Austin, Elizabeth

Jaros, M

English

Edinburgh Research Explorer

     Edinburgh Research Explorer                                      Electric field-induced shifts and lifetimes in GaAs-GaAlAsquantum wellsCitation for published version:Austin, E & Jaros, M 1985, 'Electric field-induced shifts and lifetimes in GaAs-GaAlAs quantum wells'Applied Physics Letters, vol 47, no. 3, pp. 274-276.,  10.1063/1.96189Digital Object Identifier (DOI): 10.1063/1.96189Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Publisher final version (usually the publisher pdf)Published In:Applied Physics LettersPublisher Rights Statement:© Austin, E., & Jaros, M. (1985). Electric field-induced shifts and lifetimes in GaAs-GaAlAs quantum wells.Applied Physics Letters, 47(3), 274-276doi: 10.1063/1.96189General rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 20. Feb. 2015Electric field induced shifts and lifetimes in GaAsGaAlAs quantum wellsE. J. Austin and M. Jaros  Citation: Applied Physics Letters 47, 274 (1985); doi: 10.1063/1.96189 View online: http://dx.doi.org/10.1063/1.96189 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/47/3?ver=pdfcov Published by the AIP Publishing                      This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:129.215.19.197 On: Fri, 13 Dec 2013 15:18:41IT. Kobayashi and Y. Shin<xia, J. App!. Phys. 53, 3339 (1982). zL. G. Meiners, D. L. Li1e, and D. A. Collins,J. Vac. Sci. Techno!. 16, 1458 (1979). 'L. G. Meiners, J. Vac. Sci. Techno!. IS, 1402 (1978). 4H. C. Cassey, Jr., A. Y. Cho, D. V. Lang, E. H. Nicollian, and P. W. Foy, J. App!. Phys. 50, 3484 (1979). 5y. Hirota and T. Kobayashi, 1. App!. Phys. 53, 5037 (1982). 60. J. Olego, 1. A. Baumann, M. A. Kuck, R. Schachter, C. G. Michel, and P. M. Raccah, Solid State Commun. 52, 311 (1984). 7L. M. Terman, Solid State Electron. 5, 285 (1962). "K. P. Pande and D. Gutierrez, App!. Phys. Lett. 46, 416 (1985). 9H. H. Wieder, Surf. Sci. 132, 390 (1983). lOW. E. Spicer, P. W. Chye, P. R. Skeath, C. Y. Su, and I. Lindau, 1. Vac. Sci. Techno!. 16, 1422 (1979). IIW. E. Spicer, I. Lindau, P. Skeath, and C. Y. Su, J. Vac. Sci. Techno!. 17, 1019, (1980). 12M. S. Daw, D. L. Smith, C. A. Swarts, and T. C. McGill, J. Vac. Sci. Techno!. 19, 508 (1981). 131. D. Dow and R. E. Allen, J. Vac. Sci. Techno!. 20, 659 (1982). 14A. Nedoluha, J. Vac. Sci. Techno!. 21, 429 (1982). "H. H. Wieder,lEEE Electron Device Lett. EDL-4, 108 (1983). 160. J. Olego, R. Schachter, and J. A. Baumann, App!. Phys. Lett. 45, 1127 (1984). I7E. Yamaguchi and M. Minakata, J. App!. Phys. 55, 3098 (1984). E~ectroc field induced shins and Ufetimes in GaAs-GaAIAs quantum weUs E. J. Austin and M. Jaros School 0/ PhYSics, The University. Newcastle upon Tyne NEI 7R U. United Kingdom (Received 5 March 1985; accepted for publication 14 May 1985) We have performed an exact numerical calculation for an isolated GaAs quantum well system subjected to an external electric field. This calculation predicts both the shifts and widths of the Stark resonances. The shift predictions are in agreement with recent experimental results and the width calculations allow the observed decrease in luminescence lifetime with field to be interpreted in terms of the field-induced tunneling of the carriers. Recent experimental studies '.2 of the effects of electric fields on GaAs quantum well systems have enabled both the field-induced level shifts' and the field dependence of the photoluminescence lifetime2 to be obtained. In this letter the theoretical interpretation of these results is considered with-in the framework of an exact numerical solution of the Schr6dinger equation3 for an isolated quantum well system subjected to an electric field. This calculation provides a uni-fied approach to the prediction of Stark shifts and widths and the observed decrease in luminescence lifetime with in-creasing field strength is accounted for in terms of field-in-duced quantum mechanical tunneling. Previous theoretical calculations4•5 of electric field ef-fects on quantum well systems have treated the effect of the external field on the electronic structure by approximate techniques (in particular the variation method) appropriate in the weak-field limit. In these calculations the confined states are viewed as bound states and their field-induced broadening is ignored. Such calculations correctly predict the quadratic variation of the Stark shift with field at moder-ate field intensities. However, the bound state model fails to account for the decrease in the luminescence lifetime2 with field and a more detailed calculation is required. In the envelope function approximation the rapidly varying Bloch component of the wave function is ignored and the problem reduces to the one-dimensional effective mass equation familiar from previous studies. 