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Nonlinear optical properties of doped quantum dots: Interplay between noise and carrier density
Optical Materials 69 (2017) 352e357
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Nonlinear optical properties of doped quantum dots: Interplay between noise and carrier density Aindrila Bera, Anuja Ghosh, Manas Ghosh* Department of Chemistry, Physical Chemistry Section, Visva Bharati University, Santiniketan, Birbhum, 731 235, West Bengal, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 13 January 2017 Received in revised form 25 March 2017 Accepted 26 April 2017 Available online 6 May 2017
Present work explores the profiles of a few nonlinear optical (NLO) properties of doped GaAs quantum dot (QD) with special emphasis on the role played by the carrier density under the aegis of noise. Noise term maintains a Gaussian white character and it has been introduced to the system via two different pathways; additive and multiplicative. A change of carrier density principally affects the peak height of the NLO properties. Incorporation of noise leads to some remarkable changes in the profiles of NLO properties during the variation of carrier density. These changes, however, depend on the pathway by which noise has been applied and also on the noise strength. The interplay between carrier density and noise produces some interesting outcomes that bear relevance in the related field of research. © 2017 Elsevier B.V. All rights reserved.
Keywords: Quantum dot Impurity Electro-optic effects Third-order nonlinear optical susceptibility Optical dielectric function Gaussian white noise
1. Introduction Low-dimensional semiconductor systems (LDSS) such as quantum wells (QWLs), quantum wires (QWRs) and quantum dots (QDs) are proven materials for exhibiting remarkable nonlinear optical (NLO) properties. The excellent NLO properties of LDSS are rigorously exploited in manufacturing technology-driven optoelectronic devices which simultaneously enriched the strength of fundamental physics [1e4]. Presence of impurity states in LDSS has earned ubiquitous recognition as they use to cause dramatic alteration of the optical and transport properties. Such alterations have inspired extensive inspection of the effects of shallow impurities on electronic states of LDSS, with special stress on exploring their NLO properties triggered by various external perturbations [5e46]. Presence of noise in LDSS can significantly alter the functioning of LDSS. Noise can result externally, or it may be intrinsic, originating from the changes in the structure of QD lattice in the neighborhood of impurity. It is therefore quite pertinent to inquire how presence of noise affects the NLO properties of impurity doped LDSS. In consequence, in the present work we investigate the
* Corresponding author. E-mail address: [email protected] (M. Ghosh). http://dx.doi.org/10.1016/j.optmat.2017.04.062 0925-3467/© 2017 Elsevier B.V. All rights reserved.
profiles of a few important NLO properties of doped LDSS in presence of noise namely electro-optical effect (EOE) [47e49], third-order nonlinear optical susceptibilities (TONOS) [50e60] and total optical dielectric function (TODF) [61]. Among these, EOE and TONOS represent second-order and third-order nonlinear processes, respectively. On the other hand, determination of TODF assumes importance since an extrapolated enquiry which stems from these TODF values would lead to realising the effective optical properties of the dot-matrix composite systems arising out of dielectric mismatch. To be precise, in the present work we contemplate on how the above NLO properties are influenced by the variation of carrier density (ss ), in presence of noise. A variation in carrier density happens to change the noise contribution, which, in turn, modifies the profiles of above NLO properties. However, since ss remains indifferent to the energy level separations of QD, no peak shift of above NLO properties can be expected following a change in the frequency of external electromagnetic radiation that impinges on QD. Only a change in the peak height can take place in such situation. Despite a rigorous literature survey we have not found any work that deals with the interplay between ss and noise and its influence on the NLO properties of LDSS. Recently we have explored the simultaneous role of hydrostatic pressure (HP) and temperature on several NLO properties of doped QD in presence of noise [62,63]. In these studies the focus was on understanding the interplay between HP, temperature and noise that can tailor different NLO
A. Bera et al. / Optical Materials 69 (2017) 352e357
properties. In the present investigation we invoke a similar model as used in our previous studies but shift our focus from HP and temperature to ss which is undoubtedly an entirely new aspect. Naturally, in the present communication we maintain similar values of the structural parameters (e.g. confinement potential, dopant potential etc.) as well as that of the external influences (electric field, magnetic field, noise etc.) as used erstwhile. To be specific, in the present work we consider a 2-d QD (GaAs) carrying a single electron in presence of a static electric field. The confinement is parabolic in the x y plane. An orthogonal magnetic field is present too as an additional confinement. Impurity, modeled by a Gaussian potential, has been doped into the QD system. Gaussian white noise has been externally applied to the system which initiates substantial disorder. There are two different pathways (modes) through which such introduction of disorder can be accomplished. These two modes are additive and multiplicative which differ from one another by the extent of interaction with the system. The investigation elucidates subtle interplay between carrier density and noise (which conspicuously depends on its mode of application) that eventually monitors the above NLO properties of doped QD.
2. Method The system Hamiltonian with impurity (H0 ) contains four terms and reads as
H0 ¼ H00 þ Vimp þ eFðx þ yÞ þ Vnoise :
(1)
In the above expression, the first, second, third and the fourth terms on the right hand side of the equation stand for impurity-free system containing single carrier electron, the impurity potential, the externally applied electric field having field strength F and noise contribution, respectively. The static electric field has been applied along x and y-directions. jej is the absolute value of electron charge. The noise term characterizes zero mean and spatially d-correlated Gaussian white noise (additive/multiplicative). In view of a lateral parabolic confinement in the x y plane and presence of a perpendicular magnetic field, H00 , under effective mass approximation, can be written as
H00 ¼
1 h e i2 1 iZV þ A þ m u20 x2 þ y2 ; 2m c 2
(2)
where m is the effective mass of the electron in QD and u0 is the harmonic confinement frequency. A is the vector potential which in Landau gauge becomes A ¼ ðBy; 0; 0Þ, where B is the magnetic field strength. In this gauge H00 can be further written as
c ð3Þ
H00
353
confinement frequency in the y-direction, uc ¼ meB c being the cyclotron frequency. Vimp represents the Gaussian impurity (dopant) potential and 2
cEOE ¼
8e3 ss
n 2 G4
i h i: Mij2 dij $h n4r 3 30 Z2 uij n 2 þ G2 $ uij þ n 2 þ G2
v2 v2 þ vx2 vy2
!
1 1 v þ m u20 x2 þ m U2 y2 iZuc y ; 2 2 vx
(5)
Pursuing Xie [52e54], in the present study we consider
n1 ¼ n2 ¼ n for simplicity. (3)
where the quantity Uð¼
(4)
In the above expression 3 0 is the vacuum permittivity, e is the absolute value of electron charge, ss is the carrier density, xþb y jj i; ði; j ¼ 1; 2Þ is the matrix elements of the dipole Mij ¼ ehji b moment, dij ¼ Mii Mjj , ji ðjj Þ are the eigenstates, nr is the static component of refractive index and uij ¼ ðEi Ej Þ=Z is the transition frequency, G ¼ 1=t is the relaxation rate with t as the relaxation time. Using similar approach as stated above, within second-order perturbation theory, TONOS corresponding to optical mixing between two incident light beams with frequencies n1 and n2 is given by Refs. [52e54].
# " 2ie4 ss Mij4 1 1 ð 2n1 þ n2 ; n1 ; n1 ; n2 Þ ¼ þ ; 3 i uij n1 þ G i n2 uij þ G 3 0 Z i uij 2n1 þ n2 þ G :½iðn2 n1 Þ þ G
Z2 ¼ 2m
2
can be expressed as Vimp ¼ V0 eg½ðxx0 Þ þðyy0 Þ . The relevant parameters belonging to this dopant potential are ðx0 ; y0 Þ, V0 and g1=2 . They represent the site of dopant incorporation, magnitude of the dopant potential, and the spatial region over which the impurity potential is dispersed, respectively. g can be given by g ¼ k3, where k is a constant and 3 is the static dielectric constant of the medium. The noise term of eqn. (1) can be generated by Box-Muller algorithm with necessary characteristics as mentioned before. The interaction of noise with system can be tuned in two distinct modes (pathways); additive and multiplicative. These two modes actually signify varied extents of system-noise interaction. The time€dinger equation has been solved numerically independent Schro by diagonalizing the Hamiltonian matrix (H0 ). The said matrix has been generated by the direct product basis of the harmonic oscillator eigenfunctions. The necessary convergence test has been performed and finally we have obtained the energy levels and wave functions. In view of determination of various NLO properties it is customary to explore interaction between a polarized monochromatic electromagnetic field of angular frequency n with an ensemble of QDs. It becomes a tacit assumption that the wavelength of progressive electromagnetic wave supersedes the QD dimension. Under this assumption, the wave maintains a nearly fixed amplitude throughout QD and electric dipole approximation appears to be quite judicious. Now, using standard density matrix approach and iterative procedure the expressions of various NLO properties can be obtained. Thus, the expression of EOE coefficient is given by Ref. [48].
