Structural and chemical inhomogeneities in germanium single crystals grown under conditions of constitutional supercooling

Journal of Crystal Growth 49 (1980) 612—630 © North-Holland Publishing Company

STRUCTURAL AND CHEMICAL INHOMOGENEITIES IN GERMANIUM SINGLE CRYSTALS GROWN UNDER CONDITIONS OF CONSTITUTIONAL SUPERCOOLING W. BARDSLEY and D.T.J. HURLE RSRE, St. Andrews Road, Great Malvern, Worcs. WRI4 3P5, UK

M. HART Physics Department, King’s College, Strand, London WC2R 2LS, UK

and A.R. LANG HI-I. Wills Physics Laboratory, University of Bristol, Bristol BS8 1 TL, UK Received 26 November 1979; manuscript received in final form 14 January 1980

Solute distributions and dislocation configurations in single crystal of germanium grown with a cellular interface consequent upon the existence of a zone of constitutional supercooling are described. Several solutes which dope the crystal substitutionally have been studied. Dislocations were delineated by chemical etching and by X-ray diffraction topography, whilst chemical inhomogeneities were revealed by X-ray absorption topography, autoradiography and chemical etching. It is shown that the pattern of microsegregation depends strongly on both the interface morphology (which is itself dependent on the crystal orientation) and on the freezing range of the solute-solvent system. The occvrrence of very severe microsegregation is due predominantly to the entrapment of liquid droplets in the cell boundary grooves and their subsequent migration in the imposed temperature gradient. The massive increase in dislocation density which occurs after the formation of a cellular interface is shown to be due to stresses introduced during the crystallisation following migration of the trapped liquid droplets.

1. Introduction

temperature gradient. The size and distribution of the cells were shown to be governed by the fluid flow pattern and the gradient of constitutional supercooling. The cell boundaries have relatively high concentrations of solute for solutes which have a distribution coefficient less than unity [4]. If the resultant change in the lattice parameter in the cell boundaries is sufficiently large, dislocation arrays should form to relieve the stress [5]. In this paper, we report on an examination of the solute distribution in cellular germanium using autoradiographic and X-ray absorption techniques and of the dislocation configurations using etching and X-ray diffraction topographic techniques. Although performed nearly two decades ago, this work was never submitted for publication. We now perceive that the originality of the findings stands undimmed by any subsequent duplicating study similarly using such diversity of techniques.

Two of the present authors have previously reported observations on the morphology of the germanium cellular structure obtained by growing single crystals from gallium-doped melts under conditions of constitutional supercooling [1—3]. The information was obtained principally from observation of growth striae and other features on etched surfaces [21. To summarize, it was found that the cellular structure, for all crystal orientations, was composed of an array of small facets belonging to the form {l 1 1}. The cell boundaries coincided with the re-entrant corners between {1 1 1} planes. As growth proceeded, the cell boundaries deepened, and it appeared that liquid was periodically trapped in these boundaries. The trapped liquid subsequently migrated down the crystal under the influence of the applied 612

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2. Experimental techniques The germanium single crystals, doped with gallium, indium, antimony, tin, silicon or arsenic were grown by the Czochralski technique using seed rotation rates of 70 rpm and pulling speeds in the range 7—45 pm s~.The doping concentrations were chosen such that, with the growth conditions used, a cellular structure occurred when the required fraction of melt had solidified (usually one half) [1]. Dislocation etch pits were revealed by etching {l 1 1} surfaces cut from the crystals. As described in ref. [2], three types of surface were used. These are illustrated in fig. 1, which is explained in section 3 below. The etchant used was the same as that used to delineate the cellular morphology, namely HF+HNO3 in proportion 3 at 20°C[21. A few experiments were performed in which crystals were doped with radio-gallium (half life approximately 14 h) and autoradiographs were taken by placing cut sections in contact with X-ray film. The X-ray diffraction topographs were taken with X-rays transmitted through the specimen. For this purpose, specimen slices were polished to sub-millimetre thickness and then etched in a carefully controlled fashion; for it was necessary to etch sufficiently thoroughly to remove all surface damage, but not so deeply as to cause significant variations of diffracted intensity arising from irregularities of specimen thickness and surface slopes. A wide range of diffraction behaviour was encountered, from that rep-

