Skip to main content

Defect Formation During Crystal Growth from the Melt

  • Chapter
Book cover Springer Handbook of Crystal Growth

Part of the book series: Springer Handbooks ((SHB))

Abstract

This chapter gives an overview of the important defect types and their origins during bulk crystal growth from the melt. The main thermodynamic and kinetic principles are considered as driving forces of defect generation and incorporation, respectively. Results of modeling and practical in situ control are presented. Strong emphasis is given to semiconductor crystal growth since it is from this class of materials that most has been first learned, the resulting knowledge then having been applied to other classes of material.

The treatment starts with zero-dimensional defect types, i.e., native and extrinsic point defects. Their generation and incorporation mechanisms are discussed. Micro- and macrosegregation phenomena – striations and the effect of constitutional supercooling – are added. The control of dopants by using the nonconservative growth principle is considered. One-dimensional structural disturbances – dislocations and their patterning – are discussed next. The role of high-temperature dislocation dynamics for collective interactions, such as cell structuring and bunching, is shown. In a further section second-phase precipitation and inclusion trapping are discussed. The importance of in situ stoichiometry control is underlined. Finally two special defect types are treated – faceting and twinning. First the interplay between facets and inhomogeneous dopant incorporation, then main factors of twinning including melt structure are outlined.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 309.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Abbreviations

2-D:

two-dimensional

3-D:

three-dimensional

AB:

Abrahams and Burocchi

ACRT:

accelerated crucible rotation technique

AO:

acoustooptic

BGO:

Bi12GeO20

BPS:

Burton–Prim–Slichter

BPT:

bipolar transistor

COP:

crystal-originated particle

CRSS:

critical-resolved shear stress

DD:

dislocation dynamics

DSL:

diluted Sirtl with light

EPD:

etch pit density

FEC:

full encapsulation Czochralski

FET:

field-effect transistor

GGG:

gadolinium gallium garnet

GNB:

geometrically necessary boundary

HB:

horizontal Bridgman

HBT:

heterostructure bipolar transistor

HBT:

horizontal Bridgman technique

HRTEM:

high-resolution transmission electron microscopy

HWC:

hot-wall Czochralski

IDB:

incidental dislocation boundary

IDB:

inversion domain boundary

IR:

infrared

KTN:

potassium niobium tantalate

LD:

laser diode

LEC:

liquid encapsulation Czochralski

LED:

light-emitting diode

LHPG:

laser-heated pedestal growth

LPE:

liquid-phase epitaxy

LST:

laser scattering tomography

LST:

local shaping technique

MBE:

molecular-beam epitaxy

MC:

multicrystalline

MD:

misfit dislocation

MD:

molecular dynamics

ME:

melt epitaxy

ME:

microelectronics

MESFET:

metal-semiconductor field effect transistor

MMIC:

monolithic microwave integrated circuit

MOS:

metal–oxide–semiconductor

NLO:

nonlinear optic

OSF:

oxidation-induced stacking fault

PD:

Peltier interface demarcation

PD:

photodiode

PMNT:

Pb(Mg,Nb)1-x Ti x O3

PVE:

photovoltaic efficiency

PZNT:

Pb(Zn,Nb)1-x Ti x O3

RSS:

resolved shear stress

SI:

semi-insulating

ST:

synchrotron topography

TEM:

transmission electron microscopy

THM:

traveling heater method

UV:

ultraviolet

VB:

valence band

VB:

vertical Bridgman

VCZ:

vapor pressure controlled Czochralski

VGF:

vertical gradient freeze

VLS:

vapor–liquid–solid

YAG:

yttrium aluminum garnet

References

  1. H. Föll: Defects in Crystals, Hyperscript, http://www.tf.uni-kiel.de/matwis/amat/

  2. D.T.J. Hurle, P. Rudolph: A brief history of defect formation, segregation, faceting, and twinning in melt-grown semiconductors, J. Cryst. Growth 264, 550–564 (2003)

    Article  ADS  Google Scholar 

  3. R.S. Feigelson (Ed.): 50 Years Progress in Crystal Growth (Elsevier, Amsterdam 2004) p. 109

    Google Scholar 

  4. H. Pick: Festkörperphysik, Naturwissenschaft 41, 346–354 (1954), in German

    Article  ADS  Google Scholar 

  5. S. Mahajan: Defects in semiconductors and their effects on devices, Acta Mater. 48, 137–149 (2000)

    Article  Google Scholar 

  6. I. Baumann, P. Rudolph, D. Krabe, R. Schalge: Orthoscopic investigations of the axial optical and compositional homogeneity of Czochralski grown LiNbO_3 crystals, J. Cryst. Growth 128, 903–908 (1993)

    Article  ADS  Google Scholar 

  7. P. Rudolph, M. Jurisch: Bulk growth of GaAs – An overview, J. Cryst. Growth 198/199, 325–335 (1999)

    Article  ADS  Google Scholar 

  8. N. Mainzer, E. Lakin, E. Zolotoyabko: Point-defect influence on 1/f noise in HgCdTe photodiodes, Appl. Phys. Lett. 81, 763–765 (2002)

    Article  ADS  Google Scholar 

  9. T. Fukuda, P. Rudolph, S. Uda: Fiber Crystal Growth from the Melt (Springer, Berlin 2004)

    Book  Google Scholar 

  10. V. Swaminathan, A.T. Macrander: Materials Aspects of GaAs and InP Based Structures (Prentice Hall, Upper Saddle River 1991)

    Google Scholar 

  11. H. Chen, B. Raghotharmachar, W. Vetter, M. Dudley, Y. Wang, B.J. Skromme: Effects of different defect types on the performance of devices fabricated on a 4H-SiC homoepitaxial layer, Mater. Res. Soc. Symp. Proc. 911, 1–6 (2006)

    Google Scholar 

  12. S. Miyazawa: Effect of dislocations on GaAs-MESFET Threshold voltage, and growth of dislocation-free, semi-insulating GaAs, Prog. Cryst. Growth Charact. Mater. 23, 23–71 (1991)

    Article  Google Scholar 

  13. J.R. Niklas, W. Siegel, M. Jurisch, U. Kretzer: GaAs wafer mapping by microwave-detected photoconductivity, Mater. Sci. Eng. B 80, 206–209 (2001)

