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The energetics of the relaxation of misfit strain in thin epitaxial films by means of twinning
Arfa mafer. Vol. 46, No. 4, PP. 1087-1101,1998 0 1998Acta Metallurgica Inc. Published bv Elsevier Science Ltd. All riehts reserved Printed in &eat Britain 1359-6454/98$19.00+ 0.00 PII: s1359-6454(97)00319-4
THE ENERGETICS OF THE RELAXATION OF MISFIT STRAIN IN THIN EPITAXIAL FILMS BY MEANS OF TWINNING M. DYNNA’ and A. MARTY’ ‘Department of Materials Science and Engineering, McMaster University, Hamilton, Ontario, Canada, L8S 4L7 and ‘CEAIDltpartement de Recherche Fondamentale Sur la Mat&e Condensie, SPMM/NM, 38054 Grenoble Ctdex 9, France (Received 4 September 1997; accepted 9 September 1997)
Abstract-Exact solutions for the interaction energy between infinitely long parallel twins of alternating orientation terminating at a distance h below the free surface in an elastically isotropic half-space have been obtained through the summation of pairwise interaction energies between the partial dislocations making up the twins. The results are compared with those obtained when the elastic interaction between parallel twins is approximated by the interaction between parallel dislocations of Burgers vector mb, where m is the number of partial dislocations making up the twin and b is the Burgers vector of the partial dislocations. It is shown that for twins lying on {111) planes in a half-space for which the surface normal is (001) there is an excellent agreement between the two solutions except for extremely small values of the twin spacing which are never encountered in practice. It is also shown that for this geometry the use of the approximate solution for the elastic interaction between alternating twins in calculations of the energetics of the relaxation of misfit strain via twinning gives rise to virtually the same results as the exact solution. Additional calculations are performed in order to compare the energetics of an array of alternating twins relieving misfit strain in a thin Sio.sGeo.5 film on a Ge(OO1) substrate with those of an array of 60” perfect dislocations relieving misfit strain in the same film. The role played by the twin boundary energy in the evolution of twin morphology with increasing film thickness is illustrated through an examination of the relaxation process via twinning in thin Ag(OO1) and Pt(OO1) epitaxial films which are in a state of biaxial tension. A solution is also provided for the energetics of the relaxation of misfit strain in a linear isotropic elastic half-space by means of twin-like defects in which the defect thickness is allowed to vary continuously, allowing for the determination of effective average twin thicknesses and spacings which themselves vary continuously during the relaxation process. 0 1998 Acta Metallurgica Inc.
1. INTRODUCTION
Somewhat
It is often the case that the elastic strain energy due
to the lattice mismatch between a perfectly coherent thin epitaxial film and a substrate having a different lattice parameter is so great that it is energetically favourable for a relaxation of elastic strain to take place by means of the generation of defects which act to break the state of coherency with the substrate. The most common of such defects are misfit dislocations situated at the substrate/thin film interface, and their contribution to the overall free energy of a system comprised of a substrate and a misfitting epitaxial film may be evaluated using linear elasticity theory. Exact solutions for the elastic energy of periodic arrays of misfit dislocations in both isotropic [l-3] and anisotropic [4] half-spaces have been developed, allowing for the determination of the equilibrium density of dislocations at the substrate/thin film interface as a function of film thickness.
less common
defects
which
provide
for the relaxation of elastic strain in thin eiitaxial films are twins which run through the film from the surface to the substrate/thin film interface. This mode of relaxation has been observed in a number of systems, including Si-Ge multilayers deposited on Ge(OO1) [5,6], GaAs on Si(OO1) and Si(ll1) [7], and Au-Ni and Au-Cu layers deposited on Au(001) [8]. In the latter publication, a treatment of the energy of a periodic array of alternating twins in an elastically isotropic half-space was given which was mathematically exact except for the treatment of the interaction energy between parallel twins, which was the subject of an approximation. In the present work, we give an exact closedform solution for the energy of an array of infinitely long alternating twins relieving misfit strain in an elastically isotropic half-space. The energetics of such an array of alternating twins relieving misfit strain in a thin epitaxial film are compared with those of an array of 60” perfect dis-
1087
DYNNA and MARTY:
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MISFIT STRAIN IN THIN FILMS
-Ll-
Fig. 1. A linear array of alternating twins each made up of a cluster of partial dislocations. locations addition, the role evolution thickness.
relieving misfit strain in the same film. In calculations are performed which illustrate played by twin boundary energy in the of twin morphology with increasing film
2. THE ENERGY OF PERIODIC ARRAYS
which twins comprised of m partial dislocations having a Burgers vector of magnitude b were taken to interact elastically with each other in the same way as individual dislocations having a Burgers vector of magnitude mb. If the twins form an array such as that shown in Fig. 1 (where they alternate in orientation), then following an approach previously used to calculate the interaction energy between parallel alternating dislocations in an elastically isotropic half-space [3], the interaction energy per unit area between parallel twins may be written as
+2&zoth
OF CLUSTERED DISLOCATIONS IN AN ELASTICALLY ISOTROPIC HALF-SPACE
2.1. An approximate solution for the interaction energy between twins in a linear array of alternating twins The total elastic energy per the presence of an orthogonal a thin epitaxial film having a which differs from that of the written as &tal
-E -
film
+
&in
+
unit area due to array of twins in lattice parameter substrate may be
&h (
+ e,,;,;r2V
+:(,,nh(g)
-$$(sech”(g)
_ 2n2h2 -csch’(~)
-;I
) ;($tanh(??))
-coth(g))
+csch*(g))
+ p(mb)* cos’ p 4xD ln( $tanh(
g)),
+i]
(2)
Eint,twin/tilm
+ &,twin/twin(para)+ Eint,twin/twin(perp),(1) where &lm and Etwin are the self energies per unit area of the thin film and the twins, respectively, is the interaction energy per unit &t,twin/film area between the twins and the thin film, &t,twin/twin(para) is the interaction energy per unit area between parallel twins, and &t,twin/twin(perp) is the interaction energy per unit area between the twins which are at right angles to each other. Exact expressions for all of the terms in the above equation have been given in a previous paper [8], except for that of the interaction energy per unit area between parallel twins. The latter interaction energy was the subject of an approximation in
Fig. 2. An example of an array of dislocations for which an analytical expression for the interaction energy between parallel alternating dislocations may be obtained from the sum of the self energy of twins made up of m dislocations and the interaction energy between parallel alternating twins.
where p is the shear modulus of the elastic medium, v is Poisson’s ratio, h is the film thickness, D is the spacing of the twins, m is the number of partial dislocations making up the twin, b is the magnitude of the Burgers vectors of the partial dislocations of the twin, p is the angle between the Burgers vector of a misfit dislocation and the dislocation line, and cp is the angle between the misfit dislocation slip plane normal and the normal to the free surface. The above approximation becomes exact in the limit D-P co, and is excellent whenever the spacing of the twins is much greater than the twin thickness, although it is of questionable validity when the twin spacing is of the same order as the twin thickness. It is therefore of interest to search for an exact solution for the elastic interaction between parallel twins and to see if significant differences exist between approximate and exact calculations of the equilibrium values of m and D as a function of film thickness. 2.2. An exact solution for between alternating twins
the elastic interaction
As is shown in Appendix A, an exact expression for the interaction energy per unit area between parallel twins of thickness m and spacing D may be obtained through a summation of the pairwise in-
DYNNA and MARTY:
MISFIT STRAIN IN THIN FILMS
1089
teractions between partial dislocations making up the twins and is given by &t,twin/twin(para)= ‘imbq:n;l~~~‘[In($sinh(?D!))