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u20 þ u2c Þ represents the effective
Following Vahdani, considering optical transition between two states jj0 i and jj1 i, the linear [cð1Þ ðnÞ] and the third-order nonlinear [cð3Þ ðnÞ] electric susceptibilities can be written as [61].
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cð1Þ ðnÞ ¼
2 ss M01 E01 Zn iZG
;
(6)
and
2 2 "
2
ss M01 E~ 4M01 ð3Þ c ðnÞ ¼ : E01 Zn iZG ðE01 ZnÞ2 þ ðZGÞ2 # ðM11 M00 Þ2 : ðE01 iZGÞðE01 Zn iZGÞ
(7)
As stated by Vahdani, the linear and third-order nonlinear ODFs are related to cð1Þ ðnÞ and cð3Þ ðnÞ as follows [61]: 3
ð1Þ
ðnÞ ¼ 1 þ 4pcð1Þ ðnÞ;
(8)
and 3
ð3Þ
ðnÞ ¼ 4pcð3Þ ðnÞ:
(9)
The TODF is given by 3ð
¼ 12:4, m ¼ 0:067m0 , where m0 is the free electron mass, nr ¼ 3:2, 3 0 ¼ 8:8542 1012 Fm1 , t ¼ 0:14 ps, Zu0 ¼ 250:0 meV, F ¼ 100 KV/cm, B ¼ 20:0 T, V0 ¼ 280:0 meV and r0 ¼ 0:0 nm. The parameters are suitable for GaAs QDs. Fig. 1a shows the TODF profile as a function of incident photon energy in absence of noise for five different values of ss viz. (i) 1:0 1024 m3 , (ii) 3:0 1024 m3 , (iii) 5:0 1024 m3 , (iv) 7:0 1024 m3 and (v) 9:0 1024 m3 . The plot displays steady enhancement of TODF peak height with increase in ss but without any peak shift. It needs to be mentioned that, since a variation of ss does not affect the energy levels, absence of any peak shift is quite understandable. However, above variation also does not influence the wave functions and hence does not play any role in modifying the extent of overlap between the concerned eigenstates. Thus, the said overlap, which is the most common cause behind the magnitude of peak height of several NLO properties, is not of much significance in the present context. The increase in peak height of TODF with ss , therefore, comes out to be simply a resultant of two quantities which behave in opposite ways as ss increases. A close look at eqn. (6-10) would manifest that an increase in ss leads to the enhancement of cð1Þ ðnÞ and simultaneous depletion of cð3Þ ðnÞ. The observed behavior of TODF [which is a combination of cð1Þ ðnÞ and cð3Þ ðnÞ] with variation of ss suggests a higher contribution of cð1Þ ðnÞ in comparison with cð3Þ ðnÞ. TODF profiles in presence of noise display qualitatively similar features and therefore not presented. Effect of noise can be realized through a look at Fig. 1b which exhibits the TODF profiles as a function of incoming photon energy in absence of noise [Fig. 1b (i)] and when noise is present via
3
nÞ ¼ 3 ð1Þ ðnÞ þ 3 ð3Þ ðnÞ:
(10)
3. Results and discussion The calculations are performed using the following parameters:
Fig. 1. (a) Plot of TODF vs hn at five different ss values in absence of noise: (i) 1:0 1024 m3 , (ii) 3:0 1024 m3 , (iii) 5:0 1024 m3 , (iv) 7:0 1024 m3 and (v) 9:0 1024 m3 , (b) plot of TODF vs hn with ss ¼ 9:0 1024 m3 and z ¼ 1:0 104 : (i) under noise-free condition, (ii) in presence of additive noise and (iii) in presence of multiplicative noise, (c) plot of TODF vs ss : (i) in absence of noise, (ii) in presence of additive noise with z ¼ 1:0 106 , (iii) in presence of additive noise with z ¼ 1:0 104 , (iv) in presence of additive noise with z ¼ 1:0 102 , (v) in presence of multiplicative noise with z ¼ 1:0 106 , (vi) in presence of multiplicative noise with z ¼ 1:0 104 and (vii) in presence of multiplicative noise with z ¼ 1:0 102 .
A. Bera et al. / Optical Materials 69 (2017) 352e357
355
Fig. 2. Plots of NLO properties vs hn at five different ss values in absence of noise: (i) 1:0 1024 m3 , (ii) 3:0 1024 m3 , (iii) 5:0 1024 m3 , (iv) 7:0 1024 m3 and (v) 9:0 1024 m3 ; (a) for EOE and (b) for TONOS.
Fig. 3. Plots of NLO properties vs hn with ss ¼ 9:0 1024 m3 and z ¼ 1:0 104 : (i) under noise-free condition, (ii) in presence of additive noise and (iii) in presence of multiplicative noise; (a) for EOE and (b) for TONOS.
additive [Fig. 1b(ii)] and multiplicative [Fig. 1b(iii)] modes, respectively. In all these plots ss has been kept fixed at 9:0 1024 m3 . Moreover, the noise strength z also takes a fixed value i.e. 1:0 104 . The figure clearly shows noise-induced depletion of TODF peak height from that under noise-free situation. And the depletion becomes much more prominent in presence of multiplicative noise than its additive neighbor. Moreover, presence of noise causes prominent blue-shift of TODF peaks from noise-free condition and the blue-shift becomes particularly profound in presence of multiplicative noise. Since in all these profiles ss assumes a fixed value, it is noise (including its mode of application) which causes above depletion of TODF peak height and blue-shift of TODF peaks. It can therefore be inferred that presence of noise impedes effective overlap between the pertinent eigenstates and also enlarges the energy level separation from that of noise-free condition. However, both these functions are performed with more intensity in presence of multiplicative noise than its additive relative. Fig. 1c gives us a vivid picture of interplay between noise and carrier density. In this figure we plot TODF as a function of ss in absence of noise [Fig. 1c(i)], in presence of additive noise having low [z ¼ 1:0 106 , Fig. 1c(ii)], medium [z ¼ 1:0 104 , Fig. 1c(iii)] and high [z ¼ 1:0 102 , Fig. 1c(iv)] noise strength and also in presence of multiplicative noise of low [z ¼ 1:0 106 , Fig. 1c(v)], medium [z ¼ 1:0 104 , Fig. 1c(vi)] and high [z ¼ 1:0 102 , Fig. 1c(vii)] noise strength. In conformity with previous findings we now find steady enhancement of TODF as ss increases under all conditions. Furthermore, the noise-induced depletion of TODF (from that of noise-free situation) is again observed; and the depletion becomes more severe for multiplicative noise. The plot,
however, reveals an additional feature as we find that at a given ss , TODF decreases persistently with increase in the noise strength. This feature comes out to be a common one both for additive and multiplicative noise whatsoever. Thus, an enhancement in the noise strength appears to increasingly hinder the fruitful overlap between the concerned eigenstates, regardless of its mode of application. It can be further noted that such drop in the TODF value with increase in the noise strength is much more evident for multiplicative noise than its additive analogue. Thus, application of noise, in effect, happens to enhance the effective confinement of the system which becomes responsible for the quenching of TODF. In view of its way of application, multiplicative noise enjoys greater association with the system coordinates than additive one. Thus, over the entire range of variation of carrier density, application of multiplicative noise brings about greater enhancement of system confinement and consequent suppression of TODF than that in presence of additive analogue. We now pay our attention to the other NLO properties viz. EOE and TONOS. We want to mention here that for these two NLO properties the plots which delineate the noise-carrier density interplay [cf. (Fig. 1c)] are quite similar and therefore not presented for the brevity of the manuscript. Fig. 2(aeb) display the profiles of EOE and TONOS, respectively, as a function of incident photon energy in absence of noise for five different values of ss viz. (i) 1:0 1024 m3 , (ii) 3:0 1024 m3 , (iii) 5:0 1024 m3 , (iv) 7:0 1024 m3 and (v) 9:0 1024 m3 . Similar to what has been observed for TODF [cf. (Fig. 1a)], we again find steady enhancement of peak heights with increase in ss for these two NLO properties that are devoid of any peak shift. It needs to be noted that, although the general pattern of variation of peak heights for EOE and TONOS is quite similar to that of TODF,
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A. Bera et al. / Optical Materials 69 (2017) 352e357
