- Email: [email protected]
The effects of photoexcitation on excitons in semiconductor doping superlattices
551)
Journal of Luminescence 30(198.5 550 ‘-554 North — Holland, A 01St erda rn
THE EFFECTS OF PHOTOEXCITATION ON EXCITONS IN SEMICONDUCTOR DOPING SUPERLATTICES T.L. REIN ECKE ~
I.uho,’uto’r.
fiashiiiqton. 1). C. 20375.
SI
F. CROWNE Mar~n Marietta Ri’seari/, faho,’utorjcs, Ba/moon’, Sit) 21227,
SI
Calculations of the properties of excitons in doping superlattices hase been made as a function of doping density using a variational approach. Interesting ness features are obtained when the exciton energy becomes comparable to the superlattice potential energy. These results are compared to recent experimental data on GaAs doping supcriattices.
Semiconductor doping superlattices are especially attractive for photoluminescence studies because their properties can he modified over wide ranges by photoexcitation [I]. These systems are formed by alternating layers of’ n and p doping in a bulk material. Electrons are transferred from the n layers to the p layers creating a modulated electric field and a modulated electrostatic potential for the carriers. This potential confines the carrier motion in the direction perpendicular to the layers and produces quasi-two-dimensional subbands for the electrons and holes which have an indirect gap in real space and consequently long electron hole recombination times. Under photoexcitation both the net charge in the layers and the band bending due to the superlattice potential are reduced. It is possible to reduce the superlattice potential almost to zero by photoexcitation, and in particular the band bending can become comparable to the exciton energy. In this case interesting new effects can occur as a result of the competition between the superlattice potential which favors spatial separation of the electron and hole and the Coulomb attraction between the electron and hole. We have made a series of calculations of the properties of excitons in doping superlattices. These include excitons formed from electrons and holes in the superlattice subbands, excitons with an electron or a hole in a hydrogenic state at a donor or an acceptor. and excitons in hetero-doping superlattices [2]. In the present note we summarize the results for the case of an exciton formed 00-2313/85/503.30 © Elsevier Science Publishers By. (North-Holland Physics Publishing Division)
T.L. Reinecke, F. Crowne / Excitons in semiconductor doping super/attices
581
from an electron and a hole in the superlattice subbands, and we point out the relevance of these results for photoluminescence studies of doping superlattices. Fuller details will be presented elsewhere.
Results The system is composed of alternating layers of uniform positive charge dNA whose centers are separated by distance d along the 2-axis. The electron and hole are taken to be in adjacent layers and are described by the wavefunction ZA/2) exp{ [x t((Xe Xh)2 + (Ye + Yh)) ~P(re,rh) = N~e(Z ZD/2)~h( e~NDand negative charge
—
—
—
—
—
(1)
+/,~(ZeZh)2]t/2}.
Here i=e, h are the subband wavefunctions, c~and f~are the variational parameters representing the extent of the wavefunction parallel and perpendicular to the layers, and N is a normalizing constant. The superlattice potential energy in the absence of the Coulomb interaction is parabolic, and the subband functions ~ are given straightforwardly in terms of harmonic oscillator functions with subband energies e~, i = e, h where Ee4~(41te2ND/1
—
~,
—
582
T.L. Reinecke, F.
:.:
(‘roam’
~
Exciton,c in se,n,conductor dopitiq super/attices
8
O~
0.6
Fig. I. Binding energy vs doping parameter E for several separations 4.
the superlattice potential energy and the Coulomb attraction between the electron and the hole. By direct calculation we have shown that the difference between the expectation values of the electron and hole positions goes to zero approximately exponentially with decreasing doping. As the electron and hole move together their wavefunction overlap increases, and their recombination probability increases. For a correlated system like the present one the factor in the recombination probability which represents the wavefunction
02
03
04
05
Fig. 2. Overlap matrix element vs doping parameter ~ for several separations 4.
T.L. Reinecke. F. Crowne / Excitons in semiconductor doping superlatlices
583
overlap is M2 where M=Jd3rW(r=re;r=rh).
