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Defects in the sphalerite structure
J. Phys. Chem. Solids
Pergamon
Press 1962. Vol. 23, pp. 1353-1362.
Printed in Great Britain.
DEFECTS IN THE SPHALERITE STRUCTURE D. B. HOLT C.S.I.R.
Solid State Physics Research Unit, Department of Physics, University Johannesburg, South Africa. (Received
16 February
1962; revised 4 May
of the Witwatersrand,
1962)
Abstract-Hornstra’s method for drawing the arrangements of the atoms in the cores of dislocations in covalently bonded crystals is applied to the sphalerite structure. Bond rearrangement and dislocation dissociation into an extended form are considered and the implications are discussed. A crude calculation of bond strain energy indicates that Haasen’s cracked core structure is unrealistic in germanium and in the sphalerite structure. Twinning and the electrical effects of dissociation and rearrangement are briefly discussed.
INTRODUCTION
HORNSTFU(II has shown that since covalent bonds can be regarded as directed in space, elastically stiff and brittle, it is possible to draw the atomic arrangements in the cores of dislocations of covalent crystals. Qualitative theories of bonding in semiconductors have been developed which indicate the covalent and ionic contributions to the bonding, These all stem from the work of PAULING.(~) It has been pointed out,(s) however, that this approach is rather misleading when applied to the solid state since it is based on the empirical relationship between the difference of electronegativity and the dipole moment in diatomic molecules. Theoretical objections too can be raised against this approach.(4I Many related methods have been proposed for determining the covalent and ionic contributions to the width of the forbidden energy gap in semiconductors.(5+ However, more relevant to the behaviour of interest here is a method for calculating the covalent and ionic contributions to the strengths of the bonds in sphalerite structure materials due to FOLBERTH.(*) Attempts to extract information about the degree of ionicity of bonding from quantum mechanical calculations are currently under way.(4$ 9) Thus if Hornstra’s methods were applied to such structures as sphalerite (zincblende) it would be possible to say something concerning the relative extent to which particular compounds might be expected I
to have such narrow dislocation core arrangements rather than those wider ones more characteristic of ionically bonded crystals of the same structure. --Recently the anisotropies in the [ill]-111 l] directions which follow from the lack of a centre of symmetry in the sphalerite and related structures have attracted considerable attention.(Is-Is) Out of this work has arisen a renewed interest in the existence of two kinds of “edge” dislocations(I4p aa) first pointed out by HAASENI~I)and an attempt to relate the anisotropies to bonding. (149ssS2%24)The extension of Hornstra’s analysis to deal with the core structure and the effects of dislocations and other structural defects in sphalerite seems, therefore, to be opportune. ARRANGEMEN’IS OF THE CORES OF DISLOCATIONS IN SPHALEIUTE
THJ3 ATOMIC
FRANK and NICHOLAS have obtained and listed all the Burgers vectors corresponding to stable dislocations in a number of common crystal lattices. Amongst these was the face-centred cubic structure, and much of the analysis carries over to substances such as diamond and sphalerite which have the same space lattice. Experiment shows that in the latter structures certain directions for the dislocation line are preferred. The combinations of stable Burgers vectors b and low index crystallographic line directions u give all the simple dislocations which differ in core structure.(I) The two directions b and u together
1353
1354
D.
B.
determine the slip plane of the dislocation (in the case of screw dislocations where they are parallel a plane may be arbitrarily chosen). Thus, for example, consider drawing the core of the dislocation in the sphalerite structure for which a = [liO] and which has the Burgers vector b = + [lOi]; this is a 60” dislocation@s) (so-called because of the angle between a and b).
HOLT
shown in Fig. 2. Connecting up nearest neighbours across the slip plane in a way consistent with the simplest picture of a covalent bond results in the core structure shown in Fig. 3. All the atoms with broken bonds in the core are of one type (black). Thus, for example, in InSb two types of 60” dislocations are possible. These are those in which the atoms which have broken
\
\ \
,b
\
FIG. 1. The sphalerite structure with a [ill] direction vertical. The white spheres represent atoms of one element of the compound and the black spheres the atoms of the other element. A (111) slip plane, the line direction a and the Burgers vector B of a 60” dislocation are shown.
The first step is to draw the sphalerite structure indicating the slip plane, dislocation line and Burgers vector as in Fig. 1. The second step is to choose, arbitrarily, to move the dislocation into the drawing from, say, right to left. Then(s7) the atoms above the slip plane are displaced to the right through 3 [lOi] relative to those below as
FIG. 3. The 60” dislocation in sphalerite. a and b lie along directions. The slip plane is of {ill} type.
FIG. 2. The result of moving a 60” dislocation as indicated in Fig. 1 into the crystal from right to left is to produce the relative movements of the atoms indicated by the arrows.
or dangling bonds and which constitute the row on the edge of the “extra half-plane” are all In atoms and those in which they are all Sb atoms. The former type is called the a and the latter the p form. (1s) This terminology will be extended to classes of sphalerite structure compounds other
DEFECTS
IN
THE
SPHALERITE
than III-V e.g. II-VI and I-VII by using the term CCfor the form in which the atoms of lower valency occur in the core.
STRUCTURE
1355
screw dislocation which are analogous to the a and /I types of dislocations with an edge component. Similarly the minimum Burgers circuit round a left-handed screw has only one step with a component antiparallel to the Burgers vector and this can occur in two sequences giving rise to two types of left-handed screw dislocation. Thus there are four types of screw dislocation in the sphalerite structure.
FIG. 4. The edge dislocation having the slip plane { 111 } in the sphalerite structure. a is a (112 > direction and b lies along a <110 > direction.
To most of the simple dislocations in the diamond structure there correspond a and /3 forms in the sphalerite structure. Further examples are shown in Figs. 4 and 5. However, not all dislocations have tc and fi forms. Fig. 6 shows one form of the screw dislocation in the sphalerite structure in which the Burgers vector is parallel to the positive line direction and the screw is righthanded. In the case in which the Burgers vector and line direction are antiparallel the screw is left-handed.
FIG. 6. The right-handed screw dislocation in the sphalerite structure. a and b lie along the same direction and there is no unique slip plane.
These are the simplest complete dislocations with slip plane (1111 and correspond to observed line directions in the diamond structure. Dislocations have been seen in silicon using a decoration technique and infra-red transmission microscopy. The dislocations were found to lie along (110 > directions@81 29) and along (112) directions. Dislocations lying along (112 ) directions, and
FIG. 5. The 30” dislocation. a is a <112) direction, b lies along a (110 > direction and the slip plane is of (111) type. In Fig. 6 it can be seen that of the steps round the minimum Burgers circuit l-Z-3-4-5-6 only one, namely 3-4, has a component antiparallel to the Burgers vector. This step can occur either in the sequence black-white as shown or whiteblack. Hence there are two types of right-hand
FIG. 7. The small angle boundary dislocation. a is a direction, b lies along a <110 > direction and the slip plane is of {llO} type.
D.
