JOURNAL DE PHYSIQUE
Colloque Cl, supplément au n° 10, Tome 4'6, octobre 1982 page Cl-45
GLIDE OF PARTIAL DISLOCATIONS IN SILICON
M. Heggie and R. Jones
Department of Physios, University of Exeter, Stooker Road, Exeter}
Devon, EX4 4QL} England
Résumé.- La reconstruction des dislocations partielles aboutit à des bandes
remplies séparées de bandes vides par un intervalle assez grand. Ceci conduit
à un problème pour identifier les centres responsables du comportement de la
RPE et de l'effet Hall. Nous avons envisagé que les centres forment les li-
mites entre les deux sens possibles de la reconstruction des dislocations par-
tielles, qui possèdent les caractéristiques générales des "solitons". Nous
avons trouvé que ces solitons engendrent des niveaux dans la bande interdite
qui pourraient expliquer les résultats de RPE et d'effet Hall. Nous proposons
un mécanisme de déplacement des dislocations, par l'intermédiaire des solitons
qui donne les mêmes résultats que ceux mentionnés dans la théorie de Hirsch,
et en plus, explique la différence de mobilité des partielles d'entrée et de
fuite. Les résultats théoriques et expérimentaux vont dans le sens de ce méca-
nisme et de l'existence des solitons dont il dépend.
Abstract.- Reconstruction of partial dislocations leads to filled bands
separated by a reasonably large gap from empty ones. This leads to a problem
as to the identification of centres responsible for e.s.r. and Hall effect
behaviour. We investigated the suggestion that these centres were boundaries
between the two possible senses of reconstruction of partial dislocations,
which have the general properties of solitons. We found these solitons gave
rise to states in the gap which could account for both e.s.r. and Hall effect
results. A mechanism for dislocation motion mediated by solitons is
proposed, which leads to the same results as the theory of Hirsch and, in
addition, explains the observed difference in mobility between leading and
trailing partials. Evidence from theory and experiment is presented in
support of both this mechanism and the existence of solitons on which it
1. Introduction.- In spite of much experimental effort over the last three decades
the atomic structure of electrically active centres in deformed semiconductors
has remained hidden. The central enigma arises in e.p.r. experiments, which
indicate that the paramagnetic signal associated with low temperature «0.6T m )
deformed silicon disappears upon annealing at higher temperatures or in samples
deformed at higher temperatures /1,2/. This paper aims to resolve this riddle
by interpreting dislocation behaviour in terms of point defects (solitons and their
vacancy complexes) associated with strongly reconstructed partial dislocations.
Solitons also offer a novel and credible explanation for the mechanism of
dislocation glide and its doping dependence, leading naturally to a difference in
mobility between leading and trailing partials of the same screw dislocation /3/.
2. Electronic Behaviour of Dislocations. - Dislocations in silicon have been shown
to be largely dissociated /4,5/ and in the glide set /6,7/. Most dislocations in
annealed deformed silicon are therefore combinations of 90° and 30° glide partials,
which have two suggested forms: "unreconstructed" and "reconstructed" /8/. The
former structure would give rise to a broad (>1 eV wide /9/) half-filled band which
contrasts with the experimental width (ca. 0.25 eV /10/) and leads to problems
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982107
C1-46 JOURNAL DE PHYSIQUE
with the size of the doping dependence of velocity 1111. 3n the other hand, recon-
structed dislocations, in which dangling bonds overlap sufficiently to forn bonds,
appear to be electrically inactive. However, fluctuations /12/ in the Peierls
distortion that causes reconstruction, inevitably lead to "antiphase defects"
/13/ or "solitons" in the reconstruction (figure 1) that do indeed have deep
Figure 1. Glide plane diagrams of solitons on the
0 (right) partials.
go0 (left) and 3'
Figure I . Plan de glissement contenant des s o l i t o n s
sur l e s partieZZes a' 900 (gauche) e t a'
Since the division of experimental results falls naturally between samples (pre-)
deformed at high and low temperatures, this discussion is now similarly split.
