- Email: [email protected]
Transmission coefficients for resonant tunneling in multi-barrier graded quantum well semiconductor heterostructures
5 December 1994 PHYSICS LETTERS A
ELSEVIER
Physics Letters A 195 (1994) 253-257
Transmission coefficients for resonant tunneling in multi-barrier graded quantum well semiconductor heterostructures John A. Alphonso-Gibbs, Shaune S. Allen, Steven L. Richardson Department of ElectricalEngineering and Materials ScienceResearch Center, School of Engineering, Howard University, Washington, DC 20059, USA Received 11 August 1994; acceptedfor publication 29 September 1994 Communicatedby L.J. Sham
Abstract
Resonant tunneling of electrons is studied in a new dass of multi-barrier semiconductor heterostructures which contain graded quantum wells, whose dopes can be changed by varying the alloy composition of the constituent material. Such systems, in the presence of an externally applied bias, can exhibit resonant tunneling behavior that is quite different from previously studied multi-barrier materials, because of the unique coupling of the quasi-bound states between the graded quantum wells. In this paper, we explicitly compute the electron transmission coefficient T(E) for such systems as a function of the multi-barrier geometry and internal slopes of the quantum wells to obtain novel transmission coefficient spectra and discuss their implications.
Resonant tunneling phenomena in semiconductor heterostructures continue to enjoy a significant amount of interest, since the pioneering efforts of Esaki and Tsu more than twenty years ago [ 1 ]. Recently, Kan'an, Puri, and Odagaki [2 ] proposed that double-barrier resonant tunneling structures of GaAs/ /do.sGao.sAs//dyGal_rAs//do.sGao.sAs/GaAs might be fabricated through MBE or MOCVD techniques. These systems are different from previously studied double-barrier semiconductor heterostructures because t h e / d concentration, which is denoted by y, can now be uniformly altered during growth conditions, thus changing the slope of the conduction band edge in the quantum well regions. This structure, in the absence of an applied constant electric field, is now equivalent to a semiconductor heterostructure, in which an effective static electric field has been selectively applied to the quantum well region only. The additional effect of applying a real constant electric field to the semiconductor heterostructure can now
cause an even greater change in the shape of the conduction band edge of the quantum well, and hence the resonant tunneling properties of the system, depending on whether the electric field is forward or reversed biased. Kan'an et al. [ 2] speculated that a double-barrier semiconductor heterostructure/do.16Gao.s4As //do.sGao.sAs / AlyGal_yAs //do.sGao.sAs/ GaAs could be synthesized, where the introduction of an alloy of arbitrary composition/do.t 6Gao.s4As in the region to the left of the first barrier raises the energy of that region by 0.16 eV and now makes a negatively sloped quantum well region theoretically possible (cf. Fig. 1 ). They used a transfer matrix approach using the well-developed theory of transmission lines to calculate the transmission coefficient of tunneling electrons T(E, Va) as a function of incident electron energy E and applied bias V, for an arbitrarily-shaped barrier [ 3 ]. In particular, they approximated a continuous variation in the potential energy barrier by a multistep function of N segments,
0375-9601/94/$07.00 © 1994 ElsevierScienceB.V. All rights ~served SSDI0375-9601 (94)00793-4
254
J.A. Alphonso-Gibbs et al. / Physics Letters A 195 (1994) 253-25 7
V(x)
AI 0.16Gao.s4As
AIo.sGao~As
Al3,Gal.yAs
AIo.sGao.sAs
GaAs
O.16eV
O.OcV Fig. 1. Schematic repr--~entation of conduction band edge for Alo.1~Gao.84As/Alo.sGao.~As/AlyGal_yAs/Alo.SGao.SAs/GaAs double-barrier semiconductor heterostructure with single graded quantum well in the absence of an applied constant electric field as originally suggested by Kan'an, Puff, and Odagakj in Ref. [2 ].
