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The growth of CdTe bulk crystals using the multi-tube physical vapour transport system
Journal of Crystal Growth 237–239 (2002) 1711–1715
The growth of CdTe bulk crystals using the multi-tube physical vapour transport system H.K. Sanghera, B.J. Cantwell, N.M. Aitken, A.W. Brinkman* Science Laboratories, Department of Physics, University of Durham, Rochester Building, South Road, Durham DH1 3LE, UK
Abstract The control of vapour transport is central to the growth of any crystal from the gas phase and, therefore, a novel multi-tube physical vapour transport (MTPVT) system has been developed at the University of Durham which addresses this requirement directly. This paper reports recent work including some preliminary modelling of the transport in the MTPVT system. A capillary tube is used in the MTPVT system as a flow restrictor and the mass transport rate with respect to pressure drop across the capillary calibrated. The mass of CdTe transported from the source charge to growing crystal through the crossmember capillary is investigated over two separate growth runs of the MTPVT system and the results compared with the values predicted by modelling. The CdTe crystals grown using the MTPVT system have been quality tested using defect revealing etching, photoluminescence measurements, and X-ray diffraction analysis. r 2002 Elsevier Science B.V. All rights reserved. PACS: 81.10.Bk; 06.60+W; 07.05T; 47.60+i Keywords: A2. Growth from vapor; A2. Single crystal growth; B2. Semiconducting cadmium compounds
1. Introduction In comparison with other common semiconductors, CdTe is an intrinsically brittle and mechanically weak substance, with a low stacking fault energy and critical shear stress [1,2]. As a result, relatively small stresses applied to the growing crystal can cause defects which propagate and multiply easily through the lattice, producing dislocations, twins, subgrains, etc. Clearly, there *Corresponding author. Tel.: +44-191-374-2396; fax: +44191-3747358. E-mail address: [email protected] (A.W. Brinkman).
are significant benefits to be gained by growing the crystal at as low a temperature as possible. Elimination of stress applied to the growing crystal is also essential in the production of high quality material. Oxide layers inevitably form on CdTe surfaces exposed to air and there is some evidence that these give rise to some degree of adhesion, or wetting, between the CdTe and ampoule walls [3,4]. The associated differential contraction between the container and crystal on cooling results in tensile stress (the linear thermal expansion coefficient of CdTe is B5.3 1061C1 over the temperature range 20–8001C [2], that of silica is 5.5 1071C1). It has also been shown that a radial temperature difference of just 21C within the
0022-0248/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 1 ) 0 2 3 4 0 - 5
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H.K. Sanghera et al. / Journal of Crystal Growth 237–239 (2002) 1711–1715
crystal can cause thermal stress higher than the critical resolved shear stress [5].
2. Experimental procedure The multi-tube physical vapour transport (MTPVT) crystal growth system is essentially a combination of the Rosenberger capillary restricted transport [3] and Markov designs [6,7] and is described in detail in Ref. [8]. In essence, the growth system comprises two independent vertical three zone furnaces as source and growth regions which are connected by a horizontal optically heated crossmember. The crossmember incorporates a calibrated capillary flow restrictor to regulate the throughput. Vapour pressures on either side of the capillary are measured in situ by optical absorption [9] enabling the mass transport to be monitored directly during growth. To date, two designs of growth tube have been used. In the first arrangement, there is a fixed annulus of B50 mm between a 49 mm diameter cylindrical silica pedestal and the quartz wall of the tube. In this classic Markov type configuration, a 49 mm diameter seed sits on the pedestal which is supported in turn, on three permanent indentations in the wall of the tube. In the second design, a 29 mm diameter seed sits on a quartz plug, which fits into a conical ground glass joint of approximately 3 cm in length. The gap between the plug and glass joint can be varied by inserting hardened, high purity platinum wires of calibrated diameter (typically between 25 and 150 mm) between the two. The narrow gap between crystal pedestal (and seed) and ampoule wall results in continuous removal of excess components, as well as any volatile impurities, away from the growing surface, greatly improving crystal purity and stoichiometry, it also helps to prevent the crystal from sticking to the ampoule as in closed ampoule techniques. Several crystal boules have been grown using both growth tube designs under a range of conditions. Growth has been carried out at source temperatures between 7501C and 8751C, and seed temperatures between 7001C and 8501C.
