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Quantum-confined stark effect on an exciton in a spherical quantum well of a GaAs cluster embedded in AlxGa1−xAs
Superlattices and Microstructures, Vol. 6, No. 1, 1989
23
Q U A N T U M - C O N F I N E D S T A R K E F F E C T ON AN EXCITON IN A S P H E R I C A L Q U A N T U M WELL OF A GaAs C L U S T E R E M B E D D E D IN AlxGal.xAs Yoshie Chiba and Shuhei Ohnishi F u n d a m e n t a l Research Laboratories, NEC Corporation 4-1-1 Miyazaki, Miyamae-ku, Kawasaki 213, J a p a n (Received 8 August 1988)
Cluster size dependence of the quantum-confined Stark effect on a l s exciton state in a spherical GaAs q u a n t u m well embedded in AlxGal-xAs is investigated by an exact continuous energy spectrum calculation based on Weyl's theory, within the effective-mass and the adiabatic approximations. Numerical results for the zero-field system indicate t h a t the l s exciton binding energy is about three times that of a one-dimensional q u a n t u m well and that the l s exciton state shows a drastic cluster size-dependence at small cluster radii (less than 100 atomic units (a.u.)) both of eigenvalue and exciton binding energy. It is found that u n d e r a strong electrostatic field, the resonance state for relatively larger radius (R0 = 180 a.u.) is more stable than that for small radii (R0 = 60 a.u. and 120 a.u.) because of its deeper energy level.
I. Introduction
II. Q u a n t u m Size Effect on l s Exciton State
The possibilitiy of creating new materials by designing various artificial q u a n t u m systems through the use of semiconductor heterostructures is opening a new field of material science. It is becoming possible not only to fabricate two- or oned i m e n s i o n a l s t r u c t u r e s , b u t also to f a b r i c a t e artificial structures using zero-dimensional systems called "quantum dots", which are made of a semiconductor cluster in a bulk crystal. 1-4 We can expect t h e m to have c h a r a c t e r i s t i c p h y s i c a l properties both in isolation and in the form of q u a n t u m dot a r r a y s . Such novel systems are considered among the most promising materials for a quasiatomic system with artificially controllable q u a n t u m states. From a view-point of creating a quasiatom, one of the most interesting systems is the optically generated electron-hole pair in a spherical q u a n t u m well made of a semiconducter cluster embedded in a bulk crystal. 5-7 In this paper, we present theoretical study of a l s exciton state localized in a spherical q u a n t u m well made of a GaAs cluster embedded in AlxGal_xAs bulk crystal, w i t h i n the e f f e c t i v e - m a s s a n d the a d i a b a t i c approximations. We discuss the q u a n t u m size effects on the l s eigenvalue and the exciton binding energy in Sec.II. Also, in Sec.III, we report on the external electrostatic field effects on the quantumconfined exciton in a spherical q u a n t u m well through a continuous spectrum calculation based on Weyl's theory, s-la N u m e r i c a l r e s u l t s for l s exciton states with cluster radii 60 atomic u n i t s (a.u.), 120 a.u., a n d 180 a.u. are p r e s e n t e d concerning resonance energies, exciton b i n d i n g energies, and t u n n e l i n g lifetimes.
