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SEMICONDUCTORS
H. Möller
To cite this version:
H. Möller. ENERGY OF GRAIN BOUNDARIES IN SEMICONDUCTORS. Journal de Physique
Colloques, 1982, 43 (C1), pp.C1-33-C1-43. �10.1051/jphyscol:1982106�. �jpa-00221759�
JOURNAL DE PHYSIQUE
Colloque Cl, supplément au n°10, Tome 43, octobre 1S82 page Cl-33
ENERGY OF GRAIN BOUNDARIES IN SEMICONDUCTORS H.J. Möller»
Institut für Metallphysik, Universität Göttingen, Hospitalstr. 12, 34 Göttingen, l'.R.G.
Résumé.- Les énergies des joints de flexion de coïncidence d'axe [01il ont été calculées à partir de modèles géométriques. Pour 0<^ 70.53°, les structures qui ne comportent pas de liaisons pen- dantes peuvent être considérées comme une paroi de dislocations coin. Leur énergie peut être calculée à partir de l'énergie élas- tique des dislocations individuelles. Pour des angles plus grands, les structures comportent des liaisons pendantes. Les calculs mon- trent qu'une reconstruction des liaisons dans la direction [011]
peut diminuer l'énergie totale.
Abstract.- Starting from geometrical models of the atomic structure of symmetric [011]-CSL-tilt boundaries their energies were calculated. For 0 < 70.53° structures with no dangling bonds occur which can be considered as edge dislocation arrays. Their energies can be calculated from the elastic energy of individual dislocations. For larger angles the structures contain dangling bonds. The computations show that a reconstruction of bonds along the [011] direction can lower the total energy.
1. Introduction.- Polycrystalline semiconductors and ceramics are already widely used throughout the electronic industries for a variety of purposes: solar cells, poly-silicon thin film devices, ceramic varistors to name only a few. These commercially important devices owe their electronic properties to the presence and the character of their grain boundaries. This has motivated much of the research to investigate the microstructure, micro-chemistry and the electronic properties of the grain boundaries. However, many of the fundamental questions are still open and as a result improved materials and de- vices have been developed largely by empirical processes. Nonetheless it is very desirable to investigate the basic properties of single grain boundaries since that can serve as a useful background material for the device-oriented questions.
Many of the properties of grain boundaries can only be understood if their atomic- and microstructure is known [1-3], At present, however, one can only hope to solve fundamental problems for the case of very special simple structures as they are expected for low angle and coincidence boundaries. Mainly two complementary ways of investigation can be pursued : high resolution and related electron microscopy
techniques and computer calculations of the atomic structures. For semiconducting materials with its covalent bonding the latter is especially difficult since the development of the theory of the struc- ture of defects in semiconductors is by far not completed. The most reliable results are available for point defects and surfaces in
* Present Address : Technische Universitat Hamburg-Harburg, Arbeitsbereich Halbleitertechnologie, Harburger Schlo3str. 20, 2100 Hamburg-Harburg, F.R.G.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982106
elemental semiconductors, however, the methods developed for these cases cannot be transfered immediately to dislocations and grain boundaries. Therefore some less sophisticated methods have been tried and applied to dislocations [4,51 and now to grain boundaries, too, as will be discussed in the following [6].
All calculations of the energy of an atomic arrangement as well as the interpretation of high resolution microscopy images can only be per- formed on the basis of an already approximately known atomic structure.
The simplest grain boundary structures are likely to occur for coin- cidence orientations of the neighboring grains since that leads to periodic atomic arrangements. In such cases geometric stick and ball models can sometimes be easily obtained and for a variety of coin- cidence grain boundaries such atomic structures have been proposed [7,8,91. On the basis of some of these models energy calculations have been carried out [61 and first results on [OIII tilt boundaries will be presented. This type of boundary has been chosen since most of the experimental and theoretical work has been accumulated here.
2 . Geometrical modeling.- Misorientations of the adjacent grains for which a periodic boundary structure occurs can be predicted by Boll- mann's 0-lattice theory [ l o ] . Following this method it is convenient to introduce a coordinate system which contains the rotation axis and the plane perpendicular to it. Since all of the following calculations were performed for tilt boundaries with a [OIII rotation axis we can adopt the following tetragonal unit cell to describe the diamond cubic lattice [Fig. I ] .
a, = a[1001
- -
a, =-Zj[olil -
a a = ;[0111a is the lattice constant of the conventional diamond cubic fcc unit cell to which all planes and directions will be refered unless other- wise stated [91.
