- Email: [email protected]
(n)c-Si heterojunction
ARTICLE IN PRESS
Vacuum 71 (2003) 465–469
Study on variable capacitance diode of (p)nc-Si:H/(n)c-Si heterojunction Wei Wenshenga,b,*, Wang Tianmina, Zhang Chunxib, Li Guohuac, Li Yuexiac a
Center of Material Physics & Chemistry, School of Science, Beijing University of Aeronautics & Astronautics, Beijing 100083, China b Institute of Optic-electronics, Beijing University of Aeronautics & Astronautics, Beijing 100083, China c National Key Lab. For Semiconductor Superlattice & Microstructure, Institute of Semiconductor, Chinese Academy of Science, Beijing 100083, China
Abstract Hydrogenated nanocrystalline silicon (nc-Si:H) layers of boron-doped increasing step by step was deposited on ntype crystalline silicon substrate using Plasma Enhanced Chemical Vapor Deposition (PECVD) system. After evaporating Ohm contact electrode on the side of substrate and on the side of nc-Si:H film, a structure of electrode/ (p)nc-Si:H/(n)c-Si/electrode was obtained. It is confirmed by electrical measurement such as I–V curve, C–V curve and DLTS that this is a variable capacitance diode. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: nc-Si:H film; (p)nc-Si:H/(n)c-Si heterojunction; variable capacitance diode
1. Introduction nc-Si:H film has been receiving increasing attention since novel properties such as high conductivity and visible photoluminescence at low and room temperatures due to quantum size effect. Recently, nc-Si:H film was widely used to fabricate [1–5] solar cells, large area display devices, single electron transistors, micro electromechanical system, resonant-tunneling diodes, heterojunction diode, superlattice devices and so on. The structure and electronic transport me-
chanism through interface of nc-Si:H/c-Si were active research points [6,7], an nc-Si:H/c-Si tunneling diode and an (n)nc-Si:H/(p)c-Si abrupt heterojunction diode have ever been fabricated by PECVD system and a series results was reported [3–5]. In this work, the variable capacitance diode of p-type nc-Si:H/n-type c-Si was fabricated using PECVD and its electrical behavior was characterized and discussed.
2. Preparation of the variable capacitance diode *Corresponding author. Center of Material Physics & Chemistry, School of Science, Beijing University of Aeronautics & Astronautics, Beijing 100083, China. E-mail address: [email protected] (W. Wensheng).
In order to fabricate the variable capacitance diode, n-type (100) silicon wafer with average resistivity of 0.5–1.0 O cm (NDE2 1016 cm3) was selected as substrate. A SiO2 layer around
0042-207X/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0042-207X(03)00034-4
ARTICLE IN PRESS 466
W. Wensheng et al. / Vacuum 71 (2003) 465–469
1 mm in thickness was prepared by thermal oxidation at 1020 C. The SiO2 layer was patterned by photolithography etched to make an array of square holes (30 mm 30 mm). After appropriate treatment, a fresh boron-doped nc-Si:H film of boron concentration increasing step by step was deposited using PECVD system by rf (13.56 MHz) and dc bias stimulation. A strongly hydrogen diluted silane (SiH4 diluted to about 1% in H2) was used as reactant gas. Doping was incorporated by adding B2H6 into the mixed reactant gas, while the reactant gas pressure was 0.7 Torr, the substrate’s temperature was 250 C, the power density was 0.60 W/cm2, and a negative bias of 200 V was applied in the depositing process. In order to obtain a linear slowly varied heterojunction, the doping concentration, Cp=B2H6/SiH4 vol%, was increased step by step from 0.00 to 5.00 vol%. The nc-Si:H layer out of the square holes was removed by photolithography etching leaving only the boron doped nc-Si:H layer in the hole bottoms. In the process of deposition, there were three main technological crucial factors in forming the nanophase grain in the growing film. The first one was the hydrogen dilution ratio to saline, R=H2/SiH4 vol%, which was from around
Fig. 2. HRTEM image of nc-Si:H on C-Si (25 mm 25 mm).