there is a fundamental difference between the factors that cause such variation. Equations (4) and (5) certainly reveals that, unlike TODF, there is no competing effects which are responsible for the observed behavior of peak height with ss for EOE and TONOS. For EOE the reason behind the said behavior is quite clear and straightforward. And for TONOS [cf. eqn (5)], the observed behavior can be realized as we are plotting the absolute value of it. Similar profiles in presence of noise closely resemble those in absence of noise and therefore not provided. Fig. 3(aeb) display the profiles of EOE and TONOS, respectively, as a function of incoming photon energy in absence of noise [Fig. 3a/3b (i)] and when noise is applied via additive [Fig. 3a/3b (ii)] and multiplicative [Fig. 3a/3b (iii)] modes. In all these plots ss and z assume fixed values of 9:0 1024 m3 and 1:0 104 , respectively. Similar to the features that have been envisaged previously in case of TODF [cf. (Fig. 1b)], now also we find noise-induced depletion and blue-shift of peak heights for these two NLO properties (with respect to noise-free condition). And, as observed erstwhile, both the depletion and the shift become much more conspicuous in presence of multiplicative noise in comparison with its additive neighbor. 4. Conclusion The profiles of three NLO properties such as TODF, EOE and TONOS of doped GaAs QD have been meticulously monitored giving special emphasis on the role played by the carrier density under the supervision of noise. All the NLO properties reveal steadfast enhancement with increase in the carrier density regardless of the presence of noise. Presence of noise invariably reduces the magnitude of these NLO properties and also leads to blue-shift of their peak positions (from that under noise-free situation). However, the size of drop in the peak height and the blue-shift show close linkage with the mode of application of noise and exhibit greater prominence in presence of multiplicative noise. Over the complete range of variation of carrier density; increase in the noise strength diminishes the magnitudes of the NLO properties. The findings shed light on the interplay between carrier density and noise (taking care of its mode of application) and its consequence on these NLO properties of doped QD. The results seem to carry ample significance in the related field of research. Acknowledgements The authors A. B., A. G. and M. G. thank D. S. T-F. I. S. T (Govt. of India) and U. G. C.- S. A. P (Govt. of India) for support. References lu, H. Sari, I. So € kmen, Intense laser effects [1] C.A. Duque, E. Kasapoglu, S. S¸akirog on nonlinear optical absorption and optical rectification in single quantum wells under applied electric and magnetic field, Appl. Surf. Sci. 257 (2011) 2313e2319. lu, F. Ungan, U. Yesilgul, M.E. Mora-Ramos, C.A. Duque, E. Kasapoglu, [2] S. S¸akirog € kmen, Nonlinear optical rectification and the second and third H. Sari, I. So €schl-Teller quantum well under the intense laser harmonic generation in Po field, Phys. Lett. A 376 (2012) 1875e1880. [3] C.M. Duque, R.E. Acosta, A.L. Morales, M.E. Mora-Ramos, R.L. Restrepo, J.H. Odeja, E. Kasapoglu, C.A. Duque, Optical coefficients in a semiconductor quantum ring: electric field and donor impurity effects, Opt. Mater. 60 (2016) 148e158. [4] H. Hassanabadi, G. Liu, L. Lu, Nonlinear optical rectification and the secondharmonic generation in semi-parabolic and semi-inverse quantum wells, Solid State Commun. 152 (2012) 1761e1766. [5] N. Li, K.-X. Guo, S. Shao, G.-H. Liu, Polaron effects on the optical absorption coefficients and refractive index changes in a two-dimensional quantum pseudodot system, Opt. Mater. 34 (2012) 1459e1463. [6] S. Baskoutas, E. Paspalakis, A.F. Terzis, Effects of excitons in nonlinear optical rectification in semiparabolic quantum dots, Phys. Rev. B 74 (2006) 153306.
_ Karabulut, H. S¸afak, M. Tomak, Nonlinear optical rectification in asymmet[7] I. rical semiparabolic quantum wells, Solid State Commun. 135 (2005) 735e738. €schl-Teller quan[8] H. Yıldırım, M. Tomak, Nonlinear optical properties of a Po tum well, Phys. Rev. B 72 (2005) 115340. [9] H. Tas, M. S¸ahin, The inter-sublevel optical properties of a spherical quantum dot-quantum well with and without a donor impurity, J. Appl. Phys. 112 (2012) 053717. [10] S. Yilmaz, M. S¸ahin, Third-order nonlinear absorption spectra of an impurity in a spherical quantum dot with different confining potential, Phys. Status Solidi B 247 (2010) 371e374. _ Karabulut, Ü. Atav, H. S¸afak, M. Tomak, Linear and nonlinear intersubband [11] I. optical absorptions in an asymmetric rectangular quantum well, Eur. Phys. J. B 55 (2007) 283e288. € [12] A. Ozmen, Y. Yakar, B. Çakir, Ü. Atav, Computation of the oscillator strength and absorption coefficients for the intersubband transitions of the spherical quantum dot, Opt. Commun. 282 (2009) 3999e4004. [13] Y. Yakar, B. Çakir, Calculation of linear and nonlinear optical absorption coefficients of a spherical quantum dot with parabolic potential, Opt. Commun. 283 (2010) 1795e1800. € [14] B. Çakir, Y. Yakar, A. Ozmen, Refractive index changes and absorption coefficients in a spherical quantum dot with parabolic potential, J. Lumin 132 (2012) 2659e2664. € € [15] B. Çakir, Y. Yakar, A. Ozmen, M. Ozgür Sezer, M. S¸ahin, Linear and nonlinear optical absorption coefficients and binding energy of a spherical quantum dot, Superlatt. Microstruct. 47 (2010) 556e566. [16] Z. Zeng, C.S. Garoufalis, A.F. Terzis, S. Baskoutas, Linear and nonlinear optical properties of ZnS/ZnO core shell quantum dots: effect of shell thickness, impurity, and dielectric environment, J. Appl. Phys. 114 (2013) 023510. [17] K.M. Kumar, A.J. Peter, C.W. Lee, Optical properties of a hydrogenic impurity in a confined Zn1xCdxSe/ZnSe quantum dot, Superlatt. Microstruct. 51 (2012) 184e193. [18] R. Khordad, H. Bahramiyan, Impurity position effect on optical properties of various quantum dots, Phys. E 66 (2015) 107e115. [19] M. Kirak, S. Yilmaz, M. S¸ahin, M. Gençasian, The electric field effects on the binding energies and the nonlinear optical properties of a donor impurity in a spherical quantum dot, J. Appl. Phys. 109 (2011) 094309. [20] M. Narayanan, A. John Peter, Electric field induced exciton binding energy and its non-linear optical properties in a narrow InSb/InGaxSb1x quantum dot, Superlatt. Microstruct. 51 (2012) 486e496. [21] G. Rezaei, M.R.K. Vahdani, B. Vaseghi, Nonlinear optical properties of a hydrogenic impurity in an ellipsoidal finite potential quantum dot, Curr. Appl. Phys. 11 (2011) 176e181. [22] G. Rezaei, B. Vaseghi, R. Khordad, H.A. Kenary, Optical rectification coefficient of a two-dimensional quantum pseudodot system, Phys. E 43 (2011) 1853e1856. [23] G. Rezaei, B. Vaseghi, F. Taghizadeh, M.R.K. Vahdani, M.J. Karimi, Intersubband optical absorption coefficient changes and refractive index changes in a twodimensional quantum pseudodot system, Superlatt. Microstruct. 48 (2010) 450e457. [24] M.R.K. Vahdani, G. Rezaei, Linear and nonlinear optical properties of a hydrogenic donor in lens-shaped quantum dots, Phys. Lett. A 373 (2009) 3079e3084. [25] M.J. Karimi, G. Rezaei, Effects of external electric and magnetic fields on the linear and nonlinear intersubband optical properties of finite semi-parabolic quantum dots, Phys. B 406 (2011) 4423e4428. [26] I. Karabulut, S. Baskoutas, Linear and nonlinear optical absorption coefficients and refractive index changes in spherical quantum dots: effects of impurities, electric field, size, and optical intensity, J. Appl. Phys. 103 (2008) 073512. [27] S. Baskoutas, C.S. Garoufalis, A.F. Terzis, Linear and nonlinear optical absorption coefficients in inverse parabolic quantum wells under static external electric field, Eur. Phys. J. B 84 (2011) 241e247. [28] S. Baskoutas, E. Paspalakis, A.F. Terzis, Electronic structure and nonlinear optical rectification in a quantum dot: effects of impurities and external electric field, J. Phys. : Cond. Mat. 19 (2007) 395024. [29] W. Xie, Nonlinear optical properties of a hydrogenic donor quantum dot, Phys. Lett. A 372 (2008) 5498e5500. € kmen, Effects of applied [30] E.B. Al, F. Ungan, U. Yesilgul, E. Kasapoglu, H. Sari, I. So electric and magnetic fields on the nonlinear optical properties of asymmetric GaAs/Ga1xAlxAs double inverse parabolic quantum well, Opt. Mater. 47 (2015) 1e6. €kmen, M.E. Mora-Ramos, C.A. Duque, [31] E. Kasapoglu, F. Ungan, H. Sari, I. So Donor impurity states and related optical responses in triangular quantum dots under applied electric field, Superlatt. Microstruct. 73 (2014) 171e184. [32] K.M. Kumar, A.J. Peter, C.W. Lee, Optical absorption and refractive index change of a confined exciton in a spherical quantum dot nanostructure, Eur. Phys. J. B 84 (2011) 431e438. [33] G. Liu, K.-X. Guo, H. Hassanabadi, L. Lu, Linear and nonlinear optical properties in a disk-shaped quantum dot with a parabolic potential plus a hyperbolic potential in a static magnetic field, Phys. B 407 (2012) 3676e3682. [34] E.C. Niculescu, D. Bejan, Nonlinear optical properties of GaAs pyramidal quantum dots: effects of elliptically polarized radiation, impurity, and magnetic applied fields, Phys. E 74 (2015) 51e58. [35] R.B. Mahrsia, M. Choubani, L. Bouzaïene, H. Maaref, Nonlinear optical rectification in a vertically coupled lens-shaped InAs/GaAs quantum dots with wetting layers under hydrostatic pressure and temperature, J. Alloys Compd.
