- Email: [email protected]
Role of ballistic transport in photoluminescence excitation of Si nanocrystals
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Journal of Non-Crystalline Solids 354 (2008) 2296–2299 www.elsevier.com/locate/jnoncrysol
Role of ballistic transport in photoluminescence excitation of Si nanocrystals T.V. Torchynska * ESFM-Instituto Polite´cnico Nacional, Ed. 9, U.PA.L.M., Me´xico 07738, Mexico Available online 13 February 2008
Abstract Photoluminescence of Si NCs with the size (10–300 nm) bigger than the exciton Bohr radius in the bulk Si crystals (4.8 nm) has been considered. Photoluminescence in such NC systems is analyzed from the point of view of new concept based on the effect of hot carrier ballistic transport in excitation of suboxide defect-related photoluminescence at the Si/SiOx interface. The dependence of the 1.70 eV PL band integrated intensity on Si NC sizes was numerically calculated on the base of the hot carrier ballistic PL model. The well correlation between calculated and experimental results has been obtained for Si NCs with the size from the 30–150 nm range. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Silicon; Nanocrystals; Optical spectroscopy; Atomic force and scanning tunneling microscopy; Scanning electron microscopy; Modeling and simulation; Nanoparticles; Optical properties; Luminescence
1. Introduction The interest on Si nanocrystals (NCs) was stimulated by the discovery of bright red photoluminescence (PL) of porous silicon (PSi) at the room temperature in earlier 1990th of the last century. Numerous scientific publications were issued with results of the photoluminescence (PL) investigation of Si NCs in different types of matrices [1–5]. However up to now the mechanism of bright red PL of PSi is still under discussion as well as the advantages (or disadvantages) of the application of Si NCs in optoelectronic light emitting devices are not clear. Joint investigations of optical absorption and PL spectra of PSi with different size NCs (2.0, 3.5 and 9.0 nm) have shown a gradual increase in the optical absorption coefficient near the absorption edge when the Si NC size decreases [4]. At the same time no significant size dependence of PL peak energy was observed. Latter the size dependence of peak positions of infrared (1.4–1.6 eV) PL
band has been revealed for Si NCs embedded in the SiOx [5,6]. But the intensity of this PL band was very low and essentially less than intensity of the red PL band in PSi [6]. As a rule two groups of Si NCs have to be discussed. The first group deals with Si NCs of small sizes (a 6 aB, where aB is the exciton Bohr radius equal to 4.8 nm for the bulk Si). PL in such NCs is connected with optical transitions between localized electronic states formed due to the strong quantum confinement effect. The second group deals with NCs of the big size (a > aB) from the range of 10–300 nm. Last case is known as the weak quantum confinement regime and PL for these NCs is controlled by the hot carrier ballistic effect [7]. The paper presents the comparison of PL intensity dependences, obtained experimentally in PSi and numerically calculated on the base of hot carrier ballistic PL model [3,7], versus Si NC sizes from the range of 30–300 nm.
2. Experimental details
*
Tel./fax: +52 55 5752 6008. E-mail address: [email protected]
0022-3093/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2007.10.085
PSi layers were prepared as described earlier in [3,6,7]. The size of Si NCs has been estimated using the JSM-T20 (JEOL) scanning electron (SEM) and Nanoscope
T.V. Torchynska / Journal of Non-Crystalline Solids 354 (2008) 2296–2299
2297
IIIa atomic-force (AFM) microscopes and then it is compared with earlier results obtained by the Raman scattering method [8]. PL spectra were excited by the 3.68 eV N2 laser line or by the 2.40 eV Ar ion laser line. PL spectra at 300 K were registered using the IKS-12 spectrometer with two detectors for different spectral ranges [3,6]. All PL spectra were corrected on setup spectral response.
