Dislocation cell structures in melt-grown ... - Semantic Scholar

Cryst. Res. Technol. 40, No. 1/2, 7 – 20 (2005) / DOI 10.1002/crat.200410302

Dislocation cell structures in melt-grown semiconductor compound crystals P. Rudolph* Institut für Kristallzüchtung, Max-Born-Str. 2, 12489 Berlin, Germany Received 14 October 2004, accepted 16 November 2004 Published online 1 January 2005 Key words melt growth, dislocation dynamics, cell structuring, grain boundaries, III-V, II-VI, IV-VI. PACS 81.05D, 81.10.Fq, 61.72Ef, 61.66.Hq The phenomenon of dislocation patterning during melt growth of III-V, II-VI and IV-VI semiconductor crystals is discussed. The paper is focused on the formation of cellular structures driven by the growth inherent thermo-mechanical stress. Of particular interest is the scaling of relations between cells size, dislocation density and acting shear stress. Among the materials there are characteristic similarities but also significant variations of the cell genesis. After the related compound specifics are discussed possible measures for retardation of cell patterning during growth are demonstrated. © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Dedicated to Professor Peter Paufler on the occasion of his 65th birthday.

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Introduction

One of the material problems to be still solved is the complete understanding of the high-temperature dislocation dynamics (DD) that proceeds within a growing single crystal, especially, in compounds. A characteristic consequence of the DD is the formation of ordered dislocation substructures. The selfrearrangement of the stored dislocations in cellular networks with low-angle grain boundaries during single crystal growth is typical for the most substances used. In general, dislocation cell structures consist of walls with high dislocation density separated by interiors of markedly dislocation-reduced or even dislocation-free material. Such patterns are well-known from as-grown crystalline metals (e.g. Fe, Al, Ni, Mo, see Fig.1a), metallic alloys (e.g. Fe-Si, Ti3Al, Cu-Mn, see Fig. 1b), semiconductor compounds (e.g. GaAs, GaP, SiC, CdTe, ZnSe, PbTe, see Figs. 1c-f), solid solutions (e.g. Cd1-xHgxTe, Cd1-xZnxTe, Pb1-xSnxTe, see Fig. 1g) and dielectric crystals (e.g. AgCl, KCl, NaCl, LiF, CaF2, SrTiO3, quartz, see Fig. 1h-i) independently which growth or deformation method was applied [1,2,3,4]. However, only at the first glance the images are of similar appearance. Looking more carefully some differing morphological details are noteworthy. For instance, in Mo (Fig. 1a), Cu-Mn (Fig. 1b) and GaAs (Fig. 1c) the cell interiors are nearly free of dislocations and the walls are fuzzy-like of certain thickness consisting of many tangled dislocations (see also Fig. 2). Compared to that in CdTe (Fig. 1d) and PbTe (Fig. 1e) the cell walls are very thin in the order of one dislocation row remembering classical low-angle grain boundaries. In these crystals the matrix shows mostly individual dislocations which can occasionally form a sub-cell structure (CaF2; Fig. 1i). Finally, there are crystals where cell structures are not well distinguishable in the as-grown state as in InP (Fig. 5a) or even missing if special dopants are added like Si to GaAs or Se to CdTe (Fig. 5b). Until now the reasons of such varying cell genesis are not yet finally clarified even for undoped semiconductor compounds. As it is well-known, the dislocation rearrangement into cell networks takes place under external or internal stress in the course of plastic relaxation. Cell patterning is studied best in metals under external load [5,6] but also in post-deformed elemental semiconductor single crystals, like silicon [7] and semiconductor compounds, ____________________

*Corresponding author: e-mail: [email protected] © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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P. Rudolph: Dislocation cell structures in melt-grown semiconductor compound crystals

like AIII-BVs [8,9,10,]. In growing crystals cellular substructures are formed due to the acting internal thermomechanical stress field. They are observed in melt-grown Czochralski [11], horizontal [12] and vertical Bridgman crystals [13] as well as in samples grown from solution [14] and vapour [15] (see Fig.1). However, the growth - related process of dynamical structuring (polygonization) is even in semiconductor compounds by far not yet studied with such profundity like in post-deformed specimens. Of course, this is first of all due to the still impossibility of in-situ stress measurements.

Fig. 1 Dislocation patterns formed in various crystals under differing stress conditions. (a) Mo 12% deformed at 493 K [1]; (b) Cu-Mn crystal deformed at 68.2 MPa [2]; (c) GaAs crystal grown by VCz (author’s image); (d) CdTe crystal grown by VB (authors image); (e) PbTe crystal grown by VB [33]; (f) SiC crystal grown by sublimation (courtesy of D. Siche from IKZ Berlin); (g) Cd0.96Zn0.04Te crystal grown by VB (author’s image); (h) NaCl crystal with labyrinth structure deformed by 150 MPa at T/Tm = 0.75 [3], (i) CaF2 crystal grown by VB [4].