A. Bera et al. / Optical Materials 69 (2017) 352e357 671 (2016) 200e207. [36] L. Bouzaïene, R.B. Mahrsia, M. Baira, L. Sfaxi, H. Maaref, Hydrostatic pressure and temperature effects on nonlinear optical rectification in a lens shape InAs/ GaAs quantum dot, J. Lumin 135 (2013) 271e275. [37] M.G. Barseghyan, C.A. Duque, E.C. Niculescu, A. Radu, Intense laser field effects on the linear and nonlinear optical properties in a semiconductor quantum wire with triangle cross section, Superlatt. Microstruct. 66 (2014) 10e22. [38] H.M. Baghramyan, M.G. Barseghyan, A.A. Kirakosyan, R.L. Restrepo, Linear and nonlinear optical absorption coefficients in GaAs/Ga1xAlxAs concentric double quantum rings: effects of hydrostatic pressure and aluminium concentration, J. Lumin 134 (2013) 594e599. [39] A. Hakimyfard, M.G. Barseghyan, A.A. Kirakosyan, Simultaneous effects of €schlpressure and magnetic field on intersubband optical transitions in Po Teller quantum well, Phys. E 41 (2009) 1596e1599. [40] M. Zuhair, Effect of capping layer of GaAlAs on the electronic and optical properties of GaAs spherical layer quantum dot, Phys. E 47 (2013) 275e278. [41] M. Zuhair, Hydrostatic pressure and electric-field effects on the electronic and optical properties of InAs spherical layer quantum dot, Phys. E 46 (2012) 232e235. [42] H.E. Ghazi, A. Jorio, I. Zorkani, Linear and nonlinear intra-conduction band optical absorption in (In,Ga)N/GaN spherical QD under hydrostatic pressure, Opt. Commun. 331 (2014) 73e76. [43] H.E. Ghazi, A. Jorio, I. Zorkani, Pressure-dependent of linear and nonlinear optical properties of (In,Ga)N/GaN spherical QD, Superlatt. Microstruct. 71 (2014) 211e216. [44] S.A. Mohammadi, R. Khordad, G. Rezaei, Optical properties of a semispherical quantum dot placed at the center of a cubic quantum box: optical rectification, second and third-harmonic generations, Phys. E 76 (2016) 203e208. [45] C.A. Duque, N. Porras-Montenegro, Z. Barticevic, M. Pacheco, L.E. Oliveira, Effects of applied magnetic fields and hydrostatic pressure on the optical transitions in self-assembled InAs/GaAs quantum dots, J. Phys. : Condensed Matters 18 (2006) 1877. [46] M. Pacheco, Z. Barticevic, Optical response of a quantum dot superlattice under electric and magnetic fields, Phys. Rev. B 64 (2001) 033406. [47] K.-X. Guo, T.P. Das, C.-Y. Chen, Studies on the electro-optic effects of doublelayered quantum wires in magnetic fields, Phys. B 293 (2000) 11e15. [48] L. Zhang, H.-J. Xie, Electro-optic effect in a semi-parabolic quantum well with an applied electric field, Mod. Phys. Lett. B 17 (2003) 347e354. [49] Y.-B. Yu, K.-X. Guo, Exciton effects on nonlinear electro-optic effects in semiparabolic quantum wires, Phys. E 18 (2003) 492e497.
357
[50] L. Liu, J. Li, G. Xiong, Studies of the third-order nonlinear optical susceptibility for InxGa1xN/GaN cylinder quantum dots, Phys. E 25 (2005) 466e471. [51] A. Bahari, F.R. Moghadam, Quadratic electro-optic effect and electroabsorption process in CdSeZnSCdSe structure, Phys. E 44 (2012) 782e785. [52] W. Xie, Photoionization and third-order susceptibility of a neutral donor in ZnS/InP/ZnSe core/shell spherical quantum dots, Phys. B 449 (2014) 57e60. [53] W. Xie, Third-order nonlinear optical susceptibility and photoionization of an exciton in quantum dots, Superlatt. Microstruct. 56 (2013) 8e15. [54] W. Xie, Third-order nonlinear optical susceptibility of a donor in elliptical quantum dots, Superlatt. Microstruct. 53 (2013) 49e54. [55] X. Zhang, G. Xiong, X. Feng, Well width-dependent third-order optical nonlinearities of a ZnS/CdSe cylindrical quantum dot quantum well, Phys. E 33 (2006) 120e124. [56] X. Feng, G. Xiong, X. Zhang, H. Gao, Third-order nonlinear optical susceptibilities associated with intersubband transitions in CdSe/ZnS core-shell quantum dots, Phys. B 383 (2006) 207e212. [57] M. Cristea, A. Radu, E.C. Niculescu, Electric-field effect on the third-order nonlinear optical susceptibility in inverted core-shell nanodots with dielectric confinement, J. Lumin 143 (2013) 592e599. [58] Z. Zeng, C.S. Garoufalis, S. Baskoutas, Linear and nonlinear optical susceptibilities in a laterally coupled quantum-dot-quantum-ring system, Phys. Lett. A 378 (2014) 2713e2718. [59] M. Kouhi, Electric field effect on the quadratic electro optic effects and electro absorption process in GaN/AlGaN spherical quantum dot, Optik 127 (2016) 3379e3382. [60] M. Kouhi, A. Vahedi, A. Akbarzadeh, Y. Hanifehpour, S.W. Joo, Investigation of quadratic electro-optic effects and electro-absorption process in GaN/AlGaN spherical quantum dot, Nanoscale Res. Lett. 9 (2014) 131. [61] M.R.K. Vahdani, The effect of the electronic intersubband transitions of quantum dots on the linear and nonlinear optical properties of dot-matrix system, Superlatt. Microstruct. 76 (2014) 326e338. [62] J. Ganguly, S. Saha, A. Bera, M. Ghosh, Modulating optical rectification, second and third harmonic generation of doped quantum dots: interplay between hydrostatic pressure, temperature and noise, Superlatt. Microstruct. 98 (2016) 385e399. [63] J. Ganguly, S. Saha, A. Bera, M. Ghosh, Exploring electro-optic effect and thirdorder nonlinear optical susceptibility of impurity doped quantum dots: interplay between hydrostatic pressure, temperature and noise, Opt. Commun. 387 (2017) 166e173.
