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AlxGa1−xAs double quantum wells
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Physica E 17 (2003) 222 – 224 www.elsevier.com/locate/physe
Stark e ect in the magneto-exciton energy in GaAs=Alx Ga1−x As double quantum wells E.C. Ferreiraa , J.A.P. Da Costaa , J.A.K. Freireb;∗ a Departamento
de F sica, Universidade Federal do Rio Grande do Norte, 59072-970 Natal, Rio Grande do Norte, Brazil de F sica, Centro de Ciencias Exatas, Universidade Federal do Cear a, Campus do Pici, Caixa Postal 6030, 60455-900 Fortaleza, Cear a, Brazil
b Departamento
Abstract We calculated the exciton binding energy and energy levels in GaAs=Al0:3 Ga0:7 As DQWs with non-abrupt interfaces, taking into account the electric and magnetic 3eld e ects. Numerical results for GaAs=Al0:3 Ga0:7 As double quantum wells 7 wide) show an enhancement of the e–hh exciton energy by as much as 50 meV with respect to the abrupt well system (50 A 7 interface is considered. when a 10 A ? 2002 Elsevier Science B.V. All rights reserved. PACS: 68.35.Ct; 68.65.Fg; 68.90.+g Keywords: Magneto-exciton; Gradual interfaces
Excitons in semiconductors double quantum wells (DQWs) have been the subject of several experimental and theoretical investigations due to promising applications in optical modulators and tunable free-electron lasers [1]. The main advantage over single QWs is the enhanced excitonic electro-optic response and large quantum-con3ned Stark e ect with several applications in high-speed spatial–light modulators and switches [2]. The symmetric and anti-symmetric states in DQWs are very sensitive to both electric and magnetic 3elds [3]. One of the most important problems during the growth of quantum well heterostructures is the existence of interface roughness, which controls the quality of the device.
∗ Corresponding author. Tel.: +55-85-2889932; fax: +55-852872184. E-mail address: [email protected] (J.A.K. Freire).
Continuing advances in modern crystal-growth techniques are leading to improvements in the quality of QW devices, but the interfacial Fuctuations still exist and have been reported in several works. The purpose of this work is to study the electric and magnetic 3eld e ects in the exciton properties in more real GaAs=Alx Ga1−x As DQWs, i.e., considering the existence of non-abrupt interfaces. Assuming that the exciton wave function can be decoupled as (r; zi ) = e (ze ) h (zh ) e–h (r), where i (zi ) describes the carrier motion in the DQW and e–h (r) is the relative part of the exciton wave function, the SchrHodinger equation describing the DQW con3nement can be written, within the e ective mass approximation, as 2 1 ˝ @ + Vi (zi ) − 2 @zi m⊥ i (zi ) ∓eF⊥ zi − Ei i (zi ) = 0;
1386-9477/03/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S1386-9477(02)00798-1
E.C. Ferreira et al. / Physica E 17 (2003) 222 – 224
1.7 EEXC (eV)
60
1.6 1.5
40 150
20
100
60
50
70
80 0
(k V/ cm
50 L (Å )
F
40
E
30
20
40
60
80
100
L (Å)
)
∆E (meV)
80
Fig. 1. Shift of total e–hh exciton energy (JE = E w=10 A7 − E w=0 A7 ) as a function of the QW width and of the applied electric 3eld.