3. Mass transport in the MTPVT system For modelling purposes, the salient features of the MTPVT system are illustrated in Fig. 1. In principle, the growth rate, ng ; is given by the difference in throughput between the capillary and annulus (QC QA ) and is determined by the respective dimensions and the source and growth pressures, Ps and Pg : Flow through the capillary and annulus may be molecular, viscous or a combination of the two depending on the mean free path of the gas and the diameter of the capillary, or more specifically the ratio or the two. Experience with the system has shown that under normal operation, transport through the capillary is mixed (transitional), while that through the annulus is molecular [8]. The flow rate, FM ; of a gas with molecular weight M; through a capillary of radius r and length L; for a mixed flow regime is given by [10] mol pr4 Pm DP FM ¼ t 8ZLRT p2 r3 DP 2 1 ; ð1Þ þ ð8pMRTÞ1=2 L f where R is the universal gas constant, T the absolute temperature, DP the pressure difference between the ends of the capillary, Z the fluid viscosity and Pm mean pressure across the capillary.
Capillary flow PS Sublimation
QC Pg ng
Growth
QA
Annulus flow
Fig. 1. Schematic diagram of the MTPVT system for modelling.
H.K. Sanghera et al. / Journal of Crystal Growth 237–239 (2002) 1711–1715
The friction factor, f ; is related to the fraction of gas molecules diffusely reflected from the capillary walls. In a subsidiary experiment, the flow properties of the crossmember were determined using several inert gases, yielding a mean value for f of 1.176 102. Eq. (1) can be expressed in terms of Ps and Pg respectively, by noting that DP ¼ Ps Pg ; and writing pr4 ; Kv ¼ 16ZLR
Pm ¼ ðPs þ Pg Þ=2 p 2 r3 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 : Km ¼ ð8pMRÞL f
ð2Þ
ð3Þ
The number of moles of cadmium vapour transported in a time tmax (i.e. the growth time) is then given by Z pffiffiffiffiffi 1 tmax n Kv P2s þ Km Ps Tc FM ðmolÞ ¼ Tc 0 pffiffiffiffiffio Kv P2g Km Pg Tc dt; ð4Þ where Tc is the temperature of the crossmember, and flow down the annulus has been assumed to be small. Eq. (4) indicates that transport and therefore growth rate depend quadratically on the source and growth side pressures. The sublimation of CdTe is governed by the mass action law: 1=2
PCd PTe2 ¼ KðTÞ;
ð5Þ
where PCd and PTe2 are the Cd and Te2 partial pressures, respectively and KðTÞ is the equilibrium constant. It follows from Eq. (5) that the Cd:Te2 partial pressure ratio for a stoichiometric vapour should be 2, in which case the vapour pressure of Cd over CdTe at the growth temperature, Tg ; (referred to the crystalFvapour interface) is given by the equation [11] PCd ¼ 1:26 105
2=3 6:9446 104 45:842Tg exp ; 1:9872Tg ð6Þ where PCd is in Pascal. This expression gives the equilibrium Cd partial pressure above CdTe, however, in semi-open vapour growth the conditions are not necessarily at equilibrium.
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Total transport may be estimated by assuming that the crystal grows along a linear temperature gradient, i.e. Tg ¼ T0 þ ct where
c¼
Tmax T0 ; tmax
ð7Þ
where T0 is the initial growth temperature and Tmax the final growth temperature. The results from some representative calculations are shown in Fig. 2 for a range of typical growth conditions. Unsurprisingly the model indicates that molar transport varies smoothly with T0 decreasing progressively as the difference between initial and final temperatures is reduced.