The total Hamiltonian for the exciton, in a spherical GaAs cluster embedded in AlxGal_xAs bulk under the external field F in the z direction, is given (in atomic units) by
0749-6036/89/050023 + 04 $02.00/0
Hex=He+Hhh
1
] r - rhh I
(1)
1 H =Ae+Ve(re)-Fz e e 2m*e 1
Hhh--
, Ahh+Vhh(rhh)+FZhh 2rnhh
where He (Hhh) and m*e (m*hh) are the singleparticle Hamiltonian and the position-independent effective mass for the electron (heavy hole). The p o t e n t i a l s Ve(re) a n d Vhh(rhh) d u e to t h e heterostructure are a s s u m e d to be a spherical q u a n t u m well i.e., Ve(re)=V0 e'0(R0-[rej) a n d Vhh(rhh) = V0 hh-0(R0 --]rhhl), wh'ere VO e and V0 hh are the band (Jffsets for the conduction ~nd valence bands, respectively, and 0 the step function. The effective-mass approximation is considered to be valid for cluster radius R0>60 a.u.. The last term in Eq.(1) d e s c r i b e s the C o u l o m b i n t e r a c t i o n between the electron and hole with e being the static dielectric constant. The Hamiltonian Hex is represented by using the relative and center-ofmass coordinates as 1 1 1 Hex: 2"~ ar-- ~-~AR+Vex(r , R ) - ~ - F z (2) -
-
© 1989 Academic Press Limited
Super/attices and Microstructures, Vol. 6, No. 1, 1989
24
where r=re-rhh , R=(m*ere+m*i~hrhh)/M , M = m * e + m * h h , p=m*em*hh/(m*e+m*hh), and Vex(r,R) = Ve(R + (m*hh/M)r) + Vhh(R -- (m*e/M)r). Here, we introduce the adiabatic approximation, i.e., neglect the center-of-mass motion, and consider the case of R = 0. The SchrSdinger equation reduces into a three-dimensional problem involving the relative coordinates r. I
1
Hex.B.=O=- "~pAr+Vex(r'O)- g-~r~-igz H,x,a=o
%x.R=0 = E %=,~t=0
(3a)
-025
> -030 (D >.(.9 n" [,,IJ - 0 . 3 5 Z W
-0.40
200
RADIUS
300
400
~
--T---- •
~--
> E 30
5 5 g
20
(3b)
where Vex(r,0) has two steps at [r i = (M/m*hh)R0 and ( M / m * e ) R 0 : V e x ( r , 0 ) = V 0 , e + V 0 hh ( 0 < | r l < (M/m*hh)R0), V0.hh ((M/m*hh)R0 < Irl'< (M/m*e)R0), and 0((M/m*e)R0
I00
4 0
500
g (O X W
bulk
,
0 0
,
J
exciton -,- ;
i
~
~
~
I
100 2 0 0 3 0 0 4 0 0 500 RADIUS
(a.u)
Fig. 2 Cluster radius dependence of l s exciton b i n d i n g energy in a spherical q u a n t u m well of GaAs cluster embedded in AlxGal_xAs at zero-field. binding energy of a three-dimensionally confined exciton is found to be almost three times that of onedimensional q u a n t u m well with the same size (well width)? e III. Quantum-Confined Stark Effect on ls Exciton State III-1. Calculational Framework Since, under the external electrostatic field, the carriers can t u n n e l t h r o u g h the p o t e n t i a l barrier, there are no longer any distinctly bound states but resonance states having finite lifetimes. To investigate the field effects on the system, numerical calculations were carried out through a n o n - v a r i a t i o n a l , c o n t i n u o u s - s p e c t r u m approach based on Weyl's theory, s This approach can allow one to o b t a i n not only a discrete, b u t also a continuous spectrum for the problem even in the strong-field case. Since the potential, including the non-zero field t e r m i n Eq.(3a), is no l o n g e r s p h e r i c a l l y symmetric, the e q u a t i o n c a n ' t be separated in spherical coordinates. Therefore, we use c o n v e n t i o n a l p a r a b o l i c c o o r d i n a t e s : x = (~q)l/2cos~, y = (~q)l/2sin~, z = (~ - q)/2 (~,q ~ 0, 0= < ~<2~), which make it possible to separate the field term exactly. For the spherical q u a n t u m well potential, we made a shape approximation so as to be able to separate variables, as follows
Vex(r, 0)~
U(~)+U(r])
(a. u.)
Fig. 1 Cluster radius dependence of l s exciton eigenvalue in a spherical q u a n t u m well of GaAs cluster embedded in AlxGa>xAs at zero-field.