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Optical Materials journal homepage: www.elsevier.com/locate/optmat
Nonlinear optical properties of doped quantum dots: Interplay between noise and carrier density Aindrila Bera, Anuja Ghosh, Manas Ghosh* Department of Chemistry, Physical Chemistry Section, Visva Bharati University, Santiniketan, Birbhum, 731 235, West Bengal, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 13 January 2017 Received in revised form 25 March 2017 Accepted 26 April 2017 Available online 6 May 2017
Present work explores the profiles of a few nonlinear optical (NLO) properties of doped GaAs quantum dot (QD) with special emphasis on the role played by the carrier density under the aegis of noise. Noise term maintains a Gaussian white character and it has been introduced to the system via two different pathways; additive and multiplicative. A change of carrier density principally affects the peak height of the NLO properties. Incorporation of noise leads to some remarkable changes in the profiles of NLO properties during the variation of carrier density. These changes, however, depend on the pathway by which noise has been applied and also on the noise strength. The interplay between carrier density and noise produces some interesting outcomes that bear relevance in the related field of research. © 2017 Elsevier B.V. All rights reserved.
Keywords: Quantum dot Impurity Electro-optic effects Third-order nonlinear optical susceptibility Optical dielectric function Gaussian white noise
1. Introduction Low-dimensional semiconductor systems (LDSS) such as quantum wells (QWLs), quantum wires (QWRs) and quantum dots (QDs) are proven materials for exhibiting remarkable nonlinear optical (NLO) properties. The excellent NLO properties of LDSS are rigorously exploited in manufacturing technology-driven optoelectronic devices which simultaneously enriched the strength of fundamental physics [1e4]. Presence of impurity states in LDSS has earned ubiquitous recognition as they use to cause dramatic alteration of the optical and transport properties. Such alterations have inspired extensive inspection of the effects of shallow impurities on electronic states of LDSS, with special stress on exploring their NLO properties triggered by various external perturbations [5e46]. Presence of noise in LDSS can significantly alter the functioning of LDSS. Noise can result externally, or it may be intrinsic, originating from the changes in the structure of QD lattice in the neighborhood of impurity. It is therefore quite pertinent to inquire how presence of noise affects the NLO properties of impurity doped LDSS. In consequence, in the present work we investigate the
* Corresponding author. E-mail address: [email protected] (M. Ghosh). http://dx.doi.org/10.1016/j.optmat.2017.04.062 0925-3467/© 2017 Elsevier B.V. All rights reserved.
profiles of a few important NLO properties of doped LDSS in presence of noise namely electro-optical effect (EOE) [47e49], third-order nonlinear optical susceptibilities (TONOS) [50e60] and total optical dielectric function (TODF) [61]. Among these, EOE and TONOS represent second-order and third-order nonlinear processes, respectively. On the other hand, determination of TODF assumes importance since an extrapolated enquiry which stems from these TODF values would lead to realising the effective optical properties of the dot-matrix composite systems arising out of dielectric mismatch. To be precise, in the present work we contemplate on how the above NLO properties are influenced by the variation of carrier density (ss ), in presence of noise. A variation in carrier density happens to change the noise contribution, which, in turn, modifies the profiles of above NLO properties. However, since ss remains indifferent to the energy level separations of QD, no peak shift of above NLO properties can be expected following a change in the frequency of external electromagnetic radiation that impinges on QD. Only a change in the peak height can take place in such situation. Despite a rigorous literature survey we have not found any work that deals with the interplay between ss and noise and its influence on the NLO properties of LDSS. Recently we have explored the simultaneous role of hydrostatic pressure (HP) and temperature on several NLO properties of doped QD in presence of noise [62,63]. In these studies the focus was on understanding the interplay between HP, temperature and noise that can tailor different NLO
A. Bera et al. / Optical Materials 69 (2017) 352e357
properties. In the present investigation we invoke a similar model as used in our previous studies but shift our focus from HP and temperature to ss which is undoubtedly an entirely new aspect. Naturally, in the present communication we maintain similar values of the structural parameters (e.g. confinement potential, dopant potential etc.) as well as that of the external influences (electric field, magnetic field, noise etc.) as used erstwhile. To be specific, in the present work we consider a 2-d QD (GaAs) carrying a single electron in presence of a static electric field. The confinement is parabolic in the x y plane. An orthogonal magnetic field is present too as an additional confinement. Impurity, modeled by a Gaussian potential, has been doped into the QD system. Gaussian white noise has been externally applied to the system which initiates substantial disorder. There are two different pathways (modes) through which such introduction of disorder can be accomplished. These two modes are additive and multiplicative which differ from one another by the extent of interaction with the system. The investigation elucidates subtle interplay between carrier density and noise (which conspicuously depends on its mode of application) that eventually monitors the above NLO properties of doped QD.
2. Method The system Hamiltonian with impurity (H0 ) contains four terms and reads as
H0 ¼ H00 þ Vimp þ eFðx þ yÞ þ Vnoise :
(1)
In the above expression, the first, second, third and the fourth terms on the right hand side of the equation stand for impurity-free system containing single carrier electron, the impurity potential, the externally applied electric field having field strength F and noise contribution, respectively. The static electric field has been applied along x and y-directions. jej is the absolute value of electron charge. The noise term characterizes zero mean and spatially d-correlated Gaussian white noise (additive/multiplicative). In view of a lateral parabolic confinement in the x y plane and presence of a perpendicular magnetic field, H00 , under effective mass approximation, can be written as
H00 ¼
1 h e i2 1 iZV þ A þ m u20 x2 þ y2 ; 2m c 2
(2)
where m is the effective mass of the electron in QD and u0 is the harmonic confinement frequency. A is the vector potential which in Landau gauge becomes A ¼ ðBy; 0; 0Þ, where B is the magnetic field strength. In this gauge H00 can be further written as
c ð3Þ
H00
353
confinement frequency in the y-direction, uc ¼ meB c being the cyclotron frequency. Vimp represents the Gaussian impurity (dopant) potential and 2
cEOE ¼
8e3 ss
n 2 G4
i h i: Mij2 dij $h n4r 3 30 Z2 uij n 2 þ G2 $ uij þ n 2 þ G2
v2 v2 þ vx2 vy2
!
1 1 v þ m u20 x2 þ m U2 y2 iZuc y ; 2 2 vx
(5)
Pursuing Xie [52e54], in the present study we consider
n1 ¼ n2 ¼ n for simplicity. (3)
where the quantity Uð¼
(4)
In the above expression 3 0 is the vacuum permittivity, e is the absolute value of electron charge, ss is the carrier density, xþb y jj i; ði; j ¼ 1; 2Þ is the matrix elements of the dipole Mij ¼ ehji b moment, dij ¼ Mii Mjj , ji ðjj Þ are the eigenstates, nr is the static component of refractive index and uij ¼ ðEi Ej Þ=Z is the transition frequency, G ¼ 1=t is the relaxation rate with t as the relaxation time. Using similar approach as stated above, within second-order perturbation theory, TONOS corresponding to optical mixing between two incident light beams with frequencies n1 and n2 is given by Refs. [52e54].
# " 2ie4 ss Mij4 1 1 ð 2n1 þ n2 ; n1 ; n1 ; n2 Þ ¼ þ ; 3 i uij n1 þ G i n2 uij þ G 3 0 Z i uij 2n1 þ n2 þ G :½iðn2 n1 Þ þ G
Z2 ¼ 2m
2
can be expressed as Vimp ¼ V0 eg½ðxx0 Þ þðyy0 Þ . The relevant parameters belonging to this dopant potential are ðx0 ; y0 Þ, V0 and g1=2 . They represent the site of dopant incorporation, magnitude of the dopant potential, and the spatial region over which the impurity potential is dispersed, respectively. g can be given by g ¼ k3, where k is a constant and 3 is the static dielectric constant of the medium. The noise term of eqn. (1) can be generated by Box-Muller algorithm with necessary characteristics as mentioned before. The interaction of noise with system can be tuned in two distinct modes (pathways); additive and multiplicative. These two modes actually signify varied extents of system-noise interaction. The time€dinger equation has been solved numerically independent Schro by diagonalizing the Hamiltonian matrix (H0 ). The said matrix has been generated by the direct product basis of the harmonic oscillator eigenfunctions. The necessary convergence test has been performed and finally we have obtained the energy levels and wave functions. In view of determination of various NLO properties it is customary to explore interaction between a polarized monochromatic electromagnetic field of angular frequency n with an ensemble of QDs. It becomes a tacit assumption that the wavelength of progressive electromagnetic wave supersedes the QD dimension. Under this assumption, the wave maintains a nearly fixed amplitude throughout QD and electric dipole approximation appears to be quite judicious. Now, using standard density matrix approach and iterative procedure the expressions of various NLO properties can be obtained. Thus, the expression of EOE coefficient is given by Ref. [48].
