Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 2 Part 6 pot

374 21 Diffusion in Quasicrystalline Alloys is because natural Al is monoisotopic, which forbids SIMS studies, and a suit- able radioisotope for tracer studies is missing 2 .Instead,diffusionofseveral solutes was investigated, especially in icosahedral Al-Pd-Mn. Single crystals should be used for basic studies to avoid complications due to diffusion along grain boundaries (see Chap. 32). The diffusion coefficient in a quasicrystal, like in a crystal, is a symmetric second rank tensor. Icosahedral quasicrystals are highly symmetric, diffu- sion is isotropic, and the diffusion coefficient is a scalar quantity. Decagonal quasicrystals are uniaxial and the diffusion coefficient has two principal com- ponents, like in uniaxial crystals. One component corresponds to diffusion parallel to the decagonal axis and the other one to diffusion perpendicular to the axis. A thorough diffusion study on decagonal quasicrystals requires diffusivity measurements on monocrystalline samples with two different ori- entations. Diffusion in quasicrystalline alloys has been reviewed by Nakajima and Zumkley [13] and by Mehrer et al. [14]. In this chapter, we summarise the state-of-the-art in this area, compare diffusion in quasicrystals with diffusion in related crystalline metals, and discuss possible diffusion mechanisms. 21.2.1 Icosahedral Quasicrystals Icosahedral Al-Pd-Mn: The information on the phase diagram for the ternary Al-Pd-Mn system is rather complete and has been reviewed by L ¨ uck [21]. Al-Pd-Mn was the first system for which stable icosahedral as well as decagonal quasicrystals were detected [22]. The formation of the icosahe- dral phase occurs in a ternary peritectic reaction from the melt. Conventional procedures permit the growth of single crystals [15]. The formation of the decagonal phase is sluggish, and the determination of the phase diagram in the decagonal region is difficult. Icosahedral Al-Pd-Mn contains about 70 at. % Al (see Table 21.1). The contents of transition elements Pd and Mn are about 21 and 9 at. %, respec- tively. Icosahedral Al-Pd-Mn is not a stoichiometric compound. It possesses a relatively narrow phase field, which widens with increasing temperature, but its width never exceeds a few percent [21]. Al-Pd-Mn is the quasicrystalline material for which the largest body of diffusion data is available. These data are displayed in the Arrhenius diagram of Fig. 21.3. Self-diffusion of the Mn and Pd components as well as solute diffusion of the transition elements Co, Cr, Fe, Ni, and Au and of the non- transition elements Zn, Ga, In, and Ge has been investigated. For detailed references, the reader may consult the already mentioned reviews [13, 14]. Inspection of Fig. 21.3 shows that the diffusivities can be grouped into two major categories: 2 The radiosisotope 26 Al has a half-life of 7 ×10 5 years. Its specific activity is very low, its production requires an accelerator and is very expensive. 21.2 Diffusion Properties of Quasicrystals 375 Fig. 21.3. Tracer diffusion in single-crystals of icosahedral Al-Pd-Mn according to Mehrer et al. [14]. Self-diffusion in Al is indicated as a long-dashed line – Non-transition elements are relatively fast diffusers. Their diffusivities are comparable, within one order of magnitude, to the self-diffusivity of metallic aluminium (long-dashed line). Ga and Zn were studied to mimic Al self-diffusion in the quasicrystal. Ga is isoelectronic to Al. Zn diffusion is believed to be close to Al self-diffusion, since in metallic Al it is only about a factor of 2 faster than Al [23]. – Transition elements are slow, in some cases extremely slow diffusers. For example, diffusion of Fe at 700 K is about 7 orders of magnitude slower than Ga and Zn diffusion. The diffusion enthalpies of Fe, Co, and Cr are high, almost twice as large as for Ga and Zn. The pre-exponential factors of the transition elements are two to three orders of magnitude larger than those of Zn and Ga [14]. Most of the diffusivities in Fig. 21.3 follow linear Arrhenius behaviour in the whole temperature range investigated. For Pd and Au diffusion, studied after implantation of the radioisotopes, a kink was reported in the Arrhenius diagram around 773 K by Frank and coworkers [25, 26]. The Arrhenius parameters in the high-temperature regime are similar to those of the other transition elements, whereas in the low-temperature regime very low values of the activation parameters have been reported. 