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resenting almost perfect crystal in the regions of low dislocation density, to that more characteristic of a mosaic crystal in the regions of severe microsegregation. Accordingly, contributions to contrast on the topographs arise from both “orientation contrast” and “extinction contrast”. Moreover, with germanium as the specimen, topographs could be taken under conditions of low, moderate and high absorption, by appropriate choice of wavelength and specimen thickness; and the variation of diffraction contrast under these different conditions could be exploited as an aid to characterising the lattice imperfection. (An explanation of the terms “orientation contrast” and “extinction contrast”, and an account of the procedure for taking and interpreting X-ray topographs, will be found in a review [61.)The linear coefficients of absorption, p, for AgKa, MoKo~and CuKa radiations in germanium are 18.2, 34.5 and 40.2 mm’ respectively. Hence to produce topographs with the contrast characteristics of the “low-absorption case”, i.e. pt ~ 1, requires specimen thicknesses, t, not exceeding about 55, 29 and 25 pm, respectively. The specimens used were generally a few multiples of the thickness giving pt = 1. The absorption topographs (i.e. microradiographs) were obtained with but minor modification of the experimental arrangement for taking diffraction topographs. The same fine-grain photographic emulsion (Ilford L4 nuclear emulsion) was used as a detector, with the photographic plate being placed to receive the beam directly transmitted through the specimen. The same ribbon-shaped, collimated beam was used as in the case of diffraction topography, and, in similar fashion to taking a projection topograph, the specimen was traversed back and forth during exposure in order to illuminate the whole area of interest. For radiography, the “diffracted beam slit” (which is usually placed between specimen and plate to pass only the diffracted beam) was placed between specimen and plate so as to pass only the directly transmitted beam, intercepting radiation scattered away from the direct beam. By this means, radio-

graphs with a very low background of scattered radia-

tion were obtained, though of course the procedure was slower than conventional non-scanning radiographic techniques using a broad beam. Specimens prepared for radiography had thicknesses in the range 0.4 to 1.2 mm. When taking radiographs, the speci-

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men was oriented so as not to produce a Bragg reflec-

tion. Instead, the angular adjustments available (goniometer arcs for specimen tilt, and goniometer “w-axis” for glancing-angle-adjustment) were used for lining-up impurity-rich cell walls edge-on to the mcident beam in order to enhance absorption contrast. Also, in the case when impurity droplets were distributed within the crystal, a set of radiographs taken with specimen orientations differing by 5° or 10° steps produced images from which stereo-pairs could be selected. These, viewed under twin microscopes, gave vivid three-dimensional vizualization of the droplet distribution in the crystal.

3. The dislocation arrays We have examined the dislocation arrays formed in the cellular regions of crystals with growth axes parallel to a (110) direction or a (111) direction. (These we refer to as “(110)-orientation” and “(111 )-orientation” crystals, respectively.) The dislocation etching must be performed on a surface parallel to a plane of

Ge single crystals

{l 1 l}. We refer to the planes chosen using the terminology of ref. [2], viz., as the {l1l} vert. surface (which contains the crystal growth axis), the {l 1 l} para. surface (normal to the growth axis) and the {l 1 l} oblique surface. See fig. 1. (The {l 1 l} vert. is available only with (110) crystals, and the {l 1 l} para, only with (111) crystals. The {l 11 } oblique surface makes 35°16’ with the growth surface in the case of(l 10) crystals, and 70°32’in the case of the (111) crystals.) In all of the etch patterns, X-ray diffraction topographs and absorption topographs presented which illustrate “vert.” sections, the direction of crystal growth is from top to bottom on the print. Likewise in “oblique” sections, the top of the image is closer to the seed. form

3.1. (110) oriented crystals The cellular structure on a (110) growth surface consists predominantly of a linear corrugated structure formed by alternation between facets of the two {l 1 l} planes making 35°16’ with (110) [2]. Fig. 2 is a photomicrograph of an etched {l 1 1} oblique sur-

Fig. 2. Etched {iii} oblique surface cut from a (110) crystal just after the commencement of cell formation. Cell boundaries (running left to right in the [1101 direction normal to the growth direction (110]) and dislocation etch pits are delineated. Field width 1.1 mm. ~

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Fig. 3. Etched {i ii } oblique surface of a cellular (110) crystal showiTig ~eU boundaries containing linear arrays of dislocations. Field width 1.8 mm.

Fig. 4. Magnifieci portion of fig. 3 showing a linear array of dislocations situated in the cell boundary mid-way between neighbouiing cell ridges (ridges are arrowed). Field width 0.27 mm.

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face (i.e. a surface parallel to one of the facet planes) cut in the region where regular cell boundaries first formed. The etched lines running left to right are the cell boundaries, and they lie in the [110] direction. The crystal was grown from a gallium-doped melt. The photograph shows that the dislocation density has not markedly increased over that found in the precellular region of the crystal (being typically of order 102 mm’). It further shows that there is no tendency for the dislocations to be associated particularly with the cell boundaries. This suggests that, although the cell boundaries contain an excess of galhum and are therefore selectively etched, the lattice misfit introduced by the excess gallium concentration is not yet sufficient to lead to the generation of extra dislocations (Gallium dilates the germanium lattice by approximately 7 X iO~per atomic per cent [7] )

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Fig. 5. Low-magnification micrograph showing a whole {l 11 } oblique section of a (110) crystal. Dark areas represent massive arrays of dislocations. The major axis of the elliptical section is 38 mm.

