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ZnxCd1−xSe multiple quantum wells for intersubband devices operating at short wavelengths
ARTICLE IN PRESS Journal of Crystal Growth 311 (2009) 2113–2115
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Growth and properties of wide bandgap MgSe/ZnxCd1 xSe multiple quantum wells for intersubband devices operating at short wavelengths B.S. Li a,b,1, A. Shen a,, W.O. Charles a, Q. Zhang c, M.C. Tamargo d a
Department of Electrical Engineering, The City College of the City University of New York, New York, NY 10031, USA School of Physics and Engineering, Sun Yat-Sen University, Guang Zhou 510275, China Department of Physics, The City College of the City University of New York, New York, NY 10031, USA d Department of Chemistry, The City College of the City University of New York, New York, NY 10031, USA b c
a r t i c l e in fo
abstract
Available online 17 November 2008
The authors present the design and growth of zincblende MgSe/ZnxCd1 xSe multiple quantum wells (MQWs), which have a large conduction band offset of 1.2 eV, useful for intersubband (ISB) device applications. The samples were grown on (0 0 1)-InP substrates by molecular beam epitaxy. In situ reflection high energy electron diffraction (RHEED) shows that thin MgSe layers with zincblende structure can be epitaxially grown on InP substrates. Structural degradation of MQW structures due to phase transition of MgSe from zincblende to rocksalt is suppressed and structural quality improved with the introduction of a thick ZnxCd1 xSe spacer layer. The ISB absorptions in 3.35–4.9 mm have been observed in MgSe/ZnxCd1 xSe structure. The dependence of ISB absorption on the MgSe barrier thickness is also studied in this structure. & 2008 Elsevier B.V. All rights reserved.
PACS: 73.63.Hs 78.30.Fs Keywords: A1. Intersubband absorption A3. MBE B2. II–VI B2. MgSe/ZnxCd1 xSe quantum well
1. Introduction Wide bandgap II–VI heteroepitaxy has attracted considerable attention in the last two decades due to potential applications in blue-light emission diodes and lasers. However, the fundamental aspect of p-type doping is still in severe challenge because of strong self-compensation in II–VI wide bandgap semiconductors. The recent development of II–VI CdS/ZnSe [1], (CdS/ZnSe)/BeTe [2], and ZnCdSe/ZnCdMgSe [3] quantum wells (QWs) has opened up application in mid- and near-infrared (IR) region using the intersubband (ISB) transitions within the conduction band of the QW, i.e., unipolar devices such as IR photodetectors, quantum cascade lasers (QCL), as well as ultrafast all-optical switches (UOS). Previously, it has been proposed that the II–VI-based QWs are the promising candidates for UOS at near-IR wavelength region due to the enhanced electron–phonon interaction [2]. Recently, our group has reported the ISB absorption at mid-IR region in the ZnxCdyMg1 x ySe/ZnxCd1 xdSe II–VI-based material system. The attractive characteristics of this material system are illustrated by the schematic drawing in Fig. 1. First, multi-QWs (MQWs) can be grown lattice matched to the InP substrate with good material quality. The ISB absorption, covering 4–10 mm, and electroluminescence have been observed in ZnxCdyMg1 x ySe/ Corresponding author.
E-mail addresses: [email protected] (B.S. Li), [email protected] (A. Shen). 1 Permanent address: School of Physics and Engineering, Sun Yat-Sen University, Guang Zhou 510275, China. 0022-0248/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2008.11.020
ZnxCd1 xdSe-based photodetector and quantum cascade structures, respectively [3,4]. Second, the conduction band offset (CBO) can be tuned by varying the Mg fraction in ZnxCdyMg1 x ySe and a maximum CBO is obtained when the binary MgSe is used as a barrier. Based on our previous results, the CBO is expected to be as high as 1.2 eV, which makes it possible to realize ISB absorption at near-IR region [5–7]. However, since the MgSe prefers rocksalt (RS) structure over the zincblende (ZB) structure [8,9], there is a severe challenge associated with the growth of ZB MgSe-based heterostructures. In this paper, we describe the growth charcateristics of ZB MgSe/ZnCdSe multiple QWs, in which the ISB absorptions are observed.