4•5 For a parti-cle of charge e and mass m·, initially localized in an isolated well of width 2a and depth Va' the Schrodinger equation is fl d 2r/J ---7 -(Vo+eFx)r/J=Er/J 2m· dx-'If d 2r/J ----eFxr/J = Er/J 2m. dx2 ( 1) where F is the electric field applied perpendicular to the in-terface plane. The exact solutions of ( 1) can be obtained as linear combinations of Airy functions; application of the method of phase shift analysis to the sonutions enables the Stark resonance positions and widths to be calculated. It was found3 that for mod.erate fields both the variational. and ex-act calculations predict a quadratic dependence of the Stark shift on the field; this shift can be explained by perturbation theory. The quadratic Stark shift of the ground state 10) ofa quantum system is given by ilE = e2F2 L j (k IxO) 12 , k;"O Eo-Ek (2) where the states Ik ) are the excited states of the system. An increase in the wen depth increases the binding energy E and thus decreases ilE; an increase in well width increases the spatial extent of the ground state wave function and thus the matrix elements (k IxIO). At high fields the variational bound state formalism breaks down and the predictions of the two calculations differ considerably. The exact calcula-tion predicts a decrease in the effective energy shift at high fields and the emergence of new anti resonance states from the continuum into the well.. Figure 1 gives a comparison of the predictions of the 274 Appl. Phys. Lett. 47 (3), 1 August 1985 0003-6951/85/150274-03$01 .00 @ 1985 American Institute of Physics 274  This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:129.215.19.197 On: Fri, 13 Dec 2013 15:18:412.0r-------------------::> 1.5 ~ 1.0 E UJ <l C1 0.5 ~ o 70 A log F (V cm-1 ) FIG. 1. Comparison of calculated and experimental l Stark shifts for;:: 70 A and ;::37 A wells. In the calculation the well depth was taken as 100 meV and the hole effective mass as 0.45m. in order to facilitate a comparison with the calculated shifts of Ref. I. However, the results obtained in the quadratic regime as illustrated are not particularly sensitive to the assumed well depth. The horizontal bars extend between the "experimental" and "scaled" fields of Ref. (I):(i 1j70 A data; (X X)37A data. present calculation and the experimental results I obtained for::::: 37 A and ;:::: 70 A wells. The calculations represent only the hole energy shift; because of the shallower well depth for holes this is the major contribution to the total Stark shift. For the wider wen the calculated and experimental shifts are in good agreement given the large uncertainties in the deter-mination of the experimental electric field values. However, the figure shows that the shifts predicted for the;:::: 37 A well are much too high, except for the highest field values. One possible source ofthis discrepancy is suggested by the results of more detailed (zero field) calculations6 which go beyond the envelope function approximation employed here (and in all previous Stark shift calculations). These large calcula-tions show that in narrow wells the wave function of the confined state is more extended in k space. Consequently the matrix e1ement in (2) should be written as where tPn.k is a Bloch function associated with band n and reduced wave vector k. Since the characteristic value for k' of most of the terms on the right-hand side of this matrix element is different from k on the left, the value of the matrix e1ement is substantiaBy reduced. (This quenching of the ma-trix element is familiar from studies of the breakdown ofthe effective mass approximation for impurities in semiconduc-tors 7 and leads to a systematic error on iJE as calculated here which is greatest for narrow wells.) As the fie1d strength increases, the quantum state is broadened into a resonance by the interaction with the field. This energy broadening, which is accompanied by spreading of the wave function in k space, increases rapidly with field and dominates at high field. Since the fie1d-induced broadening is fully represented in the present calculation, it is not surprising that our high-field Stark shifts agree with experiment. 