Chemical vapor transport of zinc sulfide

Journal of Crystal Growth 312 (2010) 3642–3649

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Chemical vapor transport of zinc sulfide Part I: Isotopic crystals from nearly stoichiometric vapor phase Rudolf Lauck n Max Planck Institute for Solid State Research, Heisenbergstr.1, D-70569 Stuttgart, Germany

a r t i c l e in f o

a b s t r a c t

Article history: Received 5 May 2010 Received in revised form 10 September 2010 Accepted 15 September 2010 Communicated by P. Rudolph Available online 22 September 2010

Crystals of cubic zinc sulfide with different isotopic compositions have been grown by iodine vapor transport for basic research purposes (vibrational, electronic, and thermodynamic properties). The synthesis reaction in sulfur vapor was found to be controlled by solid-state diffusion of zinc atoms through a ZnS passivation layer. Crystals up to 5 mm in length were grown from small amounts of source material. The presence of argon reduced the nucleation density and favored the formation of facets. An one-dimensional stoichiometric transport model, including iodine dissociation and considering multi-component diffusion, results in an explicit formula of the transport rate in terms of the Pe´clet number. The derivation of an iodine-based Pe´clet number allows correlation of the iodine concentration gradient with the usual inert gas based expression, and simplifies the present calculation of the partial pressures. The transport rate, the Pe´clet number, and also the growth-relevant diffusion coefficient are given as functions of the iodine concentration and the inert gas pressure. The experimental results depicted in a linear diagram are in reasonable accordance with the theory, except for single cases that can be explained possibly by foreign vapors and zinc excess, respectively. & 2010 Elsevier B.V. All rights reserved.

Dedicated to Prof. Dr. Manuel Cardona on the occasion of his 75th birthday Keywords: A1. Diffusion A1. Mass transfer A2. Growth from vapor B1. Sulfides B1. Zinc compounds

1. Introduction Zinc sulfide (ZnS) is a widely used material for optical [1] and optoelectronic applications [2–8]. In the present work, crystals of ZnS were grown for fundamental research. Especially, the mass and pressure dependencies of phonon frequencies and line widths of Raman spectra, anharmonicity effects in the lattice dynamics [9–11], the effect of isotopic mass variation on the photoluminescence spectra and the band gap, and on the electronic, vibrational, and thermodynamic properties have been studied on isotope substituted crystals [12,13]. Chemical vapor transport (CVT) has been applied as growth method in order to obtain the pure cubic phase (3C, mineral names: b-ZnS, zinc-blende, sphalerite) at temperatures below 800 1C [14]. Especially vapor phase transport with iodine as the transport agent has been the preferred classical method for growing ZnS [14–29] and also other II–VI compounds, e.g. [29–33]. Other transport agents have been used rarely [20,34,35]. Several authors have provided quantitative descriptions of the transport rate of ZnS [19,21–28]. Fundamental theoretical work on CVT in general and overviews are offered in Refs. [36–45]. Also software programs are available for the calculation of transport rates [46,47]. Different pathways of computations and also the applicability of those programs are

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discussed in Ref. [48] with regard to the growth of ZnS1 xSex crystals. In several cases, the theoretical transport rate including the convection regime is given in comparison with experimental data as a function of the total pressure [28,30]. From a practical point of view, it is advantageous if the transport rate of ZnS is calculated as well as measured as a function of the accessible iodine concentration [19,21,24,25], as has been done for other compounds, too [49–54]. Most of the authors have applied a uniform diffusion coefficient to simplify the calculations. Different simplified models are discussed in Ref. [25]. In the present approach, also an average but non-constant diffusion coefficient will be deduced under consideration of the multi-component vapor composition, and preferably relevant to the incorporation. Finally the transport rate is intended to be derived in a traceable way as an explicit function of experimentally adjustable and accessible parameters, and to be compared with the experimental results. In favor of transparency, numerical calculations are avoided as far as possible.

