In this chapter, we summarise thestate-of-the-art in this area, compare diffusion in quasicrystals with diffusion in related crystalline metals, and discuss possible diffusion mechanisms..
Trang 1374 21 Diffusion in Quasicrystalline Alloys
is because natural Al is monoisotopic, which forbids SIMS studies, and a able radioisotope for tracer studies is missing2 Instead, diffusion of severalsolutes was investigated, especially in icosahedral Al-Pd-Mn Single crystalsshould be used for basic studies to avoid complications due to diffusion alonggrain boundaries (see Chap 32)
suit-The diffusion coefficient in a quasicrystal, like in a crystal, is a symmetricsecond rank tensor Icosahedral quasicrystals are highly symmetric, diffu-sion is isotropic, and the diffusion coefficient is a scalar quantity Decagonalquasicrystals are uniaxial and the diffusion coefficient has two principal com-ponents, like in uniaxial crystals One component corresponds to diffusionparallel to the decagonal axis and the other one to diffusion perpendicular
to the axis A thorough diffusion study on decagonal quasicrystals requiresdiffusivity measurements on monocrystalline samples with two different ori-entations
Diffusion in quasicrystalline alloys has been reviewed by Nakajima andZumkley[13] and by Mehrer et al [14] In this chapter, we summarise thestate-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 byL¨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 dral phase occurs in a ternary peritectic reaction from the melt Conventionalprocedures permit the growth of single crystals [15] The formation of thedecagonal phase is sluggish, and the determination of the phase diagram inthe decagonal region is difficult
icosahe-Icosahedral Al-Pd-Mn contains about 70 at % Al (see Table 21.1) Thecontents 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 ofdiffusion 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 solutediffusion 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 detailedreferences, the reader may consult the already mentioned reviews [13, 14].Inspection of Fig 21.3 shows that the diffusivities can be grouped into twomajor categories:
2 The radiosisotope26Al has a half-life of 7× 105 years Its specific activity is verylow, its production requires an accelerator and is very expensive
Trang 221.2 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 diffusivitiesare comparable, within one order of magnitude, to the self-diffusivity ofmetallic 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 onlyabout a factor of 2 faster than Al [23]
– Transition elements are slow, in some cases extremely slow diffusers Forexample, diffusion of Fe at 700 K is about 7 orders of magnitude slowerthan Ga and Zn diffusion The diffusion enthalpies of Fe, Co, and Cr arehigh, almost twice as large as for Ga and Zn The pre-exponential factors
of the transition elements are two to three orders of magnitude largerthan those of Zn and Ga [14]
Most of the diffusivities in Fig 21.3 follow linear Arrhenius behaviour inthe whole temperature range investigated For Pd and Au diffusion, studiedafter implantation of the radioisotopes, a kink was reported in the Arrheniusdiagram around 773 K by Frank and coworkers [25, 26] The Arrheniusparameters in the high-temperature regime are similar to those of the othertransition elements, whereas in the low-temperature regime very low values
of the activation parameters have been reported
Trang 3376 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 activationvolumes were deduced:
17, 19, 20, 26) In metallic elements and alloys, vacancy-type defects areresponsible for the diffusion of matrix atoms and of substitutional solutes
In quasicrystals, as in crystalline alloys, vacancies are present in thermalequilibrium as demonstrated in positron annihilation studies by Schaeferand coworkers [28, 29] Phasons are additional point defects which arespecific to quasicrystalline materials [30] It has been suggested theoretically
by Kalugin and Katz [31] that, in addition to vacancies, phason flips mightcontribute 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 thediffusivities of those transition and non-transition elements which were alsostudied in icosahedral Al-Pd-Mn A comparison of Fig 21.3 and 21.4 supportsthe following conclusions:
