Austin, Elizabeth

Jaros, M

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E. J. Austin

M. Jaros

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Comment on ‘‘Exact calculations of quasibound states of an isolated quantum well with uniform electric field: Quantum-well Stark resonance’’

Edinburgh Research Explorer

     Edinburgh Research Explorer                                      Comment on ‘‘Exact calculations of quasibound states of an isolatedquantum well with uniform electric field: Quantum-well Starkresonance’’Citation for published version:Austin, E & Jaros, M 1988, 'Comment on ‘‘Exact calculations of quasibound states of an isolated quantumwell with uniform electric field: Quantum-well Stark resonance’’', Physical Review B (Condensed Matter),vol. B38, no. 9, pp. 6326-6327. https://doi.org//10.1103/PhysRevB.38.6326Digital Object Identifier (DOI):/10.1103/PhysRevB.38.6326Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Publisher's PDF, also known as Version of recordPublished In:Physical Review B (Condensed Matter)Publisher Rights Statement:© Austin, E., & Jaros, M. (1988). Comment on ‘‘Exact calculations of quasibound states of an isolated quantumwell with uniform electric field: Quantum-well Stark resonance’’. Physical Review B: Condensed Matter andMaterials Physics, B38(9), 6326-6327doi: /10.1103/PhysRevB.38.6326General 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. 2025PHYSICAL REVIEW B VOLUME 38, NUMBER 9 15 SEPTEMBER 1988-IIComment on "Exact calculations of quasibound states of an isolated quantum wellwith uniform electric field: Quantum-well Stark resonance"E. J. AustinDepartment ofElectrical Engineering, Imperial College ofScience and Technology, University ofLondon,Exhibition Road, London SS'72BT, United KingdomM. JarosDepartment of Theoretical Physics, The University ofNewcastle upon-Tyne-, Newcastle-upon Tyne,-NE1 7R U, United Kingdom(Received 31 March 1987; revised manuscript received 8 April 1988)It is shown that the method of Airy functions of complex argument and the scattering approachbased on a phase-shift analysis yield Stark quantum-well resonances which are in agreement in bothlow- and high-field limits. Furthermore, the scattering approach provides a unified view of reso-nances lying both below and above the semiclassical confining barrier.In a recent Rapid Communication, ' Ahn and Chuangpresented very interesting results concerning Stark reso-nances in semiconductor quantum wells. Their work isthe first study of the problem by the method of Airy func-tions of complex argument and offers universal curveswhich can be used to compute Stark shifts of confinedstates for any well parameters and over a wide range ofelectric fields. They compared their Stark shifts withthose predicted by Austin and Jaros and concluded thatthere is a discrepancy between the two methods in thehigh-field limit. This might be taken to imply that thescattering approach employed by Austin and Jaros failsto provide a useful alternative for investigations of Starkresonances. The purpose of this Comment is to point outthat no significant discrepancy exists. In fact, the scatter-ing method provides a unified view of the Starkphenomenon over a wide range of energies. Althoughmuch of the existing literature on the subject has beenpreoccupied with levels lying deep in the confining wells,there is a wealth of new field-induced effects in the hotelectron range of energies.The work of Ahn and Chuang rests on the recognitionthat the occurrence of a resonance implies the existenceof an asymptotic purely outgoing wave solution of theSchrodinger equation associated with complex energyE iI /2 (—the so-called Siegert state). Imposing the out-going wave boundary condition with the use of Airyfunctions of complex argument thus yields E and Idirectly. In general, the Stark resonances are defined inscattering theory as complex poles of the S matrix and avariety of approaches can be adopted to locate thesepoles. In