2. Experimental procedure Samples were grown on (0 0 1)-oriented semi-insulating InP substrates by molecular beam epitaxy (MBE) in a dual chamber system (Riber 2300P MBE). After a 200-nm-thick InGaAs buffer was grown in the III–VI chamber, the substrate was transferred into the II–VI growth chamber for the sample preparation. Two sets of samples are prepared. The first set consists of 15 periods of MgSe/ZnxCd1 xSe MQWs. The thicknesses of ZnxCd1 xSe well are nominally 11, 13, and 15 MLs. The MgSe barrier thickness is 20 MLs for these three samples. The other set varies the MgSe barrier thickness with well thickness of 11 MLs. The barriers of MgSe thickness are 10 and 20 MLs. The well layer had the latticematched composition, x=0.46, and uniformly doped with ZnCl2 as n-type dopant. The electron density is about (1–2) 1018 cm 3.
ARTICLE IN PRESS 2114
B.S. Li et al. / Journal of Crystal Growth 311 (2009) 2113–2115
A100 MLs of ZnxCd1 xSe spacer layer was grown after growth of every 2 QWs. The ISB absorption was measured with a multiplepass waveguide geometry using a Fourier-transform IR (FTIR) spectrometer with a wire-grid polarizer and a liquid-nitrogencool HgCdTe detector.
Fig. 1. Energy bandgap and lattice mismatch to InP for MgSe, ZnSe, and CdSe. The circles show the tunable bandgap in ZnxCdyMg1 x ySe (ZnxCd1 xSe for yellow; pink for ZnxMg1 xSe and blue for ZnxCdyMg1 x ySe). The band offset, correspondingly, can be tuned between 0 and 1.2 eV in ZnxCd1 xSe/ZnxCdyMg1 x ySe structure.
3. Results and discussion The lattice mismatch between ZB MgSe and InP substrate is about 0.2% (as shown in Fig. 1). However, theoretical calculation and experiments indicated that stable phase of MgSe is the RS structure. Thus, the critical thickness of ZB MgSe on InP is affected not only by the lattice mismatch, but also by the structural phase transition. The growth modes of these MQWs are identified in situ using the reflection high energy electron diffraction (RHEED). At the initial stage of MgSe growth on ZnxCd1 xSe buffer, clear oscillations of RHEED intensity was observed. This indicates that two-dimensional growth mode occurs at the initial stage. Fig. 2 shows a typical RHEED pattern from [11 0] and [11¯ 0] azimuths. The surface reconstruction of MgSe shows (2 1) pattern (top in Fig. 2), which is similar to the ZnxCd1 xSe under the Se-rich condition (bottom in Fig. 2), meaning the formation of ZB structure. The RHEED pattern indicates that the growth mode degraded after 4–5 periods of growth. The RHEED pattern becomes spotty, suggesting that a structural phase transition from ZB to RS took place. To avoid the structural transformation from ZB to RS during growth, we introduced a thick ZnCdSe spacer (100 MLs) after every two ZnCdSe/MgSe QWs. After two QWs growth, the surface begins to degrade slightly, but it does not fully transform to a spotty pattern. After the growth of the ZnCdSe spacer the RHEED pattern recovers, indicating that the growth mode is improved. Thus, the thick ZnCdSe spacer ensures that the onset of a transformation to RS structure does not occur, and the quality of the ZB MQW structure is improved. As shown in bottom of Fig. 2, a RHEED pattern was observed from the ZnCdSe cap layer, after the full structure growth. The sharp streaky patterns indicate that the sample has a smooth surface and it suggests that MQWs with high structural quality have been obtained. Investigations to further improve the MgSe structural quality are underway.
Fig. 2. In situ RHEED patterns of MgSe and ZnCdSe taken from (11 0) and (11¯ 0) azimuth.
ARTICLE IN PRESS B.S. Li et al. / Journal of Crystal Growth 311 (2009) 2113–2115
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Fig. 4. The dependence of MgSe barrier thickness of ISB absorption energy for MgSe/ZnxCd1 xSe QWs. Inset shows schematically the thin barriers of MgSe do not confine the electron within the quantum well.
crucial to improve the structure quality, allows the electron to escape from the QW into the spacer layer when a too thin a barrier is used. In that case, there is no ISB absorption in structure with thinner barrier. A possible solution of this problem would be to use lattice-matched quaternary spacer layers with bandgap higher than the confined electronic levels. Fig. 3. (top) Transmittance as measured for a sample with a well width of 11 MLs using p- (Tp) and s-polarized (Ts) light. (bottom) The ISB absorption spectrum, obtained by taking the negative logarithm of (Tp/Ts) ratio.