+$!coth(T)
-?$?csch2(?!$)
-i]
cosh(4zh/D) - cos(2n(m - i&/D) (m - i)2d2 p ((m - i,‘di + 4@)[1 - cos(27C(m- i)d,/D)] -
12(m - i)2d,h2 + 16h4
(4r&/D)sinh(4rrh/D) ((m - i)‘dj + 4/V)* + cosh(4zh/D) - cos(27r(m - i)u’,/D)
+ (Sz2h2/D2)( 1 - cos(27r(m - i)dp/D)cosh(4nh/D)) (cosh(4nh/D) - cos(2+ +
pmb2 sin* fl sin’ cp
47r(l - u)D n2h2
2D2
- ~)c&/D))~
[In($tanh(g))+g(tanh(g)
-coth(g))
cosh(2zhlD)
- (16n2h2/D2)cos(n(m
+
- i)dp/D)sinh(2nh/D)
cos(27r(m - i)d,/D)
((m - i)2d,Z+ 4h2)2
- cosh(4zhlD)
- i)dp/D)cosh(2nh/D)(coshz(2zh/D) (cosh(4zh/D)
pmb2 cos2 /3 ln(-$tanh(;)) 4nD
+ cos(~~(m - i)d,/D)
1 + cos(rr(m - i)d,/D)
((m - i)“dp’+ 4h2)[ 1 - cos(n(m - i)d,/D)]
+ 4(m - i)2dih2 + 48h4 + (kh/D)cos(n(m
+ cos2(7c(m - i)d,/D)
- cos(27~(m - i)dp/D))2
- 2)
1
+pbzf’
cosh(2nhlD) - cos(x(m - i)d,/D) (m - i)2d2 ’ ((m - i)‘d,2 + 4h2)[1 - cos(x(m - i)d,/D)] >
X
_ ln
cosh(2xh/D) i
+ cos(z(m
- i)d,/D)
1 + cos(n(m - i)d,/D)
where dp is the spacing of the partial dislocations in an individual twin. Because of the restrictions placed on the allowed values of m, singularities arising from the factors [I - cos(n(m - i)d,/D)] and [l - cos(27r(m - i)d,/D)] in the denominators of some of the terms of the above expression are avoided. It is worth noting that one can obtain an analytical expression for the interaction energy between parallel dislocations in dislocation arrays such as the one shown in Fig. 2 by taking the sum of equation (3) and the expression for the interaction energy between dislocations in a twin made up of m partials: !J&z(m-i)((l-ucns’p)ln(l+$) I-1 + sin* p cos*rp
i)*di
(sech2($)+csch2(~))+i]+‘~~~~_‘~)i~n~i[ln((m-
cosh(2nh/D) - cos(rc(m - i)d,/D) X
1
P
12i2d2h2 + 16h4 (i”di + 4h2)*
_ sin2 p sin2 cp4i2dih2 + 48h4 (i2d2P + 4h*)*
>I ’
(4)
’
1 < m < D/d,,
(3)
setting D = md, and replacing dp with d, where d is now a variable representing the spacing of the dislocations in the array. A comparison of the behaviour of the approximate and exact interaction energies between parallel alternating twins in a thin layer of Sii _ xGex on Ge(OO1) may be seen in Fig. 3, where calculations were performed using p = 5.64 x 10” Pa and v = 0.20 (ihe elastic constants for pure Ge[lO]), b = 2.31 A, dp = 4.00 A, fi = 90”, and cp = 54.736”. Figure 3(a) shows the approximate and exact interaction energies per unit area for both alternating twins three atomic layers thick and alternating twins 10 atomic layers thick as a function of twin spacing at a Sii _ xGex thickness of 50 nm. It may be seen that on the scale of this figure there is no appreciable difference between the two solutions. A detailed view of a portion of Fig. 3(a) for m = 10 is shown in Fig. 3(b). In this range the difference between the solutions is approximately 0.0006 Jm-‘, which is very small
1090 (a)
DYNNA
and MARTY:
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FILMS
1
I
6
m=lO
:k m=3 ,a I
I
1
I
I
I
1
500
Twin spacing (nm) (b)
m=lO
5.681
5.677
5.676 30.000
*
I
30.002
I
I
30.004
1
I
30.006
*
I
30.008
*
30.010
Twin Spacing (nm) Fig. 3(a). A comparison between the approximate and exact interaction energies as a function of the twin spacing D between parallel alternating twins for which m = 3 and WI = 10 in a 50 nm Sii _ XGe, film on Ge(OO1). No difference between the two methods of evaluating the interaction energy is visible on the scale of the figure. (b) A comparison between the approximate and exact interaction energies as a function of the twin spacing D between parallel alternating twins for which m = 10 in a 50nm Sii _ xGex film on Ge(OO1) (detail).
DYNNA and MARTY:
MISFIT STRAIN IN THIN FILMS
relative to the interaction energy itself (-5.68 Jm-*). Even when the twin spacing is as small as 3 nm in a film thickness of 50 nm, the value of the approximate interaction energy is 660.706 Jme2, while that of the exact interaction energy is 660.293 Jm-2, for a relative error in the approximate energy of only 0.06%. Only in extreme cases such as when D is just greater than md, and the twins almost touch each other does one arrive at appreciable relative errors in the approximate interaction energy. Table 1 shows the relative error in the approximate interaction energy between parallel alternating twins at D = l.Olmd, and h = 1 nm (at a given spacing the error in the approximate interaction energy is generally greatest at small values of h) for various values of the twin thickness m. When m = 3, the error is minor, but the error becomes fairly large when m = 100. It should be kept in mind that such a situation could never be encountered in practice, for it would require misfit strains and twin boundary energies which are physically unobtainable. 3. THE ENERGETICS OF THE RELAXATION OF MISFIT STRAIN IN THIN EPITAXIAL FILMS BY MEANS OF PARTIAL DISLOCATIONS AND TWINNING
A knowledge of the exact expression for the interaction energy per unit area between parallel twins of alternating orientation makes possible an exact calculation of the total energy of the system via equation (l), where &m =
E
WI”
2p(l + u)f2h (1 _ “) .
=z wb2U-ucos*/oln g D
[
47c(l - u)
mpb2 sin* b
+ 8x(1 - u) + mUcore+
(
>
r0
(m-4 I+$
(1 -ucos2p)ln (
+ sin* p ~0s’ fp
(
P
>
12i2d2h2 + 16h4 (?d; + 4h*)*
4i2d2h2 + 48h4 - sin* fl sin* cp
- 1.
(&$ + 4h2)* > + sin cp
(6)
&t,twin/olm = and &t,twln/twin(perp)=
2pvh(mb)* sin* B cos* cp
(1 - u)D2
m 3 5 10 20 100
Relative
error
1.4% 2.9% 19.1% 35.9% 57.1%
In the above equations,
where afilm and asub are the lattice parameters of the film and substrate, respectively, r. is the core cut-off radius of the partial dislocations making up the twin, U,,,, is the core energy of the partial dislocations, and YT is the twin boundary energy. By minimizing the total energy of the system with respect to m and D for a given value of h, the equilibrium twin thickness and spacing may be determined as a function of film thickness. Although crystallographic changes due to the intersection of twins (either parallel or perpendicular) are not taken into account in the expression for the total energy per unit area, the extra energy due to such changes is a relatively minor term which, if included, would not significantly alter the equilibrium values of twin thickness and spacing calculated for a given value of h. Equation (1) allows for a study of the relaxation of misfit strain by means of twinning in a wide variety of geometries, but in what follows we will limit the discussion to twinning in f.c.c. and diamond cubic films having a (001) surface normal.
relaxation in Sio,-Geo.s on Ge(OOI) via perfect and partial dislocations
r-l
x
Table 1. Relative error in the approximate value of the elastic interaction between parallel alternating twins for selected values of m at D = l.Oimd, and h = 1 nm
3.1. A comparison between the energetics of’ strain
(1 - 2u) cos2 q - 2( 1 _ v)
&z
1091
@)
A coherent Sil _ xGex solid solution deposited on a Ge(OO1) substrate is in a state of biaxial tension, for asi = 5.4309 A and aGe= 5.6577 .k [l 11. It has been argued [6] on the basis of activation energy calculations that in such layers the nucleation of partial dislocations is favoured over the nucleation of perfect 60” dislocations for values off> 0.005, and it has been shown in the same work that Si3Geg heterostructures (having an average Si content of 25%) relax through the nucleation of individual 90” partials followed by additional halfloops which form microtwins. The thickness of such microtwins will, in general, increase with increasing film thickness in order to reduce the contribution of the twin boundary energy to the overall energy of the system. This coarsening takes place even though it leads to an increase in the interaction energy between the partial dislocations acting to relieve strain in the
1092
DYNNA
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system, for the energy due to twin boundaries increases more rapidly with h than does the interaction energy between twins made up of a cluster of m partials. If, in an array of alternating twins, all of the twins are constrained to have the same thickness, then at a given film thickness there is a unique value of m which gives rise to the lowest possible overall energy for the system. This is illustrated in Fig. 4, where E,,, for a thin film of Sic.sGea.s on a Ge(OO1) substrate has been minimized with respect to the twin spacing D for values of m ranging from 2 to 9, producing a family of curves which define an envelope representing the equilibrium total energy of the system as a function of h. This envelope is shown separately in Fig. 5, which gives the equilibrium total energy directly. In determining the total energy of the system, the following parameters were used: p = 5.64 x 10” Pa, v = 0.020, j3 = 90”, rp = 54.736”, b = 2.31 8, dp = 4.00 A, f = - 0.020. The core energy UC,,, for the partial dislocation was set equal to 0.65 eV/A, which is the value given by Nandedkar and Narayan [12] for a 90” partial in Ge, while the core radius ro, which was given in Ref. [12] as being ~5 A, has been set equal to 4 A for geometrical consistency. Since the cores of the partial dislocations are in very close proximity to each other, their actual contribution to the total energy of the system may differ from that of an equivalent number
of isolated partial dislocations; the above values for the core parameters are used for illustrative purposes only. Following the approximation often made for f.c.c. crystals [lo] of taking the twin boundary energy as equal to one-half of the intrinsic stacking fault energy (which for Ge has been determined to be 0.060 JmT2 [13]), the twin boundary energy Yr has been set at 0.030 Jmm2. The curves shown in Fig. 4 were calculated using the exact solution for the interaction energy between parallel alternating twins. A similar calculation of the minimum value of E,,, for specific values of m performed using the approximate solution for the interaction energy gave rise to virtually the same results, with differences in the two total energies typically being on the order of one part in ten thousand. This behaviour was the result of the fact that the equilibrium twin spacings in the Sio,sGeo.s film were fairly large and the fact that the interaction energy between the parallel twins represented a relatively small part of the total energy of the system. In general, the difference between the two solutions for the elastic interaction of parallel alternating twins is appreciable only when m is large and D is very small. It follows from this that the equilibrium twin thickness and spacing as well as the equilibrium energy of epitaxial systems encountered in practice calculated using the approximate solution for the interaction
12
10
8
0
I
0
I
200
I
I
400
I
I
600
I
I
800
I
1000
Film Thickness (nm) Fig. 4. A family of curves corresponding to values of the twin thickness ranging from 2 to 9 atomic layers in which I$,, for Sio.sGeo.s on Ge(OO1) has been minimized with respect to the twin spacing.