2.1. High Temperature Deformed Material. - Electrical measurements /15,16,17/
reveal a half-filled band near Ev+0.4 eV,and an acceptor level at Ec-0.35 eV in
silicon. Calculations 1111 show that solitons give rise to a half-filled level
below mid-gap and that an acceptor state high in the gap arises when considering
"wide" solitons (in which the reconstruction changes only slowly from one sense tc
the other). The actual width of the soliton (and hence the position of this
acceptor level) is difficult to determine, as is the absolute position of the
half-filled level, which is very sensitive to the local atomic environment. This
strong coupling of the amphoteric level to the lattice can give rise to negative
effective-U behaviour, in which lattice distortions overcome the Coulombic
repulsion, U, between electrons. This behaviour was suggsted for dangling bonds
in amorphous materials by Anderson 1181 and for the silicon vacancy by Baraff and
co-workers 1191. Experimentally this behaviour has been proven for interstitial
boron 1201 and for the vacancy in silicon 1211. We calculate that the soliton
is more strongly coupled to the lattice than the vacancy /Ill, so negative-U
is more likely than for the vacancy and neutral solitons will be unstable with
respect to charged ones for all values of the chemical potentia1,r. This
suggestion could be tested by the photo-DLTS and photo-e.s.r. experiments of
Harris et al. 1201. It also appears that solitons are frozen-in from the high
temperature deformation, because the splitting of dislocation bands is so small
Thus it is possible to explain the properties of high-temperature-deformed
material in terms of a frozen-in concentration of negative-U solitons.
2.2 Low Temperature Deformed Material.- Contrarily, the behaviour of material
deformed at low temperatures appears to be that of positive-U centres 11,221,
although some Hall effect experiments report a small acceptor band adjacent to
the low donor band 1221. (We would attribute this to a small concentration -
1 or 2% - of solitons.) However, the principal result of the experiments of
Grazhulis and of Ossip'yan and their co-workers /22,23/ is that, approximately,
there is a donor band at Ev+0.4 eV,and an acceptor band between Ec-0.6 eV and
Ec-0.4 eV . It has been suggested Ill/ that these bands can be attributed to
vacancy-soliton complexes (figure 2. Although one-electron calculations /11/
indicate that vacancy-soliton complex levels are as strongly coupled to the
lattice as soliton levels, strong electron-electron effects due to interaction of
the dangling bond electrons on B with themselves and those in the bond CD
probably reduce the coupling and make positive-U behaviour likely. In addition,
there are indications that a considerable concentration of vacancies is produced
during low temperature deformation 110,241, leading to a high probability of
occurrence for these complexes. Annealing of the vacancy-soliton complexes at
high temperatures 12,171 would leave solitons on the dislocation lines (e.s.r.-
inactive recombination centres) after emission of vacancies.
Figure 2 Nearly axial perspec-
tive views of go0 and
3 partials and con-
taining a vacancy-
Figure 2 Perspectives presque
&aZes des partieZZes
Q 90' e t a' 30° chacme
possedmLt uv compZexe
de Z a m e e t e .
Consequently the existence of a non-equilibrium concentration of vacancy-soliton
complexes offers a possible explanation of the behaviour of low temperature
3. Mobility of Dislocations. - . The theory of dislocation velocity in the high
stress regime, where double kink nucleation is rate-limiting, is well established
1251. An "advancing" double kink is nucleated and expands under the action of
stress and thermal excitation until it reaches a critical width (determined by the
acting stress, temperature and kink-kink interaction), whereupon it is
irreversibly ripped apart by the stress (the kinks quickly traversing the distance
X, their mean free path, and then annihilating).
The formula that Hirth and Lothe /25/ derive for dislocation velocity, Vdis under
these conditions is simplest for a short dislocation segment of length L (<< X,
i.e. Vdis= bLJ, where J is the number of double kinks nucleated per unit length
of dislocation per unit time ("nucleation rate"). Approximating the kink height
and step distance with by the dislocation burgers vector, and writing the Debye
frequency as v and applied stass asu, then J can be expressed as :
J= ( ~ 6 b l k ~exp - (Fdk + W) / k ~
Fdkis the formation energy of a double kink (later we shall approximate this with
double that of the single kink, i.e. 2Fk), W is the kink migration energy and
Fdk + W = Fsp is the energy of the double kink nucleation saddle point structure.
However, in the case of longer segments (L>> X) the kink mean free path,
X = 2bexp(Fk/kT), enters the velocity formula (Vdis = bXJ) and leads, importantly,
to a modification of the exponent in Vdis i.e. Vdis = (2vab3/k~)exp-(Fk+W)/kT.