U ( t ) = VO, e t" 0 (Reff - Mt/rnhh) + VO, hh l" 0 (!:~eff - Mt / m: )
where the effective r a d i u s Reff=(2/aV'3)Ro is determined so as to m a i n t a i n the deepest q u a n t u m
25
Superlattices and Microstructures, Vol. 6, No. 1, 1989 well volume u n c h a n g e d . T h r o u g h the s h a p e approximation, the potential-barrier b o u n d a r y is slightly relaxed and the spherical q u a n t u m well changes in form into a football-like shape framed by two parabolic surfaces for the electron and heavy hole, r e s p e c t i v e l y . We p u t w ( r ) = ( ~ q ) -1/2 u(~)v(rl)exp(im~)), where m is a magnetic q u a n t u m n u m b e r . After s e p a r a t i o n of Op, two o r d i n a r y differential second-order equations, coupled by a parameter a which results from separation of the variables ~ and rl, are obtained as
[ 1''
1
21ad~2+P(~) u ( ~ , ) = 4 u ( ~ )
P(~)
(1-rn2) F~,~(u(~) + 8p~ 2
8
4
I
I
-2 I xl OmeV
I
Jk F = O0
(0<~,< °~) (4a)
1 4~
a
-6__
I xlOmeV
F-- 50
)
F =200
cx21
(XIO) _
1 d2
I
(XIO2) ! F= 50 (XI
E
~ q 2 + P ( , 1 ) v(rl)=4v(q)
(O
(1 . . . . 2)
.
1 (u(q)
Q(n)=---+~+~ 7 8pq 2 8
1 4e
-0.37
-0.36
-0.35
)
ENERGY (eV)
+°
It is seen that from the asymptotic behavior of p ( { ) - _ F U 8 ~ _ c o (~--oc) in Eq.(4a), Eq.(4a) has continuous s p e c t r u m ( _ c o < E < + c o ) and t h a t from Q 0 1 ) ~ + F q / 8 ~ + c o (q~oo), Eq.(4b) has a discrete spectrum, s,9 The ordinary second-order differential equation defined in a s e m i - i n f i n i t e region, as seen in Eqs.(4a) and (4b), is called the singular boundary value-problem, to which Weyl's limit-point limit-circle theory can be applied, u The theory allows one to o b t a i n the exact spectral function for the problem with use of a complex parameter for energy E instead of real values. For resonance state determination, it is necessary to d e t e r m i n e a set of c o u p l i n g p a r a m e t e r a a n d resonance energy E, such that it satisfies coupled boundary-value p r o b l e m s (4a) a n d (4b) simultaneously. A series of numerical calculations on Eqs.(4a) and (4b) have been simulated with reference to v a r i o u s v a l u e s along the e n e r g y parameter E versus numerous individual values for coupling parameter a. Figure 3 shows the energy spectrum of a ls exciton state for R0=120 a.u. at F = 0 kV/cm, 50 kV/cm, 100 kV/em, 150 kV/cm, and 200 kV/cm obtained by the present methodJ ,6 It is clearly seen t h a t the spectral peaks shift a n d broaden more and more as the field increases. With regard to the validity of the shape approximation for the spherical q u a n t u m well, at F = 0 kV/cm, the ls eigenvalues for the approximated potential case are found to agree very well with those for the exact spherical well case, as seen in Fig.l, with deviations being less than 2% for each cluster size. III-2. Results and Discussions In order to clarify the size dependence of the Stark effect, we present the results on ls exciton states with cluster radii 60 a.u., 120 a.u., and 180 a.u.. Table 1 presents the field dependence of the t u n n e l i n g lifetimes obtained from the uncertainty
Fig. 3 Energy spectrum for ls exciton state with cluster radius 120 a.u. at the field strengths F = 0 , 50, 100, 150, and 200 kV/cm. Insets show the spectral peaks enlarged by the energy axis. Table 1. T u n n e l i n g li etimes (ps) o l s exciton states at field intensities F = 5 0 , 100, 150, and 200 kV/cm for cluster radii R0 = 60, 120, and 180 a.u. cases, respectively. Asterisk (*) denotes that the corresponding lifetime is longer than 101° ps. R0(a.u.) F(kV/cm) 60 120 180
50 * * *
100
150
200
8 . 0 X l O ~ 4.4X10 ~ 6.3X10 ~ 5.2×102 1.2XlO ° 1.2XIO ~ 2.7X102 1.9×100 1.8XlO 1
relationship ; - / i / A E (AE being the full width at half-maximum of the spectral peak). At relatively weak field ( F < 100 kV/cm), the l s exciton states are stable enough for each Ro. In Fig. 4, we show the e l e c t r o s t a t i c field d e p e n d e n c e of r e s o n a n c e energies. It is seen that, at relatively weak fields, the energy shift is larger for larger R0. Drastic changes both in the resonance e n e r g y and the binding energy for R0 = 60 a.u. at F~100 kV/cm are understood in terms of a large t u n n e l i n g probability as shown in Table 1. The t u n n e l i n g lifetime depends mainly on the relative energy to the top of the potential barrier, which is determined by two factors, the energy level and the decrease of the potential barrier maximum, - F R o . Then, the cases of much smaller and/or much larger R0 yield less stable states due to the shallower energy level and/or larger decrease - F R o , respectively.