1356
B.
along both (110 > and (112) directions have been seen in germanium by transmission electron microscopy. The dislocation of Fig. 7 has been found in small angle boundaries in germanium.taa) No equivalent observations have as yet been made on sphalerite structure compounds. The dislocation of Fig. 7 does not have a and /3forms. DISSOCIATION VERSUS REARRANGEMENT
BOND
HORNSTRA~~)reported that in all the dislocations rearrangements of the bonds were possible which would lower the energy. VAN BLJEREN(~P 35936) has built up a theory of the movement of dislocations in the diamond structure on the assumption that the positions at which bond rearrangement has occurred act as immobile dragging points along the dislocation line. The situation in the sphalerite structure is altered by the fact that three types of bond are geometrically possible in the cores. These are A-A, A-B and B-B bonds, where A indicates an atom of the lower valence element of the compound and B an atom of the higher valence element. Consider for example the case of III-V compounds. On average each III element atom contributes 2 of an electron to each of its four tetrahedral bonds and each V element atom contributes l$ electrons per bond.(ss) Hence as a working hypothesis it may be supposed that an A-A bond in a III-V compound will contain on average 14 electrons and will be a low energy structure. Similarly an A-B containing 2 electrons is a low energy structure. But B-B bonds which would contain 2+ electrons on average, which could not therefore be paired off with opposite spins, will be of higher energy. Some re-arrangements which it has been suggested may occur in dislocations in the diamond structure(l) can be carried over into the sphalerite structure. An example is the case of the screw dislocation. By forming double bonds and so eliminating the most strained single bonds of Fig. 6 the total bond strain can, perhaps, be reduced. This is an example of bond rearrangement proper and could give rise to localized dragging points. There is, however, a second type of process, dissociation, which may lower the energy of dislocations. Consider the dissociation of the 60” dislocation into an extended form consisting of two
HOLT
partial dislocations separated by a strip of stacking fault. This can occur in two different ways in the diamond structure but in four geometrically different ways in sphalerite. Figures S(a) and (b) show the starting points. In each case a strip two atoms wide, of which the two atoms which lie in the plane of the drawing are shown, rotates into a new position and the bonds reform as indicated by the broken lines. Figures S(a) and (b) indicate the two different strips which may rotate initially. Each strip can rotate into the new position either clockwise as indicated in the drawings or anticlockwise. These two alternatives lead to indistinguishable results in the diamond structure.
AB=
% [IfId
AB,
(a)
AE-
$‘i kUor’l Cb)
k [7Jc?], Ia
EB =-
‘/s [&], lb
(cl
AE=
E'8=- ‘/s [i77], IC
+6 [77i],Id
(4 FIG. 8. Geometrically possible modes of dissociation of a 60” dislocation in sphalerite. (a) and (b) indicate the two possible double atom ribbons which might rotate initially. (c) is the extended dislocation resulting from the rotation of the atom ribbon of (a) and its successors in the clockwise sense. (d) is the extended dislocation resulting from the rotation of the atom ribbon of (b) and succeeding ribbons in the anticlockwise sense.
DEFECTS
IN
THE
SPHALERITE
However, in the sphalerite structure, if the rotation occurs in the clockwise sense the correct sequence of black-white-black-white in the vertical ( 111) direction is unaltered whereas the anticlockwise rotation results in the reversal of this sequence in the stacking fault. When the sequence remains correct the stacking fault will be said to be upright, and when it is reversed the fault will be said to be inverted. Figure 8(c) shows the extended dislocation obtained from 8(a) through the successive clockwise rotation of double atom rows to the right of that indicated in 8(a). The bonds extending up and down from the atoms in the fault are correct A-B bonds whereas the horizontal bonds in the partial dislocations at the ends of the stacking fault are wrong A-A and 3-B bonds and the latter are of high energy. Similarly, the extended dislocation in Fig. 8(d) results from 8(b) on making the rotations of successive double atom rows to the right of that in 8(a) in the anticlockwise sense. The stacking fault in this, as in all cases which appear anticlockwise as drawn, is inverted and the bonds above and below it are all wrong bonds, but the horizontal bonds in the partial dislocations are right. There are four geometrical types of extended 60” dislocations: (i) 8(c) with the stacking fault upright, (ii) 8(c) with inverted fault (not shown), (iii) 8(d) with upright fault (not shown) and (iv) 8(d) with inverted fault. A notation can be developed for these modes of dissociation which is independent of the drawings by using the sense in which the Burgers circuit is traversed. This is such as to constitute a righthanded screw with the (arbitrary) positive direction along the dislocation lines. The faulted double plane of modes of dissociation (i) and (ii) lies immediately before that of modes (iii) and (iv) in the course of the Burgers circuit about the undissociated 60” dislocation and will be referred to as plane 1. The faulting rotations of modes of dissociation (i) and (iii) are in the same sense as that in which the Burgers circuit is traversed and will be referred to as positive. Thus we may refer to the above modes respectively as: (i) 1+ (ii) l(ii) 2+ and (iv) 2-. The width, w, of the stacking fault will be given in terms of double atom ribbons. Thus for example Fig. 8(c) represents the dislocation 1 + with w = 3. Each of these four types of dissociation has both a and /3forms making eight different forms in all, but in all cases either the
STRUCTURE
1357
bonds up and down from the faulted atoms or the horizontal bonds in the partial dislocations are wrong and half of the wrong bonds are the high energy B-3 bonds. The same is true for all dislocation arrays involving partial dislocations and stacking faults in the sphalerite structure. That is, all forms of extended dislocations, Lomer-Cottrell sessiles and other combinations of partial dislocations and stacking faults on one or more planes involve either wrong bonds from the stacking faults or wrong bonds in the imperfect dislocations or, in complex arrays, both. This situation is analogous to that encountered in ordered alloys. In an ordered alloy a partial dislocation of the superlattice bounds an anti-phase domain boundary, and this involves wrong nearest neighbours. The effect of this upon the dissociation energy requires discussion. The line energy of a dislocation in anisotropic elasticity theory is given by
E = (KV/&) ln(R/ra) where R isthe distance from the dislocation line over which its stress field extends, re is a distance inside which Hooke’s law does not hold and inside which the energy is not obtainable using elasticity theory and K is an elastic energy factor, explicit expressions for which exist only for certain combinations of Burgers vector and dislocation line. Expressions for K in these cases have been given by FOREMAN. Employing these formulae and values of the elastic constants from HUNTINGTON(3s) and putting arbitrarily, but not too unrealisticalty in view of the low dislocation densities encountered R = 10-Zcm and rs = 10-T cm the values given in Table 1 are obtained for a number of dislocations in two IVB elements, two III-V compounds and a II-VI compound. On dissociation the elastic energy falls in proportion to the squares of the Burgers vectors involved, i.e. from ba = 4 for ($,O, - 4) to 6s = 6 for (4, i), -f)L plus ba = + for (4, -&, -$)R,using the notation of FRANK and ~TICHOLAS.(~~) That is, the elastic strain energy falls by one-third of its original value. The net decrease in energy produced by dissociation per unit length of dislocation is ED = *E -nlrl-n2Es-ns6~--EQwhere E is the elastic line energy of the dislocation, el is the energy per strained bond formed in the partial dislocations, ES and 6; are respectively
D.
1358
B.
HOLT
Table 1 Energy per unit length in IO8 eV/cm.
Dislocation Hornstra’s No.
-
Description
b
a
Diamond
Glide Plane
Germanium
InSb
GaAs*
ZnS
I
Screw
&z
(110)
-
EI
18.96
4.75
2.56
3.62
2.27
II
60”
&r
(110)
{Ill}
En
20.31
5.92
3.62
4.90
3.88
Edge
&z
(110)
{IOO}
Em
21.38
6.72
4.16
5.66
4.09
&?
