Influence of pulse shape in modulating excitation kinetics of impurity doped quantum dots

Superlattices and Microstructures 55 (2013) 118–130

Contents lists available at SciVerse ScienceDirect

Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices

Influence of pulse shape in modulating excitation kinetics of impurity doped quantum dots Suvajit Pal a, Manas Ghosh b,⇑ a b

Department of Chemistry, Hetampur Raj High School, Hetampur, Birbhum 731124, West Bengal, India Department of Chemistry, Physical Chemistry Section, Visva-Bharati University, Santiniketan, Birbhum 731235, West Bengal, India

a r t i c l e

i n f o

Article history: Received 8 November 2012 Received in revised form 8 December 2012 Accepted 10 December 2012 Available online 29 December 2012 Keywords: Quantum dot Impurity doping Dopant coordinate Pulsed field Pulse shape Excitation rate

a b s t r a c t Excitation in quantum dots is an important phenomenon. Realizing the importance we explore the excitation kinetics of a repulsive impurity doped quantum dot induced by an electromagnetic pulsed field of various pulse shapes. The pulsed field has been applied along both x and y directions to the doped quantum dot. The impurity potential has been assumed to have a Gaussian nature. The investigation reveals the sensitivity of the typical shape of the pulse, alongside the influences of dopant location and number of pulse towards modulating the excitation rate. At first we have concentrated on understanding the role of number of pulses fed into the system on the kinetics at three fixed dopant locations. Next, we have given our thrust in analyzing the exclusive role of dopant location on excitation kinetics at a fixed pulsed number. In both the cases the typical pulse shapes announce its role on excitation kinetics unhesitatingly through a number of observations. The present study has also indicated enough evidence of change in the mutual dominance of several factors that could favor and impede excitation accompanying the shift of dopant location. Importantly, those factors also depend severely on the pulse shape. The pulse shape interferes delicately with the interplay between impurity location and number of pulses fed into the system and emerges as an important ingredient that can engineer the excitation kinetics. Ó 2012 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +91 03463 261526, 03463 262751 6x467; fax: +91 3463 262672. E-mail address: [email protected] (M. Ghosh). 0749-6036/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.spmi.2012.12.008

S. Pal, M. Ghosh / Superlattices and Microstructures 55 (2013) 118–130

119

1. Introduction Over the couple of decades impurity doping in semiconductor materials has emerged as an useful technology that has been utilized to control opto-electronic properties of a wide range of semiconductor devices [1]. Impurity doping in quantum dots (QDs) invariably invites new perspectives and subtleties in the field of applied physics. This happens because of the interplay between various confinement sources with impurity potentials [2]. Moreover, the dopant location shows some crucial role in the said interplay and profoundly alters the electronic and optical properties of the system [3–10]. Naturally, we find a vast literature comprising of good theoretical studies of impurity states [11–15] particularly emphasizing the role of dopant location [16,17]. Investigations on fabricating the impurity states have received further impetus with the development of sophisticated experimental techniques such as molecular beam epitaxy, liquid phase epitaxy, and chemical vapor deposition. Propelled by this, there are also some excellent experimental works which encompass the mechanism and control of dopant incorporation [18–21]. The emergence of novel experimental and theoretical techniques along with the existing ones have encouraged the research on carrier dynamics in nanodevices. The time-dependent aspects in nanodevices naturally become a hotly pursued research topic. The internal transitions between impurity induced states in a QD led to research on carrier dynamics [22–24]. These transitions depend on the spatial restriction imposed by the impurity. A close scrutiny of the above dynamical features suggests us to explore the excitation of electrons strongly confined in QD’s. Analysis on this aspect deems importance since it offers us model systems for use in opto-electronic devices and as lasers. Within the purview of engineering applications excitation in QD assumes importance in optical encoding, multiplexing, photovoltaic and light emitting devices. The phenomenon also plays some notable role in the eventual population transfer among the exciton states in QD [25,26]. Recently, we have made some investigations on the excitation profiles of the doped quantum dots initiated by discontinuously reversing pulsed field [27,28]. In this paper we have attempted to investigate the excitation in the doped quantum dot exposed to a pulsed field which is continuous in nature. The motivation for the work originates from a survey of recent works which study the effect of applied electric field on doped quantum wells and dots [29–31]. Boviatsis et al. in their notable work used pulsed electromagnetic field in a double QD structure [32]. Looking at a more elaborate investigation of the pulsed field we have exploited pulses of different shapes to ascertain if a variation in the shape can modulate the target excitation. The electromagnetic pulsed field has been applied along both x and y directions to the doped quantum dot (i.e. it is polarized in x and y directions). A variation in the pulse shape basically implies a change in the pattern of energy transfer from the external field to the QD. It seems quite interesting to decipher how such a change in the mode of field-to-dot energy transfer affects the excitation kinetics. The present investigation assumes that the QD is doped with a repulsive Gaussian impurity which simulates dopant with excess electrons [33,34]. In the present work the time-dependence has been modeled by attaching a potential V(t) to the dot Hamiltonian. A consequent follow up of the dynamics of the doped dot in the time-dependent potential becomes the obvious next task to handle the problem. We have realized that alongside pulse shape, the number of pulses fed into the system from the external field (np) as well as the dopant location (r0) could play some remarkable role so far as excitation is concerned. The realization drives us rationalize thoroughly their combined influence in the context of present enquiry.