Superlattlces and Microstructures, Vo/. 6, No. 1, 1989
26 -0,25
R =60 a.u. " ' , -0.30 \ \\
- 0.55 >c.o
R=120a
u
state confined in a spherical quantum well of OaAs cluster e m b e d d e d in A l x G a l xAs u s i n g W e y l ' s theory. At zero-field, the exciton binding energy is abou[ three t i m e s t h a t of a o n e - d i m e n s i o n a l quantum well and drastic quantum size effects on the l s exciton state are found for small cluster radii with regard both to efgenvalue and to exciton binding energy. As for the quantum-confined Stark effect, it is found that the exciton state is stable up to F - 1 0 0 kV/cm and, under a strong electrostatic field, the resonance state, for smaller R0, becomes more unstable because of its shallower energy level. REFERENCES
I
II
1.
_ 0I4. 5
,
,
0
L
200
I00
ELECTRIC FIELD (kV/cm) Fig. 4 ls exciton resonance energies versus electrostatic field w i t h cluster r a d i i 60,120, and 180 a.Ll..
2. 3.
40 E v >(.9 n," w 3o z ILl
\ %
R
=120a.u.
(.9 Z Z
4.
~ %
5.
R=18Oa.u.
20
\
6.
Z
o
l--
I0 Ld
I
0
I
I00
I
200
ELECTRIC FIELD (kV/cm) Fig. 5 l s e x c i t o n b i n d i n g e n e r g i e s v e r s u s electrostatic field with cluster radii 60,120, and 180 a.u..
IV. Conclusions We have discussed the cluster size dependence of the electrostatic field effects on the l s exeiton
7. 8. 9.
The Physics and Fabricatiorz of Microstructures and Microdevices edited by M . J . K e l l y and C.Weisbuch ( S p r i n g e r - V e r l a g , H e i d e l b e r g , 1986). L.E.Brus, J. Chem. Phys. 80, 4403 (1984); IEEE J. Quantum Electronics 22, 1909(1986), and references therein. H.Watanabe in Reference 1, p. 5; H.Watanabe and T.Inoshita, Optoelectron. Dev. Technol. i , 33 (1986); T . I n o s h i t a , S . O h n i s h i , a n d A.Oshiyama, Phys. Rev. Lett. 57, 2560 (1986); T.Inoshita and H.Watanabe, in Microclusters, 281 e d i t e d by S . S u g a n o , Y . N i s h i n a , a n d S.Ohnishi (Springer-Verlag, Heidelberg, 1987) G.W.Bryant, Phys. Rev. Lett. 59, 1140 (1987); Obermayer, G.Mahler, and H.Haken, Phys. Rev. Lett. 58, 1792 (1987). S.Ohnishi and Y.Chiba, in Microclusters, 269 edited by S.Sugano, Y.Nishina, and S.Ohnishi (Springer-Verlag, Heidelberg, 1987). Y.Chiba and S.Ohnishi, to be p u b l i s h e d in Phys. Rev. B (1988). G.W.Bryant, Phys. Roy. B 37, 8763 (1988). H.Weyl, Math. Ann. 68,220 (1910). E.C.Titchmarsh, Eigenfunction Expansions
Associated with Second-Order Differential Equations (Oxford U. P., London, 1946) Vols. 1 and 2. 10. E.A.Coddington and N.Levinson, Theory of Ordinary Differential Equations (McGrow-Hill, New York, 1955), Chap. 9. 11. M . H e h e n b e r g e r , H.V.McIntosh, and E.Brandas, Phys. Rev. A tO, 1494 (1974). 12. U. Ekenberg and M. Altarelli, Phys. Roy. B 35, 7585 (1987).