{I101
EVIII
20.24
5.85
3.58
4.96
344
+r
(100)
EIX
20.56
6.04
3.59
4.92
344
1.13
1.42
1.63
I.56
1.80
-
1%
4%
19%
III
Small VIII
Bfin$;ry Dislocation
IX
-
Em/E1
-
-
-
-
Pauling’s “Amount of ionic Character”(s)
-
ss) have the decimal point misplaced.(ss) * The values for the elastic constants of GaAs quoted in HIJNTINGTON~
the additional energies per wrong A-A and B-B bond formed in the case of sphalerite structure compounds, the n’s are the increases in the numbers of the various types of bonds per cm, and Edis the energy of the stacking fault per unit length of dislocation. The values of the n’s are given in Table 2 for two dislocations. On the other hand, bond rearrangements of the types envisaged by HORNSTRA(~) will lower the energy of dislocations by ER = n4q - n5.q - %c2 - n,~; where ~1, cs and EL have the same meanings as before and where ~4 is the energy per broken bond eliminated and the n’s are the increases in the numbers of the various types of bonds per cm. Some of these energies can be roughly estimated. In germanium dangling bonds give rise to an acceptor level near the middle of the energy gap. The energy of the electron in the dangling bond must be below this and constitute a donor level. This donor level has never been detected, (3s) however. It must, therefore, lie very near the top
of the valence band. The electrons in the bond, before it was broken, lay lower in the band. As a first approximation it may be assumed that the extra energy of the electron in the broken bond therefore is cIIILX-~gv = ~4 where cmItX is the energy of the top of the valence band and cllV is the average energy of the valence electrons. The width of the valence band in germanium is 10 eV.@s) Hence if E,, = 3/S E,,,, which is obtained for the free electron model, is a reasonable approximation for germanium, then ~4 is about 4 eV and the energy of the two dangling electrons resulting from breaking a bond is about 8 eV. This value is too high. The sublimation energy of germanium is 85 kcal/mol. and this represents the energy required to break 2N bonds where N is Avogadro’s number. From this a value of 2.3 eV is obtained for the energy required to break a bond at a free surface. A rough idea of the magnitude of ~1 can be obtained from the vibrational spectrum of similar bonds. An intense infra-red band at 1265-1258 cm-l has been observed and attributed to an
DEFECTS
IN
THE
SPHALERITE
STRUCTURE
1359
Table 2 Dislocation
Hornstra’s
Dissociation
No. Description
Dissociation Mode
2f
n1
22/2 a
I
n4
fl5
7l6
n7
0
o-
ns
%42)/a
9(2)/a
4(2)w -a
%42)w a
o-
-31/(2) a
d/(2) a
Screw 1+
242
d(2) --
d(2) --
Q
a
a
w2 l-
vww
--
a
2+
2-
a
dew a
242 a
II
n3
2x4 u
2-
nz
Rearrangement
2x4
2/(2)/a
1/(2)/a
l/ww
dWw
--
a
a
a
242
1/(2) ---
1/(2) --
60” If
a
l-
a
242
d(2)w
a
a
--
“a” is the lattice spacing and w is the width of the stacking fault. per unit length of the dislocation.
Si-CHs bending vibration.@l) If this is regarded as a simple harmonic vibration, the restoring force constant can be derived. Then the energy of a vibration through 0 which is also the potential energy of a bond strained through 8 from its tetrahedral direction can be found. The angle at which the energy is 8 eV and at which the bond might be expected to break using the first of the estimates above is found to be 13”. The angle at which the energy becomes 2.3 eV is 5.4”. Since the argument has employed data concerning the stiffness of the Si-C bond which must be expected to be higher than that of the Ge-Ge bond, it is probable that the bonds in germanium will bend
a
dew a ns is the number of double bonds formed
rather farther than 5.4”. The bonds are stretched as well as bent in the Hornstra drawings and in those given here. The energy involved in stretching the bonds during bending can be calculated by using a Morse function following SWALIN.(42) This energy proves to be only 0.02 eV even for a 13” bend and so does not affect the preceding argument. Thus the pictures due to HORNSTRA(~) of the cores as containing only a few broken bonds per atomic plane are probably more realistic, at least for germanium, than the picture of the core as a crack due to HAASEN.@~) In the sphalerite structures there is an ionic contribution to the bonding in addition to the covalent contribution.
1360
D.
B.
Ionic bonding is non-directional and thus the Hornstra type of picture employed in this paper is probably applicable. Growth twins are common in diamond- and sphalerite-structure materials(54~43~ 44) and deformation twinning occurs.(45) Hence the stacking fault energies in these crystals may well be low. However, no definite evidence of stacking faults has yet been obtained in transmission electron microscope studies.(sl. as) The present evidence thus indicates that dissociation in some diamond-structure materials may lower the energy of dislocations in favoured orientations substantially, whereas in sphaleritestructure materials it would lower the energy somewhat less owing to the additional energy of the unavoidable wrong bonds which are formed. Owing to the large values of the elastic energy of the dislocations there may well be cases where ED > ER and dissociation will be preferred to the bond rearrangement process. In sphaleritestructure materials, of the four simple dislocations having slip plane { 111}, only the screw dislocation is able to rearrange its core bonds without forming wrong bonds in the process. Thus in compounds in which bond rearrangement is the preferred process it will occur most effectively along screw oriented portions of dislocation loops and elongation parallel to the Burgers vector may occur. The movement of extended dislocations should take place through the generation and movement of kinks in the partial dislocations since the simultaneous rotation of all the atoms in long rows is extremely improbable. These kinks are in the direction of motion of the dislocation, and lie in the slip plane. From Figs. 8(c) and 8(d) it can be seen that these steps will involve bond strain but neither broken bonds nor additional wrong bonds. Theories of dislocation motion involving kinks in the slip plane, have been put forward, but only for total dislocations.(46* 47) The previous discussion indicates that it may be independent kinks in the partial dislocations that are important in those materials in which the dislocations dissociate. The lowered stability of Lomer-Cottrell sessile dislocations in the sphalerite structure as compared with the diamond structure due to the additional energy of the wrong bonds involved in these arrays has implications for work-hardening theory. Unextended Lomer sessiles would break
HOLT
up into their component dislocations at considerably lower stresses than would extended LomerCottrell arrays and the sessiles would constitute weaker barriers to dislocation motion. Moreover, if the dislocations in these materials are not extended, the energy required to constrict dislocations will not play the role postulated by BASINSKI in the cutting of the forest. Consider the effect of the ionic component in the bonding in sphalerite structure materials. The lessened directionality of the bonds will favour Hornstra’s bonded core structures rather than Haasen’s cracked cores. With an increasing ionic component the bonds will become less and less directional and the dislocations will become wider. This will lead to a fall in the Peierls-Nabarro stress.(@) Wrong bonds now correspond to placing atoms carrying charges of the same sign as nearest neighbours, which requires extra electrostatic energy. TWINNING