each of which represents a single barrier with a constant potential and an effective mass [4 ]. In this paper, we have extended the suggestion of Ref. [2] to compute the transmission coefficient T(E) for multiple-barrier semiconductor heterostructures which have more than one quantum well, in the presence of an applied constant potential bias Va. In particular, we studied a Aloa6Gao.s4As/Alo. 5Gao.~As / AlyGat_yAs / Alo.sGao.sAs / Aly,Gal_y,As/Alo.sGao.sAs/GaAs triple-barrier graded quantum well semiconductor heterostructure model which now opens additional flexibility in the control of resonance tunneling behavior because of the existence of two internal quantum wells, have different A1 concentrations given by y and y', respectively (cf. Fig. 2). The coupling of the quasi-bound states in the quantum wells of these triple-barrier structures can now affect the transmission coefficient in novel ways which can be modified by varying y and y' with respect to each other in the presence of an externally applied static potential bias V~. We have solved the one-electron Schr'6dinger wave equation in the effective mass approximation for the transmission coefficient T(E) through a triple-barrier graded quantum well semiconductor heterostructure in the presence of an applied constant electric field [5] (cf. Fig. 2). We employ an exact Airy function formalism and the transfer matrix technique, as described in a previous paper [ 6 ] i, to compute T(E). Our formalism does not require the apt We should note that in our formalism we have treated the effective mass as a constant in the quantum well regions, but we find that this approximation does not seriously alter our results.
proximation of representing the continuous potential energy barrier of the conduction band edge by finite barrier s.teps and has been shown to be in excellent agreement with the approximate results [6,7 ]. We used our method to verify the results of Ref. [ 2] by calculating the transmission coefficient for a doublebarrier semiconductor heterostructure with one quantum well, where the effective electric field in the quantum well is varied by changing the AI concentration y. The transmission coefficient T(E) is then also computed as a function of the geometry of the semiconductor heterostructure, incident electron energy E, and applied bias Va. To explore the resonant tunneling behavior of our novel triple-barrier graded quantum well semiconductor heterostructure, we performed a number of model calculations using the prototype structure given in Fig. 2a, in the presence of an applied bias Va. The effective mass in the well and barrier regions were chosen to be 0.067mo and 1.087mo, respectively. The barriers were composed of Alo.5Gao.sAs with a height of 0.5 eV and a width of 20 A, while the quantum well widths were 50 A. The aluminum compositions in the well regions were continuously varied as a function of the slope parameters oq and t~2, for each quantum well, respectively. Note that the choice of t~ is arbitrary and follows the selection made in Ref. [ 2 ]. We show in Figs. 3 and 4 model calculations for the transmission coefficients for resonant tunneling through triple-barrier graded quantum well semiconductor heterostructures which are sketched in Fig. 2. In both cases, we have computed T(E) in the presence of medium (0.7 V) and low (0.4 V) voltages, with the slopes of the two inner quantum wells mod-
ZA. Alphonso-Gibbs et ai. / Physics Letters A 195 (1994) 253-25 7
255
(a) V(x)
i,AItsGaLsAJ
AluGauAJ
AIo..¢Gao.$As
AIo.16Gae.s4As
AlyGal.iAs
GaAs
0.16eV 0.0eV
AIo.16Ga l0"~v 0£4AS+A]o' eva sG~~~yGllyAs~
" ":~
GaAs
0~0eV Fig. 2. Schematic representation of conduction band edge diagram for AIo.16Gao.8~Ls/ AIo.sGa0.sAs/ AlyGam- ~Ls/ Alo.sGao.~As/ AIr Ga i - >,/ Alo.sGao.sAs/GaAstriple-barrier semiconductor heterostructure with two graded quantum wells in the absence (a) and presence (b) of a constant applied bias V.. The different aluminum concentrations in the two graded quantum well regions are denoted by y and y', respectively.
1
0.1
"~
o.1
0.05 .~. 0.01
o.ol i0.005
0.001 0.001 O.l
0.2
0.3
0.4
0.5
Energy (eV)
'
0:2'
'0:3
0'.4
0.5
Energy (eV)
Fig. 3. Calculation of transmission coefficient versus electron incident energy for triple-barrier double-well semiconductor heterostructures depicted in Fig. 2. Barrier heights and widths are 0.5 eV and 20 A, respectively. Quantum wells are 50 A in thickness. Oa've (a): oq ffi -3.6X l0 TV m -I, a2= -3.6X 107V m -I, applied voltage is 0.4 V. Curve (b): alffi-3.6X107 V m -l, a2 = - 3.6 X 107 V m - i, applied vol~ge is 0.7 V.