DYNNA and MARTY:
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1093
6
5
1
0
200
400
600
800
1000
Film Thickness (nm) Fig. 5. The equilibrium total energy of Sio.sGeo.son Ge(OO1)as a function of film thickness. energy between parallel twins differ negligibly from those determined using the exact solution for the twin interaction. In particular, the calculations previously performed for the relaxation of elastic strain via twinning in an equiatomic AuCu alloy [8] would not be altered in any significant way by the substitution of the exact energy for the elastic interaction between parallel twins. The evolution of the equilibrium value of m (obtained for a given film thickness by the selection of the curve of Fig. 4 which gave rise to the lowest minimized with respect to D at that value of Et,, thickness) is shown in Fig. 6. Transitions in the value of m occur whenever the contribution of the twin boundary energy to the overall energy of the system becomes sufficiently great to allow for a packing of partial dislocations into thicker twins. The discontinuities in the value of m present in Fig. 6 are due to the fact that in the mathematical formulation of the relaxation all of the twins acting to relieve misfit strain are constrained to have the same thickness. In practice there is no such restriction placed on the twins which act to relieve misfit strain, and twins of different thickness may coexist in a film of a given thickness. A gradual and continuous increase in the average value of the twin thickness takes place, and in principle all that is necessary for a twin of a given thickness to exist in a thin film is that the sum of the interaction energy of the twin with the epilayer and the self-energy of
the twin be less than zero. The critical thickness for a twin is given implicitly by setting equation (6) equal to the negative of equation (7) and the results of such calculations for twins relieving misfit strain in Sio.sGeo.s on Ge(OO1) are given in Table 2. Thus a twin made up of eight partial dislocations is stable in films greater than 19 nm in thickness, even though an array made up of twins eight atomic layers thick is not the energetically favoured form until the film has reached a thickness of 780 nm. Perhaps the most important feature characteristic of the relaxation of misfit strain in thin films by means of twinning is the impossibility of an asymptotic approach to a limiting value of the equilibrium energy of the system ash + 00. Because twin boundaries are always present in the thin film, the equilibrium energy of the system increases monotonically as the film thickness increases. This is in contrast to the case of relaxation of elastic strain via orthogonal arrays of alternating perfect dislocations, where a limiting value of the equilibrium energy of the system is reached ash 3 co. Thus even though it is easier to nucleate 90” partial dislocations than 60” perfect dislocations in a Sit,.sGeo.s film on Ge(OOl), and even though it may be initially energetically favourable to relieve misfit in such a film by means of 90” partial as opposed to 60” perfect dislocations, a point will be reached when the thickness of the film is large enough to make strain relaxation via 60” dislocations the ener-
1094
DYNNA and MARTY:
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9
8
6
Es
4
3
2
I
1 0
I
I
a
200
400
I
I 600
I
I 800
I
1000
Film Thickness (m-n) Fig. 6. The equilibrium value of the twin thickness m as a function of film thickness in Sio.sGeo.s on Ge(OO1).
getically preferable case. This is shown in Fig. 7, where the equilibrium energy of the system during the relaxation of misfit strain in Sio,sGeo.s on Ge(OO1) via orthogonal arrays of partial 90” dislocations and twins is compared with that of a process involving relaxation via orthogonal arrays of perfect 60” dislocations. The curve representing relaxation via partial dislocations is an extension of the curve of Fig. 5, which was calculated for twins, to include isolated partial dislocations bounded by stacking faults. The method of determining the equilibrium energy of orthogonal arrays of partial dislocations is similar to that used for perfect dislocations, with a term representing the stacking fault energy being added to the self energy of the dislocation (141. In the present case the stacking fault energy was set equal to 0.060 Jm-*, [13],
Table 2. The critical thickness for individual twins in Sio,&eo.~ on Ge(001) m
2 3 4 5 ; 8 9 10
Critical thickness (nm) 6.5 8.4 10.4 12.3 14.2 16.2 18.1 20.1 22.0
resulting in a transition from isolated partial dislocations to twins two atomic layers in thickness at a film thickness between 29 and 30nm. The curve representing relaxation via perfect dislocations was determined following the method of Ref. [3]. Figure 7 shows that the critical thickness of 90” partial dislocations (4.9 nm) is less than that of 60” perfect dislocations (6.2 nm), and that it is energetically favourable to relieve misfit by means of isolated partial dislocations up to a film thickness of 13 nm, after which relaxation via perfect dislocations is energetically preferred. Thus arrays of twins of identical thickness are never energetically preferred over arrays of 60” perfect dislocations in Sia,5Gee.s on Ge(OO1). In films encountered in practice, however, it is usually not possible for the transition from relaxation via partial to relaxation via perfect dislocations to take place because of the difficulty in nucleating perfect dislocations, and as a result the approach to full (2%) relaxation is slower than in the case of perfect dislocations. This may be seen in Fig. 8, which compares the equilibrium relaxation in Sia.sGeo.5 on Ge(OO1) by means of mechanisms involving partial and perfect dislocations. Discontinuities in the relaxation curve for the relief of misfit strain via microtwins arise from the restriction that the microtwins are all taken to be made up of the same integral number of partial dislocations; this point is addressed in Section 3.3. At
DYNNA
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FILMS
6
5
4
Partial dislocations
5’
and microtwins
a 2
Perfect dislocations
1 / I
I
0 0
I
,
I
I
a 1000
800
600
400
200
I
I
Film thickness (nm) Fig. 7. A comparison between the equilibrium total energy in SiO.sGeO.S on Ge(OO1) associated with relaxation via 60” nerfect dislocations and that associated with relaxation via 90” partial dislocations and microtwins.
Perfect dislocations 2.0
Micr otw ins>
0.5
I
I
0
I 200
I
I
400
I
I 600
8
I 800
I
I 1000
Film Thickness (nm) Fig. 8. A comparison between with relaxation via 60” perfect
the equilibrium percent relaxation in Sia sGe,,s on Ge(OO1) associated dislocations and that associated with relaxation via 90” partial dislocations and microtwins.
DYNNA
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a film thickness of 1000 nm, microtwins made up of eight partial dislocations provide for a 1.93% relaxation, while perfect dislocations allow for a 1.99% relaxation. It is to be expected that this difference in the rate of equilibrium relaxation would be reflected in the actual kinetics of relaxation in a thin film, with a process involving microtwins being considerably slower than the corresponding perfect dislocation process.
Table 3. Equilibrium values of m in Ag(OO1) and Pt(OO1) epitaxial films in a state of biaxial tension (f = - 0.020) at selected values of the twin thickness h
h bm) 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000
3.2. The influence of twin boundary energy on twin morphology
The twin boundary energy is of great importance in the energetics of the relaxation of misfit strain, and plays a key role in determining the evolution of twin morphology during the growth of films which relax via a twinning mechanism. In general, a large twin boundary energy leads to a rapid coarsening of twins as the film thickness increases, while a small twin boundary energy leads to a much slower twin coarsening. An interesting comparison is offered by the relaxation via twinning of thin Ag(OO1) and Pt(OO1) films-both metals are f.c.c., but the twin boundary energy of Pt (0.161 Jmp2) is 20 times greater than that of Ag (0.008 Jme2) [lo]. The results of calculations of the equilibrium value of m (obtained using the method of Section 3.1.) for both Ag and Pt are given in Table 3. Both metals were taken to be under a 2% tensile strain when fully coherent
m&
WPf
2 2 7 10 12 14 16 18 19 20 22 23
with the substrate, i.e. f = - 0.020. In the case of Ag, p = 3,10x 10” Pa and v = 0.412 [lo], while b = 1.67 A for an a/6(211) partial dislocation and dP= 2.89 A. We have used Cottrell’s approximation [15] for the core energy of a dislocation and set ucore= 2x(1pb2 - “)2 -
4.0 x lo-” J/m = 0.25 eV/A
10
8
400
600
Film Thickness (mn) Fig.
9. The equilibrium
value of m in Si0,5Geo,s on Ge(OO1) associated with a fictitious as a function of film thickness. Compare with Fig. 6.
twin-like
defect
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1097
2.0
1.5
1.4 0
200
400
600
800
1000
Film Thickness (nm) Fig.
IO. Equilibrium
percent
relaxation associated with the relief of misfit Ge(OO1) by means of twin-like defects.
rO=dp= 2.89 A. For Pt, cl1 = 3.461 x 10” Pa, cr2=2.507 x 10” Pa, and cM=0.765 x 10” Pa [16], which give the Voigt averages p = 6.51 x 10” Pa and v = 0.393. The magnitude of the Burgers vector of a/6(211) partial dislocations in Pt is 1.60 A, and d,, = 2.77 A. We have again set and
pb’
ucore= 241 - “)2 -- 7.2 x lo-*’ J/m = 0.45 eV/A, and r. = dp =2.77 A. It may be seen that the coarsening of twins in Pt is much more rapid than in Ag, with the equilibrium value of m in a 1 pm Pt film being 3.3 times greater than the corresponding value in a 1 pm Ag film.
strain
in Sio.sGeo5
3.3. A model for the continuous strain via twinning
relaxation
on
of misfit
The formulation of the energetics of an array of twins in a thin epitaxial film on the basis of all of the twins in the array consisting of an identical integral number of partial dislocations leads to a number of artificial effects in calculations of the equilibrium properties of the system, including discontinuities in the equilibrium relaxation as a function of the film thickness h such as those seen in Fig. 8. By allowing the thickness of the twins to vary continuously, a twin-like defect is produced which gives rise to a smooth and continuous equilibrium relaxation as a function of h. As is demonstrated in Appendix C, the self energy per unit area of a one-dimensional array of twin-like defects of thickness m and spacing D is
1098
DYNNA and MARTY:
MISFIT STRAIN IN THIN FILMS
where dr, is the spacing of the partial dislocations in the physical twin from which the twin-like defect is derived. This expression for the self energy of an array of twin-like defects in conjunction with the approximate expression for the interaction energy between parallel alternating twins (where m now varies continuously) and the other terms in equation (1) allow for the calculation of equilibrium values of m, D, and percent relaxation which vary continuously with the film thickness h. Figure 9 shows the equilibrium value of m for Si,,,Ge,,5 on Ge(OO1) where m, which may be taken to represent the average twin thickness present at equilibrium in an actual thin film, varies continuously with h. The equilibrium relaxation via twin-like defects in the same film may be seen in Fig. 10, which is similar to that of the twins in Fig. 8 but without the discontinuities, and is a good approximation of the relaxation which would be found at equilibrium when misfit strain is relieved by orthogonal arrays of twins in which no restrictions are placed on the thickness of the twins in the array.
