Analysis of the effects of applying external fields and device dimensions alterations on GaAs/AlGaAs multiple quantum well slow light devices based on excitonic population oscillation Reza Kohandani, Ashkan Zandi, and Hassan Kaatuzian* Photonics Research Laboratory (PRL), Department of Electrical Engineering, Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Avenue, Tehran 15914, Iran *Corresponding author: [email protected] Received 19 November 2013; revised 8 January 2014; accepted 9 January 2014; posted 10 January 2014 (Doc. ID 201580); published 20 February 2014
This paper demonstrates the effects of applying magnetic and electric fields and physical dimensions alterations on AlGaAs/GaAs multiple quantum well (QW) slow light devices. Physical parameters include quantum well sizes and number of quantum wells. To the best of our knowledge, this is the first analysis of the effects of both applying magnetic/electric fields and physical parameters alterations and the first suggestion for matching the prefabrication and post fabrication tuning of the slow light devices based on excitonic population oscillations. The aim of our theoretical analysis is controlling the optical properties such as central frequency, bandwidth, and slow down factor (SDF) in slow light devices based on excitonic population oscillation to achieve better tuning. To reach these purposes, first we investigate the quantum well size and number of quantum wells alteration effects. Next, we analyze the effects of applying magnetic and electric fields to the multiple quantum well structure, separately. Finally, physical parameters and applied external fields are changed for measuring frequency shift and SDF for coherent population oscillation slow light devices. The results show the available central frequency shifts in about 1.6 THz at best. Also the SDF value improvement is about one order of magnitude. These results will be applicable for optical nonlinearity enhancements, all-optical signal processing, optical communications, all-optical switches, optical modulators, and variable true delays. © 2014 Optical Society of America OCIS codes: (230.5590) Quantum-well, -wire and -dot devices; (270.1670) Coherent optical effects; (320.7130) Ultrafast processes in condensed matter, including semiconductors. http://dx.doi.org/10.1364/AO.53.001228
1. Introduction
Nonlinear and quantum optics research has demonstrated exquisite control over the speed of a pulse of light as it propagates through a material system [1]. Slow-down in the speed of a light pulse (slow light) is achieved in a wide variety of media and structures [2]. Because of its potential application to photonic devices, controlling the speed of light
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in solid state materials has drawn considerable attention. Applications include all-optical switches, optical modulators, and variable true delays [3]. There are several techniques to control the speed of light, including electromagnetically induced transparency (EIT), coherent population oscillation (CPO), stimulated Brillouin scattering, and stimulated Raman scattering [3,4]. By using EIT in cold or warm gases of three-level atoms, a small group velocity as low as 8 m∕s has been observed [1]. Slow light-based on EIT requires a long dephasing time and it is harder to implement compared with CPO.
Moreover, CPO only needs a long relaxation time and occurs at room temperature [5]. There is presently huge interest in slow light and related phenomena, and their applications in semiconductors. The advantages include compactness, the ability to tune the buffer parameters, a wide range of working temperatures, and greater compatibility with optical integrated circuits [6,7]. Semiconductors that use slow light have large material dispersion [5,8]. CPO has been widely used in semiconductor quantum wells (QWs) and quantum dots (QDs) [6,7]. The binding energy of excitons in semiconductors with multiple quantum well structures has been widely investigated. There are many reports about the effect of QW parameters on the binding energy of excitons [6,9]. Furthermore, to modify and control the optical properties of slow light device, the geometry of the device and its QW parameters can be changed before fabrication. To modify optical properties of slow light devices after fabrication, applying external magnetic and electric fields may be useful. In slow light devices based on excitonic population oscillation, external magnetic and electric fields can vary the exciton energy and control some optical properties of the device. This paper investigates the effects of changing the physical parameters and applying fields simultaneously on the optical properties of slow light devices. As the size of quantum wells varies, so too does the binding energy of excitons, causing variation of center frequency. In addition, changes in the number of quantum wells causes variation of group refractive index and slow down factor (SDF). The optimized values for the number of multiple quantum wells (MQWs) and their sizes were used to investigate the effects of applied magnetic and electric fields on device properties such as frequency shift and SDF. Results illustrate that magnetic fields affect diamagnetic shifts in exciton energy levels and electric fields cause shifts in exciton energy level. These results can be utilized for tuning the significant properties of slow light device based on excitonic population oscillation. 2. Theory
This section presents semiconductor MQW slow light devices based on CPO functional theories and considers the effects of physical parameters alterations and applying external fields on optical properties of the slow light device. A. Structure of Semiconductor MQW Slow Light Device Based on CPO
The basic working principle of slow light devices based on CPO is using the sharp variation of the refractive index to decrease the group velocity vg. The group velocity of an optical pulse in a medium with the refractive index, nω, is given by [2] vg
c c ng nω ω
dnω dω
:
(1)
The variation of the refractive index in CPO comes from the population oscillation caused by the pump and probe signals applied to a two-level system. This two-level system in semiconductor quantum well structures can be the heavy-hole (HH) exciton of QWs [2]. To investigate dispersion in two-level systems, semiconductor Bloch equations are solved. Only first C-like and HH-like quantized bands are taken into considerations [10]: ∂Pex;σ μ εt −iωex − iΓ2 N ex Pex;σ − i 12 ¯ N ex;σ ; ∂t 2h ∂N ex;σ −Γ1 N ex;σ − N 0 ex;σ − Γs N ex;σ − N ex;σ¯ ∂t μ12 εt Pex;σ ; 4 Im 2h¯
(2)
(3)
where N 0 ex;σ is the respective population difference in equilibrium. The definition of parameters which are used in these equations are represented in Table 1. The above equations are valid only in low excitation regimes and they can be solved at steady state regions. This approximation eliminates the effect of electron hole plasma screening and phase space filling. To reach linear permittivity tensor εs ωs , experienced by the signal, relevant optical equation in semiconductors should be solved [11]. Refractive index, absorption, and SDF defined, respectively, as follows: p ns ωs εs ωs ; (4) As ωs 2
ωs Imns ωs ; c
Rs ωs Rens ωs ωs
∂ Rens ωs : ∂ωs
(5)
(6)
Figure 1 shows the schematic diagram of a MQW slow light device based on excitonic population oscillation, which used as this investigation slow light device model. Figure 2 shows the absorbance, real part of refractive index, and SDF, which are plotted as function of Table 1.