376 21 Diffusion in Quasicrystalline Alloys The pressure dependence of diffusion of the non-transition element Zn and of the transition element Mn has been studied [27]. The following activation volumes were deduced: ∆V (Zn)=0.74 Ω at 776 K, (21.1) ∆V (Mn)=0.67 Ω at 1023 K . (21.2) Ω denotes the mean atomic volume of icosahedral Al-Pd-Mn, which in the case of Al-based Al-Pd-Mn is not much different from the atomic volume of Al. It is remarkable that there is no significant difference between the activation volumes of the transition and the non-transition element indicating that the diffusion mechanism of both elements is basically the same. The magnitudes and signs of these activation volumes are similar to those observed for metals, which are attributed to vacancy-mediated diffusion (see Chap. 8). Diffusion in crystalline materials is mediated by point defects (Chaps. 6, 17, 19, 20, 26). In metallic elements and alloys, vacancy-type defects are responsible for the diffusion of matrix atoms and of substitutional solutes. In quasicrystals, as in crystalline alloys, vacancies are present in thermal equilibrium as demonstrated in positron annihilation studies by Schaefer and coworkers [28, 29]. Phasons are additional point defects which are specific to quasicrystalline materials [30]. It has been suggested theoretically by Kalugin and Katz [31] that, in addition to vacancies, phason flips might contribute to some extend to self- and solute diffusion in quasicrystals. Remembering that Al is the major component in icosahedral Al-Pd-Mn, one is prompted to compare its diffusion data to those of aluminium. Fig- ure 21.4 shows an Arrhenius diagram of Al self-diffusion together with the diffusivities of those transition and non-transition elements which were also studied in icosahedral Al-Pd-Mn. A comparison of Fig. 21.3 and 21.4 supports the following conclusions: 1. Diffusion of Zn and Ga in icosahedral Al-Pd-Mn and in Al obey very similar Arrhenius laws. This suggests that Zn and Ga diffusion in icosa- hedral Al-Pd-Mn is vacancy-mediated as well and restricted to the Al- subnetwork of the quasicrystalline structure (Fig. 21.2). 2. The diffusivities of Zn and Ga are close to self-diffusion of Al. The acti- vation enthalpies and pre-exponential factors of Ga (113 kJ mol −1 ,1.2 × 10 −5 m 2 s −1 ) and Zn (121 kJ mol −1 ,2.7 ×10 −5 m 2 s −1 ) in icosahedral Al- Pd-Mn [14] and those of Al self-diffusion in metallic Al (123.5 kJ mol −1 , 1.37 ×10 −5 m 2 s −1 [24]) are very similar. This suggests that Zn and Ga diffusion provide indeed good estimates for Al self-diffusion in icosahedral Al-Pd-Mn. Al self-diffusion could not be investigated due to the lack of a suitable Al tracer (see above). 3. Diffusion of solutes in icosahedral Al-Pd-Mn and in Al are both char- acterised by a ‘wide spectrum’ of diffusivities, ranging from very slow diffusing transition elements to relatively fast diffusing non-transition el- ements. Self-diffusion and diffusion of substitutional solutes in Al are both 21.2 Diffusion Properties of Quasicrystals 377 Fig. 21.4. Self-diffusion and diffusion of solutes in Al according to [14] mediated by vacancies. The striking similarities between the spectra of solute diffusion in both materials, apart from minor differences in detail, are a strong argument in favour of a vacancy-type mechanism in icosa- hedral Al-Pd-Mn as well. As discussed in Chap. 19, the slow diffusion of transition elements in Al can be attributed to a repulsive interaction between vacancy and solute. In the low-diffusivity regime, the diffusivities of Pd and Au in icosahedral Al-Pd-Mn are distinctly higher than expected from an extrapolation of the Arrhenius-laws corresponding to the high-diffusivity regime. The low pre- exponential factors found for Pd and Au diffusion in this regime are orders of magnitude too small to be reconcilable with diffusion mechanisms operating in crystalline solids. A tentative explanation attributes this low-diffusivity regime to phason-flip assisted diffusion [14, 25, 26]. This explanation is in- timately related to the quasicrystalline structure. Hence, it cannot work for substitutional solutes, like the normal diffusers Zn or Ga, residing almost exclusively in the Al subnetwork. However, it does work for intermediate diffusers such as Au and Pd by promoting their interchange between the subnetworks. 