orientations. This test was satisfactory.
CEIPEL:
DISLOCATION
SOURCE DYNAMICS IN SILICON critical
2647
stress
Fig. 7. Critical stress vs initial length of dislocation between obstacles, i.e. the dislocation source length It may be seen that, approximately, r:,,, c L ~2
In a second test, simulations for different f( 110) dislocations were compared; apart from small numerical differences, identical results were obtained implying that the line tension is the same for, say, a screw and a 60” dislocation. In a third test, the discretization of the dislocation loop was checked from a comparison of the critical stresses for different values of L. From the linear
0
2
elasticity theory for materials with metallic bonding it follows that s,,,~ = pb/L. In this case it is expected that r’,,,, cc l/L. For Si, Fig. 7 shows that, approximately, r:,,, ,X L-‘. The difference between the two cases may be due to the geometric configuration for the maximum line tension which, in some cases for semiconductors, strongly deviates from a semicircular shape (see also Ref. [S]).
4
8 x*
IO
+ x’ [“ml”
Fig. 8. Line tension for varying x* + x’ and parameters as in Fig. 6. The * indicates the reference value of x* + x’ (see also Table 2).
2648
GEIPEL:
DISLOCATION
SOURCE
DYNAMICS
IN SILICON
line tension
50 -
E z r c
_ _
40 -
Fig. 9. Line tension
for varying
X and parameters as in Fig. 6. The * indicates x* + x’ (see also Table 2).
Figure 8 shows the influence of the stable kink-pair width on the line tension stress, i.e. on the maximum stress. It can be seen that the stable kink-pair width, which in the present simulation determines the minimum length of a dislocation segment before it grows into an adjoining Peierls valley, does not significantly affect the line tension stress, or the size of the closing loops. Thus, increasing x* + x’ by an order of magnitude leads to an increase in rlt of about 30%. The influence of the critical length X upon the line tension is shown in Fig. 9. An increase in the critical length by a factor 4, leads to a decrease in the line tension by -5O%, indicating that, in the present simulations, the critical length is the dominant quantity determining the critical stress. The origin of the temperatureand stress dependence of the simulations (compared to those in Ref. [5]) can be understood as follows: equation (6) shows that the stable kink-pair width weakly depends on temperature and stress. Further, variations in x* + x’ do not strongly affect the line
the reference
tension stress, as can be seen from Fig. 8. It may be concluded that this quantity does not explain the temperature dependence of the simulations. On the other hand, X is a strong function of temperature [see equation (4)] leading to the variation in ra shown in Fig. 9. It should be noted that only quantities that enter equations (4) and (6) affect rlt and d; thus variations in vd, W,, and S only affect the closure time of the loop, in a similar manner as in Ref. [5], but do not affect the stress- and temperature dependence of the configurations. If the critical length is set to zero, then short dislocation segments have the same velocity as longer ones. The simulations then become temperature independent and similar results as in Ref. [5] are obtained. A remark should be made about the self energy approximation, i.e. the fact that no interaction energies between different segments have been considered. In Table 1 the self- and interaction energies of a regular hexagonal loop before and after a 1% expansion are shown.
Table 1. Self and interaction energies of all dislocation segments in a regular hexagonal loop before and after a 1% expansion W,/Cq
Before (%) After (%) (wa - W) (%)
91.181 98.908 1.121
w,,+,/rl+q
w,,+z/rlq
10.935 11.044
-4.714 -4.761
wu+l/r, WJ -4.008 - 4.048
KPU;” 100.000 101.143 1.143
W,,+3= -3.970 The initial total segments), interaction of opposite
value of
loop energy Zq is 100%. W1 is the (sum of the) self energy (of the six different W,,+ I is the appropriate interaction energy of neighboring elements, W,,+ 2 is the energy of next-but-one neighboring elements and W,,+ 3is the interaction energy elements. W” - wb gives the differences of the self and total loop energies after and before expansion. WC,+ 3is the interaction energy of opposite elements where the distance is increased by 1% while the lengths of the segments are kept constant.
GEIPEL:
DISLOCATION
SOURCE
DYNAMICS
2649
IN SILICON
I pm and b = A
Table 2. Simulated dislocation sources for different temperatures and external stresser with L = T(K)
71cxl (MPa)
.x* + Y’; X (nm)
900
21 30 40 60 80 39 40 60 80 41 50 60 80 55 60 80 63 70
6.9; 1200 6.4; 1165 5.4; 1067 4.1: 920 3.5; 812 5.5; 565 5.4: 560 4.2; 481 3.5; 433 5.0; 299 4.8; 293 4.3; 215 3.6; 246 4.6; 169 4.3; 164 3 6: 147 42: IO1 4.0: 97 3.6: 92
950
1000
I050
1100
80
I: 81,(MPa); d (nm):
f (s)
21; -; ~ 27; 4553; 523 26: 4451; I31 2X;4236; 44.4 30; 4086: 22.9 39: -: ~ 39: 3596: 79.6 39; 3385; 7.60 40;3118;370 47; -; ~ 46; 2386; 5.31 46; 2282; I 95 47; 2171: 0.829 55: --; 55; 1725, 0.800 56: 1633; 0.235 63; -: ~~ 64: 1382; 0 173 64: 1409: 0.088
34; 4541: 1795(*) 32; 4446: 30 I 32; 4239: 94.1 34; 4091: 47.6 p: em: 43; 3591: 307(*) 45; 3399: 16.7 46; 31 IO: 7.X5 54: 2378::19.6(*) 54; 2281: 4.95 55; 2153: 1.82 -; -: ~~ 61; 1706: 2.58(*) 65; 1641: 0.551 6X; 1359:0.564 73: 1357: 0.225
~~~~ is the applied external stress. 71,is the maximum line tensmn during formation of the first loop, and ihril 1s the back stress of the first loop upon the nascent second half-loop: d mdicates the size of the first and second closed loops which are shown in Fig. 6; f is the closure time for the first and second loop; (*) mdlcates that there 1s no synchronous expansion of the first and second loop, i.e. the first loop expands until its back stress upon the second loop IS belo\+ a threshold level ~~~~~ - T,,- T~.,‘~) < 0.5 MPa.