Fig. 4. Calculation of transmission coefficient versus electron incident energy for triple-barrier double-well semiconductor heterostructures depicted in Fig. 2. Barrier heights and widths are 0.5 eV and 20 A, respectively. Quantum wells arc 50 ]k in thickness. Curve (a): cq = - 3 . 6 × 107 V m -~, a2= -7.2X 107 V m -t, applied voltage is 0.4 V. Curve (b): oq=-3.6X107 V m -l, a2 = - 7.2 X 10 ~ V m - i, applied voltage is 0.7 V.
ified t h r o u g h the p a r a m e t e r a~, where i = 1, 2 for the first a n d second q u a n t u m well, respectively. Note that b o t h slopes are the s a m e a n d n e g a t i v e i n Fig. 3, a n d the slope o f the s e c o n d q u a n t u m well is twice as m u c h
as the slope o f the first q u a n t u m well i n Fig. 4, alt h o u g h the slopes are still negative. T h e physics is relatively s i m p l e to u n d e r s t a n d i n these cases since the effect o f a higher bias is to shift the relative p o s i t i o n
256
J.A. Alphonso-Gibbs et al. / Physics Letters A 195 (I 994) 253-25 7
of the quasi-bound states in the quantum wells and thus shift the resonances to the left of the transmission spectra. The results have been seen and discussed for more conventional triple-barrier semiconductor heterostructures [ 6 ]. Novel features in the transmission spectra occur when the relative slopes of the inner quantum wells are arranged so that they are opposite to each other (of. Figs. 5 and 6). This effect causes a unique type of coupling between the quasi-bound states in the quantum wells and has quite a different behavior from that seen when the slopes have the same sign as be1 o.I .,~
0.01 .~
0.001 0.0001 0.00001 0
03
0.2
03
0.4
0.5
Energy (eV)
Fig. 5. Calculation of transmission coefficient versus electron incident energy for triple-barrier double-well semiconductor heterostructures depicted in Fig. 2. Barrier heights and widths are 0.5 eV and 20 A, respectively. Quantum wells are 50 A in thickness. Curve (a): cq = + 3 . 6 × 107V m -x, a2= - 3 . 6 × 107V m - l , applied voltage is 0.4 V. Curve (b): oq=3.6×107 V m - l , or2= - 3.6 X 107 V m -I , applied voltage is 0.7 V.
•d
o.1
(a;
0.01 0.001 0.0001 0
0.I
0.2
0.3
0.4
fore. In particular, for opposite signs of a, the effect of a higher applied voltage is the shifting of the resonances to the left as seen previously, but the new feature is the broadening of these resonances. This broadening of the transmission coefficients has important implications for the current density and, in particular, the peak-to-valley ratios of these systems, and these results will be discussed elsewhere [ 5 ]. In summary, we have presented model calculations that show that it is possible to create novel triple-barrier graded quantum well semiconductor heterostructures, which demonstrate new behavior in their resonant tunneling transmission spectra. In particular one can now tailor transmission spectra by both manipulating the growth conditions and composition of the materials in the quantum wells of these systems, as well as by varying the applied external voltage. The unique coupling of the positions of the quasi-bound state in these systems gives an additional degree of freedom which can "nanoengineered" for the desired structure in the resonant transmission spectra and current density, and we hope that our study will encourage further experimental efforts to synthesize such novel systems for possible applications in resonant tunneling devices. We wish to thank A.M. Kan'an and A. Purl for useful discussions throughout the course of this work. We wish to thank the Army Research Office (DAAL 0389-G-0101 ), the National Science Foundation (HRD 92 55 378), the General Electric Faculty for the Future Foundation, the Howard University Faculty Research Support Grant Program, and the Howard University Graduate School of Arts & Sciences for support of this work. Our gratitude also goes to Dr. Shirley A. Jackson for introducing us to the transfer matrix technique and to Dr. William R. Frensley for useful comments. Finally, we gratefully acknowledge supercomputing support at the Pittsburgh Supercomputing Center.
