- Email: [email protected]
Stark ladder excitonic transitions
472
Surface Science 229 f 1990) 472-475 North-Holland
STARK LADDER EXCITONIC TRANSITIONS J.A. BRUM * and F. AGULL~-RUEDA IBhcf 7X Watson Research Center, Yorkfown
**
~eig~is,
h’Y 10598,
USA
Received IO August 1989; accepted for publication 14 September 1989
We have calculated the binding energy and the transition energies of excitons for a superlattice under an external electric field applied along the growth direction. At low fields the binding energy shifts strongly showing the localization of the superlattice states inlo the Stark ladder states. For higher fields the energy levels observe a shift associated to the single quantum well Stark effect. We also discuss the contribution of superlattice fluctuations to the carrier con~nement.
The effect of an external electric field applied in bulk semiconductors attracted some attention in the past [ 11. The electric field breaks the periodicity and localizes the carriers into Stark ladder states for certain range of electric field. However, its experimental verification was not possible due to the high electric fields necessary to achieve the Stark ladder localization in the bulk semiconductors. The high quality of superlattice heterostructures grown by MBE opened the way to the observation of this effect within reasonable values of electric fields. This effect was studied theoretically [2] and observed experimentally in GaAs-Ga(Al)As superlattices [ 3,4] as well as other systems [ 5 1. In general, the optical spectra are dominated by transitions of excitonic origin. Although many of the features observed experimentally can be interpreted as band-to-band transitions, the high quality of the samples has permitted the observation of the excitonic behavior [ 61. An ideal superlattice is described as an infinite periodic sequence of alternate layers. Here, we simulate it by a finite number of quantum wells limiting their number, by numerical reasons, to a lower value than the actual samples (typically 60 wells). This procedure enhances the effect of the surface and discretize the superlattice continuum. However, be* Present address: UNICAMP, Instituto de Fisica, CP 6165, 1308 1 Campinas (SP), Brazil, and Laboratorio National de Luz Sincroton, CP 6192, 13083 Campinas (SP). Brazil. +* Permanent address: Instituto de Ciencia de Materiales (SCIS). Universidad Autonoma, 28049 Madrid, Spain. 0039~6028,/90/$03.50 0 Elsevier Science Publishers B.V. ( Nosh-Holland)
cause of the effects of the electric field, the coherence of the eigenstates is limited to a finite number of periods. We focus our attention then to the states that are localized inside the multi-quantum well structure and do not see the surface boundary conditions. We limit the superlattice in the extremes by introducing an infinite barrier at a distance L) from the last quantum well interface. This infinite barrier is chosen to be far enough so as not to modify the solutions of interest. All the eigenvalues are then bound states. They reproduce the quasi-bound states existing in the actual case with the desired accuracy. We obtain the eigensolutions for the system described above by using the effective mass approximation. The solutions of the Schrodinger equation for each layer are given by a linear combination of Airy functions. To find the eigensolutions of the multi-quantum well we use the transfer matrix technique [ 71. We neglect the difference between the band parameters in different layers, assuming those of the GaAs. We show in fig. 1 the square of the amplitudes of the eigenfunctions for the (a) electrons and (b) heavy-holes for several values of the electric field. In the infinite superlattice, the wavefunction of the adjacent wells are given by translating the center one by the appropriate number of periods with a phase shift of n between solutions of first neighbours wells and adding to the energy the equivalent number of energy steps, characteristic of the Stark ladder states [2]. That is the approach we follow here, keeping as many adjacent solutions in respect to a
J.A. Bmm, heavy
T-a7
F. A~ll~-Rued~/~tu~k
holes
lJ.Jag5-Li-L
z 4
6
70
200
IJU”
G(kV/tm)
ladder ex~itoni~ traditions
473
the excitonic states by neglecting, in a first approach, the coupling between those states, calculating the exciton formed by the electron in the nth state and the hole in the mth state with a one parameter variational wavefunction. We neglect the heavy- and lighthole coupling. The correlation between those different states is included, in a second step. We limit the basis to the 1s excitons, neglecting the excited bound states as well as the continuum exciton states. We write the general exciton state as
8(kV/Crn)
Fig. 1. Wavefunctions square amplitudes for (a) electrons and (b) heavy-holes for a 40 8, GaAs-20 8, Ga0.6A10.~,&superlattice for several values of electric field. The conduction band offset is taken as 60%, m,=0.0665 m, (where m, is the free electron mass), mhh= 0.34 m,, mth= 0.094 mO.