4. Results and discussion Two separate growth runs were selected for study, representing each of the two geometries. Run 41 was conducted using the first of the growth tube designs. Weight loss measurements showed that the total mass of cadmium telluride transported was 45.0 g, of which 42.7 g condensed on the seed, and the balance, it was assumed, was lost through the annulus. This represents a B5% loss of material. For the growth conditions, the model predicted that about 0.4 mol should have been transported through the crossmember flow restrictor. In the event, transport was only B0.2 mol, about half that expected. This discrepancy is probably a consequence of neglecting annular flow in Eq. (4) and of the implicit assumption that the partial pressures over the growing crystal were both stoichiometric and at equilibrium. Since the annulus flow was molecular, this was not the case, and the crystal was growing in a Te-rich environment. Run 45 was conducted using the second of the growth tube arrangements and the growth was carried out at higher temperatures. The total mass of CdTe transported was 87.0 g, with 84.9 g condensing on the growth side of the ampouleFthis included the small amount of material which condensed on the growth side viewing window. The number of moles of CdTe transported was 0.36. This compares with a predicted value of 0.39 mol, representing a difference of 7.9%
H.K. Sanghera et al. / Journal of Crystal Growth 237–239 (2002) 1711–1715
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Dependence of total transport on initial temperature 1.26 1.24 1003 K
Molar mass transported Fm
1.22 1023 K
1.2 1.18 1043 K
1.16 1.14
1063 K
1.12 1073 K
1.1 1.08 940
960
980
1000 1020 Initial Temperature, T0 ( K )
1040
1060
1080
Fig. 2. Calculated total molar transport as a function of initial and final growth temperatures, for a growth time of 44 h.
The CdTe crystals grown using the MTPVT system have been quality tested using defect revealing etching, photoluminescence measurements, and X-ray diffraction analysis [12]. The results of this study were encouraging, with etch pit densities comparable with melt grown material (6 104 cm2) and relatively low strain in the crystal, as shown by X-ray analysis where y 2y scans showed a broadening of 2 arcsec, corresponding to strains of the order 105. Perhaps more significantly the level of strain and dislocation density reduced with growth direction, indicating an improvement in crystal quality with growth away from the cadmium zinc (4%) telluride seed where strain levels would be expected to be high due to the lattice mismatch. The photoluminescence spectra were dominated by a narrow acceptor bound exciton peak in the near band edge region [13].
model, for example to allow for non-stoichiometric partial pressure ratios, flow down the annulus, etc., this model does provide a basis for the selection of a transport rate for a given growth temperature. The use of a semi-open restricted flow configuration offers significant advantages for the growth of ternary and multinary compounds, since these may be grown from individual binary sources each with its own flow restrictor. Problems arising from the progressive enrichment of a ternary source are thus avoided. Work in Durham is presently directed towards developing the MTPVT system for the growth of (Cd, Zn)Te.
Acknowledgements The authors wish to acknowledge the financial support of the UK Engineering and Physical Sciences Research Council.
5. Conclusion There is reasonable agreement between measured transport rates and those predicted from Eq. (4) considering the approximations made. Although there is need for refinement of the
References [1] R. Balasubramanian, R.W. Wilkox, Mat. Sci. Eng. B16 (1993) 1.
H.K. Sanghera et al. / Journal of Crystal Growth 237–239 (2002) 1711–1715 [2] H. Hartmann, R. Mach, B. Selle, Wide Gap 2–6 Compounds as Electronic Materials, in: E. Kaldis (Ed.), Current Topics in Materials Science, North-Holland, Amsterdam, 1982. [3] F. Rosenberger, M. Banish, M.B. Duval, NASA Technical Memorandum 103786, 1991. [4] R.K. Bagai, R.D.S. Yadava, B.S. Sundersheshu, G.L. Seth, M. Anandad, W.N. Borle, J. Crystal Growth 139 (1994) 258. [5] J.C. Alabert, Optical vapour pressure monitoring and mass transport control during bulk CdTe crystal growth in a novel multi-tube PVT system, Appendix A, Ph.D. Thesis, University of Durham, UK, 1998. [6] E.V. Markov, A.A. Davydov, Neo. Matter 11 (1975) 1755. [7] E.V. Markov, A.A. Davydov, Neo. Matter 7 (1971) 575.
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[8] J.T. Mullins, J. Carles, N.M. Aitken, A.W. Brinkman, J. Crystal Growth 208 (2000) 211. [9] J. Carles, J.T. Mullins, A.W. Brinkman, J. Crystal Growth 174 (1997) 740. [10] R.G. Livesey, Flow through tubes and orifices, in: J.M. Lafferty (Ed.), Foundations of Vacuum Science and Technology, Wiley, New York, 1998 (Chapter 2). [11] B. De-Largy, A. Finch, P.J. Gardner, J. Crystal Growth 61 (1983) 194. [12] N.M. Aitken, M.D.G. Potter, D.J. Buckley, J.T. Mullins, J. Carles, D.P. Halliday, K. Durose, B.K. Tanner, A.W. Brinkman, J. Crystal Growth 198/199 (1999) 984. [13] D.P. Halliday, M.D.G. Potter, J.T. Mullins, A.W. Brinkman, J. Crystal Growth 220 (2000) 30.