Fig. 3(top) shows the transmittance of a sample with a well thickness of 11 MLs in multiple passes measured by using p- (Tp) and s-polarized (Ts) light. Strong resonant absorption at ISB-T wavelength of 3.35 mm was observed when Tp is used, which disappears when Ts excitation is used. The spectrum of the ISB absorption deduced by taking ln of the transmission ratio of Tp/Ts to remove the influence of background is shown in Fig. 3 (bottom). The linewidth of the absorption peak is relatively narrow (DE=46 meV; DE/E=12%), which is typical for a bound-to-bound ISB transition within a QW. With different QW thicknesses of 11, 13 and 15 MLs, ISB absorption located at 3.35, 3.95 and 4.9 mm, covering the mid-IR 3–5 mm atmosphere window. The calculations of the energy separation between the ground state and the first excited state using the envelope function approximation indicate that the ISB absorption wavelength can be extended to near-IR region with a decrease of the QW width. As indicated above, the thickness of MgSe significantly affects the structure quality of ZnCdSe/MgSe MQWs, with thinner barriers being preferred. However, the quantum confinement effect within the QW also depends on the thickness of the barrier. Therefore, an optimum value of the barrier thickness must be considered. Fig. 4 shows the dependence of ISB absorption on the MgSe thickness in the MQW structure. Strong ISB absorption is observed in the sample with a barrier thickness of 20 MLs. However, the absorption disappears in the structure with MgSe thickness of 10 MLs. RHEED patterns and PL results (not shown here) also indicate that the samples with barrier thickness of 10 MLs have high structural quality. This suggests that the absence of ISB absorption is not related to the material quality. We explain that weak quantum confinement results in the disappearance of ISB absorption. The introduction of thick ZnCdSe spacers, which is
4. Conclusion High-quality ZB MgSe/ZnxCd1 xSe MQWs with a very large CBO have been grown by MBE. The introduction of thick ZnCdSe spacer is crucial to improve growth mode and guarantees high material quality. Strong ISB absorptions at the mid-IR region of 3–5 mm are observed, and shorter wavelengths are expected based on calculations. In this structure, barriers of MgSe must be thicker than 10 ML in order to realize the ISB absorption. We propose this is due to electron escape from the confined QW region into the thick spacer region.
Acknowledgements This work is supported in part by MIRTHE (NSF-ERC # EEC0540832) and PSC-CUNY Research Award (No. 61401-0039). References [1] M. Goppert, M. Grun, C. Maier, S. Petillon, R. Becker, A. Dinger, A. Storzum, M. Jorger, C. Klingshirn, Phys. Rev. B 65 (2002) 115334. [2] R. Akimoto, B.S. Li, K. Akita, T. Hasama, Appl. Phys. Lett. 87 (2005) 181104. [3] Kale J. Franz, William O. Charles, Aidong Shen, Anthony J. Hoffman, Mariac C. Tamargo, Claire Gmachl, Appl. Phys. Lett. 92 (2008) 121105. [4] H. Lu, A. Shen, M.C. Tamargo, C.Y. Song, H.C. Liu, S.K. Zhang, R.R. Alfano, Appl. Phys. Lett. 89 (2006) 131903. [5] M. Martin, H. Lu, X. Zhou, M.C. Tarmago, F.H. Pollak, Appl. Phys. Lett. 83 (2003) 1995. [6] M. Sohel, M. Martin, M.C. Tarmago, Appl. Phys. Lett. 85 (2004) 2794. [7] B.S. Li, A. Shen, W.O. Charles, Q. Zhang, M.C. Tamargo, Appl. Phys. Lett. 92 (2008) 261104. [8] M. Rabah, B. Abbar, Y. Al-Douri, B. Bouhafs, B. Sahraoui, Mater. Sci. Eng. B 100 (2003) 163. [9] H.M. Wang, J.H. Chang, T. Hanada, K. Arai, T. Yao, J. Crystal Growth 208 (2000) 253.