2. Method The model considers an electron subject to a harmonic confinement potential V(x, y) and a perpendicular magnetic field B. The confinement potential assumes the form Vðx; yÞ ¼ 12 m x20 ðx2 þ y2 Þ, where x0 is the harmonic confinement frequency, xc ¼ meB c being the cyclotron frequency (a measure of magnetic confinement offered by B). In the present work a magnetic field of miliTesla (mT) order has been employed. m⁄ is the effective electronic mass within the lattice of the material to be used. We have taken m⁄ = 0.5m0 and set ⁄ = e = m0 = a0 = 1. This value of m⁄ closely resembles Ge quantum dots

120

S. Pal, M. Ghosh / Superlattices and Microstructures 55 (2013) 118–130

(m⁄ = 0.55 a.u.). We have used Landau gauge [A = (By, 0, 0)] where A stands for the vector potential. The Hamiltonian in our problem reads 2

H00

 h @2 @2 ¼ þ 2m @x2 @y2

!

1 1 @ þ m x20 x2 þ m ðx20 þ x2c Þy2  ihxc y : 2 2 @x

ð1Þ

Define X2 ¼ x20 þ x2c as the effective frequency in the y-direction. The model Hamiltonian (cf. Eq. (1)) sensibly represents a 2-d quantum dot with a single carrier electron [35,36]. The form of the confinement potential conforms to kind of lateral electrostatic confinement of the electrons in the x  y plane. In real QDs the electrons are confined in 3-dimensions i.e. the carriers effectively possess a quasi-zero dimensional domain. The confinement length scales R1, R2, and R3 can, in general, be different in three spatial directions, but usually R3  R1 ’ R2. Whenever such QD’s are modeled R3 is often taken to be strictly zero and the confinement in the other two directions is described by a potential V with V(x) ? 1 for jxj ? 1, x = (x1, x2) 2 R2. x1 and x2 represent the coordinates in x and y directions, respectively. It means that, since R3 is almost zero; the potential in the other two relevant directions extends up to a length which is of the order of confinement length scales in these directions. More often than not, a parabolic confinement po  tential V ¼ 12 xjxj2 is chosen as it appears quite realistic and largely reduces computational cost [7,8,12,15,17,29]. Assuming that the z-extension could be effectively considered zero, the electronic properties in these nanostructures have been successfully described using the model of 1-electron motion in 2-d harmonic oscillator potential in the presence of a magnetic field [35,36]. Now, as the impurity perturbation is attached to the Hamiltonian (cf. Eq. (1)) it transforms to

H0 ðx; y; xc ; x0 Þ ¼ H00 ðx; y; xc ; x0 Þ þ V imp ðx0 ; y0 Þ;

ð2Þ

c½ðxx0 Þ2 þðyy0 Þ2 

where V imp ðx0 ; y0 Þ ¼ V imp ð0Þ ¼ V 0 e with c > 0 and V0 > 0 for repulsive impurity, and (x0, y0) denotes the coordinate of the impurity center. V0 is a measure of the strength of impurity potential whereas c1 determines the spatial stretch of the impurity potential. A large value of c indicates a highly quenched spatial extension of impurity potential whereas a small c accounts for spatially dispersed one. The parameter c in the impurity potential is equivalent to d12 , where d is proportional to the width of the impurity potential [33,34]. The value of c is taken to be 0.001 a.u. which corresponds to an extension of the impurity domain up to 1.41 nm. The dopant strength (V0) assumes a maximum value of 104 a.u. or 2.72 meV. The choice of the values of different quantities has been made depending on several factors. One of them being the numerical stability of our investigation. Numerical stability has been achieved if the parameters assume values within a range around the specified values. Secondly, the choice of the values has been guided by the emergence of fruitful excitation. If the parameters are kept widely away from the mentioned values, no significant excitation could be observed. Also, the mentioned values seem to provide some realization of the quantum dot from an experimental perspective. However, in the present investigation we have not explored whether a change in the size/diameter/ nature of the QD could affect the choice of values although it seems quite reasonable to expect so. The location of the impurity center has been varied from (x0, y0) = (0,0) (on-center) to (x0, y0) = (50, 50) (off-center) positions. Since the dopant is incorporated inside the dot, the radius of the dot should be a bit larger than the farthest dopant location viz. (x0, y0) = (50, 50) or 70.71 a.u. The use of such Gaussian impurity potential is quite well-known [37–39]. In view of the ongoing discussion the work of Gharaati and Khordad [40] merits mention. They introduced a new confinement potential for the spherical QD’s called Modified Gaussian Potential, MGP and showed that this potential can predict the spectral energy and wave functions of a spherical quantum dot. We write the trial wave function w(x, y) as a superposition of the product of harmonic oscillator eigenfunctions /n(ax) and /m(by) respectively as follows [27,28]:

wðx; yÞ ¼

X

C n;m /n ðaxÞ/m ðbyÞ;

ð3Þ

n;m

where Cn,m are the variational parameters and a ¼

qffiffiffiffiffiffiffiffiffi m x0 h 

and b ¼

qffiffiffiffiffiffiffi m X . In the linear variational calh 

culation, we have used an appreciably large number of basis functions (cf. Eq. (3)) with n, m = 0  20 for each of the directions (x, y). This direct product basis spans a space of (21  21) dimension. It has been verified that the basis of such size scans the 2-d space effectively completely as long