2. Gosling, T. J., Willis, J. R., Bullough, R. and Jain, S. C., J. Appl. Phys., 1993, 73, 8291. 3. Dynna, M., Okada, T. and Weatherly, G. C., Acta Metall. Mater., 1994, 42, 1661. 4. Gosling, T. J. and Willis, J. R., Phil. Mug. A, 1994, 69, 65. 5. Wegscheider, W., Eberl, K., Abstreiter, G., Cerva, H. and Oppolzer, H., Appl. Phys. Lett., 1990, 57, 1496. 6. Wegscheider, W. and Cerva, H., J. Vat. Sci. Tech. B, 1993, 11, 1056. I. Neething, J. H. and Alberts, V., J. Appl. Phys., 1994, 75, 3435. 8. Dynna, M., Marty, A., Gilles, B. and Patrat, G., Acta mater., 1997, 45, 251. 9. Wolfram, S., The Mathematics Book. 3rd edn.
Wolfram University Press, Media/Cambridge Cambridge, 1996. 10. Hirth, J. P. and Lothe, J. Theory of Dislocations. McGraw-Hill, New York, 1982. 11. Pearson, W. B., A Handbook of Lattice Spacings and Structures of Metals and Alloys. Vol. 2. Pergamon, London, 1967. 12. Nandedkar, A. S. and Narayan, J., Phil. Mag. A, 1990, 61, 873.
13. Gomez, A., Cockayne, D. J., Hirsch, P. B. and Vitek, V., Phil. Mag., 1975, 31, 105. 14. Dynna, M., Marty, A., Gilles, B. and Patrat, G., Acta mater., 1996, 44, 4417. A. H., Dislocations and Plastic Flow in Crystals. Oxford University Press, Oxford, 1953. 16. CRC Handbook of Chemistry Physics, 76th edn. CRC
15. Cottrell,
REFERENCES 1. Willis, J. R., Jain, S. C. and Bullough, R., Phil. Masg. A, 1990, 62, 115.
Press, New York, 1995.
APPENDIX A Evaluation of the elastic interaction between partial dislocations in separate twins in an elastically isotropic half-space
Consider an array of N alternating twins comprised of clusters of m partial dislocations having a spacing D occupying a length L, where L is large. Then edge effects are minor, and to an excellent approximation the interaction energy per unit length between the components of the partial dislocations in separate twins in the array which act to relieve misfit strain is m-l uint
=
(N - 1) mu,+
c
i[u(O-(rn-i)d,)
+
U(D+(m-i)d,)]
i=l
m-l + W - 2) m&o + ci=,
i[U(2D-++dp)
+
1 1
u(ZD+(rn-i)dp)]
+
,
1 < m < D/d,,
t-41)
where dp is the spacing between the partial dislocations which make up a twin. Note that the interactions between partial dislocations in the same twin are not included in this summation-they are taken into account in the calculation of the self energy of the twin as given by equation (5). In the limit of L becoming infinitely large, an exact expression for the interaction energy per unit area between those components of the partial dislocations in the array which act to relieve misfit strain is obtained: m-l
&t
= A 2
n=l
mUnD
+ c
i[U(nD-(m-i&,)
+
U(nD+(m-nd,)]
i=l
1
,
1 tm
(A21
For those components of the partial dislocations in separate twins not relieving misfit strain, the interaction energy per unit area is m-l Eint =A
mb+I)D
+2
+
m-1 mU2nD
C
+
c
c i[u((2n+l)D-(m-Qdp) i=l
i[u(2nD-(m-i)dp)
+
+
u((2,+l)D+(tA)d,)]
U(2nD+(m-Od,)]
II %
1
1 < m c D/d,,.
G43)
DYNNA
and MARTY:
Substituting the values for the pairwise relieving misfit strain,
For those components
not relieving
interaction
misfit strain,
MISFIT energies
substitution
STRAIN
[3] between
IN THIN the partial
of the pairwise
FILMS dislocations
interaction
energies
1099 gives, for the component
gives
(A51
A closed form exists for the infinite series and products appearing in equations (A4) and (A5) (as may be verified using symbolic mathematics software [9]), leading to the expression for the interaction energy per unit area between parallel alternating twins given by equation (3).
APPENDIX
B
An exact solution for the interaction energy between twins
ofsimilar orientation
An exact solution for the interaction energy between similar twins may be obtained using the general procedure outlined in Appendix A, and is given here for completeness. Since the twins are of similar orientation, one has for all components of the partial dislocations making up the twins
/&, =;
5 ?I=,
m-l
~U,D +
c ,=I
i[u(i(,D-(,-i)dp)
+
U(nD+(m-tldp)]
)
1
1
(B’)
1100
DYNNA and MARTY:
MISFIT STRAIN IN THIN FILMS
Substituting the values of the pairwise interaction energies between the partial dislocations in separate twins in the array and taking the closed form of the infinite series gives (for similar twins):
i”Fji~~_~~~~[In(~sinh(~))+~~oth(~) -!?$cs&2(~) -i]
Emt.twin/twin(para) =
pb2 sin2 j cos2 cpm-l
+ -
4n(l - u)D
i=, c.[
cosh(4nh/D) - cos(2n(m - i)d,,/D)
2 In (m - i)‘d2
p ((m - i)2d,2 + 4h2)[1 - cos(27c(m- i)d,/D)] >
(
12(m - i)2dih2 + 16h4
(4nh/D)sinh(4zh/D) ((m - i)2di + 4h2)2 ’ cosh(4nh/D) - cos(27t(m - i)d,/D)
1 +~~~~~~~2~[In(~;sinh(~))-~~ot + (8rr2h2/02)(1 - cos(2n(m - i)d,/D)cosh(4rch/D))
(cosh(4nhlD) - cos(27c(m- i)dp/D))2
pb2 sin2 fi sin2 cpm-’ +
47c(l - u)D
i=, c,[
cosh(4ah/D) - cos(27c(m - i)$/D)
1 In (m - i)2d2
p ((m - dd;
(
+ 4(m - i)2d;h2 + 48h4
(4nh/D)sinh(4nh/D)
((m - i)‘di + 4h2)’ - cosh(4zh/D) _ (8n2h2/D2)(cos(2n(m
+ 4h2)[1 - cos(2rc(m - i)d,/D)] >
- cos(2x(m - i)d,/D)
- i)dp/D)cosh(4nh/D)
1 4RD
- 1) + pmb2 cos2 /I
cosh(4nh/D) - cos(2rr(m - i)dp/D))2 (m - i)2d2 '
ln( $sinh(
y))
cosh(4xh/D) - cos(2n(m - i)d,/D) ((m - i)2d,2 + 4h2)[1 - cos(2(m - i)$/D)]
> ’ ’ < m < D’dp’
(B2)
APPENDIX C The twin-like defect
The twin-like defect can be constructed from a twin by replacing the m partial dislocations that make up a twin with an infinite number of infinitesimal dislocations having the same Burgers vector content mb. In the case of a twin, the energy per unit length may be written as
where Use,f = mpb’(1 - vcos’ /I) ln 2 + mph’ sin’ /I cos2q7-$3)+m&,, 47r(l - V) 87r(l - V) ( r0 > is the self energy per unit length of the partial dislocations making up the twin, 2yrh/sincp is the twin boundary energy per unit length, and Urni is the interaction energy per unit length between the partials making up the twin and is given m-l uint
=
1 (m - I)((:1- ”COs2 8) UI +
(sin2 B C0s2 cp)U,
+
(sin2 /3sin2 q)Us),
r=l
with pb2 In 4ir(l - U)
u, =p
and
If the partial dislocations having a Burgers vector of magnitude b are broken into iv=& partials each having a Burgers vector of magnitude A&!?=&
dp
N’
(W
DYNNA then the sum of the self-energies
and MARTY:
of the dislocations
MISFIT
STRAIN
IN THIN
FILMS
1101
tends to zero in the limit N -+ 00, for 2
lim mN(Ab)2 N-oc This includes
the core energies
of the dislocations,
= ,jimXmN
$
= 0.
0
each of which may be approximated
by
uAb2 2n(l - “)? For the interaction
energy,
we have for a given value of h Nm-I
L’i”t = C (Nm - i)(Ab)* 4, ($$I r=l
= m(d~i’(rndp
- iAx)(t)2U:nt(iAx)A.x
(C3)
In the limit as N + cc and Ax + 0, we obtain
s 4
ol”, =
0
Evaluating the above integral by equation (9).