0.5
Energy (eV)
Fig. 6. Calculation o f transmission coefficient versus electron incident energy for triple-barrier double-well semiconductor heterostructures depicted in Fig. 2. Barrier heights and widths are 0.5 eV and 20 A, respectively. Quantum wells are 50 A in thickhess. Curve (a): a l = - 3 . 6 × 107 V m -I, a 2 = 3 . 6 × l07 V m - l , applied voltage is 0.4 V. Curve (b): ~x1=-3.6×107 V m -1, a2 = 3.6 × 107 V m - i, applied voltage is 0.7 V.
References [ 1 ] L. Esaki and R. Tsu, IBM J. Res. Develop. 14 (1970) 61; R. Tsu and L. Esaki, Appl. Phys. Lett. 22 (1973) 562; L. Esaki, in: Recent topics in semiconductor physics, eds. H. Kamimura and Y. Toyozawa (World Scientific, Singapore, 1983) p. 1.
J.A. Alphonso-Gibbs et ai. / Physics Letters A 195 (1994) 253-25 7
[2] A.M. Kan'an, A. Puri and T. Odagaki, Solid State Commun. 86 (1993) 113. [3] Y. Zebda and A.M. Kan'an, J. Appl. Phys. 72 (1992) 559. [4] A.N. Khondker, M.R. Khan and A.F.M. Anwar, J. Appl. Phys. 63 (1988) 5191. [5]J.A. Alphonso-Gibbs, S.S. Allen and S.L. Richardson, unpublished.
257
[6] S.S. Allen and S.L. Richardson, Phys. Rev. B 15, to be published. [7]J.A. Alphonso-Gibbs, M.E. thesis, Howard University (1993). [8] M.O. Vassel, J. Lee and H.F. Lockwood, J. Appl. Phys. 54 (1983) 5206.
ELSEVIER
Physics Letters A 195 (1994) 253-257
Transmission coefficients for resonant tunneling in multi-barrier graded quantum well semiconductor heterostructures John A. Alphonso-Gibbs, Shaune S. Allen, Steven L. Richardson Department of ElectricalEngineering and Materials ScienceResearch Center, School of Engineering, Howard University, Washington, DC 20059, USA Received 11 August 1994; acceptedfor publication 29 September 1994 Communicatedby L.J. Sham
Abstract
Resonant tunneling of electrons is studied in a new dass of multi-barrier semiconductor heterostructures which contain graded quantum wells, whose dopes can be changed by varying the alloy composition of the constituent material. Such systems, in the presence of an externally applied bias, can exhibit resonant tunneling behavior that is quite different from previously studied multi-barrier materials, because of the unique coupling of the quasi-bound states between the graded quantum wells. In this paper, we explicitly compute the electron transmission coefficient T(E) for such systems as a function of the multi-barrier geometry and internal slopes of the quantum wells to obtain novel transmission coefficient spectra and discuss their implications.