3. Results and discussion PL spectra of PSi samples prepared at different technological conditions are presented in Fig. 1 for two excitation regimes. PL bands peaked at 1.70 and 1.90–2.00 eV have been revealed, which are characterized by different dependences on excitation light energy. Actually the 1.70 eV PL band intensity increases, but the 1.9 eV PL band intensity decreases, with the change of excitation energy from 2.40 to 3.60 eV. AFM images of PSi layers prepared at different etching current densities are shown in Fig. 2. Table 1 presents the integrated intensity of both visible PL bands and the average size of Si NCs in studied PSi estimated by AFM and SEM methods. As one can see the 1.7 eV PL band does not change its peak position versus Si NC sizes (Table 1). The intensity of the 1.7 eV PL band varies none monotonically with NC sizes and exhibits a maximum at the value equal to 38.4–40.0 nm (Table 1 and Fig. 1). The highest intensity this PL band approaches at excitation light energy of 3.60 eV. This light is absorbed in the depth of 100 nm mainly, which is comparable with PSi surface roughness and Si NC sizes on PSi surface. The intensity and peak position of the 1.9–2.0 eV PL band vary with Si NC sizes, surface area and oxidation [7]. As we have shown earlier in [3,7] the 1.7 eV PL band deals with suboxide related defects at the Si/SiOx interface and the 1.9–2.0 eV PL band is asso-
Photoemission intensity (arb. un.)
70
ciated with defects in the silicon dioxide layer on Si NC surface.
a
4
60
Fig. 2. AFM images of PSi layers prepared at time duration 10 min and etching currents 25 (a) and 75 (b) mA/cm2.
b
3
3
4 2
50 40
5 30
5
1
2
20
1 10 0 1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
Emission energy (eV) Fig. 1. PL spectra of PSi layers prepared by etching process using the current densities: 5 (1), 10 (2), 25 (3), 50 (4) and 75 (5) mA/cm2 and duration 10 min, measured at the two excitation energies: 2.40 eV (a) and 3.60 eV (b).
2298
T.V. Torchynska / Journal of Non-Crystalline Solids 354 (2008) 2296–2299
Table 1 Average NC sizes in different PSi samples measured by AFM and SEM Groups of samples
ta (min)
Ia (mA/cm2)
Average size (nm)
Standard deviation (nm)
Surface area difference (%)
Integrated intensity 1.7 eV PL band (arb. un.)
Integrated intensity 1.9–2.0 eV PL band (arb. un.)
1 2 3 4 5
10 10 10 10 10
5 10 20 60 75
88 60 40 260 357
27 37 15 52 54
4.6 5.8 8.9 11.9 58.6
30.4 51.5 68.0 12.1 2.0
10.0 17.0 23.8 55.5 34.0
4. PL model for the 1.7 eV PL band The numerical calculation based on the hot carrier ballistic PL model has been done for the 1.7 eV PL band. Ballistic (or quazi ballistic) motion of hot electrons in Si NCs toward the Si/SiOx interface can be realized if the diameter of Si NCs is small enough in comparison with the free electron path. In this case inelastic electron scattering is suppressed mainly. Taking into account the electron relaxation time, equal to 1012–1013 s, in the bulk Si conduction band, it is shown that hot carrier ballistic (or quazi ballistic) transport at PL excitation of oxide related defects is essential for Si NCs with the size of 10–100 nm [7]. The competition of three basic mechanisms: (i) electron phonon inelastic scattering, (ii) nonradiative recombination, (iii) tunneling via oxide and trapping by emission centers, has been considered at hot carrier energy relaxation in Si NCs. The 1.7 eV PL band intensity for big size Si NCs (a > aB) is estimated using the formula: I PL ¼
Z
L0 þ2r
2
P ðLÞI 0 ð1 RÞ ½1 expðaLÞg dL;
ð1Þ
L0 2r
where P(L) is the size distribution function, r is NC size dispersion, I0 is an excitation light intensity, R and a are reflection and absorption coefficients, L is a Si NC size, g is internal PL quantum efficiency in Si NCs. The parameter g is estimated using a relation: g ¼ IIred0 ¼
PL intensity (arb. units.)