Symbol μ21 N ex (EID) ωex Pex Γ2 N ex Ω Γ1 Γs σ
Definitions of the Symbols
Definition Dipole momentum between 1i, 2i Effective population difference Excitation induced dephasing Excitonic frequency Interband polarization Polarization dephasing Rabi frequency Relaxation constant Spin flip constant Spin index
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Fig. 1. Slow light device with Al0.3 Ga0.7 As∕GaAs multiple quantum well in presence of pump and signal described in [5].
detuning between signal and pump. When the signal frequency is close to the pump frequency, the induced population beating leads to a dip on the absorption spectrum. Corresponding to the absorption dip, a positive-slope variation of the refractive index is induced on the spectrum of the refractive index. This positive slope variation increases the SDFand lowers the group velocity. More details, exact definition, and value of parameters can be found in [5]. As it is clear in Fig. 2, when the detuning frequency between the pump and the signal is near zero the value of dn∕dω is high. Therefore, the group refractive index increases within this frequency and the SDF reaches the higher value. B.
Quantum Well Size Variation Effects
Changes in QW size causes variation in binding energy of exciton. According to the described model in [12], binding energy of the confined exciton in a finite quantum well is given by: Eb
E0 ; − 2 L ∕2α 2 1 − 0.5e kb w 0
(7)
where E0 is the mean value of the effective Rydberg energy for the three-dimensional exciton and Lw is the well width. The behavior of the binding energy at ground state HH1-CB1, according to quantum well width, is shown in Fig. 3. The value of parameters can be found in [12]. The behavior of binding energy will be changed for the well widths below 50 Å, because of spreading the envelop function in barrier level. Accordingly, the well widths below 50 Å have not been considered in the results. The central frequency of a slow light device based on excitonic population oscillation depends on excitonic energy. Equation (8) shows the dependence of excitonic energy on binding energy: Eex Eband gap − Ebinding energy :
(8)
Figure 4 shows the simulation results of slow light device that were explained in Section 2.A for different QW sizes. In these simulations, we neglected the effects of QW size variations on exciton oscillator 1230
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Fig. 2. (a) Absorbance, (b) real part of refractive index, and (c) SDF plotted as a function of detuning between signal and pump, due to excitonic population oscillation in GaAs/AlGaAs quantum wells described in [5].
strength (EOS). Although the effective length of quantum wells is constant, it is clear from Fig. 4 that variations in QW size can shift the central frequency of the device. As seen in Fig. 4, for 135 Å well width, the frequency shift is zero. The central frequency varies directly with well width, i.e., well width decrease causes a down shift in central frequency and well width increase causes up shift in central frequency. C.
Effects of Variation in Number of Quantum Wells
According to the described model in [5], refractive p index and SDF are in relation with 1∕ Leff .
Equation (9) shows the effective length of quantum wells relation: Leff Lwell × Number of QWs:
Fig. 3. Variations of binding energy as a function of well width for GaAs∕Al0.3 Ga0.7 As MQW.
(9)
Then, variation in number of QWs will change the slope of the refractive index and the value of the SDF. Figure 5 shows the refractive index and the slow down factor as a function of detuning for different number of QWs. The well width in Fig. 5 is 135 Å. The maximum value of SDF is the maximum value of SDF for zero detuning frequency achieved in Fig. 2(c). Figure 6 shows the maximum value of SDF for different number of QWs. This figure illustrates that when the number of quantum well decreases the maximum value of SDF increases rapidly. The well width is 135 Å. D.
Effects of Applied Magnetic Field
Magnetic fields affect slow light devices with quantum well structures by a diamagnetic shift of exciton energy levels [13]. As mentioned before, the excitonic energy presents the central frequency of a slow light device based on CPO. By applying a magnetic field, the central frequency of a device will be tunable. The magnetic field is applied parallel to the growth axis of QWs and causes alteration in the exciton energy level. This high magnetic field can be generated with a polyhelix resistive magnet [14]. The diamagnetic
Fig. 4. Frequency shift of (a) absorbance, (b) real part of refractive index, and (c) SDF due to different QW sizes.