Resonant tunneling phenomena in semiconductor heterostructures continue to enjoy a significant amount of interest, since the pioneering efforts of Esaki and Tsu more than twenty years ago [ 1 ]. Recently, Kan'an, Puri, and Odagaki [2 ] proposed that double-barrier resonant tunneling structures of GaAs/ /do.sGao.sAs//dyGal_rAs//do.sGao.sAs/GaAs might be fabricated through MBE or MOCVD techniques. These systems are different from previously studied double-barrier semiconductor heterostructures because t h e / d concentration, which is denoted by y, can now be uniformly altered during growth conditions, thus changing the slope of the conduction band edge in the quantum well regions. This structure, in the absence of an applied constant electric field, is now equivalent to a semiconductor heterostructure, in which an effective static electric field has been selectively applied to the quantum well region only. The additional effect of applying a real constant electric field to the semiconductor heterostructure can now
cause an even greater change in the shape of the conduction band edge of the quantum well, and hence the resonant tunneling properties of the system, depending on whether the electric field is forward or reversed biased. Kan'an et al. [ 2] speculated that a double-barrier semiconductor heterostructure/do.16Gao.s4As //do.sGao.sAs / AlyGal_yAs //do.sGao.sAs/ GaAs could be synthesized, where the introduction of an alloy of arbitrary composition/do.t 6Gao.s4As in the region to the left of the first barrier raises the energy of that region by 0.16 eV and now makes a negatively sloped quantum well region theoretically possible (cf. Fig. 1 ). They used a transfer matrix approach using the well-developed theory of transmission lines to calculate the transmission coefficient of tunneling electrons T(E, Va) as a function of incident electron energy E and applied bias V, for an arbitrarily-shaped barrier [ 3 ]. In particular, they approximated a continuous variation in the potential energy barrier by a multistep function of N segments,
0375-9601/94/$07.00 © 1994 ElsevierScienceB.V. All rights ~served SSDI0375-9601 (94)00793-4
254
J.A. Alphonso-Gibbs et al. / Physics Letters A 195 (1994) 253-25 7
V(x)
AI 0.16Gao.s4As
AIo.sGao~As
Al3,Gal.yAs
AIo.sGao.sAs
GaAs
O.16eV
O.OcV Fig. 1. Schematic repr--~entation of conduction band edge for Alo.1~Gao.84As/Alo.sGao.~As/AlyGal_yAs/Alo.SGao.SAs/GaAs double-barrier semiconductor heterostructure with single graded quantum well in the absence of an applied constant electric field as originally suggested by Kan'an, Puff, and Odagakj in Ref. [2 ].
each of which represents a single barrier with a constant potential and an effective mass [4 ]. In this paper, we have extended the suggestion of Ref. [2] to compute the transmission coefficient T(E) for multiple-barrier semiconductor heterostructures which have more than one quantum well, in the presence of an applied constant potential bias Va. In particular, we studied a Aloa6Gao.s4As/Alo. 5Gao.~As / AlyGat_yAs / Alo.sGao.sAs / Aly,Gal_y,As/Alo.sGao.sAs/GaAs triple-barrier graded quantum well semiconductor heterostructure model which now opens additional flexibility in the control of resonance tunneling behavior because of the existence of two internal quantum wells, have different A1 concentrations given by y and y', respectively (cf. Fig. 2). The coupling of the quasi-bound states in the quantum wells of these triple-barrier structures can now affect the transmission coefficient in novel ways which can be modified by varying y and y' with respect to each other in the presence of an externally applied static potential bias V~. We have solved the one-electron Schr'6dinger wave equation in the effective mass approximation for the transmission coefficient T(E) through a triple-barrier graded quantum well semiconductor heterostructure in the presence of an applied constant electric field [5] (cf. Fig. 2). We employ an exact Airy function formalism and the transfer matrix technique, as described in a previous paper [ 6 ] i, to compute T(E). Our formalism does not require the apt We should note that in our formalism we have treated the effective mass as a constant in the quantum well regions, but we find that this approximation does not seriously alter our results.