As growth proceeds, but still prior to the formation of liquid droplets, dislocations begin to occur in the cell boundaries. Occasionally, simple tilt boundaries have been observed in the cell boundaries; fig. 3 shows an example on a {lll} oblique surface of a gallium-doped crystal. One of the boundaries is shown at higher magnification in fig. 4. The arrows indicate the positions of the ridges of the cells which have been selectively etched. (The cause of the selective etching is discussed in ref. [2].) In order to show up the ridges the surface had to be photographed slightly out of focus with fine pencil illumination. Hence the line of dislocation etch pits, lying in the cell boundary between the peaks, is not well in focus. At later stages of growth, much denser walls of dislocations appear in the cell boundaries A low mag nification picture of the whole cross sectional area of a crystal in this state is shown in fig. 5. It is to be noted that the dislocation arrays are not uniformly dense across the growth surface, implying that the microsegregation is not uniform. This is because the presence of a cellular structure on the growth surface breaks up the previously laminar fluid flow across it [2]. Finally, as the cell boundary grooves deepen, liquid droplet entrapment occurs, and the droplets migrate slowly down the crystal (up the temperature gradient) behind the main interface. In the case of (110> oriented crystals, where the cell boundaries form vertical planes, the droplets are (approximately) confined to the cell boundaries as they move down the crystal. Then, on a {l 1 l} vert. or oblique surface, one observes irregular, looped structures which contain recrystallised material. These are termed solute trails [2]. On {l I l} vert. surfaces, the dislocation arrays associated with the solute trails appear as in fig. 6, which shows the top of an etched gallium-doped crystal taken in dark field. The dislocation arrays at the top of the crystal result from the transient occurrence of constitutional supercooling during the growth of the shoulder of the crystal where progressive reductions in melt heater power were made. (See ref. [2], section 4.) Those at the bottom of the figure are due to the occurrence of constitutional supercooling at the steady imposed growth rate. The two types of dislocation “pipe” are shown at higher magnifica.

.

tion in figs. 7a and 7b. Those in the shoulder taper to a point, whilst those in the stable cellular region (fig.

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Fig. 6. Dark field photomacrograph ofa {l 1 l} vert. section of a (110) orientation Ga-doped Ge crystal showing dislocation arrays associated with solute trials. Those in the upper (shoulder) region are due to transient increases in growth rate which occurred during the “grow-out” phase. Those at the bottom are due to the onset of constitutional supercooiing at the steady imposed growth rate. Crystal dia. at shoulder is 26 mm.

7b) are blunt and terminate with a macroscopic precipitate of gallium metal which has been etched out, The tapering of the pipes in the shoulder region (which implies a steady reduction in the size of the liquid droplet) occurs presumably because of the large fall in temperature which it suffers from the moment of its entrapment until the termination of growth: although the drop migrates steadily up the temperature gradient, its average temperature is steadily falling and therefore its equilibrium gallium concentration is steadily increasing. Since it has no source of gallium, it continuously shrinks. Note that the dislocation pipes are curved in the shoulder of the

tively lightly doped crystals grown at fast rates. In slowly-grown, heavily doped crystals, the transition occurs within a space representing an advance of only some tens of micrometres of the growth front. This abrupt transition is displayed in the X-ray diffraction topograph of a thin slice of a heavily In-doped Ge crystal (fig. 8), and in the microradiograph (fig. 16) of the same slice taken previously to thinning for the purpose of X-ray topographic study. The crystal had (110) orientation, and the slice contained the growth direction. Indexmg the growth direction [110], the plane of the slice is then (110). Topographs were taken with AgKa1, MoKa1 and CuKa1, radiations,

crystal; this is because the isotherms there are also curves, and drop trajectories run perpendicularly to the isotherms. The gradual transition from the low dislocation density of the pre-cellular region to the massive dislocation densities in the fully developed cellular structure with liquid droplet entrapment, as illustrated in figs. 2 to 5, can be discerned only in rela-

using the 220 and 004 reflections. The upper part of the crystal is nearly perfect, and, most notably in the case of CuKci1 radiation, the diffracted intensity from this perfect region is enhanced by the Borrmann effect (i.e. anomalous transmission). However, transmission is reduced locally by the strain fields of dislocations which consequently display dominantly negative contrast (i.e. appear as light lines on a dark

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Fig. 7. Higher-magnification micrograph of the {l11} vert. section in fig. 6 showing dislocation configurations associated with solute trails (a) in shouider region, (b) in lower section of crystai: note in this case the gallium inclusions at the end of the solute trails. Width of field is 1.8 mm.

