Role of Screw Dislocations in Electrolytic Crystal Growth BY VESSELINBOSTANOV BUDEVSKI STAIKOV EVGENI AND GEORGI Central Laboratory of Electrochemical Power Sources Bulgarian Academy of Sciences Sofia 1000 Bulgaria Received 10th August 1977 The theory of form and interstep distance of spirals and the growth rate of crystals under spiral growth conditions are discussed and illustrated for the electrodeposition of silver. The overvoltage dependence of the slopes of pyramids can be used for the evaluation of the rate of propagation and the specific edge energy of the spiral steps the latter being also accessible from experimental current against overvoltage curves. The coincidence of these values with values obtained from other experi- ments (two-dimensional nucleation) is a good proof of the validity of the theory of spiral growth.The mechanism of growth of singular faces intersected by screw-dislocations is completely different from that of perfect faces where the growth mechanism is con- trolled by two-dimensional nucleation. Because of the structural defect a ledge eman- ates from the point where the dislocation line intersects the surface of the crystal face. The ledge being constrained to terminate at the dislocation emergence point will wind up into a spiral as it advances during growth and will never disappear from the face. This step ensures a sufficient number of growth sites so that growth can proceed at low overpotentials making the two-dimensional nucleation mechanism an unnecessary process. The theory of spiral growth has been developed by Frank' and Burton et aL2 The basic equation for calculation of the form of the spiral in the isotropic case is the dependence of the rate of propagation of a ledge on its radius of curvature assuming that for a region of the step p =pc (where pc = &/qmOnqc is the radius of the two-dimensional nucleus at the given overvoltage rc,equivalent to the supersaturation where E J cm-l is the specific periphery energy of the nucleus and qmonC is the amount of electricity needed for the deposition of one monoatomic layer on the crystal face) the propagation rate is zero and increases to vo3as the step becomes straight (P-+ 4.Burton Cabrera and Frank found two approximations of the solution r(8) de-scribing the form of a stationary rotating circular spiral.The first approximation yields an Archimedian spiral with a radius of curvature at the centre p =pc. The ledge spacing d is in this case uniform for the whole spiral d = 4zpC= 12.6~~. This result corresponds to a constant radial propagation rate except for ledges for which p. <pc. A second approximation is derived by the same authors combining the solu- tion for small Y values where the relation (1) is taken into account with the former solution valid for large r values. This solution yields 19.9 for the factor connecting d and pc which value has been later corrected to 19.0 by Cabrera and Le~ine.~ The case of polygonized spirals has been discussed by Cabre~a,~ by Kaischew el aZ.5 84 ROLE OF SCREW DISLOCATIONS IN ELECTROLYTIC CRYSTAL GROWTH and by Chapon and Bonissent.'j In all these treatments the ledge advancement rate is considered as constant equal to that of an infinitely long step except for steps shorter than the side length I of the two-dimensional nucleus i.e.v = urn for I I and ZI = 0 for I < I,. The step distance is calculated from the period Tof rotation (at steady state) and vco:d = Tv,. For a quadrangular spiral the period T is 4 times the time lapse z needed for a new step to reach the length I,. For a constant propagation rate u = urn r = Ic/vco and d = 41,; the ledge spacings are independent of the distance from the centre in this case. For a k-cornered spiral the ledge spacing is given by d = 4kp sin2(;n/k). The assumpton u = urn for I 2I and v = 0 for I < Ic is obviously incorrect.A modification of the BCF-equation v = v,(l-l$) (2) can be used here.7 It is obvious that the time lapse z and hence the period of rotation become larger because the step adjacent to the newly growing step propagates with less speed than urn. The calculation is complicated because the propagation speeds of all the adjacent steps are also I-dependent and hence the propagation rate of the spiral is given by a system of differential equations. A numerical calculation of this system was given by Budevski Staikov and Bo~tanov,~ who found that irrespective of the form of the spiral the period of rotation of a k-cornered spiral is T = 19pC/v (3) and hence d = 19pc= 19~/~,,,,,,,~,.