10
1
1
setun
1
1
seph-Si þsetun þseNR
, where
seph-Si , setun , seNR are carrier thermalization, tunneling and nonradiative recombination times. At the estimation of a g value the next parameters have been taken: seph-Si = 1/ xph = 1012 s, seNR = 1010 s, which is typical for the bulk Si, and setun is calculated using the formula [9]: s1 tun ¼
F ðEz Þrcp N LC expð2d 0 jz Þ; jz
where d0 is oxide thickness on Si NC surface, NLC is a concentration of oxide-related luminescent centers at the Si/ SiOx interface, taking as 1.0–2.5 1020 cm3, F(E) is a frequency of interface kicking by hot carriers and rcp is a capture coefficient of luminescence centers. The numerical calculation of the PL intensity has been done for Gaussian distribution of NC sizes with the dispersion parameter r presented in Table 1. PL intensities estimated using formula (1) for different average sizes of Si NCs are presented in Fig. 3. At the calculation the concentration of luminescence centers at the Si/SiOx interface equal to 1 1020 cm3 has been used. As one can see the PL intensity decreases very fast when the average NC sizes rise up to 90–150 nm. The corresponding variation of the PL quantum efficiency g versus Si NC sizes is presented in Fig. 4(a). These dependences are calculated for two concentrations of interface related luminescence centers equal to 1 1020 and 2.5 1020 cm3. Integrated PL intensities were numerically calculated for PL spectra of the Fig. 3. Comparison of calculated integrated PL intensities with experimental values for the
20 nm 40 nm 90 nm 150 nm
1
250 nm 350 nm
0.1 0
100
ð2Þ
200
300
400
NC sizes L0 (nm) Fig. 3. Calculated PL spectra for different Si NC average sizes and for the luminescence center concentration equal to 1 1020 cm3.
T.V. Torchynska / Journal of Non-Crystalline Solids 354 (2008) 2296–2299
1.7 eV PL band versus NC sizes is shown in Fig. 4(b). As one can see the PL mechanism deals with hot carrier ballistic (or quazi ballistic) transport and PL excitation of interface related luminescence centers is essential for Si NCs with the size of 30–150 nm and the emission center concentration of 1.0–2.5 1020 cm3.
0.5
0.4
0.3
η
5. Conclusion
0.2
2
The dependence of the 1.70 eV PL band integrated intensity on Si NC sizes was numerically calculated on the base of the hot carrier ballistic PL model. The well correlation between calculated and experimental results has been obtained for Si NCs with the size from the 30– 150 nm range.
1
0.1
0.0 0
100
200
300
400
500
Si NC size, d (nm) Fig. 4(a). Variations of quantum efficiency g coefficients versus NC sizes for two luminescent center concentrations, NLC: 1 (1) and 2.5 (2) 1020 cm3.
120
Integrated PL intensity (arb. units)
2299
100
Acknowledgements The work was partially supported by the International cooperation CONACYT, Mexico–Ukraine program as well as by the SIP-IPN, Mexico. References
80 60 40 20 0 -20 0
100
200
300
400
Average NC size, L0 (nm) Fig. 4(b). Comparison of calculated (line) and experimental (points) results obtained for the integrated intensity of the 1.70 eV PL band versus Si NC sizes.
[1] F.G. Becerril Expinoza, N. Korsunska, L. Khomenkova, T.V. Torchynska, Y. Goldstein, E. Savir, J. Jedrzejewski, Phys. Stat. Sol. (C) 2 (2005) 2990. [2] E.F. Venger, T.Ya. Gorbach, L.A. Matveeva, S.V. Svechnikov, J. Exp. Theor. Phys. 89 (1999) 948. [3] T.V. Torchynska, M. Morales Rodriguez, F.G. Bacarril-Espinoza, N.E. Korsunskaya, L.Yu Khomenkova, L.V. Scherbina, Phys. Rev. B 65 (2002) 115313. [4] Y. Kanemitsu, T. Ogawa, K. Shiraishi, K. Takeda, Phys. Rev. B 48 (1993) 4883. [5] K. Nishiguchi, S. Oda, J. Appl. Phys. 92 (2002) 1399. [6] N. Korsunska, L. Khomenkova, M.K. Sheinkman, T. Stara, V. Yuhimchuk, T.V. Torchynska, A. Vivas Hernandez, J. Lumin. 115 (2005) 117. [7] T.V. Torchynska, J. Appl. Phys. 92 (2002) 4019. [8] N. Korsunska, B. Bulakh, B. Jumayev, L. Khomenkova, V. Yukhymchuk, T. Torchynska, Appl. Surf. Sci. 243 (2005) 30. [9] G.G. Qin, Mater. Res. Bull. 33 (1998) 1857.