(2)
This is shown in fig. 2 where it is seen that M2 is exponentially small for large doping and increases with decreasing doping. Discussion The present results provide a useful picture of the effects of photoexcitation on the behavior of excitons and of photoluminescence in doping superlattices. After photoexcitation the space charge of the layers is partly neutralized, and the superlattice potential decreases. An electron and a hole excited into the subbands then will move closer together under the influence of the Coulomb interaction, and their recombination probability will increase. The corresponding features in the luminescence spectra will occur at higher energies with increasing photoexcitation because of the decrease of the superlattice potential variation. The present picture is consistent with a new feature seen in recent photoluminescence studies [4] of a doping superlattice in GaAs. For low and moderate photoexcitation intensity a broad recombination band like that seen previously [5] was observed. It shifted to higher energy with increasing intensity and is attributed to recombination of electrons from the electron subband with holes at acceptors. For the highest excitation, however, a second line occurred in the spectrum at a higher energy than the first. Its energy is consistent with recombination involving an interacting electron and hole in their respective subbands or with an exciton bound at an impurity. In the picture developed here the recombination probability of electrons and holes from the subbands is increased by increasing photoexcitation because of the increased wavefunction overlap, and the probability of formation of a bound exciton complex at an impurity also is increased. As a result the recombination from both of these processes increases with photoexcitation which is consistent with what is seen experimentally. This work was supported in part by an ONR contract (T.L.R.).
References [I] G.I-I. Döhler, H. Künzel, D. Olega. K. Ploog. P. Ruden, H.). Stolz and G. Abstreiter. Phys. Rev. Lett. 47 (1981) 864. [2] G.H. Döhler and PP. Ruden, Surf. Sci. 142 (1984) 474.
584
T.L. Reinecke, F. (‘roa’ne /
Exciton,s in semiconductor doping .superlat/ice.s
[3] Yu.E. Lozvik and NV. Nishanov. Soy. Phys. Solid State 18(1976)905. [4] F. Crowne, T,L. Reinecke and B.V. Shanabrook. Proc. XVIIth tnt. Conf. Physics of Semiconductors, San Francisco. 1984. eds., D.J. Chadi and W.A. Harrison (to be published). [5] H. Jung, G.H. Döhler. H. KUnzel. K. Ploog, P. Ruden and H.J. Stolz. Solid State Commun. 43(1982)291.
Journal of Luminescence 30(198.5 550 ‘-554 North — Holland, A 01St erda rn
THE EFFECTS OF PHOTOEXCITATION ON EXCITONS IN SEMICONDUCTOR DOPING SUPERLATTICES T.L. REIN ECKE ~
I.uho,’uto’r.
fiashiiiqton. 1). C. 20375.
SI
F. CROWNE Mar~n Marietta Ri’seari/, faho,’utorjcs, Ba/moon’, Sit) 21227,
SI
Calculations of the properties of excitons in doping superlattices hase been made as a function of doping density using a variational approach. Interesting ness features are obtained when the exciton energy becomes comparable to the superlattice potential energy. These results are compared to recent experimental data on GaAs doping supcriattices.
Semiconductor doping superlattices are especially attractive for photoluminescence studies because their properties can he modified over wide ranges by photoexcitation [I]. These systems are formed by alternating layers of’ n and p doping in a bulk material. Electrons are transferred from the n layers to the p layers creating a modulated electric field and a modulated electrostatic potential for the carriers. This potential confines the carrier motion in the direction perpendicular to the layers and produces quasi-two-dimensional subbands for the electrons and holes which have an indirect gap in real space and consequently long electron hole recombination times. Under photoexcitation both the net charge in the layers and the band bending due to the superlattice potential are reduced. It is possible to reduce the superlattice potential almost to zero by photoexcitation, and in particular the band bending can become comparable to the exciton energy. In this case interesting new effects can occur as a result of the competition between the superlattice potential which favors spatial separation of the electron and hole and the Coulomb attraction between the electron and hole. We have made a series of calculations of the properties of excitons in doping superlattices. These include excitons formed from electrons and holes in the superlattice subbands, excitons with an electron or a hole in a hydrogenic state at a donor or an acceptor. and excitons in hetero-doping superlattices [2]. In the present note we summarize the results for the case of an exciton formed 00-2313/85/503.30 © Elsevier Science Publishers By. (North-Holland Physics Publishing Division)
T.L. Reinecke, F. Crowne / Excitons in semiconductor doping super/attices
581
from an electron and a hole in the superlattice subbands, and we point out the relevance of these results for photoluminescence studies of doping superlattices. Fuller details will be presented elsewhere.
