- Email: [email protected]
Effect of magnetic fields on the linear and nonlinear intersubband optical absorption coefficients and refractive index changes in square and graded quantum wells
Superlattices and Microstructures 48 (2010) 312–320
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Effect of magnetic fields on the linear and nonlinear intersubband optical absorption coefficients and refractive index changes in square and graded quantum wells E. Ozturk a,∗ , I. Sokmen b a
Department of Physics, Cumhuriyet University, 58140 Sivas, Turkiye
b
Department of Physics, Dokuz Eylül University, İzmir, Turkiye
article
info
Article history: Received 27 March 2010 Received in revised form 28 May 2010 Accepted 17 June 2010 Available online 27 July 2010 Keywords: Square quantum well Graded quantum well Intersubband transition Nonlinear absorption coefficient Nonlinear refractive index change Magnetic field
abstract In this study, both the intersubband optical absorption coefficients and the refractive index changes as dependent on the magnetic field are calculated in square and graded quantum wells. Our results show that the position and the magnitude of the linear and total absorption coefficients and refractive index changes depend on the magnetic field strength and the shape of potential. The incident optical intensity has a great effect on the total absorption and refractive index changes. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction The studies on quantum heterostructures open a new field in fundamental physics, and also offer a wide range of potential applications for optoelectronic devices. Some of these applications include semiconductor lasers [1], single-electron transistors [2], quantum computing [3], optical memories [4] and infrared photodetectors [5]. There is currently considerable interest in the optical phenomena based on the intersubband transitions in semiconductor quantum heterostructures. Due to the generally larger values of dipole matrix elements and the possibility of achieving resonance conditions, both the linear and nonlinear optical processes in these structures have been widely investigated [6–12]. A number of device applications based on the intersubband transition,
∗
Corresponding author. E-mail address: [email protected] (E. Ozturk).
0749-6036/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2010.06.015
E. Ozturk, I. Sokmen / Superlattices and Microstructures 48 (2010) 312–320
313
for example, far-infrared photo-detectors [13–17], electro-optical modulators [18–20], all optical switch [21], and infrared lasers [22,23], has been proposed and realized. Theoretical studies show that the optical properties of quantum wells mainly depend on the asymmetry of the confining potential. Such an asymmetry in the potential profile can be obtained either by applying an electric field to a symmetric quantum well or by compositionally grading the QW [24–26]. In asymmetric quantum-well structures, the changes in the absorption coefficients were predicted theoretically and confirmed experimentally to be larger than the changes that occur in conventional square QWs [27–29]. Several theoretical analyses on the changes of the absorption coefficient and refractive index associated with intersubband optical transitions in single-quantum wells and multiple-quantum well structures are presented in the literature [30–34]. In our previous studies, we have investigated the effect of a laser field on the intersubband optical transitions in square quantum wells (SQW) and graded quantum wells (GQW) in the absence of an electric field [35] and under an external electric field [36]. One of the most remarkable features of 2DEG is the intersubband optical transition between the size-quantized subbands in the same band. If the quantum well structure is subjected to a magnetic field, these transitions become very interesting. The magnetic field is an important additional parameter, since it can be applied experimentally in a well-controlled way and modifies fundamentally the electronic structure. A parallel magnetic field has a small quantitative impact on the quantum well’s energy spectrum. If a magnetic field perpendicular to the quantum well plane is applied, however the energy spectrum is changed considerably [37]. This fact directly influences the nature of electronic and optical properties in these structures. In this study we investigate the linear and nonlinear optical absorptions and the refractive index change associated with intersubband transitions within the conduction band for SQW and GQW under the magnetic field which is applied perpendicular to the growth direction. To our knowledge, this is the first study on the magnetic field dependence of the nonlinear optical properties in a SQW and GQW. 2. Theory In the effective mass approximation, the energy levels and the wave functions for electrons in a system can be obtained by solving the Schrödinger equation with a proper Hamiltonian. The onedimensional Schrödinger equation is given by
h¯ 2 d2 e2 B2 2 − ∗ 2+ z + V ( z ) ψ(z ) = E ψ(z ) ∗ 2 2m dz
2m c
(1)
where m∗ is the electron’s effective mass, e is the elementary charge, and B is the magnetic field strength applied perpendicular to the growth direction [37]. V (z ) is the confinement potential for the electron in the z-direction, and is taken as
V (z ) =
0 V
|z | ≤
0
2L0
z
V0
|z | ≤ |z | >
L0 2 L0 2 L0
for SQW for GQW
(2)
2
with V0 being the discontinuity in the conduction band edge, L0 the square and graded well width. As is known, the graded potential profiles are obtained by changing the aluminum concentration x in the Alx Ga1−x As layer. After the energies and their corresponding wave functions are obtained, the first-order refractive index changes
∆n(1) (ω) nr
and absorption coefficient (β (1) (ω)), and the third-order refractive index
314
E. Ozturk, I. Sokmen / Superlattices and Microstructures 48 (2010) 312–320
a
b
Fig. 1. Linear intersubband absorption coefficient versus the photon energy for different magnetic field values for (a) SQW and (b) GQW. The dashed curves are for B = 0.