where mi (zi ) and Vi (zi ) describes the carrier e ective mass and e ective potential in the QW with growth direction along the z-axis. The model of Freire et al. [4] is used to describe the interfacial region, being assumed that mi (zi ) and Vi (zi ) are dependent on the Al molar fraction (z) variation in the graded interface regions [4]. The in-plane exciton equation, considering an applied magnetic 3eld parallel to the QW growth direction and that e–h (r) = eim Rn; m (), can be expressed as ˝2 1 @ @ − + V e () − E Rn; m () = 0; 2 @ @ with the following in-plane e ective potential: V e () =
223
˝2 m2 e2 ˝ e2 2 2 + m + B B z 2 2 2 8 z ∞ ∞ e2 | e (ze )|2 | h (zh )|2 − d z e d zh : 4[2 + (ze − zh )2 ]1=2 0 0
The above SchrHodinger equations are solved by using a multistep method analogous to the one used in Ref. [4]. The e–hh exciton energy dependence on the QW width and on the electric 3eld is shown in Fig. 1, for
Fig. 2. Total exciton energy of the e–hh exciton as a function of the 7 wide barrier, applied magnetic 3eld B=0 T QW width, for a 40 A (symbols) and B = 10 T (curves), with interface thicknesses of 7 (- - - and ) and 10 A 7 (— and 4). 0A
7 barrier width. The energy is given in compara 40 A ison to those of a similar abrupt well structure, i.e., JE is the energy di erence between the exciton level 7 graded DQW and in the corresponding in a w = 10 A abrupt well (JE = E w=10 A7 − E w=0 A7 ). One can have a di erence of almost 50 meV in the exciton energy 7 abrupt well, as compared to the corresponding 50 A 7 when considering interfaces thicknesses of only 10 A. This di erence decreases to 30 meV (10 meV) if we include the e ect of a 50 kV=cm electric 3eld. The magneto-exciton energy as a function of the QW width is shown in Fig. 2, for a barrier width of 7 an applied magnetic 3eld of B = 0 T (symbols) 40 A, and B=10 T (curves), and for interfaces thicknesses of 7 Notice that the exciton energy broadening 0 and 10 A. due to the presence of gradual interfaces is larger in the case of zero magnetic 3eld, which can be seen by comparing the curves and symbols in Fig. 2. The 7 wide DQW, e.g., decreases from broadening in a 50 A 50 meV to only 10 meV when we consider a magnetic 3eld of B = 10 T. Gradual interfaces can be responsible for a strong broadening of the excitonic spectrum and due to its order of magnitude, it should be able to be detected by photoluminescence experiments. This work was supported by CNPq under Contract NanoSemiMat/CNPq # 550.015/01-9, and by FINEP under Contract CTPETRO/FINEP # 65.00.02.80.00.
224
E.C. Ferreira et al. / Physica E 17 (2003) 222 – 224
References [1] C. Weisbuch, H. Benisty, R. HoudrQe, J. Lumin. 85 (2000) 271. [2] W. Ossau, Surf. Sci. 174 (1986) 188.
[3] K. Leosson, J.R. Jensen, W. Langbein, J.M. Hvam, Phys. Rev. B 61 (2000) 10 322. [4] J.A.K. Freire, G.A. Farias, V.N. Freire, Solid State Commun. 106 (1998) 559.
Physica E 17 (2003) 222 – 224 www.elsevier.com/locate/physe
Stark e ect in the magneto-exciton energy in GaAs=Alx Ga1−x As double quantum wells E.C. Ferreiraa , J.A.P. Da Costaa , J.A.K. Freireb;∗ a Departamento
de F sica, Universidade Federal do Rio Grande do Norte, 59072-970 Natal, Rio Grande do Norte, Brazil de F sica, Centro de Ciencias Exatas, Universidade Federal do Cear a, Campus do Pici, Caixa Postal 6030, 60455-900 Fortaleza, Cear a, Brazil
b Departamento
Abstract We calculated the exciton binding energy and energy levels in GaAs=Al0:3 Ga0:7 As DQWs with non-abrupt interfaces, taking into account the electric and magnetic 3eld e ects. Numerical results for GaAs=Al0:3 Ga0:7 As double quantum wells 7 wide) show an enhancement of the e–hh exciton energy by as much as 50 meV with respect to the abrupt well system (50 A 7 interface is considered. when a 10 A ? 2002 Elsevier Science B.V. All rights reserved. PACS: 68.35.Ct; 68.65.Fg; 68.90.+g Keywords: Magneto-exciton; Gradual interfaces
Excitons in semiconductors double quantum wells (DQWs) have been the subject of several experimental and theoretical investigations due to promising applications in optical modulators and tunable free-electron lasers [1]. The main advantage over single QWs is the enhanced excitonic electro-optic response and large quantum-con3ned Stark e ect with several applications in high-speed spatial–light modulators and switches [2]. The symmetric and anti-symmetric states in DQWs are very sensitive to both electric and magnetic 3elds [3]. One of the most important problems during the growth of quantum well heterostructures is the existence of interface roughness, which controls the quality of the device.
∗ Corresponding author. Tel.: +55-85-2889932; fax: +55-852872184. E-mail address: [email protected] (J.A.K. Freire).