98 to 99.5. The second one was the dc negative bias Vb applied to the parallel electrodes of the deposition system. The third one was reactant gas pressure less than 1.00 Torr. Finally, in order to obtain Ohm’s contact, alloy electrodes were deposited on the back side of substrate and on the nc-Si:H film using vacuum evaporation, the structure of alloy/(p)nc-Si:H/(n)c-Si/alloy was obtained and is shown in Fig. 1. A High Resolution Transmission Electron Microscope (HRTEM) analysis was performed to check the interface of nc-Si:H/c-Si and is shown in Fig. 2, from which one can find that no buffer layer exists in the cross section of nc-Si:H/c-Si.
3. Electrical measurement and discussion
Fig. 1. Model of variable capacitance diode.
I–V characteristics of alloy/(p)nc-Si:H/(n)c-Si/ alloy were measured by HP4140B (PA meter/DC Voltage Source) at different temperatures in the range of 77–573 K. C–V measurement was performed at 1 MHz using a HP4280A Precision LCR meter on a set of these diodes. The I–V curve and C–V curve measured at 300 K is shown in Fig. 3(a) and (b), respectively. From Fig. 3(a) one can obviously find that it was a typical I–V characteristic curve of a diode, the reverse breakdown voltage and the forward turn-on voltage were
ARTICLE IN PRESS W. Wensheng et al. / Vacuum 71 (2003) 465–469
Fig. 3. (a) I–V curve at 300 K of variable capacitance diode. (b) C–V curve at 300 K of variable capacitance diode.
VR=49.90 V and VD=0.81 V, respectively. The reverse current was just several nA. Comparing Fig. 3(a) and (b), one can easily find that VDEVbi. The forward J in nc-Si:H/c-Si heterojunction can be expressed as [4]: JpexpðEac =KTÞ½expðqV =ZKTÞ 1;
ð1Þ
where Eac is the activation energy Eac=0.68 eV, and ideality factor Z=1.6. From expression (1) one can find that at low applied voltage the temperature dependence of J is mainly determined by Eac. Marsal et al. [8,9] pointed out that the current in (n)a-Si:H/(p)c-Si heterojunctions was also satisfied to expression (1), and the value of Eac is 0.88 eV. Therefore, one can see that the value of Eac in (n) nc-Si:H/(p)c-Si heterojunction is significantly smaller than that in (n)a-Si:H/(p)c-Si heterojunction. Obviously, the small value of Eac is caused by the energy band offset in the nc-Si:H layer which
467
reduces the energy gap [4], also, it is the essential factor that nc-Si:H/c-Si heterojunctions show good temperature stability. By a comparison with the previous work [3,10], one can find the difference in the transport mechanism between nc-Si:H film and nc-Si:H/cSi heterojunction. The transport mechanism of the former is the tunneling of the electrons between the neighbor nanograins. While for the latter, current is dominated by the recombination in the nc-Si:H side of the space charge region. But they are not self-contradictory. The electrical transport in nc-Si:H film is determined by only one type of carrier (electrons), and the recombination is negligible. Furthermore, the thickness of the amorphous regions is small enough to make the tunneling possible. On the other hand, in nc-Si:H/ c-Si heterojunction, especially in the nc-Si:H side of the space charge region, both electrons and holes are involved. The defect states in the amorphous Si:H tissue act as trap levels and induce the recombination of carrier couples. One can find that both nanograins and amorphous tissue play important roles in the carriers transport in heterojunction. The reverse conduction is dominated by currentgenerated in the space charge region [4, 9]. The diode has good rectification properties, 4.36 104 of rectification proportion at 71.0 V and 300 K as shown in Fig. 3(a). From Fig. 3(b) one can easily find that the capacitance, C, is dramatically changed with the applied forward voltage V when V was smaller than the interior built potential Vbi. It is well fitted with the definition of the deletion-layer-capacitance Cj of linear slowly varied heterojunction [11]: Cj ¼
1=3 qae2s ; 12ðVbi V Þ
ð2Þ
where q is the absolute value of electron charge, a is the gradient of dopant density and es is the dielectric constant. From Fig. 3(b) one can find that there exists a link point near V=0.55 V in the C–V curve, which resulted from the carrier density being sensitive to the boron-doping in the nc-Si:H film, which is shown in Fig. 4. From Fig. 3(b), according to the definition of capacitance varied
ARTICLE IN PRESS W. Wensheng et al. / Vacuum 71 (2003) 465–469
468
Fig. 4. Relation between carrier density and B2H6/SiH4.