The dislocations which bound a layer of upright { 111) twin contain wrong bonds. In indentation tests on InSb, GaSb and ZnS (111) twinning set in at about the same fraction of the melting point as did slip and this fraction was common to all these materials.(45) Thus, if the twins are upright, partial dislocations containing wrong bonds can exist and move in sphalerite structure compounds of varying degrees of ionicity of bonding. This equality in the homologous temperatures at which slip and twinning commence, would seem to indicate that it is the motion of extended dislocations which is involved in slip because then the partial dislocations involved in twinning and in the movement of extended dislocations would be the same and would have similar values of PeierlsNabarro force. Recently VENABLES~~~) has developed a dislocation‘mechanism for (111) twinning in face-centred cubic metals. The same mechanism is applicable to (111) twinning in diamond and sphalerite-structure materials. (123) twinning also results from indentation of Ge, Si, GaSb and InSb but not of ZnS.@s) This suggests that increasing ionicity of bonding inhibits the formation of this type of twin just as the increase in ionicity of bonding on going from PbS to NaCl is thought to be responsible for the inhibition of twinning.t51) It has been shown in Ge that twin boundaries
DEFECTS
IN
THE
SPHALERITE
produce no electrical effectscas) and this is to be expected too in the case of sphalerite structure materiaXs since coherent twin boundaries contain neither broken nor wrong bonds. However, incoherent twin boundaries, which consist of partial twinning dislocationscss) contain wrong bonds in sphalerite structure materials and may produce detectable electrical effects. Moreover, since coherent interfaces of upright (111) twins do not contain strained bonds either, it is not surprising that such growth twins form in profusion in InSb.(54) Hornstra’s bond arrangements will eliminate broken bonds.(l) The charge carrier scattering due to the consequent line charge and the compensating space charge cylinder(ssl XG~) would then disappear leaving only the scattering due to the iattice dilation,(5s’ together with that which is due to the stacking fault. Bond rearrangement by eliminating broken bonds and line charge(ss) may be expected, too, to decrease the difference in etching behaviour of a and /3 disloeations.os, 13) The fact that 01and 13 dislocations exhibit distinct etching and electricalcss-61) effects provides some indication that all the broken bonds in their cores have not been eliminated by rearrangement. Note added in proof Dr. J. HORNSTRA(private communication) has pointed out that the discussion of the two forms of right handed screw dislocation connected with Fig. 6 is not clear. To make a continuous helical path round the dislocation line steps l-2- . . . 6 must be followed by a step 6-l’ to the right in Fig. 6 followed by similar sequences of six steps (bonds). Of bonds which are nearest neighbours of the dislocation line (the broken line in Fig. 6) only those of the type of 6-l are not traversed in the course of the helical path. These bonds specify a direction. By alternative choice of the starting point (3) the equivalent bonds on the other side of the dislocation line (e.g. the step to the right from 3) may also be so distinguished. Both these sets of bonds have the same black-white sequence in the direction of the Burgers vector. The dislocations having the reverse sequence white-black are the other form of righthanded screw dislocation. Dr. Hornstra suggests a bond rearrangement of the core of the partial dislocation Ib of Fig. SC. “If the black atom with the dangling bond diffuses away, a new black-white bond can be formed and a dangling bond remains on the white atom. By this process the elastic strain is lowered and the unfavourable black-black bond is removed.” AcknowZedgemmts-Thanks
are due to the Council
for
STRUCTURE
1361
Scientific and Industrial Research and to Industrial Distributors (1946) Limited for financial support and to Professor F. R. N. NABARROfor helpful discussions. REFERENCES 1. HORNSTRAJ., J. Pkys. Ckem. So&& 5,129 (1958). 2. PAULING L., The Nature of #he Chemical Bond, p$ 97. (3rd Ed.) Cornell, Ithaca (1960). 3. SCANLON W. W., Solid State Physics 9. 83 (1959). 4. SHULL H., J. appl. Phys. 33, 296 (1962). 5. GOODMAN C. H. L.. I. EKectron. 1. 115 (1955)-here the graphical definition of EcoV and &on appears to be useful, but the non-equivalent electron arrangement definition does not-see MOOSER E. and PEARSONW. B., Nature, Lond. 190,406 (1961). 6. SUCHETJ. P., J. Phys. Chem. Solids 16, 265 (1960). 7. MOOSERE. and PEARSONW. B., Progr. Scmicond. 5, 103 (1960). 8. FOLBERTH0. G., Z. Nut&. 13a, 856 (1958). 9. PHILLIPSJ. C. and COHEN-M. H;, J. a&l. Phys. 33, 293 (1962). 10. ELLIS S. G., J. appl. Pkys. 30,947 (1959). 11. MOODY P. L.. GATOS H. C. and LAVINE M. C.. _1. appl. Phys. 31, 1696 (1960). 12. VENABL~ J. D. and BROUDY R. M., J. @pl. Pkys. 29, 1025 (1958). 13. GATOS H. C. and LAVINE M. C., J. Pkys. Chem. Solids 14, 169 (1960). 14. GATOS H. C, and LAVINE M. C., J. appl. Phys. 31, 743 (1960). 15. FAUST J. W. and SAGARA., J. appl. Phys. 31, 331 (1960). 16. RICHARDSJ. L. and CROCKERA. J., J. appl. Phys.
31, 611 (1960). 17. REYNOLDSD. C. and CZYZAK S. J., J. appl. Phys. 31, 94 (1960). A. J. and GATOS H. C., 18. LAVINE M. C., ROSENBERG J. uppl. Pkys. 29, 1131 (1958). 19. HANEMAN D., J. uppl. Phys. 31,217 (1960). 20. VENABLE~J. D. and BROUDY R. M., J. appf. Phys.
30, 122 (1959). 21. HAA~EN P., Acta Met. 5, 598 (1957). 22. GATOS H. C., MOODY P. L. and LAVXNEM. C., J. uppl. Pk_vs. 31, 212 (1960). 23. HOLT D. B., .I. app2.Pkvs. 31, 2231 (1960). 24. GATOS H. C:,J. iipl. Phis. 32; 1232 (1961): T. F.. Phil. Mm._ 44.I 25. FRANK F. C. and NICHOLAS _ 1213 (1953). 26. READ W. T., Phil. Mag. 45, 775 (1954). 27. BILBY B. A., Research, Land. 5, 387 (1952). 28. DASH W. C., J. appt. Phys. 27, 1193 (1956). 29. DASH W. C., Dislocations and Mechunicaal Properties of Crystals, p. 57, Wiley, New York (1957). 30. PAT~L J. R., J. appl. Pkys. 29, 170 (1958). 31. GEACH G. A., IRVINGB. A, and PHILIPS R., Researck Land. to, 411 (1957). 32. HOLT D. B. and DANGOR A. E., to be published. 33. VOGEL F. L., Acta Met. 3, 245 (1955). 34. Vrw Bum H. G., Pkysicu 24, 831 (1958).
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D.
B.
H. G., Physica 25. 775 11959). 3.5. VAN Bum 36. VAN BUEREN H. G.; Ph&ca 26; 997 (196Oj. 37. FOREMANA. J. E., Acta Met. 3. 322 (1955). $hysici 7, 213 38. HUNTINGTON- H. .B., Solid Ske (1958). 39. BARDSLEY W., Progr. Semicond. 4, 157 (1960). F., Physica 20, 801 (1954). 40. Hw 41. Chemical Applications of Spectroscopy (edited by WEST W.) p. 553. Interscience, New York (1956). 42. SWALIN R. A., J. Phys. Chem. Solids 18,290 (1961). 43. ELLIS W. C. and FAGEANT J., J. Metals, N.Y. 6,
291 (1954). 44. FRANKS J., GEACH G. A. and CHURCHMANA. T., PYOC.phys. Sot. Lond. B68, 111 (1955). A. T., GEACH G. A. and WINTON J., 45. CHURCHMAN PYOC.Roy. Sot. A238, 194 (1956). 46. HAM R. K., Private Communication. 47. GILMAN J. J., Aust. J. Phys. 13, 327 (1960). 48. BA~INSKIZ. S., Phil. Mug. 4, 393 (1959). *49. COTTRELL A. H., Dislocations and Plastic Flow in Crystals, p. 64. University Press, Oxford (1953).
HOLT
50. VENABLESJ. A., Phil. Mug. 6, 379 (1961). 51. CAHN R. W., Advanc. Phys. 3, 363 (1954). 52. BILLIG E. and RIDOUT M. S., Nature, Land. 173, 496 (1954). 53. VOTAVAE. and BERG-AN A., Acta Met. 7, 392 (1959). 54. H,&EN.P., J. Metals, N.Y. 9, 30 (1957). 55. READ W. T.. Phil. Man. 45. 1119 (1954). 56. GIBSON A. k. and PA& i. G. s‘., P&l. Mug. 3, 950 (1958). 57. BROUDY R. M. and MCCLURE J. W., J. appl. Phys. 31, 1511 (1960). 58. DEXTER D. L. and SEI~ F., Phys. Rew. 86, 964 (1952).
59. BELL R: L., private communication. 60. GATOS H. C.. FINN M. C. and LAVINE M. C.. __I. appl. Phys. 52, 1174 (1961). 61. LAVINE M. C., GATOS H. C. and FINN M. C., J. electrochem. Sot. 108, 974 (1961). 62. BATEMANT. B., MCSKIMIN H. J. and WHELAN J. M.