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u20 þ u2c Þ represents the effective
Following Vahdani, considering optical transition between two states jj0 i and jj1 i, the linear [cð1Þ ðnÞ] and the third-order nonlinear [cð3Þ ðnÞ] electric susceptibilities can be written as [61].
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A. Bera et al. / Optical Materials 69 (2017) 352e357
cð1Þ ðnÞ ¼
2 ss M01 E01 Zn iZG
;
(6)
and
2 2 "
2
ss M01 E~ 4M01 ð3Þ c ðnÞ ¼ : E01 Zn iZG ðE01 ZnÞ2 þ ðZGÞ2 # ðM11 M00 Þ2 : ðE01 iZGÞðE01 Zn iZGÞ
(7)
As stated by Vahdani, the linear and third-order nonlinear ODFs are related to cð1Þ ðnÞ and cð3Þ ðnÞ as follows [61]: 3
ð1Þ
ðnÞ ¼ 1 þ 4pcð1Þ ðnÞ;
(8)
and 3
ð3Þ
ðnÞ ¼ 4pcð3Þ ðnÞ:
(9)
The TODF is given by 3ð
¼ 12:4, m ¼ 0:067m0 , where m0 is the free electron mass, nr ¼ 3:2, 3 0 ¼ 8:8542 1012 Fm1 , t ¼ 0:14 ps, Zu0 ¼ 250:0 meV, F ¼ 100 KV/cm, B ¼ 20:0 T, V0 ¼ 280:0 meV and r0 ¼ 0:0 nm. The parameters are suitable for GaAs QDs. Fig. 1a shows the TODF profile as a function of incident photon energy in absence of noise for five different values of ss viz. (i) 1:0 1024 m3 , (ii) 3:0 1024 m3 , (iii) 5:0 1024 m3 , (iv) 7:0 1024 m3 and (v) 9:0 1024 m3 . The plot displays steady enhancement of TODF peak height with increase in ss but without any peak shift. It needs to be mentioned that, since a variation of ss does not affect the energy levels, absence of any peak shift is quite understandable. However, above variation also does not influence the wave functions and hence does not play any role in modifying the extent of overlap between the concerned eigenstates. Thus, the said overlap, which is the most common cause behind the magnitude of peak height of several NLO properties, is not of much significance in the present context. The increase in peak height of TODF with ss , therefore, comes out to be simply a resultant of two quantities which behave in opposite ways as ss increases. A close look at eqn. (6-10) would manifest that an increase in ss leads to the enhancement of cð1Þ ðnÞ and simultaneous depletion of cð3Þ ðnÞ. The observed behavior of TODF [which is a combination of cð1Þ ðnÞ and cð3Þ ðnÞ] with variation of ss suggests a higher contribution of cð1Þ ðnÞ in comparison with cð3Þ ðnÞ. TODF profiles in presence of noise display qualitatively similar features and therefore not presented. Effect of noise can be realized through a look at Fig. 1b which exhibits the TODF profiles as a function of incoming photon energy in absence of noise [Fig. 1b (i)] and when noise is present via
3
nÞ ¼ 3 ð1Þ ðnÞ þ 3 ð3Þ ðnÞ:
(10)
3. Results and discussion The calculations are performed using the following parameters:
Fig. 1. (a) Plot of TODF vs hn at five different ss values in absence of noise: (i) 1:0 1024 m3 , (ii) 3:0 1024 m3 , (iii) 5:0 1024 m3 , (iv) 7:0 1024 m3 and (v) 9:0 1024 m3 , (b) plot of TODF vs hn with ss ¼ 9:0 1024 m3 and z ¼ 1:0 104 : (i) under noise-free condition, (ii) in presence of additive noise and (iii) in presence of multiplicative noise, (c) plot of TODF vs ss : (i) in absence of noise, (ii) in presence of additive noise with z ¼ 1:0 106 , (iii) in presence of additive noise with z ¼ 1:0 104 , (iv) in presence of additive noise with z ¼ 1:0 102 , (v) in presence of multiplicative noise with z ¼ 1:0 106 , (vi) in presence of multiplicative noise with z ¼ 1:0 104 and (vii) in presence of multiplicative noise with z ¼ 1:0 102 .
A. Bera et al. / Optical Materials 69 (2017) 352e357
355
Fig. 2. Plots of NLO properties vs hn at five different ss values in absence of noise: (i) 1:0 1024 m3 , (ii) 3:0 1024 m3 , (iii) 5:0 1024 m3 , (iv) 7:0 1024 m3 and (v) 9:0 1024 m3 ; (a) for EOE and (b) for TONOS.
Fig. 3. Plots of NLO properties vs hn with ss ¼ 9:0 1024 m3 and z ¼ 1:0 104 : (i) under noise-free condition, (ii) in presence of additive noise and (iii) in presence of multiplicative noise; (a) for EOE and (b) for TONOS.
additive [Fig. 1b(ii)] and multiplicative [Fig. 1b(iii)] modes, respectively. In all these plots ss has been kept fixed at 9:0 1024 m3 . Moreover, the noise strength z also takes a fixed value i.e. 1:0 104 . The figure clearly shows noise-induced depletion of TODF peak height from that under noise-free situation. And the depletion becomes much more prominent in presence of multiplicative noise than its additive neighbor. Moreover, presence of noise causes prominent blue-shift of TODF peaks from noise-free condition and the blue-shift becomes particularly profound in presence of multiplicative noise. Since in all these profiles ss assumes a fixed value, it is noise (including its mode of application) which causes above depletion of TODF peak height and blue-shift of TODF peaks. It can therefore be inferred that presence of noise impedes effective overlap between the pertinent eigenstates and also enlarges the energy level separation from that of noise-free condition. However, both these functions are performed with more intensity in presence of multiplicative noise than its additive relative. Fig. 1c gives us a vivid picture of interplay between noise and carrier density. In this figure we plot TODF as a function of ss in absence of noise [Fig. 1c(i)], in presence of additive noise having low [z ¼ 1:0 106 , Fig. 1c(ii)], medium [z ¼ 1:0 104 , Fig. 1c(iii)] and high [z ¼ 1:0 102 , Fig. 1c(iv)] noise strength and also in presence of multiplicative noise of low [z ¼ 1:0 106 , Fig. 1c(v)], medium [z ¼ 1:0 104 , Fig. 1c(vi)] and high [z ¼ 1:0 102 , Fig. 1c(vii)] noise strength. In conformity with previous findings we now find steady enhancement of TODF as ss increases under all conditions. Furthermore, the noise-induced depletion of TODF (from that of noise-free situation) is again observed; and the depletion becomes more severe for multiplicative noise. The plot,
however, reveals an additional feature as we find that at a given ss , TODF decreases persistently with increase in the noise strength. This feature comes out to be a common one both for additive and multiplicative noise whatsoever. Thus, an enhancement in the noise strength appears to increasingly hinder the fruitful overlap between the concerned eigenstates, regardless of its mode of application. It can be further noted that such drop in the TODF value with increase in the noise strength is much more evident for multiplicative noise than its additive analogue. Thus, application of noise, in effect, happens to enhance the effective confinement of the system which becomes responsible for the quenching of TODF. In view of its way of application, multiplicative noise enjoys greater association with the system coordinates than additive one. Thus, over the entire range of variation of carrier density, application of multiplicative noise brings about greater enhancement of system confinement and consequent suppression of TODF than that in presence of additive analogue. We now pay our attention to the other NLO properties viz. EOE and TONOS. We want to mention here that for these two NLO properties the plots which delineate the noise-carrier density interplay [cf. (Fig. 1c)] are quite similar and therefore not presented for the brevity of the manuscript. Fig. 2(aeb) display the profiles of EOE and TONOS, respectively, as a function of incident photon energy in absence of noise for five different values of ss viz. (i) 1:0 1024 m3 , (ii) 3:0 1024 m3 , (iii) 5:0 1024 m3 , (iv) 7:0 1024 m3 and (v) 9:0 1024 m3 . Similar to what has been observed for TODF [cf. (Fig. 1a)], we again find steady enhancement of peak heights with increase in ss for these two NLO properties that are devoid of any peak shift. It needs to be noted that, although the general pattern of variation of peak heights for EOE and TONOS is quite similar to that of TODF,
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A. Bera et al. / Optical Materials 69 (2017) 352e357