The growth of CdTe bulk crystals using the multi-tube physical vapour transport system H.K. Sanghera, B.J. Cantwell, N.M. Aitken, A.W. Brinkman* Science Laboratories, Department of Physics, University of Durham, Rochester Building, South Road, Durham DH1 3LE, UK
Abstract The control of vapour transport is central to the growth of any crystal from the gas phase and, therefore, a novel multi-tube physical vapour transport (MTPVT) system has been developed at the University of Durham which addresses this requirement directly. This paper reports recent work including some preliminary modelling of the transport in the MTPVT system. A capillary tube is used in the MTPVT system as a flow restrictor and the mass transport rate with respect to pressure drop across the capillary calibrated. The mass of CdTe transported from the source charge to growing crystal through the crossmember capillary is investigated over two separate growth runs of the MTPVT system and the results compared with the values predicted by modelling. The CdTe crystals grown using the MTPVT system have been quality tested using defect revealing etching, photoluminescence measurements, and X-ray diffraction analysis. r 2002 Elsevier Science B.V. All rights reserved. PACS: 81.10.Bk; 06.60+W; 07.05T; 47.60+i Keywords: A2. Growth from vapor; A2. Single crystal growth; B2. Semiconducting cadmium compounds
1. Introduction In comparison with other common semiconductors, CdTe is an intrinsically brittle and mechanically weak substance, with a low stacking fault energy and critical shear stress [1,2]. As a result, relatively small stresses applied to the growing crystal can cause defects which propagate and multiply easily through the lattice, producing dislocations, twins, subgrains, etc. Clearly, there *Corresponding author. Tel.: +44-191-374-2396; fax: +44191-3747358. E-mail address: [email protected] (A.W. Brinkman).
are significant benefits to be gained by growing the crystal at as low a temperature as possible. Elimination of stress applied to the growing crystal is also essential in the production of high quality material. Oxide layers inevitably form on CdTe surfaces exposed to air and there is some evidence that these give rise to some degree of adhesion, or wetting, between the CdTe and ampoule walls [3,4]. The associated differential contraction between the container and crystal on cooling results in tensile stress (the linear thermal expansion coefficient of CdTe is B5.3 1061C1 over the temperature range 20–8001C [2], that of silica is 5.5 1071C1). It has also been shown that a radial temperature difference of just 21C within the
0022-0248/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 1 ) 0 2 3 4 0 - 5
1712
H.K. Sanghera et al. / Journal of Crystal Growth 237–239 (2002) 1711–1715
crystal can cause thermal stress higher than the critical resolved shear stress [5].
2. Experimental procedure The multi-tube physical vapour transport (MTPVT) crystal growth system is essentially a combination of the Rosenberger capillary restricted transport [3] and Markov designs [6,7] and is described in detail in Ref. [8]. In essence, the growth system comprises two independent vertical three zone furnaces as source and growth regions which are connected by a horizontal optically heated crossmember. The crossmember incorporates a calibrated capillary flow restrictor to regulate the throughput. Vapour pressures on either side of the capillary are measured in situ by optical absorption [9] enabling the mass transport to be monitored directly during growth. To date, two designs of growth tube have been used. In the first arrangement, there is a fixed annulus of B50 mm between a 49 mm diameter cylindrical silica pedestal and the quartz wall of the tube. In this classic Markov type configuration, a 49 mm diameter seed sits on the pedestal which is supported in turn, on three permanent indentations in the wall of the tube. In the second design, a 29 mm diameter seed sits on a quartz plug, which fits into a conical ground glass joint of approximately 3 cm in length. The gap between the plug and glass joint can be varied by inserting hardened, high purity platinum wires of calibrated diameter (typically between 25 and 150 mm) between the two. The narrow gap between crystal pedestal (and seed) and ampoule wall results in continuous removal of excess components, as well as any volatile impurities, away from the growing surface, greatly improving crystal purity and stoichiometry, it also helps to prevent the crystal from sticking to the ampoule as in closed ampoule techniques. Several crystal boules have been grown using both growth tube designs under a range of conditions. Growth has been carried out at source temperatures between 7501C and 8751C, and seed temperatures between 7001C and 8501C.
