Tuning diagonal components of static linear and first nonlinear polarizabilities of doped quantum dots by Gaussian white noise

Author’s Accepted Manuscript Tuning diagonal components of static linear and first nonlinear polarizabilities of doped quantum dots by Gaussian white noise Jayanta Ganguly, Manas Ghosh www.elsevier.com/locate/jpcs

PII: DOI: Reference:

S0022-3697(15)00066-9 http://dx.doi.org/10.1016/j.jpcs.2015.03.011 PCS7497

To appear in: Journal of Physical and Chemistry of Solids Received date: 21 November 2014 Revised date: 30 January 2015 Accepted date: 10 March 2015 Cite this article as: Jayanta Ganguly and Manas Ghosh, Tuning diagonal components of static linear and first nonlinear polarizabilities of doped quantum dots by Gaussian white noise, Journal of Physical and Chemistry of Solids, http://dx.doi.org/10.1016/j.jpcs.2015.03.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Tuning diagonal components of static linear and first nonlinear polarizabilities of doped quantum dots by Gaussian white noise Jayanta Gangulya and Manas Ghoshb a

∗†‡

Department of Chemistry, Brahmankhanda Basapara High School, Basapara, Birbhum 731215, West Bengal, India. b

Department of Chemistry, Physical Chemistry Section, Visva Bharati University, Santiniketan, Birbhum 731 235, West Bengal, India.

Abstract We investigate the modulation of diagonal components of static linear (αxx , αyy ) and first nonlinear (βxxx , βyyy ) polarizabilities of quantum dots by Gaussian white noise. Quantum dot is doped with impurity represented by a Gaussian potential and repulsive in nature. The study reveals the importance of mode of application of noise (additive/multiplicative) on the polarizability components. The doped system is further exposed to a static external electric field of given intensity. As important observation we have found that the strength of additive noise becomes unable to influence the polarizability components. However, the multiplicative noise influences them conspicuously and gives rise to additional interesting features. Multiplicative noise even enhances the magnitude of the polarizability components immensely. The present investigation deems importance in view of the fact that noise seriously affects the optical properties of doped quantum dot devices. Keywords: A. electronic materials, A. nanostructures, A. quantum wells, D. defects, D. optical properties

∗ † ‡

e-mail address: [email protected] Phone : (+91)(3463)261526, (3463)262751-6 (Ext. 467) Fax : +91 3463 262672

1

I.

INTRODUCTION

Development of new high-performance devices with promising optical properties has come out to be an important task of modern nanotechnology. The low-dimensional quantum systems are kind of modern devices which are highly renowned for displaying enhanced nonlinear optical effects than the bulk materials and possess widespread application in various optoelectronic devices. In consequence, we envisage an abundance of useful investigations on optical properties of these systems providing lots of information about the energy spectrum, the Fermi surface of electrons, and the value of electronic effective mass. Among the low-dimensional quantum systems, quantum dots (QD) are now recognized as prolific semiconductor optoelectronic devices. Furthermore, the presence of impurities strongly alters the optical properties of QD devices by virtue of extensive interplay between QD confinement sources and impurity potentials. Such alterations have successfully pampered new arenas of research in this field [1–11] with special emphasis on their linear and nonlinear optical properties [12–34]. External electric field has often been found to illuminate important aspects related with confined impurities. The electric field changes the energy spectrum of the carrier and controls the performance of the optoelectronic devices. Moreover, the electric field often hampers the symmetry of the system and facilitates emergence of nonlinear optical properties. Thus, the applied electric field assumes special attention in view of assiduous understanding of the optical properties of doped QDs [35–49]. Recently we have made extensive investigations of noise [50–52] as it influences the performances of QD devices. In these works we have analyzed the impact of Gaussian white noise on the diagonal components of frequency-dependent linear [50], first nonlinear [51], and the third nonlinear [52] polarizabilities of doped QD. In the present manuscript we examine the role of Gaussian white noise on the diagonal components of static linear (αxx , αyy ) and first nonlinear (βxxx , βyyy ) polarizabilities of doped QD which are still left unexplored. In the present study noise has been applied to the system additively and multiplicatively [50–52]. An external electric field of given intensity has been applied to the doped system which acts as a disturbance and generates linear and nonlinear responses. We have strived to elucidate the role of dopant location and the noise characteristics as they tailor the static diagonal polarizability components. The role of dopant site has been