changes
∆n(3) (ω,I )
nr
and absorption coefficient (β (3) (ω, I )) for the intersubband transitions between
two subbands can be clearly calculated as [8,38],
∆n(1) (ω) nr
=
1 2n2r ε0
×
|M21 |
2
m∗ k B T
ln Leff π h¯ 2 (E2 − E1 − h¯ ω)
1 + exp [(EF − E1 )/kB T ]
1 + exp [(EF − E2 )/kB T ]
(E2 − E1 − h¯ ω)2 + (h¯ /τin )2 (3) ∆n (ω, I ) µc 1 + exp [(EF − E1 )/kB T ] m∗ k B T = − 3 |M21 |2 ln nr 4nr ε0 1 + exp [(EF − E2 )/kB T ] Leff π h¯ 2 ×
(3)
I 2 2 2 [( " E2 − E1 − h¯ ω) + (h¯ /τin ) ]
(M22 − M11 )2 × 4 (E2 − E1 − h¯ ω) |M21 |2 − (E2 − E1 )2 + (h¯ /τin )2 )# ( (E2 − E1 − h¯ ω) (E2 − E1 )(E2 − E1 − h¯ ω) − (h¯ /τin )2 × − (h¯ /τin )2 [2 (E2 − E1 ) − h¯ ω] ωµc m∗ k B T 1 + exp [(EF − E1 )/kB T ] h¯ /τin |M21 |2 ln β (1) (ω) = nr 1 + exp [(EF − E2 )/kB T ] (E2 − E1 − h¯ ω)2 + (h¯ /τin )2 Leff π h¯ 2 ωµ I 1 + exp [(EF − E1 )/kB T ] m∗ k B T |M21 |2 β (3) (ω, I ) = − ln nr 2ε0 nr 1 + exp [(EF − E2 )/kB T ] Leff π h¯ 2 h¯ /τin × 2 (E2 − E1 − h¯ ω)2 + (h¯ /τin )2 " # |M22 − M11 |2 (E2 − E1 − h¯ ω)2 − (h¯ /τin )2 + 2(E2 − E1 )(E2 − E1 − h¯ ω) 2 × 4 |M21 | − . (E2 − E1 )2 + (h¯ /τin )2
(4)
(5)
(6)
Here I is the optical intensity of incident electromagnetic wave (with the angular frequency ω) that excites the structure and leads to the intersubband optical transition, µ is the permeability, T is the temperature, ω is the angular frequency of the incident photon, EF represents the Fermi energy (we use EF = 6.49 meV which corresponds to about 1.6 × 1017 cm−3 electrons), E1 and E2 denote the quantized energy levels for the initial and final states, respectively, kB is the Boltzmann constant, c is
E. Ozturk, I. Sokmen / Superlattices and Microstructures 48 (2010) 312–320
a
315
b
c
Fig. 2. Total intersubband absorption coefficient versus the photon energy for different optical intensities I = 0, 0.3, 0.6, 0.9, 1.2 MW/cm2 (from top to below) for SQW. The insets show the change of the potential profile and the energies with their squared envelope wave functions under an external magnetic field. (a) B = 0, (b) B = 15 T, (c) B = 30 T.
the speed of light in free space, Leff is the effective spatial extent of electrons, nr is the refractive index, τin is the intersubband relaxation time (where τin is a constant and used the numerical value of 0.14 ps following Ref. [38]), and the dipole matrix element is defined by
Z Mfi =
φf∗ (z ) |e| z φi (z )dz ,
(i, f = 1, 2).
(7)
The total refraction index change is given by
∆n(ω, I ) nr
=
∆n(1) (ω) nr
+
∆n(3) (ω, I ) nr
(8)
and the total absorption coefficient can be written as
β(ω, I ) = β (1) (ω) + β (3) (ω, I ).
(9)
316
E. Ozturk, I. Sokmen / Superlattices and Microstructures 48 (2010) 312–320
a
b
c
Fig. 3. Total intersubband absorption coefficient versus the photon energy for different optical intensities I = 0, 0.3, 0.6, 0.9, 1.2 MW/cm2 (from top to below) for GQW. The insets show the change of the potential profile and the energies with their squared envelope wave functions under an external magnetic field. (a) B = 0, (b) B = 15 T, (c) B = 30 T.
3. Results and discussion We have theoretically investigated the linear and nonlinear intersubband optical absorption coefficient and refractive index change for the (1–2) transition in SQW and GQW under an applied magnetic field, at T = 300 K. In this study, for numerical calculations, we have taken m∗ = 0.0665m0 (where m0 is the free electron mass), the barrier height V0 = 228 meV, the well width L0 = 110 Å, and nr = 3.2. To show clearly the magnetic field effect on the intersubband optical absorption, in Fig. 1(a) and (b) for different magnetic field values we plot the variation of the linear absorption coefficient as a function of the photon energy for the (1–2) intersubband transition, for SQW and GQW, respectively. While the electron wave function in a SQW has a symmetric character, in the GQW the electron wave function has an asymmetric character. Also, due to the asymmetric character of the electron wave function in the GQW, the carrier confinement of GQW structure is smaller compared to the results of the SQW, and for GQW the magnitude of the absorption coefficient becomes smaller than for SQW. It should be pointed out that the particle is mostly confined on the left-hand side of the GQW whereas in the SQW it moves freely in the whole well. Consequently, the subband energy levels in
E. Ozturk, I. Sokmen / Superlattices and Microstructures 48 (2010) 312–320
a
317
b
Fig. 4. Linear refractive index change as a function of photon energy for different magnetic field values for (a) SQW and (b) GQW. The dashed curves are for B = 0.