J. appl. Phys. 30, 544 (1959).
Pergamon
Press 1962. Vol. 23, pp. 1353-1362.
Printed in Great Britain.
DEFECTS IN THE SPHALERITE STRUCTURE D. B. HOLT C.S.I.R.
Solid State Physics Research Unit, Department of Physics, University Johannesburg, South Africa. (Received
16 February
1962; revised 4 May
of the Witwatersrand,
1962)
Abstract-Hornstra’s method for drawing the arrangements of the atoms in the cores of dislocations in covalently bonded crystals is applied to the sphalerite structure. Bond rearrangement and dislocation dissociation into an extended form are considered and the implications are discussed. A crude calculation of bond strain energy indicates that Haasen’s cracked core structure is unrealistic in germanium and in the sphalerite structure. Twinning and the electrical effects of dissociation and rearrangement are briefly discussed.
INTRODUCTION
HORNSTFU(II has shown that since covalent bonds can be regarded as directed in space, elastically stiff and brittle, it is possible to draw the atomic arrangements in the cores of dislocations of covalent crystals. Qualitative theories of bonding in semiconductors have been developed which indicate the covalent and ionic contributions to the bonding, These all stem from the work of PAULING.(~) It has been pointed out,(s) however, that this approach is rather misleading when applied to the solid state since it is based on the empirical relationship between the difference of electronegativity and the dipole moment in diatomic molecules. Theoretical objections too can be raised against this approach.(4I Many related methods have been proposed for determining the covalent and ionic contributions to the width of the forbidden energy gap in semiconductors.(5+ However, more relevant to the behaviour of interest here is a method for calculating the covalent and ionic contributions to the strengths of the bonds in sphalerite structure materials due to FOLBERTH.(*) Attempts to extract information about the degree of ionicity of bonding from quantum mechanical calculations are currently under way.(4$ 9) Thus if Hornstra’s methods were applied to such structures as sphalerite (zincblende) it would be possible to say something concerning the relative extent to which particular compounds might be expected I
to have such narrow dislocation core arrangements rather than those wider ones more characteristic of ionically bonded crystals of the same structure. --Recently the anisotropies in the [ill]-111 l] directions which follow from the lack of a centre of symmetry in the sphalerite and related structures have attracted considerable attention.(Is-Is) Out of this work has arisen a renewed interest in the existence of two kinds of “edge” dislocations(I4p aa) first pointed out by HAASENI~I)and an attempt to relate the anisotropies to bonding. (149ssS2%24)The extension of Hornstra’s analysis to deal with the core structure and the effects of dislocations and other structural defects in sphalerite seems, therefore, to be opportune. ARRANGEMEN’IS OF THE CORES OF DISLOCATIONS IN SPHALEIUTE
THJ3 ATOMIC
FRANK and NICHOLAS have obtained and listed all the Burgers vectors corresponding to stable dislocations in a number of common crystal lattices. Amongst these was the face-centred cubic structure, and much of the analysis carries over to substances such as diamond and sphalerite which have the same space lattice. Experiment shows that in the latter structures certain directions for the dislocation line are preferred. The combinations of stable Burgers vectors b and low index crystallographic line directions u give all the simple dislocations which differ in core structure.(I) The two directions b and u together
1353
1354
D.
B.
determine the slip plane of the dislocation (in the case of screw dislocations where they are parallel a plane may be arbitrarily chosen). Thus, for example, consider drawing the core of the dislocation in the sphalerite structure for which a = [liO] and which has the Burgers vector b = + [lOi]; this is a 60” dislocation@s) (so-called because of the angle between a and b).
HOLT
shown in Fig. 2. Connecting up nearest neighbours across the slip plane in a way consistent with the simplest picture of a covalent bond results in the core structure shown in Fig. 3. All the atoms with broken bonds in the core are of one type (black). Thus, for example, in InSb two types of 60” dislocations are possible. These are those in which the atoms which have broken
\
\ \
,b
\
FIG. 1. The sphalerite structure with a [ill] direction vertical. The white spheres represent atoms of one element of the compound and the black spheres the atoms of the other element. A (111) slip plane, the line direction a and the Burgers vector B of a 60” dislocation are shown.
The first step is to draw the sphalerite structure indicating the slip plane, dislocation line and Burgers vector as in Fig. 1. The second step is to choose, arbitrarily, to move the dislocation into the drawing from, say, right to left. Then(s7) the atoms above the slip plane are displaced to the right through 3 [lOi] relative to those below as
FIG. 3. The 60” dislocation in sphalerite. a and b lie along
FIG. 2. The result of moving a 60” dislocation as indicated in Fig. 1 into the crystal from right to left is to produce the relative movements of the atoms indicated by the arrows.
or dangling bonds and which constitute the row on the edge of the “extra half-plane” are all In atoms and those in which they are all Sb atoms. The former type is called the a and the latter the p form. (1s) This terminology will be extended to classes of sphalerite structure compounds other
DEFECTS
IN
THE
SPHALERITE
than III-V e.g. II-VI and I-VII by using the term CCfor the form in which the atoms of lower valency occur in the core.
STRUCTURE
1355
screw dislocation which are analogous to the a and /I types of dislocations with an edge component. Similarly the minimum Burgers circuit round a left-handed screw has only one step with a component antiparallel to the Burgers vector and this can occur in two sequences giving rise to two types of left-handed screw dislocation. Thus there are four types of screw dislocation in the sphalerite structure.
FIG. 4. The edge dislocation having the slip plane { 111 } in the sphalerite structure. a is a (112 > direction and b lies along a <110 > direction.
To most of the simple dislocations in the diamond structure there correspond a and /3 forms in the sphalerite structure. Further examples are shown in Figs. 4 and 5. However, not all dislocations have tc and fi forms. Fig. 6 shows one form of the screw dislocation in the sphalerite structure in which the Burgers vector is parallel to the positive line direction and the screw is righthanded. In the case in which the Burgers vector and line direction are antiparallel the screw is left-handed.
FIG. 6. The right-handed screw dislocation in the sphalerite structure. a and b lie along the same
These are the simplest complete dislocations with slip plane (1111 and correspond to observed line directions in the diamond structure. Dislocations have been seen in silicon using a decoration technique and infra-red transmission microscopy. The dislocations were found to lie along (110 > directions@81 29) and along (112) directions. Dislocations lying along (112 ) directions, and
FIG. 5. The 30” dislocation. a is a <112) direction, b lies along a (110 > direction and the slip plane is of (111) type. In Fig. 6 it can be seen that of the steps round the minimum Burgers circuit l-Z-3-4-5-6 only one, namely 3-4, has a component antiparallel to the Burgers vector. This step can occur either in the sequence black-white as shown or whiteblack. Hence there are two types of right-hand
FIG. 7. The small angle boundary dislocation. a is a
D.