Fig. 1 : Projection of the diamond cubic lattice on a (01 1 ) plane. The filled and open circles are atoms at height 0 and a/2V2, respectively.
It should be mentioned that the origin of the coordinate system does not need to be a crystal lattice point. After a rotation of one lattice with respect to another a general coincidence lattice (0- lattice) occurs for certain angles.
I we refer to the tetragonal unit cell of Fig. 1 we see that after a rotation of 0 = 2 arc tan k2/k1V2 (kl, k2-integer numbers) a vector [kl kz OItetr becomes coincident with [k, kz 01 of the unrotated lattice because (017) is a symmetry plane of tftgtfattice. A basis of
t h e 0 - l a t t i c e c a n be d e r i v e d
(common i n t e g e r f a c t o r s have t o b e s i m p l i f i e d ) .
I n g e n e r a l t h e 0 - l a t t i c e c o n t a i n s n o c r y s t a l l a t t i c e p o i n t s a t a l l i f t h e o r i g i n o f t h e c o o r d i n a t e s y s t e m h a s b e e n c h o s e n a r b i t r a r i l y . I n c a s e t h a t t h e o r i g i n i s a l a t t i c e p o i n t t h e 0 - l a t t i c e c o n t a i n s a s u p e r - l a t t i c e which c o n s i s t of a l l c r y s t a l l a t t i c e p o i n t s a t c o i n c i - d i n g p o s i t i o n s : t h e c o i n c i d e n c e s i t e l a t t i c e (CSL) .The b a s i s v e c t o r s are g i v e n by
b$SL = 2kz
- -
b F L =-
b s ( 2 )bgSL = 2k2 &2
-
The volume o f t h e CSL u n i t c e l l i s V = X a 3 / 2 , where X = 2k?
+
kZ( i f Z is e v e n i t h a s t o b e d i v i d e d bgS5) c h a r a c t e r i z e s t h e CSL f o r a g i v e n r o t a t i o n a x i s . F o r a f i x e d m i s o r i e n t a t i o n O o f t h e two l a t t i c e s , two d e g r e e s of freedom r e m a i n f o r t h e o r i e n t a t i o n o f t h e g r a i n
b o d a r y . P e r i o d i c g r a i n boundary s t r u c t u r e s o c c u r i f t h e boundary p l a n e i s a l a t t i c e p l a n e o f t h e CSL. I n t h e f o l l o w i n g o n l y symmetric tilt b o u n d a r i e s a r e c o n s i d e r e d ( u n l e s s o t h e r w i s e s t a t e d ) where t h e boundary p l a n e i s a m i r r o r p l a n e o f t h e CSL a n d c o n t a i n s t h e [ 0 1 1 ] d i r e c t i o n . ( R e l a t i v e t r a n s l a t i o n s o f t h e two l a t t i c e s a r e n o t con- s i d e r e d y e t ) . The v a r i o u s CSL and g r a i n boundary o r i e n t a t i o n s which h a v e been i n v e s t i g a t e d a r e g i v e n i n t a b l e 1 .
T a b l e 1.- CSL boundary o r i e n t a t i o n s and e n e r g i e s ( e r g / c m 2 ) f o r
1 5 33. The meaning o f y
,
y and y i s e x p l a i n e d i n t h e t e x t . (Eo i n e v ).