275 Appl. Phys. Lett., Vol. 47, No.3, 1 August 1985 u W VI Cl.. 0.5 1. a 1.5 F )( 10-5 ( V cm- 1 ) FIG. 2. Calculated tunneling lifetimes as a function of applied field for 30 A, 37 A. and 70 A wells; the other parameters used in the calculation are the same as for Fig. I. From the calculated resonance width r it is possible to obtain the corresponding lifetime 1'; the variation of lifetime with field for quantum wells of various widths is illustrated in Fig. 2. The mechanism of state decay is field-induced tun-neling out of the well and, as would be expected, this effect is greatest for narrow wells, where the particle binding energy is low. The luminescence of optically pumped quantum well systems is caused by the recombination of electron-hole pairs; nonradiative processes compete with radiatiative re-combination and the experimental lifetime 1'. is related to the radiative and nonradiative lifetimes by lire = lI1'r + lI1'nr. (3) The experimental lifetime measurements2 (performed on ;:::: 30 A wells) demonstrate that luminescence lifetime, peak luminescence intensity, and integrated luminescence aJi de-crease when the field is increased. Theoretical consider-ations suggest that 1'r should increase with field. In the enve-lope function approximation the wave function of a particle confined in a quantum weB is l/! =f(x)tPn.o(r), (4) where tPn.O (r) is a :Bloch state of zero wave vector and band index n. Sincef(x) is a slowly varying function, the transition probability for recombination can be factorized as (5) Under the influence of the electric field the maxima of the electron and hole wave functions are shifted in opposite di-rections and thus the overlap 1 if. (X)lfh (x» 12 decreases, lead-ing to a decrease in the transition probability and an increase in the radiative lifetime. However, the magnitude of this ef-fect [as previously estimated from the variational solutions of (1 )4] is much too small to account for the observed rate of E. J. Austin and M. Jaros 275  This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:129.215.19.197 On: Fri, 13 Dec 2013 15:18:41luminescence quenching. The calculation described here suggests additional field-dependent effect on the radiative lifetime. The energy broadening of the confined quantum well hole states into resonances also implies the above-men-tioned spreading of the wave function in k space and thus a decrease in its k = 0 component which will in turn reduce the transition probability (Ref. 7). This reduction is expected to be greatest for narrow quantum wells. The model adopted in the present study is too crude to provide a quantitative estimate of this effect, since the details of the band structure at the top of the valence band are ignored. Such details may be quite important because the applied field can make quan-tum well valence states associated with different nand k quasi-degenerate. Since the admixture of a higher lying state In',k') with the state In,k) is proportional to (E ... Jr. - En.Jr.' ) - 1, we might expect the transition probability to be a sensitive function of the field-induced broadening, The results presented here predict a rapid decrease of the hole lifetime with increasing applied field due to the tun-neling of carriers out of the quantum wells. The decrease in the observed luminescence lifetime follows from Eq. (4), pro-vided that the change in nonradiative lifetime is the domi-nant effect. This seems likely in view of the predicted expo-nential dependence in Fig. 2 of the hole lifetime on field strength. The increase of 1", with field explains the decrease in luminescence intensity. Comparison of Fig. 