Fig. 5. Variations of (a) real part of refractive index and (b) SDF of slow light device with three different number of quantum wells as a function of detuning. The solid line is experimental results presented by Ref. [5]. 20 February 2014 / Vol. 53, No. 6 / APPLIED OPTICS
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Fig. 7. Response of the maximum value of SDF to both the simultaneous alterations of well width and applied magnetic field. Fig. 6. Maximum value of SDF as a function of number of QWs with the well width of 135 Å.
shift of a 1 s state exciton in the presence of an applied magnetic field can be obtained as follows [13]: 0
E
Z N 1s
πz πz cos e cos h L L H0
2
H 0 dze dzh dxdy;
e2 B2 2 x y2 : 8μ c2
(10)
(11)
In the above equations, μ is the reduced mass of excitons, B is the magnetic field, L is the well width, and x; y; ze ; zh are the components of the coordinate vectors. A magnetic field can increase the excitonic energy and causes up shift in the exciton frequency by E0 ∕h¯ term where h¯ is the Planck constant. E.
Effects of Applied Electric Field
Electric fields are applied to QWs perpendicularly. This external electric field can be generated with high voltage or current drive. It causes electrons and holes to begin moving against the field direction and in the field direction, respectively, which reduces the energy of electron hole pairs. The effect of this phenomenon is a Stark shift in exciton absorption [15]. When the energy of excitons reduces, exciton frequency reduces and allows tuning of the central frequency of the slow light device. The applied electrical field causes a down shift in the central frequency.
affect the SDF. The effect of QW size on EOS has been neglected. The effective length of QW is not constant. According to the theory presented in Section 2.B, variation in the size of QWs causes changes in the central frequency of the device. Applying a magnetic field to the device and changing the well width launches two phenomena in the structure. First, the binding energy will change because of the QW size variation. Second, the excitonic energy will change due to both the applied magnetic field and the QW size alterations. The first causes a down shift of central frequency for less than 135 Å QW sizes, and an up shift of central frequency for more than 135 Å sizes. The second phenomenon causes a shift in the central frequency. Therefore, the total frequency shift comes from changes in both the applied magnetic field and QW size. Figure 8 shows the central frequency shift due to the applied magnetic field and changes in QW size. As shown in Fig. 8(c), when the magnetic field is zero and QW size increases, the central frequency changes due to variation in the binding energy. Accordingly, by
3. Results and Discussion A.
Applying a Magnetic Field to Different Sized QWs
As mentioned before, the maximum value of SDF is p in relation with 1∕ Leff , where Leff is the effective length of QW and is defined by Eq. (9). Therefore, with changes in the size of QWs, Leff changes and causes rapid alterations of the SDF. An applied magnetic field has no effect on the value of SDF of CPO based slow light devices. Figure 7 shows the variations in SDF maxima with changes of well width and applied magnetic field, simultaneously. As it is clear, the magnetic field did not 1232
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Fig. 8. Central frequency shift due to variations in (a) QW size, (b) applied magnetic field, and (c) both QW size and applied magnetic field.
applying a magnetic field and QW size changes, a diamagnetic shift in exciton energy and variation in the binding energy cause further shift in the central frequency. Therefore, the central frequency of the device will be tuned by both changing the well width before fabrication and applying magnetic field after fabrication. The maximum value of down shift in central frequency is about 0.1 THz and the maximum value for up shift is about 1.2 THz. Figure 9 shows the frequency shift of absorbance, real part of refractive index, and SDF of the slow light device, which is described in Section 2.A. These Fig. 10. Variations of the maximum values of the SDF with changes in the number of QWs and applied magnetic field.
curves are plotted for specific values of well width and magnetic field that achieved the maximum up shift and down shift frequencies. The effective length of QWs is constant. The QW size alteration affects EOS and the effects are able to change the optical properties of the slow light device. These effects have been investigated in [11]. B. Applying Magnetic Field with Different Number of QWs
Variation in the number of QWs affects the real part of refractive index and SDF. Applying a magnetic field affects the shift of central frequency. Therefore, with the change in the number of QWs and the applying magnetic field, the slope of the refractive index and the value of SDF will be changed and the central frequency will be shifted. Figure 10 shows the variation of maximum values of SDF as a function of the number of QWs and applied magnetic field. As Fig. 10 demonstrates, the applied magnetic field has no effect on the maximum value of SDF; however, changes to the number of QWs cause the maximum value of SDF to change rapidly. Figure 11 shows the frequency shift due to applying a magnetic field and changes in the number of QWs simultaneously. Frequency shift is independent of the number of QWs and only the magnetic field causes up shift in central frequency. Figure 12 shows the real part of the refractive index and SDF of a slow light device with different numbers of QWs and an applied magnetic field. The solid line is the experimental result of Ref. [5] and other curves are plotted for the four corner points of the Fig. 11(c) curve,to illustrate the results of this method. Variation of SDFand refractive index slope are due to changes in the number of QWs. Frequency shift of the central frequency is caused by diamagnetic shift of exciton energy, resultant of the applied magnetic field. C. Fig. 9. Frequency shift of (a) absorbance, (b) real part of refractive index, and (c) SDF of slow light device with different QW size and applied magnetic field. The solid line is the experimental result of Ref. [5].
Applying Electric Field with Different Size QWs
QW size variations cause change in the maximum value of SDF and variation in the refractive index slope, due to alterations of the effective length of QWs. Applying an electric field changes the energy of exciton and does not affect the SDF. 20 February 2014 / Vol. 53, No. 6 / APPLIED OPTICS
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Fig. 13. Variations of the maximum value of SDF as a function of well width and an electric field.
Fig. 11. Central frequency shift due to variation in (a) number of QWs, (b) applying magnetic field, and (c) both applied magnetic field and number of QWs.
Figure 13 shows the variations of the maximum value of SDF with the changes of well width and applied electric field. Reduction of well width forces the SDF to increase rapidly.