2. Experimental 2.1. Purification and synthesis An experimental challenge of this work was to deal with small quantities of the isotope enriched elements available from TRACE sciences international, Ontario, Canada, and EFFEKTRON, Moscow,

R. Lauck / Journal of Crystal Growth 312 (2010) 3642–3649

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Russia (Chemical purity499.9%). Highly enriched stable isotopes (I.E. ¼95.6–99.4%) were used to synthesize zinc sulfide with different isotopic compositions (nZn32S34 x S1  x with n ¼64, 68 and x ¼0, 0.5, 1). In the case of 32S, natural sulfur (NA ¼95.02%) was employed. The elements with natural isotope compositions were purchased from M.C.P. electronics, UK (C.P.¼99.9995%). As reported briefly in Ref. [9] and depicted schematically in Fig. 1a–c, several processes were performed preceding the growth: Elemental sulfur was purified by sublimation in argon atmosphere ( o0.1 mbar) at 90–150 1C in a fused silica ampoule. After zinc ( r0.2 g per run) had been etched in a solution of 1–5% nitric acid in propanol to remove the possible oxide layer, it was sublimated in the same ampoule at 350–500 1C in argon atmosphere (o1 mbar), similar to the preparation of ZnO [55]. After each sublimation, possible residues were detached by sealing off the tail of the silica ampoule. The large area of the resulting Zn layer favored the synthesis reaction with sulfur at vapor pressures from 3 to 450 mbar. For this purpose the zinc layer was heated in steps up to 800–1000 1C. As in the case of ZnSe [56], the growth of a compound passivation layer prevented the free evaporation of the metal within the ampoule, and it was controlled by solid-state diffusion of the Zn atoms through that layer. This could be demonstrated by the corresponding reaction of individual Zn droplets resulting in hollow hemispheres of ZnS, as shown in Fig. 2.

2.2. Crystal growth The portion of the ampoule with the synthesized compound was placed in the horizontal growth ampoule with an inner diameter of 10–12 mm and a length of about 350 mm (Fig. 1d). Vacuum annealing up to a maximum short-time temperature of 450–550 1C should favor the stoichiometry of the compound. Polycrystals from precursory runs were also used as source material. The process of CVT was initiated by the chemical reaction of iodine (0.1–2 mg/cm3) with the ZnS source material at 730–800 1C, followed by the diffusion of the reaction products ZnI2 and S2 along the temperature gradient of a horizontal threezone furnace. The growth of transparent (colorless, amber, and greenish) crystals took place in a temperature range 670–770 1C as a consequence of the reversed reaction. In some experiments, several small crystals ( r3.5 mm) appeared randomly distributed along a temperature gradient of 1–3 1C/cm at the ampoule wall or

Fig. 2. Demonstration of the ZnS synthesis process controlled by solid-state diffusion of zinc through the growing passivation layer at 800–900 1C in sulfur atmosphere (  100 mbar). (a) Passivation layer of 64Zn32S growing as a hemisphere on an individual 64Zn droplet at the ampoule wall, and pure small hollow hemispheres, (b) pure hollow hemispheres of 64Zn32S, when the synthesis was completed. Marker represents 0.3 mm.

on a concentric silica rod, as shown schematically in Fig. 1d. It was also possible that a single cluster of a few co- or randomly oriented crystals grew at the minimum temperature. In two cases, a single crystal of about 5 mm in length (Fig. 3) grew in the transition zone from the higher (3–8 1C/cm) to the lower temperature gradient (1–3 1C/cm), at a distance of 150–180 mm to such a cluster. Because of the higher temperature, such a single crystal was endangered to be transported onward if the source was depleted. If the experiment had been terminated too late, only a cluster was found in the ampoule region with the minimum temperature. The presence of 25–680 mbar argon reduced indeed the transport rate, but on the other hand also the nucleation density, and favored the formation of facets. The growth parameters are listed in Table 1. The grown cubic crystals have been used to study the mass and pressure dependence of phonon frequencies and line widths by Raman spectroscopy [9–11], to investigate the mass dependence of exciton energies by photoluminescence [12], and also to measure the specific heat in comparison with theoretical calculations [13].