2 with the experimental results2 shows that the nonradiative lifetimes are underestimated in our calculation. (The experimentally measured overall lifetimes are in the range 50-300 ps.) This is wen in keeping with our remarks concerning the spreading of the zero field hole wave function in k space. By adopting the envelope function effective mass Hamiltonian to repre-sent the zero field solution, we have introduced a systematic error which increases with decreasing well width and which may underestimate the lifetime by as much as an order of magnitude. In the limit of very strong fields our method3 of determining the resonance width leads to a further overesti-mate of the broadening effect due to tunneling. An additional effect found in the experimental measure-ments is an increase in luminescence lifetime with increased carrier number density. This has been interpreted in terms of exciton polarization.2 The present calculations suggest that an alternative explanation of this effect may be found in the tunneling of free carriers out of their respective wells. The resulting charge separation sets up an electric field which opposes the applied field, so the "effective field" experienced by the system is smaller at high carrier number densities. In conclusion, we have presented an exact numerical solution of the Schrooinger equation for the quantum well Stark resonance problem which predicts the origin and mag-nitudes oflevel shifts and provides an adequate explanation of the observed lifetimes. Our calculation has a conceptual advantage over previous work4•5 in that it goes beyond the limit of validity of bound state calculations and provides a unified approach to Stark shifts and widths. We thank Dr. D, A. B. Miller for supplying a preprint of Ref. 5. E. J. A. acknowledges the award of a research fellow-ship by the University of NewcastIe upon Tyne. 'C. Alibert, S, Gaillard, J, A. Brum, G. Bastard, p, Frijlink, and M, Erman, Solid State Commun, 53, 457 (19851. 2 J, A. Kash, E. E. Mendez, and H, MorkOl$, Appl. Phys, Lett, 46, 173 (19851. 3E. J. Austin and M. Jaros, Phys, Rev, B 31, 5569 (19851, 4G, Bastard, E. E. Mendez, L. L Chang, and L. £saki, Phys, Rev. B 28, 3241 (19831, sO. A. B. Miller, O. S, Chemla, T. C. Oamen, A. C. Gossard, W, Wieg-mann, T. H. Wood, and C. A. Burrus, Phys. Rev. Lett, 53, 2173 (19841. 6M, Jaros, K. B. Wong, and M, A. Gell, Phys. Rev. B 31, 1205 (1985). 7M. Jaros, Deep Levels in Semiconductors (Adam Hilger, Bristol, 19821. Metalorganic growth of epita.xial fUms of CdTe and HgCdTe on sapphire substrates w. E. Hoke, R. Traczewski, V. G. Kreismanis, R. Korenstein, and P. J. Lemonias Raytheon Company. Research Division, Lexington. Massachusetts 02173 (Received 4 March 1985; accepted for publication 17 May 1985) Specular epitaxial films ofCdTe have been grown on (0001) sapphire substrates by the metal.organic chemical vapor deposition technique. The films grow with the (1] l)A orientation and exhibit narrow x-ray linewidths. Secondary ion mass spectrometry profiles indicate negligible diffusion of aluminum from the substrates, Epitaxial films of HgCdTe have been grown on top of the CdTe layers. Initial Hall measurements are encouraging for an aU metalorganic chemical vapor deposition process for depositing HgCdTe on sapphire substrates. Epitaxial films of HgCdTe are needed for the active ele-ments in infrared detection systems. Conventionally, HgCdTe is grown on CdTe substrates which have the attrac-tive properties of a dose lattice match with HgCdTe and of immunity from autodoping ofthe HgCdTe films. However, CdTe substrates are expensive, fragile, not available in large areas, and are of lower crystalline quality compared to com-monly available substrates such as Si, GaAs, AI20 3(sap-phire), or InSb. Therefore, an alternative substrate material is desired. An attractive candidate is sapphire because of its 276 Appl. Phys. Lett, 47 (31. 1 August 1985 0003-6951/85/150276-03$01.00 @ 1985 American Institute of Physics 276  This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:129.215.19.197 On: Fri, 13 Dec 2013 15:18:41

Electric field-induced shifts and lifetimes in GaAs-GaAlAs quantum wells