    Article  Google Scholar 

  14. T.E. Schlesinger, J.E. Toney, H. Yoon, E.Y. Lee, B.A. Brunett, L. Franko, R.B. James: Cadmium zinc telluride and its use as a nuclear radiation detector material, Mater. Sci. Eng. R 32, 103–189 (2001)

    Article  Google Scholar 

  15. B.G. Ivanov, M.T. Kogan, V.M. Reiterov: Small-angle disorientation in Bridgman-Stockbarger-grown lithium fluoride crystals, J. Opt. Technol. 68, 32–34 (2001)

    Article  ADS  Google Scholar 

  16. P. Sadrabadi, P. Eisenlohr, G. Wehrhan, J. Stablein, L. Parthier, W. Blum: Evolution of dislocation structure and deformation resistance in creep exemplified on single crystals of CaF_2, Mater. Sci. Eng. A 510, 46–50 (2009)

    Article  Google Scholar 

  17. H. Halloin, P. von Ballmoos, J. Evrard, G.K. Skinner, N. Abrosimov, P. Bastie, G. Di Cocco, M. George, B. Hamelin, P. Jean, J. Knödleseder, P. Laporte, C. Badenes, P. Laurent, R.K. Smither: Performance of CLAIRE, the first balloon-borne γ-ray lens telescope, Nucl. Instrum. Methods Phys. Res. A 504, 120–125 (2003)

    Article  ADS  Google Scholar 

  18. J.R. Weertman, D. Farkas, K. Hemker, H. Kung, M. Mayo, R. Mitra, H. Van Swygenhoven: Structure and mechanical behavior of bulk nanocrystalline materials, MRS Bull. 24(2), 44–50 (1999)

    Article  Google Scholar 

  19. P. Rudolph: Non-stoichiometry related defects at the melt growth of semiconductor compound crystals – A review, Cryst. Res. Technol. 38, 542–554 (2003)

    Article  Google Scholar 

  20. P. Rudolph, C. Frank-Rotsch, U. Juda, M. Naumann, M. Neubert: Studies on dislocation patterning and bunching in semiconductor compounds (GaAs), J. Cryst. Growth 265, 331–340 (2004)

    Article  ADS  Google Scholar 

  21. F.A. Kröger: The Chemistry of Imperfect Crystals (North-Holland Publ., Amsterdam 1973)

    Google Scholar 

  22. D.I. Takamura: Point defects. In: Physical Metallurgy, ed. by R.W. Cahn (North-Holland Publ., Amsterdam 1965), Chap. XIII–XX

    Google Scholar 

  23. E. Dornberger: Prediction of OFS ring dynamics and grown-in voids in Czochralski silicon crystals. Ph.D. Thesis (Universite Catholique de Louvain, Louvain-la-Neuve 1997)

    Google Scholar 

  24. K.M. Luken, R.A. Morrow: Formation energies and charge states of native defects in GaAs: A selected compilation from the literature, Semicond. Sci. Technol. 11, 1156–1158 (1996)

    Article  ADS  Google Scholar 

  25. J.L. Rouviere, Y. Kim, J. Cunningham, J.A. Rentschler, A. Bourret, A. Ourmazd: Measuring properties of point derects by electron microscopy: The Ga vacancy in GaAs, Phys. Rev. Lett. 68, 2798–2801 (1992)

    Article  ADS  Google Scholar 

  26. L. Yujie, M. Guoli, J. Wanqi: Point defects in CdTe, J. Cryst. Growth 256, 266–275 (2003)

    Article  ADS  Google Scholar 

  27. M.A. Berding, M. van Schilfgaarde, A.T. Paxton, A. Sher: Defects in ZnTe, CdTe, and HgTe: Total energy calculations, J. Vac. Sci. Technol. A 8, 1103–1107 (1990)

    Article  ADS  Google Scholar 

  28. P. Rudolph: Fundamental studies on Bridgman growth of CdTe, Prog. Cryst. Growth Charact. Mater. 29, 275–381 (1995)

    Article  Google Scholar 

  29. R. Grill, J. Franc, P. Hoeschl, E. Belas, I. Turkevych, L. Turjanska, P. Moravec: Semiinsulating CdTe, Nucl. Instrum. Methods Phys. Res. A 487, 40–46 (2002)

    Article  ADS  Google Scholar 

  30. D.T.J. Hurle: Point defects in compound semiconductors. In: Crystal Growth – From Fundamentals to Technology, ed. by G. Müller, J.-J. Metois, P. Rudolph (Elsevier, Amsterdam 2004) pp. 323–343

    Chapter  Google Scholar 

  31. P. Rudolph: Elements of thermodynamics for the understanding and design of crystal growth processes. In: Theoretical and Technological Aspects of Crystal Growth, ed. by R. Fornari, C. Paorici (Trans Tech Publications, Switzerland 1998) pp. 1–26

    Google Scholar 

  32. V.V. Voronkov, R. Falster, F. Quast: On the properties of the intrinsic point defects in silicon: A perspective from crystal growth and wafer processing, Phys. Status Solidi (b) 222, 219–244 (2000)

    Article  ADS  Google Scholar 

  33. D.T.J. Hurle: A comprehensive thermodynamic analysis of native point defect and dopant solubilities in gallium arsenide, J. Appl. Phys. 85, 6957–7022 (1999)

    Article  ADS  Google Scholar 

  34. R. Brown, D. Maroudas, T. Sinno: Modelling point defect dynamics in the crystal growth of silicon, J. Cryst. Growth 137, 12–25 (1994)

    Article  ADS  Google Scholar 

  35. E. Dornberger, J. Virbulis, B. Hanna, R. Hoelzl, E. Daub, W. von Ammon: Silicon crystals for future requirements of 300 mm wafers, J. Cryst. Growth 229, 11–16 (2001)

    Article  ADS  Google Scholar 

  36. M.A. Berding: Native defects in CdTe, Phys. Rev. 60, 8943–8950 (1999)

    Article  ADS  Google Scholar 

  37. K.A. Jackson: Liquid metals and solidification, Am. Soc. Met. (Cleveland, Ohio 1958) 174–180

    Google Scholar 

  38. A.A. Chernov: Modern Crystallography III (Springer, Berlin 1984)

    Book  Google Scholar 

  39. I.V. Markov: Crystal Growth for Beginners (World Scientific, Singapore 1995)

    Book  Google Scholar 

  40. A.A. Chernov: Notes on interface growth kinetics 50 years after Burton, Cabrera and Frank, J. Cryst. Growth 264, 499–518 (2004)