Usually, dislocation substructures impair the crystal quality. Whereas the interior is of a constant lattice orientation across the dislocation-rich walls arises a discontinuous orientation change. As a result optical inhomogeneities do appear like in CaF2 crystals being of extreme importance for preparation of high-quality lenses for the UV photolithography [4,16]. Across semi-insulating {100} GaAs wafers, important for production of low-noise high-frequency devices, a mesoscopic resistivity variation does occur due to the accumulation of AsGa antisite defects (EL2) within the cell walls [17]. Subgrain boundaries impede also the electron transport in Cd1-xZnxTe radiation detectors [18]. Hence, the crystal grower is usually strived to find out proper measures to prevent dislocation patterning. © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Cryst. Res. Technol. 40, No. 1/2 (2005) / www.crt-journal.org

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However, there is a certain interest in crystals with mosaic structure too. For instance, crystal diffraction lenses for nuclear astrophysics show an improved reflection power when crystals with mosaicity of 20 -50 arc sec (e.g. Ge1-xSix) are used [19]. Further, in nanocrystalline materials the controlled reduction of grain size to nanometer scale lead to many interesting new properties including a great increase in strength [20]. Therefore, the further development of knowledge about collective dislocation interactions in growing crystals is of general practical relevance for both targets - repressing and provoking of the cellularity. One has to note that not all types of patterning can be attributed to DD which takes place within the crystal volume, i.e. behind the propagating crystallization front. As it is well known in case of morphological instability of a fluid-solid phase boundary, induced by constitutional supercooling, their former planar shape passes over into a characteristic cellular profile (see review of Billia and Trivedi [21], for example). As a result a lamellar-like structure with longitudinally extended walls is formed. There are some authors ascribing dislocation cell patterns in semiconductor compounds exclusively to such morphological instable interface [22,23]. However, as we clarified by depth integrating laser scattering tomography (LST) in GaAs [24] the dislocation cells are of globular type that contradicts longitudinal formed ones to be expected at cellular-shaped crystallization front. Moreover, as it is well known, dislocation cells may disappear completely if certain dopants are added, like In to GaAs [25] or Se to CdTe [26] although their presence should even promote constitutional supercooling. In the following we will concentrate on cellular patterning in the course of DD only. The phenomenon of constitutional supercooling is here not more considered. Today, an enormous number of publications on dynamical dislocation structuring is available. Reviews on fundamentals are given by Kubin [27], Amodeo and Ghoniem [28], Kratochvil [29] and Zaiser [30], for example. As mentioned already above, the prevailing part of publications deals with metals and metal alloys mostly under post-growth mechanical strain [31]. One of the first noteworthy in-situ observations of the dynamical polygonization during crystallization was carried out on monocrystalline plate-shaped Al crystals by Grange et al. [32]. They observed by real-time synchrotron x-ray topography that the formation of the dislocation substructure does not start at the propagating melt-solid interface but some millimetres behind and needs certain ripening time under thermal induced strain. Of course, there are many papers notifying the presence of cell patterns in semiconductor compound crystals as function of the growth conditions (temperature gradient, crystallization rate, doping content) but detailed quantitative analysis’s and correlations to the high-temperature DD processes are still rare. For instance, Mühlberg [33] and Kinoshita et al. [34] described an inverse correlation between cell size and cooling rate near to the growth temperatures in PbTe and Pb1-xSnxTe crystals, respectively, being obviously in relation to the acting thermo-mechanical stress. Only recently the author et al. [35,36] published first results of correlation between empirically determined network parameters, like cell size and dislocation density, with the computed thermal shear stress in undoped GaAs crystals (see Sect. 2.1.). The present paper will summarize some facts about dislocation structuring in as-grown semiconductor compounds from phenomenological point of view. It can be seen that it’s time to grasp the results by a proper theoretical framework. For that it is necessary to bring together the crystal growers with the already wellversed metal physicists in this field.

2

Experimental facts

2.1 GaAs Nominally undoped GaAs crystals with dislocation densities ρ ≥ (0.5 - 1) x 104 cm-2 grown by Liquid Encapsulated Czochralski (LEC), Vapour Pressure Controlled Czochralski (VCz), Vertical Gradient Freeze (VGF) or Horizontal Bridgman (HB) method show typical well-distinct dislocation cell structures similar to Fig. 1c [11,13,24,25,37,38]. The cells are of globular type [24] and their size decreases with increasing average dislocation density [35,37]. The observed mean cell diameters are 100 - 200 µm for 4-6" LEC crystals with typical etch pit densities (EPD) of (0.7-2) x 105 cm-2, 500 - 1000 µm for 4” VCz crystals (EPD: ~104 cm-2) and can reach values even greater than 2 mm in 4 ” VGF crystals if the EPD falls short of 5 x 103 cm-2 [13]. Usually, at dislocation densities below 5 x 103 cm-2 the cell walls begin to disintegrate into fragments. No cells do more form if the EPD falls shorter of 103 cm-2, i.e. if the mean distance between the stored dislocations exceeds ρ -1/2 ≈ 300 µm. Dislocation cells are also a characteristic feature in post-deformed GaAs specimens showing, however, much higher dislocation densities in the region of ρ = 108 – 1010 cm-2. Much work has been done on the © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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macroscopic deformation behaviour and dislocation structure of GaAs semiconductor crystals after plastic deformation at temperatures T/Tm ≤ 0.9 (Tm - melt temperature = 1511 K) [8,9,10,39,40]. Important plasticity parameters were determined by these experiments like stress-deformation curves, critical resolved shear stress, dislocation multiplication and motion rates etc. Trippelt, Klimm and Paufler [10] observed the important fact that the rearrangement of dislocations into cells begins already at lower stresses than needed for multiplication. Unfortunately, until now scaling analysis of the stress-subgrain-size relation, as it is well known for metals and dielectrics [41,42], was not yet done for post-deformed GaAs. Merely, one can estimate from the TEM images of dislocation networks, published in refs. [10,40], that cell diameters d of about 5 and 10 µm correlate with applied shear stresses τ of 30 and 14 MPa, respectively. Such proportions are quite comparable with those of post-deformed metals. In comparison with as-grown crystals, however, the cell dimensions differ markedly. Fig. 2 shows an image of dislocation cells of an as-grown 6” VCz GaAs crystal taken by x-ray synchrotron topography. The dislocations are accumulated in fuzzy walls of certain thickness (50 - 100 µm) as was likewise reported elsewhere [9,24,43,44,45]. Typically, numerous junctions and pins form a sessile dislocation jungle (see also refs. [24,44]) which is rather stable against post-growth annealing [46,49]. The analysis of the full width at half maximum (FWHM) of the x-ray rocking curve (XRC) over standard GaAs wafers revealed that only very small mean tilt angles around 10 arc sec do exist between the cells [13, 47,48].