3. One-dimensional transport model 3.1. Flow equations Fig. 1. Schematic diagram of the process sequence. (a) Purification of sulfur by sublimation, (b) sublimation of zinc, (c) synthesis via vapor phase, and (d) crystal growth. Details are described in the text.

The steady state model is based on the applicable assumption of a stoichiometric compound with a narrow homogeneity range and without iodine incorporation. The total pressure is assumed

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to be constant throughout the ampoule, and two temperatures Ts (source) and Tc (crystallization) are defined. The experimental conditions (horizontal ampoule, diameterr12 mm, total pressure o1 bar) justify the assumption of diffusion-limited transport without thermal convection [19,22,28,57]. The chemical vapor transport is considered as a combination of chemical transport reactions for calculating the equilibrium partial pressures with one-dimensional physical vapor transport. Five species a are assumed to be present: ZnI2, S2, I2, I [25,27], and in addition Ar, with the corresponding partial pressures pa. The molar flow density ja of each component can be written as ja ¼

u D dpa ðxÞ pa ðxÞ RT RT dx

ð1Þ

with the advective drift velocity u (Stefan wind), the average diffusion coefficient D (discussed in the next section), the universal gas constant R and the average absolute temperature T [25,40,41,57–65]. The length scale is x, with the transport distance L. The total pressure P can be assumed to be constant along x P ¼ pZnI2 þpS2 þ pI2 þ pI þ pAr ¼ const:

where j is the molar flow density with regard to ZnS. Without iodine incorporation, the total iodine flux density is given by jZnI2 þ jI2 þ 0:5jI ¼ 0,

ð4Þ

and the total resulting argon flow density is jAr ¼ 0:

ð5Þ

Using Eq. (5), integration of Eq. (1) for a ¼Ar from x ¼0 (source) to x ¼L (crystal) gives an expression for the Stefan wind u, namely pc uL ¼ ln Ar D psAr

ð6aÞ

wherein L is the transport distance, and the indices c and s denote the values at the crystal surface and the source, respectively. Using only Eq. (6a), i.e. without application of Eqs. (2)–(4), Eq. (1) can be integrated for all other species, too. Thus ja ¼

D psa pcAr pca psAr pcAr ln s : RTL pcAr psAr pAr

ð7aÞ

ð2Þ

The stoichiometric incorporation is described by jZnI2 ¼ 2jS2 ¼ j,

ð3Þ

This equation is comparable with flux equations in the Ge–iodine system [63], and can be derived also from the Stefan–Maxwell equations for a ternary system with two transported vapor species and one stagnant gas [66, chapter 3, p. 93]. Due to iodine dissociation, other formulas of CVT and physical vapor transport (PVT) from literature [40,41,64,65] are non-transferable. Eq. (6a) is also the definition of the Pe´clet number NPe [67–69], i.e. the ratio of the advective to the diffusive flux, and can be written as NPe ¼ ln

pcAr : psAr

ð6bÞ

Applying Eq. (6b), it is obvious to express Eq. (7a) in terms of the Pe´clet number NPe. Hence ja ¼

Fig. 3. Cubic single crystal of 68Zn32S0.5 34S0.5 grown at 740 1C in 525 mbar argon, with iodine concentration of 0.73 mg/cm3 (experiment No. 7). The crystal curvature seems to be a clue of the beginning decomposition reaction.

DNPe psa exp NPe pca : RTL expNPe 1

ð7bÞ

It should be emphasized that the validity of Eqs. (7a) and (7b) is not restricted to the stoichiometric vapor phase. Furthermore, Eq. (7b) seems to be the more general formula because NPe can be connected not only with Ar but also with other gas species, as shown in the following: If the iodine-containing species, i.e. ZnI2, I2 and I, are considered, a further sum of the corresponding partial

Table 1 Growth parameters. Experiment No.