23
Q U A N T U M - C O N F I N E D S T A R K E F F E C T ON AN EXCITON IN A S P H E R I C A L Q U A N T U M WELL OF A GaAs C L U S T E R E M B E D D E D IN AlxGal.xAs Yoshie Chiba and Shuhei Ohnishi F u n d a m e n t a l Research Laboratories, NEC Corporation 4-1-1 Miyazaki, Miyamae-ku, Kawasaki 213, J a p a n (Received 8 August 1988)
Cluster size dependence of the quantum-confined Stark effect on a l s exciton state in a spherical GaAs q u a n t u m well embedded in AlxGal-xAs is investigated by an exact continuous energy spectrum calculation based on Weyl's theory, within the effective-mass and the adiabatic approximations. Numerical results for the zero-field system indicate t h a t the l s exciton binding energy is about three times that of a one-dimensional q u a n t u m well and that the l s exciton state shows a drastic cluster size-dependence at small cluster radii (less than 100 atomic units (a.u.)) both of eigenvalue and exciton binding energy. It is found that u n d e r a strong electrostatic field, the resonance state for relatively larger radius (R0 = 180 a.u.) is more stable than that for small radii (R0 = 60 a.u. and 120 a.u.) because of its deeper energy level.
I. Introduction
II. Q u a n t u m Size Effect on l s Exciton State
The possibilitiy of creating new materials by designing various artificial q u a n t u m systems through the use of semiconductor heterostructures is opening a new field of material science. It is becoming possible not only to fabricate two- or oned i m e n s i o n a l s t r u c t u r e s , b u t also to f a b r i c a t e artificial structures using zero-dimensional systems called "quantum dots", which are made of a semiconductor cluster in a bulk crystal. 1-4 We can expect t h e m to have c h a r a c t e r i s t i c p h y s i c a l properties both in isolation and in the form of q u a n t u m dot a r r a y s . Such novel systems are considered among the most promising materials for a quasiatomic system with artificially controllable q u a n t u m states. From a view-point of creating a quasiatom, one of the most interesting systems is the optically generated electron-hole pair in a spherical q u a n t u m well made of a semiconducter cluster embedded in a bulk crystal. 5-7 In this paper, we present theoretical study of a l s exciton state localized in a spherical q u a n t u m well made of a GaAs cluster embedded in AlxGal_xAs bulk crystal, w i t h i n the e f f e c t i v e - m a s s a n d the a d i a b a t i c approximations. We discuss the q u a n t u m size effects on the l s eigenvalue and the exciton binding energy in Sec.II. Also, in Sec.III, we report on the external electrostatic field effects on the quantumconfined exciton in a spherical q u a n t u m well through a continuous spectrum calculation based on Weyl's theory, s-la N u m e r i c a l r e s u l t s for l s exciton states with cluster radii 60 atomic u n i t s (a.u.), 120 a.u., a n d 180 a.u. are p r e s e n t e d concerning resonance energies, exciton b i n d i n g energies, and t u n n e l i n g lifetimes.