there is a fundamental difference between the factors that cause such variation. Equations (4) and (5) certainly reveals that, unlike TODF, there is no competing effects which are responsible for the observed behavior of peak height with ss for EOE and TONOS. For EOE the reason behind the said behavior is quite clear and straightforward. And for TONOS [cf. eqn (5)], the observed behavior can be realized as we are plotting the absolute value of it. Similar profiles in presence of noise closely resemble those in absence of noise and therefore not provided. Fig. 3(aeb) display the profiles of EOE and TONOS, respectively, as a function of incoming photon energy in absence of noise [Fig. 3a/3b (i)] and when noise is applied via additive [Fig. 3a/3b (ii)] and multiplicative [Fig. 3a/3b (iii)] modes. In all these plots ss and z assume fixed values of 9:0 1024 m3 and 1:0 104 , respectively. Similar to the features that have been envisaged previously in case of TODF [cf. (Fig. 1b)], now also we find noise-induced depletion and blue-shift of peak heights for these two NLO properties (with respect to noise-free condition). And, as observed erstwhile, both the depletion and the shift become much more conspicuous in presence of multiplicative noise in comparison with its additive neighbor. 4. Conclusion The profiles of three NLO properties such as TODF, EOE and TONOS of doped GaAs QD have been meticulously monitored giving special emphasis on the role played by the carrier density under the supervision of noise. All the NLO properties reveal steadfast enhancement with increase in the carrier density regardless of the presence of noise. Presence of noise invariably reduces the magnitude of these NLO properties and also leads to blue-shift of their peak positions (from that under noise-free situation). However, the size of drop in the peak height and the blue-shift show close linkage with the mode of application of noise and exhibit greater prominence in presence of multiplicative noise. Over the complete range of variation of carrier density; increase in the noise strength diminishes the magnitudes of the NLO properties. The findings shed light on the interplay between carrier density and noise (taking care of its mode of application) and its consequence on these NLO properties of doped QD. The results seem to carry ample significance in the related field of research. Acknowledgements The authors A. B., A. G. and M. G. thank D. S. T-F. I. S. T (Govt. of India) and U. G. C.- S. A. P (Govt. of India) for support. References lu, H. Sari, I. So € kmen, Intense laser effects [1] C.A. Duque, E. Kasapoglu, S. S¸akirog on nonlinear optical absorption and optical rectification in single quantum wells under applied electric and magnetic field, Appl. Surf. Sci. 257 (2011) 2313e2319. lu, F. Ungan, U. Yesilgul, M.E. Mora-Ramos, C.A. Duque, E. Kasapoglu, [2] S. S¸akirog € kmen, Nonlinear optical rectification and the second and third H. Sari, I. So €schl-Teller quantum well under the intense laser harmonic generation in Po field, Phys. Lett. A 376 (2012) 1875e1880. [3] C.M. Duque, R.E. Acosta, A.L. Morales, M.E. Mora-Ramos, R.L. Restrepo, J.H. Odeja, E. Kasapoglu, C.A. Duque, Optical coefficients in a semiconductor quantum ring: electric field and donor impurity effects, Opt. Mater. 60 (2016) 148e158. [4] H. Hassanabadi, G. Liu, L. Lu, Nonlinear optical rectification and the secondharmonic generation in semi-parabolic and semi-inverse quantum wells, Solid State Commun. 152 (2012) 1761e1766. [5] N. Li, K.-X. Guo, S. Shao, G.-H. Liu, Polaron effects on the optical absorption coefficients and refractive index changes in a two-dimensional quantum pseudodot system, Opt. Mater. 34 (2012) 1459e1463. [6] S. Baskoutas, E. Paspalakis, A.F. Terzis, Effects of excitons in nonlinear optical rectification in semiparabolic quantum dots, Phys. Rev. B 74 (2006) 153306.
_ Karabulut, H. S¸afak, M. Tomak, Nonlinear optical rectification in asymmet[7] I. rical semiparabolic quantum wells, Solid State Commun. 135 (2005) 735e738. €schl-Teller quan[8] H. Yıldırım, M. Tomak, Nonlinear optical properties of a Po tum well, Phys. Rev. B 72 (2005) 115340. [9] H. Tas, M. S¸ahin, The inter-sublevel optical properties of a spherical quantum dot-quantum well with and without a donor impurity, J. Appl. Phys. 112 (2012) 053717. [10] S. Yilmaz, M. S¸ahin, Third-order nonlinear absorption spectra of an impurity in a spherical quantum dot with different confining potential, Phys. Status Solidi B 247 (2010) 371e374. _ Karabulut, Ü. Atav, H. S¸afak, M. Tomak, Linear and nonlinear intersubband [11] I. optical absorptions in an asymmetric rectangular quantum well, Eur. Phys. J. B 55 (2007) 283e288. € [12] A. Ozmen, Y. Yakar, B. Çakir, Ü. Atav, Computation of the oscillator strength and absorption coefficients for the intersubband transitions of the spherical quantum dot, Opt. Commun. 282 (2009) 3999e4004. [13] Y. Yakar, B. Çakir, Calculation of linear and nonlinear optical absorption coefficients of a spherical quantum dot with parabolic potential, Opt. Commun. 283 (2010) 1795e1800. € [14] B. Çakir, Y. Yakar, A. Ozmen, Refractive index changes and absorption coefficients in a spherical quantum dot with parabolic potential, J. Lumin 132 (2012) 2659e2664. € € [15] B. Çakir, Y. Yakar, A. Ozmen, M. Ozgür Sezer, M. S¸ahin, Linear and nonlinear optical absorption coefficients and binding energy of a spherical quantum dot, Superlatt. Microstruct. 47 (2010) 556e566. [16] Z. Zeng, C.S. Garoufalis, A.F. Terzis, S. Baskoutas, Linear and nonlinear optical properties of ZnS/ZnO core shell quantum dots: effect of shell thickness, impurity, and dielectric environment, J. Appl. Phys. 114 (2013) 023510. [17] K.M. Kumar, A.J. Peter, C.W. Lee, Optical properties of a hydrogenic impurity in a confined Zn1xCdxSe/ZnSe quantum dot, Superlatt. Microstruct. 51 (2012) 184e193. [18] R. Khordad, H. Bahramiyan, Impurity position effect on optical properties of various quantum dots, Phys. E 66 (2015) 107e115. [19] M. Kirak, S. Yilmaz, M. S¸ahin, M. Gençasian, The electric field effects on the binding energies and the nonlinear optical properties of a donor impurity in a spherical quantum dot, J. Appl. Phys. 109 (2011) 094309. [20] M. Narayanan, A. John Peter, Electric field induced exciton binding energy and its non-linear optical properties in a narrow InSb/InGaxSb1x quantum dot, Superlatt. Microstruct. 51 (2012) 486e496. [21] G. Rezaei, M.R.K. Vahdani, B. Vaseghi, Nonlinear optical properties of a hydrogenic impurity in an ellipsoidal finite potential quantum dot, Curr. Appl. Phys. 11 (2011) 176e181. [22] G. Rezaei, B. Vaseghi, R. Khordad, H.A. Kenary, Optical rectification coefficient of a two-dimensional quantum pseudodot system, Phys. E 43 (2011) 1853e1856. [23] G. Rezaei, B. Vaseghi, F. Taghizadeh, M.R.K. Vahdani, M.J. Karimi, Intersubband optical absorption coefficient changes and refractive index changes in a twodimensional quantum pseudodot system, Superlatt. Microstruct. 48 (2010) 450e457. [24] M.R.K. Vahdani, G. Rezaei, Linear and nonlinear optical properties of a hydrogenic donor in lens-shaped quantum dots, Phys. Lett. A 373 (2009) 3079e3084. [25] M.J. Karimi, G. Rezaei, Effects of external electric and magnetic fields on the linear and nonlinear intersubband optical properties of finite semi-parabolic quantum dots, Phys. B 406 (2011) 4423e4428. [26] I. Karabulut, S. Baskoutas, Linear and nonlinear optical absorption coefficients and refractive index changes in spherical quantum dots: effects of impurities, electric field, size, and optical intensity, J. Appl. Phys. 103 (2008) 073512. [27] S. Baskoutas, C.S. Garoufalis, A.F. Terzis, Linear and nonlinear optical absorption coefficients in inverse parabolic quantum wells under static external electric field, Eur. Phys. J. B 84 (2011) 241e247. [28] S. Baskoutas, E. Paspalakis, A.F. Terzis, Electronic structure and nonlinear optical rectification in a quantum dot: effects of impurities and external electric field, J. Phys. : Cond. Mat. 19 (2007) 395024. [29] W. Xie, Nonlinear optical properties of a hydrogenic donor quantum dot, Phys. Lett. A 372 (2008) 5498e5500. € kmen, Effects of applied [30] E.B. Al, F. Ungan, U. Yesilgul, E. Kasapoglu, H. Sari, I. So electric and magnetic fields on the nonlinear optical properties of asymmetric GaAs/Ga1xAlxAs double inverse parabolic quantum well, Opt. Mater. 47 (2015) 1e6. €kmen, M.E. Mora-Ramos, C.A. Duque, [31] E. Kasapoglu, F. Ungan, H. Sari, I. So Donor impurity states and related optical responses in triangular quantum dots under applied electric field, Superlatt. Microstruct. 73 (2014) 171e184. [32] K.M. Kumar, A.J. Peter, C.W. Lee, Optical absorption and refractive index change of a confined exciton in a spherical quantum dot nanostructure, Eur. Phys. J. B 84 (2011) 431e438. [33] G. Liu, K.-X. Guo, H. Hassanabadi, L. Lu, Linear and nonlinear optical properties in a disk-shaped quantum dot with a parabolic potential plus a hyperbolic potential in a static magnetic field, Phys. B 407 (2012) 3676e3682. [34] E.C. Niculescu, D. Bejan, Nonlinear optical properties of GaAs pyramidal quantum dots: effects of elliptically polarized radiation, impurity, and magnetic applied fields, Phys. E 74 (2015) 51e58. [35] R.B. Mahrsia, M. Choubani, L. Bouzaïene, H. Maaref, Nonlinear optical rectification in a vertically coupled lens-shaped InAs/GaAs quantum dots with wetting layers under hydrostatic pressure and temperature, J. Alloys Compd.
