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Influence of binding energy on dipole moment, polarizability and self-polarization effect of impurity doped quantum dots: Role of noise
Accepted Manuscript Influence of binding energy on dipole moment, polarizability and self-polarization effect of impurity doped quantum dots: Role of noise Anuja Ghosh, Aindrila Bera, Manas Ghosh PII: DOI: Reference:
S0009-2614(17)30364-0 http://dx.doi.org/10.1016/j.cplett.2017.04.042 CPLETT 34731
To appear in:
Chemical Physics Letters
Received Date: Accepted Date:
17 March 2017 12 April 2017
Please cite this article as: A. Ghosh, A. Bera, M. Ghosh, Influence of binding energy on dipole moment, polarizability and self-polarization effect of impurity doped quantum dots: Role of noise, Chemical Physics Letters (2017), doi: http://dx.doi.org/10.1016/j.cplett.2017.04.042
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 proof before it is published in its final 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.
Influence of binding energy on dipole moment, polarizability and self-polarization effect of impurity doped quantum dots: Role of noise Anuja Ghosh, Aindrila Bera and Manas Ghosh
∗ †‡
Department of Chemistry, Physical Chemistry Section, Visva Bharati University, Santiniketan, Birbhum 731 235, West Bengal, India.
Abstract Present study inspects the profiles of electric dipole moment (DPM), polarizability (αp ) and self-polarization effect (SPE) of doped GaAs quantum dots (QDs) in presence of noise. Special stress has been given on understanding the role of binding energy (BE). Noise term maintains a Gaussian white character and it has been introduced to the system additively and multiplicatively. Application of noise affects the above properties noticeably with conspicuous dependence on the pathway of application. The findings reveal feasible routes to tune the above aspects of doped QD system through expedient adjustment of BE, particularly in presence of noise. Keywords: quantum dot; impurity; binding energy; dipole moment; polarizability; self-polarization effect; Gaussian white noise
∗ † ‡
e-mail address:[email protected] Phone:(+91)(3463)261526, (3463)262751-6 (Ext. 467) Fax : +91 3463 262672
1
I.
INTRODUCTION
Low-dimensional semiconductor systems (LDSS) such as quantum wells (QWLs), quantum wires (QWRs) and quantum dots (QDs) are frequently invoked as candidates for appreciating and testing many important physical concepts and technological applications. Presence of impurities in LDSS has become a rule rather than an exception. Impurity affects the probability distribution of electrons and consequently the spatial disposition of energy levels. The squeezed spatial freedom in LDSS drives the electrons spend a considerable amount of time near impurity. Such closeness enhances their mutual interaction reflected through increase in the binding energy (BE) of the system. In this way, presence of impurities results in alterations in the thermal, transport, optical and electrical properties of LDSS that deserve mention, particularly at low temperatures. As an obvious outcome, we find rich abundance of important studies on several properties of LDSS containing impurity [1–27]. Presence of electric field in LDSS can cause polarization of carrier distribution, shift in the energy of the eigenstates and alteration of effective confinement potential. As a result, the energy spectrum of the confined states of the carriers is also changed accordingly. In consequence, the electronic and optical properties of LDSS are also changed with promising scope of development of novel optoelectronic devices. All such alterations have profound impact on the nonlinear optical (NLO) properties of LDSS. Determination of electric dipole moment, polarizability and self-polarization effect (SPE) of LDSS is of prime importance as it sheds light on the carrier dynamics and NLO properties of them. Thus, there exist a wealth of studies linked with electric dipole moment [28–30], polarizability [31–46] and SPE [47–49] of LDSS. SPE is a measure of the extent to which the well-potential influences impurity. In this case the electronic wave function is affected by the resultant effect of impurity and the well potential. In effect, the well potential (confinement potential) drives the shifting of electronic probability density distribution with respect to the site of dopant incorporation. The resulting polarization is called SPE. It needs to be mentioned now that presence of noise in LDSS can significantly alter its functioning. Noise can result externally, or it may be inherent, originating from the variations in the QD structure in the neighborhood of impurity. It is therefore quite pertinent to inquire how application of noise affects various properties of LDSS containing impurity.