GQW structure are larger than that of the SQW. By considering the variation of the energy difference, E2 − E1 , it should be pointed out that by applying the magnetic field we can obtain a blue shift in the intersubband optical transitions. The linear intersubband absorption coefficient of SQW and GQW changes in energy and magnitude with increasing magnetic field strength over a wide range of the magnetic field’s amplitude. In Figs. 2 and 3 for SQW and GQW, respectively, we show the total absorption coefficient β(ω, I ) as a function of the photon’s energy for five different values of exciting optical intensity I, without and with an applied magnetic field. The insets display the change of the potential profile (the dashed lines) and the energies with their squared envelope wave functions (the solid curves) of the ground and the second subband under a magnetic field. There is no shift at the resonant peak position with the change of incident optical intensity. The total absorption coefficient β(ω, I ) is significantly reduced with increasing optical intensity by the negative β (3) (ω, I ) contribution. Therefore, the contribution of both the linear and the nonlinear terms should be considered in the calculation of β(ω, I ) near the resonance frequency (E2 − E1 ∼ = h¯ ω), especially at higher intensity values. These resonance frequency values are approximately 1.21 × 1014 , 1.25 × 1014 , 1.38 × 1014 s−1 for Fig. 2(a),(b),(c), and are approximately 1.11 × 1014 , 1.17 × 1014 , 1.32 × 1014 s−1 for Fig. 3(a),(b),(c), respectively. The resonant peak of the total absorption coefficient can be bleached at sufficiently high incident optical intensities. In our study, the bleaching begins approximately at the incident optical intensity value of 0.8 MW/cm2 for SQW and GQW. The bleaching effect can be clearly observed when I = 0.9 MW/cm2 , the resonant peak is significantly split up into two peaks due to the strong bleaching effect. This is due to the negative contribution of the third-order nonlinear term. From these figures, we can conclude that the calculation of the total changes in the optical absorption coefficients using only the linear term may not be correct for systems, because these calculations strongly depends on the third-order nonlinear term β (3) (ω, I ), especially with a high optical intensity. This effect was also observed in GaAs/AlGaAs double triangular quantum wells [26] in the absence of a laser field. In our calculations, the magnitude and position of the absorption coefficient depend on the shape of potential and the magnetic field strength. Such a dependence of the exciting optical intensity on the shape of potential can be very useful for several potential device applications. As seen from insets of Figs. 2 and 3, by increasing the magnetic field the geometric confinement of the electron increases and the penetration into the potential barriers decreases. This penetration modifies the subband dispersion relations and causes a change in the overlap function between the ground and second subband. Thus, for B = 30 T the magnitude of the absorption coefficients becomes smaller than for B = 15 T. Despite the variation of the energy difference between E2 − E1 for both SQW and GQW increases with the magnetic field, the
318
E. Ozturk, I. Sokmen / Superlattices and Microstructures 48 (2010) 312–320
a
b
c
Fig. 5. Total refractive index change as a function of photon energy for different optical intensities I = 0, 0.3, 0.6, 0.9, 1.2 MW/cm2 (from top to below) for SQW. (a) B = 0, (b) B = 15 T, (c) B = 30 T.
GQW structure changes faster than SQW by increasing the magnetic field. The probabilities of finding the electrons for different magnetic field values are given in Table 1. Fig. 4(a) and (b) show the linear refractive index changes as a function of photon energy for different magnetic field values for SQW and GQW, respectively. From these figures, it can be seen that the linear refractive index change is related to the magnetic field strength and the shape of potential. As the magnetic field rises, the linear refractive index changes have been reduced in magnitude and also shifted towards higher energies. The main reason for this resonance shift is the increment in energy interval of two different electronic states between which an optical transition occurs. In Figs. 5 and 6, for different magnetic field values we have shown the total refractive index changes as a function of the photon energy for five different incident optical intensities of 0, 0.3, 0.6, 0.9 and 1.2 MW/cm2 for SQW and GQW, respectively. As can be seen from the curves, the total refractive index changes will be reduced as the incident optical intensity increases. A higher optical intensity will cause the nonlinear term to increase, (see Eq. (4)), but the linear term does not change with intensity. Because these two terms are in opposite in sign, any increase in the magnitude of nonlinear term will also increase the difference between them, but reduce the total refractive index change. Therefore, if it is desired to obtain a larger change in the refractive index, then a relatively weaker incident optical
E. Ozturk, I. Sokmen / Superlattices and Microstructures 48 (2010) 312–320
a
319
b
c
Fig. 6. Total refractive index change as a function of photon energy for different optical intensities I = 0, 0.3, 0.6, 0.9, 1.2 MW/cm2 (from top to below) for GQW. (a) B = 0, (b) B = 15 T, (c) B = 30 T.
intensity should be employed. By increasing the magnetic field the total refractive index changes have been reduced in magnitude and also shifted towards higher energies, and for GQW the magnitude of the total refractive index change is smaller than for SQW. 4. Summary The magnetic field effect on the linear and third-order nonlinear intersubband absorption coefficient and the refractive index change in SQW and GQW within the framework of the effective mass approximation have been investigated. Our results show that the linear changes in the absorption coefficient and refractive index are not related to the incident optical intensity, whereas the incident optical intensity has a great influence on the third-order nonlinear change, as is expected from theoretical expressions. Moreover, both for SQW and GQW the total absorption coefficient and refractive index change will be reduced as the incident optical intensity increases. We have shown that by increasing the magnetic field strength it can control the carrier confinement in SQW and GQW structures. We can observe that the increasing magnetic field strength changes the separation between subbands, thus the energy differences, the linear and nonlinear absorption peaks and refraction index change modifies in magnitude and position as the magnetic field value increases.