    Article  ADS  Google Scholar 

  41. K. Fujiwara, K. Nakajima, T. Ujihara, N. Usami, G. Sazaki, H. Hasegawa, S. Mizoguchi, K. Nakajima: In situ observations of crystal growth behavior of silicon melt, J. Cryst. Growth 243, 275–282 (2002)

    Article  ADS  Google Scholar 

  42. S. Arai, S. Tsukimoto, S. Muto, H. Saka: Direct observation of the atomic structure in a solid-liquid interface, Microsc. Microanal. 6, 358–361 (2000)

    ADS  Google Scholar 

  43. T. Motooka, K. Nishihira, R. Oshima, H. Nishizawa, F. Hori: Atomic diffusion at solid/liquid interface of silicon: transition layer and defect formation, Phys. Rev. B 65, 813041–813044 (2002)

    Article  Google Scholar 

  44. R. Falster, V. Voronkov: Engineering of intrinsic point defects in silicon wafers and crystals, Mater. Sci. Eng. B 73, 87–94 (2000)

    Article  Google Scholar 

  45. F. Rosenberger: Fundamentals of Crystal Growth I (Springer, Berlin 1979)

    Book  Google Scholar 

  46. S. Erdei, F.W. Ainger: Trends in the growth of stoichiometric single crystals, J. Cryst. Growth 174, 293–300 (1997)

    Article  ADS  Google Scholar 

  47. M. Jurisch, H. Wenzl: Workshop on Simulations in Crystal Growth (DGKK, Memmelsdorf 2002)

    Google Scholar 

  48. W. Dreyer, F. Duderstadt: On the modelling of semi-insulating GaAs including surface tension and bulk stresses (EMS) (Weierstraß-Institut, Berlin 2004), Treatise No. 995

    MATH  Google Scholar 

  49. P. Rudolph, F.-M. Kießling: Growth and characterization of GaAs crystals produced by the VCz method without boric oxide encapsulation, J. Cryst. Growth 292, 532–537 (2006)

    Article  ADS  Google Scholar 

  50. P. Rudolph, F.-M. Kießling: The horizontal Bridgman method, Cryst. Res. Technol. 23, 1207–1224 (1988)

    Article  Google Scholar 

  51. E. Monberg: Bridgman and related growth techniques. In: Handbook of Crystal Growth, Vol. 2a, ed. by D.T.J. Hurle (Elsevier, North-Holland 1994), Chap. 2

    Google Scholar 

  52. P. Rudolph, S. Kawasaki, S. Yamashita, S. Yamamoto, Y. Usuki, Y. Konagaya, S. Matada, T. Fukuda: Attempts to growth of undoped CdTe single crystals with high electrical resistivity, J. Cryst. Growth 161, 28–33 (1996)

    Article  ADS  Google Scholar 

  53. S. Oda, M. Yamamoto, M. Seiwa, G. Kano, T. Inoue, M. Mori, R. Shimakura, M. Oyake: Defects in and device properties of semi-insulating GaAs, Semicond. Sci. Technol. 7, 215–223 (1992)

    Article  ADS  Google Scholar 

  54. J. Nishizawa: Stoichometry control for growth of III–V crystals, J. Cryst. Growth 99, 1–8 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  55. J. Nishizawa, Y. Oyama: Stoichiometry of III–V compounds, Mater. Sci. Eng. R 12, 273–426 (1994)

    Article  Google Scholar 

  56. M. Neubert, P. Rudolph: Growth of semi-insulating GaAs crystals in low-temperature gradients by using the vapour pressure controlled Czochralski method (VCz), Prog. Cryst. Growth Charact. Mater. 43, 119–185 (2001)

    Article  Google Scholar 

  57. F.-M. Kießling, P. Rudolph, M. Neubert, U. Juda, M. Naumann, W. Ulrici: Growth of GaAs crystals from Ga-rich melts by the VCz method without liquid encapsulation, J. Cryst. Growth 269, 218–228 (2004)

    Article  ADS  Google Scholar 

  58. U.A. Borovlev, N.V. Ivannikova, V.N. Shlegel, Y.V. Vasiliev, V.A. Gusev: Progress in growth of large sized BGO crystals by the low-thermal-gradient Cz technique, J. Cryst. Growth 229, 305–311 (2001)

    Article  ADS  Google Scholar 

  59. E. Pfeiffer: Untersuchungen zur Optimierung der Züchtungstechnologie von PbMoO3-Einkristallen nach der Czochralski-Methode. Ph.D. Thesis (Humboldt-University, Berlin 1990), in German

    Google Scholar 

  60. E. Pfeiffer, P. Rudolph: German patent DD 290–226 (1989)

    Google Scholar 

  61. K. Hein, E. Buhrig (Eds.): Kristallisation aus Schmelzen (Verlag für Grundstoffindustrie, Leipzig 1983), in German

    Google Scholar 

  62. P. Rudolph, U. Rinas, K. Jacobs: Systematic steps towards exactly stoichiometric and uncompensated CdTe Bridgman crystals, J. Cryst. Growth 138, 249–254 (1994)

    Article  ADS  Google Scholar 

  63. E. Northrup, S.B. Zhang: Dopant and defect energetics: Si in GaAs, Phys. Rev. B 47, 6791–6794 (1993)

    Article  ADS  Google Scholar 

  64. H. Zimmermann, R. Boyn, C. Michel, P. Rudolph: Absorption-calibrated determination of impurity concentrations in CdTe from excitonic photoluminescence, Phys. Status Solidi (a) 118, 225–234 (1990)

    Article  ADS  Google Scholar 

  65. J.A. Burton, R.C. Prim, W.P. Slichter: The distribution of solute in crystals grown from the melt, J. Chem. Phys. 21, 1987–1991 (1953)

    Article  ADS  Google Scholar 

  66. V.G. Levich: Physicochemical Hydrodynamics (Prentice-Hall, Englewood Cliffs 1961)

    Google Scholar 

  67. A. Ostrogorsky, G. Müller: A model of effective segregation coefficient, accounting for convection in the solute layer at the growth interface, J. Cryst. Growth 121, 587–598 (1992)