(mdp - x) $ 0
*C~:,,(x) dx
(C4)
P
leads to the self energy per unit area of a one-dimensional
array
of twin-like
defects given
THE ENERGETICS OF THE RELAXATION OF MISFIT STRAIN IN THIN EPITAXIAL FILMS BY MEANS OF TWINNING M. DYNNA’ and A. MARTY’ ‘Department of Materials Science and Engineering, McMaster University, Hamilton, Ontario, Canada, L8S 4L7 and ‘CEAIDltpartement de Recherche Fondamentale Sur la Mat&e Condensie, SPMM/NM, 38054 Grenoble Ctdex 9, France (Received 4 September 1997; accepted 9 September 1997)
Abstract-Exact solutions for the interaction energy between infinitely long parallel twins of alternating orientation terminating at a distance h below the free surface in an elastically isotropic half-space have been obtained through the summation of pairwise interaction energies between the partial dislocations making up the twins. The results are compared with those obtained when the elastic interaction between parallel twins is approximated by the interaction between parallel dislocations of Burgers vector mb, where m is the number of partial dislocations making up the twin and b is the Burgers vector of the partial dislocations. It is shown that for twins lying on {111) planes in a half-space for which the surface normal is (001) there is an excellent agreement between the two solutions except for extremely small values of the twin spacing which are never encountered in practice. It is also shown that for this geometry the use of the approximate solution for the elastic interaction between alternating twins in calculations of the energetics of the relaxation of misfit strain via twinning gives rise to virtually the same results as the exact solution. Additional calculations are performed in order to compare the energetics of an array of alternating twins relieving misfit strain in a thin Sio.sGeo.5 film on a Ge(OO1) substrate with those of an array of 60” perfect dislocations relieving misfit strain in the same film. The role played by the twin boundary energy in the evolution of twin morphology with increasing film thickness is illustrated through an examination of the relaxation process via twinning in thin Ag(OO1) and Pt(OO1) epitaxial films which are in a state of biaxial tension. A solution is also provided for the energetics of the relaxation of misfit strain in a linear isotropic elastic half-space by means of twin-like defects in which the defect thickness is allowed to vary continuously, allowing for the determination of effective average twin thicknesses and spacings which themselves vary continuously during the relaxation process. 0 1998 Acta Metallurgica Inc.
1. INTRODUCTION
Somewhat
It is often the case that the elastic strain energy due
to the lattice mismatch between a perfectly coherent thin epitaxial film and a substrate having a different lattice parameter is so great that it is energetically favourable for a relaxation of elastic strain to take place by means of the generation of defects which act to break the state of coherency with the substrate. The most common of such defects are misfit dislocations situated at the substrate/thin film interface, and their contribution to the overall free energy of a system comprised of a substrate and a misfitting epitaxial film may be evaluated using linear elasticity theory. Exact solutions for the elastic energy of periodic arrays of misfit dislocations in both isotropic [l-3] and anisotropic [4] half-spaces have been developed, allowing for the determination of the equilibrium density of dislocations at the substrate/thin film interface as a function of film thickness.
less common
defects
which
provide
for the relaxation of elastic strain in thin eiitaxial films are twins which run through the film from the surface to the substrate/thin film interface. This mode of relaxation has been observed in a number of systems, including Si-Ge multilayers deposited on Ge(OO1) [5,6], GaAs on Si(OO1) and Si(ll1) [7], and Au-Ni and Au-Cu layers deposited on Au(001) [8]. In the latter publication, a treatment of the energy of a periodic array of alternating twins in an elastically isotropic half-space was given which was mathematically exact except for the treatment of the interaction energy between parallel twins, which was the subject of an approximation. In the present work, we give an exact closedform solution for the energy of an array of infinitely long alternating twins relieving misfit strain in an elastically isotropic half-space. The energetics of such an array of alternating twins relieving misfit strain in a thin epitaxial film are compared with those of an array of 60” perfect dis-
1087
DYNNA and MARTY:
1088
MISFIT STRAIN IN THIN FILMS
-Ll-
Fig. 1. A linear array of alternating twins each made up of a cluster of partial dislocations. locations addition, the role evolution thickness.
relieving misfit strain in the same film. In calculations are performed which illustrate played by twin boundary energy in the of twin morphology with increasing film
2. THE ENERGY OF PERIODIC ARRAYS
which twins comprised of m partial dislocations having a Burgers vector of magnitude b were taken to interact elastically with each other in the same way as individual dislocations having a Burgers vector of magnitude mb. If the twins form an array such as that shown in Fig. 1 (where they alternate in orientation), then following an approach previously used to calculate the interaction energy between parallel alternating dislocations in an elastically isotropic half-space [3], the interaction energy per unit area between parallel twins may be written as
+2&zoth
OF CLUSTERED DISLOCATIONS IN AN ELASTICALLY ISOTROPIC HALF-SPACE
2.1. An approximate solution for the interaction energy between twins in a linear array of alternating twins The total elastic energy per the presence of an orthogonal a thin epitaxial film having a which differs from that of the written as &tal
-E -
film
+
&in
+
unit area due to array of twins in lattice parameter substrate may be
&h (
+ e,,;,;r2V
+:(,,nh(g)
-$$(sech”(g)
_ 2n2h2 -csch’(~)
-;I
) ;($tanh(??))
-coth(g))
+csch*(g))
+ p(mb)* cos’ p 4xD ln( $tanh(
g)),
+i]
(2)
Eint,twin/tilm
+ &,twin/twin(para)+ Eint,twin/twin(perp),(1) where &lm and Etwin are the self energies per unit area of the thin film and the twins, respectively, is the interaction energy per unit &t,twin/film area between the twins and the thin film, &t,twin/twin(para) is the interaction energy per unit area between parallel twins, and &t,twin/twin(perp) is the interaction energy per unit area between the twins which are at right angles to each other. Exact expressions for all of the terms in the above equation have been given in a previous paper [8], except for that of the interaction energy per unit area between parallel twins. The latter interaction energy was the subject of an approximation in
Fig. 2. An example of an array of dislocations for which an analytical expression for the interaction energy between parallel alternating dislocations may be obtained from the sum of the self energy of twins made up of m dislocations and the interaction energy between parallel alternating twins.
where p is the shear modulus of the elastic medium, v is Poisson’s ratio, h is the film thickness, D is the spacing of the twins, m is the number of partial dislocations making up the twin, b is the magnitude of the Burgers vectors of the partial dislocations of the twin, p is the angle between the Burgers vector of a misfit dislocation and the dislocation line, and cp is the angle between the misfit dislocation slip plane normal and the normal to the free surface. The above approximation becomes exact in the limit D-P co, and is excellent whenever the spacing of the twins is much greater than the twin thickness, although it is of questionable validity when the twin spacing is of the same order as the twin thickness. It is therefore of interest to search for an exact solution for the elastic interaction between parallel twins and to see if significant differences exist between approximate and exact calculations of the equilibrium values of m and D as a function of film thickness. 2.2. An exact solution for between alternating twins
the elastic interaction
As is shown in Appendix A, an exact expression for the interaction energy per unit area between parallel twins of thickness m and spacing D may be obtained through a summation of the pairwise in-
DYNNA and MARTY:
MISFIT STRAIN IN THIN FILMS
1089
teractions between partial dislocations making up the twins and is given by &t,twin/twin(para)= ‘imbq:n;l~~~‘[In($sinh(?D!))