1356
B.
along both (110 > and (112) directions have been seen in germanium by transmission electron microscopy. The dislocation of Fig. 7 has been found in small angle boundaries in germanium.taa) No equivalent observations have as yet been made on sphalerite structure compounds. The dislocation of Fig. 7 does not have a and /3forms. DISSOCIATION VERSUS REARRANGEMENT
BOND
HORNSTRA~~)reported that in all the dislocations rearrangements of the bonds were possible which would lower the energy. VAN BLJEREN(~P 35936) has built up a theory of the movement of dislocations in the diamond structure on the assumption that the positions at which bond rearrangement has occurred act as immobile dragging points along the dislocation line. The situation in the sphalerite structure is altered by the fact that three types of bond are geometrically possible in the cores. These are A-A, A-B and B-B bonds, where A indicates an atom of the lower valence element of the compound and B an atom of the higher valence element. Consider for example the case of III-V compounds. On average each III element atom contributes 2 of an electron to each of its four tetrahedral bonds and each V element atom contributes l$ electrons per bond.(ss) Hence as a working hypothesis it may be supposed that an A-A bond in a III-V compound will contain on average 14 electrons and will be a low energy structure. Similarly an A-B containing 2 electrons is a low energy structure. But B-B bonds which would contain 2+ electrons on average, which could not therefore be paired off with opposite spins, will be of higher energy. Some re-arrangements which it has been suggested may occur in dislocations in the diamond structure(l) can be carried over into the sphalerite structure. An example is the case of the screw dislocation. By forming double bonds and so eliminating the most strained single bonds of Fig. 6 the total bond strain can, perhaps, be reduced. This is an example of bond rearrangement proper and could give rise to localized dragging points. There is, however, a second type of process, dissociation, which may lower the energy of dislocations. Consider the dissociation of the 60” dislocation into an extended form consisting of two
HOLT
partial dislocations separated by a strip of stacking fault. This can occur in two different ways in the diamond structure but in four geometrically different ways in sphalerite. Figures S(a) and (b) show the starting points. In each case a strip two atoms wide, of which the two atoms which lie in the plane of the drawing are shown, rotates into a new position and the bonds reform as indicated by the broken lines. Figures S(a) and (b) indicate the two different strips which may rotate initially. Each strip can rotate into the new position either clockwise as indicated in the drawings or anticlockwise. These two alternatives lead to indistinguishable results in the diamond structure.
AB=
% [IfId
AB,
(a)
AE-
$‘i kUor’l Cb)
k [7Jc?], Ia
EB =-
‘/s [&], lb
(cl
AE=
E'8=- ‘/s [i77], IC
+6 [77i],Id
(4 FIG. 8. Geometrically possible modes of dissociation of a 60” dislocation in sphalerite. (a) and (b) indicate the two possible double atom ribbons which might rotate initially. (c) is the extended dislocation resulting from the rotation of the atom ribbon of (a) and its successors in the clockwise sense. (d) is the extended dislocation resulting from the rotation of the atom ribbon of (b) and succeeding ribbons in the anticlockwise sense.
DEFECTS
IN
THE
SPHALERITE
However, in the sphalerite structure, if the rotation occurs in the clockwise sense the correct sequence of black-white-black-white in the vertical ( 111) direction is unaltered whereas the anticlockwise rotation results in the reversal of this sequence in the stacking fault. When the sequence remains correct the stacking fault will be said to be upright, and when it is reversed the fault will be said to be inverted. Figure 8(c) shows the extended dislocation obtained from 8(a) through the successive clockwise rotation of double atom rows to the right of that indicated in 8(a). The bonds extending up and down from the atoms in the fault are correct A-B bonds whereas the horizontal bonds in the partial dislocations at the ends of the stacking fault are wrong A-A and 3-B bonds and the latter are of high energy. Similarly, the extended dislocation in Fig. 8(d) results from 8(b) on making the rotations of successive double atom rows to the right of that in 8(a) in the anticlockwise sense. The stacking fault in this, as in all cases which appear anticlockwise as drawn, is inverted and the bonds above and below it are all wrong bonds, but the horizontal bonds in the partial dislocations are right. There are four geometrical types of extended 60” dislocations: (i) 8(c) with the stacking fault upright, (ii) 8(c) with inverted fault (not shown), (iii) 8(d) with upright fault (not shown) and (iv) 8(d) with inverted fault. A notation can be developed for these modes of dissociation which is independent of the drawings by using the sense in which the Burgers circuit is traversed. This is such as to constitute a righthanded screw with the (arbitrary) positive direction along the dislocation lines. The faulted double plane of modes of dissociation (i) and (ii) lies immediately before that of modes (iii) and (iv) in the course of the Burgers circuit about the undissociated 60” dislocation and will be referred to as plane 1. The faulting rotations of modes of dissociation (i) and (iii) are in the same sense as that in which the Burgers circuit is traversed and will be referred to as positive. Thus we may refer to the above modes respectively as: (i) 1+ (ii) l(ii) 2+ and (iv) 2-. The width, w, of the stacking fault will be given in terms of double atom ribbons. Thus for example Fig. 8(c) represents the dislocation 1 + with w = 3. Each of these four types of dissociation has both a and /3forms making eight different forms in all, but in all cases either the
STRUCTURE
1357
bonds up and down from the faulted atoms or the horizontal bonds in the partial dislocations are wrong and half of the wrong bonds are the high energy B-3 bonds. The same is true for all dislocation arrays involving partial dislocations and stacking faults in the sphalerite structure. That is, all forms of extended dislocations, Lomer-Cottrell sessiles and other combinations of partial dislocations and stacking faults on one or more planes involve either wrong bonds from the stacking faults or wrong bonds in the imperfect dislocations or, in complex arrays, both. This situation is analogous to that encountered in ordered alloys. In an ordered alloy a partial dislocation of the superlattice bounds an anti-phase domain boundary, and this involves wrong nearest neighbours. The effect of this upon the dissociation energy requires discussion. The line energy of a dislocation in anisotropic elasticity theory is given by
E = (KV/&) ln(R/ra) where R isthe distance from the dislocation line over which its stress field extends, re is a distance inside which Hooke’s law does not hold and inside which the energy is not obtainable using elasticity theory and K is an elastic energy factor, explicit expressions for which exist only for certain combinations of Burgers vector and dislocation line. Expressions for K in these cases have been given by FOREMAN. Employing these formulae and values of the elastic constants from HUNTINGTON(3s) and putting arbitrarily, but not too unrealisticalty in view of the low dislocation densities encountered R = 10-Zcm and rs = 10-T cm the values given in Table 1 are obtained for a number of dislocations in two IVB elements, two III-V compounds and a II-VI compound. On dissociation the elastic energy falls in proportion to the squares of the Burgers vectors involved, i.e. from ba = 4 for ($,O, - 4) to 6s = 6 for (4, i), -f)L plus ba = + for (4, -&, -$)R,using the notation of FRANK and ~TICHOLAS.(~~) That is, the elastic strain energy falls by one-third of its original value. The net decrease in energy produced by dissociation per unit length of dislocation is ED = *E -nlrl-n2Es-ns6~--EQwhere E is the elastic line energy of the dislocation, el is the energy per strained bond formed in the partial dislocations, ES and 6; are respectively
D.
1358
B.
HOLT
Table 1 Energy per unit length in IO8 eV/cm.
Dislocation Hornstra’s No.
-
Description
b
a
Diamond
Glide Plane
Germanium
InSb
GaAs*
ZnS
I
Screw
&z
(110)
-
EI
18.96
4.75
2.56
3.62
2.27
II
60”
&r
(110)
{Ill}
En
20.31
5.92
3.62
4.90
3.88
Edge
&z
(110)
{IOO}
Em
21.38
6.72
4.16
5.66
4.09
&?