Fig. 5. DLTS of variable capacitance diode.
coefficient, g [12]: g¼
Cmax Cmin : 2ðCmax þ Cmin Þ
ð3Þ
One can calculate g=0.41 of this diode, that means it is much more than g=0.167 of a linear slowly varied single crystal silicon heterojunction. This differs from classical behavior and provides evidence of quantum effect of the capacitance [13,14]. The carrier density on the side of nc-Si:H film varied with the increase step by step of the boron dopant in the nc-Si:H layer as shown in Fig. 4, so that this p–n junction is not an abrupt one. From the Deep Level Transient Spectroscopy (DLTS) which is shown in Fig. 5, one can easily find that there were smooth peaks but not sharp ones in the curves of transient capacitance signal at different rate window time t, it is just confirmed that this is a real slowly varied p–n junction in the interface of nc-Si:H/c-Si. Because there is amorphous component of Si:Hn (1pnp3) about 50 vol% in the ncSi:H layer, the gap states are composed of deep energy levels distributed continuously in Si:H. When the temperature rises up to a value, the number of holes trapped by deep energy levels is changed dramatically, as well as the space charges in the barrier of heterojunction are changed obviously, thus resulting in a peak of transient capacitance. According to the method of DLTS [15,16], the spectrum of DLTS is contributed by the overlap of related gap states, but not from a single deep energy level, so as to the peaks of DLTS’ spectra at different t are those slowly
Fig. 6. C3R–V curve of variable capacitance diode (VRbVbi).
varied broad peaks but not those sharp peaks. Also, according Ref. [15], the temperature T corresponding to the peak of DLTS’ spectrum decreased with increasing of t, as shown in Fig. 5. From Fig. 6, one can find C3p VR nearly when VR is far larger than Vbi. It is tested that this heterojunction is a linear slowly varied junction according to Ref. [11].
4. Conclusion A structure of alloy/(p)nc-Si:H/(n)c-Si/alloy was fabricated. It is tested by HRTEM that there was no buffer layer in the cross section. Also it was confirmed by electric measurement of I–V, C–V and DLTS that the structure is a variable capacitance diode.
ARTICLE IN PRESS W. Wensheng et al. / Vacuum 71 (2003) 465–469
Acknowledgements This work was supported by the National Foundation of Natural Science of China (No. 59982002) and it is a Doctoral Stations Fund Project Sponsored by National Educational Administration (No. 200220006037). We are also grateful to Prof. He Yuliang for fruitful discussion.
References [1] Yue GZ, Han DX, Williamson DL, Yang J, Kenneth L, Subhendu GH. Appl Phys Lett 2000;77:3185–7. [2] Mamura YI, Ataka S, Takasaki Y, Kusano C, Hirai T, Maruyama E. Appl Phys Lett 1979;35:349–51. [3] He YL, Hu GY, Liu M, Xu GY. Phys Rev (B) 1999;59:15352–7. [4] Xu GY, Wang TM. Semicond Sci Tech 2000;15:613–8.
469
[5] Xu GY, Liu M, Wang TM, et al. J Phys: Condens Matter 1999;11:8495–8. [6] Ye QY, Tsu R, Edward HN. Phys Rev (B) 1991;44: 1806–11. [7] Birkholz M, Selle B, Fuhs W, et al. Phys Rev (B) 2001;64:085402. [8] Marsal LE, Pallares J, Correig X, Calderer J, Alcubilla R. J Appl Phys 1996;79:8493. [9] Mimura H, Hatanaka Y. J Appl Phys 1992;71:2316. [10] He YL, Wei YY, Zheng GZ, Yu MB, Liu M. J Appl Phys 1997;82:3408. [11] Sze SM. Semi conductor devices—Physics and Technology. Beijing: Science Press, 1985. p. 92 (in Chinese). [12] Liu G, Yu YH, Shi JQ, et al. Devices of Semiconductor. Beijing: Publishing House of Electronics Industry, 2000. p. 269 (in Chinese). + [13] Buttiker M. J Phys: Condens Matter 1993;5:9361. [14] Hou JG, Yang JL, et al. Phys Rev Lett 2001;86:5321–3. [15] Lang DV, Cohen JD, Harbison JP. Phys Rev (B) 1982;8:5285–319. [16] Lang DV. J Appl Phys 1974;7:3023–32.