Note that now the length dependence (Vdis0cL) for short segments (which has been
observed in the electron microscope /26,27/) has disappeared and the exponent is
reduced. It is these observations that have allowed Hirssh 1261 and Louchet1271
independently to estimate Fkz0.4 eV , W=1.2 eV and X=4000A at 6000 C. This low
formation energy and the high kink migration energy would be expected from a
model where dangling bonds are eliminated from kinks by reconstruction.
The most important aspect of dislocation motion in silicon is its doping
dependence, which has been measured by George and co-workers 1281 and by others
/29,30 311. Hirsch's theory 1321 for this effect assumes the presence of deep
levels (Ed and Ea for donor and acceptor, respectively) in the kinks give rise
to three different contributions to the total dislocation velocity (v*, vo, v-
being due, respectively, to ~ositive,neutral and negative kinks). Although
chemical equilibrium is achieved between the stable kinks and the electron gas,
each differently charged kink will go through a different saddle point. Hirsch
allows for this by writing in, but subsequently neglecting, changes in migration
W, between these charge states. However, the effect of these changes
is such that, mathematically, it appears as i f chemical equilibrium were
achieved at the saddle point. Thus Hirsch's equations for v+, vO, v- can be
written in terms of the saddle point levels Ed, Ea, as below, provided that
chemical equilibrium can be attained at some stage.
JOURNAL DE PHYSIQUE
v+/vO = exp (Ed -r+eV ) /kT and v-/vo = exp ( p - Ea-eV )/kT
It is readily seen that the changes in velocity with arise from the energy
saving in the saddle point structures due apparently to carrier trapping by the
levels Ed and Ea. SchrSter 1131 has fitted this theory to the results of
George et al. and obtained Ed=Ev+0.67 eV , neglecting A W m and eV (the
correction due to charging of the dislocation line). A better fit/ll/ can be
achieved by including, in a simple way, eV and using a more appropriate
variation of the band gap with temperature. This leads to levels Ed=Ev+0.34 eV
and Ea=Ev+0.55 eV .
The two "atomistic" models for dislocation motion presented by Hirsch 1321 based
on kinks with dangling bonds("Hirsch kinks") and by Jones 1331 invoking
reconstructed kinks each have weaknesses. Objections to the former are
(a) the migration energy changes, AWm, with charging cannot be neglected and
(b) kinks associated with dangling bonds would have higher formation energies
than the reported 0.4 eV .
The reconstructed kink model encounters the
problems ( ) that equilibration of the electron gas with the saddle point levels
is unlikely and (b) that the saddle point levels always appear above mid-gap
according to theoretical calculations Ill/, contrary to experiment.
The soliton theory of dislocation motion 1111 satisfies each of these objections.
The following four points contain the essence of the theory:
(a) Double kinks are preferentially (practically solely) nucleated at
dislocation sites occupied by solitons.
(b) Solitons are either not bound to or only weakly bound to kinks, which have
a higher migration energy than solitons.
( ) Solitons are in chemical equilibrium with the electron gas.
(d) Single kinks can only move when there is a soliton coincident with them,
i.e. only kink-soliton complexes are mobile (c.f. Hirsch kinks).
Figure 3. Double kink nucleation on a go0 partial.
Figure 3. ~ u c Z $ a t i o nd'un double d$crochemen.t s t i r c r r t i e Z L e a\ 90'
The proposed mechanism for the initial stages of double kink nucleation is
shown in figures 3 and 4 for the 90° and 300 partial, respectively. "Attack" of
atom C by the dangling bond on atom A leads to double kink nucleation, whereas
attack of atom B by atom A leads to soliton migration along the partial. (Note
that the axes of the ~artialsin these glide plane diagrams are denoted by heavy
lines for the reconstruction bonds in the 90° partial and shading of the
pentagonal cells typical of the 30° partial.) Subsequent steps illustrate
the soliton migrating away, leaving a recognisable double kink. The fact that
Figure 4. Double kink nucleation on a 30° ~artial.