Fig. 2 X-ray synchrotron topogram of a 6-inch semi-insulating VCz GaAs crystal grown at IKZ Berlin (the image was taken by T. Tuomi from the University of Helsinki at the Hasylab-DESY facility in Hamburg [44]).

As has been demonstrated by the Burgers vector analysis of Schlossmacher and Urban [49] the predominant dislocation types forming the cellular structures in as-grown GaAs are of 30°- and 60°-configurations (70 - 80 %) as well as screw dislocations (7 - 11%). The major fraction of the dislocations forming the cellular structure are still lying on {111} glide planes in glissile configurations. However, as was shown by Tuomi et al. [44] also differing glide systems like for edge dislocations within the {110} planes have been detected. Similar to that in post-deformed samples the dislocation in the walls belongs to the basal glide system {111} too [9]. Obviously, glide processes play an important role for the DD at both high and low temperatures. Of course, for the formation of spatial cellular networks the participation of three-dimensional mechanisms, like cross slip and climb, is required. First attempts to correlate the cell dimensions d in undoped GaAs crystals with the dislocation density ρ and acting thermo-mechanical stress during growth were done by Rudolph et al. [35,36]. Initially, a rough scaling of the mean cell diameters versus mean dislocation densities by using the sporadic literature reports for LEC, VCz and VGF GaAs crystals was carried out (see Fig. 3 in ref. [35]). Then more precise EPD measurements along crystallographic directions of 6-inch wafers were added [36]. Fig. 3 shows the d(ρ-1/2) plot combining both the literature data and a characteristic measurement along the [110] direction. For comparison the results of Langford‘s [50] mechanical deformation experiments on α-iron crystals, representing a typical correlation in metals [51], are added. As can be seen, the GaAs points fit the function for iron quite well which was given in a general form by Holt [51] as d = k´ ρ -1/2

(1)

with k´ factor of proportionality being for metals in the range of 10 - 15. Considering that the real content of dislocations is about two times higher than the EPD a somewhat enhanced k´ -value of ~ 20 should be derived from our as-grown GaAs results (see Fig. 3). © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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Fig. 3 Cell diameter d vs. dislocation spacing ρ-1/2 compiled from different GaAs crystals grown by various methods [35]. Selected values along the [110] direction of a 6” VCz wafer are included. For comparison the results of Langford‘s [50] mechanical deformation experiments on αiron crystals, representing the typical correlation in metals [41,51], are added.

To correlate the cell size with the acting thermo-mechanical stress for the first time was made use of the global computer simulated von Mises stresses [35] and radial resolved shear stress fields within the growing crystal slightly behind the growing interface [36]. It is important to note, that the calculations were based on the timedependent crystal geometry in strong correlation to the real position of the crystallized fractions where the wafers for the cell structure analysis were taken from. In Fig. 4 the dependence of the measured cell dimensions of a typical 6-inch VCz GaAs wafer from the calculated maximal shear stress values along the [110] direction is inserted. Also the rough estimations obtained from the calculated von Mises stresses for different VCz crystals [35] are added. For comparison, various data from post-deformation experiments on LEC GaAs [10,40], InP [52] and metallic crystals [41,51] are compiled too. The extrapolations of some dielectrics are given as broken lines [42,53]. As can be seen up to a cell size of about 700 µm, correlating with a dislocation density of ~104 cm-2 (see Fig. 3) and shear stress τ of ~ 1 MPa, the GaAs values fall into the function d = K G b τ -1

(2)

with K another proportionality factor than in eq. (1), G the shear modulus of Young (estimated after the elastic constants at 1500 K [54] to be 29 GPa) and b the Burgers vector (= ½ = 0.4 nm). The well-known eq. (2) was termed by Kuhlmann-Wilsdorf [55] as “law of similitude” because of the self-similar behaviour of the dislocation patterns during each type of load. According to these first results in Fig. 4 K-values between 20 and 40 can be deduced for as-grown GaAs so far the curves falls into the d ~τ -1 slope. Taking into account the Taylor relation τ =αGbρ½