378 21 Diffusion in Quasicrystalline Alloys Icosahedral Zn-Mg-RE: Zn-Mg-RE alloys (RE = rare earth element) are the prototype of non-Al-based quasicrystalline alloys. In these phases, Zn is the base element with a content of about 60 at. % (Table 21.1). These qua- sicrystals are considered to belong to the Frank-Kaspar type phases, which are characterised by an electron to atom ratio of about 2.1. Their structureal units are Bergman clusters [32], which are formed by dense stacking of tetra- hedra, relating them to the Frank-Kaspar phases [33]. Three quasicrystalline phases have been found: a face-centered icosahedral (fci) phase, a simple cu- bic icosahedral (si) phase, and a phase with decagonal structure. Crystalline structures which are related to the quasicrystalline phases are found as well (for references see, e.g., [16]). Tracer diffusion experiments of 65 Zn have been performed on icosahe- dral Zn 64.2 Mg 26.4 Ho 9.4 and Zn 60.7 Mg 30.6 Y 8.7 quasicrystals [34] grown by the top-seeded solution-growth technique [16]. Diffusion data are displayed in Fig. 21.5 together with tracer diffusion of 65 Zn in a related crystalline Zn- Mg-Y phase with hexagonal structure. It is not surprising that Zn diffusion Fig. 21.5. Self-diffusion of 65 Zn in icosahedral Zn 64.2 Mg 26.4 Ho 9.4 and Zn 60.7 Mg 30.6 Y 8.7 quasicrystals and in a related hexagonal phase (h-ZnMgY) ac- cording to Galler et al. [34]. Dashed lines: self-diffusion in Zn parallel and per- pendicular to its hexagonal axis; dotted line: Zn diffusion in icosahedral Al-Pd-Mn 21.2 Diffusion Properties of Quasicrystals 379 in the closely related Zn-Mg-Ho and Zn-Mg-Y quasicrystals proceeds at al- most identical rates. Its diffusion is, however, slower than self-diffusion in metallic Zn. This difference can be attributed only partly to the different melting temperatures of Zn (693 K) and Zn-Mg-Ho (863 K). In a tempera- ture scale normalised to the respective melting temperatures, Zn self-diffusion in hexagonal Zn is still about one order of magnitude faster. Zn diffusion in icosahedral Al-Pd-Mn (dotted line) proceeds about one order of magnitude faster than Zn diffusion in the Zn-based quasicrystals. A comparison between Zn diffusion in the Zn-based quasicrystals and the ternary crystalline com- pound with similar composition reveals similar diffusivities of Zn diffusion in particular for diffusion perpendicular to the hexagonal axis. These facts and the magnitude of the pre-exponential factors for Zn diffusion in both quasicrystals (Zn-Mg-Ho: 3.9×10 −3 m 2 s −1 ;Zn-Mg-Y:3.3×10 −3 m 2 s −1 )are hints that diffusion in these materials is vacancy-mediated as well. 21.2.2 Decagonal Quasicrystals Decagonal phases belong to the category of two-dimensional quasicrystals. They exhibit quasiperiodic order in layers perpendicular to the decagonal axis, whereas they are periodic parallel to the decagonal axis [35]. The ternary systems Al-CoAl-NiAl and Al-CoAl-CuAl are characterised by the formation of decagonal phases; no icosahedral phases are observed [21]. The field of the decagonal phase is elongated with respect to a widely varying Ni to Co ratio. In contrast, only a slight variation of the Al content is possible without leaving the phase field with quasicrystalline order. Diffusion of several tracers has been studied on oriented single crystals parallel and perpendicular to the decagonal axis and also in polycrystals. Diffusion data for decagonal Al-Ni-Co are summarised in Fig. 21.6. Self- diffusion of both minority components (solid lines) has been studied on single crystals of Al 72.6 Ni 10.5 Co 16.9 using 57 Co and 63 Ni as radiotracers by Khoukaz et al. [37, 38]. Diffusion of 63 Ni has been studied by the same authors studied on polycrystals of the composition Al 70.2 Ni 15.1 Co 14.7 .Data obtained by Nakajima and coworkers for diffusion of 60 Co in decagonal Al 72.2 Ni 11.8 Co 16 are shown as dashed lines [13, 36]. Employing SIMS pro- filing, diffusion of Ga has been studied to mimic