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Growth and properties of wide bandgap MgSe/ZnxCd1 xSe multiple quantum wells for intersubband devices operating at short wavelengths B.S. Li a,b,1, A. Shen a,, W.O. Charles a, Q. Zhang c, M.C. Tamargo d a
Department of Electrical Engineering, The City College of the City University of New York, New York, NY 10031, USA School of Physics and Engineering, Sun Yat-Sen University, Guang Zhou 510275, China Department of Physics, The City College of the City University of New York, New York, NY 10031, USA d Department of Chemistry, The City College of the City University of New York, New York, NY 10031, USA b c
a r t i c l e in fo
abstract
Available online 17 November 2008
The authors present the design and growth of zincblende MgSe/ZnxCd1 xSe multiple quantum wells (MQWs), which have a large conduction band offset of 1.2 eV, useful for intersubband (ISB) device applications. The samples were grown on (0 0 1)-InP substrates by molecular beam epitaxy. In situ reflection high energy electron diffraction (RHEED) shows that thin MgSe layers with zincblende structure can be epitaxially grown on InP substrates. Structural degradation of MQW structures due to phase transition of MgSe from zincblende to rocksalt is suppressed and structural quality improved with the introduction of a thick ZnxCd1 xSe spacer layer. The ISB absorptions in 3.35–4.9 mm have been observed in MgSe/ZnxCd1 xSe structure. The dependence of ISB absorption on the MgSe barrier thickness is also studied in this structure. & 2008 Elsevier B.V. All rights reserved.
PACS: 73.63.Hs 78.30.Fs Keywords: A1. Intersubband absorption A3. MBE B2. II–VI B2. MgSe/ZnxCd1 xSe quantum well
1. Introduction Wide bandgap II–VI heteroepitaxy has attracted considerable attention in the last two decades due to potential applications in blue-light emission diodes and lasers. However, the fundamental aspect of p-type doping is still in severe challenge because of strong self-compensation in II–VI wide bandgap semiconductors. The recent development of II–VI CdS/ZnSe [1], (CdS/ZnSe)/BeTe [2], and ZnCdSe/ZnCdMgSe [3] quantum wells (QWs) has opened up application in mid- and near-infrared (IR) region using the intersubband (ISB) transitions within the conduction band of the QW, i.e., unipolar devices such as IR photodetectors, quantum cascade lasers (QCL), as well as ultrafast all-optical switches (UOS). Previously, it has been proposed that the II–VI-based QWs are the promising candidates for UOS at near-IR wavelength region due to the enhanced electron–phonon interaction [2]. Recently, our group has reported the ISB absorption at mid-IR region in the ZnxCdyMg1 x ySe/ZnxCd1 xdSe II–VI-based material system. The attractive characteristics of this material system are illustrated by the schematic drawing in Fig. 1. First, multi-QWs (MQWs) can be grown lattice matched to the InP substrate with good material quality. The ISB absorption, covering 4–10 mm, and electroluminescence have been observed in ZnxCdyMg1 x ySe/ Corresponding author.
E-mail addresses: [email protected] (B.S. Li), [email protected] (A. Shen). 1 Permanent address: School of Physics and Engineering, Sun Yat-Sen University, Guang Zhou 510275, China. 0022-0248/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2008.11.020
ZnxCd1 xdSe-based photodetector and quantum cascade structures, respectively [3,4]. Second, the conduction band offset (CBO) can be tuned by varying the Mg fraction in ZnxCdyMg1 x ySe and a maximum CBO is obtained when the binary MgSe is used as a barrier. Based on our previous results, the CBO is expected to be as high as 1.2 eV, which makes it possible to realize ISB absorption at near-IR region [5–7]. However, since the MgSe prefers rocksalt (RS) structure over the zincblende (ZB) structure [8,9], there is a severe challenge associated with the growth of ZB MgSe-based heterostructures. In this paper, we describe the growth charcateristics of ZB MgSe/ZnCdSe multiple QWs, in which the ISB absorptions are observed.