Vacuum 71 (2003) 465–469
Study on variable capacitance diode of (p)nc-Si:H/(n)c-Si heterojunction Wei Wenshenga,b,*, Wang Tianmina, Zhang Chunxib, Li Guohuac, Li Yuexiac a
Center of Material Physics & Chemistry, School of Science, Beijing University of Aeronautics & Astronautics, Beijing 100083, China b Institute of Optic-electronics, Beijing University of Aeronautics & Astronautics, Beijing 100083, China c National Key Lab. For Semiconductor Superlattice & Microstructure, Institute of Semiconductor, Chinese Academy of Science, Beijing 100083, China
Abstract Hydrogenated nanocrystalline silicon (nc-Si:H) layers of boron-doped increasing step by step was deposited on ntype crystalline silicon substrate using Plasma Enhanced Chemical Vapor Deposition (PECVD) system. After evaporating Ohm contact electrode on the side of substrate and on the side of nc-Si:H film, a structure of electrode/ (p)nc-Si:H/(n)c-Si/electrode was obtained. It is confirmed by electrical measurement such as I–V curve, C–V curve and DLTS that this is a variable capacitance diode. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: nc-Si:H film; (p)nc-Si:H/(n)c-Si heterojunction; variable capacitance diode
1. Introduction nc-Si:H film has been receiving increasing attention since novel properties such as high conductivity and visible photoluminescence at low and room temperatures due to quantum size effect. Recently, nc-Si:H film was widely used to fabricate [1–5] solar cells, large area display devices, single electron transistors, micro electromechanical system, resonant-tunneling diodes, heterojunction diode, superlattice devices and so on. The structure and electronic transport me-
chanism through interface of nc-Si:H/c-Si were active research points [6,7], an nc-Si:H/c-Si tunneling diode and an (n)nc-Si:H/(p)c-Si abrupt heterojunction diode have ever been fabricated by PECVD system and a series results was reported [3–5]. In this work, the variable capacitance diode of p-type nc-Si:H/n-type c-Si was fabricated using PECVD and its electrical behavior was characterized and discussed.
2. Preparation of the variable capacitance diode *Corresponding author. Center of Material Physics & Chemistry, School of Science, Beijing University of Aeronautics & Astronautics, Beijing 100083, China. E-mail address: [email protected] (W. Wensheng).
In order to fabricate the variable capacitance diode, n-type (100) silicon wafer with average resistivity of 0.5–1.0 O cm (NDE2 1016 cm3) was selected as substrate. A SiO2 layer around
0042-207X/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0042-207X(03)00034-4
ARTICLE IN PRESS 466
W. Wensheng et al. / Vacuum 71 (2003) 465–469
1 mm in thickness was prepared by thermal oxidation at 1020 C. The SiO2 layer was patterned by photolithography etched to make an array of square holes (30 mm 30 mm). After appropriate treatment, a fresh boron-doped nc-Si:H film of boron concentration increasing step by step was deposited using PECVD system by rf (13.56 MHz) and dc bias stimulation. A strongly hydrogen diluted silane (SiH4 diluted to about 1% in H2) was used as reactant gas. Doping was incorporated by adding B2H6 into the mixed reactant gas, while the reactant gas pressure was 0.7 Torr, the substrate’s temperature was 250 C, the power density was 0.60 W/cm2, and a negative bias of 200 V was applied in the depositing process. In order to obtain a linear slowly varied heterojunction, the doping concentration, Cp=B2H6/SiH4 vol%, was increased step by step from 0.00 to 5.00 vol%. The nc-Si:H layer out of the square holes was removed by photolithography etching leaving only the boron doped nc-Si:H layer in the hole bottoms. In the process of deposition, there were three main technological crucial factors in forming the nanophase grain in the growing film. The first one was the hydrogen dilution ratio to saline, R=H2/SiH4 vol%, which was from around
Fig. 2. HRTEM image of nc-Si:H on C-Si (25 mm 25 mm).