Ref. 2 it was proposed to combine the use ofAiry functions with the method of phase-shift analysis.In particular, the phase shift was used to obtain the Starkresonance position in two ways: from the Breit-Wignerformula and from the position of the maximum of the(field-induced) change in the density of states. These re-sults are displayed in Fig. 3 of Ref. 2. The Breit-Wignerresonance position shows anomalous (positive) Stark shiftat high fields, whereas the density-of-states peak shows aconsistent negative shift. Rescaling of our results for thedensity of states in accordance with the method of Ref. Ishows that these are in agreement with those presented inFig. 3 of that wark. Thus the two calculations are quiteconsistent: the complex-Airy-function method yields thesame position of the maximum of the change in the densi-ty of states as that predicted by the phase-shift method.The discrepancy in the resonance positions obtainedfrom the Breit-Wigner formula in the high-field regime isneither new nor surprising. The formula represents alow-field sharp-resonance approximation and is known tohave a limited validity when applied to broad resonances.Our results showing the deviation of the simple Breit-Wigner fit from the peak of the density of states mark theimportant and physically meaningful point where thesharp-resonance approximation breaks down. Althoughthe turn-around behavior cannot be observed directly as apositive shift in the energy of the corresponding quantumstate, the related change in the density of states associat-ed with the antiresonance at higher energies should beobservable.Both the phase shift and the complex-Airy-functioncalculations exhibit a remarkable feature which has notbeen emphasized before and which is related to the factthat the quadratic shift as predicted from perturbationtheory prevails even at high fields. This means that theparticle is becoming trapped in a triangular well whosedepth increases linearly with the applied field. Evidencefor this interpretation comes from our calculations ofhigh-field lifetimes and line shapes which indicate thatthe system fails to "remember" its zero-field structure.Although the complex-Airy-function method offers adirect determination of the Stark resonance positions, byfocusing on the Siegert solution it discards off-I'esonanceinformation. The phase-shift method involves a morelengthy calculation of the resonance parameters but givessolutions of the scattering problem at all energies. It isthen possible to relate the width of the lowest Stark reso-nance to the ant;iresonance features in the range of higherenergies. The relationship between the resonance and an-tiresonance widths and positions gives a quantitativemeaning to observables such as line shapes and transit38 6326 1988 The American Physical Society38 COMMENTS 6327times. Such relations become particularly interesting insuperlattices where quantum states below as well asabove the semiclassical barrier are grouped into bandsseparated by forbidden gaps. For example, we have re-cently extended our calculations to systems of two to fivequantum wells with overlapping wave functions. It tran-spires that the position and bandwidths of the quasi-confined quantum states in the hot electron range are alsoaffected by the electric field. Under certain cir-cumstances wave-function localization effects causehigh-energy states to have smaller width than their low-energy counterparts. Such anomalous field-induced lo-calization corresponds to sharp features in the firstderivative of the phase shift with respect to energy whichis equivalent to the time delay (i.e., field-induced changesin the time of flight). It is difficu1t to conceive of otherthan scattering techniques that would provide a suitablemeans for dealing with this class of problems.D. Ahn and S. L. Chuang, Phys. Rev. B 34, 9034 (1986).E. J. Austin and M. Jaros, Phys. Rev. B 31, 5569 (1985).E. J. Austin and M. Jaros, J. Appl. Phys. 62, 558 (1987).