( * ) These o r i g R ? a t i % F d i d nogi$eld low e n e r g y s t r u c t u r e s .Z 0 P l a n e a p p comp ' d i s l
33a 20.05 (1 44) 1044 1028 1028
19 26.53 (1 33) 702 695 695
27 31.59 ( 2 5 5 )
- -
9 809 38.94 (1 22) 786 649 649
11 50.48 (233)
- -
73033b 58.99 (455)
- -
6 603 70.53 (1 1 7 ) 30 1.06 30
17 86.63 (433) 853+654E0
- -
17* 93.37 (322)
- - -
3 109.47 (271 ) 933+327E0
- -
33b* 121.47 (522)
- - -
11 129.52 (311) 1092+654E 1015+327E0
-
9 141.06 (471) 1077+6543:
- -
27
*
148.41 (511 1- - -
19
*
153.47 (611)- - -
33a* 159.95 (81 1 )
- - -
A c c o r d i n g t o t h e a r b i t r a r i n e s s o f t h e o r i g i n o f t h e c o o r d i n a t e s y s t e m v a r i o u s CSL may b e c o n s t r u c t e d w i t h t h e same u n i t c e l l b u t d i f f e r e n t g r a i n boundary s t r u c t u r e s and o n l y a n e n e r g y c a l c u l a t i o n c a n show which CSL ( f o r a g i v e n r o t a t i o n a n g l e 0 ) y i e l d s t h e boundary w i t h t h e l o w e s t e n e r g y . The d i f f e r e n t CSL c a n be t r a n s f o r m e d i n t o e a c h o t h e r by a t r a n s l a t i o n v e c t o r . So it i s a s p e c i a l p r o p e r t y o f t h e C0111 r o t a - t i o n t h a t t h e same CSL c a n a l s o b e c r e a t e d by a r o t a t i o n a o f
-
4 = 1 8 0 - 6 and a r e l a t i v e s h i f t o f t h e two l a t t i c e s by 3 < I l l > .
Fig 2: t = 3 (0 = 70.53O )
coincidence site lattice with different internal coordinates.
The filled and open squares are atoms of the rotated lattice. Coindicence points are encircled.
Grain boundary studies in cubic metals have shown that certain low energy atomic configurations ("structural units") which have been identified in low energy CSL boundaries are also found in arbitrary boundaries. Close to a coincidence orientation the CSL structure is maintained by the incorporation of appropriate dislocation networks, at larger dev,iations the structure is made up by a mixture of
structural units of neighboring CSL boundaries (eventually plus some dislocations and/or steps). Experimental results in semiconductors point towards the same direction. Therefore it seems necessary to get an overview of the possible relative lattice translations (and Burgers vectors) that maintain a certain grain boundary structure. Bollmann has shown that for each 0-lattice a DSC lattice may be constructed that meets this condition. For the [ 0 1 1 1 rotation under consideration here it can be described by a body centered orthorhombic unit cell with
Dislocatiorsin CSL boundaries whose Burgers vector b is not from the corresponding DSC lattice destroy the structure and must be connected with a structural fault.
Geometrical models were constructed for the CSL orientations of table 1 [ 9 ] . For the following considerations two typical structures will be presented here, the structure of a E = 33 (Fig. 3) and of a 1 = 1 1
(Fig. 4 ) boundary.
[ o i l ]
t
[ioolFig. 3: [Oil] projection of the ( 1 = 33) symmetric tilt boundary,
0 = 20.05~.
Fig. 4: [ 0 1 1 ] projection of t k T t = 1 1 )
,
symmetrictilt boundary, 0 = 129.52~.
The first boundary contains no dangling bonds and consists of an array of structural units
-
a seven and five membered ring of atoms-
which has-been identified as a Lomer dislocation with a Burgers vector
b =
"
[Oil-] and a (100) glide plane. Since all of the CSL boundaries- in t8e range 0 < 70.53 (.I = 3) with X = 1
+
2 n2 (n = 2-
4) contain the same type of defect at different spacings, it has been proposed[ 9 1 that even arbitrary boundaries in this misorientation range can be built from this edqe dislocation, a point which will be investigated later.
The structures of the CSL boundaries with 0 r 70.53~ all contained broken bonds, however, a possible reconstruction of bonds along the
[Oil] axis has to be taken into account.
On the basis of some of the proposed models computer calculations of the grain boundary energy were carried out. Since it is a special property of the covalently bonded materials that the wave functions
forming a bond are strongly localized computations demand that one has to determine in advance how the atoms are connected with each other.
The procedure described next therefore does not alter the essential features of a structure unless one begins from a different model.
2. Calculation of grain boundary energies.-
2.1 Boundaries with no broken bonds.- The method used for the energy calculation has been derived earlier [I21 and applied successfully to the determination of the core structure of dislocations in
germanium and silicon [4,5,13]. It uses a pair potential function whi&
allows the computation of the interaction energy of an atom with its four neighbors in the deformed crystal. Keating has derived the following approximation [141.