Fig. 12. Effects of variation in number of QWs and applying a magnetic field on (a) real part of refractive index and (b) SDF of slow light device. The solid line is the experimental result of Ref. [5]. 1234
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As mentioned in Section 2.B., changing the well width allows for central frequency tuning. Also, applying an electric field causes shifts in the central frequency of the device. Changing QW size and applying an electric field affect (i) the binding energy, due to QW size alteration, which causes variation in the central frequency, and (ii) electro-absorption, due to both QW size changes and applied electric field, which shifts the central frequency. Therefore, the total frequency shift comes from both changes in the applied electric field and QW size. Figure 14 shows the frequency shift caused by an applied electric field and changing QW size. When the electric field is zero and QW size increases, the frequency shift decreases because of the variation in binding energy. An applied electric field affects exciton absorption, which causes a shift of the central frequency.
Fig. 14. Frequency shift of the central frequency due to variation in size of QWs and applying an electric field. (a) Constant electric field, (b) constant well width, and (c) variation in both electric field and well width.
Therefore, the central frequency can be tuned by changing the well width during the fabrication process and applying an electric field after the fabrication process. As the simulations show, the maximum value of down shift of the central frequency is about 1.5 THz and the maximum value of up shift is about 0.1 THz. Figure 15 shows the frequency shift of absorbance, real part of refractive index, and SDF of a slow light device, described in Section 2.A. These curves are plotted for specific values of well width and electric field that present the best possible results. The effective length of QWs is constant.
D.
Applying Electric Field with Different Number of QWs
Variation in number of QWs causes variation in real part of refractive index and SDF. The effect of applying an electric field is a frequency shift of the central frequency. Therefore, applying an electric field with a different number of QWs causes variations in the slope of refractive index, the value of SDF, and the shift of central frequency. Figure 16 shows the variation of the maximum value of the SDF as a function of well width and electric field. As Fig. 16 demonstrates, the applied electric field has no effect on the maximum value of SDF, but the maximum value of SDF varies with changes in the number of QWs. Figure 17 shows the shift of central frequency due to applying an electric field and changing the number of QWs. The frequency shift is independent of the
Fig. 16. Variations of the SDF as a function of the number of QWs with an applied electric field.
Fig. 15. Frequency shift of (a) absorbance, (b) real part of refractive index, and (c) SDF of a slow light device with different QW sizes and an applied electric field. The solid line is the experimental result of Ref. [5].
Fig. 17. Shift of central frequency caused by (a) changes in the number of QWs with constant electric field, (b) different amounts of applied electric field with a constant number of QWs, and (c) variation in both the applied electrical field and the number of QWs. 20 February 2014 / Vol. 53, No. 6 / APPLIED OPTICS
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the maximum value of SDF. Several methods have been proposed for tuning and improving the optical properties of these devices. These methods are the first suggestions for matching the prefabrication and post fabrication tuning of slow devices based on excitonic population oscillations. According to the best results, the central frequency shift was in the range of 1.6 THz and the SDF value improvement was about 1 order of magnitude. These methods can be used to overcome the bandwidth limitation of slow light devices used in photonic applications. By using these methods, the bandwidth, the slowdown factor, and the central frequency of the slow light device can be tuned to a wide range of frequencies and reach higher values of SDF. These results can be used in optical nonlinearity enhancements, all-optical signal processing applications, and optical communications. References
Fig. 18. Effects of variation in numberof QWs and applying an electric field on (a) real part of refractive index and (b) SDFof a slow light device. The solid line is the experimental result of Ref. [5].
number of QWs and only the effect of an applied electric field causes a shift in the central frequency. Figure 18 shows the absorbance spectra and SDF of a slow light device with different QW layers and an applied electric field. The curves are plotted for the four corner points of the Fig. 17(c) curve. Variations to both the SDF and refractive index slope are caused by changes in the number of QWs. The shift of central frequency comes from the applied electric field. These results allow a higher SDF value and appropriate central frequency to be achieved by changing the number of QWs during the fabrication process and applying an electric field during device functioning, respectively. 4. Conclusions