1

2

3

4

5

6

7

8

9

10

11

Source temperature Ts (1C) Growth temperature Tc (1C) Transport distance L (cm) Transported mass m (mg) Growth period t (h) Iodine conc. co (mg/cm3) Argon pressure poAr (mbar) Total pressure P (mbar) Diffusion coeff. D (cm2/s) Standard coeff. D0 (cm2/s)

773 702 34 200 191 0.12 679 730 0.779 0.057

762 693 20 231 172 0.12 670 720 0.774 0.057

777 710 18 220 105 0.20 656 740 0.720 0.053

785 717 17 240 124 0.185 645 725 0.754 0.053

785 715 17 46 142 0.175 602 677 0.802 0.053

777 711 16 220 105 0.20 658 742 0.719 0.053

777 729 22 293 110 0.73 525 818 0.450 0.036

775 745 15 210 168 1.20 479 957 0.325 0.030

736 705 13 325 274 1.57 26.5 598 0.291 0.018

784 700 23 256 158 0.50 634 836 0.513 0.043

782 719 11 206 68 0.44 472 652 0.645 0.041

4.5

2

2

3.5

2

1 6

7

Appearance of crystals, maximum dimension (mm) Individual single crystal Several separated crystals 2 Cluster, unidirectional Cluster, randomly oriented 5

5 7 5 0

The total pressure P and the diffusion coefficients D and D are calculated values.

6

8

10

R. Lauck / Journal of Crystal Growth 312 (2010) 3642–3649

stoichiometric condition of the partial pressures

pressures can be defined by piod ¼ pZnI2 þpI2 þ0:5pI :

ð8Þ

This relation is another description of the iodine conservation in terms of the vapor pressures, but piod, like pAr, is a function of x. Taking into account only the iodine-containing species, i.e. ZnI2, I2 and I, combination of Eqs. (1), (4), and (8) results in a second expression of the Pe´clet number as an analogical function of the crystal-to-source ratio of the iodine content as given by Eq. (8) NPe ¼ ln

pciod : psiod

ð9aÞ

Also, it follows from Eqs. (6b) and (9a) that NPe ¼ ln

pciod þ pcAr : psiod þ psAr

ð9bÞ

Consequently, Eq. (7b) can be applied even if no inert gas is present, except in the case of sublimation. Due to the two expressions of NPe in Eq. (6b) and (9a), the argon pressure gradient is coupled with the pressure gradient of the iodine content, and hence with the thermodynamics of the chemical reactions. This relation seems to be remarkable, but in fact it is not too astonishing, because iodine is not incorporated, and both the inert gas and the overall iodine are effectively stagnant, i.e. the total fluxes are zero. Therefore, this quinternary system can be considered as a pseudo-ternary system with two transported (i.e. incorporated) species and a stagnant gas mixture of ZnI2, I2, I, and Ar. It is notable that ZnI2 acts on the one hand as the molecule transporting the Zn atom to the crystal surface, and that it also contributes to the stagnant iodine content, zeroing the total iodine flux density corresponding to Eq. (4). If the partial pressure profile is assumed to be nearly linear due to the low transport rate [67], the pressure poAr of argon introduced into the ampoule is equal to the average partial pressure ðpcAr þpsAr Þ=2, and the initial pressure poiod of the added iodine is equal to ðpciod þpsiod Þ=2, where piod is defined by Eq. (8). Hence, it follows from Eqs. (6b) and (9a) that poAr pc ps ¼ cAr ¼ sAr : o piod piod piod

ð10Þ

The initial iodine pressure poiod can be calculated from the ideal gas law with poiod ¼

co RT MI2

ð11Þ

where co is the concentration (in mg/cm3) of the initially added iodine in the ampoule, and MI2 is the molar mass of the I2 molecule. The molar transport rate of zinc sulfide is given by Eq. (7b) for a ¼ZnI2. If MZnS is the molar mass, the corresponding mass flow J (in mass of ZnS per unit time and unit area, e.g. mg/hcm2) is J ¼ MZnS jZnI2 :

3645

ð12Þ

For the calculation of the transport rate J, knowledge of the diffusion coefficient and the partial pressures pa at the crystal (c), at the source (s), and the average values (at half transport distance) is required. First, the diffusion coefficient is treated.