Continuing advances in modern crystal-growth techniques are leading to improvements in the quality of QW devices, but the interfacial Fuctuations still exist and have been reported in several works. The purpose of this work is to study the electric and magnetic 3eld e ects in the exciton properties in more real GaAs=Alx Ga1−x As DQWs, i.e., considering the existence of non-abrupt interfaces. Assuming that the exciton wave function can be decoupled as (r; zi ) = e (ze ) h (zh ) e–h (r), where i (zi ) describes the carrier motion in the DQW and e–h (r) is the relative part of the exciton wave function, the SchrHodinger equation describing the DQW con3nement can be written, within the e ective mass approximation, as 2 1 ˝ @ + Vi (zi ) − 2 @zi m⊥ i (zi ) ∓eF⊥ zi − Ei i (zi ) = 0;
1386-9477/03/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S1386-9477(02)00798-1
E.C. Ferreira et al. / Physica E 17 (2003) 222 – 224
1.7 EEXC (eV)
60
1.6 1.5
40 150
20
100
60
50
70
80 0
(k V/ cm
50 L (Å )
F
40
E
30
20
40
60
80
100
L (Å)
)
∆E (meV)
80
Fig. 1. Shift of total e–hh exciton energy (JE = E w=10 A7 − E w=0 A7 ) as a function of the QW width and of the applied electric 3eld.
where mi (zi ) and Vi (zi ) describes the carrier e ective mass and e ective potential in the QW with growth direction along the z-axis. The model of Freire et al. [4] is used to describe the interfacial region, being assumed that mi (zi ) and Vi (zi ) are dependent on the Al molar fraction (z) variation in the graded interface regions [4]. The in-plane exciton equation, considering an applied magnetic 3eld parallel to the QW growth direction and that e–h (r) = eim Rn; m (), can be expressed as ˝2 1 @ @ − + V e () − E Rn; m () = 0; 2 @ @ with the following in-plane e ective potential: V e () =
223
˝2 m2 e2 ˝ e2 2 2 + m + B B z 2 2 2 8 z ∞ ∞ e2 | e (ze )|2 | h (zh )|2 − d z e d zh : 4[2 + (ze − zh )2 ]1=2 0 0
The above SchrHodinger equations are solved by using a multistep method analogous to the one used in Ref. [4]. The e–hh exciton energy dependence on the QW width and on the electric 3eld is shown in Fig. 1, for
Fig. 2. Total exciton energy of the e–hh exciton as a function of the 7 wide barrier, applied magnetic 3eld B=0 T QW width, for a 40 A (symbols) and B = 10 T (curves), with interface thicknesses of 7 (- - - and ) and 10 A 7 (— and 4). 0A
7 barrier width. The energy is given in compara 40 A ison to those of a similar abrupt well structure, i.e., JE is the energy di erence between the exciton level 7 graded DQW and in the corresponding in a w = 10 A abrupt well (JE = E w=10 A7 − E w=0 A7 ). One can have a di erence of almost 50 meV in the exciton energy 7 abrupt well, as compared to the corresponding 50 A 7 when considering interfaces thicknesses of only 10 A. This di erence decreases to 30 meV (10 meV) if we include the e ect of a 50 kV=cm electric 3eld. The magneto-exciton energy as a function of the QW width is shown in Fig. 2, for a barrier width of 7 an applied magnetic 3eld of B = 0 T (symbols) 40 A, and B=10 T (curves), and for interfaces thicknesses of 7 Notice that the exciton energy broadening 0 and 10 A. due to the presence of gradual interfaces is larger in the case of zero magnetic 3eld, which can be seen by comparing the curves and symbols in Fig. 2. The 7 wide DQW, e.g., decreases from broadening in a 50 A 50 meV to only 10 meV when we consider a magnetic 3eld of B = 10 T. Gradual interfaces can be responsible for a strong broadening of the excitonic spectrum and due to its order of magnitude, it should be able to be detected by photoluminescence experiments. This work was supported by CNPq under Contract NanoSemiMat/CNPq # 550.015/01-9, and by FINEP under Contract CTPETRO/FINEP # 65.00.02.80.00.
224
E.C. Ferreira et al. / Physica E 17 (2003) 222 – 224
References [1] C. Weisbuch, H. Benisty, R. HoudrQe, J. Lumin. 85 (2000) 271. [2] W. Ossau, Surf. Sci. 174 (1986) 188.
[3] K. Leosson, J.R. Jensen, W. Langbein, J.M. Hvam, Phys. Rev. B 61 (2000) 10 322. [4] J.A.K. Freire, G.A. Farias, V.N. Freire, Solid State Commun. 106 (1998) 559.