(3)

with α a constant in the range 0.2 - 1.0 and combining eqs. (1) and (3) becomes the relation between subgrain size and stress in normalized form d/b =α k´(G/τ)m =K(G/τ)m

(4)

with α k´ = K = 23 and m = 1 for the prevailing number of materials investigated by post-deformation [41]. Taking for GaAs an estimated values of k´ = 20 (Fig. 3) and K = 20 - 40 the value of α becomes between 1 and 2 that seems to be somewhat overestimated compared to the deformation experiments including the well fitting points of post-deformed GaAs and InP in Fig.4. However, one has to consider that even at lower dislocation densities like in as-grown crystals the logarithmic term must apply for α ~ ln(ρ 1/2b)-1 [56] shifting this value towards a somewhat higher number compared to materials with very large ρ. When looking in Fig. 4 at the region where the resolved shear stresses τ fall below 1 MPa (i.e. τ / G b < 0.1 µm-1), that correlates with cell sizes > 700 µm and ρ < 104 cm-2, it is obvious that the slope of the d(τ) plot is changed and the exponent becomes m < -1. In fact, for numerous GaAs wafers taken from the top of the © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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cylindrical section of different 6” VCz crystals an identical proportionality of d ∝ τ -0.15 was obtained. As long as this phenomenon proves to be correct a more detailed discussion is needed (see also Sect. 4). But before that much more experimental statistics is required.

Fig. 4 Measured dislocation cell size d vs. calculated normalized resolved shear stress τ /Gb along the [110] direction of a 6-inch VCz GaAs wafer (empty squares) and estimated from von Mises stress modelling in VCz GaAs crystals (white squares) in comparison with the data of post-deformed GaAs [10,40], InP [52] and metals [51]. The slopes for some postdeformed dielectrics [42] and the mean dependence with K = 23 after Raj and Pharr [41] are added.

As it is already well established, the cell structure disappears when specific dopants, e.g. In or Si of higher concentrations (> 1018 cm-3), are added to growing GaAs crystals [25,45]. However, when doped crystals are post-deformed by high-enough mechanical stress the cell structure can be evoked again. Jimenex-Melendo et al. [8] described such appearance if specimens doped with 5 x 1019 In atoms per cm3 are strained at 512 °C by more than 5 %. Surprisingly, then the cell walls are of similar entangled and diffuse morphology like in undoped as-grown crystals (compare present Fig. 2 with Fig. 1 in ref. [8], for example) except for the cell size and boundary tilt angle which are of about 1 µm and 5´, respectively. Recently, the author’s team demonstrated that in undoped GaAs crystals the cell structure can be effectively restrained when the crystal is growing under near stoichiometric crystallization conditions (see Fig. 5c) [36].

2.2 GaP and InP As it was shown in ref. [35] also in growing undoped GaP crystals the stored dislocations rearrange into cell structures. Similar to GaAs LEC crystals typical mean cell sizes of 100 - 200 µm were detected. The cell walls, however, are often not closed completely and the dislocations seem to be not so tangled throughout the crystal like in GaAs. Moriya [57] mentioned that even in GaP crystals do not form dislocation networks. He used, however, for LST analysis sulphur-doped material only. The dislocation behaviour in GaP crystals has been widely investigated by post-deformation experiments (see [58,59,60], for example) but, unfortunately, the analysis was not directed on the dislocation cell structuring. This is, obviously, due to the use of mostly S-doped material in which the cell formation process is restrained similar to doped GaAs (Sect. 2.1). The appearance of characteristic slip bands are reported [58] which prevail the plastic deformation in such specimens at temperatures as low as 700 °C. A very important observation is given by Wagner, Paufler and Rotsch [60]. They found that all deformed samples exhibit indications of dislocation climb already at temperatures above 600 °C and attributed phosphor vacancies (VP) to the dominant assisting intrinsic point defect. Information’s about dislocation patterning in InP crystals are relatively rare and somewhat inconsistent in the literature. Compared to GaAs often a missing cell structure is reported even if the dislocation density exceeds 105 cm-2 [35,61,62] (see Fig. 5a). Some authors revealed silhouettes of non-completed cells in asgrown material [63,64,65]. Shimizu et al. [12] measured in crystals grown under non-stoichiometric conditions FWHM values of the XRC between 10 and 100 arc sec which they attributed to a mosaic structure with © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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dislocation-enriched boundaries. Although the situation concerning dislocation structuring in undoped asgrown InP crystals is not yet completely clarified one can note that the cell formation is much more restrained compared with GaAs and GaP and cannot take place in InP at relatively low shear stress values below 10 MPa as can be assumed for standard crystal growth conditions [66]. Generally, like in another III-Vs, certain doping elements at high concentrations (e.g. S, Zn > 1018 cm-3) can prevent the cell formation process totally due to the minimization of the dislocation movement [67]. Fe, however, cannot be used as effective mobility “stopper” because of its low solubility limit of ~ 1017 cm-3 in InP [68]. Well-distinguishable networks of grains with dimensions of 2 - 10 µm can be forced in undoped InP crystals by post-growth mechanical bend higher than 20 MPa and temperatures over 800 °C [52]. From such experiments Geibel [52] determined the relation between cell dimension and acting normal stress σ in undoped InP as d = 22 G b σ -1. The data are inserted in Fig. 4 well fitting those of another deformed materials (note multiplying the normal stress σ by the Schmidt factor Φ ≈ 0.5 the acting shear stress τ becomes somewhat reduced).