3 a 4 2 8 4
E - - - (gi ~ ~ - r 2 ) ~ + - 1
harm 1 6ro i=l r.r.+-r2)2
o 8r; (-I-] 3 o
This expression takes into consideration small deviations of the equilibrium distance (r ) of two atoms and distortions of the equili- brium bond angles (second term, r . difference vector between the atom 0
and its i-th neighbor). It contaThs two free parameters (a, B) which have to be fitted to the elastic moduli of the material. Since large deformations of bond angles and distances are to be expected in the vicinity of the boundary plane anharmonic terms have to be added to improve the calculations. A modified form of Keating's anharmonic potential developed by Koizumi and Ninomiya has finally been used [I51
3 a 4
E = E harm
+ -
16rg i=l (r.r. -r2) exp%
4OY
(gigi-rg)-1-1 o
That introduces three more parameters (y,6,~) which are determined from the elastic modulus of the third order. They are given in reference [ 14,15 1 for germanium and silicon.
The total energy of a given assembly of atoms is the sum of the inter- action energies of all atoms. For the beginning atoms far away from the boundary plane were fixed whereas all other atoms were shifted in the direction of the largest energy gradient until the minimum con- figuration was reached. Usually it was sufficient to relax only a few atoms in the vicinity of the boundary. In a second step relative translations of the two crystal lattices perpendicular and parallel to the boundary plane were introduced. For the symmetric boundary
structures (1 = 33, 19, 3) only translations perpendicular to the boundary plane were necessary to reach the equilibrium configuration whereas for the asymmetric structure of the E = 9 boundary [7,9] also
a small displacement of about 5% of the periodicity length parallel to the plane was needed. The computed relative energies (related to the area of the plane) ycomD are given in table 1.
The calculations show &at the relaxation of the atoms reduces the total energy by about 40%. Earlier Mijller [ 9 ] has approximated the' boundary energies by just taking into account the energy of the bonds which connect the crystals
E = n Eo
+
E l Z sin2~ P P i i' (6)
where E is the mean energy of a broken bond, n their number, a. the distortyon angle of the i-th bond in the boundary plane. The se6ond t e n of this equation is derived from the elastic shear energy per unit volume with El = 7.2 eV for silicon, however, does not take into account atom relaxation. If one reduces El by about 40% one yields suprisingly similar values to the computer calculated boundary energies (table 1). With this adjusted parameter E1 the energies of some of the remaining grain boundaries were approximated.
2.2 Boundaries with broken bonds.- The geometric model of the X = 1 1 boundary (Fig. 4) shows a structure with two bonds per periodicity length. The dangling bonds point either up- or downwards. The energy calculation of structures with an isolated bond was performed in the following way. The single bond was saturated by an atom which was allowed to adjust so that the distortion energy of this bond could be kept as small as possible. The total energy is then the sum of the minimized boundary energy plus the energy of a dangling bond (= Eo/2).
Unfortunately there is still a great uncertainty as tothe absolut value of Eo, Marklund 151 gives E = 1 eV, Jones 141 E = 2 eV and Mijller [ 91 E = 1.8 eV. ~herefore~in some cases it may0be impossible to decide whych reconstructed structure has to be favored.
For Z = 1 1 two possibilities of reconstructing the dangling bonds were tested, a partial saturation of every second bond in the symmetry plans of the boundary along the [0111 direction and an alternating arrange- m,ent of the remaining dangling bonds, and the total reconstruction of all bonds. Table 2 gives the results for silicon which show that in any case the partial reconstruction yields the lowest energy.
Table 2.- Total energy ycomp of the t = 1 1 (0 = 129.52~) boundary (Eo is eV).
1
ycomp [ erg/cm2I I
Table 2.- Total enc _vLt,y (Eo is eV).
ycomp [ erg/cm2
I I
3. Ener y of periodic dislocation arrays.- It is tempting to treat f s v m m e t r i c u l a r rancre 0 I 38.94
no reconstruction partial reconstruction total reconstruction
(t =
9)
as an array of Lomer edge dislocation as has been suggested-
earlier. That should be possible because the Burgers vector
2
= ~ [ 0 1 1 1 is a crystal lattice vector and therefore belongs to the DSC latgices which can be constructed for each orientation angle 0.1238
+
327 (2E0) 1015+
327 (lEo) 1837+
0 EoSince the Read and Shockley equation [I71 for the energy (per unit area)
05
a small angle boundary is an approximation for small angles( 0 5 10 ) one first has to generalize the equation in order to compare with computer calculated CSL boundary energies.
