- Email: [email protected]
Optical manifestation of magnetoexcitons in near-surface quantum wells
Applied Surface Science 212–213 (2003) 127–130
Optical manifestation of magnetoexcitons in near-surface quantum wells B. Flores-Desirena, F. Pe´rez-Rodrı´guez* Instituto de Fı´sica, Universidad Auto´noma de Puebla, Apdo. Post. J-48, Puebla, Pue. 72570, Mexico
Abstract The optical response of excitons in quantum wells, close to the sample boundary and under the action of a strong magnetic field perpendicular to their plane, is investigated theoretically. Solving the system of coupled equations for the coherent electronhole interband amplitude and the electromagnetic field, reflectivity spectra for such nanostructures are calculated. The effect of the interaction of magnetoexcitons with the sample surface on the resonance structure of reflectivity spectra is analyzed. These optical spectra are also affected by the phase change of the electromagnetic wave as it propagates in the cap layer, overlying the quantum well. # 2003 Elsevier Science B.V. All rights reserved. PACS: 71.35.Ji; 78.67.De Keywords: Magnetoexcitons; Quantum wells
1. Introduction Magnetoexcitons in nanostructures have been the focus of attention of several research groups in recent years. In particular, excitons in quantum wells subjected to a sufficiently strong transverse magnetic field have been studied [1–6]. In such systems the discrete energy spectrum of excitons results from the size quantization of the electron and hole in the direction perpendicular to the quantum-well plane and the action of the transverse magnetic field (Landau quantization). The Coulomb attraction between the electron and hole is practically suppressed by both the * Corresponding author. Tel.: þ52-222-245-76-45; fax: þ52-222-244-89-47. E-mail address: [email protected] (F. Pe´rez-Rodrı´guez).
confining potential and the strong transverse magnetic field. As was shown in the set of works [1–3], exciton resonances in spectra of photoluminescence excitation and magnetophotoluminescence for near-surface quantum wells undergo a blue shift as the thickness of the cap layer is decreased. The blue shift of magnetoexciton levels was attributed to two factors: the first one is associated with the effect of the high vacuum potential barrier, which in this case is close to the quantum well; the second factor is connected to the influence of the image potential, which appears because of the dielectric mismatch at the sample surface and characterizes the interaction of the electron and hole with their image charges. In this work we study theoretically the coupling of magnetoexcitons in near-surface quantum wells with the incident light. The developed theory is based on
0169-4332/03/$ – see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0169-4332(03)00035-7
B. Flores-Desirena, F. Pe´ rez-Rodrı´guez / Applied Surface Science 212–213 (2003) 127–130
128
the Stahl’s real-space density-matrix approach [7] and allows for calculations of optical properties (reflectivity, tansmissivity and absorption) for a great variety of quantum well structures in a unified way. Here, we present theoretical reflectivity spectra for samples as those used in [1–3].
Eq. (2) has the form: Vim ðre ; rh Þ ¼
e 2 es 1 4pe0 es es þ 1 2 3 1 61 1 1 7 4 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5: 4 ze zh 2 r2 þ ðze þ zh Þ
2. Theory
(4)
Let us consider a semiconductor quantum-well structure (cap layer/quantum well/substrate) occupying the space z > 0. The optical properties of excitons in this semiconductor nanostructure can be described within the Stahl’s density matrix approach [7], which leads to the set of differential equations for the coherent amplitude Yðre ; rh Þ of the electron and hole with coordinates re and rh : ~ vc ðrÞEðRÞ; hðo þ inÞÞYðre ; rh Þ ¼ M ðHvc
(1)
where Hvc is the two-band Hamiltonian with gap Eg , which can be expressed as: 2
2
2
2
h @ h @ þ Ve ðze Þ 2me;z @z2e 2mh;z @z2h 1 ði hr~r;e þ eAe Þ2 þ Vh ðzh Þ þ 2me;jj 1 þ ði hr~r;h eAh Þ2 2mh;jj
Hvc ¼ Eg
e2 þ Vim ðre ; rh Þ: 4pe0 es r
The excitonic polarization P is related to the coherent amplitude Y by the expression: Z ~ vc ðrÞYðR; rÞ dr: P¼2 M (5) The polaritonic fields E and P in Eqs. (1) and (5) obey Maxwell equations: r r E e1
o2 o2 E¼ P; 2 c e0 c 2
(6)