This paper theoretically analyzed the effects of physical parameters alterations and applying external fields on a AlGaAs/GaAs multiple QW semiconductor slow light device. The effects of variations of QW size, QW numbers, and applying magnetic/electric fields include changing the optical properties of the slow light device, such as the central frequency and
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1. D. J. Gauthier, A. L. Gaeta, and R. W. Boyd, “Slow light: from basics to future prospects,” Photonics Spectra 40, 44–50 (2006). 2. J. B. Khurgin and R. S. Tucker, Slow Light Science and Applications (CRC, 2009). 3. W. Yan, T. Wang, X. M. Li, and Y. J. Jin, “Electromagnetically induced transparency and theoretical slow light in semiconductor multiple quantum wells,” Appl. Phys. B 108, 515–519 (2012). 4. H. Kaatuzian, Photonics, 2nd ed. (AUT, 2009), Vol. 2, in Persian. 5. S. W. Chang, S. L. Chuang, P. C. Ku, C. J. Chang-Hasnian, P. Palinginis, and H. Wang, “Slow light using excitonic population oscillation,” Phys. Rev. B 70, 235333 (2004). 6. D. Sun and P. C. Ku, “Slow light using P-doped semiconductor heterostructures for high-bandwidth nonlinear signal processing,” J. Lightwave Technol. 26, 3811–3817 (2008). 7. C. J. Chang-Hasnian, P. C. Ku, J. Kim, and S. L. Chuang, “Variable optical buffer using slow light in semiconductor nanostructures,” Proc. IEEE 91, 1884–1897 (2003). 8. B. Pesala, Z. Y. Chen, A. V. Uskov, and C. Chang-Hasnain, “Experimental demonstration of slow and superluminal light in semiconductor optical amplifiers,” Opt. Express 14, 12968–12975 (2006). 9. M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultra-slow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999). 10. R. S. Knox, Theory of Excitons (Academic, 1963). 11. H. Kaatuzian, H. Shokri Kojori, A. Zandi, and M. Ataei, “Analysis of quantum well size alteration effects on slow light device based on excitonic population oscillation,” Opt. Quantum Electron. 45, 947–959 (2013). 12. H. Mathieu, P. Lefebvre, and P. Christol, “Simple analytical method for calculating exciton binding energies in semiconductor quantum wells,” Phys. Rev. B 46, 4092–4101 (1992). 13. M. Bugajski, W. Kuszko, and K. Regifiski, “Diamagnetic shift of exciton energy levels in GaAs-Ga1-xAlxAs quantum wells,” Solid State Commun. 60, 669–673 (1986). 14. J. C. Maan, G. Belle, A. Fasolino, M. Altarelli, and K. Ploog, “Magneto-optical determination of exciton binding energy in GaAs-Ga1-xAlxAs quantum wells,” Phys. Rev. B 30, 2253–2256 (1984). 15. D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, and C. A. Burrus, “Band-edge electro absorption in quantum well structure: the quantum confined Stark shift,” Phys. Rev. Lett. 53, 2173–2176 (1984).
This paper demonstrates the effects of applying magnetic and electric fields and physical dimensions alterations on AlGaAs/GaAs multiple quantum well (QW) slow light devices. Physical parameters include quantum well sizes and number of quantum wells. To the best of our knowledge, this is the first analysis of the effects of both applying magnetic/electric fields and physical parameters alterations and the first suggestion for matching the prefabrication and post fabrication tuning of the slow light devices based on excitonic population oscillations. The aim of our theoretical analysis is controlling the optical properties such as central frequency, bandwidth, and slow down factor (SDF) in slow light devices based on excitonic population oscillation to achieve better tuning. To reach these purposes, first we investigate the quantum well size and number of quantum wells alteration effects. Next, we analyze the effects of applying magnetic and electric fields to the multiple quantum well structure, separately. Finally, physical parameters and applied external fields are changed for measuring frequency shift and SDF for coherent population oscillation slow light devices. The results show the available central frequency shifts in about 1.6 THz at best. Also the SDF value improvement is about one order of magnitude. These results will be applicable for optical nonlinearity enhancements, all-optical signal processing, optical communications, all-optical switches, optical modulators, and variable true delays. © 2014 Optical Society of America OCIS codes: (230.5590) Quantum-well, -wire and -dot devices; (270.1670) Coherent optical effects; (320.7130) Ultrafast processes in condensed matter, including semiconductors. http://dx.doi.org/10.1364/AO.53.001228
1. Introduction
Nonlinear and quantum optics research has demonstrated exquisite control over the speed of a pulse of light as it propagates through a material system [1]. Slow-down in the speed of a light pulse (slow light) is achieved in a wide variety of media and structures [2]. Because of its potential application to photonic devices, controlling the speed of light
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in solid state materials has drawn considerable attention. Applications include all-optical switches, optical modulators, and variable true delays [3]. There are several techniques to control the speed of light, including electromagnetically induced transparency (EIT), coherent population oscillation (CPO), stimulated Brillouin scattering, and stimulated Raman scattering [3,4]. By using EIT in cold or warm gases of three-level atoms, a small group velocity as low as 8 m∕s has been observed [1]. Slow light-based on EIT requires a long dephasing time and it is harder to implement compared with CPO.