appreciate formation
For parallel segments, the interaction energies are positive, while for anti-parallel segments the interaction energies are negative. It can be seen that the difference in the self energies before and after expansion is comparable to the difference in the total loop energies. Therefore the self-energy approximation leads to comparable results to those where all interactions are considered. However, when the lengths of the dislocation segments are kept constant, the interaction energies of opposite elements decrease, as can be seen from the table. In the present simulations, all interactions have to be considered, to
-
4400
-
4200
-
I
II
I
.
outer
loops
upon
the
5.2. Results Table 2 shows the stressand temperature dependence of the critical stress (defined as the maximum line tension stress) and other corresponding parameters (such as the kink-pair width, .Y* + s’. the critical length, X, the time taken for a complete loop to form, t, the loop size, (1, the temperature, T. and the applied stress. Go,.
size
loop 4600
the impact of of new loops.
’
I
I”
,
,
I
40
8
,
,
,
I
f
,
,
80
(1
100
Text ?JPol Fig. IO. Size of first closed loop for varying zexlwith T = 900 K. It can be seen that the loop size decreases with increasing stress.
2650
DISLOCATION
GEIPEL: ,:_I
,
,
,
I
,
SOURCE
DYNAMICS
FO", Di,sloyti,on,in
IN SILICON
Si<,lll,> ,
,
,
,
, 1
I
A Il.-
-5000
-
b’_ \_ I
-10000
I
I
- 10000
Fig. 11. Simulation
I
I -5000
I
I
I
in Table 2 can be
??The geometric shape of a closed dislocation loop is close to a regular hexagon (see, e.g., Fig. 6). This is in agreement with experimental observations (Fig. 3).
800
900
I
b
0 x [nm]
as in Fig: 6 for r = 80 MPa, T= 6, it can be seen that d decreases
The results of the simulations summarized as follows:
I
s
3
I
I 5000
I
I
I
1000 K, L = 1 ,um. From a comparison slightly with increasing stress.
1
_ 10000
with Fig.
??The size of the first closed loop decreases with increasing stress. This can be seen from Fig. 10, or from a comparison of Figs 6 and 11 where the applied stress has been increased from 60 MPa to 80 MPa. This also happens in the simulations for metals (compare Figs 1 and 2).
1000
1100
1200
T WI Fig. 12. Size of first closed loop for varying T with 7 = 80 MPa. It can be seen that the loop size decreases with increasing temperature. However, for T = 1175 K, it is rent = 80 MPa, and if ~~~~ z ~~~~~~ the loop size increases because of the increasing closing time.
GEIPEL:
DISLOCATION
SOURCE
60’ 10000~
'
r
c
'
0
0
8
a
DYNAMICS
Dislocation
1
n
'
r
'
1
1
a
*
c
8
1
265 I
IN SILICON
in Si
a
r
'
8
3
1
I
r
’
s
(
n
1
5000 -
A IkI_ hl
O-
V
7
24 x
-5000
-
-100001
-10000
Fig. 13. Simulation
-5000
0 x [nm]
5000
as in Fig. 6 for 5 = 60 MPa, T = 950 K, L = 1 pm. A comparison that d increases with decreasing time.