2
Recently, we have explored SPE of doped QD in presence of noise in a different context [50]. In the present study we explore the electric dipole moment (µ), polarizability (αp ) and self-polarization effect (SPE) of GaAs QD containing impurity under the governance of noise with special emphasis on the role played by binding energy (BE). Despite a rigorous literature survey we could not find such type of study in presence of noise. However, effect of BE on polarizability of LDSS has been studied by Morales et al. [31] and Zounoubi et al. [41] whereas similar study on SPE of LDSS has been conducted by Erdogan et al. [49]. At this point of discussion it needs to be mentioned that the present study differs in some respects from other studies cited above. For instance, in their work Morales et al. have considered the effects of both electric field and hydrostatic pressure on polarizability of GaAs − (Ga, Al)As quantum well [31]. Moreover, Zounoubi et al. have put special emphasis on understanding the role of magnetic field on the polarizability of cylindrical GaAs QD [41]. The investigation bears importance as the application of magnetic field to a crystal changes the dimensionality of electronic levels and leads to a redistribution of the density of states. However, in the present communication the effect of hydrostatic pressure has not been considered and no special stress has been applied in understanding the exhaustive role played by magnetic field on polarizability. In case of SPE, Erdogan et al. inspected the role of energy density for square and cylindrical GaAs/AlAs quantum well-wires under the simultaneous presence of electric and magnetic field [49]. Present study, however, refrains from studying the effect of energy density on SPE. The main point of difference being the fact that the present work incorporates noise-effect on polarizability and SPE which has not been explored by above workers. Present enquiry represents a 2-d QD (GaAs ) containing a single electron. The QD is exposed to a static electric field. The confinement can be described by a parabolic potential in the x −y plane. A perpendicular magnetic field is also present providing additional
confinement. The QD is doped with impurity represented by a Gaussian potential. The system is further subjected to externally introduced Gaussian white noise that produces considerable disorder in the system. We can conceive two distinct roadways (modes) through which noise can be introduced to the system. These two modes are known as additive and multiplicative and they can be discriminated on the basis of strength of system-noise interaction. The investigation illuminates how subtly noise (with obvious emphasis on its mode of application) supervises the electric dipole moment, polarizability and SPE of doped 3
QD when BE varies over a range. To be more specific, our focus lies on monitoring the profiles of above quantities particularly as a function of BE in two different types of noisy environments under given strengths of electric and magnetic fields.
II.
METHOD
The system Hamiltonian containing impurity (H0 ) is a combination of four terms and can be given by H0 = H00 + Vimp + |e|F (x + y) + Vnoise ,
(1)
where H00 , Vimp , |e|F (x + y) and Vnoise represent the impurity-free Hamiltonian, the impurity potential, the applied static electric field (along x and y axes) of strength F and the noise term, respectively. |e| is the absolute value of electron charge. The noise term follows a Gaussian distribution and is endowed with zero average and spatial δ-correlation conditions. Moreover, depending upon the mode of application, noise may be either additive or multiplicative. Subject to the lateral parabolic confinement in the x−y plane and an orthogonal magnetic field, the effective mass approximation gives H00 =
1 h e i2 1 ∗ 2 2 −i~∇ + A + m ω0 (x + y 2 ). 2m∗ c 2
(2)
m∗ and ω0 are the effective mass of the electron and the harmonic confinement frequency, respectively. A denotes the vector potential and in Landau gauge it reads A = (By, 0, 0), where B is the magnetic field strength. In this gauge H00 can alternatively be given by 2 ~2 ∂ ∂2 1 1 ∂ 0 H0 = − ∗ + 2 + m∗ ω02 x2 + m∗ Ω2 y 2 − i~ωc y . (3) 2 2m ∂x ∂y 2 2 ∂x p Ω(= ω02 + ωc2 ) indicates the effective confinement frequency in the y-direction and ωc (= eB ) m∗ c
represents the cyclotron frequency which visibly depends on magnetic field strength.
In the present study the impurity (dopant) potential (Vimp ) is represented by a Gaussian 2 2 function and is given by V = V e−γ [(x−x0 ) +(y−y0 ) ] . The important parameters pertaining imp
0
to this dopant potential are (x0 , y0 ), V0 and γ −1/2 . They stand for the location of dopant inclusion, the strength of the dopant potential, and the spatial domain over which the impurity potential is disseminated, respectively. γ can be viewed as γ = kε, where k is a constant 4
and ε is the dielectric constant of the medium. We have invoked Box-Muller algorithm to generate the characteristic noise term having strength ζ. We have constructed the Hamiltonian matrix (H0 ) in the direct product basis of the harmonic oscillator eigenfunctions. The said matrix is then diagonalized to obtain the energy eigenvalues and the eigenfunctions. In this context the required convergence test has been carried out. Polarizability (αp ) and dipole moment (µ) are given by: αp = −
e [hψ|x + y|ψiF 6=0 − hψ|x + y|ψiF =0 ] , F
(4)
where ψ is the wave function of the system and µ = αp F.