3.2. Diffusion coefficient The stoichiometric incorporation is described by Eq. (3). Due to different diffusion coefficients in a multi-component system, the

pZnI2 ¼ 2pS2

ð13Þ

is violated. That is, the molecule ZnI2, with a lower diffusion coefficient than S2, needs a steeper pressure gradient to achieve a stoichiometric incorporation conforming to Eq. (3). But if an adequate average diffusion coefficient of the participants in the incorporation, namely ZnI2 and S2, is defined, it is possible henceforth to apply Eq. (13) approximately. Such an average diffusion coefficient is not constant but depends on the vapor composition, too. In a ternary system with the gas species A, B2, and the inert gas C (dissociative sublimation of AB), the average diffusion coefficient D can be defined by the combination of the transport rate given by Faktor et al. [40,41] with the first Gilliland equation [66,70,71], which contains the binary diffusion coefficients DA/C of A in C and DB2 =C of B2 in C. Thus 1 2 1 ¼ þ : D 3DA=C 3DB2 =C

ð14Þ

It seems to be a consequent approach to transform this relation from the ternary system into the current quinternary (pseudoternary) system. In analogy to Eq. (14), it is obvious to define an average diffusion coefficient of the participants in the incorporation, and to write 1 2 1 ¼ þ m D 3Dm 3D ZnI2 S2

ð15Þ

where Dm a are the effective diffusion coefficients of a ¼ ZnI2 and S2 in the multi-component system (denoted by the index m), which are calculated by Wilke’s equation [66,72–74] X pb ð1pa =PÞ ¼ , ð16Þ m Da PDab baa taking account of the binary diffusion coefficients Dab (b ¼I2, I, Ar, S2 and ZnI2, respectively). The special averaging by Eq. (15) can be considered as an expedient approach because it yields a diffusion coefficient which is relevant to crystal growth. On the other hand, a certain inaccuracy with regard to the partial pressures and flow rates of the other species is accepted as a compromise. In addition, the temperature and pressure dependence of diffusion coefficients [37,40,64,75] is expressed by   P0 T n Dab ¼ D0ab ð17Þ P T0 with the standard values P0 ¼1 bar, T0 ¼273 K, the average exponent n ¼1.76 and the standard binary diffusion coefficients (at P0 ¼1 bar and T0 ¼273 K) D0ab of Ref. [76]: D0S2 =I2 ¼ 0:031 cm2 =s,

D0S2 =I ¼ 0:052 cm2 =s,

D0S2 =Ar ¼ 0:094cm2 =s:

These values are comparable to those in Ref. [25]. Also from that Ref. [76], the standard binary values of CdI2 in S2, I2, I, and in Ar are taken to calculate the other ones using Graham’s law [41]  0:5 Mg , ð18Þ D0ab ¼ D0ag Mb where Mb and Mg are the corresponding molar masses. In certain circumstances, Graham’s law can provide diffusion coefficients closer to empirical values ( o6%) than the hard-sphere model [41]. It is an additional advantage that the iodide molecules ZnI2 and CdI2 are similar and that the molecular masses are not extremely different [33]. Thus D0ZnI2 =I2 ffi0:013cm2 =s, D0ZnI2 =Ar ffi 0:055 cm2 =s,

D0ZnI2 =I ffi 0:026cm2 =s, and D0S2 =ZnI2  D0ZnI2 =S2 ffi0:029 cm2 =s:

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R. Lauck / Journal of Crystal Growth 312 (2010) 3642–3649

3.3. Calculation of partial pressures and transport rate The ZnS–I system is described by two reactions [25,27] ZnSðsÞ þI2 ðgÞ ¼ ZnI2 ðgÞ þ 12S2 ðgÞ,

ð19Þ

and I2 ðgÞ ¼ 2IðgÞ, and the corresponding equilibrium constants [25,27,77] pffiffiffiffiffiffiffi pZnI2 pS2 K1 ðTÞ ¼ p I2