3. Mass transport in the MTPVT system For modelling purposes, the salient features of the MTPVT system are illustrated in Fig. 1. In principle, the growth rate, ng ; is given by the difference in throughput between the capillary and annulus (QC QA ) and is determined by the respective dimensions and the source and growth pressures, Ps and Pg : Flow through the capillary and annulus may be molecular, viscous or a combination of the two depending on the mean free path of the gas and the diameter of the capillary, or more specifically the ratio or the two. Experience with the system has shown that under normal operation, transport through the capillary is mixed (transitional), while that through the annulus is molecular [8]. The flow rate, FM ; of a gas with molecular weight M; through a capillary of radius r and length L; for a mixed flow regime is given by [10] mol pr4 Pm DP FM ¼ t 8ZLRT p2 r3 DP 2 1 ; ð1Þ þ ð8pMRTÞ1=2 L f where R is the universal gas constant, T the absolute temperature, DP the pressure difference between the ends of the capillary, Z the fluid viscosity and Pm mean pressure across the capillary.
Capillary flow PS Sublimation
QC Pg ng
Growth
QA
Annulus flow
Fig. 1. Schematic diagram of the MTPVT system for modelling.
H.K. Sanghera et al. / Journal of Crystal Growth 237–239 (2002) 1711–1715
The friction factor, f ; is related to the fraction of gas molecules diffusely reflected from the capillary walls. In a subsidiary experiment, the flow properties of the crossmember were determined using several inert gases, yielding a mean value for f of 1.176 102. Eq. (1) can be expressed in terms of Ps and Pg respectively, by noting that DP ¼ Ps Pg ; and writing pr4 ; Kv ¼ 16ZLR
Pm ¼ ðPs þ Pg Þ=2 p 2 r3 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 : Km ¼ ð8pMRÞL f
ð2Þ
ð3Þ
The number of moles of cadmium vapour transported in a time tmax (i.e. the growth time) is then given by Z pffiffiffiffiffi 1 tmax n Kv P2s þ Km Ps Tc FM ðmolÞ ¼ Tc 0 pffiffiffiffiffio Kv P2g Km Pg Tc dt; ð4Þ where Tc is the temperature of the crossmember, and flow down the annulus has been assumed to be small. Eq. (4) indicates that transport and therefore growth rate depend quadratically on the source and growth side pressures. The sublimation of CdTe is governed by the mass action law: 1=2
PCd PTe2 ¼ KðTÞ;
ð5Þ
where PCd and PTe2 are the Cd and Te2 partial pressures, respectively and KðTÞ is the equilibrium constant. It follows from Eq. (5) that the Cd:Te2 partial pressure ratio for a stoichiometric vapour should be 2, in which case the vapour pressure of Cd over CdTe at the growth temperature, Tg ; (referred to the crystalFvapour interface) is given by the equation [11] PCd ¼ 1:26 105
2=3 6:9446 104 45:842Tg exp ; 1:9872Tg ð6Þ where PCd is in Pascal. This expression gives the equilibrium Cd partial pressure above CdTe, however, in semi-open vapour growth the conditions are not necessarily at equilibrium.
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Total transport may be estimated by assuming that the crystal grows along a linear temperature gradient, i.e. Tg ¼ T0 þ ct where
c¼
Tmax T0 ; tmax
ð7Þ
where T0 is the initial growth temperature and Tmax the final growth temperature. The results from some representative calculations are shown in Fig. 2 for a range of typical growth conditions. Unsurprisingly the model indicates that molar transport varies smoothly with T0 decreasing progressively as the difference between initial and final temperatures is reduced.
4. Results and discussion Two separate growth runs were selected for study, representing each of the two geometries. Run 41 was conducted using the first of the growth tube designs. Weight loss measurements showed that the total mass of cadmium telluride transported was 45.0 g, of which 42.7 g condensed on the seed, and the balance, it was assumed, was lost through the annulus. This represents a B5% loss of material. For the growth conditions, the model predicted that about 0.4 mol should have been transported through the crossmember flow restrictor. In the event, transport was only B0.2 mol, about half that expected. This discrepancy is probably a consequence of neglecting annular flow in Eq. (4) and of the implicit assumption that the partial pressures over the growing crystal were both stoichiometric and at equilibrium. Since the annulus flow was molecular, this was not the case, and the crystal was growing in a Te-rich environment. Run 45 was conducted using the second of the growth tube arrangements and the growth was carried out at higher temperatures. The total mass of CdTe transported was 87.0 g, with 84.9 g condensing on the growth side of the ampouleFthis included the small amount of material which condensed on the growth side viewing window. The number of moles of CdTe transported was 0.36. This compares with a predicted value of 0.39 mol, representing a difference of 7.9%
H.K. Sanghera et al. / Journal of Crystal Growth 237–239 (2002) 1711–1715
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Dependence of total transport on initial temperature 1.26 1.24 1003 K
Molar mass transported Fm
1.22 1023 K
1.2 1.18 1043 K
1.16 1.14
1063 K
1.12 1073 K
1.1 1.08 940
960
980
1000 1020 Initial Temperature, T0 ( K )
1040
1060
1080
Fig. 2. Calculated total molar transport as a function of initial and final growth temperatures, for a growth time of 44 h.