1 Diffusion of Zn and Ga in icosahedral Al-Pd-Mn and in Al obey verysimilar 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 vation enthalpies and pre-exponential factors of Ga (113 kJ mol−1 , 1.2 ×
acti-10−5m2s−1) and Zn (121 kJ mol−1 , 2.7 × 10 −5m2s−1) in icosahedral
Al-Pd-Mn [14] and those of Al self-diffusion in metallic Al (123.5 kJ mol−1,
1.37 × 10 −5m2s−1 [24]) are very similar This suggests that Zn and Ga
diffusion provide indeed good estimates for Al self-diffusion in icosahedralAl-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 acterised by a ‘wide spectrum’ of diffusivities, ranging from very slowdiffusing transition elements to relatively fast diffusing non-transition el-ements Self-diffusion and diffusion of substitutional solutes in Al are both
Trang 4char-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 ofsolute 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 interactionbetween vacancy and solute
In the low-diffusivity regime, the diffusivities of Pd and Au in icosahedralAl-Pd-Mn are distinctly higher than expected from an extrapolation of theArrhenius-laws corresponding to the high-diffusivity regime The low pre-exponential factors found for Pd and Au diffusion in this regime are orders ofmagnitude too small to be reconcilable with diffusion mechanisms operating
in crystalline solids A tentative explanation attributes this low-diffusivityregime to phason-flip assisted diffusion [14, 25, 26] This explanation is in-timately related to the quasicrystalline structure Hence, it cannot work forsubstitutional solutes, like the normal diffusers Zn or Ga, residing almostexclusively in the Al subnetwork However, it does work for intermediatediffusers such as Au and Pd by promoting their interchange between thesubnetworks
Trang 5378 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 isthe 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, whichare characterised by an electron to atom ratio of about 2.1 Their structurealunits are Bergman clusters [32], which are formed by dense stacking of tetra-hedra, relating them to the Frank-Kaspar phases [33] Three quasicrystallinephases have been found: a face-centered icosahedral (fci) phase, a simple cu-bic icosahedral (si) phase, and a phase with decagonal structure Crystallinestructures which are related to the quasicrystalline phases are found as well(for references see, e.g., [16])
Tracer diffusion experiments of 65Zn have been performed on dral Zn64.2Mg26.4Ho9.4and Zn60.7Mg30.6Y8.7 quasicrystals [34] grown by thetop-seeded solution-growth technique [16] Diffusion data are displayed inFig 21.5 together with tracer diffusion of 65Zn in a related crystalline Zn-Mg-Y phase with hexagonal structure It is not surprising that Zn diffusion
icosahe-Fig 21.5. Self-diffusion of 65Zn in icosahedral Zn64.2Mg26.4Ho9.4 and
Zn60.7Mg30.6Y8.7 quasicrystals and in a related hexagonal phase (h-ZnMgY)
ac-cording to Galler et al [34] Dashed lines: self-diffusion in Zn parallel and pendicular to its hexagonal axis; dotted line: Zn diffusion in icosahedral Al-Pd-Mn
Trang 6per-21.2 Diffusion Properties of Quasicrystals 379
in the closely related Zn-Mg-Ho and Zn-Mg-Y quasicrystals proceeds at most identical rates Its diffusion is, however, slower than self-diffusion inmetallic Zn This difference can be attributed only partly to the differentmelting 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
al-in hexagonal Zn is still about one order of magnitude faster Zn diffusion al-inicosahedral Al-Pd-Mn (dotted line) proceeds about one order of magnitudefaster than Zn diffusion in the Zn-based quasicrystals A comparison between
Zn diffusion in the Zn-based quasicrystals and the ternary crystalline pound with similar composition reveals similar diffusivities of Zn diffusion
com-in particular for diffusion perpendicular to the hexagonal axis These factsand the magnitude of the pre-exponential factors for Zn diffusion in both