(5)
However, during the calculation of dipole moment we consider the dimensionless quantity µ/µ0 , where µ0 = ea0 and a0 is the effective Bohr radius. And SPE of impurity doped QD, on the other hand, can be given by [50] P = −hψ| (x − x0 ) |ψi + hψ 0 | (x − x0 ) |ψ 0 i − hψ| (y − y0 ) |ψi + hψ 0 | (y − y0 ) |ψ 0 i, e
(6)
where ψ is the wave function describing the system and ψ 0 is the wave function in absence of confinement effects. BE (EB ) for the ground state is defined as the difference between the ground state energies in absence and in presence of impurity and is given by EB = E0 − E,
(7)
where E0 is the ground state energy in absence of impurity and E is the same in presence of impurity.
III.
RESULTS AND DISCUSSION
The calculations are performed using ε = 12.4, m∗ = 0.067m0 , where m0 is the free electron mass. The values of the other parameters are: ~ω0 = 250.0 meV, F = 100 KV/cm, B = 20.0 T, V0 = 280.0 meV, r0 = 0.0 nm, and ζ = 1.0 × 10−2 . The parameters are suitable for GaAs QDs.
5
A.
Polarizability (αp ):
Fig. 1 evinces the change of αp with BE when noise is absent [fig. 1(i)] and when noise is introduced via additive [fig. 1(ii)] and multiplicative [fig. 1(iii)] pathways, respectively. In absence of noise αp decreases steadily with increase in BE [31, 41] although the rate of (i) (ii) (iii)
1.0 0.8
5
αP x 10 Å
3
(i)
0.6 0.4
(ii)
0.2 (iii)
0.0 0
50
100
150
200
250
BE (meV)
FIG. 1: Plots of αp vs BE: (i) without noise, (ii) noise applied in additive mode and (iii) noise applied in multiplicative mode.
decrease falls at large BE (> 150 meV). However, in presence of noise, initially αp decreases with increase in BE up to BE ∼ 100 meV beyond which polarizability does not change any further with increase in BE. The behavior divulges progressive compression in the spatial spread of wave function with increase in BE. In absence of noise, the extent of compression falls a bit at high value of BE. However, in presence of noise, the compression terminates after a typical value of BE. Fig. 1 also reveals that the presence of noise causes drastic reduction in the αp value from that of noise-free condition. In other words, application of noise causes significant shrinkage of spatial extension of wave function. The role of mode of application of noise becomes also evident from the above plot. This is because of the fact that the extent of drop in the αp value has been found to be more with multiplicative noise than its additive analogue. This is because of the fact that, multiplicative noise - by virtue of its mode of application - undergoes more close connection with the system than its additive counterpart leading to greater suppression of spatial spread of wave function.
6
B.
Dipole moment (DPM):
Fig. 2 exhibits the analogous profile for DPM consisting of similar features. It can (i) (ii) (iii)
7.0
µ/µ0
5.6
(i)
4.2 2.8
(ii)
1.4 (iii)
0.0 0
50
100
150
200
250
BE (meV)
FIG. 2: Plots of µ/µ0 vs BE: (i) without noise, (ii) noise applied in additive mode and (iii) noise applied in multiplicative mode.
therefore be inferred that, in absence of noise, the dot-impurity separation undergoes steady decline with increase in BE. The extent of decline, slows down a little bit at high BE value (> 150 meV). In presence of noise the said decline ceases beyond BE ∼ 100 meV. Moreover, presence of noise happens to cause a drastic reduction in the dot-impurity separation and DPM falls severely from that of noise-free value. And the reduction in DPM (from noise-free state) becomes more severe in presence of multiplicative noise than its additive relative.
C.
Self-polarization effect (SPE):
Fig. 3 evinces the change of SPE with BE when noise is absent [fig. 3(i)] and when noise is introduced via additive [fig. 3(ii)] and multiplicative [fig. 3(iii)] pathways, respectively. Both in absence and presence of noise SPE decreases moderately in a steady way with increase in BE [49]. The behavior divulges progressive compression in the spatial elongation of wave function with increase in BE under all conditions. However, the magnitude of SPE is much reduced in presence of noise than under noise-free condition. In other words, it is again revealed that the incorporation of noise induces considerable shrinkage of spatial extension of wave function. As found earlier, also in the present case multiplicative noise causes greater diminish of SPE than its additive analogue (from noise-free situation). 7
80.0 (i)
P/e (Å)
60.0 (ii)
40.0
20.0
(iii)
(i) (ii) (iii)
0.0 0
50
100
150
200
250
BE (meV)
FIG. 3: Plots of SPE vs BE: (i) without noise, (ii) noise applied in additive mode and (iii) noise applied in multiplicative mode. IV.