The CdTe crystals grown using the MTPVT system have been quality tested using defect revealing etching, photoluminescence measurements, and X-ray diffraction analysis [12]. The results of this study were encouraging, with etch pit densities comparable with melt grown material (6 104 cm2) and relatively low strain in the crystal, as shown by X-ray analysis where y 2y scans showed a broadening of 2 arcsec, corresponding to strains of the order 105. Perhaps more significantly the level of strain and dislocation density reduced with growth direction, indicating an improvement in crystal quality with growth away from the cadmium zinc (4%) telluride seed where strain levels would be expected to be high due to the lattice mismatch. The photoluminescence spectra were dominated by a narrow acceptor bound exciton peak in the near band edge region [13].
model, for example to allow for non-stoichiometric partial pressure ratios, flow down the annulus, etc., this model does provide a basis for the selection of a transport rate for a given growth temperature. The use of a semi-open restricted flow configuration offers significant advantages for the growth of ternary and multinary compounds, since these may be grown from individual binary sources each with its own flow restrictor. Problems arising from the progressive enrichment of a ternary source are thus avoided. Work in Durham is presently directed towards developing the MTPVT system for the growth of (Cd, Zn)Te.
Acknowledgements The authors wish to acknowledge the financial support of the UK Engineering and Physical Sciences Research Council.
5. Conclusion There is reasonable agreement between measured transport rates and those predicted from Eq. (4) considering the approximations made. Although there is need for refinement of the
References [1] R. Balasubramanian, R.W. Wilkox, Mat. Sci. Eng. B16 (1993) 1.
H.K. Sanghera et al. / Journal of Crystal Growth 237–239 (2002) 1711–1715 [2] H. Hartmann, R. Mach, B. Selle, Wide Gap 2–6 Compounds as Electronic Materials, in: E. Kaldis (Ed.), Current Topics in Materials Science, North-Holland, Amsterdam, 1982. [3] F. Rosenberger, M. Banish, M.B. Duval, NASA Technical Memorandum 103786, 1991. [4] R.K. Bagai, R.D.S. Yadava, B.S. Sundersheshu, G.L. Seth, M. Anandad, W.N. Borle, J. Crystal Growth 139 (1994) 258. [5] J.C. Alabert, Optical vapour pressure monitoring and mass transport control during bulk CdTe crystal growth in a novel multi-tube PVT system, Appendix A, Ph.D. Thesis, University of Durham, UK, 1998. [6] E.V. Markov, A.A. Davydov, Neo. Matter 11 (1975) 1755. [7] E.V. Markov, A.A. Davydov, Neo. Matter 7 (1971) 575.
1715
[8] J.T. Mullins, J. Carles, N.M. Aitken, A.W. Brinkman, J. Crystal Growth 208 (2000) 211. [9] J. Carles, J.T. Mullins, A.W. Brinkman, J. Crystal Growth 174 (1997) 740. [10] R.G. Livesey, Flow through tubes and orifices, in: J.M. Lafferty (Ed.), Foundations of Vacuum Science and Technology, Wiley, New York, 1998 (Chapter 2). [11] B. De-Largy, A. Finch, P.J. Gardner, J. Crystal Growth 61 (1983) 194. [12] N.M. Aitken, M.D.G. Potter, D.J. Buckley, J.T. Mullins, J. Carles, D.P. Halliday, K. Durose, B.K. Tanner, A.W. Brinkman, J. Crystal Growth 198/199 (1999) 984. [13] D.P. Halliday, M.D.G. Potter, J.T. Mullins, A.W. Brinkman, J. Crystal Growth 220 (2000) 30.