2

critically explored because of its sensitivity in shaping the optical properties of doped QDs. In their previous works Karabulut and Baskoutas [24], and Baskoutas et al. [35] highlighted the importance of off-center impurities and introduced a novel numerical method (P M M, potential morphing method). It needs to be mentioned that in one of our recent works [53] we have studied the frequency-dependent linear and nonlinear polarizabilities of doped QD whence the dopant was propagating under damped condition. However, in the present work damping as well as dopant propagation are not at all considered. Thus the environment is completely different. Moreover, frequency-dependent polarizabilities [53] bear widely different characteristics from that of static polarizabilities (in the present study). However, the basic methodologies in both the works are similar with some different equations arising out of different perturbing environments. The present analysis reveals the finer details in the profiles of aforesaid polarizability components as a result of intricate interplay between noise and the effective confinement potential. The effective confinement potential intimately depends on the site of dopant incorporation and thus the latter tunes the overall pattern of the polarizability components. Apart from this, the significance of mode of application of noise (additive/multiplicative) to the doped system has also been congruously addressed in the present manuscript.

II.

METHOD

Our model Hamiltonian represents a 2-d quantum dot with single carrier electron laterally confined (parabolic) in the x − y plane. The confinement potential reads V (x, y) = 21 m∗ ω02 (x2 + y 2 ), where ω0 is the harmonic confinement frequency. The parabolic confinement potential has found extensive usage in various studies on QDs [1, 4, 5, 8, 9, 14, 22, 37], particularly in the study of optical properties of doped QDs by C ¸ akir et al. [? ]. A perpendicular magnetic field (B ∼ mT in the present work) is also present as an additional confinement. Using the effective mass approximation we can write the Hamiltonian of the system as H00 =

1 h e i2 1 ∗ 2 2 −i~∇ + A + m ω0 (x + y 2 ). 2m∗ c 2

(1)

In the above equation m∗ stands for the effective electronic mass within the lattice of the material. The value of m∗ has been chosen to be 0.067m0 resembling GaAs quantum dots. 3

We have set ~ = e = m0 = a0 = 1 and perform our calculations in atomic unit. In Landau gauge [A = (By, 0, 0)] (A being the vector potential), the Hamiltonian transforms to H00 ωc =

eB m∗ c

1 ∂ ~2 ∂ 2 ∂2 1 = − ∗ ( 2 + 2 ) + m∗ ω02 x2 + m∗ (ω02 + ωc2 )y 2 − i~ωc y , 2m ∂x ∂y 2 2 ∂x

(2)

being the cyclotron frequency. Ω2 = ω02 + ωc2 can be viewed as the effective

frequency in the y-direction. We now introduce impurity (dopant) to QD and the dopant is represented by a Gaussian potential [54–56]. To be specific, in the present case we write the impurity potential as 2 2 V = V e−ξ[(x−x0 ) +(y−y0 ) ] . Choice of positive values for ξ and V gives rise to repulsive imp

0

0

impurity. Among various parameters of impurity potential (x0 , y0 ) denotes the dopant coordinate, V0 is a measure of strength of impurity potential, and ξ −1 determines the spatial stretch of impurity potential. Recently Khordad and his coworkers introduced a new type of confinement potential for spherical QD’s called Modified Gaussian Potential, MGP [57, 58]. The Hamiltonian of the doped system reads H0 = H00 + Vimp .