Fig. 5 Etched surfaces of crystals with missing or depressed cellular structure. a – LEC - grown undoped InP (IKZ Berlin; see also ref. [35]), b –VB -grown CdZn0.6Se0.4 crystal (author’s image; see also ref. [70]), c – undoped VCz GaAs grown under in-situ controlled stoichiometric conditions (IKZ Berlin; see also ref. [36]).

2.3 CdTe, Cd1-xZnxTe, and CdTe1-xSex Typical etch pits images of {111} surfaces of CdTe and Cd1-xZnxTe (x = 0.03) crystals grown by the VB method are shown in Figs.1d and 1g, respectively. The presence of cellular structures is obvious. Mean cell dimensions in the range of 100 - 200 µm at EPDs of 5 x 104 and 2 x 104 cm-2, respectively, have been detected in CdTe crystals grown in axial temperature gradients ≥ 20 K cm-1 [69,70,71]. However, the characteristic features differ from those in GaAs. The matrix contains numerous isolated dislocations and the cell walls are formed much sharper consisting of only one row of dislocation pits. Identical dislocation arrangements are © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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described for vapour-grown samples too [15]. Durose and Russel [15] and Sabinina et al. [72] investigated the cell wall structure in vapour- and melt-grown CdTe samples by transmission electron microscopy and observed that the dislocations which constitute the boundaries are nearly all parallel and most have the same Burgers vector (see Fig. 1 in ref. [15]). Such behaviour is well-known from the standard type of polygonized low-angle grain boundaries containing only the excess dislocations of similar Burgers vector after the annihilation process is completed. In fact, the relatively long growth and cooling periods contribute in CdTe crystals much more effectively to the substructure ripening [15] than in GaAs. Also the larger disorientation angle between the neighbouring cells refers to a typical polygonized grain-boundary structure. Tilt angles of 60 - 120 arc sec [26,70,73,74] and even 18 arc min [15] were reported for melt- and vapour-grown CdTe crystals, respectively. Obviously, compared with GaAs the substructure in CdTe and related solid solutions reacts much more sensitively to variations of the stress field acting during crystal growth. For instance, Lay et al. [75] found that under low thermo-mechanical stress in small temperature gradients < 10 K cm-1 the dislocation density was reduced down to 104 cm-2 and no any features of subgrain boundaries could be detected. This result was confirmed by numerous further authors summarized in ref. [70]. Also in the II-VI crystals the dislocation structuring is prevented by doping with isoelectric components Zn and Se, for example. It was observed that the tendency to form low-angle grain boundaries can be much more depressed if Se is added instead of Zn. Absolutely subgrain-free VB CdTe1-xSex mixed crystals with low EPD (< 8 x 103 cm-2) and FWHM of XRC = 6.5 arc sec have been reported at x = 0.4 [26,70] (Fig. 5b).

2.4 PbTe and Pb1-xSnxTe Fig. 1e shows a typical image of an etched (100) surface of a VB-grown PbTe crystal. A characteristic lowangle grain boundary structure with sharp cell walls, almost identical with CdTe (Fig. 1d), is perceptible. Also in these crystals the grain interiors contain many incidental dislocations which were almost not observed in GaAs (Fig. 1c). However, markedly higher disorientation angles in the range of 2´- 30´, in some cases even up to 3°, have been ascertained by x-ray topography [33]. Against it the grain sizes are comparable with those in low-stressed GaAs being in the range of 300 µm - 1.5 mm. After Kinoshita and Sugii [76] the grain extension can reach even several millimetres when special measures for reduction of the radial and axial temperature gradients were applied. In such case the tilt angle between the subgrains was also found to decrease to 1´- 2´. Surprisingly, a considerable higher dislocation density than in III-V and II-VI crystals have been reported for VB-grown PbTe and Pb1-xSnxTe (105 - 106 cm-2). Mühlberg [33] investigated the relation between grain dimensions d and growth conditions in VB PbTe. Identical studies were carried out by Kinoshita and Sugii [76] on Pb0.8Sn0.2Te mixed crystals showing nearly the same dislocation structure. Both authors observed a correlation to the inverse root of the cooling rate as d = a(v⋅ G)-1/2 ~ τ -1/2

(5)

with v the crystallization velocity, G the axial temperature gradient and a the proportionality factor of about 102. It was unimportant whether G was chosen great and v small, or vice versa [33]. It is noteworthy that no dependence on the G/v ratio was found which is responsible for the appearance of constitutional supercooling and, hence, a growing-in cell formation mechanism. Therefore, one can conclude that the analysed subgrain structure in PbTe and Pb0.8Sn0.2Te crystals is caused by dynamic polygonization and not by morphological instability of the growing interface. Taking into account the direct proportionality between the stress tensors and the cooling rate v⋅ G, as it is shown by Indenbom [77], becomes the relation d ∝ τ -0.5 which does not fit the universal eq. (2) whereupon d ∝ τ -1. However, also in the case of GaAs grown under very low stresses a differing slope with exponent m below - 1 was obtained by our first modelling results (Sect. 2.1 and Fig. 4).