L e t u s c o n s i d e r t h e r e f o r e t h e p e r i o d i c a r r a y of edge d i s l o c a t i o n s o f F i g . 5.which form a symmetric tilt boundary. For l a r g e r a n g l e s one c a n n o t n e g l e c t anymore t h a t b o t h l a t t i c e s a r e t i l t e d away from t h e
boundary p l a n e by 8/2 and
F i g . 5 a c c o r d i n g l y t h e same i s t r u e f o r
t h e d i s l o c a t i o n s and t h e i r Burgers v e c t o r . The d i f f e r e n t s p a c i n q s Do and 2D a r e i n t r o d u c e d
B / 2
-.
I2 9
- I I h e r e t o a c c o u n t a l s o f o r bounda- r i e s w i t h n o t e v e n l y spaced d i s - l o c a t i o n a r r a y s (e.9. .I= 3 3 ) .I I The energy ( p e r u n i t a r e a ) ydisl
I
s t o r e d i n such a d i s l o c a t i o nI\ I I a r r a y h a s been c a l c u l a t e d
\ !
f o l l o w i n g a procedure g i v e n i nhr h p kk hk hY
[ 1 8 1 . I t y i e l d s-
yo s i n 0 / 2 { c o s 2 [no c o t hno -
l n ( 2 s i n h go) 1 +' d i s l - (7
w i t h yo = p b / 2 ~ r ( l - v ) ,
no=
( 2 a r s i n 8 / 2 ) / b , v P o i s s o n ' s r a t i o and r t h e c o r e r a d i u s of t h e edge d i s l o c a t i o n .I f t h e d i s l o c a t i o n s a r e n o t e v e n l y s p a c e d t h e n yc i s n o t z e r o and be- comes
-
1 cosh n + IY C
-
2 yo {ln c o s h g:-c + s i n h n0(cosh no-c)where c = c o s a Do/D
.
E q u a t i o n s 7 , 8 c o n t a i n two p a r a m e t e r s y and r , i f we c o n s i d e r yo a f r e e p a r a m e t e r f o r t h e b e g i n n i n g . ~ i t t ? n ~ t h i s e q u a t i o n t o t h e computer c a l c u l a t e d g r a i n boundary e n e r g i e s of .I= 33, 19 and 9 we o b t a i n t h e f o l l o w i n g v a l u e s ( t a b l e 3 )
Table 3.- Comparison of computer c a l c u l a t e d g r a i n boundary e n e r g i e s and e n e r g i e s of p e r i o d i c d i s l o c a t i o n a r r a y s . A l l e n e r g i e s i n [erg/cm21.
The comparison shows an e x c e l l e n t agreement between t h e f i t t e d para- m e t e r y and t h e v a l u e o f ~ b / Z i ~ ( l = v ) i f one u s e s t h e Burgers v e c t o r of t h e flomer d i s l o c a t i o n b=a/2 [ 0 1 1 ] . The v a l u e r = 0.5b f o r t h e c u t - o f f r a d i u s i s r e a s o n a b l e . Although t h e d i s l o c a t i o n c o r e s b e g i n t o o v e r l a p f o r 0 2 26.53 (X = 19) t h e e q u a t i o n ( 7 ) s t i l l seems t o be v a l i d f o r l a r q e r a n g l e s , however, a s l i g h t l y r e d u c e d c o r e r a d i u s r e s u l t s i n t h e c a s e of t h e .I= 9 boundary.
With t h e h e l p o f e q u a t i o n 7 and t h e v a l u e s f o r yo and r / b one i s a b l e now t o c a l c u l a t e e n e r g i e s of more p e r i o d i c d i s l o c a t i o n g r a i n boundaries.
Two t y p e s of s p a c i n g s have been c o n s i d e r e d (n
>
3) ( a ) w i t h Do/D = n/n+l i f n i s odd,( b ) and Do/D = 1 i f n i s even.
I n b o t h c a s e s we have a CSL boundary w i t h E = 1+2 n 2 and O = 2 a r c t a n l / n d 2 . The c a l c u l a t e d e n e r g i e s a r e g i v e n i n F i g . 6 .