+$!coth(T)
-?$?csch2(?!$)
-i]
cosh(4zh/D) - cos(2n(m - i&/D) (m - i)2d2 p ((m - i,‘di + 4@)[1 - cos(27C(m- i)d,/D)] -
12(m - i)2d,h2 + 16h4
(4r&/D)sinh(4rrh/D) ((m - i)‘dj + 4/V)* + cosh(4zh/D) - cos(27r(m - i)u’,/D)
+ (Sz2h2/D2)( 1 - cos(27r(m - i)dp/D)cosh(4nh/D)) (cosh(4nh/D) - cos(2+ +
pmb2 sin* fl sin’ cp
47r(l - u)D n2h2
2D2
- ~)c&/D))~
[In($tanh(g))+g(tanh(g)
-coth(g))
cosh(2zhlD)
- (16n2h2/D2)cos(n(m
+
- i)dp/D)sinh(2nh/D)
cos(27r(m - i)d,/D)
((m - i)2d,Z+ 4h2)2
- cosh(4zhlD)
- i)dp/D)cosh(2nh/D)(coshz(2zh/D) (cosh(4zh/D)
pmb2 cos2 /3 ln(-$tanh(;)) 4nD
+ cos(~~(m - i)d,/D)
1 + cos(rr(m - i)d,/D)
((m - i)“dp’+ 4h2)[ 1 - cos(n(m - i)d,/D)]
+ 4(m - i)2dih2 + 48h4 + (kh/D)cos(n(m
+ cos2(7c(m - i)d,/D)
- cos(27~(m - i)dp/D))2
- 2)
1
+pbzf’
cosh(2nhlD) - cos(x(m - i)d,/D) (m - i)2d2 ’ ((m - i)‘d,2 + 4h2)[1 - cos(x(m - i)d,/D)] >
X
_ ln
cosh(2xh/D) i
+ cos(z(m
- i)d,/D)
1 + cos(n(m - i)d,/D)
where dp is the spacing of the partial dislocations in an individual twin. Because of the restrictions placed on the allowed values of m, singularities arising from the factors [I - cos(n(m - i)d,/D)] and [l - cos(27r(m - i)d,/D)] in the denominators of some of the terms of the above expression are avoided. It is worth noting that one can obtain an analytical expression for the interaction energy between parallel dislocations in dislocation arrays such as the one shown in Fig. 2 by taking the sum of equation (3) and the expression for the interaction energy between dislocations in a twin made up of m partials: !J&z(m-i)((l-ucns’p)ln(l+$) I-1 + sin* p cos*rp
i)*di
(sech2($)+csch2(~))+i]+‘~~~~_‘~)i~n~i[ln((m-
cosh(2nh/D) - cos(rc(m - i)d,/D) X
1
P
12i2d2h2 + 16h4 (i”di + 4h2)*
_ sin2 p sin2 cp4i2dih2 + 48h4 (i2d2P + 4h*)*
>I ’
(4)
’
1 < m < D/d,,
(3)
setting D = md, and replacing dp with d, where d is now a variable representing the spacing of the dislocations in the array. A comparison of the behaviour of the approximate and exact interaction energies between parallel alternating twins in a thin layer of Sii _ xGex on Ge(OO1) may be seen in Fig. 3, where calculations were performed using p = 5.64 x 10” Pa and v = 0.20 (ihe elastic constants for pure Ge[lO]), b = 2.31 A, dp = 4.00 A, fi = 90”, and cp = 54.736”. Figure 3(a) shows the approximate and exact interaction energies per unit area for both alternating twins three atomic layers thick and alternating twins 10 atomic layers thick as a function of twin spacing at a Sii _ xGex thickness of 50 nm. It may be seen that on the scale of this figure there is no appreciable difference between the two solutions. A detailed view of a portion of Fig. 3(a) for m = 10 is shown in Fig. 3(b). In this range the difference between the solutions is approximately 0.0006 Jm-‘, which is very small
1090 (a)
DYNNA
and MARTY:
MISFIT
STRAIN
IN THIN
FILMS
1
I
6
m=lO
:k m=3 ,a I
I
1
I
I
I
1
500
Twin spacing (nm) (b)
m=lO
5.681
5.677
5.676 30.000
*
I
30.002
I
I
30.004
1
I
30.006
*
I
30.008
*
30.010
Twin Spacing (nm) Fig. 3(a). A comparison between the approximate and exact interaction energies as a function of the twin spacing D between parallel alternating twins for which m = 3 and WI = 10 in a 50 nm Sii _ XGe, film on Ge(OO1). No difference between the two methods of evaluating the interaction energy is visible on the scale of the figure. (b) A comparison between the approximate and exact interaction energies as a function of the twin spacing D between parallel alternating twins for which m = 10 in a 50nm Sii _ xGex film on Ge(OO1) (detail).
DYNNA and MARTY:
MISFIT STRAIN IN THIN FILMS
relative to the interaction energy itself (-5.68 Jm-*). Even when the twin spacing is as small as 3 nm in a film thickness of 50 nm, the value of the approximate interaction energy is 660.706 Jme2, while that of the exact interaction energy is 660.293 Jm-2, for a relative error in the approximate energy of only 0.06%. Only in extreme cases such as when D is just greater than md, and the twins almost touch each other does one arrive at appreciable relative errors in the approximate interaction energy. Table 1 shows the relative error in the approximate interaction energy between parallel alternating twins at D = l.Olmd, and h = 1 nm (at a given spacing the error in the approximate interaction energy is generally greatest at small values of h) for various values of the twin thickness m. When m = 3, the error is minor, but the error becomes fairly large when m = 100. It should be kept in mind that such a situation could never be encountered in practice, for it would require misfit strains and twin boundary energies which are physically unobtainable. 3. THE ENERGETICS OF THE RELAXATION OF MISFIT STRAIN IN THIN EPITAXIAL FILMS BY MEANS OF PARTIAL DISLOCATIONS AND TWINNING
A knowledge of the exact expression for the interaction energy per unit area between parallel twins of alternating orientation makes possible an exact calculation of the total energy of the system via equation (l), where &m =
E
WI”
2p(l + u)f2h (1 _ “) .
=z wb2U-ucos*/oln g D
[
47c(l - u)
mpb2 sin* b
+ 8x(1 - u) + mUcore+
(
>
r0
(m-4 I+$
(1 -ucos2p)ln (
+ sin* p ~0s’ fp
(
P
>
12i2d2h2 + 16h4 (?d; + 4h*)*
4i2d2h2 + 48h4 - sin* fl sin* cp
- 1.
(&$ + 4h2)* > + sin cp
(6)
&t,twin/olm = and &t,twln/twin(perp)=
2pvh(mb)* sin* B cos* cp
(1 - u)D2
m 3 5 10 20 100
Relative
error
1.4% 2.9% 19.1% 35.9% 57.1%
In the above equations,
where afilm and asub are the lattice parameters of the film and substrate, respectively, r. is the core cut-off radius of the partial dislocations making up the twin, U,,,, is the core energy of the partial dislocations, and YT is the twin boundary energy. By minimizing the total energy of the system with respect to m and D for a given value of h, the equilibrium twin thickness and spacing may be determined as a function of film thickness. Although crystallographic changes due to the intersection of twins (either parallel or perpendicular) are not taken into account in the expression for the total energy per unit area, the extra energy due to such changes is a relatively minor term which, if included, would not significantly alter the equilibrium values of twin thickness and spacing calculated for a given value of h. Equation (1) allows for a study of the relaxation of misfit strain by means of twinning in a wide variety of geometries, but in what follows we will limit the discussion to twinning in f.c.c. and diamond cubic films having a (001) surface normal.
relaxation in Sio,-Geo.s on Ge(OOI) via perfect and partial dislocations
r-l
x
Table 1. Relative error in the approximate value of the elastic interaction between parallel alternating twins for selected values of m at D = l.Oimd, and h = 1 nm
3.1. A comparison between the energetics of’ strain
(1 - 2u) cos2 q - 2( 1 _ v)
&z
1091
@)
A coherent Sil _ xGex solid solution deposited on a Ge(OO1) substrate is in a state of biaxial tension, for asi = 5.4309 A and aGe= 5.6577 .k [l 11. It has been argued [6] on the basis of activation energy calculations that in such layers the nucleation of partial dislocations is favoured over the nucleation of perfect 60” dislocations for values off> 0.005, and it has been shown in the same work that Si3Geg heterostructures (having an average Si content of 25%) relax through the nucleation of individual 90” partials followed by additional halfloops which form microtwins. The thickness of such microtwins will, in general, increase with increasing film thickness in order to reduce the contribution of the twin boundary energy to the overall energy of the system. This coarsening takes place even though it leads to an increase in the interaction energy between the partial dislocations acting to relieve strain in the
1092
DYNNA
and MARTY:
MISFIT STRAIN IN THIN FILMS
system, for the energy due to twin boundaries increases more rapidly with h than does the interaction energy between twins made up of a cluster of m partials. If, in an array of alternating twins, all of the twins are constrained to have the same thickness, then at a given film thickness there is a unique value of m which gives rise to the lowest possible overall energy for the system. This is illustrated in Fig. 4, where E,,, for a thin film of Sic.sGea.s on a Ge(OO1) substrate has been minimized with respect to the twin spacing D for values of m ranging from 2 to 9, producing a family of curves which define an envelope representing the equilibrium total energy of the system as a function of h. This envelope is shown separately in Fig. 5, which gives the equilibrium total energy directly. In determining the total energy of the system, the following parameters were used: p = 5.64 x 10” Pa, v = 0.020, j3 = 90”, rp = 54.736”, b = 2.31 8, dp = 4.00 A, f = - 0.020. The core energy UC,,, for the partial dislocation was set equal to 0.65 eV/A, which is the value given by Nandedkar and Narayan [12] for a 90” partial in Ge, while the core radius ro, which was given in Ref. [12] as being ~5 A, has been set equal to 4 A for geometrical consistency. Since the cores of the partial dislocations are in very close proximity to each other, their actual contribution to the total energy of the system may differ from that of an equivalent number
of isolated partial dislocations; the above values for the core parameters are used for illustrative purposes only. Following the approximation often made for f.c.c. crystals [lo] of taking the twin boundary energy as equal to one-half of the intrinsic stacking fault energy (which for Ge has been determined to be 0.060 JmT2 [13]), the twin boundary energy Yr has been set at 0.030 Jmm2. The curves shown in Fig. 4 were calculated using the exact solution for the interaction energy between parallel alternating twins. A similar calculation of the minimum value of E,,, for specific values of m performed using the approximate solution for the interaction energy gave rise to virtually the same results, with differences in the two total energies typically being on the order of one part in ten thousand. This behaviour was the result of the fact that the equilibrium twin spacings in the Sio,sGeo.s film were fairly large and the fact that the interaction energy between the parallel twins represented a relatively small part of the total energy of the system. In general, the difference between the two solutions for the elastic interaction of parallel alternating twins is appreciable only when m is large and D is very small. It follows from this that the equilibrium twin thickness and spacing as well as the equilibrium energy of epitaxial systems encountered in practice calculated using the approximate solution for the interaction
12
10
8
0
I
0
I
200
I
I
400
I
I
600
I
I
800
I
1000
Film Thickness (nm) Fig. 4. A family of curves corresponding to values of the twin thickness ranging from 2 to 9 atomic layers in which I$,, for Sio.sGeo.s on Ge(OO1) has been minimized with respect to the twin spacing.