S. Pal, M. Ghosh / Superlattices and Microstructures 55 (2013) 118–130

121

as monitoring the observables under investigation is concerned. A convergence test run by us with still greater number of basis functions confirmed our observation. The general expression for the matrix elements of H00 in the chosen basis are as follows:



H00



n;m;n0 ;m0

   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 1 n0 þ x0 þ m0 þ x20 þ x2c dn;n0 dm;m0  ihxc 2 2 "(rffiffiffiffi ) (rffiffiffiffiffiffiffiffiffiffiffiffiffiffi )# rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffi a n0 n0 þ 1 m0 þ 1 m0 dn0 þ1;n : dm0 þ1;m þ  dn0 1;n  dm0 1;m : 2 2 b 2 2

¼ h

ð4Þ

And the matrix elements of Vimp are given by [27]

ðV imp Þn;m;n0 ;m0 ¼ V 0  D1  D2 

0 0 minðn;n X Þminðm;m X Þ

k¼0

f ðk; n; n0 Þ  gðl; m; m0 Þ:

ð5Þ

l¼0

where 0

f ðk; n; n0 Þ ¼ 2k  k!  n C k  n C k  ð1  a2 Þ

nþn0 k 2

 Hnþn0 2k ða1 q1 Þ;

and 0

gðl; m; m0 Þ ¼ 2l  l!  m C l  m C l  ð1  b2 Þ

mþm0 l 2

 Hmþm0 2l ðb1 q2 Þ:

1=2

1=2

The other relevant quantities are D1 ¼ Ak1dp1 , D2 ¼ Bk2dp2 , A ¼ nþn0 a 0 1=2 , B ¼ mþm0 b 0 1=2 , d21 ¼ a2 þ c, ð2 ð2 n!n !pÞ m!m !pÞ   cx2 ðd2 cÞ cy2 ðd2 cÞ , and k2 ¼ exp  0 d22 , q1 ¼ cdx10 , q2 ¼ cdy20 , a ¼ da1 , b ¼ db2 . And Hn(x) d22 ¼ b2 þ c, k1 ¼ exp  0 d21 1

2

stands for the Hermite polynomials of nth order. The pth eigenstate of the system in this representation can be written as

X

wp ðx; yÞ ¼

C ij;p f/i ðaxÞ/j ðbyÞg;

ð6Þ

ij

where i, j are the appropriate quantum numbers, respectively and (ij) are composite indices specifying the direct product basis. The external pulsed electric field e(t) is now switched on with

ex ðtÞ ¼ ex ð0ÞSðtÞ sinðmx tÞ for x-direction, and analogously,

ey ðtÞ ¼ ey ð0ÞSðtÞ sinðmy tÞ for y-direction. ex(0) and ey(0) are the field intensities along x and y directions at t = 0 and mx and my being the oscillation frequencies in x and y directions, respectively. Generally speaking, the pulsed field e(t) has been considered to be characterized by intensity modulated by a pulse-shape function S(t) where,

eðtÞ ¼ eð0ÞSðtÞ sinðmtÞ:

ð7Þ

The pulse has a peak field strength e(0), and a fixed frequency m. We have considered four different pulse shape functions, namely, sinusoidal, Gaussian, triangular, and saw-tooth which assume the following analytical expressions: 2

SðtÞ ¼ sin



pt Tp

 ;

Sinusoidal pulse

SðtÞ ¼ exp½kðt  T p Þ2 ; Gaussian pulse

2t SðtÞ ¼ 1 

1 

; Triangular pulse; and Tp   t SðtÞ ¼ a 1  ; Saw—tooth pulse: Tp

122

S. Pal, M. Ghosh / Superlattices and Microstructures 55 (2013) 118–130

where Tp stands for pulse duration time. Thus Tp, or equivalently np (the number of pulses), appears to be a key control parameter. Fig. 1a–d depict the profiles of five consecutive pulses of different shapes as a function of time for a ready comparison. As the pulsed field begins interacting with the dot electron, the time-dependent part of the Hamiltonian reads

VðtÞ ¼ ex ðtÞ  x  jej þ ey ðtÞ  y  jej:

ð8Þ

Now the time-dependent Hamiltonian reads

HðtÞ ¼ H0 þ VðtÞ:

ð9Þ

The matrix element involving any two arbitrary eigenstates p and q of H0 due to V(t) reads

Vðp; qÞ ¼ hwp jVðtÞjwq i ¼ ex ð0ÞSðtÞ sinðmx tÞ

XX  C ij;p C kl;q h/i ðaxÞjxj/k ðaxÞidjl ij

kl

XX  þ ey ð0ÞSðtÞ sinðmy tÞ C ij;p C kl;q h/j ðbyÞjyj/l ðbyÞidik ; ij

ð10Þ

kl

where

h/i ðaxÞjxj/k ðaxÞi ¼

"rffiffiffiffiffiffiffiffiffiffiffiffi # rffiffiffi kþ1 k dk;i þ dk1;i ; 2 2 a 1

ð11Þ

and

(c)

1.0

1.0

0.8

0.8

0.6

0.6

S (t)

S (t)

(a)

0.4

0.4

0.2

0.2

0.0

0.0 0

5

10

15

20

0

25

5

10

15

20

25

15

20

25

t (a.u.)

t (a.u.)