??The size of the first closed loop increases with decreasing temperature. This is shown in Fig. 12, and can also be seen from a comparison of Figs 6 and 13 where the temperature has been decreased by 50 K. However, in Fig. 12 it can also be seen that in the case size increases with where zeX1- L,,~, the loop increasing temperature. In this case, the effective stress, ~~~~ - tit, is small leading to a large closure time which finally leads to a bigger loop size also. ??With increasing stress and increasing temperature, there is a substantial increase in dislocation velocity which is indicated by the decreasing closure time for the first loop. ??In general, the size of the second loop is smaller than the size of the first loop due to the repulsion of the outer loop. However, the repelling force of the outer loop increases the closure time for the inner loop also. Therefore, in some cases, the second loop is larger than the first loop which stems from numerical errors in the computations. ??The dislocation density increases with increasing stress and increasing temperature; the second loop stress and acquires a larger area with decreasing temperature. This indicates a lower decreasing density with decreasing stress dislocation and temperature. This is in accord with decreasing experimental expectations that high stresses should correlate with high dislocation densities and high temperatures. ??For an applied stress slightly greater than the critical stress, the back stress of the already existing
8
j
1 10000
with Fig. 6 shobs
loops determines when the operation of a FrankRead source stops. ??The critical stress increases roughly linearly with increasing temperature, which can be seen in Fig. 14 (solid line). A comment should be made about the last result. It is very important to distinguish between the so-called yield stress in a constant strain-rate experiment and the microscopic critical stress. The macroscopic yield stress is a global property arising from a number of active sources; for an overview on yield stress properties refer to Refs [l1. 121. It is difficult to relate the yield stress to the critical stress for the operation of a single source. Therefore the above result does not mean that it is more difficult for a dislocation to move at higher temperatures. something which would be contrary to experimental observations. For low temperatures the mobility of dislocations is low and there are no time constraints for the solid line, i.e. the closing time for the loop is very high for low temperatures. Figure 14 also shows the case where the closing time of the loop is kept below a line; threshold closing time (tLhr= 0. I s dashed tihr = 0.01 s dotted line). It can be seen that for the low-temperature range the critical stress is decreasing with increasing temperature. Because yield stress experiments are performed for finite (small) times the dashed and dotted curves reflect the behaviour of the materials in common constant strain rate
2652
GEIPEL:
DISLOCATION critical
b r i 0 a n i I ’ I a n n”I
800
SOURCE DYNAMICS IN SILICON stress
I ( ’ I * 8 I #“I
(L= 1 OOOnm) G( I I ( “I 8 0 n ’ 1 ’ 8 1 0 mn c ”
600 -j
Fig. 14. Critical stress vs T with L = 1 lrn. It can be seen that, approximately, 7cnloc T. The dashed (dotted) line shows the case where the closing time of the loop is kept below a threshold value fthr= 1 s (&h,= 0.1 s).
experiments better for the low-T range rather than the solid line. 6.
CONCLUSIONS
The aim of this paper was to simulate the operation of a Frank-Read source for dislocation multiplication in directionally bonded materials, e.g. tetrahedrally bonded semiconductors. As an example, the source dislocation was taken to be a perfect 60” dislocation in Si. However, the present simulations can be extended to other dislocations including partials by changing the appropriate Burgers vector and, for the latter case, introducing an additional term in the back stress, y/b, where y is the (intrinsic) stacking fault energy and b is the magnitude of the Burgers vector of the leading partial dislocation. In Si, for which y - 60 mJ/m*, this leads to an additional back stress of - 200 MPa. For the 30” and 90” partial dislocations, the numerical values for critical stress, etc., are similar to those for a 60” perfect dislocation (Table 2). Thus, at T = 1000 K, the minimum stress required for the operation of a l/6(112) partial dislocation can be estimated to be rmln- 250 MPa. Assuming that deformation twinning occurs by the operation of a partial Frank-Read source [13], this would be the minimum stress required for twin formation. However, this estimate assumes a stationary trailing partial and the repelling force between the leading and trailing partial has not been taken into account. In the case where the distance between the leading and trailing partial is small (large enough, however, for a faulted loop to form), this
repelling force may reduce the effective critical stress considerably. Therefore the above-estimated value for the operation of a partial dislocation (the critical stress for twinning) is probably too high, and the “correct” value may be much smaller. Twinning occurs predominantly at lower temperatures below the brittle-ductile transition temperature. According to the model of Ref. [13], for twin formation the leading and trailing partials must have different mobilities, something that is favored at lower temperatures. On the other hand, dislocation loops have to close, i.e. a critical stress has to be overcome. The present simulations indicate that the critical stress decreases with decreasing temperature, making closed loops easier to form at lower temperatures. This supports the proposed twinning model. The twinning model of Ref. [13] has also been extended to explain polytypic transformations in certain semiconductors with very low stacking-fault energies, e.g. SIC [ 141.The present algorithm can also be used for this purpose. If the applied stress is slightly higher than the critical stress, the back stress due to (a small number of) already existing dislocation loops upon a nascent loop (which is much smaller than the line tension stress) brings the operation of the Frank-Read source to a temporal stop. Hence, the number of closed loops in this model is highly temperature- and stress sensitive. The present algorithm can be used to make quantitative estimates of the stress and temperature required for polytypic transformations. In fact, the present results for a perfect dislocation source already indicate that
GEIPEL:
DISLOCATION
SOURCE
such phase transformations should be highly sensitive to temperature and stress. The extension of the algorithm to partial dislocations, including the consideration of the stacking-fault energy and the repelling force between the leading and trailing partials, is currently in progress. Acknowledgemenrs-A large part of this work was performed during my PhD research at the Department of Materials Science and Engineering, Case Western Reserve University, Cleveland OH, which was supported by grant #FG02-93ER45496 from the Department of Energy. I would like to acknowledge fruitful discussions with Pirouz Pirouz (DMSE) and Ladislas Kubin (CNRSONERA, Chitillon. France); many improvements to the manuscript were initiated by them.
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