chosen center well as it is necessary to describe the correlation among them induced by the Coulombic interaction. Back to fig. 1, we observe the strong confinement of the heavy-holes at weak electric fields. For the parameters used here the heavy-holes are practically confined inside one well at 10 kV/cm. On the other hand, for the same electric field, the electrons are spread out over many wells. At higher electric fields we observe the Stark shift. With the parameters of fig. 1 the electrons are never entirely localized in one well, beginning to escape to the superlattice continuum before the full localization is achieved (in our case, for F= 110 kV/cm we observe already an interaction between the Stark ladder state and the superlatticecontinuum). The light-holes (not shown here) ionize at lower fields than the electrons and the heavy-holes as a combined effect of the lower barrier and the lighter mass. The Coulombic interaction couples all these states, reordering them to the lower energy configuration. A first calculation of the excitonic state in a superlattice was performed by Chomette et al. [ 81. More recently, Galbraith and Duggan [9] studied the exciton in a double quantum well in the presence of an external electric field. Chu and Chang [ 111 calculated the superlattice exciton showing the fo~ation of the saddle-point excitonic structure. Here, we choose to emphasize the effects of the Stark ladder localization to the electron correlation between the adjacent states. For that, we calculate
The ,%_‘s are obtained by minimizing the excitonic energy with A,,, and by neglecting the coupling among different states. To include it, we have to solve the following system of equations
=O, n,m,n’,m’ = 1,2 ,,.., N. (2) As it was pointed out, the wavefunctions iv,, and @, are periodic with a difference of phase and we neglect the solutions that show surface effects. We can take N as large as it is necessary to include the correlation among the different states induced by the Coulombic interaction. The eigensolutions are obtained by numerical diagonalization. We plot in fig. 2 the exciton binding energy for the decoupled approximation as a function of the electric field for a 40 A GaAs-40 A Gao.6A10_4As superlattice. The piot is truncated at high fields when the Stark ladder states begin to interact with the continuum and the concept of quasi-bound state is no longer valid. As it was pointed out before, the light-hole excitons are swept out of the well at weaker fields than the heavy-holes. Two mechanisms are responsible for the variation in the binding energy: the localization and the intra-well Stark shift. At Iow fields, the intrawell exciton shows a sharp shift to higher binding energies as a consequence of the localization, saturating at about 40 kV/cm for the heavy-hole exciton. This sharp transition gives us a measure of the coherence of the superlattice states. It is only at high
474
J.A. hum, F. Agulld-Rueda/Stark ladder excitomc transitions
J
1
Elect::
Fig. 2. Exciton binding energies for a 40 A GaAs-40 A Ga0.6AI, lines) and light- (dashed lines) eters are the
field
fields that the Stark shift predominates and the binding energies decrease [ 10 1. This latter feature cannot be observed for the light-hole excitons since they begin to interact with the superlattice continuum before being localized inside the single well. Even for the heavy-hole excitons, the binding energy never reaches completely the single quantum well limit, since the Stark shift is already significative before the localization is fully achieved. It is surprising to observe that both light- and heavy-hole excitons have close binding energies at low fields, with the light-hole exciton saturating at higher values. For the crossed excitons, the binding energy first increases with the electric field due to the localization of the states. For intermediary and higher fields (and consequent further localization) it decreases, showing the spatial localization in different wells for the electrons and holes. As it is expected, for higher values of An the maximum and the saturation of the binding energy occur at lower values. For clarity, we only show the An<0 transitions. The An> 0 one present a similar behavior, with their binding energies slightly higher. In fig. 3 we plot the respective oscillator strengths. Their features follow the same qualitative behavior as their binding energies, with
40
Electric
(k8e/cm)
as a function of the electric field 40A~ superlattice for heavy- (full hole excitons. The other paramsame as fig. 1.