Moreover, CPO only needs a long relaxation time and occurs at room temperature [5]. There is presently huge interest in slow light and related phenomena, and their applications in semiconductors. The advantages include compactness, the ability to tune the buffer parameters, a wide range of working temperatures, and greater compatibility with optical integrated circuits [6,7]. Semiconductors that use slow light have large material dispersion [5,8]. CPO has been widely used in semiconductor quantum wells (QWs) and quantum dots (QDs) [6,7]. The binding energy of excitons in semiconductors with multiple quantum well structures has been widely investigated. There are many reports about the effect of QW parameters on the binding energy of excitons [6,9]. Furthermore, to modify and control the optical properties of slow light device, the geometry of the device and its QW parameters can be changed before fabrication. To modify optical properties of slow light devices after fabrication, applying external magnetic and electric fields may be useful. In slow light devices based on excitonic population oscillation, external magnetic and electric fields can vary the exciton energy and control some optical properties of the device. This paper investigates the effects of changing the physical parameters and applying fields simultaneously on the optical properties of slow light devices. As the size of quantum wells varies, so too does the binding energy of excitons, causing variation of center frequency. In addition, changes in the number of quantum wells causes variation of group refractive index and slow down factor (SDF). The optimized values for the number of multiple quantum wells (MQWs) and their sizes were used to investigate the effects of applied magnetic and electric fields on device properties such as frequency shift and SDF. Results illustrate that magnetic fields affect diamagnetic shifts in exciton energy levels and electric fields cause shifts in exciton energy level. These results can be utilized for tuning the significant properties of slow light device based on excitonic population oscillation. 2. Theory
This section presents semiconductor MQW slow light devices based on CPO functional theories and considers the effects of physical parameters alterations and applying external fields on optical properties of the slow light device. A. Structure of Semiconductor MQW Slow Light Device Based on CPO
The basic working principle of slow light devices based on CPO is using the sharp variation of the refractive index to decrease the group velocity vg. The group velocity of an optical pulse in a medium with the refractive index, nω, is given by [2] vg
c c ng nω ω
dnω dω
:
(1)
The variation of the refractive index in CPO comes from the population oscillation caused by the pump and probe signals applied to a two-level system. This two-level system in semiconductor quantum well structures can be the heavy-hole (HH) exciton of QWs [2]. To investigate dispersion in two-level systems, semiconductor Bloch equations are solved. Only first C-like and HH-like quantized bands are taken into considerations [10]: ∂Pex;σ μ εt −iωex − iΓ2 N ex Pex;σ − i 12 ¯ N ex;σ ; ∂t 2h ∂N ex;σ −Γ1 N ex;σ − N 0 ex;σ − Γs N ex;σ − N ex;σ¯ ∂t μ12 εt Pex;σ ; 4 Im 2h¯
(2)
(3)
where N 0 ex;σ is the respective population difference in equilibrium. The definition of parameters which are used in these equations are represented in Table 1. The above equations are valid only in low excitation regimes and they can be solved at steady state regions. This approximation eliminates the effect of electron hole plasma screening and phase space filling. To reach linear permittivity tensor εs ωs , experienced by the signal, relevant optical equation in semiconductors should be solved [11]. Refractive index, absorption, and SDF defined, respectively, as follows: p ns ωs εs ωs ; (4) As ωs 2
ωs Imns ωs ; c
Rs ωs Rens ωs ωs
∂ Rens ωs : ∂ωs
(5)
(6)
Figure 1 shows the schematic diagram of a MQW slow light device based on excitonic population oscillation, which used as this investigation slow light device model. Figure 2 shows the absorbance, real part of refractive index, and SDF, which are plotted as function of Table 1.
Symbol μ21 N ex (EID) ωex Pex Γ2 N ex Ω Γ1 Γs σ
Definitions of the Symbols
Definition Dipole momentum between 1i, 2i Effective population difference Excitation induced dephasing Excitonic frequency Interband polarization Polarization dephasing Rabi frequency Relaxation constant Spin flip constant Spin index
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Fig. 1. Slow light device with Al0.3 Ga0.7 As∕GaAs multiple quantum well in presence of pump and signal described in [5].
detuning between signal and pump. When the signal frequency is close to the pump frequency, the induced population beating leads to a dip on the absorption spectrum. Corresponding to the absorption dip, a positive-slope variation of the refractive index is induced on the spectrum of the refractive index. This positive slope variation increases the SDFand lowers the group velocity. More details, exact definition, and value of parameters can be found in [5]. As it is clear in Fig. 2, when the detuning frequency between the pump and the signal is near zero the value of dn∕dω is high. Therefore, the group refractive index increases within this frequency and the SDF reaches the higher value. B.
Quantum Well Size Variation Effects
Changes in QW size causes variation in binding energy of exciton. According to the described model in [12], binding energy of the confined exciton in a finite quantum well is given by: Eb
E0 ; − 2 L ∕2α 2 1 − 0.5e kb w 0
(7)
where E0 is the mean value of the effective Rydberg energy for the three-dimensional exciton and Lw is the well width. The behavior of the binding energy at ground state HH1-CB1, according to quantum well width, is shown in Fig. 3. The value of parameters can be found in [12]. The behavior of binding energy will be changed for the well widths below 50 Å, because of spreading the envelop function in barrier level. Accordingly, the well widths below 50 Å have not been considered in the results. The central frequency of a slow light device based on excitonic population oscillation depends on excitonic energy. Equation (8) shows the dependence of excitonic energy on binding energy: Eex Eband gap − Ebinding energy :
(8)
Figure 4 shows the simulation results of slow light device that were explained in Section 2.A for different QW sizes. In these simulations, we neglected the effects of QW size variations on exciton oscillator 1230
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Fig. 2. (a) Absorbance, (b) real part of refractive index, and (c) SDF plotted as a function of detuning between signal and pump, due to excitonic population oscillation in GaAs/AlGaAs quantum wells described in [5].
strength (EOS). Although the effective length of quantum wells is constant, it is clear from Fig. 4 that variations in QW size can shift the central frequency of the device. As seen in Fig. 4, for 135 Å well width, the frequency shift is zero. The central frequency varies directly with well width, i.e., well width decrease causes a down shift in central frequency and well width increase causes up shift in central frequency. C.
Effects of Variation in Number of Quantum Wells
According to the described model in [5], refractive p index and SDF are in relation with 1∕ Leff .
Equation (9) shows the effective length of quantum wells relation: Leff Lwell × Number of QWs:
Fig. 3. Variations of binding energy as a function of well width for GaAs∕Al0.3 Ga0.7 As MQW.
(9)
Then, variation in number of QWs will change the slope of the refractive index and the value of the SDF. Figure 5 shows the refractive index and the slow down factor as a function of detuning for different number of QWs. The well width in Fig. 5 is 135 Å. The maximum value of SDF is the maximum value of SDF for zero detuning frequency achieved in Fig. 2(c). Figure 6 shows the maximum value of SDF for different number of QWs. This figure illustrates that when the number of quantum well decreases the maximum value of SDF increases rapidly. The well width is 135 Å. D.