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4 Fig. 8. X-ray diffraction topograph of a vert. section of a (110) orientation In-doped Giinystáfsllowing the sudden onset of a cellular structure accompanied by solute trails and high density of dislocations. CuKcc1 radiation, reflection 220, diffraction vector parallel to growth direction. Width of image is 20 mm.

background on the trails, photographic In the lower part, where solute droplets plate). and a high dislocation density are present, the benefit of anomalous transmission is lost; and in the CuKa case the overall diffracted intensity is less than in the upper part. (In fig. 8 the lower, imperfect part has been reproduced with added density in order to bring out details of the patt of solute trails.) In the case of AgKa 1 radiation, the lower value of pt and the greater contrastproducing response to a given strain gradient of this shorter wavelength combine to produce a strong excess diffracted intensity from the solute trails. In the case of the intermediate wavelength, MoKa1, the average diffracted intensities from the upper and lower parts of the slice are quite similar, Neither diffraction topographs nor absorption topographs resolve any zone of regular cellular growth, without droplet entrapment, above the level of transition to the droplet entrapment regime. The dislocations in the upper part give no indication of lying in cell boundaries. They lie dominantly in the (111) and (111) slip planes that make 35°16’ with (110). At the top centre of fig. 8 the dislocation den-

2. In the outer regions, sity is only 40 lines mm nearer the about transition level, the density increases to about 120 lines mm2. This slip configuration doubtless arises from relief of long range stresses induced in the upper part of the crystal by the lower, dropletcontaining zone. At the transition horizon, a high dislocation density surrounds the tops of individual solute trails, but does not extend back from the trail top up into the grown crystal further than about 50 pm, on average. In this specimen, lengths of solute trails range from 0.5 to 1. mm. Returning to the case of lighter doping, we see compared in fig. 9a and 9b the diffraction topograph of a thin slice and the etch pattern on an adjacent, parallel surface with {l 1 1} vert. orientation in a lightly gallium-doped (110) oriented crystal. On the etch pattern we see, above, weak contrast from the surface relief resulting from etching the cellular structure, and, below, strong contrast from dislocation etch pits and etched-out gallium-rich regions. We see corresponding features in the X-ray diffraction topograph, and can derive additional information from the latter. From the solute trails themselves and

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the matrix immediately surrounding them the diffraction contrast is intense and the dislocation density is too high for individual lines to be resolved. Within the solute trails we see some white dots and flecks. These indicate where the gallium-rich crystal has been cornpletely etched through, or, in the case of some trails, where the recrystallized material is misoriented so much as to fail to satisfy the Bragg condition. The topograph confirms that the upper part of the crystal is virtually dislocation-free, and in the lower part it shows the internal arrangement of dislocations which

a Fig. 9a.

have glided outwards from the solute trails. Fig. 10 is an enlargement of part of fig. 9a just above the centre and about ~ the image width from the left-hand edge. It shows the arrays of coaxial prismatic loops punched out in (110) directions from the solute trail. The loop diameters are in the 5 to 10 pm range. (They appear as solid dots rather than loops in the topograph partly because they are viewed obliquely and partly because of limited resolution.) In this crystal there is a tendency for loops to radiate dominantly from the top of trails, i.e. from centres of dila-

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Fig. 9. Correspondence between etch pattern and X-ray diffraction topograph of ft 1 i} vert. sections of a (110) orientation Gadoped Ge crystal grown with light doping at a fast rate. Width of crystal section —14 mm. (a) X-ray topograph, AgK~ 1radiation, reflection 220. diffraction vector vertical, (b) optical micrograph of adjacent etched section showing dislocation etch pits.

tional stress associated with the first material to recrystallize behind the migrating drop. Here the temperature would have been at its highest in the range over which recrystallization took place, and the ger. manium matrix yield stress correspondingly lowest. As fig. 9a shows, the etching of this thin specimen resulted in the accidental formation of a large hole at about midlevel in the specimen. The hole is surrounded by Pendellosung interference fringes which are the X-ray analogue of the equal-thickness extinc-

tion contours of transmission electron microscopy (cf. fig. 5a of ref. [8]). The X-ray Pendellosung penod in this reflection is 20 pm. However caution is needed in interpreting changes of order of Pendellosung interference directly in terms of changes in specimen thickness. The order of interference is very sensitive to strain gradients: lattice plane curvature increases the order of interference, at constant specimen thickness [9]. Now in fig. 9a where we see, particularly in the area above the hole, displacements of

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Fig. 10. Enlargement of a region in fig. 9a just above centre and and about one-third of image width across from the left-hand margin. Observe dislocation loops punched out along (110) from solute trail on right. Field width 0.9 mm.