where e1 is the high-frequency dielectric constant. The system of Eqs. (1)–(6) is complemented by the boundary condition of vanishing of the coherent amplitude (Y ¼ 0), when the electron or the hole is at the sample surface (ze ¼ 0 or zh ¼ 0), and the condition of continuity for tangential components of the electric and magnetic fields of the electromagnetic wave. In solving Eqs. (1)–(6), we have employed the shell model [8,9] for the transition dipole density:
(2)
Here, we use the following notation: E is the electric field of the electromagnetic wave of frequency o, ~ vc the interband-transition dipole density, n a pheM nomenological damping factor, r the magnitude of r re rh ¼ ð~ r; zÞ, R the radius vector of the exciton center-of-mass. For simplicity, we shall assume that the effective masses for the electron (me;z , me;jj ) and hole (mh;z , mh;jj ) as well as the low-frequency dielectric constant es do not vary in the whole sample. The quantities Ve ðze Þ and Vh ðzh Þ are the confining potentials with a steplike form. The vector potential A in Eq. (2) is given by: re;h ; (3) Ae;h ¼ 12 B0 ~ where B0 is the external magnetic field parallel to the growth direction (B0 jj^z). The image potential Vim in
~ ~ vc ðrÞ ¼ M 0 dðr rd ÞdðzÞ; M 2pr
(7)
with rd ! 0. Besides, the coherent amplitude Y was expanded in an orthogonal basis formed by products of electron and hole wave functions in the z direction, and eigenfunctions of the magnetic Hamiltonian in the x–y plane. The Coulomb interaction and the image potential produce off-diagonal terms by mixing the basis states. This procedure is similar to that used in [1–4], but in our case leads to an algebraic system of equations for the coefficients of the expansion for Y and the amplitudes of the electric field E both inside the sample and in the vacuum. With solving numerically this system of equations, which will be shown elsewhere because of its cumbersome form, reflectivity spectra are obtained.
B. Flores-Desirena, F. Pe´ rez-Rodrı´guez / Applied Surface Science 212–213 (2003) 127–130
129
Fig. 1. Normal-incidence reflectivity spectra (a–h) for quantum wells with thickness LQW ¼ 50 — in a magnetic field B0 ¼ 8 T. The ˚. corresponding cap-layer thickness Lcap is (a) 30, (b) 50, (c) 200, (d) 400, (e) 600, (f) 800, (g) 1000, (h) 1200 A
3. Results and discussion Here we present theoretical normal-incidence reflectivity spectra (Fig. 1) for GaAs/ InGaAs/ GaAs quantum wells as those considered in [1–3]. The heterostructures have near-surface quantum wells of the same thickness LQW ¼ 50 — and cap layers with different thickness Lcap ¼ 30, 50, 200, 400, 600, 800, ˚ . The depths for the electron and 1000 and 1200 A heavy-hole (hh) confining potential wells are 113 and 75 meV, respectively. Other sample parameters used in the calculations are: e1 ¼ es ¼ 12:5, me;z ¼ me;k ¼ 0:067m0 (m0 is the free electron mass), mhh;z ¼ 0:35m0 , mhh;k ¼ 0:2m0 , hn ¼ 1 meV. So, there are two confined hh subbands in the valence band and only one confined electron subband in the conduction band. As is seen in Fig. 1, the reflectivity spectra exhibit resonances associated to the e1-hh1 magnetoexcitons with angular momentum projection equal to zero (m ¼ 0). The resonant coupling of only s quasi-2d excitons with the incident light is owing to the chosen
~ vc ðrÞ, which is form of the interband dipole density M here given by the shell model [Eq. (7)]. With decreasing the cap layer thickness Lcap , the reflectivity resonances are shifted to the higher energies [compare subfigures (c, b, a)]. This result agrees with the photoluminescence excitation measurements carried out in [1–3], where a blue shift of peaks connected to s exciton transitions is reported. As was commented in the Section 1, this shift is an effect of the interaction of the exciton with the sample surface and, hence, can be observed only in the case of near-surface quantum wells. The variation of the cap layer thickness can alter not only the positions of the exciton resonances in the reflectivity, but also their lineshape. Indeed, for a small cap-layer thickness Lcap 200 — the reflectivity resonances are maxima. As Lcap is increased, the reflectivity resonances become minima [see panel (e) for Lcap ¼ ˚ ]. Further increase of Lcap makes the lineshape 600 A gradually return to its initial form [compare panel (h) with (c)]. Such a cyclic behavior of the resonance lineshape is due to the phase change of the electromagnetic field as it propagates in the cap layer [10].