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     Edinburgh Research Explorer                                      Electric field-induced shifts and lifetimes in GaAs-GaAlAsquantum wellsCitation for published version:Austin, E & Jaros, M 1985, 'Electric field-induced shifts and lifetimes in GaAs-GaAlAs quantum wells',Applied Physics Letters, vol. 47, no. 3, pp. 274-276. https://doi.org/10.1063/1.96189Digital Object Identifier (DOI):10.1063/1.96189Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Publisher's PDF, also known as Version of recordPublished In:Applied Physics LettersPublisher Rights Statement:© Austin, E., & Jaros, M. (1985). Electric field-induced shifts and lifetimes in GaAs-GaAlAs quantum wells.Applied Physics Letters, 47(3), 274-276doi: 10.1063/1.96189General rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 29. Nov. 2025Electric field induced shifts and lifetimes in GaAsGaAlAs quantum wellsE. J. Austin and M. Jaros  Citation: Applied Physics Letters 47, 274 (1985); doi: 10.1063/1.96189 View online: http://dx.doi.org/10.1063/1.96189 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/47/3?ver=pdfcov Published by the AIP Publishing                      This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:129.215.19.197 On: Fri, 13 Dec 2013 15:18:41IT. Kobayashi and Y. Shin<xia, J. App!. Phys. 53, 3339 (1982). zL. G. Meiners, D. L. Li1e, and D. A. Collins,J. Vac. Sci. Techno!. 16, 1458 (1979). 'L. G. Meiners, J. Vac. Sci. Techno!. IS, 1402 (1978). 4H. C. Cassey, Jr., A. Y. Cho, D. V. Lang, E. H. Nicollian, and P. W. Foy, J. App!. Phys. 50, 3484 (1979). 5y. Hirota and T. Kobayashi, 1. App!. Phys. 53, 5037 (1982). 60. J. Olego, 1. A. Baumann, M. A. Kuck, R. Schachter, C. G. Michel, and P. M. Raccah, Solid State Commun. 52, 311 (1984). 7L. M. Terman, Solid State Electron. 5, 285 (1962). "K. P. Pande and D. Gutierrez, App!. Phys. Lett. 46, 416 (1985). 9H. H. Wieder, Surf. Sci. 132, 390 (1983). lOW. E. Spicer, P. W. Chye, P. R. Skeath, C. Y. Su, and I. Lindau, 1. Vac. Sci. Techno!. 16, 1422 (1979). IIW. E. Spicer, I. Lindau, P. Skeath, and C. Y. Su, J. Vac. Sci. Techno!. 17, 1019, (1980). 12M. S. Daw, D. L. Smith, C. A. Swarts, and T. C. McGill, J. Vac. Sci. Techno!. 19, 508 (1981). 131. D. Dow and R. E. Allen, J. Vac. Sci. Techno!. 20, 659 (1982). 14A. Nedoluha, J. Vac. Sci. Techno!. 21, 429 (1982). "H. H. Wieder,lEEE Electron Device Lett. EDL-4, 108 (1983). 160. J. Olego, R. Schachter, and J. A. Baumann, App!. Phys. Lett. 45, 1127 (1984). I7E. Yamaguchi and M. Minakata, J. App!. Phys. 55, 3098 (1984). E~ectroc field induced shins and Ufetimes in GaAs-GaAIAs quantum weUs E. J. Austin and M. Jaros School 0/ PhYSics, The University. Newcastle upon Tyne NEI 7R U. United Kingdom (Received 5 March 1985; accepted for publication 14 May 1985) We have performed an exact numerical calculation for an isolated GaAs quantum well system subjected to an external electric field. This calculation predicts both the shifts and widths of the Stark resonances. The shift predictions are in agreement with recent experimental results and the width calculations allow the observed decrease in luminescence lifetime with field to be interpreted in terms of the field-induced tunneling of the carriers. Recent experimental studies '.2 of the effects of electric fields on GaAs quantum well systems have enabled both the field-induced level shifts' and the field dependence of the photoluminescence lifetime2 to be obtained. In this letter the theoretical interpretation of these results is considered with-in the framework of an exact numerical solution of the Schr6dinger equation3 for an isolated quantum well system subjected to an electric field. This calculation provides a uni-fied approach to the prediction of Stark shifts and widths and the observed decrease in luminescence lifetime with in-creasing field strength is accounted for in terms of field-in-duced quantum mechanical tunneling. Previous theoretical calculations4•5 of electric field ef-fects on quantum well systems have treated the effect of the external field on the electronic structure by approximate techniques (in particular the variation method) appropriate in the weak-field limit. In these calculations the confined states are viewed as bound states and their field-induced broadening is ignored. Such calculations correctly predict the quadratic variation of the Stark shift with field at moder-ate field intensities. However, the bound state model fails to account for the decrease in the luminescence lifetime2 with field and a more detailed calculation is required. In the envelope function approximation the rapidly varying Bloch component of the wave function is ignored and the problem reduces to the one-dimensional effective mass equation familiar from previous studies. 