Figure 4 . Nucle'ation d'un double d e ' c r o c h e r e n t s u r p a r t i ~ l l e 2 30'
atom B is closer to A than is atom C leads to the proposition that soliton
motion is easier than kink migration and that solitons are not well bound to
Figure 5 follows the energy of a dislocation site as it undergoes double kink
nucleation. The abscissa is the "reaction coordinate", which maps the
complicated atomic movements involved in double kink nucleation and expansion onto
one variable, RC. Special intermediates in the process are shown diagrammatically
above the plot, drawing the reconstructed dislocation in the glide plane as a
line and the soliton as a *. The first step is the appearance of a soliton (by
migration from another site), with a probability of occurrence per unit length
of dislocation PS, which is proportional to the linear concentration of solitons.
Subsequent steps involve double kink nucleation at the soliton (with
probability per unit time, Js C e p - '
%x(w /k~), followed by emission of the soliton
leaving a reconstructed double kink. %us we can formulate an expression for J,
the nucleation rate: J = PsJscC exp (-Fs/kT) exp (-~s/~/k~)exp - (Fdk+Wdk)/kT.
Figure 5. Plot of a minimum energy contour for
double kink nucleation with (above)
. .. accompanying schematic of the process.
: - - A.--- Figure 5. Graphique 0% contour minimwn dl&e'rgie
pour nuclZation double dzcrochemen t
avec (ci-dessusl s c h 6 ~ a t i ~ u e
-& ------- 3--
The dotted line of figure 5 shows the higher energy process of double kink
nucleation in the absence of solitons, and it is plain to see that the r8le of
the soliton has been to reduce the migration energy, Wdk, because the saddle point
energy is lowered. It is this structure, the double-kink-soliton saddle point,
that gives rise to the levels Ed and E effective in doping experiments, because
J W ~ X P - ( F ~ + W/ k ~ ~ h e r e
FS+W,6=Fsp, the formation energy of the double-kink-
soliton saddle point.
Theoretical tight-binding one-electron calculations /11/ of the deep states of
these saddle points on the 90° and 30° partials (the structures of figures 3b and
4b) give a half-filled level between Ev+0.24 eV and Ev+0.30 eV , which is in
good agreement with the donor levels Ev+0.28eV or Ev+0.34 eV obtained in fitting
experimental results (see earlier). The position of the acceptor level depends
on the Coulombic repulsion, U, between two electons in the deep state, which would
have to be 0.39 eV to agree with Schroter's fitting or 0.21 eV to agree with ours
(neglecting differences in structure between charge states). The latter value is
commensurate with the U calculated by Baraff et al. 1191 for the vacancy, i.e.
0.25 eV. Furthermore, our calculations also indicate that the sensitivity of the
saddle point level to movements of atom A (in the direction Q marked in figure 1 )
is five times smaller than that of the soliton. We attribute reduced sensitivity
to the extra coupling of the dangling bond with orbitals on atom C and conclude
that positive effective-U would be most probable in this case, since the Coulomb
effect would outweigh that of the lattice distortion.
Although we have couChedour explanation and calculations in terms of double kinks
of the smallest width, extension of both to double kinks of critical width, which
would be more appropriate, is not in principle difficult. Theoretical
calculations on isolated kinks /11/ reveal little difference from double kinks.
Finally, there is experimental evidence /3/ due to Wessel and Alexander that the
leading and+trailingpartials of a dissociated dislocation have different
mobilities. Comparing measurements of the width of unusually wide dissociated
6 and screw dislocations, they conclude that leading and trailing 90° partials
have similar mobilities, but leading and trailing 30° ~artialsbehave as though
there were approximately a 0.1 evdifference in p-type material (leading partials
being faster) and none in n-type material. This effect can be explained by a
difference in deep levels between saddle points on leading and on trailing 3 0'
partials (i.e. Es t-ESpl) of 0.05 e V . We made theoretical calculations on
saddle points on !eadlng partials (shown in figures 3 and 4) and on trailing
C 1-50 JOURNAL DE PHYSIQUE
parttals, where the normal stacking region (marked with an N and the stacking
fault (marked with an S) were interchanged. We found Es t-Espl = - 0.06 e V for
the 30° partial and 0 0 e V for the 90° partial, which age in good agreement
with experiment but for the sign in the case of the 3 partial. Regarding this
error as a consequence of an inaccurate calculation, it seems likely that this
is a credible explanation of the leading/trailing mobility difference.