Effects of Applied Magnetic Field
Magnetic fields affect slow light devices with quantum well structures by a diamagnetic shift of exciton energy levels [13]. As mentioned before, the excitonic energy presents the central frequency of a slow light device based on CPO. By applying a magnetic field, the central frequency of a device will be tunable. The magnetic field is applied parallel to the growth axis of QWs and causes alteration in the exciton energy level. This high magnetic field can be generated with a polyhelix resistive magnet [14]. The diamagnetic
Fig. 4. Frequency shift of (a) absorbance, (b) real part of refractive index, and (c) SDF due to different QW sizes.
Fig. 5. Variations of (a) real part of refractive index and (b) SDF of slow light device with three different number of quantum wells as a function of detuning. The solid line is experimental results presented by Ref. [5]. 20 February 2014 / Vol. 53, No. 6 / APPLIED OPTICS
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Fig. 7. Response of the maximum value of SDF to both the simultaneous alterations of well width and applied magnetic field. Fig. 6. Maximum value of SDF as a function of number of QWs with the well width of 135 Å.
shift of a 1 s state exciton in the presence of an applied magnetic field can be obtained as follows [13]: 0
E
Z N 1s
πz πz cos e cos h L L H0
2
H 0 dze dzh dxdy;
e2 B2 2 x y2 : 8μ c2
(10)
(11)
In the above equations, μ is the reduced mass of excitons, B is the magnetic field, L is the well width, and x; y; ze ; zh are the components of the coordinate vectors. A magnetic field can increase the excitonic energy and causes up shift in the exciton frequency by E0 ∕h¯ term where h¯ is the Planck constant. E.
Effects of Applied Electric Field
Electric fields are applied to QWs perpendicularly. This external electric field can be generated with high voltage or current drive. It causes electrons and holes to begin moving against the field direction and in the field direction, respectively, which reduces the energy of electron hole pairs. The effect of this phenomenon is a Stark shift in exciton absorption [15]. When the energy of excitons reduces, exciton frequency reduces and allows tuning of the central frequency of the slow light device. The applied electrical field causes a down shift in the central frequency.
affect the SDF. The effect of QW size on EOS has been neglected. The effective length of QW is not constant. According to the theory presented in Section 2.B, variation in the size of QWs causes changes in the central frequency of the device. Applying a magnetic field to the device and changing the well width launches two phenomena in the structure. First, the binding energy will change because of the QW size variation. Second, the excitonic energy will change due to both the applied magnetic field and the QW size alterations. The first causes a down shift of central frequency for less than 135 Å QW sizes, and an up shift of central frequency for more than 135 Å sizes. The second phenomenon causes a shift in the central frequency. Therefore, the total frequency shift comes from changes in both the applied magnetic field and QW size. Figure 8 shows the central frequency shift due to the applied magnetic field and changes in QW size. As shown in Fig. 8(c), when the magnetic field is zero and QW size increases, the central frequency changes due to variation in the binding energy. Accordingly, by
3. Results and Discussion A.
Applying a Magnetic Field to Different Sized QWs
As mentioned before, the maximum value of SDF is p in relation with 1∕ Leff , where Leff is the effective length of QW and is defined by Eq. (9). Therefore, with changes in the size of QWs, Leff changes and causes rapid alterations of the SDF. An applied magnetic field has no effect on the value of SDF of CPO based slow light devices. Figure 7 shows the variations in SDF maxima with changes of well width and applied magnetic field, simultaneously. As it is clear, the magnetic field did not 1232
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Fig. 8. Central frequency shift due to variations in (a) QW size, (b) applied magnetic field, and (c) both QW size and applied magnetic field.
applying a magnetic field and QW size changes, a diamagnetic shift in exciton energy and variation in the binding energy cause further shift in the central frequency. Therefore, the central frequency of the device will be tuned by both changing the well width before fabrication and applying magnetic field after fabrication. The maximum value of down shift in central frequency is about 0.1 THz and the maximum value for up shift is about 1.2 THz. Figure 9 shows the frequency shift of absorbance, real part of refractive index, and SDF of the slow light device, which is described in Section 2.A. These Fig. 10. Variations of the maximum values of the SDF with changes in the number of QWs and applied magnetic field.
curves are plotted for specific values of well width and magnetic field that achieved the maximum up shift and down shift frequencies. The effective length of QWs is constant. The QW size alteration affects EOS and the effects are able to change the optical properties of the slow light device. These effects have been investigated in [11]. B. Applying Magnetic Field with Different Number of QWs
Variation in the number of QWs affects the real part of refractive index and SDF. Applying a magnetic field affects the shift of central frequency. Therefore, with the change in the number of QWs and the applying magnetic field, the slope of the refractive index and the value of SDF will be changed and the central frequency will be shifted. Figure 10 shows the variation of maximum values of SDF as a function of the number of QWs and applied magnetic field. As Fig. 10 demonstrates, the applied magnetic field has no effect on the maximum value of SDF; however, changes to the number of QWs cause the maximum value of SDF to change rapidly. Figure 11 shows the frequency shift due to applying a magnetic field and changes in the number of QWs simultaneously. Frequency shift is independent of the number of QWs and only the magnetic field causes up shift in central frequency. Figure 12 shows the real part of the refractive index and SDF of a slow light device with different numbers of QWs and an applied magnetic field. The solid line is the experimental result of Ref. [5] and other curves are plotted for the four corner points of the Fig. 11(c) curve,to illustrate the results of this method. Variation of SDFand refractive index slope are due to changes in the number of QWs. Frequency shift of the central frequency is caused by diamagnetic shift of exciton energy, resultant of the applied magnetic field. C. Fig. 9. Frequency shift of (a) absorbance, (b) real part of refractive index, and (c) SDF of slow light device with different QW size and applied magnetic field. The solid line is the experimental result of Ref. [5].