CONCLUSION
The profiles of polarizability, dipole moment and self-polarization effect of doped GaAs QD have been deciphered under the aegis of noise with special emphasis on the role played by binding energy. In absence of noise αp decreases steadily with increase in BE although the rate of decrease falls at large BE (> 150 meV). However, in presence of noise, initially αp decreases with increase in BE up to BE ∼ 100 meV beyond which polarizability does not change any further. Moreover, presence of noise causes drastic reduction in the αp value from that of noise-free condition. Also, the extent of drop in the αp value has been found to be more with multiplicative noise than its additive analogue. DPM reveals similar features as that of αp . SPE, on the other hand, decreases moderately in a steady way with increase in BE both in absence and presence of noise. Furthermore, multiplicative noise causes greater fall of SPE than its additive analogue. Morales et al. have found quite similar observation since polarizability follows the BE variation very closely, i.e., a smallest BE ensures that the system is in its maximum deformation leading to the highest value of polarizability [31]. The observations of Zounoubi et al. also appear to be in agreement as they have reported that an increase in the magnetic field strength increases BE and strongly reduces the polarizability [41]. In case of SPE, Erdogan et al. have reported that SPE decreases linearly as BE increases [49]. In the present study we also find similar linear drop in SPE as BE increases. However, the drop becomes slightly nonlinear at high BE value. On the whole, present work reveals that a variation of BE, in effect, controls the effective confinement of the system. 8
The effective confinement governs the spatial elongation of wave function and the extent of dot-impurity separations which are reflected through the profiles of polarizability, dipole moment and SPE. Application of noise chiefly modifies the magnitude of polarizability, dipole moment and SPE from that of noise-free situation. However, the nature of such modification manifestly depends on the mode of introduction of noise. The results seem to carry ample significance in the related field of research.
V.
ACKNOWLEDGEMENTS
The authors A. G., A. B. 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.
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10
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11
1.0
(i) noise-free (ii) additive noise (iii) multiplicative noise
0.8
5
P x 10 Å
3
(i)
0.6 0.4
(ii)
0.2 (iii)
0.0 0
50
100
150
BE (meV)
200
250
Research Highlights
Dipole moment, polarizability and self-polarization effect (SPE) of doped quantum dot are studied.
The dot is subjected to Gaussian white noise.
Special emphasis is put on role of binding energy (BE).
Interplay between noise and BE prominently affects above properties.
S0009-2614(17)30364-0 http://dx.doi.org/10.1016/j.cplett.2017.04.042 CPLETT 34731
To appear in:
Chemical Physics Letters
Received Date: Accepted Date:
17 March 2017 12 April 2017
Please cite this article as: A. Ghosh, A. Bera, M. Ghosh, Influence of binding energy on dipole moment, polarizability and self-polarization effect of impurity doped quantum dots: Role of noise, Chemical Physics Letters (2017), doi: http://dx.doi.org/10.1016/j.cplett.2017.04.042
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 proof before it is published in its final 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.
Influence of binding energy on dipole moment, polarizability and self-polarization effect of impurity doped quantum dots: Role of noise Anuja Ghosh, Aindrila Bera and Manas Ghosh
∗ †‡
Department of Chemistry, Physical Chemistry Section, Visva Bharati University, Santiniketan, Birbhum 731 235, West Bengal, India.
Abstract Present study inspects the profiles of electric dipole moment (DPM), polarizability (αp ) and self-polarization effect (SPE) of doped GaAs quantum dots (QDs) in presence of noise. Special stress has been given on understanding the role of binding energy (BE). Noise term maintains a Gaussian white character and it has been introduced to the system additively and multiplicatively. Application of noise affects the above properties noticeably with conspicuous dependence on the pathway of application. The findings reveal feasible routes to tune the above aspects of doped QD system through expedient adjustment of BE, particularly in presence of noise. Keywords: quantum dot; impurity; binding energy; dipole moment; polarizability; self-polarization effect; Gaussian white noise
∗ † ‡
e-mail address:[email protected] Phone:(+91)(3463)261526, (3463)262751-6 (Ext. 467) Fax : +91 3463 262672
1
I.