quasicrystals (Zn-Mg-Ho: 3.9 ×10 −3m2s−1 ; Zn-Mg-Y: 3.3 ×10 −3m2s−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 decagonalaxis, whereas they are periodic parallel to the decagonal axis [35] The ternarysystems Al-CoAl-NiAl and Al-CoAl-CuAl are characterised by the formation
of decagonal phases; no icosahedral phases are observed [21] The field ofthe decagonal phase is elongated with respect to a widely varying Ni to Coratio In contrast, only a slight variation of the Al content is possible withoutleaving the phase field with quasicrystalline order
Diffusion of several tracers has been studied on oriented single crystalsparallel 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 onsingle crystals of Al72.6Ni10.5Co16.9 using 57Co and 63Ni as radiotracers byKhoukaz et al [37, 38] Diffusion of 63Ni has been studied by the sameauthors studied on polycrystals of the composition Al70.2Ni15.1Co14.7 Dataobtained by Nakajima and coworkers for diffusion of60Co in decagonalAl72.2Ni11.8Co16 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 diffusiondepends only weakly on composition Diffusion of Co and Ni in both principaldirections obey Arrrhenius laws It is remarkable that no significant deviationfrom Arrhenius behaviour has been detected down to the lowest tempera-tures In a temperature scale normalised with the melting temperatures, thediffusivities 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
Trang 7380 21 Diffusion in Quasicrystalline Alloys
Fig 21.6 Self-diffusion of65Ni,60Co, and57Co in decagonal Al-Ni-Co tals from [14, 34, 36] Monte Carlo simulations of Al diffusion are also shown [40]
quasicrys-element diffuses significantly faster than the transition quasicrys-elements Diffusivitiesdeduced 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 Alself-diffusion
The combination of crystalline and quasicrystalline order makes decagonalAl-Ni-Co particularly interesting from the viewpoint of diffusion mechanisms.Quasicrystalline order exists only in layers perpendicular to the decagonalaxis, whereas periodic (crystalline) order prevails parallel to the decagonalaxis 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 toquasiperiodic order and phason-mediated diffusion should cause a diffusionanisotropy 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 60Co diffusion in Al72.2Ni11.8Co16 the anisotropy is even opposite tothe expectation for phason-dominated diffusion in the whole temperature
Trang 8References 381range [13] This is even more so for Ga diffusion, where diffusion parallel todecagonal 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 ofuniaxial metals and the decagonal quasicrystals provides additional evidencefor 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 Masumoto, Mater Trans Japan Inst
Met 31, 98 (1990)
9 Z.P Luo, S Zhang, Y Tang, D Zhao, Scripta Metal et Mat 28, 1513 (1993)
10 H.-U Nissen, C Beeli, Electron Microscopy and Surface Investigations of
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11 J.-M Dubois, Bulk and Surface Properties of Quasicrystalline Materials and
their Potential Applications, Chap 28 in [4]
12 J.-M Dubois, Useful Quasicrystals, World Scientific Publ Comp., Singapore,
2005
13 H Nakajima, Th Zumkley, Defect and Diffusion Forum 194–199, 789 (2001)
14 H Mehrer, R Galler, W Frank, R Bl¨uher, A Strohm, Diffusion in
Quasicrys-tals, Chap 4.1 in [5]
15 M Feuerbacher, C Thomas, K Urban, Single Quasicrystal Growth, Chap 1.1
in [5]
16 R Sterzel, E Uhrig, E Dahlmann, A Langsdorf, W Assmus, Preparation
of Zn-Mg-RE Quasicrystals and Related Compounds (RE = Y, Ho, Er, Dy),
Chap 1.3 in [5]
17 P Gille, R.-U Barz, L.M Zhang, Growth of Decagonal Al-Ni-Co and Al-Co-Cu
Quasicrystals by the Czochralski Method, Chap 1.5 in [5]
18 D Levine, P.J Steinhardt, Phys Rev Lett 53, 2477 (1984)
19 M Boudard, M de Boissieu, H Vincent, G Heger, C Beeli, H.-U Nissen, R
Ibberson, M Audier, J.-M Dubois, C Janot, J Phys: Cond Matter 4, 10149
(1992)
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20 C Janot, Phys Rev B 73, 181 (1996)
21 R L¨uck, Production of Quasicrystalline Alloys and Phase Diagrams, p 222
in [4], 2002
22 C Beeli, H.-U Nissen, J Robadey, Philos Mag Lett 63, 87 (1991)