    Article  ADS  Google Scholar 

  68. W.A. Tiller: Principles of solidification. In: The Art and Science of Growing Crystals, ed. by J.J. Gilman (Wiley, New York 1963), Chap. 15

    Google Scholar 

  69. N. Sato, M. Kakimoto, Y. Kadota: The carbon and boron concentration control in GaAs crystals grown by liquid encapsulated Czochralski method. In: Semi-Insulating III–V Materials, ed. by A. Milnes, C. Miner (Hilger, Bristol 1990)

    Google Scholar 

  70. M. Jurisch, F. Börner, T. Bünger, S. Eichler, T. Flade, U. Kretzer, A. Köhler, J. Stenzenberger, B. Weinert: LEC- and VGF-growth of SI GaAs single crystals – Recent developments and current issues, J. Cryst. Growth 275, 283–291 (2005)

    Article  ADS  Google Scholar 

  71. K. Jacob, C. Frank, M. Neubert, P. Rudolph, W. Ulrici, M. Jurisch, J. Korb: A study on carbon incorporation in semi-insulating GaAs crystals grown by the vapor pressure controlled Czochralski technique (VCz), Cryst. Res. Technol. 35, 1163–1171 (2000)

    Article  Google Scholar 

  72. S. Eichler, A. Seidl, F. Börner, U. Kretzer, B. Weinert: A combined carbon and oxygen segregation model for the LEC growth of SI GaAs, J. Cryst. Growth 247, 69–76 (2003)

    Article  ADS  Google Scholar 

  73. J. Bohm, A. Lüdge, W. Schröder: Crystal growth by floating zone melting. In: Handbook of Crystal Growth, Vol. 2a, ed. by D.T.J. Hurle (Elsevier, North-Holland 1994), Chap. 4

    Google Scholar 

  74. R. Triboulet: The travelling heater method (THM) for Hg1-xCdxTe and related materials, Prog. Cryst. Growth Charact. Mater. 28, 85–144 (1994)

    Article  Google Scholar 

  75. C. Genzel, P. Gille, I. Hähnert, F.-M. Kießling, P. Rudolph: Structural perfection of (Hg,Cd)Te grown by THM, J. Cryst. Growth 101, 232–236 (1990)

    Article  ADS  Google Scholar 

  76. H.J. Koh, Y. Furukawa, P. Rudolph, T. Fukuda: Oxide mixed crystals grown by heater-immersed zone melting method with multi-capillary holes, J. Cryst. Growth 149, 236–240 (1995)

    Article  ADS  Google Scholar 

  77. R.S. Feigelson: Pulling optical fibers, J. Cryst. Growth 79, 669–680 (1986)

    Article  ADS  Google Scholar 

  78. T. Fukuda, P. Rudolph, S. Uda (Eds.): Fiber Crystal Growth from the Melt (Springer, Berlin 2004)

    Google Scholar 

  79. K.M. Kim, A.F. Witt, H.C. Gatos: Crystal growth from the melt under destabilizing thermal gradients, J. Electrochem. Soc. 119, 1218–1222 (1972)

    Article  Google Scholar 

  80. H.J. Scheel: Theoretical and technological solutions of the striation problem, J. Cryst. Growth 287, 214–223 (2006)

    Article  ADS  Google Scholar 

  81. E. Bauser: Atomic mechanisms of LPE. In: Handbook of Crystal Growth, Vol. 3b, ed. by D.T.J. Hurle (Elsevier, North-Holland 1994), Chap. 20

    Google Scholar 

  82. J. Barthel, M. Jurisch: Oszillation der Erstarrungsgeschwindigkeit beim Kristallwachstum aus der Schmelze mit rotierendem Keimkristall, Kristall und Technik 8, 199–206 (1973), in German

    Article  Google Scholar 

  83. D.T.J. Hurle, E. Jakeman: Effects of fluctuations on measurement of distribution coefficient by directional solidification, J. Cryst. Growth 5, 227–232 (1969)

    Article  ADS  Google Scholar 

  84. D.T.J. Hurle, R.W. Series: Use of magnetic field in melt growth. In: Handbook of Crystal Growth, Vol. 2a, ed. by D.T.J. Hurle (Elsevier, North-Holland 1994), Chap. 5

    Google Scholar 

  85. H.J. Scheel: Accelerated crucible rotation: A novel stirring technique in high-temperature solution growth, J. Cryst. Growth 13/14, 560–565 (1972)

    Article  ADS  Google Scholar 

  86. G.N. Kozhemyakin: Imaging of convection in a Czochralski crucible under ultrasound waves, J. Cryst. Growth 257, 237–244 (2003)

    Article  ADS  Google Scholar 

  87. E. Gilioli, J.L. Weyher, L. Zanotti, C. Mucchino: Growth striations in GaAs as revealed by DSL photoetching, Mater. Sci. Forum 203, 13–17 (1996)

    Article  Google Scholar 

  88. J.L. Weyher, P.J. van der Wel, G. Frigerio, C. Mucchino: DSL photoetching and high spatial resolution PL study of growth striations in undoped semi-insulating LEC-grown GaAs, Proceedings of the 6th Conference on Semi-Insulating III–V (1990) pp. 161–166

    Google Scholar 

  89. R.T. Gray, M.F. Larrousse, W.R. Wilcox: Diffusional decay of striations, J. Cryst. Growth 92, 530–542 (1988)

    Article  ADS  Google Scholar 

  90. B. Billia, R. Trivedi: Pattern formation in crystal growth. In: Handbook of Crystal Growth, Vol. 1b, ed. by D.T.J. Hurle (Elsevier, North-Holland 1994), Chap. 14

    Google Scholar 

  91. W.A. Tiller, K.A. Jackson, J.W. Rutter, B. Chalmers: The redistribution of solute atoms during the solidification of metals, Acta Metallurg. 1, 428–437 (1953)

    Article  Google Scholar 

  92. P. Dold, K.W. Benz: Rotating magnetic fields: Fluid flow and ctrystal growth applications, Prog. Cryst. Growth Charact. Mater. 38, 7–38 (1999)

    Article  Google Scholar 

  93. C. Stelian, Y. Delannoy, Y. Fautrelle, T. Duffar: Solute segregation in directional solidification of GaInSb concentrated alloys under alternating magnetic fields, J. Cryst. Growth 266, 207–215 (2004)