A. Bera et al. / Optical Materials 69 (2017) 352e357 671 (2016) 200e207. [36] L. Bouzaïene, R.B. Mahrsia, M. Baira, L. Sfaxi, H. Maaref, Hydrostatic pressure and temperature effects on nonlinear optical rectification in a lens shape InAs/ GaAs quantum dot, J. Lumin 135 (2013) 271e275. [37] M.G. Barseghyan, C.A. Duque, E.C. Niculescu, A. Radu, Intense laser field effects on the linear and nonlinear optical properties in a semiconductor quantum wire with triangle cross section, Superlatt. Microstruct. 66 (2014) 10e22. [38] H.M. Baghramyan, M.G. Barseghyan, A.A. Kirakosyan, R.L. Restrepo, Linear and nonlinear optical absorption coefficients in GaAs/Ga1xAlxAs concentric double quantum rings: effects of hydrostatic pressure and aluminium concentration, J. Lumin 134 (2013) 594e599. [39] A. Hakimyfard, M.G. Barseghyan, A.A. Kirakosyan, Simultaneous effects of €schlpressure and magnetic field on intersubband optical transitions in Po Teller quantum well, Phys. E 41 (2009) 1596e1599. [40] M. Zuhair, Effect of capping layer of GaAlAs on the electronic and optical properties of GaAs spherical layer quantum dot, Phys. E 47 (2013) 275e278. [41] M. Zuhair, Hydrostatic pressure and electric-field effects on the electronic and optical properties of InAs spherical layer quantum dot, Phys. E 46 (2012) 232e235. [42] H.E. Ghazi, A. Jorio, I. Zorkani, Linear and nonlinear intra-conduction band optical absorption in (In,Ga)N/GaN spherical QD under hydrostatic pressure, Opt. Commun. 331 (2014) 73e76. [43] H.E. Ghazi, A. Jorio, I. Zorkani, Pressure-dependent of linear and nonlinear optical properties of (In,Ga)N/GaN spherical QD, Superlatt. Microstruct. 71 (2014) 211e216. [44] S.A. Mohammadi, R. Khordad, G. Rezaei, Optical properties of a semispherical quantum dot placed at the center of a cubic quantum box: optical rectification, second and third-harmonic generations, Phys. E 76 (2016) 203e208. [45] C.A. Duque, N. Porras-Montenegro, Z. Barticevic, M. Pacheco, L.E. Oliveira, Effects of applied magnetic fields and hydrostatic pressure on the optical transitions in self-assembled InAs/GaAs quantum dots, J. Phys. : Condensed Matters 18 (2006) 1877. [46] M. Pacheco, Z. Barticevic, Optical response of a quantum dot superlattice under electric and magnetic fields, Phys. Rev. B 64 (2001) 033406. [47] K.-X. Guo, T.P. Das, C.-Y. Chen, Studies on the electro-optic effects of doublelayered quantum wires in magnetic fields, Phys. B 293 (2000) 11e15. [48] L. Zhang, H.-J. Xie, Electro-optic effect in a semi-parabolic quantum well with an applied electric field, Mod. Phys. Lett. B 17 (2003) 347e354. [49] Y.-B. Yu, K.-X. Guo, Exciton effects on nonlinear electro-optic effects in semiparabolic quantum wires, Phys. E 18 (2003) 492e497.
357
[50] L. Liu, J. Li, G. Xiong, Studies of the third-order nonlinear optical susceptibility for InxGa1xN/GaN cylinder quantum dots, Phys. E 25 (2005) 466e471. [51] A. Bahari, F.R. Moghadam, Quadratic electro-optic effect and electroabsorption process in CdSeZnSCdSe structure, Phys. E 44 (2012) 782e785. [52] W. Xie, Photoionization and third-order susceptibility of a neutral donor in ZnS/InP/ZnSe core/shell spherical quantum dots, Phys. B 449 (2014) 57e60. [53] W. Xie, Third-order nonlinear optical susceptibility and photoionization of an exciton in quantum dots, Superlatt. Microstruct. 56 (2013) 8e15. [54] W. Xie, Third-order nonlinear optical susceptibility of a donor in elliptical quantum dots, Superlatt. Microstruct. 53 (2013) 49e54. [55] X. Zhang, G. Xiong, X. Feng, Well width-dependent third-order optical nonlinearities of a ZnS/CdSe cylindrical quantum dot quantum well, Phys. E 33 (2006) 120e124. [56] X. Feng, G. Xiong, X. Zhang, H. Gao, Third-order nonlinear optical susceptibilities associated with intersubband transitions in CdSe/ZnS core-shell quantum dots, Phys. B 383 (2006) 207e212. [57] M. Cristea, A. Radu, E.C. Niculescu, Electric-field effect on the third-order nonlinear optical susceptibility in inverted core-shell nanodots with dielectric confinement, J. Lumin 143 (2013) 592e599. [58] Z. Zeng, C.S. Garoufalis, S. Baskoutas, Linear and nonlinear optical susceptibilities in a laterally coupled quantum-dot-quantum-ring system, Phys. Lett. A 378 (2014) 2713e2718. [59] M. Kouhi, Electric field effect on the quadratic electro optic effects and electro absorption process in GaN/AlGaN spherical quantum dot, Optik 127 (2016) 3379e3382. [60] M. Kouhi, A. Vahedi, A. Akbarzadeh, Y. Hanifehpour, S.W. Joo, Investigation of quadratic electro-optic effects and electro-absorption process in GaN/AlGaN spherical quantum dot, Nanoscale Res. Lett. 9 (2014) 131. [61] M.R.K. Vahdani, The effect of the electronic intersubband transitions of quantum dots on the linear and nonlinear optical properties of dot-matrix system, Superlatt. Microstruct. 76 (2014) 326e338. [62] J. Ganguly, S. Saha, A. Bera, M. Ghosh, Modulating optical rectification, second and third harmonic generation of doped quantum dots: interplay between hydrostatic pressure, temperature and noise, Superlatt. Microstruct. 98 (2016) 385e399. [63] J. Ganguly, S. Saha, A. Bera, M. Ghosh, Exploring electro-optic effect and thirdorder nonlinear optical susceptibility of impurity doped quantum dots: interplay between hydrostatic pressure, temperature and noise, Opt. Commun. 387 (2017) 166e173.