Pendellosung fringes where they pass over cell boundaries, naive interpretation in terms of thickness contours would infer that the cell boundaries stand up as ridges. But this is not so. The observed Pendellosung fringe shift is dominated by the effects of strain, and must be due to lattice curvature produced by coherency stresses between cell boundary material and the

surrounding germanium matrix. (There are no dislocations present to relieve this stress.) 3.2. (111)oriented crystals The dislocation configurations seen for any other orientation of growth can be rationalised in terms of

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Fig. 11. Macrograph of an etched {i 1 i} para. surface of a boron-doped Si crystal showing a large central {i 1 1}facet with dislocation arrays at the sites of macro-steps. The three longer sides of the concentric, alternating hexagons correspond to the macrosteps which make acute-angled re-entrant corners with the main parallel to the interface. Field dia. 25 mm.

the structures described above for (110) oriented crystals. An exception occurs for (111) oriented crystals, where two distinct types of structure are observed. The first type is illustrated by fig. 11 which is actually of a {l ll} para. surface of a boron-doped silicon crystal. However, germanium crystals show precisely the same phenomenon. The interface consists of a large central, alternating hexagonal facet bounded by sets of macro-steps formed by the (111) growth plane and the {1 11 } planes at 70°32’ and 109°28’ to it. (The evidence for occurrence of step risers in both these orientations is given in section 3 (iii) of ref. [2] -) Microsegregation and liquid entrapment occur at both the acute-angled and obtuseangled re-entrant corners with the result that the step edges are delineated by dislocation arrays as shown in fig. 11. On an oblique {1 1 1} section, the dislocation arrays are as shown in fig. 12 which illustrates a galhum-doped Ge crystal. The vertical rows of etch pit arrays arise from the solute trails left by migrating gallium droplets as can be seen from fig. 13 which is a f 1 10} vert. section of the same crystal. Often, the

macrosteps sweep slowly across the interfaceas growth proceeds, producing inclined rows of solute trails. The symmetrical configuration shown in fig. 11 is obtamed only with slow rotation rates; at high rotation rates, the pattern is distorted by the fluid flow. The other type of (Ill) structure is illustrated in fig. 14 which is a dark field macrograph of a {1 1 l} para. Section of a gallium-doped crystal which was rotated at 70 rpm. Here the dislocation arrays form a multi-start spiral. (The spiral pattern of fluid flow is discussed in section 3(v) of ref. [21.)The difference between the two structures is thought to be due to a difference in the macroscopic shape of the freezingpoint isotherm. The hexagonal type occurs when the isotherm is convex to the melt. In this case there exists a central (111) facet on the solid—liquid interface prior to the onset of constitutional supercooling [101. This facet remains after the occurrence of constitutional supercooling, the stepped structure forming only where the angle between the isotherm and the (Ill) plane exceeds a critical value. The spiral structure forms when the isotherm is concave to the

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}

Fig. 12. Etched till oblique section of a (1 11> axis Ga-doped Ge crystal showing dislocation arrays generated in the vicinity of solute trails formed at a macro-step. Field width 1.8 mm.

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Fig. 13. Etched {i vert. surface of a (111) axis Ga-doped Ge crystal showing solute trails emanating from the trace of a re-entrant at a macro-step. Field width 1.2 mm.

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ft

Fig. 14. Dark fie’d macrograph .of an etched. 1 ij para. surface of a Ga-doped Ge crystal grown with solid-liquid interface concave to the melt. Note the multiple spiral arm~.Crystal section dia. —21 mm.

melt. Then an annular area of facet e~dstsaround the periphery of the growing surface. The spiral dislocation network is interpreted as arising from the microsegregation which occurs when a number of macrogrowth sheets originating at the annular facet grow independently of each other towards the centre of the crystal, rejecting gallium as they grow. The spiral form is imparted by the overall solute pattern imposed by the crystal rotation. The spiral dislocation pattern was first observed, but not explained, by Esaki [11].

4. The solute distribution The solute distribution in the (111) spiral structure was studied by autoradiography on radio-galhum-doped crystals. An autoradiograph and a dislocation-etched {l 1 1) para. section of a (111) crystal similar to the one shown in fig. 14 is shown in fig. 15. The direct correlation between the regions of high dislocation density and high gallium concentration can be seen.