Al diffusion by Galler et al. [34]. Co diffusion data are available for two slightly different alloys. Co diffusion depends only weakly on composition. Diffusion of Co and Ni in both principal directions obey Arrrhenius laws. It is remarkable that no significant deviation from Arrhenius behaviour has been detected down to the lowest tempera- tures. In a temperature scale normalised with the melting temperatures, the diffusivities of Co and Ni in decagonal Al-Ni-Co and the (vacancy-mediated) self-diffusivities of metallic Co and Ni [39] are fairly close to each other. Dif- fusion of Ga is several orders of magnitude faster than diffusion of the tran- sition elements Co and Ni. As in icosahedral Al-Pd-Mn, the non-transition 380 21 Diffusion in Quasicrystalline Alloys Fig. 21.6. Self-diffusion of 65 Ni, 60 Co, and 57 Co in decagonal Al-Ni-Co quasicrys- tals from [14, 34, 36]. Monte Carlo simulations of Al diffusion are also shown [40] element diffuses significantly faster than the transition elements. Diffusivities deduced from molecular dynamic simulations for Al self-diffusion in Al-Ni-Co by G ¨ ahler and Hocker [40] also shown in Fig. 21.6. The magnitude of the Al diffusivities and the sign of the diffusion anisotropy are similar to those of Ga diffusion, supporting the view that Ga is indeed suitable to mimic Al self-diffusion. The combination of crystalline and quasicrystalline order makes decagonal Al-Ni-Co particularly interesting from the viewpoint of diffusion mechanisms. Quasicrystalline order exists only in layers perpendicular to the decagonal axis, whereas periodic (crystalline) order prevails parallel to the decagonal axis. If a diffusion mechanism would dominate, which is specific to quasiperi- odic order, one should expect that diffusion within the layers perpendicular to the decagonal axis is faster than parallel to it. Phasons are specific to quasiperiodic order and phason-mediated diffusion should cause a diffusion anisotropy with slower diffusion in the direction of the decagonal axis. Fig- ure 21.6 shows that for Co and Ni diffusion the anisotropy is very small. For 60 Co diffusion in Al 72.2 Ni 11.8 Co 16 the anisotropy is even opposite to the expectation for phason-dominated diffusion in the whole temperature References 381 range [13]. This is even more so for Ga diffusion, where diffusion parallel to decagonal axis is clearly faster than perpendicular to the axis. The magnitude of the diffusion anisotropy Co and Ni in decagonal Al-Ni- Co is similar to the anisotropies reported for uniaxial metals (see Chap. 17). For self-diffusion in the hexagonal metals Be, Mg, Zn, and Cd and for tetrag- onal In and Sn, the diffusivities parallel and perpendicular to the axis, differ by not more than a factor of 2 (see Chap. 17 and [39]). Whereas in Zn and Cd, diffusion parallel to the hexagonal axis is slightly faster, the opposite is true in Be and Mg. The similarities between the small diffusion anisotropies of uniaxial metals and the decagonal quasicrystals provides additional evidence for vacancy-mediated diffusion in decagonal Al-Ni-Co. References 1. D. Shechtman, I.A. Blech, J.W. Cahn, Phys. Rev. Lett. 53, 1951 (1984) 2. C. Janot, Quasicrystals – a Primer, Clarendon Press, Oxford, 1994 3. Z. Stadnik (Ed.), Physical Properties of Quasicrystals, Springer-Verlag, 1999 4. J B. Suck, M. Schreiber, P. H¨aussler (Eds.), Quasicrystals – an Introduction to Structure, Physical Properties, and Applications, Springer-Series in Materials Science, Vol. 55, 2002 5. H R. Trebin (Ed.), Quasicrystals – Structure and Physical Properties, Wiley- VCH, 2003 6. A.P. Tsai, A. Inoue, T. Masumoto, Jap. J. Appl. Phys. 36, L1505 (1987) 7. A.P. Tsai, A. Inoue, T. Masumoto, Acta Met. 37, 1443 (1989) 8. A.P. Tsai, A. Inoue, Y. Yokoyama, T. 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Wurtzite 1. 120 , indirect 0 .66 3, indirect Gallium phosphide (GaP) Gallium arsenide (GaAs) Gallium antimonide (GaSb) Indium phosphide (InP) Indium arsenide (InAs) Indium antimonide (InSb) Zinc blende Zinc blende Zinc blende Zinc blende Zinc blende Zinc blende 0.543095 0. 564 613 a: 0.3111, c: 0.4978 0.545 12 0. 565 32 0 .60 959 0.5 868 7 0 .60 584 0 .64 794 3.7, direct 2. 261 , indirect 1.435, direct 0. 