4•5 For a parti-cle of charge e and mass m·, initially localized in an isolated well of width 2a and depth Va' the Schrodinger equation is fl d 2r/J ---7 -(Vo+eFx)r/J=Er/J 2m· dx-'If d 2r/J ----eFxr/J = Er/J 2m. dx2 ( 1) where F is the electric field applied perpendicular to the in-terface plane. The exact solutions of ( 1) can be obtained as linear combinations of Airy functions; application of the method of phase shift analysis to the sonutions enables the Stark resonance positions and widths to be calculated. It was found3 that for mod.erate fields both the variational. and ex-act calculations predict a quadratic dependence of the Stark shift on the field; this shift can be explained by perturbation theory. The quadratic Stark shift of the ground state 10) ofa quantum system is given by ilE = e2F2 L j (k IxO) 12 , k;"O Eo-Ek (2) where the states Ik ) are the excited states of the system. An increase in the wen depth increases the binding energy E and thus decreases ilE; an increase in well width increases the spatial extent of the ground state wave function and thus the matrix elements (k IxIO). At high fields the variational bound state formalism breaks down and the predictions of the two calculations differ considerably. The exact calcula-tion predicts a decrease in the effective energy shift at high fields and the emergence of new anti resonance states from the continuum into the well.. Figure 1 gives a comparison of the predictions of the 274 Appl. Phys. Lett. 47 (3), 1 August 1985 0003-6951/85/150274-03$01 .00 @ 1985 American Institute of Physics 274  This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:129.215.19.197 On: Fri, 13 Dec 2013 15:18:412.0r-------------------::> 1.5 ~ 1.0 E UJ <l C1 0.5 ~ o 70 A log F (V cm-1 ) FIG. 1. Comparison of calculated and experimental l Stark shifts for;:: 70 A and ;::37 A wells. In the calculation the well depth was taken as 100 meV and the hole effective mass as 0.45m. in order to facilitate a comparison with the calculated shifts of Ref. I. However, the results obtained in the quadratic regime as illustrated are not particularly sensitive to the assumed well depth. The horizontal bars extend between the "experimental" and "scaled" fields of Ref. (I):(i 1j70 A data; (X X)37A data. present calculation and the experimental results I obtained for::::: 37 A and ;:::: 70 A wells. The calculations represent only the hole energy shift; because of the shallower well depth for holes this is the major contribution to the total Stark shift. For the wider wen the calculated and experimental shifts are in good agreement given the large uncertainties in the deter-mination of the experimental electric field values. However, the figure shows that the shifts predicted for the;:::: 37 A well are much too high, except for the highest field values. One possible source ofthis discrepancy is suggested by the results of more detailed (zero field) calculations6 which go beyond the envelope function approximation employed here (and in all previous Stark shift calculations). These large calcula-tions show that in narrow wells the wave function of the confined state is more extended in k space. Consequently the matrix e1ement in (2) should be written as where tPn.k is a Bloch function associated with band n and reduced wave vector k. Since the characteristic value for k' of most of the terms on the right-hand side of this matrix element is different from k on the left, the value of the matrix e1ement is substantiaBy reduced. (This quenching of the ma-trix element is familiar from studies of the breakdown ofthe effective mass approximation for impurities in semiconduc-tors 7 and leads to a systematic error on iJE as calculated here which is greatest for narrow wells.) As the fie1d strength increases, the quantum state is broadened into a resonance by the interaction with the field. This energy broadening, which is accompanied by spreading of the wave function in k space, increases rapidly with field and dominates at high field. Since the fie1d-induced broadening is fully represented in the present calculation, it is not surprising that our high-field Stark shifts agree with experiment. 