ð20Þ

ð21Þ

and K2 ðTÞ ¼

p2I , pI2

which are calculated using the relation   DH DS þ , KðTÞ ¼ exp  RT R

ð22Þ

ð23Þ

and the thermochemical parameters from Ref. [78], where DH and DS are the changes of the enthalpy and the entropy of reactions, respectively. R is the gas constant, and T is the absolute temperature. The combination of Eqs. (2), (8), (10), (13), and (21)–(23) results in the equation 0 ¼ 2K1 ðPpð3 þ2qÞÞ4ð1 þqÞp1:5 ð2 þ qÞð2K1 K2 p1:5 Þ0:5 ,

ð24Þ

with p  pS2 and q  poAr =poiod . The constant total pressure P is determined by application of those equations above to the average temperature (Tc +Ts)/2 at half transport distance, under the assumption of approximately linear partial pressure gradients (Section 3.1). The solution of Eq. (24) by Newton’s method using EXCEL 97 yields the equilibrium pressures of sulfur (S2) at crystal surface (Tc) and source (Ts). Consequently, also the partial pressures of the other species are calculated from Eqs. (8), (10), (13), (21), and (22). The Pe´clet number of Eqs. (6b), (9a) and (9b), respectively, is illustrated in Fig. 4 as a function of the initial iodine concentration for several argon pressures. In all cases, even for poAr ¼ 0, NPe is much less than one, and therefore the partial pressure gradients are nearly linear [67]. The growth-relevant average diffusion coefficient D, defined by Eq. (15), is derived using the Eqs. (15)–(17) and the binary standard values of Section 3.2. In the same way, the corresponding average standard value D0, defined at 1 bar and 273 K, is deduced from the standard binary diffusion coefficients. On the other hand, D0 can be determined vice versa through the universally valid Eq. (17) if the total pressure, the temperature, and the coefficient D are already known. Both D and D0 are given as

Fig. 4. The Pe´clet number of the chemical vapor transport of ZnS as a function of the initial iodine concentration co for different argon pressures poAr . Temperatures Ts ¼1050 K, Tc ¼1000 K.

Fig. 5. Growth-relevant average diffusion coefficient in the multi-component ZnS–I system as a function of the initial iodine concentration co for different argon pressures poAr . (a) Temperature and pressure dependent diffusion coefficient D at 1025 K, (b) the corresponding pseudo-standard diffusion coefficient D0.

functions of the initial iodine concentration for several argon pressures (Fig. 5). In the case of poAr ¼ 0, the pressure-dependent curve of Fig. 5a is similar to conventional ones [33,49]. Unlike the standard binary coefficients, the so-called standard coefficient D0 cannot be considered as constant, but it varies by a factor of 3 (Fig. 5b), because the diffusivities of the species are significantly different. D0 increases with increasing content of argon (low atomic mass), and decreases with increasing concentration of iodine (large and heavy molecule [25]). However, D0 is independent of the total pressure but it depends on the mole fractions due to Eq. (16). Therefore, D0 can be considered as a pseudo-standard coefficient. The ordinate intersection point (at the maximum value of D0 ¼0.0635 cm2/s) of the four argon related curves in Fig. 5b corresponds to the average standard coefficient of ZnI2 and S2 in pure argon gas. In the literature, a curve similar to Fig. 5b, but without the effect of argon, is given for the ZnTe–iodine system [54]. Specific values of mean [25,50,54] and effective [79] standard coefficients D0 (at 1 bar and 273 K) of different argon-free systems lie in the range of 0.01–0.04 cm2/s, which is comparable to the present calculation. In addition, the partial pressure distribution is calculated in a similar way as described in Refs. [40,57–62,80]. For the present system, the combination of Eqs. (6a) and (6b) with Eq. (1), and the integration from x ¼0 (source) to x¼X ( rL) yield   RTL XNPe XNPe pa ðXÞ ¼ ja þpa ð0Þexp : ð25Þ 1exp DNPe L L An example of partial pressure profiles is illustrated in Fig. 6. The linearity of the gradients corresponds to a Pe´clet number of NPe ¼ 0.044.