proximation of representing the continuous potential energy barrier of the conduction band edge by finite barrier s.teps and has been shown to be in excellent agreement with the approximate results [6,7 ]. We used our method to verify the results of Ref. [ 2] by calculating the transmission coefficient for a doublebarrier semiconductor heterostructure with one quantum well, where the effective electric field in the quantum well is varied by changing the AI concentration y. The transmission coefficient T(E) is then also computed as a function of the geometry of the semiconductor heterostructure, incident electron energy E, and applied bias Va. To explore the resonant tunneling behavior of our novel triple-barrier graded quantum well semiconductor heterostructure, we performed a number of model calculations using the prototype structure given in Fig. 2a, in the presence of an applied bias Va. The effective mass in the well and barrier regions were chosen to be 0.067mo and 1.087mo, respectively. The barriers were composed of Alo.5Gao.sAs with a height of 0.5 eV and a width of 20 A, while the quantum well widths were 50 A. The aluminum compositions in the well regions were continuously varied as a function of the slope parameters oq and t~2, for each quantum well, respectively. Note that the choice of t~ is arbitrary and follows the selection made in Ref. [ 2 ]. We show in Figs. 3 and 4 model calculations for the transmission coefficients for resonant tunneling through triple-barrier graded quantum well semiconductor heterostructures which are sketched in Fig. 2. In both cases, we have computed T(E) in the presence of medium (0.7 V) and low (0.4 V) voltages, with the slopes of the two inner quantum wells mod-
ZA. Alphonso-Gibbs et ai. / Physics Letters A 195 (1994) 253-25 7
255
(a) V(x)
i,AItsGaLsAJ
AluGauAJ
AIo..¢Gao.$As
AIo.16Gae.s4As
AlyGal.iAs
GaAs
0.16eV 0.0eV
AIo.16Ga l0"~v 0£4AS+A]o' eva sG~~~yGllyAs~
" ":~
GaAs
0~0eV Fig. 2. Schematic representation of conduction band edge diagram for AIo.16Gao.8~Ls/ AIo.sGa0.sAs/ AlyGam- ~Ls/ Alo.sGao.~As/ AIr Ga i - >,/ Alo.sGao.sAs/GaAstriple-barrier semiconductor heterostructure with two graded quantum wells in the absence (a) and presence (b) of a constant applied bias V.. The different aluminum concentrations in the two graded quantum well regions are denoted by y and y', respectively.
1
0.1
"~
o.1
0.05 .~. 0.01
o.ol i0.005
0.001 0.001 O.l
0.2
0.3
0.4
0.5
Energy (eV)
'
0:2'
'0:3
0'.4
0.5
Energy (eV)
Fig. 3. Calculation of transmission coefficient versus electron incident energy for triple-barrier double-well semiconductor heterostructures depicted in Fig. 2. Barrier heights and widths are 0.5 eV and 20 A, respectively. Quantum wells are 50 A in thickness. Oa've (a): oq ffi -3.6X l0 TV m -I, a2= -3.6X 107V m -I, applied voltage is 0.4 V. Curve (b): alffi-3.6X107 V m -l, a2 = - 3.6 X 107 V m - i, applied vol~ge is 0.7 V.
Fig. 4. Calculation of transmission coefficient versus electron incident energy for triple-barrier double-well semiconductor heterostructures depicted in Fig. 2. Barrier heights and widths are 0.5 eV and 20 A, respectively. Quantum wells arc 50 ]k in thickness. Curve (a): cq = - 3 . 6 × 107 V m -~, a2= -7.2X 107 V m -t, applied voltage is 0.4 V. Curve (b): oq=-3.6X107 V m -l, a2 = - 7.2 X 10 ~ V m - i, applied voltage is 0.7 V.
ified t h r o u g h the p a r a m e t e r a~, where i = 1, 2 for the first a n d second q u a n t u m well, respectively. Note that b o t h slopes are the s a m e a n d n e g a t i v e i n Fig. 3, a n d the slope o f the s e c o n d q u a n t u m well is twice as m u c h
as the slope o f the first q u a n t u m well i n Fig. 4, alt h o u g h the slopes are still negative. T h e physics is relatively s i m p l e to u n d e r s t a n d i n these cases since the effect o f a higher bias is to shift the relative p o s i t i o n
256
J.A. Alphonso-Gibbs et al. / Physics Letters A 195 (I 994) 253-25 7
of the quasi-bound states in the quantum wells and thus shift the resonances to the left of the transmission spectra. The results have been seen and discussed for more conventional triple-barrier semiconductor heterostructures [ 6 ]. Novel features in the transmission spectra occur when the relative slopes of the inner quantum wells are arranged so that they are opposite to each other (of. Figs. 5 and 6). This effect causes a unique type of coupling between the quasi-bound states in the quantum wells and has quite a different behavior from that seen when the slopes have the same sign as be1 o.I .,~
0.01 .~
0.001 0.0001 0.00001 0
03
0.2
03
0.4
0.5
Energy (eV)
Fig. 5. Calculation of transmission coefficient versus electron incident energy for triple-barrier double-well semiconductor heterostructures depicted in Fig. 2. Barrier heights and widths are 0.5 eV and 20 A, respectively. Quantum wells are 50 A in thickness. Curve (a): cq = + 3 . 6 × 107V m -x, a2= - 3 . 6 × 107V m - l , applied voltage is 0.4 V. Curve (b): oq=3.6×107 V m - l , or2= - 3.6 X 107 V m -I , applied voltage is 0.7 V.