(3)

We have employed a variational recipe to solve the time-independent Schr¨odinger equation and the trial function ψ(x, y) has been constructed as a superposition of the product of harmonic oscillator eigenfunctions [50–52] φn (px) and φm (qy) respectively, as ψ(x, y) =

X

Cn,m φn (px)φm (qy),

(4)

n,m

where Cn,m are the variational parameters and p =

q

m∗ ω0 ~

and q =

q

m∗ Ω . ~

The general

expressions for the matrix elements of H00 and Vimp in the chosen basis have been derived [50–52]. In the linear variational calculation, requisite number of basis functions have been exploited after performing the convergence test. And H0 is diagonalized in the direct product basis of harmonic oscillator eigenfunctions. With the application of noise the time-dependent Hamiltonian becomes H(t) = H0 + V1 (t).

(5)

The noise consists of random term (σ(t)) which follows a Gaussian distribution (produced by Box-Muller algorithm) having strength µ. It is characterized by the equations [50–52]: hσ(t)i = 0, 4

(6)

the zero average condition, and hσ(t)σ(t0 )i = 2µδ(t − t0 ),

(7)

the two-time correlation condition where the correlation time is negligible. The Gaussian white noise has been administered additively [V1 (t) = σ(t)] as well as multiplicatively [V1 (t) = σ(t)(x + y)] [50–52]. Experimentally, external noise can be generated by using a function generator (Hewlett-Packard 33120A) and its characteristics, Gaussian distribution and zero mean can be maintained [59]. The external noise could be introduced multiplicatively using a circuit that enables to drive the nonlinear element by using the voltage from an external source [60]. Hence the findings of current work can be made experimentally realizable and relevant. The external static electric field V2 of strength  is now applied externally where V2 = x .x + y .y,

(8)

x and y are the field intensities along x and y directions. Now the time-dependent Hamiltonian reads H(t) = H0 + V1 (t) + V2 .

(9)

The matrix elements due to V1 (t) and V2 can be readily derived [50–52]. The evolving wave function can now be expressed by a superposition of the eigenstates of H0 , i.e. ψ(x, y, t) =

X

aq (t)ψq .

(10)

q

The associated time-dependent Schr¨odinger equation (TDSE) has now been solved numerically to obtain ψ(x, y, t). For the numerical solution we have invoked 6-th order Runge-Kutta-Fehlberg method with a time step size 4t = 0.01 a.u. on verifying the numerical stability of the integrator. The time-dependent superposition coefficients [aq (t)] has been used to calculate the time-average energy of the dot hEi [50–52]. We have determined the energy eigenvalues for various combinations of x and y and used them to compute the diagonal components of linear and nonlinear polarizabilities by the following relations obtained by numerical differentiation. 4 1 5 [hE(2x )i + hE(−2x )i] , αxx 2x = hE(0)i − [hE(x )i + hE(−x )i] + 2 3 12 5

(11)

5 4 1 αyy 2y = hE(0)i − [hE(y )i + hE(−y )i] + [hE(2y )i + hE(−2y )i] , 2 3 12

(12)

βxxx 3x = [E(x , 0) − E(−x , 0)] −

1 [E(2x , 0) − E(−2x , 0)] , 2

(13)

βyyy 3y = [E(y , 0) − E(−y , 0)] −

1 [E(2y , 0) − E(−2y , 0)] . 2

(14)

,

III. A.

RESULTS AND DISCUSSION Effect of Additive Noise:

We begin with the plots of αxx and αyy as functions of dopant location (r0 ) [fig. 1a]. Both 12.0

b)12.0 αxx αyy

βxxx βyyy

-5

8.0

8.0

β (a.u.) x 10

α (a.u.) x 10

-3

a)

4.0

4.0 0.0 -4.0 -8.0 -12.0

0.0

-16.0 0

10

20

30

40

0

50

r0 (a.u.)

10

20

30

40

50

r0 (a.u.)