R. Lauck / Journal of Crystal Growth 312 (2010) 3642–3649

Fig. 6. Partial pressure gradients for an example of iodine transport of ZnS at Ts ¼1050 K, and Tc ¼1000 K. Transport rate J¼ 3.52 mg/h cm2, total pressure P ¼75 mbar, poAr ¼ 30 mbar, iodine concentration co ¼ 0.1 mg/cm3, diffusion coefficient D (1025 K)¼ 4.2 cm2/s, standard diffusion coefficient D0 ¼0.031 cm2/s, Pe´clet number NPe ¼0.044.

The transport rate J (in mg/hcm2) is calculated from Eq. (12), making use of Eq. (7a) or (7b), and the calculated partial pressures above. Thus, the transport rate is given as a function of the iodine concentration for several argon pressures, as shown in the linear diagram of Fig. 7. At high argon pressures, the transport rate increases continuously with increasing iodine concentration. At low argon pressures and at low iodine concentration, it increases sharply up to a maximum value, especially in the case without inert gas. This maximum is evident in the literature [19,24,25,51,52]. Subsequently, the rate decreases in this case with increasing iodine concentration. The reason for this behavior is the interplay between the chemical reaction as the driving force and the physical effect of diffusion reducing the transport rate with increasing total vapor pressure, i.e. with increasing iodine vapor pressure, too. In all cases, the rate becomes less sensitive to high iodine concentration. Therefore, a high iodine concentration is not required to gain a considerable transport rate.

4. Comparison between theory and experiment The experimental transport rate is given as the transported mass divided by the growth period and the cross section surface. The transport rates of several runs in comparison with the calculated ones are depicted in a linear diagram accentuating the deviations (Fig. 8). In most cases, the experimental results are fairly in line with the theoretical predictions, especially in comparison to the literature. It is evident that the influence of the different isotope masses taken into account is small. The maximum difference of the masses is 6% (68Zn to 64Zn, and 34S to 32 S), and that of the corresponding diffusion coefficients is 3%. Furthermore, the deviations of the two parameters weaken each other in Eq. (12). It should be noted that the present growth experiments were performed within the framework of a service order to produce several separated crystals with different isotopic compositions along a temperature gradient, and not to analyze the chemical vapor transport. This circumstance implicates the disadvantage of higher uncertainties of the growth parameters, such as the iodine concentration, the growth temperature, the growth period if the source was depleted, and in particular the effective transport distance due to the spatial distribution of the crystals (Fig. 1d). The calculated data also involve error bars because they are based

3647

on experimental parameters. Unfortunately, the experimental results cannot be illustrated in a conventional manner like the theoretical curves of Fig. (7), because the corresponding growth parameters are different among each other (Table 1). A noticeable deviation of the experimental value by 40% higher than the theoretical one is present for the experiment No. 7, and could be caused possibly by thermal convection [19,23,28,30,33,50,53,57,65,81]. Upon opening the ampoule in that case, the total pressure at room temperature appeared to be higher than one bar by reason of unknown vapor. If the experimental rate is clearly lower than the calculated one, as in experiments 10 and 11, also foreign gases could be the reason [52,71,76,82–84], increasing the total vapor pressure and decreasing the transport rate. In addition, a foreign vapor can also affect the chemical equilibrium. When the ampoule of experiment 11 was opened, a distinct smell of hydrogen sulfide was noticed. On the other hand, it should be mentioned that such deviations would be almost unremarkable in a usual logarithmic diagram, except for the fifth experiment. In the latter case the discrepancy (Jexp is only 15% of Jcalc) is so extreme that a chemical interference may be assumed in terms of zinc excess originated from an incomplete synthesis. In this case, the sublimation of zinc had been too fast, and small droplets (like in Fig. 2a) were observed afterward. In an additional special experiment, it was noticed that the intentional addition of elemental zinc impeded the transport.