DYNNA and MARTY:
MISFIT STRAIN IN THIN FILMS
1093
6
5
1
0
200
400
600
800
1000
Film Thickness (nm) Fig. 5. The equilibrium total energy of Sio.sGeo.son Ge(OO1)as a function of film thickness. energy between parallel twins differ negligibly from those determined using the exact solution for the twin interaction. In particular, the calculations previously performed for the relaxation of elastic strain via twinning in an equiatomic AuCu alloy [8] would not be altered in any significant way by the substitution of the exact energy for the elastic interaction between parallel twins. The evolution of the equilibrium value of m (obtained for a given film thickness by the selection of the curve of Fig. 4 which gave rise to the lowest minimized with respect to D at that value of Et,, thickness) is shown in Fig. 6. Transitions in the value of m occur whenever the contribution of the twin boundary energy to the overall energy of the system becomes sufficiently great to allow for a packing of partial dislocations into thicker twins. The discontinuities in the value of m present in Fig. 6 are due to the fact that in the mathematical formulation of the relaxation all of the twins acting to relieve misfit strain are constrained to have the same thickness. In practice there is no such restriction placed on the twins which act to relieve misfit strain, and twins of different thickness may coexist in a film of a given thickness. A gradual and continuous increase in the average value of the twin thickness takes place, and in principle all that is necessary for a twin of a given thickness to exist in a thin film is that the sum of the interaction energy of the twin with the epilayer and the self-energy of
the twin be less than zero. The critical thickness for a twin is given implicitly by setting equation (6) equal to the negative of equation (7) and the results of such calculations for twins relieving misfit strain in Sio.sGeo.s on Ge(OO1) are given in Table 2. Thus a twin made up of eight partial dislocations is stable in films greater than 19 nm in thickness, even though an array made up of twins eight atomic layers thick is not the energetically favoured form until the film has reached a thickness of 780 nm. Perhaps the most important feature characteristic of the relaxation of misfit strain in thin films by means of twinning is the impossibility of an asymptotic approach to a limiting value of the equilibrium energy of the system ash + 00. Because twin boundaries are always present in the thin film, the equilibrium energy of the system increases monotonically as the film thickness increases. This is in contrast to the case of relaxation of elastic strain via orthogonal arrays of alternating perfect dislocations, where a limiting value of the equilibrium energy of the system is reached ash 3 co. Thus even though it is easier to nucleate 90” partial dislocations than 60” perfect dislocations in a Sit,.sGeo.s film on Ge(OOl), and even though it may be initially energetically favourable to relieve misfit in such a film by means of 90” partial as opposed to 60” perfect dislocations, a point will be reached when the thickness of the film is large enough to make strain relaxation via 60” dislocations the ener-
1094
DYNNA and MARTY:
MISFIT STRAIN IN THIN FILMS
9
8
6
Es
4
3
2
I
1 0
I
I
a
200
400
I
I 600
I
I 800
I
1000
Film Thickness (m-n) Fig. 6. The equilibrium value of the twin thickness m as a function of film thickness in Sio.sGeo.s on Ge(OO1).
getically preferable case. This is shown in Fig. 7, where the equilibrium energy of the system during the relaxation of misfit strain in Sio,sGeo.s on Ge(OO1) via orthogonal arrays of partial 90” dislocations and twins is compared with that of a process involving relaxation via orthogonal arrays of perfect 60” dislocations. The curve representing relaxation via partial dislocations is an extension of the curve of Fig. 5, which was calculated for twins, to include isolated partial dislocations bounded by stacking faults. The method of determining the equilibrium energy of orthogonal arrays of partial dislocations is similar to that used for perfect dislocations, with a term representing the stacking fault energy being added to the self energy of the dislocation (141. In the present case the stacking fault energy was set equal to 0.060 Jm-*, [13],
Table 2. The critical thickness for individual twins in Sio,&eo.~ on Ge(001) m
2 3 4 5 ; 8 9 10
Critical thickness (nm) 6.5 8.4 10.4 12.3 14.2 16.2 18.1 20.1 22.0
resulting in a transition from isolated partial dislocations to twins two atomic layers in thickness at a film thickness between 29 and 30nm. The curve representing relaxation via perfect dislocations was determined following the method of Ref. [3]. Figure 7 shows that the critical thickness of 90” partial dislocations (4.9 nm) is less than that of 60” perfect dislocations (6.2 nm), and that it is energetically favourable to relieve misfit by means of isolated partial dislocations up to a film thickness of 13 nm, after which relaxation via perfect dislocations is energetically preferred. Thus arrays of twins of identical thickness are never energetically preferred over arrays of 60” perfect dislocations in Sia,5Gee.s on Ge(OO1). In films encountered in practice, however, it is usually not possible for the transition from relaxation via partial to relaxation via perfect dislocations to take place because of the difficulty in nucleating perfect dislocations, and as a result the approach to full (2%) relaxation is slower than in the case of perfect dislocations. This may be seen in Fig. 8, which compares the equilibrium relaxation in Sia.sGeo.5 on Ge(OO1) by means of mechanisms involving partial and perfect dislocations. Discontinuities in the relaxation curve for the relief of misfit strain via microtwins arise from the restriction that the microtwins are all taken to be made up of the same integral number of partial dislocations; this point is addressed in Section 3.3. At
DYNNA
and MARTY:
MISFIT
STRAIN
IN THIN
1095
FILMS
6
5
4
Partial dislocations
5’
and microtwins
a 2
Perfect dislocations
1 / I
I
0 0
I
,
I
I
a 1000
800
600
400
200
I
I
Film thickness (nm) Fig. 7. A comparison between the equilibrium total energy in SiO.sGeO.S on Ge(OO1) associated with relaxation via 60” nerfect dislocations and that associated with relaxation via 90” partial dislocations and microtwins.
Perfect dislocations 2.0
Micr otw ins>
0.5
I
I
0
I 200
I
I
400
I
I 600
8
I 800
I
I 1000
Film Thickness (nm) Fig. 8. A comparison between with relaxation via 60” perfect
the equilibrium percent relaxation in Sia sGe,,s on Ge(OO1) associated dislocations and that associated with relaxation via 90” partial dislocations and microtwins.
DYNNA
1096
and MARTY:
MISFIT STRAIN IN THIN FILMS
a film thickness of 1000 nm, microtwins made up of eight partial dislocations provide for a 1.93% relaxation, while perfect dislocations allow for a 1.99% relaxation. It is to be expected that this difference in the rate of equilibrium relaxation would be reflected in the actual kinetics of relaxation in a thin film, with a process involving microtwins being considerably slower than the corresponding perfect dislocation process.
Table 3. Equilibrium values of m in Ag(OO1) and Pt(OO1) epitaxial films in a state of biaxial tension (f = - 0.020) at selected values of the twin thickness h
h bm) 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000
3.2. The influence of twin boundary energy on twin morphology
The twin boundary energy is of great importance in the energetics of the relaxation of misfit strain, and plays a key role in determining the evolution of twin morphology during the growth of films which relax via a twinning mechanism. In general, a large twin boundary energy leads to a rapid coarsening of twins as the film thickness increases, while a small twin boundary energy leads to a much slower twin coarsening. An interesting comparison is offered by the relaxation via twinning of thin Ag(OO1) and Pt(OO1) films-both metals are f.c.c., but the twin boundary energy of Pt (0.161 Jmp2) is 20 times greater than that of Ag (0.008 Jme2) [lo]. The results of calculations of the equilibrium value of m (obtained using the method of Section 3.1.) for both Ag and Pt are given in Table 3. Both metals were taken to be under a 2% tensile strain when fully coherent
m&
WPf
2 2 7 10 12 14 16 18 19 20 22 23
with the substrate, i.e. f = - 0.020. In the case of Ag, p = 3,10x 10” Pa and v = 0.412 [lo], while b = 1.67 A for an a/6(211) partial dislocation and dP= 2.89 A. We have used Cottrell’s approximation [15] for the core energy of a dislocation and set ucore= 2x(1pb2 - “)2 -
4.0 x lo-” J/m = 0.25 eV/A
10
8
400
600
Film Thickness (mn) Fig.
9. The equilibrium
value of m in Si0,5Geo,s on Ge(OO1) associated with a fictitious as a function of film thickness. Compare with Fig. 6.
twin-like
defect
DYNNA
and MARTY:
MISFIT
STRAIN
IN THIN
FILMS
1097
2.0
1.5
1.4 0
200
400
600
800
1000
Film Thickness (nm) Fig.
IO. Equilibrium
percent
relaxation associated with the relief of misfit Ge(OO1) by means of twin-like defects.
rO=dp= 2.89 A. For Pt, cl1 = 3.461 x 10” Pa, cr2=2.507 x 10” Pa, and cM=0.765 x 10” Pa [16], which give the Voigt averages p = 6.51 x 10” Pa and v = 0.393. The magnitude of the Burgers vector of a/6(211) partial dislocations in Pt is 1.60 A, and d,, = 2.77 A. We have again set and
pb’
ucore= 241 - “)2 -- 7.2 x lo-*’ J/m = 0.45 eV/A, and r. = dp =2.77 A. It may be seen that the coarsening of twins in Pt is much more rapid than in Ag, with the equilibrium value of m in a 1 pm Pt film being 3.3 times greater than the corresponding value in a 1 pm Ag film.
strain
in Sio.sGeo5
3.3. A model for the continuous strain via twinning
relaxation
on
of misfit
The formulation of the energetics of an array of twins in a thin epitaxial film on the basis of all of the twins in the array consisting of an identical integral number of partial dislocations leads to a number of artificial effects in calculations of the equilibrium properties of the system, including discontinuities in the equilibrium relaxation as a function of the film thickness h such as those seen in Fig. 8. By allowing the thickness of the twins to vary continuously, a twin-like defect is produced which gives rise to a smooth and continuous equilibrium relaxation as a function of h. As is demonstrated in Appendix C, the self energy per unit area of a one-dimensional array of twin-like defects of thickness m and spacing D is
1098
DYNNA and MARTY:
MISFIT STRAIN IN THIN FILMS
where dr, is the spacing of the partial dislocations in the physical twin from which the twin-like defect is derived. This expression for the self energy of an array of twin-like defects in conjunction with the approximate expression for the interaction energy between parallel alternating twins (where m now varies continuously) and the other terms in equation (1) allow for the calculation of equilibrium values of m, D, and percent relaxation which vary continuously with the film thickness h. Figure 9 shows the equilibrium value of m for Si,,,Ge,,5 on Ge(OO1) where m, which may be taken to represent the average twin thickness present at equilibrium in an actual thin film, varies continuously with h. The equilibrium relaxation via twin-like defects in the same film may be seen in Fig. 10, which is similar to that of the twins in Fig. 8 but without the discontinuities, and is a good approximation of the relaxation which would be found at equilibrium when misfit strain is relieved by orthogonal arrays of twins in which no restrictions are placed on the thickness of the twins in the array.