F i g . 6: E n e r g i e s of p e r i o d i c d i s l o c a t i o n g r a i n boundaries, open c i r c l e s : c a l c u l a t e d v a l u e s from e q u a t i o n 7 , c l o s e d c i r c l e s : computer c a l c u l a t e d v a l u e s .
The e n e r g i e s c a l c u l a t e d h e r e a r e v a l i d f o r t e m p e r a t u r e s T = 0 . I n o r d e r t o compare w i t h e x p e r i m e n t a l l y d e t e r m i n e d v a l u e s one h a s t o t a k e i n t o a c c o u n t , t h a t t h e e n t r o p y of t h e bcundary i s l i k e l y t o r e d u c e t h e energy o f an a r b i t r a r y boundary more t h a n t h a t of a more w e r f e c t CSL boundary. T h e r e f o r e f r e e e n e r s y c u r v e s v e r s u s m i s o r i e n t a t i o n O a r e smoothed o u t and o n l y deep c u s p s i n t h e y(O) c u r v e a r e e x p e c t e d t o remain, i n o u r c a s e p r o b a b l y t h e c u s p s a t O ( Z = 19.9 and 3 ) .
We may a l s o c o n s i d e r t h e s t r u c t u r e of t h e g r a i n b o u n d a r i e s t r e a t e d s o f a r a s a m i x t u r e o f s t r u c t u r a l u n i t s of t h e a d j a c e n t CSL b o u n d a r i e s , i n o u r c a s e t h e p e r f e c t l a t t i c e s t r u c t u r e and t h e t = 1 9 s t r u c t u r e
(Lomer d i s l o c a t i o n ) . The q u e s t i o n a r i s e s whether it may a l s o be p o s s i b l e t o c o n s t r u c t boundary s t r u c t u r e s i n t h e a n g u l a r range 38.94 < O < 70.53 i n t h e same manner.
An a r b i t r a r y a n g l e of O = 50° h a s been chosen and it can a c t u a l l y been shown ( F i g . 7 ) t h a t t h e s t r u c t u r e i s a m i x t u r e of t h e Lomer d i s l o c a t i o n s t r u c t u r e w i t h t h e s t a c k i n g f a u l t s t r u c t u r e o f t h e t w i n boundary. From e q u a t i o n 7 it i s a g a i n p o s s i b l e t o c a l c u l a t e t h e e n e r g y of t h e p e r i o d i c d i s l o c a t i o n a r r a y . The r e s u l t i s g i v e n i n F i g . 6 f o r (n 2 0 )
( a ) Do/D = (3+n) / ( 4 + n ) i f n i s even
( 5 ) Do/D = 1 i f n i s odd,
and O = 2 a r c t a n ( (2+n) / (4+n) d2) )
.
W e have d e m o n s t r a t e d t h a t a t l e a s t f o r t h e [ 0 1 1 ] tilt boundary w i t h O 5 70.53 t h e boundary s t r u c t u r e s can e i t h e r be t r e a t e d a s a d i s l o c a t i m a r r a y o r a s a m i x t u r e o f s t r u c t u r a l u n i t s of n e i g h b o r i n g CS l a t t i c e s . The l a t t e r view seems more a p p r o p r i a t e i f one t a k e s i n t o account r e s u l t s g a i n e d from t h e s t u d y o f g r a i n b o u n d a r i e s i n c u b i c m e t a l s .
R e f e r e n c e s
[ I ] CUNNINGHAM B . , AST D . , G r a i n B o u n d a r i e s i n S e m i c o n d u c t o r s , P r o c e e d i n g s o f t h e M a t e r i a l s R e s e a r c h S o c . , 5 ( 1 9 8 1 1 ,
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[ 2 1 PAPON A.M., PETIT M . , SILVESTRE, G . , BACMANN, J . J . , I b i d . p . 2 7
1 3 1 FONTAINE C . , SMITH D.A., I b i d , p . 39 [ 4 ] JONES R . , P h i l . Mag.
2
( 1 9 7 7 ) 6 7 7[ 5 1 MARKLUND S . , P h y s . S t a t . S o l . (b)
85
( 1 9 7 8 ) 6 7 3[ 6 1 SCHULZ A . , D i p l o m a T h e s i s ( 1 9 8 2 ) , U n i v e r s i t y o f G o t t i n g e n [ 7 ] KOHN S . A . , Am. M i n e r a l .