    Article  ADS  Google Scholar 

  94. V. Socoliuc, D. Vizman, B. Fischer, J. Friedrich, G. Müller: 3D numerical simulation of Rayleigh-Bénard convection in an electrically conducting melt acted on by a travelling magnetic field, Magnetohydrodynamics 39, 187–200 (2003)

    ADS  Google Scholar 

  95. W.W. Mullins, R.F. Sekerka: Stability of planar interface during solidification of a dilute alloy, J. Appl. Phys. 35, 444–451 (1964)

    Article  ADS  Google Scholar 

  96. S.R. Coriell, G.B. McFadden: Morphological stability. In: Handbook of Crystal Growth, Vol. 1b, ed. by D.T.J. Hurle (Elsevier, North-Holland 1994), Chap. 12

    Google Scholar 

  97. H. Wenzl, W.A. Oates, K. Mika: Defect thermodynamics and phase diagrams in compound crystals. In: Handbook of Crystal Growth, Vol. 1a, ed. by D.T.J. Hurle (Elsevier, North-Holland 1994), Chap. 3

    Google Scholar 

  98. P. Schlossmacher, K. Urban: Dislocations and precipitates in gallium arsenide, J. Appl. Phys. 71, 620–629 (1992)

    Article  ADS  Google Scholar 

  99. S. Brochard, J. Rabier, J. Grilhé: Nucleation of partial dislocations from a surface-step in semiconductors: a first approach of the mobility effect, Eur. Phys. J. Appl. Phys. 2, 99–105 (1998)

    Article  ADS  Google Scholar 

  100. G. Grange, C. Jourdan, A.L. Coulet, J. Gastaldi: Observation of the melting-solidification process of an Al crystal by synchrotron x-ray topography, J. Cryst. Growth 72, 748–752 (1985)

    Article  ADS  Google Scholar 

  101. K. Sumino, I. Yonenaga: Interactions of impurities with dislocations: Mechanical effects, Solid State Phenom. 85/86, 145–176 (2002)

    Article  Google Scholar 

  102. E. Nadgorny: Dislocation dynamics and mechanical properties of crystals. In: Progress in Materials Science, Vol. 31, ed. by J.W. Christian, P. Haasen, T.B. Massalski (Pergamon, Oxford 1988)

    Google Scholar 

  103. R.J. Amodeo, N.M. Ghoniem: Dislocation dynamics. I. A proposed methodology for deformation micromechanics; Dislocation dynamics. II. Applications to the formation of persistent slip bands, planar arrays, and dislocation cells, Phys. Rev. B 41, 6958–6976 (1990)

    Google Scholar 

  104. R.N. Nabarro: Steady-state diffusional creep, Philos. Mag. A 16, 231–238 (1967)

    Article  ADS  Google Scholar 

  105. E. Billig: Some defects in crystals grown from the melt, Proc. R. Soc. Lond. Ser. A 235, 37–55 (1956)

    Article  ADS  Google Scholar 

  106. V.L. Indenbom: Ein Beitrag zur Entstehung von Spannungen und Versetzungen beim Kristallwachstum, Kristall und Technik 14, 493–507 (1979), in German

    Article  Google Scholar 

  107. A.S. Jordan, A.R. von Neida, R. Caruso: The theory and practice of dislocation reduction in GaAs and InP, J. Cryst. Growth 70, 555–573 (1984)

    Article  ADS  Google Scholar 

  108. N. Miyazaki, H. Uchida, S. Hagihara, T. Munakata, T. Fukuda: Thermal stress analysis of bulk single crystal during Czochralski growth (comparison between anisotropic analysis and isotropic analysis), J. Cryst. Growth 113, 227–241 (1991)

    Article  ADS  Google Scholar 

  109. S. Motakef, A.F. Witt: Thermoelastic analysis of GaAs in LEC growth configuration: I. Effect of liquid encapsulation on thermal stresses, J. Cryst. Growth 80, 37–50 (1987)

    Article  ADS  Google Scholar 

  110. C.T. Tsai, A.N. Gulluoglu, C.S. Hartley: A crystallographic methodology for modeling dislocation dynamics in GaAs crystals grown from melt, J. Appl. Phys. 73, 1650–1656 (1993)

    Article  ADS  Google Scholar 

  111. J. Völkl: Stress in cooling crystals. In: Handbook of Crystal Growth, Vol. 2b, ed. by D.T.J. Hurle (Elsevier, North-Holland 1994), Chap. 14

    Google Scholar 

  112. R. Lohonka, G. Vanderschaeve, J. Kratochvil: Modelling of the plastic behaviour of III-V compound semiconductors during compressive tests, Mater. Sci. Eng. A 337, 50–58 (2002)

    Article  Google Scholar 

  113. K. Sumino, I. Yonenaga: Interactions of impurities with dislocations: mechanical effects, Solid State Phenom. 85/86, 145–176 (2002)

    Article  Google Scholar 

  114. S. Grondet, T. Duffar, F. Louchet, F. Theodore, N. Van Den Bogaert, J.L. Santailler: A visco-plastic model of the deformation of InP during LEC growth taking into accound dislocation annihilation, J. Cryst. Growth 252, 92–101 (2003)

    Article  ADS  Google Scholar 

  115. S. Pendurti, V. Prasad, H. Zhang: Modelling dislocation generation in high pressure Czochralski growth of InP single crystals: parts I and II, Modelling Simul. Mater. Sci. Eng. 13, 249–297 (2005)

    Google Scholar 

  116. N. Miyazaki, Y. Matsuura, D. Imahase: Thermal stress analysis of lead molybdate single crystal during growth process: Discussion on relation between thermal stress and crystal quality, J. Cryst. Growth 289, 659–662 (2006)

    Article  ADS  Google Scholar 

  117. M. Zaiser: Dislocation patterns in crystalline solids – phenomenology. In: Crystal Growth – From Theory to Technology, ed. by G. Müller, J.-J. Metois, P. Rudolph (Elsevier, Amsterdam 2004) pp. 215–238