3

Discussion and Conclusions

From the above results arise three questions - i) how the cellular structures do form?, ii) which are the genetic differences responsible for modified cell morphologies in various compounds?, and iii) how such cell patterns can be prevented during crystal growth? To come to the point, to date there exists not yet a commonly accepted approach towards completely theoretical understanding of the genesis of dislocation patterning [30]. Further, © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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careful quantitative analysis’s of the correlation between cell structuring and growth conditions as well as proper empirical and computational statistics are still rare in the field of semiconductor compounds. Therefore, the following conclusions are rather of speculative character yet.

3.1 Origins of cell patterning As can be seen from the GaAs data in Fig. 3 the d(ρ -1/2)-function of eq. (1) remains correct up to relatively long dislocation spacing of several hundred micrometers being of some magnitudes of order larger than in post-deformed specimens [41,51]. That means the cell patterning process is really of large-range order which is still working in growing compound crystals with relatively low dislocation densities (in GaAs down to ~ 5 x 103 cm-2). Considering the model of Holt [51] whereupon eq. (1) characterizes the cut-off area inside of which dislocation interactions can take place effectively the critical length scale extends in low-dislocation GaAs up to several mm when k´ is in the range of 10 - 20 (Fig. 3). Obviously, such a wide-ranging interaction sphere is also the cause for the functional dependence between cell dimension and relatively low acting shear stress in growing GaAs crystals which fits the universal relation d ∝ τ -1 in eq. (2) up to cell sizes of about 700 µm (Fig. 4). A number of theories have been proposed to account for the stress dependence of the subgrain size as it is summarized in refs. [6,30,41], for example. Until now the question is whether the cell patterning is driven energetically or by a self-organizing process in frames of equilibrium or non-equilibrium thermodynamics, respectively. There are well-known facts to be said for energy-related processes. In classical sense the driving force for subgrain formation is the reduction in strain energy resulting from the clustering (i.e. mutual field screening) of dislocations in cell and subgrain boundaries. However, the process of plasticity cannot explained exclusively by equilibrium thermodynamics due to the presence of typical preconditions for irreversibility such as stress and temperature gradients in both post-deformation and growth processes. Even a growing crystal which remains during its cooling down phase for a relatively long time within high-temperature thermomechanical stress and thermal gradient fields one has to treat as thermodynamically open system with continuous import and export of entropy. As a result, a rate of entropy is produced within the crystal evoking self-ordered patterning of the already presented or currently generated dislocations. In fact, when looking at the huge cells or grains which are formed in GaAs (Sect. 2.1, Fig. 2) or PbTe crystals (Sect. 2.4, Fig. 1e) down to very low acting thermo-mechanical stresses, even below the critical resolved shear stress τCRSS ≈ 0.5 MPa (the region τ/Gb < 4 x 10-2 µm-1 in Fig. 4), an explanation by energy minimization alone is hardly possible. From kinetic point of view glide cannot more assumed to be the prevailing mechanism for dislocation movements into cell walls at shear stresses lower than 0.5 MPa (such values were calculated for the core region in 6” VCz GaAs crystals growing under optimised low- temperature gradients but showing still dislocation cells [36]). Probably, diffusion-controlled climb remains to be the decisive process of patterning as it is known from the high-temperature low-stress creep regime which is dominated by diffusional flow [53]. Edward et al. [78] proposed a dynamic model by adopting interchange parameters of creep leading in comparison with eq. (2) to a modified exponent m = 4/n in the d ∝ τ -m relation [41]. In fact there were carried out high-temperature low-stress tensile creep experiments on Al where d = 5000 µm and m = 0.54 have been obtained [79]. Even an exponent of m = 0.13 was reported for creep tests on Nb [80]. Possible, this is the approach for understanding of the decreased slope in the d(τ)-curve at very low stresses as was observed for GaAs (Fig. 4) and PbTe [eq. (5)]. However, more experimental investigations and theoretical deepening are required.

3.2 Cell genesis vs. material specifics As it is well known under conditions of externally or internally acting stresses dislocations exhibit movements like glide and climb. The glide velocity is proportional to a power of the shear stress τm as vgl = vo (τ /τo)m exp (-Q / kBT)

(6)

where τo is 1 MPa and vo, m and Q are material constants, kB is the Boltzman konstant and T the absolute temperature. Recently, Sumino and Yonenaga [67] compiled the magnitudes of these parameters for the III-V compounds. They showed that there are certainly differences between the velocities of 60°(α), 60°(β) and screw dislocations within a given crystal but there are no significant distinctions between undoped materials © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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which would explain the existing and missing cell structure in as-grown GaAs (Fig. 2) and InP (Fig. 5a), respectively. It is noteworthy that the formation of spatial cellular patterns is only possible when three-dimensional dislocation movements like cross-glide and climb can take place. Djemel et al. [9] pointed out that cross glide is the major step in formation of the microstructure in post-deformed GaAs. Recently, also Madec et al. [81] ascribed the pattern formation dynamics priority to cross-slip mechanism which, however, can proceed effectively only in case of relatively high stacking fault energy. Even in semiconductor compounds with zinc blende structure containing characteristic partial dislocations (Shokley partials) [67] cross-slip can be restrained due to a large equilibrium stacking fault distance between them which is inversely related to the stacking fault energy γSF as [82] dSh = G a2 / (24 π γSF)