(b)

(d)

1.00

1.0

0.8

S (t)

S (t)

0.99 0.6

0.4

0.98

0.2 0.97 0.0 0

5

10

15

t (a.u.)

20

25

0

5

10

t (a.u.)

Fig. 1. Plot of pulse shape function S(t) against t for five successive pulses: (a) sinusoidal pulse, (b) Gaussian pulse, (c) triangular pulse, and (d) saw-tooth pulse.

S. Pal, M. Ghosh / Superlattices and Microstructures 55 (2013) 118–130

"rffiffiffiffiffiffiffiffiffiffi # rffiffiffi 1 lþ1 l dl;j þ dl1;j : h/j ðbyÞjyj/l ðbyÞi ¼ b 2 2

123

ð12Þ

H0 is diagonal in the {w} basis. Under the external perturbation, the evolving wave function is described by a linear combination of the eigenstates of H0.

wðx; y; tÞ ¼

X aq ðtÞwq ;

ð13Þ

q

In order to determine the time-dependent superposition coefficients we need to solve the timedependent Schrödinger equation (TDSE).

@w ¼ Hw or equivalently @t iha_q ðtÞ ¼ Haq ðtÞ;

ih

ð14Þ

with the initial conditions ap(0) = 1, aq(0) = 0, for all q – p, where p may be the ground or any other excited states of H0. The TDSE in the direct product basis (cf. Eq. (14)) has been integrated by the 6th order Runge–Kutta–Fehlberg method with a time step size 4t = 0.01 a.u. and the numerical stability of the integrator has been checked. The quantity Pk(t) = jak(t)j2 indicates the population of kth state of H0 at time t. There occurs a continuous growth and decay in the ground state population [P0(t)] during the time evolution. Naturally, the quantity Q(t) = 1  P0(t) serves as a measure of excitation. In consequence, the quantity Rex ðtÞ ¼ dQ serves as the time-dependent rate of excitation. We have calculated dt RT the time-average rate of excitation ½hRex i ¼ T1 0 Rex ðtÞdt with T being the total time of dynamic evolution (100 ps in the present investigation). The dopant location, the number of pulses fed into the system, and obviously the particular shape of the pulse have been found to influence this rate in a coupled manner and shed light on some important features of excitation profile. 3. Results and discussion Let us now start our discussions on the finer details of the dynamical aspects. We are already preoccupied with the anticipation that the dopant location (r0) could interplay with np subtly to influence the excitation profile. The anticipation originates from a survey of the notable works of Baskoutas et al. [16], Karabulut and Baskoutas [41], and others [2,6,42] related to dopant located at off-center position which are worth-mentioning. Particularly, Karabulut, Baskoutas and their coworkers studied the offcenter impurities invoking an accurate numerical method (PMM, potential morphing method). Figs. 2a, 2b, 2c display the instantaneous excitation [Q(t)] as a function of time for different pulse shapes for on-center (r0 = 0.0 a.u.), near off-center (r0 = 28.28 a.u.), and far off-center (r0 = 70.71 a.u.) dopant locations, respectively. In all these plots the number of pulse has been kept equal to 7. A close look at these three plots does not reveal any significant role played by the dopant coordinate so far as instantaneous excitation is concerned. In all the three dopant coordinates, more or less, the sinusoidal pulse exhibits smooth undulatory pattern, the Gaussian pulse shows abrupt jumps and falls, whereas the triangular and saw-tooth pulses, after an initial rise, continue to make highly irregular appearances. Thus, we feel that, it needs somewhat more sincere inspection in order to realize the role of dopant location on excitation kinetics comprehensively. In view of this, however, simple monitoring of instantaneous excitation turns out to be inadequate and it needs a more critical analysis. Thus, we realize that it is necessary to determine the time-average excitation rate (hRexi) to assess the role of dopant location or more importantly, the nature of interplay between the dopant position and the pulse number. As the dopant is introduced at a greater distance from the dot confinement center (r0 = 0) the confines of electric (x0) and magnetic (xc) origins naturally become weak and favor excitation. In order to visualize the role played by the spatial b imp jw i as a function of radial stretch of impurity potential we have plotted the matrix elements hw0 j V 0 position of impurity (r0) (Fig. 3). The plot nicely delineates the overlap of the impurity potential with the ground state wave function jw0i. As expected, the plot reveals a continually decreasing dot-impurity overlap with gradual shift of the dopant away from the dot confinement center. However, at large

124

S. Pal, M. Ghosh / Superlattices and Microstructures 55 (2013) 118–130

(i)

1.0

0.8

0.6

Q (t)

(iv) 0.4

(iii)

(ii)

(i) (ii) (iii) (iv)

0.2

0.0

t (a.u.) Fig. 2a. Plot of Q(t) against t with V0 = 1.0  106 a.u., c = 0.001 a.u., and np = 7 at on-center (r0 = 0.0 a.u.) dopant location for different pulse shapes: (i) Gaussian, (ii) sinusoidal, (iii) saw-tooth, and (iv) triangular.