Journal of Non-Crystalline Solids 354 (2008) 2296–2299 www.elsevier.com/locate/jnoncrysol
Role of ballistic transport in photoluminescence excitation of Si nanocrystals T.V. Torchynska * ESFM-Instituto Polite´cnico Nacional, Ed. 9, U.PA.L.M., Me´xico 07738, Mexico Available online 13 February 2008
Abstract Photoluminescence of Si NCs with the size (10–300 nm) bigger than the exciton Bohr radius in the bulk Si crystals (4.8 nm) has been considered. Photoluminescence in such NC systems is analyzed from the point of view of new concept based on the effect of hot carrier ballistic transport in excitation of suboxide defect-related photoluminescence at the Si/SiOx interface. The dependence of the 1.70 eV PL band integrated intensity on Si NC sizes was numerically calculated on the base of the hot carrier ballistic PL model. The well correlation between calculated and experimental results has been obtained for Si NCs with the size from the 30–150 nm range. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Silicon; Nanocrystals; Optical spectroscopy; Atomic force and scanning tunneling microscopy; Scanning electron microscopy; Modeling and simulation; Nanoparticles; Optical properties; Luminescence
1. Introduction The interest on Si nanocrystals (NCs) was stimulated by the discovery of bright red photoluminescence (PL) of porous silicon (PSi) at the room temperature in earlier 1990th of the last century. Numerous scientific publications were issued with results of the photoluminescence (PL) investigation of Si NCs in different types of matrices [1–5]. However up to now the mechanism of bright red PL of PSi is still under discussion as well as the advantages (or disadvantages) of the application of Si NCs in optoelectronic light emitting devices are not clear. Joint investigations of optical absorption and PL spectra of PSi with different size NCs (2.0, 3.5 and 9.0 nm) have shown a gradual increase in the optical absorption coefficient near the absorption edge when the Si NC size decreases [4]. At the same time no significant size dependence of PL peak energy was observed. Latter the size dependence of peak positions of infrared (1.4–1.6 eV) PL
band has been revealed for Si NCs embedded in the SiOx [5,6]. But the intensity of this PL band was very low and essentially less than intensity of the red PL band in PSi [6]. As a rule two groups of Si NCs have to be discussed. The first group deals with Si NCs of small sizes (a 6 aB, where aB is the exciton Bohr radius equal to 4.8 nm for the bulk Si). PL in such NCs is connected with optical transitions between localized electronic states formed due to the strong quantum confinement effect. The second group deals with NCs of the big size (a > aB) from the range of 10–300 nm. Last case is known as the weak quantum confinement regime and PL for these NCs is controlled by the hot carrier ballistic effect [7]. The paper presents the comparison of PL intensity dependences, obtained experimentally in PSi and numerically calculated on the base of hot carrier ballistic PL model [3,7], versus Si NC sizes from the range of 30–300 nm.
2. Experimental details
*
Tel./fax: +52 55 5752 6008. E-mail address: [email protected]
0022-3093/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2007.10.085
PSi layers were prepared as described earlier in [3,6,7]. The size of Si NCs has been estimated using the JSM-T20 (JEOL) scanning electron (SEM) and Nanoscope
T.V. Torchynska / Journal of Non-Crystalline Solids 354 (2008) 2296–2299
2297
IIIa atomic-force (AFM) microscopes and then it is compared with earlier results obtained by the Raman scattering method [8]. PL spectra were excited by the 3.68 eV N2 laser line or by the 2.40 eV Ar ion laser line. PL spectra at 300 K were registered using the IKS-12 spectrometer with two detectors for different spectral ranges [3,6]. All PL spectra were corrected on setup spectral response.