130
B. Flores-Desirena, F. Pe´ rez-Rodrı´guez / Applied Surface Science 212–213 (2003) 127–130
In summary, we have applied the Stahl’s real-space density-matrix approach for calculating optical properties of magnetoexcitons in near-surface quantum wells. Specifically, the correlation of the position and lineshape of exciton resonances in reflectivity spectra with the thickness of the cap layer, overlying the quantum well, was investigated. Besides, it was established that the reflectivity resonances undergo a blue shift as the cap-layer thickness is decreased. This result agrees with the photoluminescence measurements reported in [1–3] and shows the usefulness of reflectivity spectra for studying the behavior of excitons in near-surface quantum wells. Acknowledgements This work was partially supported by CONACYT (Mexico) under grant 36047-E.
References [1] L.V. Kulik, V.D. Kulakovskii, M. Bayer, A. Forchel, N.A. Gippius, S.G. Tikhodeev, Phys. Rev. B 54 (1996) R2335. [2] A.L. Yablonskii, A.B. Dzyubenko, S.G. Tikhodeev, L.V. Kulik, V.D. Kulakovskii, JETP Lett. 64 (1996) 51. [3] N.A. Gippius, A.L. Yablonskii, A.B. Dzyubenko, S.G. Tikhodeev, I.V. Kulik, V.D. Kulakovskii, A. Forchel, J. Appl. Phys. 83 (1998) 5410. [4] F. Vera, Z. Barticevic, J. Appl. Phys. 83 (1998) 7720. [5] A. Getter, I.E. Perakis, Phys. Rev. B 60 (1999) 16027. [6] G. Martı´nez, R. Aguilar, P.H. Herna´ ndez, G.H. Cocoletzi, Phys. Rev. B 62 (2000) 6916. [7] A. Stahl, I. Balslev, Electrodynamics of the Semiconductor Band Edge, Springer, Berlin, 1987. [8] G. Czaijkowski, A. Tredicucci, Nuovo Cimento D 14 (1992) 1203. [9] D. Merbach, E. Scho¨ ll, W. Ebeling, P. Michler, J. Gutowski, Phys. Rev. B 58 (1998) 10709. [10] V. Srinivas, Y.J. Chen, C.E.C. Wood, Phys. Rev. B 48 (1993) 12300.
Optical manifestation of magnetoexcitons in near-surface quantum wells B. Flores-Desirena, F. Pe´rez-Rodrı´guez* Instituto de Fı´sica, Universidad Auto´noma de Puebla, Apdo. Post. J-48, Puebla, Pue. 72570, Mexico
Abstract The optical response of excitons in quantum wells, close to the sample boundary and under the action of a strong magnetic field perpendicular to their plane, is investigated theoretically. Solving the system of coupled equations for the coherent electronhole interband amplitude and the electromagnetic field, reflectivity spectra for such nanostructures are calculated. The effect of the interaction of magnetoexcitons with the sample surface on the resonance structure of reflectivity spectra is analyzed. These optical spectra are also affected by the phase change of the electromagnetic wave as it propagates in the cap layer, overlying the quantum well. # 2003 Elsevier Science B.V. All rights reserved. PACS: 71.35.Ji; 78.67.De Keywords: Magnetoexcitons; Quantum wells
1. Introduction Magnetoexcitons in nanostructures have been the focus of attention of several research groups in recent years. In particular, excitons in quantum wells subjected to a sufficiently strong transverse magnetic field have been studied [1–6]. In such systems the discrete energy spectrum of excitons results from the size quantization of the electron and hole in the direction perpendicular to the quantum-well plane and the action of the transverse magnetic field (Landau quantization). The Coulomb attraction between the electron and hole is practically suppressed by both the * Corresponding author. Tel.: þ52-222-245-76-45; fax: þ52-222-244-89-47. E-mail address: [email protected] (F. Pe´rez-Rodrı´guez).
confining potential and the strong transverse magnetic field. As was shown in the set of works [1–3], exciton resonances in spectra of photoluminescence excitation and magnetophotoluminescence for near-surface quantum wells undergo a blue shift as the thickness of the cap layer is decreased. The blue shift of magnetoexciton levels was attributed to two factors: the first one is associated with the effect of the high vacuum potential barrier, which in this case is close to the quantum well; the second factor is connected to the influence of the image potential, which appears because of the dielectric mismatch at the sample surface and characterizes the interaction of the electron and hole with their image charges. In this work we study theoretically the coupling of magnetoexcitons in near-surface quantum wells with the incident light. The developed theory is based on
0169-4332/03/$ – see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0169-4332(03)00035-7
B. Flores-Desirena, F. Pe´ rez-Rodrı´guez / Applied Surface Science 212–213 (2003) 127–130
128
the Stahl’s real-space density-matrix approach [7] and allows for calculations of optical properties (reflectivity, tansmissivity and absorption) for a great variety of quantum well structures in a unified way. Here, we present theoretical reflectivity spectra for samples as those used in [1–3].