1.0

0.8

(iii)

Q (t)

0.6

0.4

0.2

(i)

(ii) (i) (ii) (iii) (iv)

(iv)

0.0

t (a.u.) Fig. 2b. Plot of Q(t) against t with V0 = 1.0  106 a.u., c = 0.001 a.u., and np = 7 at near off-center (r0 = 28.28 a.u.) dopant location for different pulse shapes: (i) Gaussian, (ii) sinusoidal, (iii) saw-tooth, and (iv) triangular.

r0 the overlap settles to an otherwise steady value. Thus, quite naturally, the excitation process is influenced by a variation of dopant location. Let us now have a close look at the plot that delineates the time-average excitation rate (hRexi) as a function of (np) for three different dopant locations; on-center (r0 = 0.0 a.u.), near off-center (r0 = 28.28 a.u.), and far off-center (r0 = 70.71 a.u.) (Figs. 4a, 4b, 4c). Each of the plots contains the excitation rate profiles for different pulse shapes. In these plots we have varied np from 1 to 20. This range of np is expected to reveal the influence of pulse on the extent of excitation. The plots clearly evince the dependence of excitation rate on number of pulse mingled with diversities arising out of dopant location. For an on-center dopant, hRexi has been found to pass through a minima for all pulse shapes, however, the location as well as the depth of the minima depends on the shape (Fig. 4a). The minimization in the excitation rate takes place at around np 8, np 5, np 7, and np 8 for sinusoidal, triangular,

S. Pal, M. Ghosh / Superlattices and Microstructures 55 (2013) 118–130

125

1.0

0.8

Q (t)

0.6

0.4

(iii)

(iv)

(i) (ii) (iii) (iv)

(i)

0.2

(ii)

0.0

t (a.u.) Fig. 2c. Plot of Q(t) against t with V0 = 1.0  106 a.u., c = 0.001 a.u., and np = 7 at far off-center (r0 = 70.71 a.u.) dopant location for different pulse shapes: (i) Gaussian, (ii) sinusoidal, (iii) saw-tooth, and (iv) triangular.

20

15

10

5

0

0

10

20

30

40

50

60

70

b imp jw i against r0 with V0 = 1.0  106 a.u. and c = 0.001 a.u. Fig. 3. Plot of hw0 j V 0

Gaussian, and saw-tooth pulses, respectively. Also, whereas the minima is manifestly deep for sinusoidal and saw-tooth pulses, it is quite moderate for remaining two pulse shapes. Moreover, for all the pulse shapes, we envisage a saturation in the excitation rate beyond np 10 signifying kind of compromise between the influences that favor and oppose the excitation process. The specific values of nps beyond which the saturation begins also depend on the nature of pulse shape and read approximately 14, 11, 11, 13 for sinusoidal, triangular, Gaussian, and saw-tooth pulses, respectively. In the small np domain, an increase in np supplies more and more energy to the system to a very limited extent. Since the strong confinement resists any attempt to promote excitation, it does so more enthusiastically as np increases up to a small value forcing the excitation rate to a minimum. After the minima the large amount of energy input outweighs the confinement strength and the rate begins to rise. At very large np, the excitation process gets accompanied by considerable deexcitation and a compromise between them leads to saturation. Overall, from the above discussions it comes out that

126

S. Pal, M. Ghosh / Superlattices and Microstructures 55 (2013) 118–130

(i)

1.9 1.8

(iii) 1.7

(ii)

1.6 1.5

(iv)

(i) (ii) (iii) (iv)

1.4 1.3 0

2

4

6

8

10

12

14

Fig. 4a. Plot of hRexi vs. np for on-center (r0 = 0.0 a.u.) dopant location for different pulse shapes: (i) sinusoidal, (ii) triangular, (iii) Gaussian, and (iv) saw-tooth.

1.9

(ii)

1.8

(i) (iii)

1.7

(iv)

1.6 1.5

(i) (ii) (iii) (iv)

1.4 1.3 0

2

4

6

8

10

12

14

Fig. 4b. Plot of hRexi vs. np for near off-center (r0 = 28.28 a.u.) dopant location for different pulse shapes: (i) sinusoidal, (ii) triangular, (iii) Gaussian, and (iv) saw-tooth.

in the on-center dopant location, the pulse shape indeed marks its signature on the excitation kinetics. By and large, the excitation kinetics follows more or less similar pattern with variation of np depending on the factors that foment and impede excitation for all the pulse shapes. However, the nuances of pulse shape effects on excitation kinetics are exhibited via the subtle variations in the np values corresponding to location of minima, the onset of saturation and also examining the depth of minima. At near off-center dopant location (r0 = 28.28 a.u.) the similar plots carry some interesting features (Fig. 4b). Although for all the pulse shapes we again observe minimization in hRexi followed by saturation as a function of np as before, but the extent of minimization has been severely depleted except for the Gaussian pulse. Interestingly, for the Gaussian shaped pulse the minimum appears to be deeper now. The locations of minimization being np 6, np 4, np 8, and np 7 for sinusoidal, triangular, Gaussian, and saw-tooth pulses, respectively. The saturation np values now become approximately

S. Pal, M. Ghosh / Superlattices and Microstructures 55 (2013) 118–130

127

(i)

2.0 1.8

(ii) 1.6

(iv) (iii)

1.4

(i) (ii) (iii) (iv)

1.2 1.0 0

2

4

6

8

10

12

14

Fig. 4c. Plot of hRexi vs. np for far off-center (r0 = 70.71 a.u.) dopant location for different pulse shapes: (i) sinusoidal, (ii) triangular, (iii) Gaussian, and (iv) saw-tooth.