320
E. Ozturk, I. Sokmen / Superlattices and Microstructures 48 (2010) 312–320
Table 1 The probabilities of finding of the electrons as dependent on different magnetic field values in the well width (L0 = 110 Å) (with respect to the parameters which used in this paper). B (T)
The probabilities of finding of the electrons SQW
0 15 30
GQW
E1
E2
E1
E2
0.971 0.973 0.980
0.865 0.875 0.899
0.922 0.934 0.955
0.686 0.746 0.822
Thus, the modulation of the absorption coefficients and refraction index changes which can be suitable for good performance optical modulators and various infrared optical device applications can be easily obtained by tuning the magnetic field strength and the shape of the potential. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]
[35]
[36] [37] [38]
N. Kristaedter, et al., Appl. Phys. Lett. 69 (1996) 1226. E. Leobandung, L. Guo, S.Y. Chou, Appl. Phys. Lett. 67 (1995) 2338. D. Loss, D.P. Divicenzo, Phys. Rev. A 57 (1998) 120. K. Imamura, et al., Jpn. J. Appl. Phys. 34 (1995) L1445. X. Jiang, S.S. Li, M.Z. Tidrow, Physica E 5 (1999) 27. L. Zhang, H.J. Xie, Phys. Rev. B 68 (2003) 235315. İ. Karabulut, Ü. Atav, H. Şafak, Phys. Rev. B 72 (2005) 207301; İ. Karabulut, H. Şafak, M. Tomak, Solid State Commun. 135 (2005) 735. S. Ünlü, İ. Karabulut, H. Şafak, Physica E 33 (2006) 319. R. Dingle, W. Wiegman, C.H. Henry, Phys. Rev. Lett. 33 (1974) 827. A. Harwitt, J.S. Harris Jr., Appl. Phys. Lett. 50 (1987) 685. D.D. Yang, F.H. Julien, P. Boucaud, J.M. Lourtioz, R. Planel, Phot. Tech. Lett. IEEE 2 (1990) 181. A. Fenigstein, A. Fraenkel, E. Finkman, G. Bahir, S.E. Schacham, Appl. Phys. Lett. 66 (1995) 2513. L.C. West, S.J. Eglash, Appl. Phys. Lett. 46 (1985) 1156. R.J. Turton, M. Jaros, Appl. Phys. Lett. 47 (1989) 1986. F. Capasso, K. Mohammed, A.Y. Cho, Appl. Phys. Lett. 48 (1986) 478. K.W. Gossen, S.A. Lyon, Appl. Phys. Lett. 47 (1985) 289. K.K. Choi, B.F. Levine, C.G. Bethea, J. Walker, R.J. Malik, Appl. Phys. Lett. 50 (1987) 1814. D. Ahn, S.L. Chuang, Phys. Rev. B 35 (1987) 4149. D. Ahn, S.L. Chuang, Phys. Rev. B 34 (1986) 9034. R.P.G. Karunasiri, Y.J. Mii, K.L. Wang, IEEE Electron Dev. Lett. 11 (1990) 227. S. Noda, T. Uemura, T. Yamashita, A. Sasaki, J. Appl. Phys. 68 (1990) 6529. R.F. Kazarinov, R.A. Suris, Sov. Phys. Semicond. 5 (1971) 707. J. Faist, F. Capasso, D.L. Sivco, C. Sirtori, A.L. Hutchinson, A.Y. Cho, Science 264 (1994) 553. E. Ozturk, I. Sokmen, Superlattices Microstruct. 41 (2007) 36. I. Karabulut, U. Atav, H. Safak, M. Tomak, Eur. Phys. J. B 55 (2007) 283. B. Chen, K.-X. Guo, R.-Z. Wang, Z.-H. Zhang, Z.-L. Liu, Solid State Commun. 149 (2009) 310. D.A.B. Miller, D.S. Chemla, S. Schmitt-Rink, Phys. Rev. B 33 (1986) 6976. M.N. Islam, R.L. Hillman, D.A.B. Miller, D.S. Chemla, A.C. Gossard, J.H. English, Appl. Phys. Lett. 50 (1987) 1098. P.F. Yuh, K.L. Wang, IEEE J. Quantum Electron. QE-25 (1989) 1671. Y. Kan, et al., IEEE J. Quantum Electron. 23 (1987) 2167. S.L. Chuang, D. Ahn, J. Appl. Phys. 65 (1989) 2822. J.J. Shi, S.H. Pan, Superlattices Microstruct. 17 (1995) 91. L.M. Burileanu, E.C. Niculescu, N. Eseanu, A. Radu, Physica E 41 (2009) 856. E. Kasapoglu, I. Sokmen, Physica B 403 (2008) 3746; E. Kasapoglu, I. Sokmen, Phys. Lett. A 372 (2007) 56; E. Kasapoglu, H. Sari, I. Sokmen, Phys. Stat. Sol. (b) 242 (2005) 2468. E. Ozturk, H. Sari, I. Sokmen, Eur. Phys. J. Appl. Phys. 35 (2006) 1; E. Ozturk, H. Sari, I. Sokmen, Appl. Phys. A 80 (2005) 541; E. Ozturk, H. Sari, I. Sokmen, Solid State Commun. 132 (2004) 497. E. Ozturk, H. Sari, I. Sokmen, J. Phys. D: Appl.Phys. 38 (2005) 935. E. Kasapoglu, I. Sokmen, Physica E 27 (2005) 198; E. Kasapoglu, H. Sari, I. Sokmen, Superlattices Microstruct. 29 (2001) 25. D. Ahn, S.L. Chuang, IEEE J. Quantum Electron. QE-23 (1987) 2196.