I
0
120
Fig. 3. Oscillator
80
field
120
(kV/cm)
strengths for the exciton states of fig. 2 as a function of the electric field.
the crossed excitons loosing their oscillator strength quite sharply. Finally, in fig. 4 we show the transition energies for the same structure of fig. 2. The effect of the binding energy can only be observed at low fields, when the sharp transition takes place. At
1.64
0
20
Electric Fig. 4. Transition
40
60
field
(kV/cm)
energies for the structure of the electric field.
of fig. 2 as a function
J.A. Brum, F. Agulld-Rueda/Stark ladder excitonic transitions
higher fields, the main deviation from the Stark ladder behavior,E= (An+ An’) eF(LGaAs+LGaxAII_xAS), comes from the intra-well Stark shift. Contrary to the single quantum well, here the Stark shift is more significant for the light-hole than to the heavy-hole excitons. The inclusion of the coupling between the different states changes the binding energy although does not alter the main qualitative features. We can separate the correlation between the states in two differentcategories: n-m#n’-m’andn-m=n’-m’. The former has a small contribution being only significant at very low fields (in our case, I4 kV/cm) what is consistent with the results obtained by Galbraith and Duggan [ 91. The latter contribution is more important since they are degenerate before the coupling. The general effect is to concentrate the oscillator strength to one transition which shows a weaker binding energy. This effect disappears when the localization in one well is achieved. In other words, the effect is only sizeable before the saturation of the binding energy observed in fig. 2. Finally, we consider the fluctuations in the superlattice structure (Al concentration in the barrier and the layers width). The effect of disorder in superlattices with and without electric field has been considered by several authors [ 12 1. Here, we calculated the eigenstates in a 25 quantum well structure, with the layers varying by one monolayer and the Al concentration in the barrier by 0.05. The variation in the above parameters is given by generated random numbers. Although the wavefunctions change the shape from one well to the other, accusing the lost of periodicity, they keep the main features. In particular, we only observe a significant further localization induced by the fluctuations at low electric fields. We do not expect a significant change in the spectra other than the broadening of the structures.
475
A comparison with the experimental results was described elsewhere [ 61. We obtain a semi-quantitative agreement: although the absolute value of the binding energies for the intra-well exciton are lower than the measured one, the sharp transitions and the electric field to which they occur are in very good agreement with the experimental results.
We are in debt with E.E. Mendez and G. Bastard for helpful discussions. Part of this work was supported by the Army Research Office. One of us (F.A.R.) acknowledges the support of the Institute of Materials Science of Madrid (Spain ).
References
[ I] See for example, J. Heirichs and R.O. Jones, J. Phys. C 5 (1972) 2149; B. Velicky and J. Sak, Phys. Status Solidi 16 ( 1966) 147. [2] J. Bleuse, G. Bastard and P. Voisin, Phys. Rev. Lett. 60 (1988) 220. [ 31 E.E. Mendez, F. Agullo-Rueda and J.M. Hong, Phys. Rev. Lett. 60 (1988) 2426. [4] P. Voisin, J. Bleuse, C. Bouche, S. Gaillard, C. Alibert and A. Regreny, Phys. Rev. Lett. 6 1 ( 1988) 1639. J. Bleuse, P. Voisin, M. Allovon and M. Quillec, Appl. Phys. Lett. 53 (1988) 2632. [6 F. Agullo-Rueda, E.E. Mendez, J.A. Brum and J.M. Hong, Proc. 14th Int. Conf. on Modulated Semiconductor Structures, Ann Arbor (MI), July 17-20, 1989. [7 P.W.A. McLLroy, J. Appl. Phys. 59 (1986) 3532. 18 A. Chomette, B. Lambert, B. Deveaud, F. Clerot, A. Regreny and G. Bastard, Europhys. Lett. 4 ( 1987) 46 I. 19 I. Galbraith and D. Duggan, preprint. [la J.A. Brum and G. Bastard, Phys. Rev. B 31 (1985) 3893. H. Chu and Y.-C. Chang, Phys. Rev. B 36 ( 1987) 2946. [ll [12 See for example, J.V. Jose, G. Monsivais and J. Flares, Phys. Rev. B 31 (1985) 6906; H.X. Jiangand J.Y. Lin, J. Appl. Phys. 63 (1988) 1984.