2. Experimental procedure Samples were grown on (0 0 1)-oriented semi-insulating InP substrates by molecular beam epitaxy (MBE) in a dual chamber system (Riber 2300P MBE). After a 200-nm-thick InGaAs buffer was grown in the III–VI chamber, the substrate was transferred into the II–VI growth chamber for the sample preparation. Two sets of samples are prepared. The first set consists of 15 periods of MgSe/ZnxCd1 xSe MQWs. The thicknesses of ZnxCd1 xSe well are nominally 11, 13, and 15 MLs. The MgSe barrier thickness is 20 MLs for these three samples. The other set varies the MgSe barrier thickness with well thickness of 11 MLs. The barriers of MgSe thickness are 10 and 20 MLs. The well layer had the latticematched composition, x=0.46, and uniformly doped with ZnCl2 as n-type dopant. The electron density is about (1–2) 1018 cm 3.
ARTICLE IN PRESS 2114
B.S. Li et al. / Journal of Crystal Growth 311 (2009) 2113–2115
A100 MLs of ZnxCd1 xSe spacer layer was grown after growth of every 2 QWs. The ISB absorption was measured with a multiplepass waveguide geometry using a Fourier-transform IR (FTIR) spectrometer with a wire-grid polarizer and a liquid-nitrogencool HgCdTe detector.
Fig. 1. Energy bandgap and lattice mismatch to InP for MgSe, ZnSe, and CdSe. The circles show the tunable bandgap in ZnxCdyMg1 x ySe (ZnxCd1 xSe for yellow; pink for ZnxMg1 xSe and blue for ZnxCdyMg1 x ySe). The band offset, correspondingly, can be tuned between 0 and 1.2 eV in ZnxCd1 xSe/ZnxCdyMg1 x ySe structure.
3. Results and discussion The lattice mismatch between ZB MgSe and InP substrate is about 0.2% (as shown in Fig. 1). However, theoretical calculation and experiments indicated that stable phase of MgSe is the RS structure. Thus, the critical thickness of ZB MgSe on InP is affected not only by the lattice mismatch, but also by the structural phase transition. The growth modes of these MQWs are identified in situ using the reflection high energy electron diffraction (RHEED). At the initial stage of MgSe growth on ZnxCd1 xSe buffer, clear oscillations of RHEED intensity was observed. This indicates that two-dimensional growth mode occurs at the initial stage. Fig. 2 shows a typical RHEED pattern from [11 0] and [11¯ 0] azimuths. The surface reconstruction of MgSe shows (2 1) pattern (top in Fig. 2), which is similar to the ZnxCd1 xSe under the Se-rich condition (bottom in Fig. 2), meaning the formation of ZB structure. The RHEED pattern indicates that the growth mode degraded after 4–5 periods of growth. The RHEED pattern becomes spotty, suggesting that a structural phase transition from ZB to RS took place. To avoid the structural transformation from ZB to RS during growth, we introduced a thick ZnCdSe spacer (100 MLs) after every two ZnCdSe/MgSe QWs. After two QWs growth, the surface begins to degrade slightly, but it does not fully transform to a spotty pattern. After the growth of the ZnCdSe spacer the RHEED pattern recovers, indicating that the growth mode is improved. Thus, the thick ZnCdSe spacer ensures that the onset of a transformation to RS structure does not occur, and the quality of the ZB MQW structure is improved. As shown in bottom of Fig. 2, a RHEED pattern was observed from the ZnCdSe cap layer, after the full structure growth. The sharp streaky patterns indicate that the sample has a smooth surface and it suggests that MQWs with high structural quality have been obtained. Investigations to further improve the MgSe structural quality are underway.
Fig. 2. In situ RHEED patterns of MgSe and ZnCdSe taken from (11 0) and (11¯ 0) azimuth.
ARTICLE IN PRESS B.S. Li et al. / Journal of Crystal Growth 311 (2009) 2113–2115
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Fig. 4. The dependence of MgSe barrier thickness of ISB absorption energy for MgSe/ZnxCd1 xSe QWs. Inset shows schematically the thin barriers of MgSe do not confine the electron within the quantum well.
crucial to improve the structure quality, allows the electron to escape from the QW into the spacer layer when a too thin a barrier is used. In that case, there is no ISB absorption in structure with thinner barrier. A possible solution of this problem would be to use lattice-matched quaternary spacer layers with bandgap higher than the confined electronic levels. Fig. 3. (top) Transmittance as measured for a sample with a well width of 11 MLs using p- (Tp) and s-polarized (Ts) light. (bottom) The ISB absorption spectrum, obtained by taking the negative logarithm of (Tp/Ts) ratio.