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Effect of magnetic fields on the linear and nonlinear intersubband optical absorption coefficients and refractive index changes in square and graded quantum wells E. Ozturk a,∗ , I. Sokmen b a
Department of Physics, Cumhuriyet University, 58140 Sivas, Turkiye
b
Department of Physics, Dokuz Eylül University, İzmir, Turkiye
article
info
Article history: Received 27 March 2010 Received in revised form 28 May 2010 Accepted 17 June 2010 Available online 27 July 2010 Keywords: Square quantum well Graded quantum well Intersubband transition Nonlinear absorption coefficient Nonlinear refractive index change Magnetic field
abstract In this study, both the intersubband optical absorption coefficients and the refractive index changes as dependent on the magnetic field are calculated in square and graded quantum wells. Our results show that the position and the magnitude of the linear and total absorption coefficients and refractive index changes depend on the magnetic field strength and the shape of potential. The incident optical intensity has a great effect on the total absorption and refractive index changes. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction The studies on quantum heterostructures open a new field in fundamental physics, and also offer a wide range of potential applications for optoelectronic devices. Some of these applications include semiconductor lasers [1], single-electron transistors [2], quantum computing [3], optical memories [4] and infrared photodetectors [5]. There is currently considerable interest in the optical phenomena based on the intersubband transitions in semiconductor quantum heterostructures. Due to the generally larger values of dipole matrix elements and the possibility of achieving resonance conditions, both the linear and nonlinear optical processes in these structures have been widely investigated [6–12]. A number of device applications based on the intersubband transition,
∗
Corresponding author. E-mail address: [email protected] (E. Ozturk).
0749-6036/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2010.06.015
E. Ozturk, I. Sokmen / Superlattices and Microstructures 48 (2010) 312–320
313
for example, far-infrared photo-detectors [13–17], electro-optical modulators [18–20], all optical switch [21], and infrared lasers [22,23], has been proposed and realized. Theoretical studies show that the optical properties of quantum wells mainly depend on the asymmetry of the confining potential. Such an asymmetry in the potential profile can be obtained either by applying an electric field to a symmetric quantum well or by compositionally grading the QW [24–26]. In asymmetric quantum-well structures, the changes in the absorption coefficients were predicted theoretically and confirmed experimentally to be larger than the changes that occur in conventional square QWs [27–29]. Several theoretical analyses on the changes of the absorption coefficient and refractive index associated with intersubband optical transitions in single-quantum wells and multiple-quantum well structures are presented in the literature [30–34]. In our previous studies, we have investigated the effect of a laser field on the intersubband optical transitions in square quantum wells (SQW) and graded quantum wells (GQW) in the absence of an electric field [35] and under an external electric field [36]. One of the most remarkable features of 2DEG is the intersubband optical transition between the size-quantized subbands in the same band. If the quantum well structure is subjected to a magnetic field, these transitions become very interesting. The magnetic field is an important additional parameter, since it can be applied experimentally in a well-controlled way and modifies fundamentally the electronic structure. A parallel magnetic field has a small quantitative impact on the quantum well’s energy spectrum. If a magnetic field perpendicular to the quantum well plane is applied, however the energy spectrum is changed considerably [37]. This fact directly influences the nature of electronic and optical properties in these structures. In this study we investigate the linear and nonlinear optical absorptions and the refractive index change associated with intersubband transitions within the conduction band for SQW and GQW under the magnetic field which is applied perpendicular to the growth direction. To our knowledge, this is the first study on the magnetic field dependence of the nonlinear optical properties in a SQW and GQW. 2. Theory In the effective mass approximation, the energy levels and the wave functions for electrons in a system can be obtained by solving the Schrödinger equation with a proper Hamiltonian. The onedimensional Schrödinger equation is given by
h¯ 2 d2 e2 B2 2 − ∗ 2+ z + V ( z ) ψ(z ) = E ψ(z ) ∗ 2 2m dz
2m c
(1)
where m∗ is the electron’s effective mass, e is the elementary charge, and B is the magnetic field strength applied perpendicular to the growth direction [37]. V (z ) is the confinement potential for the electron in the z-direction, and is taken as
V (z ) =
0 V
|z | ≤
0
2L0
z
V0
|z | ≤ |z | >
L0 2 L0 2 L0
for SQW for GQW
(2)
2
with V0 being the discontinuity in the conduction band edge, L0 the square and graded well width. As is known, the graded potential profiles are obtained by changing the aluminum concentration x in the Alx Ga1−x As layer. After the energies and their corresponding wave functions are obtained, the first-order refractive index changes
∆n(1) (ω) nr
and absorption coefficient (β (1) (ω)), and the third-order refractive index
314
E. Ozturk, I. Sokmen / Superlattices and Microstructures 48 (2010) 312–320
a
b
Fig. 1. Linear intersubband absorption coefficient versus the photon energy for different magnetic field values for (a) SQW and (b) GQW. The dashed curves are for B = 0.