275 Appl. Phys. Lett., Vol. 47, No.3, 1 August 1985 u W VI Cl.. 0.5 1. a 1.5 F )( 10-5 ( V cm- 1 ) FIG. 2. Calculated tunneling lifetimes as a function of applied field for 30 A, 37 A. and 70 A wells; the other parameters used in the calculation are the same as for Fig. I. From the calculated resonance width r it is possible to obtain the corresponding lifetime 1'; the variation of lifetime with field for quantum wells of various widths is illustrated in Fig. 2. The mechanism of state decay is field-induced tun-neling out of the well and, as would be expected, this effect is greatest for narrow wells, where the particle binding energy is low. The luminescence of optically pumped quantum well systems is caused by the recombination of electron-hole pairs; nonradiative processes compete with radiatiative re-combination and the experimental lifetime 1'. is related to the radiative and nonradiative lifetimes by lire = lI1'r + lI1'nr. (3) The experimental lifetime measurements2 (performed on ;:::: 30 A wells) demonstrate that luminescence lifetime, peak luminescence intensity, and integrated luminescence aJi de-crease when the field is increased. Theoretical consider-ations suggest that 1'r should increase with field. In the enve-lope function approximation the wave function of a particle confined in a quantum weB is l/! =f(x)tPn.o(r), (4) where tPn.O (r) is a :Bloch state of zero wave vector and band index n. Sincef(x) is a slowly varying function, the transition probability for recombination can be factorized as (5) Under the influence of the electric field the maxima of the electron and hole wave functions are shifted in opposite di-rections and thus the overlap 1 if. (X)lfh (x» 12 decreases, lead-ing to a decrease in the transition probability and an increase in the radiative lifetime. However, the magnitude of this ef-fect [as previously estimated from the variational solutions of (1 )4] is much too small to account for the observed rate of E. J. Austin and M. Jaros 275  This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:129.215.19.197 On: Fri, 13 Dec 2013 15:18:41luminescence quenching. The calculation described here suggests additional field-dependent effect on the radiative lifetime. The energy broadening of the confined quantum well hole states into resonances also implies the above-men-tioned spreading of the wave function in k space and thus a decrease in its k = 0 component which will in turn reduce the transition probability (Ref. 7). This reduction is expected to be greatest for narrow quantum wells. The model adopted in the present study is too crude to provide a quantitative estimate of this effect, since the details of the band structure at the top of the valence band are ignored. Such details may be quite important because the applied field can make quan-tum well valence states associated with different nand k quasi-degenerate. Since the admixture of a higher lying state In',k') with the state In,k) is proportional to (E ... Jr. - En.Jr.' ) - 1, we might expect the transition probability to be a sensitive function of the field-induced broadening, The results presented here predict a rapid decrease of the hole lifetime with increasing applied field due to the tun-neling of carriers out of the quantum wells. The decrease in the observed luminescence lifetime follows from Eq. (4), pro-vided that the change in nonradiative lifetime is the domi-nant effect. This seems likely in view of the predicted expo-nential dependence in Fig. 2 of the hole lifetime on field strength. The increase of 1", with field explains the decrease in luminescence intensity. Comparison of Fig. 2 with the experimental results2 