The solute distribution in tin and indium-doped, (110) oriented crystals was studied by taking X-ray absorption topographs of thin crystal sections. Solute trails terminating in indium metal precipitates can clearly be seen in fig. 16. Fig. 17 is an absorption topograph of a tin-doped crystal. The tin concentration in the solute trail arrowed was determined by densitometric measurements to be approximately 0.73 wt% Sn, and the width of the image of the solute trail to be approximately 40 microns. The latter figure is somewhat greater than that found by etching solute trails in this crystal. The difference is probably due to lack of optimum alignment of the specimen: a significant reduction from the maximum sharpness of solute trail images can be produced by a change in crystal orientation of no more than a few degrees. It follows that the figure of 0.73 wt% Sn represents the minimum value. Assuming that the trapped droplet was in local thermodynamic equiibrium with the surrounding crystal at the moment of entrapment, we deduce, from the solidus composition/temperature data of Trumbore [12], that its temperature at the moment of entrapment was

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~830°C. The axial temperature gradient in the crystal was about 10 deg mm~and therefore the root of the cell boundary groove must have been not less than 10 mm behind the main solid—liquid interface. The large number of trapped liquid droplets can be gauged from figs. 16 and 17. Indeed under severe conditions of constitutional supercooling, a large fraction of the already-grown crystal is re-crystallized as a result of the motion of the “snow-storm” of liquid droplets descending through the crystal A close examination of solute trail patterns on etched crystal sections show that some portions of the crystal have been re-crystallized several times by the successive para section of radio-Ga-doped Ge crystal grown with a solid-liquid interface concave to the melt. (a) Autoradiograph showing spiral regions of very high gallium Fig. 15. {tii}

concentration, (b) Etched adjacent section showing spiral arrays of dislocation etch pits.

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X-ray absorption topograph of a vert. section of a 110 orientatation In-doped Ge crystal showing solute trails terminating with In inclusions, Topograph taken with specimen thickness —400 ~m, and the X-ray generator (copper target) operated at 15 kV constant potential. This specimen was subsequently thinned to —150 ~smfor the taking of X-ray diffraction topographs, cf. fig. 8. Note in top centre a localised region of heavier incorporation of solute, associated with the “grow-out” phase. This region was confined to material which was etched away before the diffraction topographs were taken. This absorption topograph (and fig. 17) are printed to reproduce contrast as seen on the original radiographs, i.e. material more X-ray opaque appears light. Hence In droplets appear as white dots. (The straight, vertical light and dark bands, seen especially on the left-hand side, are experimental artefacts.)

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Fig. 17. X-ray absorption topograph of a vert. section of a <111> orientation Sn-doped Ge crystal showing solute trails. The solute trail arrowed is estimated to contain —‘0.73 wt% Sn. Specimen thickness 1.2 mm, width of field 8.5 mm. XX-ray tube (molybdenum target) operated at 40 kV constant potential.

passage of liquid droplets. The crystal remains essentially single, however; but, by X-ray topographic criteria, highly imperfect.

5. Discussion

The striking feature of the segregation behaviour is the myriad of solute trails and trapped droplets which form almost immediately after the onset of constitutional supercooling when the solute concentration is relatively high. It would appear to be characteristic of diamond-cubic semiconductors, but not of metals. This difference probably reflects the very low solidstate diffusion coefficients for substitutional impurities in the diamond structure. Lateral solid state diffusion can effectively raise the solidus temperature in the cell boundary grooves and thereby cause freezing of the groove material. This can occur on a significant scale only in metals. In diamond-cubic semiconduc-

tors, very deep cell boundary grooves develop. This situation occurs most readily when the cell boundary is aligned parallel to the temperature gradient and when the freezing range of the binary melt is sufficiently great (as in the case of gallium in germanium). Solute trails are believed to occur as the result of climb by temperature gradient zone melting of liquid droplets and cylinders. (The cylinders have their long axes parallel to the growth interface: fig. 13 is an example of a section through cylinders, intersecting their axes perpendicularly.) How do those droplets and cylinders form? Firstly, consider (110) oriented crystals in which the liquid-filled channels are vertical. Apparently, a quasi-periodic pinching-off at the root of the channels occurs, with consequent entrapment of liquid at the root. We have no theory to describe this phenomenon: we just have the evidence of its occurrence. In cell walls, the instability may be dominantly in the direction perpendicular to the growth interface, leading to the entrapment of cylinders; or it

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W. Bardsley et al. /Inhomogeneities in Ge single crystals