{I101
EVIII
20.24
5.85
3.58
4.96
344
+r
(100)
EIX
20.56
6.04
3.59
4.92
344
1.13
1.42
1.63
I.56
1.80
-
1%
4%
19%
III
Small VIII
Bfin$;ry Dislocation
IX
-
Em/E1
-
-
-
-
Pauling’s “Amount of ionic Character”(s)
-
ss) have the decimal point misplaced.(ss) * The values for the elastic constants of GaAs quoted in HIJNTINGTON~
the additional energies per wrong A-A and B-B bond formed in the case of sphalerite structure compounds, the n’s are the increases in the numbers of the various types of bonds per cm, and Edis the energy of the stacking fault per unit length of dislocation. The values of the n’s are given in Table 2 for two dislocations. On the other hand, bond rearrangements of the types envisaged by HORNSTRA(~) will lower the energy of dislocations by ER = n4q - n5.q - %c2 - n,~; where ~1, cs and EL have the same meanings as before and where ~4 is the energy per broken bond eliminated and the n’s are the increases in the numbers of the various types of bonds per cm. Some of these energies can be roughly estimated. In germanium dangling bonds give rise to an acceptor level near the middle of the energy gap. The energy of the electron in the dangling bond must be below this and constitute a donor level. This donor level has never been detected, (3s) however. It must, therefore, lie very near the top
of the valence band. The electrons in the bond, before it was broken, lay lower in the band. As a first approximation it may be assumed that the extra energy of the electron in the broken bond therefore is cIIILX-~gv = ~4 where cmItX is the energy of the top of the valence band and cllV is the average energy of the valence electrons. The width of the valence band in germanium is 10 eV.@s) Hence if E,, = 3/S E,,,, which is obtained for the free electron model, is a reasonable approximation for germanium, then ~4 is about 4 eV and the energy of the two dangling electrons resulting from breaking a bond is about 8 eV. This value is too high. The sublimation energy of germanium is 85 kcal/mol. and this represents the energy required to break 2N bonds where N is Avogadro’s number. From this a value of 2.3 eV is obtained for the energy required to break a bond at a free surface. A rough idea of the magnitude of ~1 can be obtained from the vibrational spectrum of similar bonds. An intense infra-red band at 1265-1258 cm-l has been observed and attributed to an
DEFECTS
IN
THE
SPHALERITE
STRUCTURE
1359
Table 2 Dislocation
Hornstra’s
Dissociation
No. Description
Dissociation Mode
2f
n1
22/2 a
I
n4
fl5
7l6
n7
0
o-
ns
%42)/a
9(2)/a
4(2)w -a
%42)w a
o-
-31/(2) a
d/(2) a
Screw 1+
242
d(2) --
d(2) --
Q
a
a
w2 l-
vww
--
a
2+
2-
a
dew a
242 a
II
n3
2x4 u
2-
nz
Rearrangement
2x4
2/(2)/a
1/(2)/a
l/ww
dWw
--
a
a
a
242
1/(2) ---
1/(2) --
60” If
a
l-
a
242
d(2)w
a
a
--
“a” is the lattice spacing and w is the width of the stacking fault. per unit length of the dislocation.
Si-CHs bending vibration.@l) If this is regarded as a simple harmonic vibration, the restoring force constant can be derived. Then the energy of a vibration through 0 which is also the potential energy of a bond strained through 8 from its tetrahedral direction can be found. The angle at which the energy is 8 eV and at which the bond might be expected to break using the first of the estimates above is found to be 13”. The angle at which the energy becomes 2.3 eV is 5.4”. Since the argument has employed data concerning the stiffness of the Si-C bond which must be expected to be higher than that of the Ge-Ge bond, it is probable that the bonds in germanium will bend
a
dew a ns is the number of double bonds formed
rather farther than 5.4”. The bonds are stretched as well as bent in the Hornstra drawings and in those given here. The energy involved in stretching the bonds during bending can be calculated by using a Morse function following SWALIN.(42) This energy proves to be only 0.02 eV even for a 13” bend and so does not affect the preceding argument. Thus the pictures due to HORNSTRA(~) of the cores as containing only a few broken bonds per atomic plane are probably more realistic, at least for germanium, than the picture of the core as a crack due to HAASEN.@~) In the sphalerite structures there is an ionic contribution to the bonding in addition to the covalent contribution.
1360
D.
B.
Ionic bonding is non-directional and thus the Hornstra type of picture employed in this paper is probably applicable. Growth twins are common in diamond- and sphalerite-structure materials(54~43~ 44) and deformation twinning occurs.(45) Hence the stacking fault energies in these crystals may well be low. However, no definite evidence of stacking faults has yet been obtained in transmission electron microscope studies.(sl. as) The present evidence thus indicates that dissociation in some diamond-structure materials may lower the energy of dislocations in favoured orientations substantially, whereas in sphaleritestructure materials it would lower the energy somewhat less owing to the additional energy of the unavoidable wrong bonds which are formed. Owing to the large values of the elastic energy of the dislocations there may well be cases where ED > ER and dissociation will be preferred to the bond rearrangement process. In sphaleritestructure materials, of the four simple dislocations having slip plane { 111}, only the screw dislocation is able to rearrange its core bonds without forming wrong bonds in the process. Thus in compounds in which bond rearrangement is the preferred process it will occur most effectively along screw oriented portions of dislocation loops and elongation parallel to the Burgers vector may occur. The movement of extended dislocations should take place through the generation and movement of kinks in the partial dislocations since the simultaneous rotation of all the atoms in long rows is extremely improbable. These kinks are in the direction of motion of the dislocation, and lie in the slip plane. From Figs. 8(c) and 8(d) it can be seen that these steps will involve bond strain but neither broken bonds nor additional wrong bonds. Theories of dislocation motion involving kinks in the slip plane, have been put forward, but only for total dislocations.(46* 47) The previous discussion indicates that it may be independent kinks in the partial dislocations that are important in those materials in which the dislocations dissociate. The lowered stability of Lomer-Cottrell sessile dislocations in the sphalerite structure as compared with the diamond structure due to the additional energy of the wrong bonds involved in these arrays has implications for work-hardening theory. Unextended Lomer sessiles would break
HOLT
up into their component dislocations at considerably lower stresses than would extended LomerCottrell arrays and the sessiles would constitute weaker barriers to dislocation motion. Moreover, if the dislocations in these materials are not extended, the energy required to constrict dislocations will not play the role postulated by BASINSKI in the cutting of the forest. Consider the effect of the ionic component in the bonding in sphalerite structure materials. The lessened directionality of the bonds will favour Hornstra’s bonded core structures rather than Haasen’s cracked cores. With an increasing ionic component the bonds will become less and less directional and the dislocations will become wider. This will lead to a fall in the Peierls-Nabarro stress.(@) Wrong bonds now correspond to placing atoms carrying charges of the same sign as nearest neighbours, which requires extra electrostatic energy. TWINNING