1
Surface Science 229 f 1990) 472-475 North-Holland
STARK LADDER EXCITONIC TRANSITIONS J.A. BRUM * and F. AGULL~-RUEDA IBhcf 7X Watson Research Center, Yorkfown
**
~eig~is,
h’Y 10598,
USA
Received IO August 1989; accepted for publication 14 September 1989
We have calculated the binding energy and the transition energies of excitons for a superlattice under an external electric field applied along the growth direction. At low fields the binding energy shifts strongly showing the localization of the superlattice states inlo the Stark ladder states. For higher fields the energy levels observe a shift associated to the single quantum well Stark effect. We also discuss the contribution of superlattice fluctuations to the carrier con~nement.
The effect of an external electric field applied in bulk semiconductors attracted some attention in the past [ 11. The electric field breaks the periodicity and localizes the carriers into Stark ladder states for certain range of electric field. However, its experimental verification was not possible due to the high electric fields necessary to achieve the Stark ladder localization in the bulk semiconductors. The high quality of superlattice heterostructures grown by MBE opened the way to the observation of this effect within reasonable values of electric fields. This effect was studied theoretically [2] and observed experimentally in GaAs-Ga(Al)As superlattices [ 3,4] as well as other systems [ 5 1. In general, the optical spectra are dominated by transitions of excitonic origin. Although many of the features observed experimentally can be interpreted as band-to-band transitions, the high quality of the samples has permitted the observation of the excitonic behavior [ 61. An ideal superlattice is described as an infinite periodic sequence of alternate layers. Here, we simulate it by a finite number of quantum wells limiting their number, by numerical reasons, to a lower value than the actual samples (typically 60 wells). This procedure enhances the effect of the surface and discretize the superlattice continuum. However, be* Present address: UNICAMP, Instituto de Fisica, CP 6165, 1308 1 Campinas (SP), Brazil, and Laboratorio National de Luz Sincroton, CP 6192, 13083 Campinas (SP). Brazil. +* Permanent address: Instituto de Ciencia de Materiales (SCIS). Universidad Autonoma, 28049 Madrid, Spain. 0039~6028,/90/$03.50 0 Elsevier Science Publishers B.V. ( Nosh-Holland)
cause of the effects of the electric field, the coherence of the eigenstates is limited to a finite number of periods. We focus our attention then to the states that are localized inside the multi-quantum well structure and do not see the surface boundary conditions. We limit the superlattice in the extremes by introducing an infinite barrier at a distance L) from the last quantum well interface. This infinite barrier is chosen to be far enough so as not to modify the solutions of interest. All the eigenvalues are then bound states. They reproduce the quasi-bound states existing in the actual case with the desired accuracy. We obtain the eigensolutions for the system described above by using the effective mass approximation. The solutions of the Schrodinger equation for each layer are given by a linear combination of Airy functions. To find the eigensolutions of the multi-quantum well we use the transfer matrix technique [ 71. We neglect the difference between the band parameters in different layers, assuming those of the GaAs. We show in fig. 1 the square of the amplitudes of the eigenfunctions for the (a) electrons and (b) heavy-holes for several values of the electric field. In the infinite superlattice, the wavefunction of the adjacent wells are given by translating the center one by the appropriate number of periods with a phase shift of n between solutions of first neighbours wells and adding to the energy the equivalent number of energy steps, characteristic of the Stark ladder states [2]. That is the approach we follow here, keeping as many adjacent solutions in respect to a
J.A. Bmm, heavy
T-a7
F. A~ll~-Rued~/~tu~k
holes
lJ.Jag5-Li-L
z 4
6
70
200
IJU”
G(kV/tm)
ladder ex~itoni~ traditions
473
the excitonic states by neglecting, in a first approach, the coupling between those states, calculating the exciton formed by the electron in the nth state and the hole in the mth state with a one parameter variational wavefunction. We neglect the heavy- and lighthole coupling. The correlation between those different states is included, in a second step. We limit the basis to the 1s excitons, neglecting the excited bound states as well as the continuum exciton states. We write the general exciton state as
8(kV/Crn)
Fig. 1. Wavefunctions square amplitudes for (a) electrons and (b) heavy-holes for a 40 8, GaAs-20 8, Ga0.6A10.~,&superlattice for several values of electric field. The conduction band offset is taken as 60%, m,=0.0665 m, (where m, is the free electron mass), mhh= 0.34 m,, mth= 0.094 mO.