3. Results and discussion PL spectra of PSi samples prepared at different technological conditions are presented in Fig. 1 for two excitation regimes. PL bands peaked at 1.70 and 1.90–2.00 eV have been revealed, which are characterized by different dependences on excitation light energy. Actually the 1.70 eV PL band intensity increases, but the 1.9 eV PL band intensity decreases, with the change of excitation energy from 2.40 to 3.60 eV. AFM images of PSi layers prepared at different etching current densities are shown in Fig. 2. Table 1 presents the integrated intensity of both visible PL bands and the average size of Si NCs in studied PSi estimated by AFM and SEM methods. As one can see the 1.7 eV PL band does not change its peak position versus Si NC sizes (Table 1). The intensity of the 1.7 eV PL band varies none monotonically with NC sizes and exhibits a maximum at the value equal to 38.4–40.0 nm (Table 1 and Fig. 1). The highest intensity this PL band approaches at excitation light energy of 3.60 eV. This light is absorbed in the depth of 100 nm mainly, which is comparable with PSi surface roughness and Si NC sizes on PSi surface. The intensity and peak position of the 1.9–2.0 eV PL band vary with Si NC sizes, surface area and oxidation [7]. As we have shown earlier in [3,7] the 1.7 eV PL band deals with suboxide related defects at the Si/SiOx interface and the 1.9–2.0 eV PL band is asso-
Photoemission intensity (arb. un.)
70
ciated with defects in the silicon dioxide layer on Si NC surface.
a
4
60
Fig. 2. AFM images of PSi layers prepared at time duration 10 min and etching currents 25 (a) and 75 (b) mA/cm2.
b
3
3
4 2
50 40
5 30
5
1
2
20
1 10 0 1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
Emission energy (eV) Fig. 1. PL spectra of PSi layers prepared by etching process using the current densities: 5 (1), 10 (2), 25 (3), 50 (4) and 75 (5) mA/cm2 and duration 10 min, measured at the two excitation energies: 2.40 eV (a) and 3.60 eV (b).
2298
T.V. Torchynska / Journal of Non-Crystalline Solids 354 (2008) 2296–2299
Table 1 Average NC sizes in different PSi samples measured by AFM and SEM Groups of samples
ta (min)
Ia (mA/cm2)
Average size (nm)
Standard deviation (nm)
Surface area difference (%)
Integrated intensity 1.7 eV PL band (arb. un.)
Integrated intensity 1.9–2.0 eV PL band (arb. un.)
1 2 3 4 5
10 10 10 10 10
5 10 20 60 75
88 60 40 260 357
27 37 15 52 54
4.6 5.8 8.9 11.9 58.6
30.4 51.5 68.0 12.1 2.0
10.0 17.0 23.8 55.5 34.0
4. PL model for the 1.7 eV PL band The numerical calculation based on the hot carrier ballistic PL model has been done for the 1.7 eV PL band. Ballistic (or quazi ballistic) motion of hot electrons in Si NCs toward the Si/SiOx interface can be realized if the diameter of Si NCs is small enough in comparison with the free electron path. In this case inelastic electron scattering is suppressed mainly. Taking into account the electron relaxation time, equal to 1012–1013 s, in the bulk Si conduction band, it is shown that hot carrier ballistic (or quazi ballistic) transport at PL excitation of oxide related defects is essential for Si NCs with the size of 10–100 nm [7]. The competition of three basic mechanisms: (i) electron phonon inelastic scattering, (ii) nonradiative recombination, (iii) tunneling via oxide and trapping by emission centers, has been considered at hot carrier energy relaxation in Si NCs. The 1.7 eV PL band intensity for big size Si NCs (a > aB) is estimated using the formula: I PL ¼
Z
L0 þ2r
2
P ðLÞI 0 ð1 RÞ ½1 expðaLÞg dL;
ð1Þ
L0 2r
where P(L) is the size distribution function, r is NC size dispersion, I0 is an excitation light intensity, R and a are reflection and absorption coefficients, L is a Si NC size, g is internal PL quantum efficiency in Si NCs. The parameter g is estimated using a relation: g ¼ IIred0 ¼
PL intensity (arb. units.)