FIG. 1: Plot of polarizability components vs r0 with additive noise: (a) α components and (b) β components.

the profiles display distinct minima at r0 ∼ 20.0 a.u. A dopant located in the vicinity of dot confinement center is subjected to stringent confinement while undergoing strong dot-impurity repulsive interaction. The repulsive interaction supersedes the confinement and leads to large value of linear response at on and near off-center locations. On the other hand, a far off-center dopant experiences a diminished repulsive force and the lack of confinement makes it rather flexible. The flexibility promotes emergence of large linear response at far off-center locations. The observed minimization at a typical dopant 6

location indicates absolute dominance of factors that attempt to diminish the dispersive nature of the system over the reverse ones. Fig. 1b depicts the similar profile for the βxxx and βyyy components as functions of r0 . Both the components exhibit distinct maxima (in absolute sense) again at r0 ∼ 20.0 a.u. However, at far off-center dopant location (r0 ≥∼ 40.0 a.u.) both the components saturate to some extent. The findings strongly indicate that the dot-dopant interaction heavily depends on dopant location and run in conformity with the important works of Karabulut and Baskoutas [24], and Baskoutas et al. [35] in related context. A change in dopant location basically alters the ’effective confinement potential’ of the system. The alteration occurs as the dopant incorporated around a particular location (r0 ∼ 20.0 a.u.) undergoes typical interaction with the dot confinement center. Introduction of noise makes the scenario further complicated and affects the linear and nonlinear polarizability components in evidently different ways. Hence, the overall impact of introduction of noise on the polarizabilities reveals heavy dependence on their orders. Thus, we envisage minimization of linear polarizability through a highly depressed dispersive character of the system while maximization of first nonlinear polarizability through profound augmentation of its asymmetric nature. In the present discussion the asymmetric nature of system becomes relevant as long as second order polarizability is concerned. The saturation at far off-center location indicates sort of compromise between several parameters that affect the dispersive and asymmetric nature of the system. The additive noise acts as the perfect driving force that initiates the said compromise. However, the strength of additive noise fails to make an impact on the polarizability profiles [50–52].

B.

Effect of Multiplicative Noise:

We now illustrate the influence of white multiplicative noise on the direct components of static linear (αxx , αyy ) and first nonlinear (βxxx , βyyy ) polarizability. The multiplicative noise brings about two remarkable changes in the profiles of above components. Firstly, there occurs a tremendous enhancement in the magnitudes of the polarizability components by several orders, and secondly, noise strength commences to play anchoring role in shaping them. It can therefore be argued that the direct coupling of noise to the system coordinate is responsible for these remarkable changes. Fig. 2a depicts the plots of 7

1.8 a)

b)

2.4

βxxx (a.u.) x 10

-7

-13

1.6

αxx (a.u.) x 10

2.8

1.4

1.2

2.0 1.6 1.2

1.0

0.8 0

100

200

300

µ (a.u.) x 10

400

0

500

7

100

200

300

µ (a.u.) x 10

400

500

7

FIG. 2: Plot of polarizability components vs µ with multiplicative noise: (a) αxx and (b) βxxx .

αxx as a function of noise strength (µ). The component increases monotonically with increase in µ and culminates in saturation as µ ≥∼ 3.0 × 10−5 a.u. An increase in µ enhances the dispersive nature of the system which results in monotonic increase of α component. The saturation of αxx in high noise strength seems quite contrary to expectation and may have some different background. A very high value of µ beyond ∼ 3.0 × 10−5 a.u. invites strong system-noise interaction which could enhance the effective confinement of the system. The enhanced confinement passivates the noise effect leading to saturation. The profile of αyy component appears quite similar and therefore not presented. Fig. 2b delineates the similar plot for βxxx component against µ which consists of a maxima that is clearly discernible. The maximization in βxxx takes place nearly at µ ∼ 2.0 × 10−5 a.u. The enhancement of nonlinear polarizability of QD devices is of utmost technological importance and the observed behavior indicates development of maximum asymmetric character of the doped system at a particular noise strength. Beyond maximization, at µ ∼ 3.5 × 10−5 a.u. we observe onset of saturation in βxxx which suggests kind of negotiation between noise and the effective confinement strength. The βyyy component exhibits nearly similar behavior and therefore not presented. Finally we inquire the role of dopant location (r0 ) on the polarizability components in presence of multiplicative noise [fig. 3]. Fig. 3a shows the variation of αxx and αyy for different locations of the dopant. Both the components increase slowly with r0 up to ∼ 20.0 a.u. As the dopant is shifted further, the components make a pronounced jump and finally settles at r0 ∼ 40.0 a.u. In the near off-center locations the strong confinement 8