The total Hamiltonian for the exciton, in a spherical GaAs cluster embedded in AlxGal_xAs bulk under the external field F in the z direction, is given (in atomic units) by
0749-6036/89/050023 + 04 $02.00/0
Hex=He+Hhh
1
] r - rhh I
(1)
1 H =Ae+Ve(re)-Fz e e 2m*e 1
Hhh--
, Ahh+Vhh(rhh)+FZhh 2rnhh
where He (Hhh) and m*e (m*hh) are the singleparticle Hamiltonian and the position-independent effective mass for the electron (heavy hole). The p o t e n t i a l s Ve(re) a n d Vhh(rhh) d u e to t h e heterostructure are a s s u m e d to be a spherical q u a n t u m well i.e., Ve(re)=V0 e'0(R0-[rej) a n d Vhh(rhh) = V0 hh-0(R0 --]rhhl), wh'ere VO e and V0 hh are the band (Jffsets for the conduction ~nd valence bands, respectively, and 0 the step function. The effective-mass approximation is considered to be valid for cluster radius R0>60 a.u.. The last term in Eq.(1) d e s c r i b e s the C o u l o m b i n t e r a c t i o n between the electron and hole with e being the static dielectric constant. The Hamiltonian Hex is represented by using the relative and center-ofmass coordinates as 1 1 1 Hex: 2"~ ar-- ~-~AR+Vex(r , R ) - ~ - F z (2) -
-
© 1989 Academic Press Limited
Super/attices and Microstructures, Vol. 6, No. 1, 1989
24
where r=re-rhh , R=(m*ere+m*i~hrhh)/M , M = m * e + m * h h , p=m*em*hh/(m*e+m*hh), and Vex(r,R) = Ve(R + (m*hh/M)r) + Vhh(R -- (m*e/M)r). Here, we introduce the adiabatic approximation, i.e., neglect the center-of-mass motion, and consider the case of R = 0. The SchrSdinger equation reduces into a three-dimensional problem involving the relative coordinates r. I
1
Hex.B.=O=- "~pAr+Vex(r'O)- g-~r~-igz H,x,a=o
%x.R=0 = E %=,~t=0
(3a)
-025
> -030 (D >.(.9 n" [,,IJ - 0 . 3 5 Z W
-0.40
200
RADIUS
300
400
~
--T---- •
~--
> E 30
5 5 g
20
(3b)
where Vex(r,0) has two steps at [r i = (M/m*hh)R0 and ( M / m * e ) R 0 : V e x ( r , 0 ) = V 0 , e + V 0 hh ( 0 < | r l < (M/m*hh)R0), V0.hh ((M/m*hh)R0 < Irl'< (M/m*e)R0), and 0((M/m*e)R0
I00
4 0
500
g (O X W
bulk
,
0 0
,
J
exciton -,- ;
i
~
~
~
I
100 2 0 0 3 0 0 4 0 0 500 RADIUS
(a.u)
Fig. 2 Cluster radius dependence of l s exciton b i n d i n g energy in a spherical q u a n t u m well of GaAs cluster embedded in AlxGal_xAs at zero-field. binding energy of a three-dimensionally confined exciton is found to be almost three times that of onedimensional q u a n t u m well with the same size (well width)? e III. Quantum-Confined Stark Effect on ls Exciton State III-1. Calculational Framework Since, under the external electrostatic field, the carriers can t u n n e l t h r o u g h the p o t e n t i a l barrier, there are no longer any distinctly bound states but resonance states having finite lifetimes. To investigate the field effects on the system, numerical calculations were carried out through a n o n - v a r i a t i o n a l , c o n t i n u o u s - s p e c t r u m approach based on Weyl's theory, s This approach can allow one to o b t a i n not only a discrete, b u t also a continuous spectrum for the problem even in the strong-field case. Since the potential, including the non-zero field t e r m i n Eq.(3a), is no l o n g e r s p h e r i c a l l y symmetric, the e q u a t i o n c a n ' t be separated in spherical coordinates. Therefore, we use c o n v e n t i o n a l p a r a b o l i c c o o r d i n a t e s : x = (~q)l/2cos~, y = (~q)l/2sin~, z = (~ - q)/2 (~,q ~ 0, 0= < ~<2~), which make it possible to separate the field term exactly. For the spherical q u a n t u m well potential, we made a shape approximation so as to be able to separate variables, as follows
Vex(r, 0)~
U(~)+U(r])
(a. u.)
Fig. 1 Cluster radius dependence of l s exciton eigenvalue in a spherical q u a n t u m well of GaAs cluster embedded in AlxGa>xAs at zero-field.