98 to 99.5. The second one was the dc negative bias Vb applied to the parallel electrodes of the deposition system. The third one was reactant gas pressure less than 1.00 Torr. Finally, in order to obtain Ohm’s contact, alloy electrodes were deposited on the back side of substrate and on the nc-Si:H film using vacuum evaporation, the structure of alloy/(p)nc-Si:H/(n)c-Si/alloy was obtained and is shown in Fig. 1. A High Resolution Transmission Electron Microscope (HRTEM) analysis was performed to check the interface of nc-Si:H/c-Si and is shown in Fig. 2, from which one can find that no buffer layer exists in the cross section of nc-Si:H/c-Si.
3. Electrical measurement and discussion
Fig. 1. Model of variable capacitance diode.
I–V characteristics of alloy/(p)nc-Si:H/(n)c-Si/ alloy were measured by HP4140B (PA meter/DC Voltage Source) at different temperatures in the range of 77–573 K. C–V measurement was performed at 1 MHz using a HP4280A Precision LCR meter on a set of these diodes. The I–V curve and C–V curve measured at 300 K is shown in Fig. 3(a) and (b), respectively. From Fig. 3(a) one can obviously find that it was a typical I–V characteristic curve of a diode, the reverse breakdown voltage and the forward turn-on voltage were
ARTICLE IN PRESS W. Wensheng et al. / Vacuum 71 (2003) 465–469
Fig. 3. (a) I–V curve at 300 K of variable capacitance diode. (b) C–V curve at 300 K of variable capacitance diode.
VR=49.90 V and VD=0.81 V, respectively. The reverse current was just several nA. Comparing Fig. 3(a) and (b), one can easily find that VDEVbi. The forward J in nc-Si:H/c-Si heterojunction can be expressed as [4]: JpexpðEac =KTÞ½expðqV =ZKTÞ 1;
ð1Þ
where Eac is the activation energy Eac=0.68 eV, and ideality factor Z=1.6. From expression (1) one can find that at low applied voltage the temperature dependence of J is mainly determined by Eac. Marsal et al. [8,9] pointed out that the current in (n)a-Si:H/(p)c-Si heterojunctions was also satisfied to expression (1), and the value of Eac is 0.88 eV. Therefore, one can see that the value of Eac in (n) nc-Si:H/(p)c-Si heterojunction is significantly smaller than that in (n)a-Si:H/(p)c-Si heterojunction. Obviously, the small value of Eac is caused by the energy band offset in the nc-Si:H layer which
467
reduces the energy gap [4], also, it is the essential factor that nc-Si:H/c-Si heterojunctions show good temperature stability. By a comparison with the previous work [3,10], one can find the difference in the transport mechanism between nc-Si:H film and nc-Si:H/cSi heterojunction. The transport mechanism of the former is the tunneling of the electrons between the neighbor nanograins. While for the latter, current is dominated by the recombination in the nc-Si:H side of the space charge region. But they are not self-contradictory. The electrical transport in nc-Si:H film is determined by only one type of carrier (electrons), and the recombination is negligible. Furthermore, the thickness of the amorphous regions is small enough to make the tunneling possible. On the other hand, in nc-Si:H/ c-Si heterojunction, especially in the nc-Si:H side of the space charge region, both electrons and holes are involved. The defect states in the amorphous Si:H tissue act as trap levels and induce the recombination of carrier couples. One can find that both nanograins and amorphous tissue play important roles in the carriers transport in heterojunction. The reverse conduction is dominated by currentgenerated in the space charge region [4, 9]. The diode has good rectification properties, 4.36 104 of rectification proportion at 71.0 V and 300 K as shown in Fig. 3(a). From Fig. 3(b) one can easily find that the capacitance, C, is dramatically changed with the applied forward voltage V when V was smaller than the interior built potential Vbi. It is well fitted with the definition of the deletion-layer-capacitance Cj of linear slowly varied heterojunction [11]: Cj ¼
1=3 qae2s ; 12ðVbi V Þ
ð2Þ
where q is the absolute value of electron charge, a is the gradient of dopant density and es is the dielectric constant. From Fig. 3(b) one can find that there exists a link point near V=0.55 V in the C–V curve, which resulted from the carrier density being sensitive to the boron-doping in the nc-Si:H film, which is shown in Fig. 4. From Fig. 3(b), according to the definition of capacitance varied
ARTICLE IN PRESS W. Wensheng et al. / Vacuum 71 (2003) 465–469
468
Fig. 4. Relation between carrier density and B2H6/SiH4.