may occur in the direction parallel to the interface as well, leading to the formation of discrete drops. At the nodes where three cell walls meet, drops are produced, as would be expected from the geometry there; and these drops are large. Consider next (ill> oriented crystals In these, there are sheets of liquid trailing behind re-entrant corners formed between a segment of main facet (Ill), and a step riser As pre viously described, these macro steps sweep across the interface The plane of the liquid sheet contains the (100) direction common to the two {ilI} planes which form mam facet and step riser respectively, and its inclination to the growth interface is of course that of the trajectory of the re-entrant (which in turn is determined by local relative growth rates on mam facet and on step riser) With a fairly shallow inclina tion there is a strong component of the axial temper ature gradient across the sheet of liquid from one face to the other Consequently, the liquid will tend to dissolve into the solid at its hotter surface. In this situation, instability of thickness of the wafer of liquid is not surprising. There results a quasi-periodic breaking-off of liquid-filled cylinders from the wafer root. These cylinders then move away from the plane of the wafer, up the temperature gradient, as shown clearly in fig. 13. The length of the solute trails depends on the magnitude of the temperature gradient in the crystal and the time for which the .gradient is maintained before quenching to room temperature occurs. A rough estimate of the expected length of trail is derived in the Appendix and shown to be in reasonable agreement with the observed lengths which are in the range 10 to 1 mm. In contrast to the systems considered in this paper, we note that in systems where a eutectic forms at a temperature only a little below the melting point of the crystal matrix, the eutectic reaction will occur in the cell boundary groove and progressively spread over the whole growth surface as the bulk composition approaches the eutectic composition. This behaviour is common in the growth of Ill-V compounds, examples being the systems InSb—NiSb [131 and InP—Cr [14]. Turning from discussion of compositional inhomogeneities to consideration of structural defects, we note a striking feature, the ease with which low-dislocation-density p~crystals can be produced. Patel, Tramposch and Chaudhuri [15] found that germa-

nium doped with 1026 Ga atoms m3 could be grown substantially dislocation-free both by pulling and by horizontal growth. Our crystals have not been entirely dislocation-free in the non-cellular region: we have not, however, generally doped to this high level. Patel et al [15] ascribe the freedom from disloca tions to a lowered vacancy concentration and a lower dislocation mobility in material which is extrinsically p type at the melting pomt Dikhoff [16] has also ob tamed pulled crystals of Ga-doped Ge which are dis location free In the course of their work Patelet al observed m horizontally-grown crystals of Ga doped Ge dislocation configurations similar to those shown here in figs 6 and 7, and such they also attributed to the effects of constitutional supercoolmg Our exper lments, represented for example by figs 9a and 9b show that the enhanced dislocation content that occurs m the cellular region of lightly doped material arises from the “extra’ dislocation population in close physical association with the solute trails. By contrast, in some experiments with As and Sb-doped Ge we have found that the dislocation density increases with increasing doping level in the absence of a cellular structure. (This observation is in agreement with the ideas of Patel et al. concerning the different effects of p and n-type doping.) Furthermore, when a cellular structure has formed with these dopants, the extra dislocations tend not to retain a simple pattern of slip, as in the case of Ga-doped Ge, but instead form a strongly polygonized substructure. Summarizing our findings on the relationship between dislocation density and cellular structure, we have observed that the dislocation multiplication that occurs in the presence of a developed cellular structure derives predominantly from the relief, by plastic flow, of the stresses generated by the re-crystallization which proceeds behind migrating droplets. In the case of a heavily doped crystal, the degradation of crystal quality upon the onset of constitutional supercooling is no less than catastrophic.

Appendix Length of solute trails

It is probable, at least for temperatures not too far below the melting point of pure germanium, that the

W. Bardsley eta!. / Inhomogeneities in Ge single crystals

velocity of the trapped droplet is limited by diffusion of germanium across the droplet itself. The boundary between the liquid and solid phases will therefore be in, or approximately in, local thermodynamic equiibrium. A rigorous solution of the coupled heat and particle transport equations subject to the constraints imposed at the phase boundary (whose shape would have to be determined as a free-boundary problem) would be extremely difficult. However, a one-dimensional calculation should not be seriously in error, provided that the diameter of the droplet (or cylinder) is not so small that interface kinetic effects become important or the Gibbs—Thomson effect produces a marked influence on the condition of thermodynamic equilibrium. This latter complication should be avoided since the diameters of the droplets are of the order of 50 /lm. These and other complicating factors have been considered in detail by Cline and Anthony [17]. Taking the coordinate x parallel to the macroscopic gradient, steady of the droplet thermal with respect to thethe matrix is velocity thus givenu by: D dCL

f u dt =D f (G/m) dt,

(3)

°

where we have neglected the (small) activation energy for liquid diffusion and have taken D to be constant. The droplet temperature will fall with time according to the relation (4)

T(t)=110)—(V—u)Gt,

where V is crystal growth rate. Since V is much larger than v, we will neglect the latter in comparison with the former and will use the approximation (V v) V in what follows. In eq. (4), T(0) is the temperature of the droplet at the moment of its entrapment, and will be below the melting point, T*, of pure germanium (T* = 937°C).Assuming ideal behaviour in the Ga—Ge systern [19], the slope, m = dT/dCL, of the liquidus curve is given by 2) r L /1 1 Vi m— (RT exp [R L (5) —