10
1
1
setun
1
1
seph-Si þsetun þseNR
, where
seph-Si , setun , seNR are carrier thermalization, tunneling and nonradiative recombination times. At the estimation of a g value the next parameters have been taken: seph-Si = 1/ xph = 1012 s, seNR = 1010 s, which is typical for the bulk Si, and setun is calculated using the formula [9]: s1 tun ¼
F ðEz Þrcp N LC expð2d 0 jz Þ; jz
where d0 is oxide thickness on Si NC surface, NLC is a concentration of oxide-related luminescent centers at the Si/ SiOx interface, taking as 1.0–2.5 1020 cm3, F(E) is a frequency of interface kicking by hot carriers and rcp is a capture coefficient of luminescence centers. The numerical calculation of the PL intensity has been done for Gaussian distribution of NC sizes with the dispersion parameter r presented in Table 1. PL intensities estimated using formula (1) for different average sizes of Si NCs are presented in Fig. 3. At the calculation the concentration of luminescence centers at the Si/SiOx interface equal to 1 1020 cm3 has been used. As one can see the PL intensity decreases very fast when the average NC sizes rise up to 90–150 nm. The corresponding variation of the PL quantum efficiency g versus Si NC sizes is presented in Fig. 4(a). These dependences are calculated for two concentrations of interface related luminescence centers equal to 1 1020 and 2.5 1020 cm3. Integrated PL intensities were numerically calculated for PL spectra of the Fig. 3. Comparison of calculated integrated PL intensities with experimental values for the
20 nm 40 nm 90 nm 150 nm
1
250 nm 350 nm
0.1 0
100
ð2Þ
200
300
400
NC sizes L0 (nm) Fig. 3. Calculated PL spectra for different Si NC average sizes and for the luminescence center concentration equal to 1 1020 cm3.
T.V. Torchynska / Journal of Non-Crystalline Solids 354 (2008) 2296–2299
1.7 eV PL band versus NC sizes is shown in Fig. 4(b). As one can see the PL mechanism deals with hot carrier ballistic (or quazi ballistic) transport and PL excitation of interface related luminescence centers is essential for Si NCs with the size of 30–150 nm and the emission center concentration of 1.0–2.5 1020 cm3.
0.5
0.4
0.3
η
5. Conclusion
0.2
2
The dependence of the 1.70 eV PL band integrated intensity on Si NC sizes was numerically calculated on the base of the hot carrier ballistic PL model. The well correlation between calculated and experimental results has been obtained for Si NCs with the size from the 30– 150 nm range.
1
0.1
0.0 0
100
200
300
400
500
Si NC size, d (nm) Fig. 4(a). Variations of quantum efficiency g coefficients versus NC sizes for two luminescent center concentrations, NLC: 1 (1) and 2.5 (2) 1020 cm3.
120
Integrated PL intensity (arb. units)
2299
100
Acknowledgements The work was partially supported by the International cooperation CONACYT, Mexico–Ukraine program as well as by the SIP-IPN, Mexico. References
80 60 40 20 0 -20 0
100
200
300
400
Average NC size, L0 (nm) Fig. 4(b). Comparison of calculated (line) and experimental (points) results obtained for the integrated intensity of the 1.70 eV PL band versus Si NC sizes.
[1] F.G. Becerril Expinoza, N. Korsunska, L. Khomenkova, T.V. Torchynska, Y. Goldstein, E. Savir, J. Jedrzejewski, Phys. Stat. Sol. (C) 2 (2005) 2990. [2] E.F. Venger, T.Ya. Gorbach, L.A. Matveeva, S.V. Svechnikov, J. Exp. Theor. Phys. 89 (1999) 948. [3] T.V. Torchynska, M. Morales Rodriguez, F.G. Bacarril-Espinoza, N.E. Korsunskaya, L.Yu Khomenkova, L.V. Scherbina, Phys. Rev. B 65 (2002) 115313. [4] Y. Kanemitsu, T. Ogawa, K. Shiraishi, K. Takeda, Phys. Rev. B 48 (1993) 4883. [5] K. Nishiguchi, S. Oda, J. Appl. Phys. 92 (2002) 1399. [6] N. Korsunska, L. Khomenkova, M.K. Sheinkman, T. Stara, V. Yuhimchuk, T.V. Torchynska, A. Vivas Hernandez, J. Lumin. 115 (2005) 117. [7] T.V. Torchynska, J. Appl. Phys. 92 (2002) 4019. [8] N. Korsunska, B. Bulakh, B. Jumayev, L. Khomenkova, V. Yukhymchuk, T. Torchynska, Appl. Surf. Sci. 243 (2005) 30. [9] G.G. Qin, Mater. Res. Bull. 33 (1998) 1857.