Fig. 5. DLTS of variable capacitance diode.
coefficient, g [12]: g¼
Cmax Cmin : 2ðCmax þ Cmin Þ
ð3Þ
One can calculate g=0.41 of this diode, that means it is much more than g=0.167 of a linear slowly varied single crystal silicon heterojunction. This differs from classical behavior and provides evidence of quantum effect of the capacitance [13,14]. The carrier density on the side of nc-Si:H film varied with the increase step by step of the boron dopant in the nc-Si:H layer as shown in Fig. 4, so that this p–n junction is not an abrupt one. From the Deep Level Transient Spectroscopy (DLTS) which is shown in Fig. 5, one can easily find that there were smooth peaks but not sharp ones in the curves of transient capacitance signal at different rate window time t, it is just confirmed that this is a real slowly varied p–n junction in the interface of nc-Si:H/c-Si. Because there is amorphous component of Si:Hn (1pnp3) about 50 vol% in the ncSi:H layer, the gap states are composed of deep energy levels distributed continuously in Si:H. When the temperature rises up to a value, the number of holes trapped by deep energy levels is changed dramatically, as well as the space charges in the barrier of heterojunction are changed obviously, thus resulting in a peak of transient capacitance. According to the method of DLTS [15,16], the spectrum of DLTS is contributed by the overlap of related gap states, but not from a single deep energy level, so as to the peaks of DLTS’ spectra at different t are those slowly
Fig. 6. C3R–V curve of variable capacitance diode (VRbVbi).
varied broad peaks but not those sharp peaks. Also, according Ref. [15], the temperature T corresponding to the peak of DLTS’ spectrum decreased with increasing of t, as shown in Fig. 5. From Fig. 6, one can find C3p VR nearly when VR is far larger than Vbi. It is tested that this heterojunction is a linear slowly varied junction according to Ref. [11].
4. Conclusion A structure of alloy/(p)nc-Si:H/(n)c-Si/alloy was fabricated. It is tested by HRTEM that there was no buffer layer in the cross section. Also it was confirmed by electric measurement of I–V, C–V and DLTS that the structure is a variable capacitance diode.
ARTICLE IN PRESS W. Wensheng et al. / Vacuum 71 (2003) 465–469
Acknowledgements This work was supported by the National Foundation of Natural Science of China (No. 59982002) and it is a Doctoral Stations Fund Project Sponsored by National Educational Administration (No. 200220006037). We are also grateful to Prof. He Yuliang for fruitful discussion.
References [1] Yue GZ, Han DX, Williamson DL, Yang J, Kenneth L, Subhendu GH. Appl Phys Lett 2000;77:3185–7. [2] Mamura YI, Ataka S, Takasaki Y, Kusano C, Hirai T, Maruyama E. Appl Phys Lett 1979;35:349–51. [3] He YL, Hu GY, Liu M, Xu GY. Phys Rev (B) 1999;59:15352–7. [4] Xu GY, Wang TM. Semicond Sci Tech 2000;15:613–8.
469
[5] Xu GY, Liu M, Wang TM, et al. J Phys: Condens Matter 1999;11:8495–8. [6] Ye QY, Tsu R, Edward HN. Phys Rev (B) 1991;44: 1806–11. [7] Birkholz M, Selle B, Fuhs W, et al. Phys Rev (B) 2001;64:085402. [8] Marsal LE, Pallares J, Correig X, Calderer J, Alcubilla R. J Appl Phys 1996;79:8493. [9] Mimura H, Hatanaka Y. J Appl Phys 1992;71:2316. [10] He YL, Wei YY, Zheng GZ, Yu MB, Liu M. J Appl Phys 1997;82:3408. [11] Sze SM. Semi conductor devices—Physics and Technology. Beijing: Science Press, 1985. p. 92 (in Chinese). [12] Liu G, Yu YH, Shi JQ, et al. Devices of Semiconductor. Beijing: Publishing House of Electronics Industry, 2000. p. 269 (in Chinese). + [13] Buttiker M. J Phys: Condens Matter 1993;5:9361. [14] Hou JG, Yang JL, et al. Phys Rev Lett 2001;86:5321–3. [15] Lang DV, Cohen JD, Harbison JP. Phys Rev (B) 1982;8:5285–319. [16] Lang DV. J Appl Phys 1974;7:3023–32.