     Edinburgh Research Explorer                                      Comment on ‘‘Exact calculations of quasibound states of anisolated quantum well with uniform electric field: Quantum-wellStark resonance’’Citation for published version:Austin, E & Jaros, M 1988, 'Comment on ‘‘Exact calculations of quasibound states of an isolated quantumwell with uniform electric field: Quantum-well Stark resonance’’' Physical Review B: Condensed Matter andMaterials Physics, vol B38, no. 9, pp. 6326-6327., /10.1103/PhysRevB.38.6326Digital Object Identifier (DOI):/10.1103/PhysRevB.38.6326Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Publisher final version (usually the publisher pdf)Published In:Physical Review B: Condensed Matter and Materials PhysicsPublisher Rights Statement:© Austin, E., & Jaros, M. (1988). Comment on ‘‘Exact calculations of quasibound states of an isolated quantumwell with uniform electric field: Quantum-well Stark resonance’’. Physical Review B: Condensed Matter andMaterials Physics, B38(9), 6326-6327doi: /10.1103/PhysRevB.38.6326General 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. 2015PHYSICAL REVIEW B VOLUME 38, NUMBER 9 15 SEPTEMBER 1988-IIComment on "Exact calculations of quasibound states of an isolated quantum wellwith uniform electric field: Quantum-well Stark resonance"E. J. AustinDepartment ofElectrical Engineering, Imperial College ofScience and Technology, University ofLondon,Exhibition Road, London SS'72BT, United KingdomM. JarosDepartment of Theoretical Physics, The University ofNewcastle upon-Tyne-, Newcastle-upon Tyne,-NE1 7R U, United Kingdom(Received 31 March 1987; revised manuscript received 8 April 1988)It is shown that the method of Airy functions of complex argument and the scattering approachbased on a phase-shift analysis yield Stark quantum-well resonances which are in agreement in bothlow- and high-field limits. Furthermore, the scattering approach provides a unified view of reso-nances lying both below and above the semiclassical confining barrier.In a recent Rapid Communication, ' Ahn and Chuangpresented very interesting results concerning Stark reso-nances in semiconductor quantum wells. Their work isthe first study of the problem by the method of Airy func-tions of complex argument and offers universal curveswhich can be used to compute Stark shifts of confinedstates for any well parameters and over a wide range ofelectric fields. They compared their Stark shifts withthose predicted by Austin and Jaros and concluded thatthere is a discrepancy between the two methods in thehigh-field limit. This might be taken to imply that thescattering approach employed by Austin and Jaros failsto provide a useful alternative for investigations of Starkresonances. The purpose of this Comment is to point outthat no significant discrepancy exists. In fact, the scatter-ing method provides a unified view of the Starkphenomenon over a wide range of energies. Althoughmuch of the existing literature on the subject has beenpreoccupied with levels lying deep in the confining wells,there is a wealth of new field-induced effects in the hotelectron range of energies.The work of Ahn and Chuang rests on the recognitionthat the occurrence of a resonance implies the existenceof an asymptotic purely outgoing wave solution of theSchrodinger equation associated with complex energyE iI /2 (—the so-called Siegert state). Imposing the out-going wave boundary condition with the use of Airyfunctions of complex argument thus yields E and Idirectly. In general, the Stark resonances are defined inscattering theory as complex poles of the S matrix and avariety of approaches can be adopted to locate thesepoles. In Ref. 2 it was proposed to combine the use ofAiry functions with the method of phase-shift analysis.In particular, the phase shift was used to obtain the Starkresonance position in two ways: from the Breit-Wignerformula and from the position of the maximum of the(field-induced) change in the density of states. These re-sults are displayed in Fig. 3 of Ref. 2. The Breit-Wignerresonance position shows anomalous (positive) Stark shiftat high fields, whereas the density-of-states peak shows aconsistent negative shift. Rescaling of our results for thedensity of states in accordance with the method of Ref. Ishows that these are in agreement with those presented inFig. 3 of that wark. Thus the two calculations are quiteconsistent: the complex-Airy-function method yields thesame position of the maximum of the change in the densi-ty of states as that predicted by the phase-shift method.The discrepancy in the resonance positions obtainedfrom the Breit-Wigner formula in the high-field regime isneither new nor surprising. The formula represents alow-field sharp-resonance approximation and is known tohave a limited validity when applied to broad resonances.Our results showing the deviation of the simple Breit-Wigner fit from the peak of the density of states mark theimportant and physically meaningful point where thesharp-resonance approximation breaks down. Althoughthe turn-around behavior cannot be observed directly as apositive shift in the energy of the corresponding quantumstate, the related change in the density of states associat-ed with the antiresonance at higher energies should beobservable.Both the phase shift and the complex-Airy-functioncalculations exhibit a remarkable feature which has notbeen emphasized before and which is related to the factthat the quadratic shift as predicted from perturbationtheory prevails even at high fields. This means that theparticle is becoming trapped in a triangular well whosedepth increases linearly with the applied field. Evidencefor this interpretation comes from our calculations ofhigh-field lifetimes and line shapes which indicate thatthe system fails to "remember" its zero-field structure.Although the complex-Airy-function method offers adirect determination of the Stark resonance positions, byfocusing on the Siegert solution it discards off-I'esonanceinformation. The phase-shift method involves a morelengthy calculation of the resonance parameters but givessolutions of the scattering problem at all energies. It isthen possible to relate the width of the lowest Stark reso-nance to the ant;iresonance features in the range of higherenergies. The relationship between the resonance and an-tiresonance widths and positions gives a quantitativemeaning to observables such as line shapes and transit38 6326 1988 The American Physical Society38 COMMENTS 6327times. Such relations become particularly interesting insuperlattices where quantum states below as well asabove the semiclassical barrier are grouped into bandsseparated by forbidden gaps. For example, we have re-cently extended our calculations to systems of two to fivequantum wells with overlapping wave functions. It tran-spires that the position and bandwidths of the quasi-confined quantum states in the hot electron range are alsoaffected by the electric field. Under certain cir-cumstances wave-function localization effects causehigh-energy states to have smaller width than their low-energy counterparts. Such anomalous field-induced lo-calization corresponds to sharp features in the firstderivative of the phase shift with respect to energy whichis equivalent to the time delay (i.e., field-induced changesin the time of flight). It is difficu1t to conceive of otherthan scattering techniques that would provide a suitablemeans for dealing with this class of problems.D. Ahn and S. L. Chuang, Phys. Rev. B 34, 9034 (1986).E. J. Austin and M. Jaros, Phys. Rev. B 31, 5569 (1985).E. J. Austin and M. Jaros, J. Appl. Phys. 62, 558 (1987).

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Comment on ‘‘Exact calculations of quasibound states of an isolated quantum well with uniform electric field: Quantum-well Stark resonance’’

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