The dislocations which bound a layer of upright { 111) twin contain wrong bonds. In indentation tests on InSb, GaSb and ZnS (111) twinning set in at about the same fraction of the melting point as did slip and this fraction was common to all these materials.(45) Thus, if the twins are upright, partial dislocations containing wrong bonds can exist and move in sphalerite structure compounds of varying degrees of ionicity of bonding. This equality in the homologous temperatures at which slip and twinning commence, would seem to indicate that it is the motion of extended dislocations which is involved in slip because then the partial dislocations involved in twinning and in the movement of extended dislocations would be the same and would have similar values of PeierlsNabarro force. Recently VENABLES~~~) has developed a dislocation‘mechanism for (111) twinning in face-centred cubic metals. The same mechanism is applicable to (111) twinning in diamond and sphalerite-structure materials. (123) twinning also results from indentation of Ge, Si, GaSb and InSb but not of ZnS.@s) This suggests that increasing ionicity of bonding inhibits the formation of this type of twin just as the increase in ionicity of bonding on going from PbS to NaCl is thought to be responsible for the inhibition of twinning.t51) It has been shown in Ge that twin boundaries
DEFECTS
IN
THE
SPHALERITE
produce no electrical effectscas) and this is to be expected too in the case of sphalerite structure materiaXs since coherent twin boundaries contain neither broken nor wrong bonds. However, incoherent twin boundaries, which consist of partial twinning dislocationscss) contain wrong bonds in sphalerite structure materials and may produce detectable electrical effects. Moreover, since coherent interfaces of upright (111) twins do not contain strained bonds either, it is not surprising that such growth twins form in profusion in InSb.(54) Hornstra’s bond arrangements will eliminate broken bonds.(l) The charge carrier scattering due to the consequent line charge and the compensating space charge cylinder(ssl XG~) would then disappear leaving only the scattering due to the iattice dilation,(5s’ together with that which is due to the stacking fault. Bond rearrangement by eliminating broken bonds and line charge(ss) may be expected, too, to decrease the difference in etching behaviour of a and /3 disloeations.os, 13) The fact that 01and 13 dislocations exhibit distinct etching and electricalcss-61) effects provides some indication that all the broken bonds in their cores have not been eliminated by rearrangement. Note added in proof Dr. J. HORNSTRA(private communication) has pointed out that the discussion of the two forms of right handed screw dislocation connected with Fig. 6 is not clear. To make a continuous helical path round the dislocation line steps l-2- . . . 6 must be followed by a step 6-l’ to the right in Fig. 6 followed by similar sequences of six steps (bonds). Of bonds which are nearest neighbours of the dislocation line (the broken line in Fig. 6) only those of the type of 6-l are not traversed in the course of the helical path. These bonds specify a direction. By alternative choice of the starting point (3) the equivalent bonds on the other side of the dislocation line (e.g. the step to the right from 3) may also be so distinguished. Both these sets of bonds have the same black-white sequence in the direction of the Burgers vector. The dislocations having the reverse sequence white-black are the other form of righthanded screw dislocation. Dr. Hornstra suggests a bond rearrangement of the core of the partial dislocation Ib of Fig. SC. “If the black atom with the dangling bond diffuses away, a new black-white bond can be formed and a dangling bond remains on the white atom. By this process the elastic strain is lowered and the unfavourable black-black bond is removed.” AcknowZedgemmts-Thanks
are due to the Council
for
STRUCTURE
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Scientific and Industrial Research and to Industrial Distributors (1946) Limited for financial support and to Professor F. R. N. NABARROfor helpful discussions. REFERENCES 1. HORNSTRAJ., J. Pkys. Ckem. So&& 5,129 (1958). 2. PAULING L., The Nature of #he Chemical Bond, p$ 97. (3rd Ed.) Cornell, Ithaca (1960). 3. SCANLON W. W., Solid State Physics 9. 83 (1959). 4. SHULL H., J. appl. Phys. 33, 296 (1962). 5. GOODMAN C. H. L.. I. EKectron. 1. 115 (1955)-here the graphical definition of EcoV and &on appears to be useful, but the non-equivalent electron arrangement definition does not-see MOOSER E. and PEARSONW. B., Nature, Lond. 190,406 (1961). 6. SUCHETJ. P., J. Phys. Chem. Solids 16, 265 (1960). 7. MOOSERE. and PEARSONW. B., Progr. Scmicond. 5, 103 (1960). 8. FOLBERTH0. G., Z. Nut&. 13a, 856 (1958). 9. PHILLIPSJ. C. and COHEN-M. H;, J. a&l. Phys. 33, 293 (1962). 10. ELLIS S. G., J. appl. Pkys. 30,947 (1959). 11. MOODY P. L.. GATOS H. C. and LAVINE M. C.. _1. appl. Phys. 31, 1696 (1960). 12. VENABL~ J. D. and BROUDY R. M., J. @pl. Pkys. 29, 1025 (1958). 13. GATOS H. C. and LAVINE M. C., J. Pkys. Chem. Solids 14, 169 (1960). 14. GATOS H. C, and LAVINE M. C., J. appl. Phys. 31, 743 (1960). 15. FAUST J. W. and SAGARA., J. appl. Phys. 31, 331 (1960). 16. RICHARDSJ. L. and CROCKERA. J., J. appl. Phys.
31, 611 (1960). 17. REYNOLDSD. C. and CZYZAK S. J., J. appl. Phys. 31, 94 (1960). A. J. and GATOS H. C., 18. LAVINE M. C., ROSENBERG J. uppl. Pkys. 29, 1131 (1958). 19. HANEMAN D., J. uppl. Phys. 31,217 (1960). 20. VENABLE~J. D. and BROUDY R. M., J. appf. Phys.
30, 122 (1959). 21. HAA~EN P., Acta Met. 5, 598 (1957). 22. GATOS H. C., MOODY P. L. and LAVXNEM. C., J. uppl. Pk_vs. 31, 212 (1960). 23. HOLT D. B., .I. app2.Pkvs. 31, 2231 (1960). 24. GATOS H. C:,J. iipl. Phis. 32; 1232 (1961): T. F.. Phil. Mm._ 44.I 25. FRANK F. C. and NICHOLAS _ 1213 (1953). 26. READ W. T., Phil. Mag. 45, 775 (1954). 27. BILBY B. A., Research, Land. 5, 387 (1952). 28. DASH W. C., J. appt. Phys. 27, 1193 (1956). 29. DASH W. C., Dislocations and Mechunicaal Properties of Crystals, p. 57, Wiley, New York (1957). 30. PAT~L J. R., J. appl. Pkys. 29, 170 (1958). 31. GEACH G. A., IRVINGB. A, and PHILIPS R., Researck Land. to, 411 (1957). 32. HOLT D. B. and DANGOR A. E., to be published. 33. VOGEL F. L., Acta Met. 3, 245 (1955). 34. Vrw Bum H. G., Pkysicu 24, 831 (1958).
1362
D.
B.
H. G., Physica 25. 775 11959). 3.5. VAN Bum 36. VAN BUEREN H. G.; Ph&ca 26; 997 (196Oj. 37. FOREMANA. J. E., Acta Met. 3. 322 (1955). $hysici 7, 213 38. HUNTINGTON- H. .B., Solid Ske (1958). 39. BARDSLEY W., Progr. Semicond. 4, 157 (1960). F., Physica 20, 801 (1954). 40. Hw 41. Chemical Applications of Spectroscopy (edited by WEST W.) p. 553. Interscience, New York (1956). 42. SWALIN R. A., J. Phys. Chem. Solids 18,290 (1961). 43. ELLIS W. C. and FAGEANT J., J. Metals, N.Y. 6,
291 (1954). 44. FRANKS J., GEACH G. A. and CHURCHMANA. T., PYOC.phys. Sot. Lond. B68, 111 (1955). A. T., GEACH G. A. and WINTON J., 45. CHURCHMAN PYOC.Roy. Sot. A238, 194 (1956). 46. HAM R. K., Private Communication. 47. GILMAN J. J., Aust. J. Phys. 13, 327 (1960). 48. BA~INSKIZ. S., Phil. Mug. 4, 393 (1959). *49. COTTRELL A. H., Dislocations and Plastic Flow in Crystals, p. 64. University Press, Oxford (1953).
HOLT
50. VENABLESJ. A., Phil. Mug. 6, 379 (1961). 51. CAHN R. W., Advanc. Phys. 3, 363 (1954). 52. BILLIG E. and RIDOUT M. S., Nature, Land. 173, 496 (1954). 53. VOTAVAE. and BERG-AN A., Acta Met. 7, 392 (1959). 54. H,&EN.P., J. Metals, N.Y. 9, 30 (1957). 55. READ W. T.. Phil. Man. 45. 1119 (1954). 56. GIBSON A. k. and PA& i. G. s‘., P&l. Mug. 3, 950 (1958). 57. BROUDY R. M. and MCCLURE J. W., J. appl. Phys. 31, 1511 (1960). 58. DEXTER D. L. and SEI~ F., Phys. Rew. 86, 964 (1952).
59. BELL R: L., private communication. 60. GATOS H. C.. FINN M. C. and LAVINE M. C.. __I. appl. Phys. 52, 1174 (1961). 61. LAVINE M. C., GATOS H. C. and FINN M. C., J. electrochem. Sot. 108, 974 (1961). 62. BATEMANT. B., MCSKIMIN H. J. and WHELAN J. M.
J. appl. Phys. 30, 544 (1959).