43
( 1 9 6 3 ) 2 6 3181 HORNSTRA S . , P h y s i c a ~ ( 1 9 5 9 ) 4 0 9 , I b i d .
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( 1 9 6 0 ) 1 9 8 [ 9 1 MoLLER H . J . , P h i l . Mag. A43 ( 1 9 8 1 ) 1 0 4 51 1 0 1 BOLLMANN W . , C r y s t a l D e f e c t s a n d C r y s t a l l i n e I n t e r f a c e s ( B e r l i n : S p r i n g e r V e r l a g 1 9 7 0 )
[ I l l BALLUFE'I R.W., BROKMAN A . , KING A.H., A c t a M e t .
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( 1 9 8 2 ) 1 4 5 3 [ I 2 1 HAYDOCKR., H E I N E V . , KELLYJ.M., J . P h y s . ( 1 9 7 2 ) 2 8 4 5 ,I b i d ( 1 9 7 5 ) 2 5 9 1
1 1 3 1 VETH H . , PhD T h e s i s 1 9 8 2 , U n i v e r s i t y o f G o t t i n g e n
[ I 4 1 KEATING P . N . , P h y s . R e v .
145
( 1 9 6 6 ) 6 3 7 , I b i d .149
( 1 9 6 6 ) 6 7 4 [ I 5 1 KOIZUMI H . , NINOMIYA T . , J . P h y s . S o c . J a p a n2
( 1 9 7 8 ) 8 9 8 [ I 6 1 BOURRET A . , DESSEAUX S . , P h i l . Mag.A39
( 1 9 7 9 ) 4 0 5 a n d 4 1 3 1 1 7 1 READW.T., SHOCKLEY W . , P h y s . R e v .2
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DISCUSSION
M . HEGG1E.- The Keating p o t e n t i a l cannot be f i t t e d t o phonon d i s p e r s i o n c u r v e s
throughout t h e B r i l l o u i n zone. The v a l u e s of t h e Keating parameters vary by up t o a f a c t o r 3 between d i f f e r e n t a u t h o r s . How can you p l a c e confidence i n your c a l c u l a - t i o n s ?
H . J . MOLLER.- The s e t of parameters used was t h e one which i s most a p p r o p r i a t e f o r
t h e s e kind o f c a l c u l a t i o n s . Unpublished r e s u l t s by H . Veth on t h e c o r e s t r u c t u r e o f d i s l o c a t i o n s i n germanium and s i l i c o n showed q u i t e good agreement with experiments.
R.C. POND.- Have you considered t h e p o s s i b i l i t y of removing i n d i v i d u a l atoms from g r a i n boundary s t r u c t u r e s i n o r d e r t o reduce t h e i r energy. I n m a t e r i a l s w i t h a double atom b a s i s t h i s t y p e of r e l a x a t i o n l e a d s t o s t r u c t u r e s which cannot be o b t a i n e d a l t e r n a t i v e l y by r e l a t i v e displacement.
H . J . MOLLER.- Yes, t h i s h a s been c o n s i d e r e d i n g e n e r a l . However, no comparative
c a l c u l a t i o n s f o r a s p e c i f i c s t r u c t u r e have been c a r r i e d o u t y e t .
H . F . MA TAR^.- Which p o r t i o n o f t h e e n e r g y i s more i m p o r t a n t w i t h i n y o u r c a l c u l a - t i o n of t h e c u s p s : l a t t i c e r e c o n s t r u c t i o n (no d a n g l i n g bonds) o r lowered s t r a i n e n e r g y ? I n t h e Read-Shockley e n e r g y e n v e l o p e , we g e n e r a l l y i n t e r p r e t e d c u s p s a s due t o t w i n r e l a t i o n s o r mainly t o e l i m i n a t i o n o f d a n g l i n g bonds.
H. J. M ~ L L E R . - The c a l c u l a t i o n o f x 1 1 g r a i n boundary e n e r g y shows t h a t a bond r e c o n s t r u c t i o n c a n r a i s e t h e s t r a i n e n e r g y c o n s i d e r a b l y and t h e r e f o r e may be u n f a v o r a b l e . I n g e n e r a l b o t h p o r t i o n s o f t h e e n e r g y a r e e q u a l l y i m p o r t a n t and o n l y t h e d e t a i l e d c a l c u l a t i o n can show what s t r u c t u r e o c c u r s .
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