11, 8, 12, and 10 for the same sequence of pulse shapes. The observations appear to comply with the weakened dot confinement effect in the near off-center location of the dopant in comparison with the on-center location. The skirmish between this weakened confinement and the energy input also becomes less significant so far as hRexi is concerned. Thus, the extent of minimization has been substantially reduced. The observations at near off-center location thus categorically separate Gaussian shaped pulse from remaining ones towards excitation. Thus, the role of pulse shape in modulating the excitation kinetics becomes more prominent in the near off-center dopant location in comparison with the on-center counterpart. More importantly, notwithstanding the shift of the dopant away from the dot confinement center, the particular Gaussian shape of the pulse seems to preserve the confinement strength as it is. The emergence of minima completely disappears at far off-center dopant location (r0 = 70.71 a.u.) except for the saw-tooth pulse (Fig. 4c). For sinusoidal, Gaussian, and triangular pulse shape functions hRexi exhibits a nearly steady character over entire range of np values. At this far off-center location, because of virtually insignificant confinement the excitation rate increases continually with np without appearance of any minima and culminates in steady value. To be specific, the sinusoidal pulse exhibits a very faint minima at np 3 reminiscent of its behavior at on and near off-center locations. It shows saturation at np 7. hRex i increases with np for both triangular and Gaussian pulses at low np and finally undergoes saturation at np 6 for both of them. However, the rise in hRexi with np from its initial value to the saturation value is much more pronounced for the Gaussian pulse than the triangular pulse. The exceptional candidate in this dopant location is the saw-tooth pulse which evinces minimization and saturation in hRexi at np 10 and np 14, respectively. Thus, in the far off-center dopant location, it is the saw-tooth pulse that strives to preserve the confinement strength. However, owing to the heavily depleted dot-impurity overlap at this dopant location the saw-tooth pulse becomes partially successful in its endeavor. Table 1 summarizes the finer details of the effects of pulse shape on excitation kinetics for ready viewing. In the preceding sections we have given our thrust mainly on analyzing the role of np on excitation kinetics at three different dopant locations. Now we turn our attention towards investigating the influence of dopant locations more exclusively on the excitation kinetics. For this purpose we have varied dopant coordinate (r0) over a range for fixed values of np. Earlier we came across that the excitation rate exhibits stable behavior as soon as np P 15 for all pulse shapes. The stability guided us to fix np equal to 15 and vary r0 over a range. Fig. 5 depicts the hRexi vs. r0 plots for different pulse shapes at np = 15. It needs to be realized now that the effect of gradual shift of dopant from on to off-center

128

S. Pal, M. Ghosh / Superlattices and Microstructures 55 (2013) 118–130

Table 1 The details of pulse shape function S(t) along with np values corresponding to minimization (np,min) and saturation (np, sat) and depth of minima. S(t) On-center dopant Sinusoidal Triangular Gaussian Saw-tooth Near off-center dopant Sinusoidal Triangular Gaussian Saw-tooth

np,min

Far off-center dopant Sinusoidal Triangular Gaussian Saw-tooth

np,sat

Depth of minima

8 5 7 8

14 11 11 13

Large Moderate Moderate Large

6 4 8 7

11 8 12 10

Small Small Large Moderate

3 – – 10

7 6 6 14

Very feeble – – Moderate

locations is not absolutely straightforward, rather it is self-contradictory. On one hand such shift weakens the strength of dot confinement thereby favoring excitation, but, on the other hand it also reduces dot-impurity repulsive interaction (because of decreasing overlap) and excitation is unfavored. Owing to these two opposite influences we expect appearance of maxima and minima in the excitation kinetics with a variation of dopant coordinate. The maxima or minima simply indicates a changeover in the dominance of one influence over its rival. The said shift of dopant mingled with typical shapes of various pulses reveal diverse patterns of evolution of excitation kinetics which are worth-mentioning. A close look at Fig. 5 exposes the said diversity elegantly; the sinusoidal and saw-tooth pulse exhibit maxima at r0 = 26.67 a.u. and r0 = 40.50 a.u., respectively, the triangular pulse shows a shallow minima at r0 = 37.31 a.u., and the Gaussian pulse, starting from a rather low value, suddenly jumps to a high value at r0 = 26.32 a.u. However, for all the pulses, at very far off-center locations (r0 P 55 a.u.) we envisage a saturation in the excitation rate proclaiming ample balance between the rival factors that affect excitation.

2.0

(i)

1.8

(ii) (iv)

1.6

(i) (ii) (iii) (iv)

(iii)

1.4

0

10

20

30

40

50

60

70

Fig. 5. Plot of hRexi vs. r0 at np = 15 for different pulse shapes: (i) sinusoidal, (ii) triangular, (iii) Gaussian, and (iv) saw-tooth.