4. Conclusions. - The behaviour of high temperature deformed material is
dominated by negative-U recombination centres, which are defects in the
reconstruction of partial dislocations (i.e. solitons). The generation of
vacancies during low-temperature deformation gives rise to vacancy-soliton
complexes, which are positive-U centres. Dislocation glide is "catalysed" by
solitons, which give the correct doping dependence and predict in a qualitative
way the difference in mobility between leading and trailing partials. The rGle
of vacancy-solitons in dislocation motion is uncertain, but in velocity experiments
where they might be present they could behave as pinning points (in fact pinning
due to jogs, which are related to vacancy-solitoncomplexes,has been observed
. - Malcolm Heggie acknowledges the financial support of the
S.E.R.C. and we thank H. Alexander, A. Claesson, P.B. Hirsch, F. Louchet,
S. Marklund, Yu. Ossipl~an A. Ourmazd for useful discussions.
SCHR~~TER SCHEIBE E., SCHON H., J. Micros. 1 8 (1980) 23
/2/ GRAZHULIS V., OSSIP'YAN YU., Soviet Phys.: JETP 2 (1971) 623
+ WEBER E., ALEXANDER H., J. de Physique4Q (1979) C6-101
/3/ WESSEL K., ALEXANDER H., Phil Mag. (1977) 1523
/4/ COCKAYNE D., HONS A., J. de Physique a (1979) C6-11
/5/ GOMEZ A . , COCKAYNE D., HIRSCH P., VITEK V., Phil. Mag.& (1975) 105
/6/ OLSEN A., SPENCE J., Phil Mas. 43 (1981) 945
ANSTTS G., HIRSCH P., HUMPHREYS C., HUTCHISON J , OURMAD A,,
Inst. Phys. Conf. Ser.60 (1981) 15
/8/ MARKLUND S., Phys. Stat. Sol. (b) 92 (1979) 83
/9/ NORTHRUP J., COHEN M., CHELIKOWSKY J., OLSEN A., SPENCE J.,
Phys. Rev. B 2 (1981) 4623
/lo/ KIMERLING L., PATEL J., BENTON J., FREELAND P.,
Inst. Phys. Conf. Ser. 2 (1980) 401
/11/ HEGGIE M., P p Thesis (1982) University of Exeter
/12/ RICE M., STRASSLER S., Sol. Stat. C o m s . u (1973) 125
1131 HIRSCH P., J. M i c r o s . 3 (1980) 3
/14/ JONES R., Inst. Phys. Conf. Ser. a (1981) 45
/15/ SCHROTER W., J. de Physique (1979) C6-51
1161 MANTOVANI S., MAZZEGA E., J. de Physique 40 (1979) C6-63
1171 KIMERLING L., PATEL J., J. de Physique a(l979) C6-67
1181 ANDERSON P.; Phys. Rev. Lett. 34 (1975) 953
/19/ BARAFF G., KANE E., SCHLUTER M., Phys. Rev. Ba(l980) 3563, 5662
1201 HARRIS R., NEWTON J., WATKINS G., Phys. Rev. Lett.s(l980) 1271
/21/ WATKINS G., CHATTERJEE A., HARRIS R., Inst. Phys. Conf. Ser. 2 (1980)199
1221 GRAZHULIS V., KVEDER W., MUKHINA V., Phys. Stat. Sol. (a) 42 (1977) 407
12.31 OSSIP'YAN YU., Cryst. Res. Tech. 16 (1981) 239
1241 EREMENKOV V., NIKITENKO V., Y A K I M ~ E., Pis'ma Eksp. Teor. Fiz.26 (1977) 72
1251 HIRTH J.,LoTHEJ., "Theory of Dislocations'' McGraw Hill (1968) 497
1261 HIRSCH P., OURMAZD A., PIROUZ P., Inst. Phys. Conf. Ser. 60 (1981) 29
/27/ LOUCHET F., Inst. Phys. Conf. Ser. (1981) 35
1281 GEORGE A . , ESCARAVAGE C., CHAMPIER G., SCHROTER W.,
Phys. Stat. Sol. (a) (1972) 483
/29/ EROFEEV V., NIKITENKO V., OSVENSKII V., Phys. Stat. Sol. (a) 3 (1969) 79
/30/ KULKARNI S., WILLIAMS W., J. Appl. Phys. 41 (1976) 4318
1311 PATEL J., FRISCH H., Appl. Phys. Letters U (1975) 118
1321 HIRSCH P., J. de Physique 40 (1979) C6-117
/33/ JONES R , Phil. Mag. B 2 (1980) 213