(4) The situation becomes more complicated in the casz of two dislocations located at a distance comparable with I,. If the two dislocations have the same sign and their distance is close to I they are acting as one dislocation with a double step height the slope of the produced pyramid is increased. If L >Z, d remains unchanged as does the slope.z When the two dislocations have opposite signs (fig. 1) they produce only one step both ends of which terminate at the two emergence points of the dislocations. If the distance of the two dislocations is smaller than I this step becomes inactive. For larger distances the dislocation pair produce loops of steps. The period of pro- duction of loops was calculated by Nanev s using approximation methods. The results of the exact numerical solution calculated for the two cases represented in fig.1are given in fig. 2. It is seen that for low supersaturations where Zc approaches the value of the dislocation distance L the step distance becomes infinitely high (be- cause of the very low advancement rate in the initial stage) while with increasing super- saturation i. e. decreasing I,-value the period of loop production decreases rapidly bringing the d/I value below the 9.5 margin (note that I = 2p in this case of quad- rangular symmetry). With higher supersaturations the d/Z,-value increases to 9.5 because two non-interacting spirals are formed. The growth pattern becomes even more complicated if more than two dislocations are interacting and especially if the component of the Burgers vector normal to the face is higher than the height of a monoatomic step.From these considerations it follows that a uniform step density for all spirals of growth on a crystal face can only be expected when simple non-interacting spirals are produced. VESSELIN BOSTANOV EVGENI BUDEVSKI AND GEORGI STAIKOV phase 4 phase 3 phase 2 1 -phase ' 1 phase 0 (initiai) phase & = phase 4 (final) \J \I \ I I phase 4 1 I phaise 1 phase 0 .(initiat)'= b 'I FIG.1 .-Interaction of pairs of screw-dislocationswith opposite sign. (a)Dislocations situatedalong a line parallel to the direction of closest packing (Le. normal to that of lowest propagation rate); distance L = 21,. (b) Dislocationsalong a line at an angle of 45" to the directionof closest packing; L = I,.86 ROLE OF SCREW DISLOCATIONS IN ELECTROLYTIC CRYSTAL GROWTH 1 2 3 4 5 L /I FIG.2.-Relative step distance d/I,of the pyramid of growth of conjugated pairs of screw-dislocations as function of dislocation distance L/lc. 0-dislocations situated as shown in fig. l(a). O-dis-locations situated as shown in fig. l(b). (The transition from phase 3 to phase 4-the final phase. in the case fig. l(b) is assumed to proceed with a velocity of the concave angle steps equal to urn). GROWTH MORPHOLOGY OF IMPERFECT FACES On a face intersected by screw-dislocations a pyramidal growth pattern is observed. On a cubic face the pyramids are rectangular (fig. 3) while on an octahedral face they are triangular (e.g.fig 6). The pyramids of growth are obviously the macroscopic picture of the spirals of growth. There is much evidence for this statement (i) the slope of the pyramids depends on supersaturation. Changing the supersaturation the slope is also changed. (ii) The pyramids appear always on the same site of the face. At anodic dissolution an etch pit is produced at these sites. The regular pyramidal pattern of growth is observed on faces with a high degree of perfection intersected by only few dislocations. In some cases pyramids with differ- ent slopes are observed (fig. 4). The ratio of the slopes can be easily calculated from some geometrical parameters e.g.,the angle between the valley line of the two pyramids and the edge of the pyramid.A slope ratio of 1.26 was calculated in the case of fig. 4(u). For the three pyramids of fig. 4(b) all of them having different slopes ratios of 1:1.5 :2.25 have been found. In fig. 5 a variety of pyramids with different slopes are observed with typical slope ratios ranging between 1.3 to 1.5. The picture sequence in fig. 6 shows the develop- ment of a pyramid with higher slope than the basic one fig. 6 (u)-(d). The slope ratio here is -2. In later stages (e)-(A) the sequence shows a sudden flattening of the top of one of the pyramids and the appearance of a series of new pyramidal tops which have been overgrown by the basic pyramid and have become active as this pyramid dis- appears. These are few selected examples of the growth morphology in simple cases.The FIG.3-Pyramids of growth on a cubic silver single crystal face. a b FIG.4.-Pyramids of growth with different slopes. (a) Slope ratio of the two pyramids is 1 :1.26. (b)The slope ratio of the three pyramids is tan al:tan uz = 1.5 and tan q:tan a3 = 2.25. A pulsat-ing current of growth is used in case (b)to demonstrate direction and velocity of propagation of the spiral steps. FIG.5.-Pyramids of growth on cubic face. Slopes of the marked pyramids are tan u1:tan uz = 1.5 tan all:tan a4 = 1.3 and tan a5:tan a(;= 1.9. [Toface page 86 FIG.6.The development of the growth process of pyramids on an octahedral single crystal face. Slope ratio of the two upper pyramids in (d) 1 :2. A flattening of the top of the lower left pyramid is observed in (e) resulting in the appearance of a series of new pyramids.[Toface page 87 FIG.7.-Separated pyramids of growth obtained by injecting a certain amount of electricity on a previously levelled face. [Toface page 87 VESSELIN BOSTANOV EVGENI BUDEVSKI AND GEORGI STAIKOV 87 morphological pattern becomes more complicated and impossible to interpret in the general case. SLOPES OF PYRAMIDS OF GROWTH The slope of a pyramid of growth determined by the step height h and the step distance d tan C( = uh,/d = uq,,,h,q,/l9e (5) is proportional to the overvoltage qc. h is the height of a monoatomic layer and u the number of layers forming the ledge. This slope against overvoltage dependence can be easily investigated using the technique developed by Nanev Budevski and Kais~hew.