S. Pal, M. Ghosh / Superlattices and Microstructures 55 (2013) 118–130

129

In the ongoing discussion it needs to be realized that the impurities localized at the far off-center locations are expected to be close to the QD surface. Such proximity to the surface is expected to modify the observed results. However, in our present investigation we did not consider the surface effects. In this context the work of Zeng et al. is notable [43]. Based on their work we can anticipate that a change in the alignment of the magnetic field, coupled with the shape, size and the dopant location at QD surface, could significantly alter the excitation kinetics induced by the pulsed field. The alteration takes place since the binding energy of the surface impurities does not change in the same way as that of the on-center impurity when the magnetic field is tilted. On the other hand, for the growth direction magnetic field, the change of the binding energy of off-center impurity is similar to that of an on-center impurity. Thus, a change in the dopant location from on to off-center locations would undoubtedly invite significantly new features if surface effects are considered. 4. Conclusions The excitation kinetics of repulsive impurity doped quantum dots triggered by pulsed field of different pulse shapes reveals noteworthy features. The dopant location and the number of pulses fed into the system, along with the typical shape of the pulse display prominent roles in modulating the excitation rate. An on-center dopant location is endowed with a minimization of excitation rate at some particular np values. When np becomes quite large, we encounter saturation in the excitation rate for all pulse shapes. However, the specific role of pulse shape is revealed through the differences in the np values corresponding to excitation minima, saturation and also through the varied depths of the excitation minima. The above features pertaining to typical shape of the pulse undergo modification with shift of the dopant to near off-center location and in the far off-center location even the excitation minima disappears for triangular and Gaussian pulses. The near and far off-center dopant locations characteristically isolate Gaussian and Saw-tooth pulses, respectively, from other pulse shapes. Moreover, analysis made to understand the exclusive role played by the dopant coordinate on excitation kinetics divulges pulse shape dependence to a great extent. The said dependence occurs owing to kind of changeover in the mutual dominance between the factors that favor and oppose the excitation process accompanying the dopant shift, and the changeover depends manifestly on pulse shape. At very far off-center locations the excitation kinetics reaches saturation for all pulse shapes. We expect that the results obtained might reveal some important technological applications of doped quantum dot nanomaterials. Acknowledgements The authors S.P. 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 partial financial support. Thanks are also to Dr. Rahul Sharma, Department of Chemistry, St. Xavier’s College, Kolkata, for his sincere cooperation. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

C.P. Poole Jr., F.J. Owens, Introduction to Nanotechnology, Wiley, New York, 2003. J.L. Movilla, J. Planelles, Off-centering of hydrogenic impurities in quantum dots, Phys. Rev. B. 71 (2005) 075319. 7p. M.J. Kelly, Low-Dimensional Semiconductors, Oxford University Press, Oxford, 1995. W. Xie, Nonlinear optical properties of an off-center donor in a quantum dot under applied magnetic field, Solid. State. Commun. 151 (2011) 545–549. W. Xie, Impurity effects on optical property of a spherical quantum dot in the presence of an electric field, Physica B 405 (2010) 3436–3440. W. Xie, Optical properties of an off-center hydrogenic impurity in a spherical quantum dot with Gaussian potential, Superlattices Microstruct. 48 (2010) 239–247. Y. Yakar, B. Çakír, A. Özmen, Calculation of linear and nonlinear optical absorption coefficients of a spherical quantum dot with parabolic potential, Opt. Commun. 283 (2010) 1795–1800. A.J. Peter, Polarizabilities of shallow donors in spherical quantum dots with parabolic confinement, Phys. Lett. A 355 (2006) 59–62. K.M. Kumar, A.J. Peter, C.W. Lee, Optical properties of a hydrogenic impurity in a confined Zn1xCdxSe/ZnSe spherical quantum dot, Superlattices Microstruct. 51 (2012) 184–193.