chosen center well as it is necessary to describe the correlation among them induced by the Coulombic interaction. Back to fig. 1, we observe the strong confinement of the heavy-holes at weak electric fields. For the parameters used here the heavy-holes are practically confined inside one well at 10 kV/cm. On the other hand, for the same electric field, the electrons are spread out over many wells. At higher electric fields we observe the Stark shift. With the parameters of fig. 1 the electrons are never entirely localized in one well, beginning to escape to the superlattice continuum before the full localization is achieved (in our case, for F= 110 kV/cm we observe already an interaction between the Stark ladder state and the superlatticecontinuum). The light-holes (not shown here) ionize at lower fields than the electrons and the heavy-holes as a combined effect of the lower barrier and the lighter mass. The Coulombic interaction couples all these states, reordering them to the lower energy configuration. A first calculation of the excitonic state in a superlattice was performed by Chomette et al. [ 81. More recently, Galbraith and Duggan [9] studied the exciton in a double quantum well in the presence of an external electric field. Chu and Chang [ 111 calculated the superlattice exciton showing the fo~ation of the saddle-point excitonic structure. Here, we choose to emphasize the effects of the Stark ladder localization to the electron correlation between the adjacent states. For that, we calculate
The ,%_‘s are obtained by minimizing the excitonic energy with A,,, and by neglecting the coupling among different states. To include it, we have to solve the following system of equations
=O, n,m,n’,m’ = 1,2 ,,.., N. (2) As it was pointed out, the wavefunctions iv,, and @, are periodic with a difference of phase and we neglect the solutions that show surface effects. We can take N as large as it is necessary to include the correlation among the different states induced by the Coulombic interaction. The eigensolutions are obtained by numerical diagonalization. We plot in fig. 2 the exciton binding energy for the decoupled approximation as a function of the electric field for a 40 A GaAs-40 A Gao.6A10_4As superlattice. The piot is truncated at high fields when the Stark ladder states begin to interact with the continuum and the concept of quasi-bound state is no longer valid. As it was pointed out before, the light-hole excitons are swept out of the well at weaker fields than the heavy-holes. Two mechanisms are responsible for the variation in the binding energy: the localization and the intra-well Stark shift. At Iow fields, the intrawell exciton shows a sharp shift to higher binding energies as a consequence of the localization, saturating at about 40 kV/cm for the heavy-hole exciton. This sharp transition gives us a measure of the coherence of the superlattice states. It is only at high
474
J.A. hum, F. Agulld-Rueda/Stark ladder excitomc transitions
J
1
Elect::
Fig. 2. Exciton binding energies for a 40 A GaAs-40 A Ga0.6AI, lines) and light- (dashed lines) eters are the
field
fields that the Stark shift predominates and the binding energies decrease [ 10 1. This latter feature cannot be observed for the light-hole excitons since they begin to interact with the superlattice continuum before being localized inside the single well. Even for the heavy-hole excitons, the binding energy never reaches completely the single quantum well limit, since the Stark shift is already significative before the localization is fully achieved. It is surprising to observe that both light- and heavy-hole excitons have close binding energies at low fields, with the light-hole exciton saturating at higher values. For the crossed excitons, the binding energy first increases with the electric field due to the localization of the states. For intermediary and higher fields (and consequent further localization) it decreases, showing the spatial localization in different wells for the electrons and holes. As it is expected, for higher values of An the maximum and the saturation of the binding energy occur at lower values. For clarity, we only show the An<0 transitions. The An> 0 one present a similar behavior, with their binding energies slightly higher. In fig. 3 we plot the respective oscillator strengths. Their features follow the same qualitative behavior as their binding energies, with
40
Electric
(k8e/cm)
as a function of the electric field 40A~ superlattice for heavy- (full hole excitons. The other paramsame as fig. 1.