Fig. 7. Transport rate of ZnS as a function of the initial iodine concentration co for different argon pressures poAr . Transport distance L¼20 cm, temperatures Ts ¼ 1050 K, Tc ¼ 1000 K.

Fig. 8. Comparison of experimental transport rates (triangle) with theoretical ones (circle), in order of increasing theoretical values. ZnS crystals have been grown with the natural isotopic composition (experiment No. 1, 2, 8, 9, and 10), and with the tailored isotopic compositions 64Zn32S (No. 11), 64Zn34S (No. 3, 5, 6), 68Zn32S (No. 4), and 68Zn32S0.5 34S0.5 (No. 7).

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The non-stoichiometry of the ZnS–I system will be treated in a following paper (part II). In the literature, it is not unusual to find considerable discrepancies and variations between experimental and theoretical transport rates, very often attenuated by a logarithmic presentation [21,23,25,50,52,57]. Convincing explanations have been given in a few cases. It has been suggested that inaccuracy in thermochemical data [25] and surface interactions [21,85] can play a role. Pa"osz has explained the drastic suppression of the growth rate of ZnS by the presence of ZnSO4 in the commercial material [24]. However, in the present experiments possible oxides were separated after sublimation of zinc. Besides unknown vapors and non-stoichiometry, inaccurate values of the diffusion coefficient or the application of an unsuitable transport approach could give reasons for driftdeviations. The author of the present publication also applied another more simplified model (i.e. ignored iodine dissociation, constant D0 ¼ 0.05 cm2/s, pAr and piod are constant along the transport distance), which resulted in evidently higher theoretical transport rates, by a factor of about 2, than in the present work.

5. Summary After purification of the elements by sublimation, the synthesis of ZnS was favored by the formation of a large-area zinc layer and governed by the solid-state diffusion of the zinc atoms through the growing passivation layer. The method of the ‘‘all-in-one’’ ampoule is advantageous in the case of limited amounts of substances. Chemical vapor transport with iodine was successfully applied to grow faceted cubic crystals with tailored isotopic compositions along a temperature gradient from small quantities of source material. These crystals have been suitable for different measurements (Raman scattering, photoluminescence, heat capacity). Consistent with the Stefan–Maxwell equations, the present traceable modeling of the diffusion-limited transport, including iodine dissociation, results in an explicit and general expression of the transport rate in terms of the Pe´clet number. The derivation of a second iodine-based expression of the Pe´clet number allows correlation of the partial pressure gradient of argon with that of the total iodine content, and simplifies the determination of the partial pressures in the present work. A special averaging of the effective diffusion coefficients in the multi-component system results in a growth-relevant expression. The corresponding pseudo-standard coefficient is not constant but varies by a factor of 3. Both these relevant diffusion coefficients and the transport rate are shown as functions of the initial iodine concentration and the argon pressure, i.e. experimentally adjustable and accessible parameters. In addition, the Pe´clet number is illustrated as a function of these variables. The experimental transport rates depicted in a linear diagram are in reasonable agreement with the stoichiometric model and justify the presented diffusion approach. Exceptions are attributed to foreign vapors and zinc excess, respectively.

Acknowledgements The author would like to thank primarily M. Cardona for the request to grow ZnS crystals with different isotopic compositions, and for his friendly promotion and sympathetic sponsorship of the present work. He is very grateful to M. Konuma, R.K. Kremer, and S. Bayrakci for stimulating discussions and critical reading of the manuscript, and to C. Busch for helpful assistance in the application of EXCEL. Thanks are also due to K.-H. Bender for

drawing Fig. 1. In addition, he would express his thanks to S. Lauck for the essential PC-support, and for the invitation to the 3. piano concert of S. Rachmaninoff performed by B. Glemser and the ‘‘Badische Philharmonie Pforzheim’’ where the idea for the formulation of Eq. (7b) was born. The author would like to apologize profusely to the reader if any citation is ignored.

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