changes
∆n(3) (ω,I )
nr
and absorption coefficient (β (3) (ω, I )) for the intersubband transitions between
two subbands can be clearly calculated as [8,38],
∆n(1) (ω) nr
=
1 2n2r ε0
×
|M21 |
2
m∗ k B T
ln Leff π h¯ 2 (E2 − E1 − h¯ ω)
1 + exp [(EF − E1 )/kB T ]
1 + exp [(EF − E2 )/kB T ]
(E2 − E1 − h¯ ω)2 + (h¯ /τin )2 (3) ∆n (ω, I ) µc 1 + exp [(EF − E1 )/kB T ] m∗ k B T = − 3 |M21 |2 ln nr 4nr ε0 1 + exp [(EF − E2 )/kB T ] Leff π h¯ 2 ×
(3)
I 2 2 2 [( " E2 − E1 − h¯ ω) + (h¯ /τin ) ]
(M22 − M11 )2 × 4 (E2 − E1 − h¯ ω) |M21 |2 − (E2 − E1 )2 + (h¯ /τin )2 )# ( (E2 − E1 − h¯ ω) (E2 − E1 )(E2 − E1 − h¯ ω) − (h¯ /τin )2 × − (h¯ /τin )2 [2 (E2 − E1 ) − h¯ ω] ωµc m∗ k B T 1 + exp [(EF − E1 )/kB T ] h¯ /τin |M21 |2 ln β (1) (ω) = nr 1 + exp [(EF − E2 )/kB T ] (E2 − E1 − h¯ ω)2 + (h¯ /τin )2 Leff π h¯ 2 ωµ I 1 + exp [(EF − E1 )/kB T ] m∗ k B T |M21 |2 β (3) (ω, I ) = − ln nr 2ε0 nr 1 + exp [(EF − E2 )/kB T ] Leff π h¯ 2 h¯ /τin × 2 (E2 − E1 − h¯ ω)2 + (h¯ /τin )2 " # |M22 − M11 |2 (E2 − E1 − h¯ ω)2 − (h¯ /τin )2 + 2(E2 − E1 )(E2 − E1 − h¯ ω) 2 × 4 |M21 | − . (E2 − E1 )2 + (h¯ /τin )2
(4)
(5)
(6)
Here I is the optical intensity of incident electromagnetic wave (with the angular frequency ω) that excites the structure and leads to the intersubband optical transition, µ is the permeability, T is the temperature, ω is the angular frequency of the incident photon, EF represents the Fermi energy (we use EF = 6.49 meV which corresponds to about 1.6 × 1017 cm−3 electrons), E1 and E2 denote the quantized energy levels for the initial and final states, respectively, kB is the Boltzmann constant, c is
E. Ozturk, I. Sokmen / Superlattices and Microstructures 48 (2010) 312–320
a
315
b
c
Fig. 2. Total intersubband absorption coefficient versus the photon energy for different optical intensities I = 0, 0.3, 0.6, 0.9, 1.2 MW/cm2 (from top to below) for SQW. The insets show the change of the potential profile and the energies with their squared envelope wave functions under an external magnetic field. (a) B = 0, (b) B = 15 T, (c) B = 30 T.
the speed of light in free space, Leff is the effective spatial extent of electrons, nr is the refractive index, τin is the intersubband relaxation time (where τin is a constant and used the numerical value of 0.14 ps following Ref. [38]), and the dipole matrix element is defined by
Z Mfi =
φf∗ (z ) |e| z φi (z )dz ,
(i, f = 1, 2).
(7)
The total refraction index change is given by
∆n(ω, I ) nr
=
∆n(1) (ω) nr
+
∆n(3) (ω, I ) nr
(8)
and the total absorption coefficient can be written as
β(ω, I ) = β (1) (ω) + β (3) (ω, I ).
(9)
316
E. Ozturk, I. Sokmen / Superlattices and Microstructures 48 (2010) 312–320
a
b
c
Fig. 3. Total intersubband absorption coefficient versus the photon energy for different optical intensities I = 0, 0.3, 0.6, 0.9, 1.2 MW/cm2 (from top to below) for GQW. The insets show the change of the potential profile and the energies with their squared envelope wave functions under an external magnetic field. (a) B = 0, (b) B = 15 T, (c) B = 30 T.
3. Results and discussion We have theoretically investigated the linear and nonlinear intersubband optical absorption coefficient and refractive index change for the (1–2) transition in SQW and GQW under an applied magnetic field, at T = 300 K. In this study, for numerical calculations, we have taken m∗ = 0.0665m0 (where m0 is the free electron mass), the barrier height V0 = 228 meV, the well width L0 = 110 Å, and nr = 3.2. To show clearly the magnetic field effect on the intersubband optical absorption, in Fig. 1(a) and (b) for different magnetic field values we plot the variation of the linear absorption coefficient as a function of the photon energy for the (1–2) intersubband transition, for SQW and GQW, respectively. While the electron wave function in a SQW has a symmetric character, in the GQW the electron wave function has an asymmetric character. Also, due to the asymmetric character of the electron wave function in the GQW, the carrier confinement of GQW structure is smaller compared to the results of the SQW, and for GQW the magnitude of the absorption coefficient becomes smaller than for SQW. It should be pointed out that the particle is mostly confined on the left-hand side of the GQW whereas in the SQW it moves freely in the whole well. Consequently, the subband energy levels in
E. Ozturk, I. Sokmen / Superlattices and Microstructures 48 (2010) 312–320
a
317
b
Fig. 4. Linear refractive index change as a function of photon energy for different magnetic field values for (a) SQW and (b) GQW. The dashed curves are for B = 0.