23 A.D Le Claire, G Neumann, Diffusion of Impurities in Metallic Elements, Chap 3 in: Diffusion in Solid Metals and Alloys, H Mehrer (Vol Ed.), Landolt
B¨ornstein, New Series, Group III: Crystal and Solid State Physics, Vol 26,Springer-Verlag, 1990
24 R Messer, S Dais, D Wolf, in: Proc 18th Ampere Congress, P.S Allen, E.R.Andrew, C.A Bates (Eds.), Nottingham, England, 1974
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Trang 10Part IV
Diffusion in Semiconductors
Trang 1122 General Remarks on Semiconductors
The present chapter and the two subsequent ones deal with diffusion in theelemental semiconductors Si and Ge Semiconducting materials play a major
rˆole in high-tech equipment used in industry and in daily life Silicon (Si)
is the most important semiconductor for the fabrication of microelectronicdevices such as memory and processor chips for computers and solar cells forenergy production in photovoltaic devices Germanium (Ge) constitutes the
base material for γ-radiation detectors Gallium arsenide (GaAs) and other
compounds of group III and group V elements of the periodic table are mainlyused in opto-electronic and high-frequency devices such as solid-state lasers incompact-disc players and receivers in cellular phones SiC is a semiconductorwith large band gap and has potential for applications in devices that mustoperate at high temperatures, high frequencies, and under irradiation
We remind the reader that both Si and Ge crystallise in the diamondstructure Many III-V compounds like GaAs occur in the zinc blende struc-ture, which is closely related to the diamond structure In both structures,the Bravais lattice is face-centered cubic and the structure can be created
by the translation of two atoms at the positions (0,0,0) and a4(1,1,1), where
a is the cubic lattice parameter In the diamond structure, both positions
are occupied by the same type of atoms; in the zinc blende structure, thebasis 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)
Trang 12386 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 [1]
parameter at 300 K in eV,
in nm type of band gap
Germanium (Ge) Diamond 0.564613 0.663, indirectGallium nitride (GaN) Wurtzite a: 0.3111,
c: 0.4978 3.7, directGallium phosphide (GaP) Zinc blende 0.54512 2.261, indirectGallium arsenide (GaAs) Zinc blende 0.56532 1.435, directGallium antimonide (GaSb) Zinc blende 0.60959 0.72, directIndium phosphide (InP) Zinc blende 0.58687 1.351, directIndium arsenide (InAs) Zinc blende 0.60584 0.35, directIndium antimonide (InSb) Zinc blende 0.64794 0.280, direct
coordination number is 4, each atom is surrounded by a tetrahedron of fourneighbouring atoms, and the bonding is covalent The crystal structure, thelattice constant, the ambient temperature band gap energy, and the type ofband gap are listed for the elemental group IV semiconductors and for someIII-V compound semiconductors in Table 22.1 We note that both Si and Geare semiconductors with indirect band gaps, whereas most compound semi-conductors have direct band gaps We also note that the packing density ofatoms in semiconductor structures is considerably lower than in close-packedmetals Silicon-carbide (SiC) crystallises in various polytypes and has a veryhigh melting temperature of 2545◦C Depending on the polytype the band
gap lies between 2.39 and 3.26 eV
22.1 ‘Semiconductor Age’ and Diffusion
Periods of mankind are named after materials: stone age, bronze age, andiron age In the 1970s the number of publications about semiconductors out-numbered for the first time those about steels and some people started todenote the present period as the ‘semiconductor age’ This development wasinitiated in 1945, when a research group at the Bell Telephone Laboratories
in the United States was established to focus on a better understanding ofsemiconductors Vacuum tube technology had fully matured at that time but
it had become also clear that the short life and the high power consumption oftubes would limit further progress in telephony and other electronic endeav-ours The transistor effect was discovered in 1947 by William B Shockley,John Bardeen and Walter H Brattain, three members of the Bell Labs
group, who received the Nobel prize in physics in 1956 ‘ for their studies