~ On previously levelled face intersected by a small number of screw- dislocations a short current pulse is applied.As separated regular pyramids are formed (fig. 7) the amount of electricity injected can be assumed to be evenly dis- tributed among them. Thus the volumes of these pyramids are known their bases can be directly measured so that the slopes are easily calculated. Fig. 8 represents data obtained by this technique. The straight line is in agree- u) c g d -6 C (n 5 c -v 10% 01 -G. E u2 e 5' x u) -15 0 -0 I -2oz d' -302 3 ;L I I I 1 20 40 60 overvoltage / 10-3 v FIG.&-Slope of the pyramids (tan a) as function of overvoltage.The slope is given in step height units in the right ordinate. ment with eqn (5). The value of E calculated from these data assuming u = 1 is E = 0.98 x J cm-I in a fairly good agreement with previous results (E = 2 x from measurements on dislocation-free face^).^ RATE OF PROPAGATION OF SPIRAL LEDGES The propagation velocity of the spiral ledges is an important kinetic parameter. Being monoatomic the single steps of the pyramids are invisible. The following technique has been used10 to get information for this parameter it has been demon- strated that the slope of the pyramids of growth depends on overvoltage. If therefore 88 ROLE OF SCREW DISLOCATIONS IN ELECTROLYTIC CRYSTAL GROWTH during growth at constant overvoltage the overvoltage is changed for a short period a condensation of the spiral ledges at the centre results which begins to propagate with a constant speed from the top of the pyramid downwards as a stripe with a different slope (Le.with a different shade as observed by phase-contrast microscopy). A stripe produced by this technique is observed on the octahedral face represented in fig. 9. A linear relation between velocity and overpotential has been ObservedlO giving for the rate constant K = 0.92 cm s-l V-l in good agreement with values obtained from propagation rates of monoatomic layers.11 A linear relation between step velocity and overvoltage indicates a linear relation between current and step length. i = icLy (6) where K is connected to K by K = Kvqmon (7) The linear current-step length relation (6) has been experimentally verified in other experiments.ll CURRENT AGAINST OVERVOLTAGE RELATION At steady-state growth conditions the spirals of growth have uniform ledge spacings producing a uniformly stepped surfhce with a step density L = l/d or from eqn (4) Ls = 4mon TC/l9&* (8) For small yc values the current density is proportional to rc.With eqn (6) this equation yields i = ~q~~~~~~/19~ (9) = By:. 1 I I I 1 0.05 0.10 015 0.20 j-1/2/ (A cm-2) 1/2 FIG.10.-Steady-state current-overvoltage relation in a AEi-lj2 against Pplot for separation of the concentration polarization and the ohmic potential drop from the crystallization overpotential according to eqn (10).FIG,9.-Stripes produced by a short increase in the overpotential on a growing octahedral face. The photograph was taken about one second after the pulse application. [Toface page 88 VESSELIN BOSTANOV EVGENI BUDEVSKI AND GEORGI STAIKOV 89 To verify the theoretically expected square of overvoltage dependence of the cur- rent at steady-state conditions we have to separate the ohmic drop and the concentra- tion polarization terms from the overall voltage change AE of the cell. For this pur- pose we may assume that for the low potential changes observed in the discussed experiments both these terms depend linearly on current density while the part of the polarization which drives the ions across the double layer to the crystal lattice depends on the square root of the current [see eqn (9)].AE = i(Rc + RQ)+ (i/B)'I2 A straight line is obtained by plotting AEi-ll2 as function of the square root of i as seen in fig. 10. The ordinate intercept gives the value of the constant B in the i against qc2 relation. From this constant and knowledge of K the value of the specific edge energy E can be estimated IC can be taken either from monoatomic layer1' or more correctly from the spiral ledge propagation rate measurements which have been de- scribed.I0 The value of E is 2.4 x J crn-l in good agreement with the value of E from other experiment^.^ The experiments described are not only in accord with some basic deductions of the theory of spiral growth but represent a very good quantitative verification of the Burton Cabrera and Frank's theory for the case of electrocrystallization.F. C. Frank Disc. Faraday Soc. 1949 5,48,67. W. K. Burton N. Cabrera and F. C. Frank Phil. Trans. A 1951,243,299. N. Cabrera and M. M. Levine Phil. Mag. 1956,1,450. N. Cabrera Structure and Properties of Solid Surfaces (Chicago Univ. Press Chicago 1953) p. 295. R. Kaischew E. Budevski and J. Malinovski 2.phys. Chem. 1955,204,348. C. Chapon and A. Bonissent J. Cryst. Growth 1973,18 103. 'E. Budevski G. Staikov and V. Bostanov J. Cryst. Growth 1975,29 316. * C. N. Nanev KristaN und Technik 1977 12 587. R. Kaischew and E. Budevski Contemporary Phys. 1967 8,489. lo V. Bostanov R. Roussinova and E. Budevski Izuest. Otd. Khim. Nauk. Bulg. Acad. Nauk. 1969,2 885. '* V. Bostanov R Roussinova and E. Budevski Chem.-Ingr. Tech. 1973 45 179.