(7)

with a - the lattice parameter. Additionally, stacking faults impede the intersections of dislocations being essentially for sessile jungle formation. Hence, it can be assumed that the differences of the γSF values (see Table 1) play a decisive role for the divergent cell formation genesis in growing III-V compounds. In comparison to GaAs in InP crystals a reduced cross-glide mechanism can be proposed due to its much lower stacking fault energy. In CdTe, however, the comparable low magnitude of γSF is not of influence on the grain structure formation process (Fig. 1d). Probably, in this material (and in PbTe too; see Fig. 1e) cross-glide is insignificant for dynamical polygonization and climb in combination with high-mobility glide are much more responsible. There are two facts to be said for it: i) the morphology of grains in CdTe and PbTe is not globularly shaped like in GaAs but rather stretched with features of two-dimensionality [15,33,74] referring to a prevailing glide dynamics, and ii) due to the high ionicity in II-VIs and IV-VIs the dislocations carry a large electrical charge leading to their much higher mobility in comparison to III-Vs [83]. Additionally, a higher self-diffusivity and larger content of high-temperature intrinsic point defects is typically for CdTe and PbTe expressed in larger phase extends δx in the phase diagram (i.e. larger deviations from stoichiometry) than in III-Vs (Table 1). Such characteristics promote vacancy- and/or interstitial-assisted dislocation climb. The phenomenological expression for the climb velocity is [28] vcl = vo (τ / G) = A (Di / b) (GΩ / kBT) cj (γSF / Gb)2 (τ / G)

(8) 3

where vo is the characteristic climb velocity, A is a constant on the order 10 , Di is the self-diffusion coefficient, Ω is the atomic volume, and cj is the concentration of jogs. Further terms were already introduced in eqs. (1) (7). From Table 1 can be seen that in InP also the climb processes should be reduced due to the smallest existence region in the phase diagram (see also ref. [87]), i.e. lowest native point defect reservoir of all III-Vs. In fact, concerning the deformation experiments of Wagner et al. [60] at 600 °C the probability to climb drops in the order GaAs > GaP > InP. From all these material facts it is now quite understandable why undoped asgrown InP crystals do not exhibit indications of pronounced cellular patterning. Further discussions about the interaction between point defects and dislocation movement in III-Vs are given by Siethoff et al. [39, 84]. Table 1 Material properties important for dislocation dynamics in undoped semiconductor compounds compiled from the current literature, e.g. [54,60,85,86,87,88] and textbooks. For comparison the data of copper are added. b - Burgers vector, G - shear modulus, fi - ionicity factor, δx - maximal compound phase extensions in the phase diagram, τCRSS - critical resolved shear stress, γSF - stacking fault energy, DA,B -self-diffusion coefficient for the A or B component, respectively, ~Tm - close to the melting point. Material Structure b [nm] G [GPa] fi δx τCRSS (~Tm) [MPa] γSF [10-7 J cm-2] DA,B (~Tm) [cm2 s-1]

Cu fcc 0.25 47 0 0 0.02 78 5 x 10-9

GaAs ZB 0.40 29 0.310 2 x 10-4 0.3 - 0.5 55 2 x 10-13 (As)

© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

GaP ZB 0.39 58 0.327 2-3 x 10-4 41 1.7 x 10-13 (Ga)

InP ZB 0.42 23.6 0.421 5 x 10-5 0.6 18 1.7 x 10-11 (P)

CdTe ZB 0.46 20 0.720 1-3 x 10-4 0.2 11 6 x 10-9 (Cd)

PbTe NaCl 0.46 13.4 ~ 0.80 1 x 10-3

1.4 x 10-10 (Pb)


17

There remains one important question. What are the reasons for the markedly differing wall morphologies in the cell structures of GaAs (GaP) and CdTe (PbTe) as discussed in Sects. 2.1 - 2.4 and demonstrated in Figs. 1d, 1e and 2? Whereas in as-grown GaAs crystals the cell walls are diffuse and fuzzy similar to the ones observed in metals after deformation at medium temperatures in CdTe well-developed grain boundaries consisting of single-dislocation rows are appearing. From today point of view one can assume that such features refer to different ripening levels within the framework of polygonization. Obviously, in GaAs the high tendency to dislocation screening by pronounced sessile junction formation [9,24,25] leads to an entangled dislocation jungle within the walls being stable against post-annealing. Such morphology remembers the socalled incidental dislocation boundaries (IDBs) which are assumed to be a result of statistical mutual trapping of dislocations. On the other hand CdTe and PbTe show typical characteristics of geometrically necessary boundaries (GNBs) with different activity of slip systems on each side of the boundary [89]. Generally, the interchange between dislocation movement and precipitates (As in GaAs) needs to be considered in the framework of the high-temperature DD concepts for semiconductor compounds too. Usually, all compound materials are grown from melts with excess of one of the components (As in GaAs, Ga in GaP, In in InP, Te in CdTe) due to the deviation of both the congruent melting point and equilibrium between partial pressures (pA = pBn = pmin) from the stoichiometry. As a result possible dislocation pinning and bowing out on precipitate nuclei can be take place. Such process can also considerable modify the cell structure genesis between the materials. Features of dislocation pinning on obstacles in III-V-compounds have been described by numerous authors, e.g. [8,25].