U ( t ) = VO, e t" 0 (Reff - Mt/rnhh) + VO, hh l" 0 (!:~eff - Mt / m: )
where the effective r a d i u s Reff=(2/aV'3)Ro is determined so as to m a i n t a i n the deepest q u a n t u m
25
Superlattices and Microstructures, Vol. 6, No. 1, 1989 well volume u n c h a n g e d . T h r o u g h the s h a p e approximation, the potential-barrier b o u n d a r y is slightly relaxed and the spherical q u a n t u m well changes in form into a football-like shape framed by two parabolic surfaces for the electron and heavy hole, r e s p e c t i v e l y . We p u t w ( r ) = ( ~ q ) -1/2 u(~)v(rl)exp(im~)), where m is a magnetic q u a n t u m n u m b e r . After s e p a r a t i o n of Op, two o r d i n a r y differential second-order equations, coupled by a parameter a which results from separation of the variables ~ and rl, are obtained as
[ 1''
1
21ad~2+P(~) u ( ~ , ) = 4 u ( ~ )
P(~)
(1-rn2) F~,~(u(~) + 8p~ 2
8
4
I
I
-2 I xl OmeV
I
Jk F = O0
(0<~,< °~) (4a)
1 4~
a
-6__
I xlOmeV
F-- 50
)
F =200
cx21
(XIO) _
1 d2
I
(XIO2) ! F= 50 (XI
E
~ q 2 + P ( , 1 ) v(rl)=4v(q)
(O
(1 . . . . 2)
.
1 (u(q)
Q(n)=---+~+~ 7 8pq 2 8
1 4e
-0.37
-0.36
-0.35
)
ENERGY (eV)
+°
It is seen that from the asymptotic behavior of p ( { ) - _ F U 8 ~ _ c o (~--oc) in Eq.(4a), Eq.(4a) has continuous s p e c t r u m ( _ c o < E < + c o ) and t h a t from Q 0 1 ) ~ + F q / 8 ~ + c o (q~oo), Eq.(4b) has a discrete spectrum, s,9 The ordinary second-order differential equation defined in a s e m i - i n f i n i t e region, as seen in Eqs.(4a) and (4b), is called the singular boundary value-problem, to which Weyl's limit-point limit-circle theory can be applied, u The theory allows one to o b t a i n the exact spectral function for the problem with use of a complex parameter for energy E instead of real values. For resonance state determination, it is necessary to d e t e r m i n e a set of c o u p l i n g p a r a m e t e r a a n d resonance energy E, such that it satisfies coupled boundary-value p r o b l e m s (4a) a n d (4b) simultaneously. A series of numerical calculations on Eqs.(4a) and (4b) have been simulated with reference to v a r i o u s v a l u e s along the e n e r g y parameter E versus numerous individual values for coupling parameter a. Figure 3 shows the energy spectrum of a ls exciton state for R0=120 a.u. at F = 0 kV/cm, 50 kV/cm, 100 kV/em, 150 kV/cm, and 200 kV/cm obtained by the present methodJ ,6 It is clearly seen t h a t the spectral peaks shift a n d broaden more and more as the field increases. With regard to the validity of the shape approximation for the spherical q u a n t u m well, at F = 0 kV/cm, the ls eigenvalues for the approximated potential case are found to agree very well with those for the exact spherical well case, as seen in Fig.l, with deviations being less than 2% for each cluster size. III-2. Results and Discussions In order to clarify the size dependence of the Stark effect, we present the results on ls exciton states with cluster radii 60 a.u., 120 a.u., and 180 a.u.. Table 1 presents the field dependence of the t u n n e l i n g lifetimes obtained from the uncertainty
Fig. 3 Energy spectrum for ls exciton state with cluster radius 120 a.u. at the field strengths F = 0 , 50, 100, 150, and 200 kV/cm. Insets show the spectral peaks enlarged by the energy axis. Table 1. T u n n e l i n g li etimes (ps) o l s exciton states at field intensities F = 5 0 , 100, 150, and 200 kV/cm for cluster radii R0 = 60, 120, and 180 a.u. cases, respectively. Asterisk (*) denotes that the corresponding lifetime is longer than 101° ps. R0(a.u.) F(kV/cm) 60 120 180
50 * * *
100
150
200
8 . 0 X l O ~ 4.4X10 ~ 6.3X10 ~ 5.2×102 1.2XlO ° 1.2XIO ~ 2.7X102 1.9×100 1.8XlO 1
relationship ; - / i / A E (AE being the full width at half-maximum of the spectral peak). At relatively weak field ( F < 100 kV/cm), the l s exciton states are stable enough for each Ro. In Fig. 4, we show the e l e c t r o s t a t i c field d e p e n d e n c e of r e s o n a n c e energies. It is seen that, at relatively weak fields, the energy shift is larger for larger R0. Drastic changes both in the resonance e n e r g y and the binding energy for R0 = 60 a.u. at F~100 kV/cm are understood in terms of a large t u n n e l i n g probability as shown in Table 1. The t u n n e l i n g lifetime depends mainly on the relative energy to the top of the potential barrier, which is determined by two factors, the energy level and the decrease of the potential barrier maximum, - F R o . Then, the cases of much smaller and/or much larger R0 yield less stable states due to the shallower energy level and/or larger decrease - F R o , respectively.