Eq. (2) has the form: Vim ðre ; rh Þ ¼
e 2 es 1 4pe0 es es þ 1 2 3 1 61 1 1 7 4 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5: 4 ze zh 2 r2 þ ðze þ zh Þ
2. Theory
(4)
Let us consider a semiconductor quantum-well structure (cap layer/quantum well/substrate) occupying the space z > 0. The optical properties of excitons in this semiconductor nanostructure can be described within the Stahl’s density matrix approach [7], which leads to the set of differential equations for the coherent amplitude Yðre ; rh Þ of the electron and hole with coordinates re and rh : ~ vc ðrÞEðRÞ; hðo þ inÞÞYðre ; rh Þ ¼ M ðHvc
(1)
where Hvc is the two-band Hamiltonian with gap Eg , which can be expressed as: 2
2
2
2
h @ h @ þ Ve ðze Þ 2me;z @z2e 2mh;z @z2h 1 ði hr~r;e þ eAe Þ2 þ Vh ðzh Þ þ 2me;jj 1 þ ði hr~r;h eAh Þ2 2mh;jj
Hvc ¼ Eg
e2 þ Vim ðre ; rh Þ: 4pe0 es r
The excitonic polarization P is related to the coherent amplitude Y by the expression: Z ~ vc ðrÞYðR; rÞ dr: P¼2 M (5) The polaritonic fields E and P in Eqs. (1) and (5) obey Maxwell equations: r r E e1
o2 o2 E¼ P; 2 c e0 c 2
(6)
where e1 is the high-frequency dielectric constant. The system of Eqs. (1)–(6) is complemented by the boundary condition of vanishing of the coherent amplitude (Y ¼ 0), when the electron or the hole is at the sample surface (ze ¼ 0 or zh ¼ 0), and the condition of continuity for tangential components of the electric and magnetic fields of the electromagnetic wave. In solving Eqs. (1)–(6), we have employed the shell model [8,9] for the transition dipole density:
(2)
Here, we use the following notation: E is the electric field of the electromagnetic wave of frequency o, ~ vc the interband-transition dipole density, n a pheM nomenological damping factor, r the magnitude of r re rh ¼ ð~ r; zÞ, R the radius vector of the exciton center-of-mass. For simplicity, we shall assume that the effective masses for the electron (me;z , me;jj ) and hole (mh;z , mh;jj ) as well as the low-frequency dielectric constant es do not vary in the whole sample. The quantities Ve ðze Þ and Vh ðzh Þ are the confining potentials with a steplike form. The vector potential A in Eq. (2) is given by: re;h ; (3) Ae;h ¼ 12 B0 ~ where B0 is the external magnetic field parallel to the growth direction (B0 jj^z). The image potential Vim in
~ ~ vc ðrÞ ¼ M 0 dðr rd ÞdðzÞ; M 2pr
(7)
with rd ! 0. Besides, the coherent amplitude Y was expanded in an orthogonal basis formed by products of electron and hole wave functions in the z direction, and eigenfunctions of the magnetic Hamiltonian in the x–y plane. The Coulomb interaction and the image potential produce off-diagonal terms by mixing the basis states. This procedure is similar to that used in [1–4], but in our case leads to an algebraic system of equations for the coefficients of the expansion for Y and the amplitudes of the electric field E both inside the sample and in the vacuum. With solving numerically this system of equations, which will be shown elsewhere because of its cumbersome form, reflectivity spectra are obtained.