Fig. 3(top) shows the transmittance of a sample with a well thickness of 11 MLs in multiple passes measured by using p- (Tp) and s-polarized (Ts) light. Strong resonant absorption at ISB-T wavelength of 3.35 mm was observed when Tp is used, which disappears when Ts excitation is used. The spectrum of the ISB absorption deduced by taking ln of the transmission ratio of Tp/Ts to remove the influence of background is shown in Fig. 3 (bottom). The linewidth of the absorption peak is relatively narrow (DE=46 meV; DE/E=12%), which is typical for a bound-to-bound ISB transition within a QW. With different QW thicknesses of 11, 13 and 15 MLs, ISB absorption located at 3.35, 3.95 and 4.9 mm, covering the mid-IR 3–5 mm atmosphere window. The calculations of the energy separation between the ground state and the first excited state using the envelope function approximation indicate that the ISB absorption wavelength can be extended to near-IR region with a decrease of the QW width. As indicated above, the thickness of MgSe significantly affects the structure quality of ZnCdSe/MgSe MQWs, with thinner barriers being preferred. However, the quantum confinement effect within the QW also depends on the thickness of the barrier. Therefore, an optimum value of the barrier thickness must be considered. Fig. 4 shows the dependence of ISB absorption on the MgSe thickness in the MQW structure. Strong ISB absorption is observed in the sample with a barrier thickness of 20 MLs. However, the absorption disappears in the structure with MgSe thickness of 10 MLs. RHEED patterns and PL results (not shown here) also indicate that the samples with barrier thickness of 10 MLs have high structural quality. This suggests that the absence of ISB absorption is not related to the material quality. We explain that weak quantum confinement results in the disappearance of ISB absorption. The introduction of thick ZnCdSe spacers, which is
4. Conclusion High-quality ZB MgSe/ZnxCd1 xSe MQWs with a very large CBO have been grown by MBE. The introduction of thick ZnCdSe spacer is crucial to improve growth mode and guarantees high material quality. Strong ISB absorptions at the mid-IR region of 3–5 mm are observed, and shorter wavelengths are expected based on calculations. In this structure, barriers of MgSe must be thicker than 10 ML in order to realize the ISB absorption. We propose this is due to electron escape from the confined QW region into the thick spacer region.
Acknowledgements This work is supported in part by MIRTHE (NSF-ERC # EEC0540832) and PSC-CUNY Research Award (No. 61401-0039). References [1] M. Goppert, M. Grun, C. Maier, S. Petillon, R. Becker, A. Dinger, A. Storzum, M. Jorger, C. Klingshirn, Phys. Rev. B 65 (2002) 115334. [2] R. Akimoto, B.S. Li, K. Akita, T. Hasama, Appl. Phys. Lett. 87 (2005) 181104. [3] Kale J. Franz, William O. Charles, Aidong Shen, Anthony J. Hoffman, Mariac C. Tamargo, Claire Gmachl, Appl. Phys. Lett. 92 (2008) 121105. [4] H. Lu, A. Shen, M.C. Tamargo, C.Y. Song, H.C. Liu, S.K. Zhang, R.R. Alfano, Appl. Phys. Lett. 89 (2006) 131903. [5] M. Martin, H. Lu, X. Zhou, M.C. Tarmago, F.H. Pollak, Appl. Phys. Lett. 83 (2003) 1995. [6] M. Sohel, M. Martin, M.C. Tarmago, Appl. Phys. Lett. 85 (2004) 2794. [7] B.S. Li, A. Shen, W.O. Charles, Q. Zhang, M.C. Tamargo, Appl. Phys. Lett. 92 (2008) 261104. [8] M. Rabah, B. Abbar, Y. Al-Douri, B. Bouhafs, B. Sahraoui, Mater. Sci. Eng. B 100 (2003) 163. [9] H.M. Wang, J.H. Chang, T. Hanada, K. Arai, T. Yao, J. Crystal Growth 208 (2000) 253.