Superlattlces and Microstructures, Vo/. 6, No. 1, 1989
26 -0,25
R =60 a.u. " ' , -0.30 \ \\
- 0.55 >c.o
R=120a
u
state confined in a spherical quantum well of OaAs cluster e m b e d d e d in A l x G a l xAs u s i n g W e y l ' s theory. At zero-field, the exciton binding energy is abou[ three t i m e s t h a t of a o n e - d i m e n s i o n a l quantum well and drastic quantum size effects on the l s exciton state are found for small cluster radii with regard both to efgenvalue and to exciton binding energy. As for the quantum-confined Stark effect, it is found that the exciton state is stable up to F - 1 0 0 kV/cm and, under a strong electrostatic field, the resonance state, for smaller R0, becomes more unstable because of its shallower energy level. REFERENCES
I
II
1.
_ 0I4. 5
,
,
0
L
200
I00
ELECTRIC FIELD (kV/cm) Fig. 4 ls exciton resonance energies versus electrostatic field w i t h cluster r a d i i 60,120, and 180 a.Ll..
2. 3.
40 E v >(.9 n," w 3o z ILl
\ %
R
=120a.u.
(.9 Z Z
4.
~ %
5.
R=18Oa.u.
20
\
6.
Z
o
l--
I0 Ld
I
0
I
I00
I
200
ELECTRIC FIELD (kV/cm) Fig. 5 l s e x c i t o n b i n d i n g e n e r g i e s v e r s u s electrostatic field with cluster radii 60,120, and 180 a.u..
IV. Conclusions We have discussed the cluster size dependence of the electrostatic field effects on the l s exeiton
7. 8. 9.
The Physics and Fabricatiorz of Microstructures and Microdevices edited by M . J . K e l l y and C.Weisbuch ( S p r i n g e r - V e r l a g , H e i d e l b e r g , 1986). L.E.Brus, J. Chem. Phys. 80, 4403 (1984); IEEE J. Quantum Electronics 22, 1909(1986), and references therein. H.Watanabe in Reference 1, p. 5; H.Watanabe and T.Inoshita, Optoelectron. Dev. Technol. i , 33 (1986); T . I n o s h i t a , S . O h n i s h i , a n d A.Oshiyama, Phys. Rev. Lett. 57, 2560 (1986); T.Inoshita and H.Watanabe, in Microclusters, 281 e d i t e d by S . S u g a n o , Y . N i s h i n a , a n d S.Ohnishi (Springer-Verlag, Heidelberg, 1987) G.W.Bryant, Phys. Rev. Lett. 59, 1140 (1987); Obermayer, G.Mahler, and H.Haken, Phys. Rev. Lett. 58, 1792 (1987). S.Ohnishi and Y.Chiba, in Microclusters, 269 edited by S.Sugano, Y.Nishina, and S.Ohnishi (Springer-Verlag, Heidelberg, 1987). Y.Chiba and S.Ohnishi, to be p u b l i s h e d in Phys. Rev. B (1988). G.W.Bryant, Phys. Roy. B 37, 8763 (1988). H.Weyl, Math. Ann. 68,220 (1910). E.C.Titchmarsh, Eigenfunction Expansions
Associated with Second-Order Differential Equations (Oxford U. P., London, 1946) Vols. 1 and 2. 10. E.A.Coddington and N.Levinson, Theory of Ordinary Differential Equations (McGrow-Hill, New York, 1955), Chap. 9. 11. M . H e h e n b e r g e r , H.V.McIntosh, and E.Brandas, Phys. Rev. A tO, 1494 (1974). 12. U. Ekenberg and M. Altarelli, Phys. Roy. B 35, 7585 (1987).