2. Gosling, T. J., Willis, J. R., Bullough, R. and Jain, S. C., J. Appl. Phys., 1993, 73, 8291. 3. Dynna, M., Okada, T. and Weatherly, G. C., Acta Metall. Mater., 1994, 42, 1661. 4. Gosling, T. J. and Willis, J. R., Phil. Mug. A, 1994, 69, 65. 5. Wegscheider, W., Eberl, K., Abstreiter, G., Cerva, H. and Oppolzer, H., Appl. Phys. Lett., 1990, 57, 1496. 6. Wegscheider, W. and Cerva, H., J. Vat. Sci. Tech. B, 1993, 11, 1056. I. Neething, J. H. and Alberts, V., J. Appl. Phys., 1994, 75, 3435. 8. Dynna, M., Marty, A., Gilles, B. and Patrat, G., Acta mater., 1997, 45, 251. 9. Wolfram, S., The Mathematics Book. 3rd edn.
Wolfram University Press, Media/Cambridge Cambridge, 1996. 10. Hirth, J. P. and Lothe, J. Theory of Dislocations. McGraw-Hill, New York, 1982. 11. Pearson, W. B., A Handbook of Lattice Spacings and Structures of Metals and Alloys. Vol. 2. Pergamon, London, 1967. 12. Nandedkar, A. S. and Narayan, J., Phil. Mag. A, 1990, 61, 873.
13. Gomez, A., Cockayne, D. J., Hirsch, P. B. and Vitek, V., Phil. Mag., 1975, 31, 105. 14. Dynna, M., Marty, A., Gilles, B. and Patrat, G., Acta mater., 1996, 44, 4417. A. H., Dislocations and Plastic Flow in Crystals. Oxford University Press, Oxford, 1953. 16. CRC Handbook of Chemistry Physics, 76th edn. CRC
15. Cottrell,
REFERENCES 1. Willis, J. R., Jain, S. C. and Bullough, R., Phil. Masg. A, 1990, 62, 115.
Press, New York, 1995.
APPENDIX A Evaluation of the elastic interaction between partial dislocations in separate twins in an elastically isotropic half-space
Consider an array of N alternating twins comprised of clusters of m partial dislocations having a spacing D occupying a length L, where L is large. Then edge effects are minor, and to an excellent approximation the interaction energy per unit length between the components of the partial dislocations in separate twins in the array which act to relieve misfit strain is m-l uint
=
(N - 1) mu,+
c
i[u(O-(rn-i)d,)
+
U(D+(m-i)d,)]
i=l
m-l + W - 2) m&o + ci=,
i[U(2D-++dp)
+
1 1
u(ZD+(rn-i)dp)]
+
,
1 < m < D/d,,
t-41)
where dp is the spacing between the partial dislocations which make up a twin. Note that the interactions between partial dislocations in the same twin are not included in this summation-they are taken into account in the calculation of the self energy of the twin as given by equation (5). In the limit of L becoming infinitely large, an exact expression for the interaction energy per unit area between those components of the partial dislocations in the array which act to relieve misfit strain is obtained: m-l
&t
= A 2
n=l
mUnD
+ c
i[U(nD-(m-i&,)
+
U(nD+(m-nd,)]
i=l
1
,
1 tm
(A21
For those components of the partial dislocations in separate twins not relieving misfit strain, the interaction energy per unit area is m-l Eint =A
mb+I)D
+2
+
m-1 mU2nD
C
+
c
c i[u((2n+l)D-(m-Qdp) i=l
i[u(2nD-(m-i)dp)
+
+
u((2,+l)D+(tA)d,)]
U(2nD+(m-Od,)]
II %
1
1 < m c D/d,,.
G43)
DYNNA
and MARTY:
Substituting the values for the pairwise relieving misfit strain,
For those components
not relieving
interaction
misfit strain,
MISFIT energies
substitution
STRAIN
[3] between
IN THIN the partial
of the pairwise
FILMS dislocations
interaction
energies
1099 gives, for the component
gives
(A51
A closed form exists for the infinite series and products appearing in equations (A4) and (A5) (as may be verified using symbolic mathematics software [9]), leading to the expression for the interaction energy per unit area between parallel alternating twins given by equation (3).
APPENDIX
B
An exact solution for the interaction energy between twins
ofsimilar orientation
An exact solution for the interaction energy between similar twins may be obtained using the general procedure outlined in Appendix A, and is given here for completeness. Since the twins are of similar orientation, one has for all components of the partial dislocations making up the twins
/&, =;
5 ?I=,
m-l
~U,D +
c ,=I
i[u(i(,D-(,-i)dp)
+
U(nD+(m-tldp)]
)
1
1
(B’)
1100
DYNNA and MARTY:
MISFIT STRAIN IN THIN FILMS
Substituting the values of the pairwise interaction energies between the partial dislocations in separate twins in the array and taking the closed form of the infinite series gives (for similar twins):
i”Fji~~_~~~~[In(~sinh(~))+~~oth(~) -!?$cs&2(~) -i]
Emt.twin/twin(para) =
pb2 sin2 j cos2 cpm-l
+ -
4n(l - u)D
i=, c.[
cosh(4nh/D) - cos(2n(m - i)d,,/D)
2 In (m - i)‘d2
p ((m - i)2d,2 + 4h2)[1 - cos(27c(m- i)d,/D)] >
(
12(m - i)2dih2 + 16h4
(4nh/D)sinh(4zh/D) ((m - i)2di + 4h2)2 ’ cosh(4nh/D) - cos(27t(m - i)d,/D)
1 +~~~~~~~2~[In(~;sinh(~))-~~ot + (8rr2h2/02)(1 - cos(2n(m - i)d,/D)cosh(4rch/D))
(cosh(4nhlD) - cos(27c(m- i)dp/D))2
pb2 sin2 fi sin2 cpm-’ +
47c(l - u)D
i=, c,[
cosh(4ah/D) - cos(27c(m - i)$/D)
1 In (m - i)2d2
p ((m - dd;
(
+ 4(m - i)2d;h2 + 48h4
(4nh/D)sinh(4nh/D)
((m - i)‘di + 4h2)’ - cosh(4zh/D) _ (8n2h2/D2)(cos(2n(m
+ 4h2)[1 - cos(2rc(m - i)d,/D)] >
- cos(2x(m - i)d,/D)
- i)dp/D)cosh(4nh/D)
1 4RD
- 1) + pmb2 cos2 /I
cosh(4nh/D) - cos(2rr(m - i)dp/D))2 (m - i)2d2 '
ln( $sinh(
y))
cosh(4xh/D) - cos(2n(m - i)d,/D) ((m - i)2d,2 + 4h2)[1 - cos(2(m - i)$/D)]
> ’ ’ < m < D’dp’
(B2)
APPENDIX C The twin-like defect
The twin-like defect can be constructed from a twin by replacing the m partial dislocations that make up a twin with an infinite number of infinitesimal dislocations having the same Burgers vector content mb. In the case of a twin, the energy per unit length may be written as
where Use,f = mpb’(1 - vcos’ /I) ln 2 + mph’ sin’ /I cos2q7-$3)+m&,, 47r(l - V) 87r(l - V) ( r0 > is the self energy per unit length of the partial dislocations making up the twin, 2yrh/sincp is the twin boundary energy per unit length, and Urni is the interaction energy per unit length between the partials making up the twin and is given m-l uint
=
1 (m - I)((:1- ”COs2 8) UI +
(sin2 B C0s2 cp)U,
+
(sin2 /3sin2 q)Us),
r=l
with pb2 In 4ir(l - U)
u, =p
and
If the partial dislocations having a Burgers vector of magnitude b are broken into iv=& partials each having a Burgers vector of magnitude A&!?=&
dp
N’
(W
DYNNA then the sum of the self-energies
and MARTY:
of the dislocations
MISFIT
STRAIN
IN THIN
FILMS
1101
tends to zero in the limit N -+ 00, for 2
lim mN(Ab)2 N-oc This includes
the core energies
of the dislocations,
= ,jimXmN
$
= 0.
0
each of which may be approximated
by
uAb2 2n(l - “)? For the interaction
energy,
we have for a given value of h Nm-I
L’i”t = C (Nm - i)(Ab)* 4, ($$I r=l
= m(d~i’(rndp
- iAx)(t)2U:nt(iAx)A.x
(C3)
In the limit as N + cc and Ax + 0, we obtain
s 4
ol”, =
0
Evaluating the above integral by equation (9).
(mdp - x) $ 0
*C~:,,(x) dx
(C4)
P
leads to the self energy per unit area of a one-dimensional
array
of twin-like
defects given