shows that the nonradiative lifetimes are underestimated in our calculation. (The experimentally measured overall lifetimes are in the range 50-300 ps.) This is wen in keeping with our remarks concerning the spreading of the zero field hole wave function in k space. By adopting the envelope function effective mass Hamiltonian to repre-sent the zero field solution, we have introduced a systematic error which increases with decreasing well width and which may underestimate the lifetime by as much as an order of magnitude. In the limit of very strong fields our method3 of determining the resonance width leads to a further overesti-mate of the broadening effect due to tunneling. An additional effect found in the experimental measure-ments is an increase in luminescence lifetime with increased carrier number density. This has been interpreted in terms of exciton polarization.2 The present calculations suggest that an alternative explanation of this effect may be found in the tunneling of free carriers out of their respective wells. The resulting charge separation sets up an electric field which opposes the applied field, so the "effective field" experienced by the system is smaller at high carrier number densities. In conclusion, we have presented an exact numerical solution of the Schrooinger equation for the quantum well Stark resonance problem which predicts the origin and mag-nitudes oflevel shifts and provides an adequate explanation of the observed lifetimes. Our calculation has a conceptual advantage over previous work4•5 in that it goes beyond the limit of validity of bound state calculations and provides a unified approach to Stark shifts and widths. We thank Dr. D, A. B. Miller for supplying a preprint of Ref. 5. E. J. A. acknowledges the award of a research fellow-ship by the University of NewcastIe upon Tyne. 'C. Alibert, S, Gaillard, J, A. Brum, G. Bastard, p, Frijlink, and M, Erman, Solid State Commun, 53, 457 (19851. 2 J, A. Kash, E. E. Mendez, and H, MorkOl$, Appl. Phys, Lett, 46, 173 (19851. 3E. J. Austin and M. Jaros, Phys, Rev, B 31, 5569 (19851, 4G, Bastard, E. E. Mendez, L. L Chang, and L. £saki, Phys, Rev. B 28, 3241 (19831, sO. A. B. Miller, O. S, Chemla, T. C. Oamen, A. C. Gossard, W, Wieg-mann, T. H. Wood, and C. A. Burrus, Phys. Rev. Lett, 53, 2173 (19841. 6M, Jaros, K. B. Wong, and M, A. Gell, Phys. Rev. B 31, 1205 (1985). 7M. Jaros, Deep Levels in Semiconductors (Adam Hilger, Bristol, 19821. Metalorganic growth of epita.xial fUms of CdTe and HgCdTe on sapphire substrates w. E. Hoke, R. Traczewski, V. G. Kreismanis, R. Korenstein, and P. J. Lemonias Raytheon Company. Research Division, Lexington. Massachusetts 02173 (Received 4 March 1985; accepted for publication 17 May 1985) Specular epitaxial films ofCdTe have been grown on (0001) sapphire substrates by the metal.organic chemical vapor deposition technique. The films grow with the (1] l)A orientation and exhibit narrow x-ray linewidths. Secondary ion mass spectrometry profiles indicate negligible diffusion of aluminum from the substrates, Epitaxial films of HgCdTe have been grown on top of the CdTe layers. Initial Hall measurements are encouraging for an aU metalorganic chemical vapor deposition process for depositing HgCdTe on sapphire substrates. Epitaxial films of HgCdTe are needed for the active ele-ments in infrared detection systems. Conventionally, HgCdTe is grown on CdTe substrates which have the attrac-tive properties of a dose lattice match with HgCdTe and of immunity from autodoping ofthe HgCdTe films. However, CdTe substrates are expensive, fragile, not available in large areas, and are of lower crystalline quality compared to com-monly available substrates such as Si, GaAs, AI20 3(sap-phire), or InSb. Therefore, an alternative substrate material is desired. An attractive candidate is sapphire because of its 276 Appl. Phys. Lett, 47 (31. 1 August 1985 0003-6951/85/150276-03$01.00 @ 1985 American Institute of Physics 276  This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. 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Electric field-induced shifts and lifetimes in GaAs-GaAlAs quantum wells

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