    Chapter  Google Scholar 

  118. L. Kubin: Collective defect behavior under stress, Science 312, 864–865 (2006)

    Article  Google Scholar 

  119. F.R.N. Nabarro, M.S. Duesbery (Eds.): Dislocations in Solids, Vol. 11 (North-Holland, Amsterdam 2002)

    Google Scholar 

  120. M. Naumann, P. Rudolph, M. Neubert, J. Donecker: Dislocation studies in VCz GaAs by laser scattering tomography, J. Cryst. Growth 231, 22–33 (2001)

    Article  ADS  Google Scholar 

  121. T. Tuomi, L. Knuuttila, J. Riikonen, P.J. McNally, W.-M. Chen, J. Kanatharana, M. Neubert, P. Rudolph: Synchrotron x-ray topography of undoped VCz GaAs crystals, J. Cryst. Growth 237, 350–355 (2002)

    Article  ADS  Google Scholar 

  122. P. Rudolph: Dislocation patterning in semiconductor compounds, Cryst. Res. Technol. 40, 7–20 (2005)

    Article  Google Scholar 

  123. W. Pantleon: The evolution of disorientations for several types of boundaries, Mater. Sci. Eng. A 319–321, 211–215 (2001)

    Article  Google Scholar 

  124. G. Grange, C. Jourdan, A.L. Coulet, J. Gastaldi: Observation of the melting-solidification process of an Al crystal by synchrotron x-ray topography, J. Cryst. Growth 72, 748–752 (1985)

    Article  ADS  Google Scholar 

  125. B. Jakobson, H.F. Poulsen, U. Lienert, J. Almer, S.D. Shastri, H.O. Sørensen, C. Gundlach, W. Pantleon: Formation and subdivision of deformation structures during plastic deformation, Science 312, 889–892 (2006)

    Article  ADS  Google Scholar 

  126. C. Frank-Rotsch, U. Juda, F.-M. Kießling, P. Rudolph: Dislocation patterning during crystal growth of semiconductor compounds (GaAs), Mater. Sci. Technol. 21, 1450–1454 (2005)

    Article  Google Scholar 

  127. D.L. Holt: Dislocation cell formation in metals, J. Appl. Phys. 41, 3197–3201 (1970)

    Article  ADS  Google Scholar 

  128. J. P. Poirier: Creep of Crystals – High-Temperature Deformation Processes in Metals, Ceramics, Cambridge Earth Science Series (Cambridge Univ. Press, Cambridge 1985)

    Google Scholar 

  129. P. Rudolph, C. Frank-Rotsch, U. Juda, F.-M. Kießling: Scaling of dislocation cells in GaAs crystals by global numeric simulation and their restraints by in situ control of stoichiometry, Mater. Sci. Eng. A 400/401, 170–174 (2005)

    Article  Google Scholar 

  130. S.V. Raj, G.M. Pharr: A compilation and analysis of data for the stress dependence of the subgrain size, Mater. Sci. Eng. 81, 217–237 (1986)

    Article  Google Scholar 

  131. B. Birkmann, J. Stenzenberger, M. Jurisch, J. Härtwig, V. Alex, G. Müller: Investigations of residual dislocations in VGF-grown Si-doped GaAs, J. Cryst. Growth 276, 335–346 (2005)

    Article  ADS  Google Scholar 

  132. G. Müller, P. Schwesig, B. Birkmann, J. Härtwig, S. Eichler: Types and origin of dislocations in large GaAs and InP bulk crystals with very low dislocation densities, Phys. Status Solidi (a) 202, 2870–2879 (2005)

    Article  ADS  Google Scholar 

  133. B. Devincre, L.P. Kubin: Mesoscopic simulations of dislocations and plasticity, Mater. Sci. Eng. A 234–236, 8–14 (1997)

    Article  Google Scholar 

  134. I.V. Sabinina, A.K. Gutakovski, T.I. Milenov, N.N. Lykakh, Y.G. Sidorov, M.M. Gospodinov: Melt growth of CdTe crystals and transmission electron microscopic, Cryst. Res. Technol. 26, 967–972 (1991)

    Article  Google Scholar 

  135. L. Parthier, C. Poetsch, K. Pöhl, J. Stäblein, G. Wehrhan: About the influence of lattice-defects on the optical homogeneity of CaF_2 crystals for use in high performance microlithography, Gemeinsame Jahrestagung der DGK und DGKK, Jena, Referate (Oldenburg, München 2004) p. 5

    Google Scholar 

  136. S.V. Raj, I.S. Iskovitz, A.D. Freed: Modeling the role of dislocation substructure during class M and exponential creep, NASA Technical Memorandum 106986, 1–77 (1995)

    Google Scholar 

  137. P. Rudolph, N. Schäfer, T. Fukuda: Crystal growth of ZnSe from the melt, Mater. Sci. Eng. R 15, 85–133 (1995)

    Article  Google Scholar 

  138. P. Rudolph, K. Umetsu, H.J. Koh, T. Fukuda: Correlation between ZnSe crystal growth conditions from melt and generation of large-angle grain boundaries and twins, Jpn. J. Appl. Phys. 33, 1991–1994 (1994)

    Article  ADS  Google Scholar 

  139. J.P. Tower, R. Tobin, P.J. Perah, R.M. Ware: Interface shape and crystallinity in LEC GaAs, J. Cryst. Growth 114, 665–675 (1991)

    Article  ADS  Google Scholar 

  140. P. Rudolph, A. Engel, I. Schentke, A. Grochocki: Distribution and genesis of inclusions in CdTe and (Cd,Zn)Te single crystals grown by the Bridgman method and by the travelling heater method, J. Cryst. Growth 147, 297–304 (1995)

    Article  ADS  Google Scholar 

  141. H. Klapper: Generation and propagation of dislocations during crystal growth, Mater. Chem. Phys. 66, 101–109 (2000)

    Article  Google Scholar 

  142. M. Shibata, T. Suzuki, S. Kuma, T. Inada: LEC growth of large GaAs single crystals, J. Cryst. Growth 128, 439–443 (1993)

    Article  ADS  Google Scholar 

  143. F.-C. Wang, M.F. Rau, J. Kurz, M.F. Ehman, D.D. Liao, R. Carter: Correlation of growth phenomena to electrical properties of gnarl defects in GaAs. In: Defect Recognition and Image Processing in III–V Compounds II, ed. by E.R. Weber (Elsevier, Amsterdam 1987) p. 117

    Google Scholar 

  144. J. Kratochvil: Self-organization model of localization of cyclic strain into PSBs and formation of dislocation wall structure, Mater. Sci. Eng. A 309/310, 331–335 (2001)