1~F

—

—

‘

where the(33.2 gas constant and L the heatthe of fusion Rof isGe MJ kg~atom_i). In latent fact, over

where D is the diffusion coefficient for Ge in the liquid Ga—Ge droplet, CL is the Ge concentration in the droplet, C~is the Ge concentration in the crystal (Cs 1), m is the slope of the liquidus curve and dTL/dx is the temperature gradient in the droplet. This latter quantity is related to the temperature gradient in the bulk of the crystal (G) by [18]:

temperature range of present interest, which runs from T(0) down to the droplet temperature (‘700°C) when the crystal is snatched from the melt and quenched, m changes by only a few percent. We adopt the mean value, m 380°per unit atomic fraction for the range 937—700°C.Combining eqs. (3) and (4), and taking G to be constant over the range of Tand x involved, gives

dTL

— -

—

dTL

0

t

(1)

u=~

D

t

X

629

~

3Ks G 2Ks +KL

for a spherical droplet, dTL dx

=

2Ks KS+KL

G

xo

D

f~ (_ ~)

J’G T(O)

dT.

(6)

m

Tf is the temperature of the droplet immediately prior to quenching, which we derive from TrT(0)—Gx~,

(7)

for a cylindrical droplet, where Ks, KL are respectively the thermal conductivflies of crystal and droplet. With sufficient accuracy for present purposes we assume Ks = KL. Thus the total distance moved (x 0) by a cylindrical droplet in the time (t) for which the crystal remains growing after entrapment and before quenching to room ternperature is:

where Xt is the length of crystal grown after entrapment of the droplet. From eqs. (6) and (7) we have /DG\

X0—)Xt.

(8)

Taking values2 appropriate to heavily G doped Ge [1], si, flO) 900°Cand 10°mm~, D = 10.2 mm

630

W. Bardsley eta!.

/ Inhomogeneities in Ge single crystals

and considering the crystal shown in fig. 6 (for which V 5.3 pm 5~ and Xt = 20 mm), we predict the value x0 = 1 mm. This compares well with a measured value (fig. 7b) of 0.9 mm. In the case of the crystal shown in fig. 13, the relevant parameters are V = 25.4 pm s-i and Xt = 17 mm, so that the predicted x0 = 0.18 mm. Measurement on fig. 13 gives x0 = 0.25 mm, in fair agreement with prediction.

References [11 W. Bardsiey, J.M. Cailan, HA. Chedzey and D.T.J. Hurie, Solid State Electron. 3 (1961) 142. [2J W. Bardsiey, J.S. Boulton and D.T.J. Rune, Solid State Electron. 5 (1962) 395. [3] W. Bardsley, J.B. Mullin and D.T.J. Hurie, ISI Pubiication No. P110 (Iron and Steel Institute, UK, 1968) PP. 93—101. [41J.W. Rutter and B. Chalmers, Can. J. Phys. 31(1953)15. [51 W.A. Tiller, J. AppI. Phys. 29 (1958)611. [61 A.R. Lang, in: Diffraction and Imaging Techniques in Material Science, 2nd rev. ed., Eds. S. Ameiinckx, R.

Gevers and J. Van Landuyt (North-Holland, Amsterdam, 1978) pp. 623—7 14. E.S. Greiner and P. Briedt, J. Metals—Trans. AIME (1955) 187. 181 PB. Hirsch, A. Howie and M.J. Whelan, Phil. Trans. Roy. Soc. (London) A252 (1960) 499. [91M. Hart, Z. Physik 189 (1966) 269. [101 K.F. Hulme and J.B. Mullin, Phil. Mag. 4 (i959) 1286. [lij L. Esaki, in: Solid State Physics in Electronics and Telecommunications, Vol. 1, Part 1, Eds. M. Desirant and J.L. Michiels (Academic Press, New York, 1960) p. 5i4.

171

[12] F.A. Trumbore, Bell System Tech. J. 39 (1960) 205. [131D.T.J. Hurle and J.D. Hunt, ISI Publication No. PilO (iron and Steel Institute, UK, i968) pp. 162—166. 1141 B.W. Straughan, D.T.J. Hurie, K. Lloyd and J.B. Mullin, J. Crystai Growth 21(1974)117. [15] J.R. Patel, R.F. Tramposch and A.R. Chaudhuri, in: Metallurgy of Elemental and Compound Semiconductors, Met. Soc. Conf., Voi. 12, Ed. RU. Grubei (Interscience, New York, 1961) pp. 45—63. [161 J.A.M. Dikhoff, Philips Tech. Rev. 25 (1964) 195. [17J H.E. Ciine and T.R. Anthony, J. Appi. Phys. 49(1976) 2777. [18] H.S. Carsiaw and J.C. Jaeger, The Conduction of Heat in Solids, 2nded. (Clarendon, Oxford, 1959) p. 426. [19] C.D. Thurmond and M. Kowalchik, Bell System Tech. J. 39 (1960) 169.