GQW structure are larger than that of the SQW. By considering the variation of the energy difference, E2 − E1 , it should be pointed out that by applying the magnetic field we can obtain a blue shift in the intersubband optical transitions. The linear intersubband absorption coefficient of SQW and GQW changes in energy and magnitude with increasing magnetic field strength over a wide range of the magnetic field’s amplitude. In Figs. 2 and 3 for SQW and GQW, respectively, we show the total absorption coefficient β(ω, I ) as a function of the photon’s energy for five different values of exciting optical intensity I, without and with an applied magnetic field. The insets display the change of the potential profile (the dashed lines) and the energies with their squared envelope wave functions (the solid curves) of the ground and the second subband under a magnetic field. There is no shift at the resonant peak position with the change of incident optical intensity. The total absorption coefficient β(ω, I ) is significantly reduced with increasing optical intensity by the negative β (3) (ω, I ) contribution. Therefore, the contribution of both the linear and the nonlinear terms should be considered in the calculation of β(ω, I ) near the resonance frequency (E2 − E1 ∼ = h¯ ω), especially at higher intensity values. These resonance frequency values are approximately 1.21 × 1014 , 1.25 × 1014 , 1.38 × 1014 s−1 for Fig. 2(a),(b),(c), and are approximately 1.11 × 1014 , 1.17 × 1014 , 1.32 × 1014 s−1 for Fig. 3(a),(b),(c), respectively. The resonant peak of the total absorption coefficient can be bleached at sufficiently high incident optical intensities. In our study, the bleaching begins approximately at the incident optical intensity value of 0.8 MW/cm2 for SQW and GQW. The bleaching effect can be clearly observed when I = 0.9 MW/cm2 , the resonant peak is significantly split up into two peaks due to the strong bleaching effect. This is due to the negative contribution of the third-order nonlinear term. From these figures, we can conclude that the calculation of the total changes in the optical absorption coefficients using only the linear term may not be correct for systems, because these calculations strongly depends on the third-order nonlinear term β (3) (ω, I ), especially with a high optical intensity. This effect was also observed in GaAs/AlGaAs double triangular quantum wells [26] in the absence of a laser field. In our calculations, the magnitude and position of the absorption coefficient depend on the shape of potential and the magnetic field strength. Such a dependence of the exciting optical intensity on the shape of potential can be very useful for several potential device applications. As seen from insets of Figs. 2 and 3, by increasing the magnetic field the geometric confinement of the electron increases and the penetration into the potential barriers decreases. This penetration modifies the subband dispersion relations and causes a change in the overlap function between the ground and second subband. Thus, for B = 30 T the magnitude of the absorption coefficients becomes smaller than for B = 15 T. Despite the variation of the energy difference between E2 − E1 for both SQW and GQW increases with the magnetic field, the
318
E. Ozturk, I. Sokmen / Superlattices and Microstructures 48 (2010) 312–320
a
b
c
Fig. 5. Total refractive index change as a function of photon energy for different optical intensities I = 0, 0.3, 0.6, 0.9, 1.2 MW/cm2 (from top to below) for SQW. (a) B = 0, (b) B = 15 T, (c) B = 30 T.
GQW structure changes faster than SQW by increasing the magnetic field. The probabilities of finding the electrons for different magnetic field values are given in Table 1. Fig. 4(a) and (b) show the linear refractive index changes as a function of photon energy for different magnetic field values for SQW and GQW, respectively. From these figures, it can be seen that the linear refractive index change is related to the magnetic field strength and the shape of potential. As the magnetic field rises, the linear refractive index changes have been reduced in magnitude and also shifted towards higher energies. The main reason for this resonance shift is the increment in energy interval of two different electronic states between which an optical transition occurs. In Figs. 5 and 6, for different magnetic field values we have shown the total refractive index changes as a function of the photon energy for five different incident optical intensities of 0, 0.3, 0.6, 0.9 and 1.2 MW/cm2 for SQW and GQW, respectively. As can be seen from the curves, the total refractive index changes will be reduced as the incident optical intensity increases. A higher optical intensity will cause the nonlinear term to increase, (see Eq. (4)), but the linear term does not change with intensity. Because these two terms are in opposite in sign, any increase in the magnitude of nonlinear term will also increase the difference between them, but reduce the total refractive index change. Therefore, if it is desired to obtain a larger change in the refractive index, then a relatively weaker incident optical
E. Ozturk, I. Sokmen / Superlattices and Microstructures 48 (2010) 312–320
a
319
b
c
Fig. 6. Total refractive index change as a function of photon energy for different optical intensities I = 0, 0.3, 0.6, 0.9, 1.2 MW/cm2 (from top to below) for GQW. (a) B = 0, (b) B = 15 T, (c) B = 30 T.
intensity should be employed. By increasing the magnetic field the total refractive index changes have been reduced in magnitude and also shifted towards higher energies, and for GQW the magnitude of the total refractive index change is smaller than for SQW. 4. Summary The magnetic field effect on the linear and third-order nonlinear intersubband absorption coefficient and the refractive index change in SQW and GQW within the framework of the effective mass approximation have been investigated. Our results show that the linear changes in the absorption coefficient and refractive index are not related to the incident optical intensity, whereas the incident optical intensity has a great influence on the third-order nonlinear change, as is expected from theoretical expressions. Moreover, both for SQW and GQW the total absorption coefficient and refractive index change will be reduced as the incident optical intensity increases. We have shown that by increasing the magnetic field strength it can control the carrier confinement in SQW and GQW structures. We can observe that the increasing magnetic field strength changes the separation between subbands, thus the energy differences, the linear and nonlinear absorption peaks and refraction index change modifies in magnitude and position as the magnetic field value increases.