Results The system is composed of alternating layers of uniform positive charge dNA whose centers are separated by distance d along the 2-axis. The electron and hole are taken to be in adjacent layers and are described by the wavefunction ZA/2) exp{ [x t((Xe Xh)2 + (Ye + Yh)) ~P(re,rh) = N~e(Z ZD/2)~h( e~NDand negative charge
—
—
—
—
—
(1)
+/,~(ZeZh)2]t/2}.
Here i=e, h are the subband wavefunctions, c~and f~are the variational parameters representing the extent of the wavefunction parallel and perpendicular to the layers, and N is a normalizing constant. The superlattice potential energy in the absence of the Coulomb interaction is parabolic, and the subband functions ~ are given straightforwardly in terms of harmonic oscillator functions with subband energies e~, i = e, h where Ee4~(41te2ND/1
—
~,
—
582
T.L. Reinecke, F.
:.:
(‘roam’
~
Exciton,c in se,n,conductor dopitiq super/attices
8
O~
0.6
Fig. I. Binding energy vs doping parameter E for several separations 4.
the superlattice potential energy and the Coulomb attraction between the electron and the hole. By direct calculation we have shown that the difference between the expectation values of the electron and hole positions goes to zero approximately exponentially with decreasing doping. As the electron and hole move together their wavefunction overlap increases, and their recombination probability increases. For a correlated system like the present one the factor in the recombination probability which represents the wavefunction
02
03
04
05
Fig. 2. Overlap matrix element vs doping parameter ~ for several separations 4.
T.L. Reinecke. F. Crowne / Excitons in semiconductor doping superlatlices
583
overlap is M2 where M=Jd3rW(r=re;r=rh).
(2)
This is shown in fig. 2 where it is seen that M2 is exponentially small for large doping and increases with decreasing doping. Discussion The present results provide a useful picture of the effects of photoexcitation on the behavior of excitons and of photoluminescence in doping superlattices. After photoexcitation the space charge of the layers is partly neutralized, and the superlattice potential decreases. An electron and a hole excited into the subbands then will move closer together under the influence of the Coulomb interaction, and their recombination probability will increase. The corresponding features in the luminescence spectra will occur at higher energies with increasing photoexcitation because of the decrease of the superlattice potential variation. The present picture is consistent with a new feature seen in recent photoluminescence studies [4] of a doping superlattice in GaAs. For low and moderate photoexcitation intensity a broad recombination band like that seen previously [5] was observed. It shifted to higher energy with increasing intensity and is attributed to recombination of electrons from the electron subband with holes at acceptors. For the highest excitation, however, a second line occurred in the spectrum at a higher energy than the first. Its energy is consistent with recombination involving an interacting electron and hole in their respective subbands or with an exciton bound at an impurity. In the picture developed here the recombination probability of electrons and holes from the subbands is increased by increasing photoexcitation because of the increased wavefunction overlap, and the probability of formation of a bound exciton complex at an impurity also is increased. As a result the recombination from both of these processes increases with photoexcitation which is consistent with what is seen experimentally. This work was supported in part by an ONR contract (T.L.R.).
References [I] G.I-I. Döhler, H. Künzel, D. Olega. K. Ploog. P. Ruden, H.). Stolz and G. Abstreiter. Phys. Rev. Lett. 47 (1981) 864. [2] G.H. Döhler and PP. Ruden, Surf. Sci. 142 (1984) 474.
584
T.L. Reinecke, F. (‘roa’ne /
Exciton,s in semiconductor doping .superlat/ice.s
[3] Yu.E. Lozvik and NV. Nishanov. Soy. Phys. Solid State 18(1976)905. [4] F. Crowne, T,L. Reinecke and B.V. Shanabrook. Proc. XVIIth tnt. Conf. Physics of Semiconductors, San Francisco. 1984. eds., D.J. Chadi and W.A. Harrison (to be published). [5] H. Jung, G.H. Döhler. H. KUnzel. K. Ploog, P. Ruden and H.J. Stolz. Solid State Commun. 43(1982)291.