3.3 Steps to prevent dislocation patterning Generally, as has been shown in Sects. 2.1. – 2.4., independently on the materials used, the dislocation patterning can be prevented by doping very effectively. No cell structuring was observed in GaAs doped with In or Si at concentrations > 1018 cm-3whereas after Sumino and Yonenaga [67] immobilization of dislocations becomes stronger in Si-doped GaAs than in In (and Zn)-doped GaAs. Such effect is due to the impurity gettering at the dislocation core rising with temperature because of the increasing diffusion rate. In consequence, the yield stress is enhanced by dislocation locking. Jiminez-Melendo et al. [90] observed a decrease in the stacking fault energy when GaAs was doped with In. They explained this effect by Suzuki segregation or direct interaction between solute atoms and stacking fault making cross-slip events more unlikely. No low-angle grains were found in CdTe and PbTe crystals when solution hardening by mixing components Se (x > 0.4, see Fig. 5b) and Sn (x > 0.15) was provided, respectively. However, there is the wellknown drawback of segregation when dopands are added to the melt. On the one hand appears the danger of morphological interface instability by constitutional supercooling and, thus, low growth rates or high temperature gradients are required for its prevention. On the other hand, for device applications doped and mixed crystals cannot always replace undoped and binary material because of the sensitivity of physical parameter verification by adding foreign atoms. Obviously, the best way to exclude dislocation patterning is the reduction of the dislocation density by minimization of the thermo-mechanical stress. In undoped GaAs was observed that at ρ values < 5 x 10-3 cm-2 the cell structure began to disappear. However, for compound crystals with larger diameters over 100 mm the obtainment of such low dislocation densities is not yet solved empirically when hardening dopants are not added. Hence, the current efforts are directed on homogenisation of the thermal field in the growing crystals in order to reduce the dislocation multiplication and their mobility by minimizing the thermo-mechanical stress. Most hopeful is the VGF growth technique already producing III-V-crystals with dislocation densities ≤ 104 cm-2 when the ingots are growing in very low axial temperature gradients < 5 K cm-1 (e.g. [13]). Also in undoped 4” VCz GaAs crystals a nearly homogeneous EPD distribution without cell pattern was obtained when the propagating melt-solid interface was slightly convex without any concave parts during the whole growth process [35]. However one has to consider, that only very low growth rates < 5 mm h-1 are permissible under such condition. Another possible way to prevent cell patterning is the minimization of the intrinsic point defect content by in-situ control of stoichiometry during growth. The stoichiometry is regulated by the partial pressure of the volatile component (As, P) over the melt applying an extra heated source within the growth chamber. Recently the author and his co-workers demonstrated by using the VCz arrangement without boric oxide encapsulant that the cellular structure dissolves when the GaAs crystal is growing from proper controlled Ga-rich melt composition [36]. A detailed description of this method is given elsewhere [91]. Fig. 5c shows an etched © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

18


sample grown by this technique. However, a markedly Ga-enrichment in the melt is required to guarantee the growth of a stoichiometric GaAs crystal composition making this method also susceptible for morphological instable growth and incorporation of Ga inclusions. Shimizu et al. [12] controlled the stoichiometry in HBgrown InP crystals. They found that the dislocation domains were reduced when the phosphorous counter pressure above the melt was in equilibrium with near stoichiometric growth, i.e. 25 atm. However, the authors leave open whether a constitutional supercooling or polygonization is responsible for growing-in of the dislocation substructure. The non-conservative dislocation reactions, important for spatial cell structure formation, can be also decreased by the growth at temperatures much lower than the melting point. For instance, completely substructure-free PbTe crystals have been grown at 550 °C (Tm = 924 °C) by the Travelling Heater Method (THM) from a Te-rich melt-solution zone [92]. This method has been also successfully tested for numerous further materials, among them GaAs, GaP, InP and CdTe [93]. Again, only very low growth rates in the range of 1 - 5 mm d-1are possible to apply. Also an enhanced probability of solvent inclusions have been reported. Finally, one has to mentioned the principle of solid state recrystallization. Usually, positioning the asgrown crystal with dislocation substructure for a certain time in a temperature gradient or mechanical stress field near to the melting point (strained anneal technique) an increase of the grain dimensions or even a total healing up is occurred in numerous materials like metals and oxides [93]. It is also known from ZnSe and CdTe that long post-growth annealing of the wafers leads to an increase of the cell dimensions and, therefore, improvement of structural quality [94,95]. However, in some compounds like GaAs the post-growth annealing does not effectively influence the cell structure morphology (see e.g. [46]) due to the high rigidity of the dislocation jungle within the cell walls. But more special experimental investigations in this direction would to be recommendable. Acknowledgements The author wish to thank the co-workers of IKZ Berlin M. Neubert, Ch. Frank-Rotsch, F.-M. Kiessling, M. Czupalla, M. Pietsch, B. Lux, Th. Wurche, U. Juda, M. Naumann for crystal growth, preparation and characterization. He is grateful for numerous interesting discussions with M. Jurisch, B. Weinert and St. Eichler from Freiberger Compound Materials GmbH, H. Leipner from the Martin-Luther-University in Halle, M. Zaiser from the University of Edinburgh in UK, G. Gottstein from the University in Aachen, W. Pantleon from Risø Nat. Lab. in Denmark, and L. P. Kubin from LEM CNRS-ONERA in France.

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