72, direct 1.351,... and DV DV contain contributions of several charge states [2, 3]: eq eq eq eq CI = CI 0 + CI + + CI 2+ + (23 .3) eq eq eq eq CV = CV + + CV 0 + CV − , (23 .4) and eq eq eq eq CI DI = CI 0 DI 0 + CI + DI + + CI 2+ DI 2+ + eq CV DV = eq CV + DV + + eq CV 0 DV 0 + eq CV − DV − + (23 .5) (23 .6) I 0 , I + , I 2+ , V + , V 0 , V − , denote charge states of self-interstitials and vacancies Unfortunately,... Compounds: Recent Progress, Defect and Diffusion Forum 194–199, 68 7–7 02 (20 00) 19 H Bracht, Diffusion and Intrinsic Point-Defect Properties in Silicon, MRS Bulletin, 22 27 June 20 00 20 T.Y Tan, U G¨sele, Diffusion in Semiconductors, in: Diffusion in Condensed o Matter – Methods, Materials, Models, P Heitjans, J K¨rger (Eds.), Springera Verlag, 20 05, p 165 23 Self-diffusion in Elemental Semiconductors Compared with... are occupied by the same type of atoms; in the zinc blende structure, the basis is formed by two different types of atoms (Fig 22 .1) In both cases the Fig 22 .1 Diamond structure of Si and Ge (right) and zinc blende structure (left) 3 86 22 General Remarks on Semiconductors Table 22 .1 Crystal structure, lattice parameters, and band gaps of Si, Ge, and of some III-V compound semiconductors according to Shaw... the significance of semiconductors [2 5] 22 .2 Specific Features of Semiconductor Diffusion The advent of microelectronic and optoelectronic devices has strongly stimulated the interest in diffusion processes in semiconductors, because solid- state diffusion is used as a fundamental process in their manufacture Elements 390 22 General Remarks on Semiconductors from different parts of the periodic table have... Landolt-B¨rnstein, New Series, Group III, Vol 22 b, Springero Verlag, Berlin, 1989 A Seeger, K.P Chik, Diffusion Mechanisms and Point Defects in Silicon and Germanium, Phys Stat Sol 29 , 455–5 42 (1 968 ) D Shaw, Self- and Impurity Diffusion in Ge and Si, Phys Stat Sol (b) 72, 11–39 (1975) A.F Willoughby, Atomic Diffusion in Semiconductors, Rep Progr Phys 41, 166 5–1705 (1978) References 393 14 W Frank, U G¨sele,... in metals (see Chap 5) 2 The open structure of the diamond and zinc blende lattice favours interstitial incorporation of self- and foreign atoms For this reason, interstitial 22 .2 Specific Features of Semiconductor Diffusion 3 4 5 6 7 8 9 391 diffusion and interstitial-substitutional exchange diffusion of hybrid solutes via the dissociative or kick-out mechanisms, (see Chaps 6 and 25 ) are more prominent... precise window patterns in Fig 22 .2 Resistivity of various materials and silicon of various doping levels 22 .2 Specific Features of Semiconductor Diffusion 389 the oxide and in turn provide precise control of the areas in which diffusion would occur This combined process of photolithography and oxide masking (photoresist process) has since been developed to a precision that can be controlled to sub-micrometer... commonly agreed (see, e.g., [4 6] ) that self-diffusion in Ge occurs by a vacancy mechanism This interpretation is among other observations (see also Chaps 24 and 25 ) supported by the following ones: – Measurements of the dependence of the diffusion coefficient on the isotope mass (isotope effect) have been performed by Campbell using the isotope 23 .2 Germanium 399 Fig 23 .2 Self-diffusion coefficients of Ge... and Germao nium, in: Diffusion in Crystalline Solids, G.E Murch, A.S Nowick (Vol Eds.), Academic Press Inc., 1984 15 P.M Fahey, P.B Griffin, J.D Plummer, Point Defects and Dopant Diffusion in Silicon, Rev Mod Phys 61 , 28 9–384 (1989) 16 N.A Stolwijk, Atomic Transport in Semiconductors: Diffusion Mechanisms and Chemical Trends, Defect and Diffusion Forum 95–98, 895–9 16 (1993) 17 S.M Hu, Mater Sci Eng R13, 105 . Alloys 20 . C. Janot, Phys. Rev. B 73, 181 (19 96) 21 . R. L¨uck, Production of Quasicrystalline Alloys and Phase Diagrams,p .22 2 in [4], 20 02 22. C. Beeli, H U. Nissen, J. Robadey, Philos. Mag. Lett. 63 ,. Tsai, H. Nakajima, Mat. Sci. A 29 4 29 6, 7 02 (20 00) 37. C. Khoukaz, R. Galler, H. Mehrer, P.C. Canfield, I.R. Fisher, M. Feuerbacher, Mater. Sci. Eng. A 29 4 29 6, 69 7 (20 00) 38. C. Khoukaz, R. Galler,. III: Crystal and Solid State Physics, Vol. 26 , Springer-Verlag, 1990 40. F. G¨ahler, S. Hocker, J. Non-Cryst. Solids 334–335, 308 (20 04) Part IV Diffusion in Semiconductors 22 General Remarks