    Article  Google Scholar 

  145. O. Politano, J.M. Salazar: A 3D mesoscopic approach for discrete dislocation dynamics, Mater. Sci. Eng. A 309/310, 261–264 (2001)

    Article  Google Scholar 

  146. H. Ono: Dislocation reactions and lineage formation in liquid encapsulated Czochralski grown GaAs crystals, J. Cryst. Growth 89, 209–219 (1988)

    Article  ADS  Google Scholar 

  147. T. Duffar, P. Dusserre, F. Picca, S. Lacroix, N. Giacometti: Bridgman growth without crucible contact using the dewetting phenomenon, J. Cryst. Growth 211, 434–439 (2000)

    Article  ADS  Google Scholar 

  148. P. Rudolph, M. Czupalla, C. Frank-Rotsch, U. Juda, F.-M. Kießling, M. Neubert, M. Pietsch: Semi-insulating 4–6-inch GaAs crystals grown in low temperature gradients by the VCz method, J. Ceram. Proc. Res. 4, 1–8 (2003)

    Google Scholar 

  149. M. Althaus, K. Sonnenberg, E. Küssel, R. Naeven: Some new design features for vertical Bridgman furnaces and the investigation of small angle grain boundaries developed during VB growth of GaAs, J. Cryst. Growth 166, 566–571 (1996)

    Article  ADS  Google Scholar 

  150. T. Kawase, Y. Hagi, M. Tasumi, K. Fujita, R. Nakai: Low-dislocation-density and low-residual-strain semi-insulating GaAs grown by vertical boat method. In: 1996 IEEE Semiconducting and Semi-insulating Materials Conference, IEEE SIMC-9, Toulouse 1996, ed. by C. Fontaine (IEEE, Piscataway 1996) pp. 275–278

    Google Scholar 

  151. T. Bünger, D. Behr, S. Eichler, T. Flade, W. Fliegel, M. Jurisch, A. Kleinwechter, U. Kretzer, T. Steinegger, B. Weinert: Development of a vertical gradient freeze process for low EPD GaAs substrates, Mater. Sci. Eng. B 80, 5–9 (2001)

    Article  Google Scholar 

  152. G. Müller, B. Birkmann: Optimization of VGF-growth of GaAs crystals by the aid of numerical modelling, J. Cryst. Growth 237–239, 1745–1751 (2002)

    Article  Google Scholar 

  153. P. Rudolph, F. Matsumoto, T. Fukuda: Studies on interface curvature during vertical Bridgman growth of InP in a flat-bottom container, J. Cryst. Growth 158, 43–48 (1996)

    Article  ADS  Google Scholar 

  154. U. Sahr, I. Grant, G. Müller: Growth of S-doped 2′′ InP-crystals by the vertical gradient freeze technique. In: Indium Phosphide and Related Materials, 2001. IPRM. IEEE International Conference on 14–18 May 2001 in Nara, Japan, pp. 533–536

    Google Scholar 

  155. F.-M. Kießling, P. Rudolph, M. Neubert, U. Juda, M. Naumann, W. Ulrici: Growth of GaAs crystals from Ga-rich melts by the VCz method without liquid encapsulation, J. Cryst. Growth 269, 218–228 (2004)

    Article  ADS  Google Scholar 

  156. P. Rudolph, F.-M. Kießling: Growth and characterization of GaAs crystals produced by the VCz method without boric oxide encapsulation, J. Cryst. Growth 292, 532–537 (2006)

    Article  ADS  Google Scholar 

  157. B. Bakó, I. Groma, G. Györgyi, G. Zimányi: Dislocation patterning: The role of climb in meso-scale simulations, Comput. Mater. Sci. 38, 22–28 (2006)

    Article  Google Scholar 

  158. J.M. Parsey, F.A. Thiel: A new apparatus for the controlled growth of single crystals by horizontal Bridgman techniques, J. Cryst. Growth 73, 211–220 (1985)

    Article  ADS  Google Scholar 

  159. K. Tomizawa, K. Sassa, Y. Shimanuki: J. Nishizawa, Growth of low dislocation density GaAs by as pressure-controlled Czochralski method, J. Electrochem. Soc. 131, 2394–2397 (1984)

    Article  Google Scholar 

  160. P. Schlossmacher, K. Urban, H. Rüfer: Dislocations and precipitates in gallium arsenide, J. Appl. Phys. 71, 620–629 (1992)

    Article  ADS  Google Scholar 

  161. T. Steinegger, M. Naumann, M. Jurisch, J. Donecker: Precipitate engineering in GaAs studied by laser scattering tomography, Mater. Sci. Eng. B 80, 215–219 (2001)

    Article  Google Scholar 

  162. M. Mühlberg, D. Hesse: TEM precipitation studies in Te-rich as-grown PbTe single crystals, Phys. Status Solidi (a) 76, 513–524 (1983)

    Article  ADS  Google Scholar 

  163. K.-T. Wilke, J. Bohm: Kristallzüchtung (H. Deutsch, Thun, Frankfurt 1988) p. 356, in German

    Google Scholar 

  164. L. Su, Y. Dong, W. Yang, T. Sun, Q. Wang, J. Xu, G. Zhao: Growth, characterization and optical quality of CaF_2 single crystals grown by the temperature gradient technique, Mater. Res. Bull. 40, 619–628 (2005)

    Article  Google Scholar 

  165. P. Rudolph, M. Neubert, M. Mühlberg: Defects in CdTe Bridgman monocrystals caused by nonstoichiometric growth conditions, J. Cryst. Growth 128, 582–587 (1993)

    Article  ADS  Google Scholar 

  166. R. Fornari, C. Frigeri, R. Gleichmann: Structural and electrical properties of n-type bulk gallium arsenide grown from non-stoichiometric melts, J. Electron. Mater. 18, 185–189 (1989)

    Article  ADS  Google Scholar 

  167. R.S. Rai, S. Mahajan, S. McDevitt, D.J. Johnson: Characterisation of CdTe, (Cd,Zn)Te, and Cd(Te,Se) single crystals by transmission electron microscopy, J. Vac. Sci. Technol. B 9, 1892–1896 (1991)