Applying Electric Field with Different Size QWs
QW size variations cause change in the maximum value of SDF and variation in the refractive index slope, due to alterations of the effective length of QWs. Applying an electric field changes the energy of exciton and does not affect the SDF. 20 February 2014 / Vol. 53, No. 6 / APPLIED OPTICS
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Fig. 13. Variations of the maximum value of SDF as a function of well width and an electric field.
Fig. 11. Central frequency shift due to variation in (a) number of QWs, (b) applying magnetic field, and (c) both applied magnetic field and number of QWs.
Figure 13 shows the variations of the maximum value of SDF with the changes of well width and applied electric field. Reduction of well width forces the SDF to increase rapidly.
Fig. 12. Effects of variation in number of QWs and applying a magnetic field on (a) real part of refractive index and (b) SDF of slow light device. The solid line is the experimental result of Ref. [5]. 1234
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As mentioned in Section 2.B., changing the well width allows for central frequency tuning. Also, applying an electric field causes shifts in the central frequency of the device. Changing QW size and applying an electric field affect (i) the binding energy, due to QW size alteration, which causes variation in the central frequency, and (ii) electro-absorption, due to both QW size changes and applied electric field, which shifts the central frequency. Therefore, the total frequency shift comes from both changes in the applied electric field and QW size. Figure 14 shows the frequency shift caused by an applied electric field and changing QW size. When the electric field is zero and QW size increases, the frequency shift decreases because of the variation in binding energy. An applied electric field affects exciton absorption, which causes a shift of the central frequency.
Fig. 14. Frequency shift of the central frequency due to variation in size of QWs and applying an electric field. (a) Constant electric field, (b) constant well width, and (c) variation in both electric field and well width.
Therefore, the central frequency can be tuned by changing the well width during the fabrication process and applying an electric field after the fabrication process. As the simulations show, the maximum value of down shift of the central frequency is about 1.5 THz and the maximum value of up shift is about 0.1 THz. Figure 15 shows the frequency shift of absorbance, real part of refractive index, and SDF of a slow light device, described in Section 2.A. These curves are plotted for specific values of well width and electric field that present the best possible results. The effective length of QWs is constant.
D.
Applying Electric Field with Different Number of QWs
Variation in number of QWs causes variation in real part of refractive index and SDF. The effect of applying an electric field is a frequency shift of the central frequency. Therefore, applying an electric field with a different number of QWs causes variations in the slope of refractive index, the value of SDF, and the shift of central frequency. Figure 16 shows the variation of the maximum value of the SDF as a function of well width and electric field. As Fig. 16 demonstrates, the applied electric field has no effect on the maximum value of SDF, but the maximum value of SDF varies with changes in the number of QWs. Figure 17 shows the shift of central frequency due to applying an electric field and changing the number of QWs. The frequency shift is independent of the
Fig. 16. Variations of the SDF as a function of the number of QWs with an applied electric field.
Fig. 15. Frequency shift of (a) absorbance, (b) real part of refractive index, and (c) SDF of a slow light device with different QW sizes and an applied electric field. The solid line is the experimental result of Ref. [5].
Fig. 17. Shift of central frequency caused by (a) changes in the number of QWs with constant electric field, (b) different amounts of applied electric field with a constant number of QWs, and (c) variation in both the applied electrical field and the number of QWs. 20 February 2014 / Vol. 53, No. 6 / APPLIED OPTICS
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the maximum value of SDF. Several methods have been proposed for tuning and improving the optical properties of these devices. These methods are the first suggestions for matching the prefabrication and post fabrication tuning of slow devices based on excitonic population oscillations. According to the best results, the central frequency shift was in the range of 1.6 THz and the SDF value improvement was about 1 order of magnitude. These methods can be used to overcome the bandwidth limitation of slow light devices used in photonic applications. By using these methods, the bandwidth, the slowdown factor, and the central frequency of the slow light device can be tuned to a wide range of frequencies and reach higher values of SDF. These results can be used in optical nonlinearity enhancements, all-optical signal processing applications, and optical communications. References
Fig. 18. Effects of variation in numberof QWs and applying an electric field on (a) real part of refractive index and (b) SDFof a slow light device. The solid line is the experimental result of Ref. [5].
number of QWs and only the effect of an applied electric field causes a shift in the central frequency. Figure 18 shows the absorbance spectra and SDF of a slow light device with different QW layers and an applied electric field. The curves are plotted for the four corner points of the Fig. 17(c) curve. Variations to both the SDF and refractive index slope are caused by changes in the number of QWs. The shift of central frequency comes from the applied electric field. These results allow a higher SDF value and appropriate central frequency to be achieved by changing the number of QWs during the fabrication process and applying an electric field during device functioning, respectively. 4. Conclusions
This paper theoretically analyzed the effects of physical parameters alterations and applying external fields on a AlGaAs/GaAs multiple QW semiconductor slow light device. The effects of variations of QW size, QW numbers, and applying magnetic/electric fields include changing the optical properties of the slow light device, such as the central frequency and
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