130

S. Pal, M. Ghosh / Superlattices and Microstructures 55 (2013) 118–130

[10] R. Khordad, Diamagnetic susceptibility of hydrogenic donor impurity in a V-groove GaAs/Ga1xAlxAs quantum wire, Eur. Phys. J.B. 78 (2010) 399–404. [11] B. Gülveren, Ü. Atav, M. Tomak, Electronic properties of a large quantum dot at a finite temperature, Physica E 28 (2005) 482–490. [12] B. Gülveren, Ü. Atav, M. Sahin, M. Tomak, A parabolic quantum dot with N electrons and an impurity, Physica E 30 (2005) 143–149. [13] E. Räsänen, J. Könemann, R.J. Puska, M.J. Haug, R.M. Nieminen, Impurity effects in quantum dots: toward quantitative modeling, Phys. Rev. B 70 (2004) 115308. 6p. [14] M. Aichinger, S.A. Chin, E. Krotscheck, E. Räsänen, Effects of geometry and impurities on quantum rings in magnetic fields, Phys. Rev. B 73 (2006) 195310. 8p. [15] S. Elagoz, R. Amca, S. Kutlu, I. Sökmen, Shallow impurity binding energy in lateral parabolic confinement under an external magnetic field, Superlattices Microstruct. 44 (2008) 802–808. [16] 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. Mater. 19 (2007) 395024. 9p. [17] S. Baskoutas, A.F. Terzis, E. Voutsinas, Binding energy of donor states in a quantum dot with parabolic confinement, J. Comput. Theor. Nanosci. 1 (2004) 317–321. [18] S.V. Nistor, M. Stefan, L.C. Nistor, E. Goovaerts, G. VanTendeloo, Incorporation and localization of substitutional Mn2+ ions in cubic ZnS quantum dots, Phys. Rev. B 81 (2010) 035336. 6p. [19] S.V. Nistor, L.C. Nistor, M. Stefan, C.D. Mateescua, R. Birjega, N. Solovieva, M. Nikl, Synthesis and characterization of Mn2+ doped ZnS nanocrystals self-assembled in a tight mesoporous structure, Superlattices Microstruct. 46 (2009) 306–311. [20] M. Stefan, S.V. Nistor, D. Ghica, C.D. Mateescu, M. Nikl, R. Kucerkova, Substitutional and surface Mn2+ centers in cubic ZnS: Mn nanocrystals. A correlated EPR and photoluminescence study, Phys. Rev. B 83 (2011) 045301. 11p. [21] S.V. Nistor, D. Ghica, L.C. Nistor, M. Stefan, C.D. Mateescu, Local structure at Mn2+ ions in vacuum annealed small cubic ZnS nanocrystals self-assembled into a mesoporous structure, J. Nanosci. Nanotechnol. 11 (2011) 9296–9303. [22] C.A. Duque, N. Porras-Montenegro, Z. Barticevic, M. Pacheco, L.E. Oliveira, Electron-hole transitions in self-assembled InAs/ GaAs quantum dots: effects of applied magnetic fields and hydrostatic pressure, Microelectron. J. 36 (2005) 231–233. [23] 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.: Condens. Matter. 18 (2006) 1877– 1884. [24] E. Voutsinas, J. Boviatsis, A. Fountoulakis, Control of a double quantum dot structure by stimulated Raman adiabatic passage, Phys. Status Solidi C 4 (2007) 439–441. [25] E. Paspalakis, A.F. Terzis, in: Proceedings of the 5th WSEAS International Conference on Microelectronics, Nanoelectronics, Optoelectronics, Prague, Czech Republic, March 12–14, 2006, pp. 44–49. [26] A. Fountoulakis, A.F. Terzis, E. Paspalakis, Coherent single-electron transfer in coupled quantum dots, J. Appl. Phys. 106 (2009) 074305. 8p. [27] N.K. Datta, S. Ghosh, M. Ghosh, Effect of impurity strength and impurity domain on excitation of doped quantum dot induced by discontinuously reversing pulsed field, J. Appl. Phys. 108 (2010) 104305. 7p. [28] N.K. Datta, S. Ghosh, M. Ghosh, Excitations in doped quantum dot driven by discontinuously and randomly reversing electric field: Influence of impurity, Chem. Phys. 378 (2010) 66–72. [29] S. Baskoutas, A.F. Terzis, Binding energy of hydrogenic impurity states in an inverse parabolic quantum well under static external fields, Eur. Phys. J.B. 69 (2009) 237–244. [30] S. Baskoutas, A.F. Terzis, Binding energy of hydrogenic impurity states in an inverse parabolic quantum well under electric field, Physica E 40 (2008) 1367–1370. [31] W. Xie, Q. Xie, Electric field effects of hydrogenic impurity in a disc-like quantum dot, Physica B 404 (2009) 1625–1628. [32] J. Boviatsis, E. Voutsinas, A. Fountoulakis, Entanglement of two interacting electrons in a driven double quantum dot structure, Superlattices Microstruct. 41 (2007) 396–399. [33] V. Halonen, P. Hyvönen, P. Pietiläinen, T. Chakraborty, Effects of scattering centers on the energy spectrum of a quantum dot, Phys. Rev. B 53 (1996) 6971–6974. [34] V. Halonen, P. Pietiläinen, T. Chakraborty, Optical-absorption spectra of quantum dots and rings with a repulsive scattering centre, Europhys. Lett. 33 (1996) 337–382. [35] L. Jacak, P. Hawrylak, A. Wojos, Quantum Dots, Springer-Verlag, Berlin, 1998. [36] T. Chakraborty, Quantum Dots-A Survey of the Properties of Artificial Atoms, Elsevier, Amsterdam, 1999. [37] J. Adamowski, A. Kwas´niowski, B. Szafran, LO-phonon-induced screening of electronelectron interaction in D centres and quantum dots, J. Phys.: Cond. Mater. 17 (2005) 4489–4500. [38] S. Bednarek, B. Szafran, K. Lis, J. Adamowski, Modeling of electronic properties of electrostatic quantum dots, Phys. Rev. B 68 (2003) 155333. 9p. [39] B. Szafran, S. Bednarek, J. Adamowski, Parity symmetry and energy spectrum of excitons in coupled self-assembled quantum dots, Phys. Rev. B 64 (2001) 125301. 10p. [40] A. Gharaati, R. Khordad, A new confinement potential in spherical quantum dots: modified Gaussian potential, Superlattices Microstruct. 48 (2010) 276–287. [41] I. Karabulut, S. Baskoutas, Second and third harmonic generation susceptibilities of spherical quantum dots: effects of impurities, electric field and size, J. Comput. Theor. Nanosci. 6 (2009) 153–156. [42] W. Xie, Binding energy of an off-center hydrogenic donor in a spherical Gaussian quantum dot, Physica B 403 (2008) 2828– 2831. [43] Z. Zeng, C.S. Garoufalis, S. Baskoutas, A.F. Terzis, Tuning the binding energy of surface impurities in cylindrical GaAs/AlGaAs quantum dots by a tilted magnetic field, J. Appl. Phys. 112 (2012) 064326. 5p.