    Article  Google Scholar 

  168. K. Sonnenberg: Defect studies in GaAs by NIR-microscopy with different contrast techniques, IFF Bull. 51, 14–55 (1997)

    Google Scholar 

  169. J. Shen, D.K. Aidun, L. Regel, W.R. Wilcox: Characterization of precipitates in CdTe and Cd1-xZnxTe grown by vertical Bridgman-Stockbarger technique, J. Cryst. Growth 132, 250–260 (1993)

    Article  ADS  Google Scholar 

  170. H.G. Brion, C. Mewes, I. Hahn, U. Schäufele: Infrared contrast of inclusions in CdTe, J. Cryst. Growth 134, 281–286 (1993)

    Article  ADS  Google Scholar 

  171. R.J. Dinger, I.L. Fowler: Te inclusions in CdTe grown from a slowly cooled Te solution and by traveling solvent method, Rev. Phys. Appl. 12, 135–139 (1977)

    Article  Google Scholar 

  172. M. Salk, M. Fiederle, K.W. Benz, A.S. Senchenkov, A.V. Egorov, D.G. Matioukhin: CdTe and CdTe_0.9Se_0.1 crystals grown by the travelling heater method using a rotating magnetic field, J. Cryst. Growth 138, 161–167 (1994)

    Article  ADS  Google Scholar 

  173. J. Donecker, B. Lux, P. Reiche: Use of optical diffraction effects in crystals for growth characterization, J. Cryst. Growth 166, 303–308 (1996)

    Article  ADS  Google Scholar 

  174. K.F. Hulme, J.B. Mullin: Facets and anomalous solute distributions in InSb crystals, Philos. Mag. 41, 1286–1288 (1959)

    Article  ADS  Google Scholar 

  175. M.T. Santos, C. Marin, E. Dieguez: Morphology of Bi12GeO20 crystals grown along the (111) directions by the Czochralski method, J. Cryst. Growth 160, 283–288 (1996)

    Article  ADS  Google Scholar 

  176. P. Reiche, J. Bohm, H. Hermoneit, D. Schultze, P. Rudolph: Effect of an electrical field on the growth of lithium niobate single crystals, J. Cryst. Growth 108, 759–764 (1991)

    Article  ADS  Google Scholar 

  177. J.B. Mullin, K.F. Hulme: Orientation-dependent distribution coefficients in melt-grown InSb crystals, J. Phys. Chem. Solids 17, 1–6 (1960)

    Article  ADS  Google Scholar 

  178. Y. Liu, A. Virozub, S. Brandon: Facetting during directional growth of oxides from the melt: coupling between thermal fields, kinetics and melt/crystal interface shapes, J. Cryst. Growth 205, 333–353 (1999)

    Article  ADS  Google Scholar 

  179. O. Weinstein, S. Brandon: Dynamics of partially faceted melt–crystal interfaces III: Three-dimensional computational approach and calculations, J. Cryst. Growth 284, 235–253 (2005)

    Article  ADS  Google Scholar 

  180. M. Shibata, Y. Sasaki, T. Inada, S. Kuma: Observation of edge-facets in 〈100〉 InP crystals grown by LEC method, J. Cryst. Growth 102, 557–561 (1990)

    Article  ADS  Google Scholar 

  181. H. Gottschalk, G. Patzer, H. Alexander: Stacking fault energy and ionicity of cubic III–V compounds, Phys. Status Solidi (a) 45, 207–217 (1978)

    Article  ADS  Google Scholar 

  182. D.T.J. Hurle: A mechanism for twin formation during Czochralski and encapsulated vertical Bridgman growth of III–V compound semiconductors, J. Cryst. Growth 147, 239–250 (1995)

    Article  ADS  Google Scholar 

  183. E. Billig: Some defects in crystals grown from the melt, Proc. R. Soc. Lond. Ser. A 235, 37–55 (1956)

    Article  ADS  Google Scholar 

  184. V.V. Voronkov: Structure of crystal surfaces, Sov. Phys. Cryst. 19, 573 (1975), (see also [6.179])

    Google Scholar 

  185. J. Amon, F. Dumke, G. Müller: Influence of the crucible shape on the formation of facets and twins in the growth of GaAs by the vertical gradient freeze technique, J. Cryst. Growth 187, 1–8 (1998)

    Article  ADS  Google Scholar 

  186. H. Chung, M. Dudley, D.J. Larson, D.T.J. Hurle, D.F. Bliss, V. Prassad: The mechanism of growth-twin formation in zincblende crystals: New insights from a study of magnetic liquid encapsulated Czochralski-grown InP single crystals, J. Cryst. Growth 187, 9–17 (1998)

    Article  ADS  Google Scholar 

  187. M. Dudley, B. Raghothamachar, Y. Guo, X.R. Huang, H. Chung, D.T.J. Hurle, D.F. Bliss: The influence of polarity on twinning in zincblende structure crystals: new insights from a study of magnetic liquid encapsulated, Czochralski grown InP single crystals, J. Cryst. Growth 192, 1–10 (1998)

    Article  ADS  Google Scholar 

  188. Y. Hosokawa, Y. Yabuhara, R. Nakai, K. Fujita: Development of 4-inch diameter InP single crystal with low dislocation density using VCZ method, Indium Phosphide and Related Materials, 1998, IPRM IEEE International Conference on 11–15 May 1998 in Tsukuba, Japan, pp. 34–37

    Google Scholar 

  189. C.P. Khattak, F. Schmid: Growth of CdTe crystals by the heat exchanger method (HEM), SPIE 1106, 47–55 (1989)

    ADS  Google Scholar 

  190. A.F. Witt: Growth of CdTe under controlled heat transfer conditions, Final Report, DAAG No. 29–82-K-0119 (Mater. Proc. Center M.I.T., Cambridge 1986)

    Google Scholar 

  191. P. Rudolph: Travelling magnetic fields applied to bulk crystal growth from the melt: The step from basic research to industrial scale, J. Cryst. Growth 310, 1298–1306 (2008)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Peter Rudolph .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag

About this chapter

Cite this chapter

Rudolph, P. (2010). Defect Formation During Crystal Growth from the Melt. In: Dhanaraj, G., Byrappa, K., Prasad, V., Dudley, M. (eds) Springer Handbook of Crystal Growth. Springer Handbooks. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74761-1_6

Download citation

Publish with us

Policies and ethics