•d
o.1
(a;
0.01 0.001 0.0001 0
0.I
0.2
0.3
0.4
fore. In particular, for opposite signs of a, the effect of a higher applied voltage is the shifting of the resonances to the left as seen previously, but the new feature is the broadening of these resonances. This broadening of the transmission coefficients has important implications for the current density and, in particular, the peak-to-valley ratios of these systems, and these results will be discussed elsewhere [ 5 ]. In summary, we have presented model calculations that show that it is possible to create novel triple-barrier graded quantum well semiconductor heterostructures, which demonstrate new behavior in their resonant tunneling transmission spectra. In particular one can now tailor transmission spectra by both manipulating the growth conditions and composition of the materials in the quantum wells of these systems, as well as by varying the applied external voltage. The unique coupling of the positions of the quasi-bound state in these systems gives an additional degree of freedom which can "nanoengineered" for the desired structure in the resonant transmission spectra and current density, and we hope that our study will encourage further experimental efforts to synthesize such novel systems for possible applications in resonant tunneling devices. We wish to thank A.M. Kan'an and A. Purl for useful discussions throughout the course of this work. We wish to thank the Army Research Office (DAAL 0389-G-0101 ), the National Science Foundation (HRD 92 55 378), the General Electric Faculty for the Future Foundation, the Howard University Faculty Research Support Grant Program, and the Howard University Graduate School of Arts & Sciences for support of this work. Our gratitude also goes to Dr. Shirley A. Jackson for introducing us to the transfer matrix technique and to Dr. William R. Frensley for useful comments. Finally, we gratefully acknowledge supercomputing support at the Pittsburgh Supercomputing Center.
0.5
Energy (eV)
Fig. 6. Calculation o f transmission coefficient versus electron incident energy for triple-barrier double-well semiconductor heterostructures depicted in Fig. 2. Barrier heights and widths are 0.5 eV and 20 A, respectively. Quantum wells are 50 A in thickhess. Curve (a): a l = - 3 . 6 × 107 V m -I, a 2 = 3 . 6 × l07 V m - l , applied voltage is 0.4 V. Curve (b): ~x1=-3.6×107 V m -1, a2 = 3.6 × 107 V m - i, applied voltage is 0.7 V.
References [ 1 ] L. Esaki and R. Tsu, IBM J. Res. Develop. 14 (1970) 61; R. Tsu and L. Esaki, Appl. Phys. Lett. 22 (1973) 562; L. Esaki, in: Recent topics in semiconductor physics, eds. H. Kamimura and Y. Toyozawa (World Scientific, Singapore, 1983) p. 1.
J.A. Alphonso-Gibbs et ai. / Physics Letters A 195 (1994) 253-25 7
[2] A.M. Kan'an, A. Puri and T. Odagaki, Solid State Commun. 86 (1993) 113. [3] Y. Zebda and A.M. Kan'an, J. Appl. Phys. 72 (1992) 559. [4] A.N. Khondker, M.R. Khan and A.F.M. Anwar, J. Appl. Phys. 63 (1988) 5191. [5]J.A. Alphonso-Gibbs, S.S. Allen and S.L. Richardson, unpublished.
257
[6] S.S. Allen and S.L. Richardson, Phys. Rev. B 15, to be published. [7]J.A. Alphonso-Gibbs, M.E. thesis, Howard University (1993). [8] M.O. Vassel, J. Lee and H.F. Lockwood, J. Appl. Phys. 54 (1983) 5206.