B. Flores-Desirena, F. Pe´ rez-Rodrı´guez / Applied Surface Science 212–213 (2003) 127–130
129
Fig. 1. Normal-incidence reflectivity spectra (a–h) for quantum wells with thickness LQW ¼ 50 — in a magnetic field B0 ¼ 8 T. The ˚. corresponding cap-layer thickness Lcap is (a) 30, (b) 50, (c) 200, (d) 400, (e) 600, (f) 800, (g) 1000, (h) 1200 A
3. Results and discussion Here we present theoretical normal-incidence reflectivity spectra (Fig. 1) for GaAs/ InGaAs/ GaAs quantum wells as those considered in [1–3]. The heterostructures have near-surface quantum wells of the same thickness LQW ¼ 50 — and cap layers with different thickness Lcap ¼ 30, 50, 200, 400, 600, 800, ˚ . The depths for the electron and 1000 and 1200 A heavy-hole (hh) confining potential wells are 113 and 75 meV, respectively. Other sample parameters used in the calculations are: e1 ¼ es ¼ 12:5, me;z ¼ me;k ¼ 0:067m0 (m0 is the free electron mass), mhh;z ¼ 0:35m0 , mhh;k ¼ 0:2m0 , hn ¼ 1 meV. So, there are two confined hh subbands in the valence band and only one confined electron subband in the conduction band. As is seen in Fig. 1, the reflectivity spectra exhibit resonances associated to the e1-hh1 magnetoexcitons with angular momentum projection equal to zero (m ¼ 0). The resonant coupling of only s quasi-2d excitons with the incident light is owing to the chosen
~ vc ðrÞ, which is form of the interband dipole density M here given by the shell model [Eq. (7)]. With decreasing the cap layer thickness Lcap , the reflectivity resonances are shifted to the higher energies [compare subfigures (c, b, a)]. This result agrees with the photoluminescence excitation measurements carried out in [1–3], where a blue shift of peaks connected to s exciton transitions is reported. As was commented in the Section 1, this shift is an effect of the interaction of the exciton with the sample surface and, hence, can be observed only in the case of near-surface quantum wells. The variation of the cap layer thickness can alter not only the positions of the exciton resonances in the reflectivity, but also their lineshape. Indeed, for a small cap-layer thickness Lcap 200 — the reflectivity resonances are maxima. As Lcap is increased, the reflectivity resonances become minima [see panel (e) for Lcap ¼ ˚ ]. Further increase of Lcap makes the lineshape 600 A gradually return to its initial form [compare panel (h) with (c)]. Such a cyclic behavior of the resonance lineshape is due to the phase change of the electromagnetic field as it propagates in the cap layer [10].
130
B. Flores-Desirena, F. Pe´ rez-Rodrı´guez / Applied Surface Science 212–213 (2003) 127–130
In summary, we have applied the Stahl’s real-space density-matrix approach for calculating optical properties of magnetoexcitons in near-surface quantum wells. Specifically, the correlation of the position and lineshape of exciton resonances in reflectivity spectra with the thickness of the cap layer, overlying the quantum well, was investigated. Besides, it was established that the reflectivity resonances undergo a blue shift as the cap-layer thickness is decreased. This result agrees with the photoluminescence measurements reported in [1–3] and shows the usefulness of reflectivity spectra for studying the behavior of excitons in near-surface quantum wells. Acknowledgements This work was partially supported by CONACYT (Mexico) under grant 36047-E.
References [1] L.V. Kulik, V.D. Kulakovskii, M. Bayer, A. Forchel, N.A. Gippius, S.G. Tikhodeev, Phys. Rev. B 54 (1996) R2335. [2] A.L. Yablonskii, A.B. Dzyubenko, S.G. Tikhodeev, L.V. Kulik, V.D. Kulakovskii, JETP Lett. 64 (1996) 51. [3] N.A. Gippius, A.L. Yablonskii, A.B. Dzyubenko, S.G. Tikhodeev, I.V. Kulik, V.D. Kulakovskii, A. Forchel, J. Appl. Phys. 83 (1998) 5410. [4] F. Vera, Z. Barticevic, J. Appl. Phys. 83 (1998) 7720. [5] A. Getter, I.E. Perakis, Phys. Rev. B 60 (1999) 16027. [6] G. Martı´nez, R. Aguilar, P.H. Herna´ ndez, G.H. Cocoletzi, Phys. Rev. B 62 (2000) 6916. [7] A. Stahl, I. Balslev, Electrodynamics of the Semiconductor Band Edge, Springer, Berlin, 1987. [8] G. Czaijkowski, A. Tredicucci, Nuovo Cimento D 14 (1992) 1203. [9] D. Merbach, E. Scho¨ ll, W. Ebeling, P. Michler, J. Gutowski, Phys. Rev. B 58 (1998) 10709. [10] V. Srinivas, Y.J. Chen, C.E.C. Wood, Phys. Rev. B 48 (1993) 12300.