I
0
120
Fig. 3. Oscillator
80
field
120
(kV/cm)
strengths for the exciton states of fig. 2 as a function of the electric field.
the crossed excitons loosing their oscillator strength quite sharply. Finally, in fig. 4 we show the transition energies for the same structure of fig. 2. The effect of the binding energy can only be observed at low fields, when the sharp transition takes place. At
1.64
0
20
Electric Fig. 4. Transition
40
60
field
(kV/cm)
energies for the structure of the electric field.
of fig. 2 as a function
J.A. Brum, F. Agulld-Rueda/Stark ladder excitonic transitions
higher fields, the main deviation from the Stark ladder behavior,E= (An+ An’) eF(LGaAs+LGaxAII_xAS), comes from the intra-well Stark shift. Contrary to the single quantum well, here the Stark shift is more significant for the light-hole than to the heavy-hole excitons. The inclusion of the coupling between the different states changes the binding energy although does not alter the main qualitative features. We can separate the correlation between the states in two differentcategories: n-m#n’-m’andn-m=n’-m’. The former has a small contribution being only significant at very low fields (in our case, I4 kV/cm) what is consistent with the results obtained by Galbraith and Duggan [ 91. The latter contribution is more important since they are degenerate before the coupling. The general effect is to concentrate the oscillator strength to one transition which shows a weaker binding energy. This effect disappears when the localization in one well is achieved. In other words, the effect is only sizeable before the saturation of the binding energy observed in fig. 2. Finally, we consider the fluctuations in the superlattice structure (Al concentration in the barrier and the layers width). The effect of disorder in superlattices with and without electric field has been considered by several authors [ 12 1. Here, we calculated the eigenstates in a 25 quantum well structure, with the layers varying by one monolayer and the Al concentration in the barrier by 0.05. The variation in the above parameters is given by generated random numbers. Although the wavefunctions change the shape from one well to the other, accusing the lost of periodicity, they keep the main features. In particular, we only observe a significant further localization induced by the fluctuations at low electric fields. We do not expect a significant change in the spectra other than the broadening of the structures.
475
A comparison with the experimental results was described elsewhere [ 61. We obtain a semi-quantitative agreement: although the absolute value of the binding energies for the intra-well exciton are lower than the measured one, the sharp transitions and the electric field to which they occur are in very good agreement with the experimental results.
We are in debt with E.E. Mendez and G. Bastard for helpful discussions. Part of this work was supported by the Army Research Office. One of us (F.A.R.) acknowledges the support of the Institute of Materials Science of Madrid (Spain ).
References
[ I] See for example, J. Heirichs and R.O. Jones, J. Phys. C 5 (1972) 2149; B. Velicky and J. Sak, Phys. Status Solidi 16 ( 1966) 147. [2] J. Bleuse, G. Bastard and P. Voisin, Phys. Rev. Lett. 60 (1988) 220. [ 31 E.E. Mendez, F. Agullo-Rueda and J.M. Hong, Phys. Rev. Lett. 60 (1988) 2426. [4] P. Voisin, J. Bleuse, C. Bouche, S. Gaillard, C. Alibert and A. Regreny, Phys. Rev. Lett. 6 1 ( 1988) 1639. J. Bleuse, P. Voisin, M. Allovon and M. Quillec, Appl. Phys. Lett. 53 (1988) 2632. [6 F. Agullo-Rueda, E.E. Mendez, J.A. Brum and J.M. Hong, Proc. 14th Int. Conf. on Modulated Semiconductor Structures, Ann Arbor (MI), July 17-20, 1989. [7 P.W.A. McLLroy, J. Appl. Phys. 59 (1986) 3532. 18 A. Chomette, B. Lambert, B. Deveaud, F. Clerot, A. Regreny and G. Bastard, Europhys. Lett. 4 ( 1987) 46 I. 19 I. Galbraith and D. Duggan, preprint. [la J.A. Brum and G. Bastard, Phys. Rev. B 31 (1985) 3893. H. Chu and Y.-C. Chang, Phys. Rev. B 36 ( 1987) 2946. [ll [12 See for example, J.V. Jose, G. Monsivais and J. Flares, Phys. Rev. B 31 (1985) 6906; H.X. Jiangand J.Y. Lin, J. Appl. Phys. 63 (1988) 1984.
1