a)

b) αxx αyy

βxxx βyyy

-13

2.8

3.6

β (a.u.) x 10

α (a.u.) x 10

-7

4.0

3.2 2.8 2.4

2.4 2.0 1.6 1.2

0

10

20

30

40

50

0

r0 (a.u.)

10

20

30

40

50

r0 (a.u.)

FIG. 3: Plot of polarizability components vs r0 with multiplicative noise: (a) α components and (b) β components.

hinders the linear optical response suppressing the influence of noise and dot-impurity repulsive interaction. Accordingly, the α components exhibit sluggish enhancement for such dopants. The confinement falls as the dopant shifts beyond 20.0 a.u. and α components rise abruptly. This lack of confinement is utilized by multiplicative noise to increases the dispersive character of the system revealed through sudden enhancement of linear polarizability components. The saturation commencing at r0 ∼ 40.0 a.u. indicates a nearly perfect balance between the effective confinement and the noise strength. Fig. 3b evinces the variation of βxxx and βyyy with r0 . Interestingly, only in this occasion we could find noticeable difference in the profiles of βxxx and βyyy . The βxxx component has been found to increase monotonically with r0 whereas βyyy exhibits distinct minima at r0 ∼ 13.0 a.u. However, at far off-center locations (r0 ≥∼ 43.0 a.u.) both the components display sort of settlement. We can therefore infer that the multiplicative noise can differentiate nonlinear polarizability components depending on the direction of applied field. The said difference can be attributed to coupling of multiplicative noise with system coordinates which in turn affects the effective confinement potential. Moreover, the intrinsic non-equivalence of two β components is also responsible for the observed differences in their profiles. Thus, while noise promotes βxxx smoothly (revealed through its persistent increase with r0 ), it reduces the asymmetric nature of the system to a minimum at the typical value of r0 ∼ 13.0 a.u. pertinent to the βyyy component.

9

IV.

CONCLUSIONS

Modulation of diagonal components of static linear and first nonlinear polarizabilities of impurity doped quantum dots have been investigated under the aegis of Gaussian white noise. The polarizability components are found to be strongly driven by noise characteristics. In case of additive noise β components exhibit maximization at a dopant location of r0 ∼ 20.0 a.u. which signals possibility of immense technological application. Interestingly, the noise strength declines to affect the profiles of α and β components in presence of additive noise. Application of multiplicative noise enhances the polarizability components by several orders of magnitudes and now noise strength (µ) plays some remarkable role in fabricating the optical properties. In view of this a controlled adjustment of noise strength could be of significant technological importance. The intrinsic non-equivalence of β components becomes clearly discernible as a function of r0 . Now, the βxxx and βyyy components behave in manifestly different manners with emergence of minimization for the βyyy component. The overall study reveals that Gaussian white noise could be an important probe to fine-tune the linear and nonlinear polarizabilities of the doped QD system. The noise brings about several significant features in the profiles of linear and nonlinear polarizabilities as a function of two potentially important parameters, viz. the noise strength and the dopant location. However, multiplicative noise comes out to be more important than its additive counterpart owing to its more subtle influence on the polarizabilities.

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Figure Captions Fig. 1: Plot of polarizability components vs r0 with additive noise: (a) α components and (b) β components. Fig. 2: Plot of polarizability components vs µ with multiplicative noise: (a) αxx and (b) βxxx . Fig. 3: Plot of polarizability components vs r0 with multiplicative noise: (a) α components and (b) β components.

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