320
E. Ozturk, I. Sokmen / Superlattices and Microstructures 48 (2010) 312–320
Table 1 The probabilities of finding of the electrons as dependent on different magnetic field values in the well width (L0 = 110 Å) (with respect to the parameters which used in this paper). B (T)
The probabilities of finding of the electrons SQW
0 15 30
GQW
E1
E2
E1
E2
0.971 0.973 0.980
0.865 0.875 0.899
0.922 0.934 0.955
0.686 0.746 0.822
Thus, the modulation of the absorption coefficients and refraction index changes which can be suitable for good performance optical modulators and various infrared optical device applications can be easily obtained by tuning the magnetic field strength and the shape of the potential. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]
[35]
[36] [37] [38]
N. Kristaedter, et al., Appl. Phys. Lett. 69 (1996) 1226. E. Leobandung, L. Guo, S.Y. Chou, Appl. Phys. Lett. 67 (1995) 2338. D. Loss, D.P. Divicenzo, Phys. Rev. A 57 (1998) 120. K. Imamura, et al., Jpn. J. Appl. Phys. 34 (1995) L1445. X. Jiang, S.S. Li, M.Z. Tidrow, Physica E 5 (1999) 27. L. Zhang, H.J. Xie, Phys. Rev. B 68 (2003) 235315. İ. Karabulut, Ü. Atav, H. Şafak, Phys. Rev. B 72 (2005) 207301; İ. Karabulut, H. Şafak, M. Tomak, Solid State Commun. 135 (2005) 735. S. Ünlü, İ. Karabulut, H. Şafak, Physica E 33 (2006) 319. R. Dingle, W. Wiegman, C.H. Henry, Phys. Rev. Lett. 33 (1974) 827. A. Harwitt, J.S. Harris Jr., Appl. Phys. Lett. 50 (1987) 685. D.D. Yang, F.H. Julien, P. Boucaud, J.M. Lourtioz, R. Planel, Phot. Tech. Lett. IEEE 2 (1990) 181. A. Fenigstein, A. Fraenkel, E. Finkman, G. Bahir, S.E. Schacham, Appl. Phys. Lett. 66 (1995) 2513. L.C. West, S.J. Eglash, Appl. Phys. Lett. 46 (1985) 1156. R.J. Turton, M. Jaros, Appl. Phys. Lett. 47 (1989) 1986. F. Capasso, K. Mohammed, A.Y. Cho, Appl. Phys. Lett. 48 (1986) 478. K.W. Gossen, S.A. Lyon, Appl. Phys. Lett. 47 (1985) 289. K.K. Choi, B.F. Levine, C.G. Bethea, J. Walker, R.J. Malik, Appl. Phys. Lett. 50 (1987) 1814. D. Ahn, S.L. Chuang, Phys. Rev. B 35 (1987) 4149. D. Ahn, S.L. Chuang, Phys. Rev. B 34 (1986) 9034. R.P.G. Karunasiri, Y.J. Mii, K.L. Wang, IEEE Electron Dev. Lett. 11 (1990) 227. S. Noda, T. Uemura, T. Yamashita, A. Sasaki, J. Appl. Phys. 68 (1990) 6529. R.F. Kazarinov, R.A. Suris, Sov. Phys. Semicond. 5 (1971) 707. J. Faist, F. Capasso, D.L. Sivco, C. Sirtori, A.L. Hutchinson, A.Y. Cho, Science 264 (1994) 553. E. Ozturk, I. Sokmen, Superlattices Microstruct. 41 (2007) 36. I. Karabulut, U. Atav, H. Safak, M. Tomak, Eur. Phys. J. B 55 (2007) 283. B. Chen, K.-X. Guo, R.-Z. Wang, Z.-H. Zhang, Z.-L. Liu, Solid State Commun. 149 (2009) 310. D.A.B. Miller, D.S. Chemla, S. Schmitt-Rink, Phys. Rev. B 33 (1986) 6976. M.N. Islam, R.L. Hillman, D.A.B. Miller, D.S. Chemla, A.C. Gossard, J.H. English, Appl. Phys. Lett. 50 (1987) 1098. P.F. Yuh, K.L. Wang, IEEE J. Quantum Electron. QE-25 (1989) 1671. Y. Kan, et al., IEEE J. Quantum Electron. 23 (1987) 2167. S.L. Chuang, D. Ahn, J. Appl. Phys. 65 (1989) 2822. J.J. Shi, S.H. Pan, Superlattices Microstruct. 17 (1995) 91. L.M. Burileanu, E.C. Niculescu, N. Eseanu, A. Radu, Physica E 41 (2009) 856. E. Kasapoglu, I. Sokmen, Physica B 403 (2008) 3746; E. Kasapoglu, I. Sokmen, Phys. Lett. A 372 (2007) 56; E. Kasapoglu, H. Sari, I. Sokmen, Phys. Stat. Sol. (b) 242 (2005) 2468. E. Ozturk, H. Sari, I. Sokmen, Eur. Phys. J. Appl. Phys. 35 (2006) 1; E. Ozturk, H. Sari, I. Sokmen, Appl. Phys. A 80 (2005) 541; E. Ozturk, H. Sari, I. Sokmen, Solid State Commun. 132 (2004) 497. E. Ozturk, H. Sari, I. Sokmen, J. Phys. D: Appl.Phys. 38 (2005) 935. E. Kasapoglu, I. Sokmen, Physica E 27 (2005) 198; E. Kasapoglu, H. Sari, I. Sokmen, Superlattices Microstruct. 29 (2001) 25. D. Ahn, S.L. Chuang, IEEE J. Quantum Electron. QE-23 (1987) 2196.