INTRODUCTION
Low-dimensional semiconductor systems (LDSS) such as quantum wells (QWLs), quantum wires (QWRs) and quantum dots (QDs) are frequently invoked as candidates for appreciating and testing many important physical concepts and technological applications. Presence of impurities in LDSS has become a rule rather than an exception. Impurity affects the probability distribution of electrons and consequently the spatial disposition of energy levels. The squeezed spatial freedom in LDSS drives the electrons spend a considerable amount of time near impurity. Such closeness enhances their mutual interaction reflected through increase in the binding energy (BE) of the system. In this way, presence of impurities results in alterations in the thermal, transport, optical and electrical properties of LDSS that deserve mention, particularly at low temperatures. As an obvious outcome, we find rich abundance of important studies on several properties of LDSS containing impurity [1–27]. Presence of electric field in LDSS can cause polarization of carrier distribution, shift in the energy of the eigenstates and alteration of effective confinement potential. As a result, the energy spectrum of the confined states of the carriers is also changed accordingly. In consequence, the electronic and optical properties of LDSS are also changed with promising scope of development of novel optoelectronic devices. All such alterations have profound impact on the nonlinear optical (NLO) properties of LDSS. Determination of electric dipole moment, polarizability and self-polarization effect (SPE) of LDSS is of prime importance as it sheds light on the carrier dynamics and NLO properties of them. Thus, there exist a wealth of studies linked with electric dipole moment [28–30], polarizability [31–46] and SPE [47–49] of LDSS. SPE is a measure of the extent to which the well-potential influences impurity. In this case the electronic wave function is affected by the resultant effect of impurity and the well potential. In effect, the well potential (confinement potential) drives the shifting of electronic probability density distribution with respect to the site of dopant incorporation. The resulting polarization is called SPE. It needs to be mentioned now that presence of noise in LDSS can significantly alter its functioning. Noise can result externally, or it may be inherent, originating from the variations in the QD structure in the neighborhood of impurity. It is therefore quite pertinent to inquire how application of noise affects various properties of LDSS containing impurity.
2
Recently, we have explored SPE of doped QD in presence of noise in a different context [50]. In the present study we explore the electric dipole moment (µ), polarizability (αp ) and self-polarization effect (SPE) of GaAs QD containing impurity under the governance of noise with special emphasis on the role played by binding energy (BE). Despite a rigorous literature survey we could not find such type of study in presence of noise. However, effect of BE on polarizability of LDSS has been studied by Morales et al. [31] and Zounoubi et al. [41] whereas similar study on SPE of LDSS has been conducted by Erdogan et al. [49]. At this point of discussion it needs to be mentioned that the present study differs in some respects from other studies cited above. For instance, in their work Morales et al. have considered the effects of both electric field and hydrostatic pressure on polarizability of GaAs − (Ga, Al)As quantum well [31]. Moreover, Zounoubi et al. have put special emphasis on understanding the role of magnetic field on the polarizability of cylindrical GaAs QD [41]. The investigation bears importance as the application of magnetic field to a crystal changes the dimensionality of electronic levels and leads to a redistribution of the density of states. However, in the present communication the effect of hydrostatic pressure has not been considered and no special stress has been applied in understanding the exhaustive role played by magnetic field on polarizability. In case of SPE, Erdogan et al. inspected the role of energy density for square and cylindrical GaAs/AlAs quantum well-wires under the simultaneous presence of electric and magnetic field [49]. Present study, however, refrains from studying the effect of energy density on SPE. The main point of difference being the fact that the present work incorporates noise-effect on polarizability and SPE which has not been explored by above workers. Present enquiry represents a 2-d QD (GaAs ) containing a single electron. The QD is exposed to a static electric field. The confinement can be described by a parabolic potential in the x −y plane. A perpendicular magnetic field is also present providing additional
confinement. The QD is doped with impurity represented by a Gaussian potential. The system is further subjected to externally introduced Gaussian white noise that produces considerable disorder in the system. We can conceive two distinct roadways (modes) through which noise can be introduced to the system. These two modes are known as additive and multiplicative and they can be discriminated on the basis of strength of system-noise interaction. The investigation illuminates how subtly noise (with obvious emphasis on its mode of application) supervises the electric dipole moment, polarizability and SPE of doped 3
QD when BE varies over a range. To be more specific, our focus lies on monitoring the profiles of above quantities particularly as a function of BE in two different types of noisy environments under given strengths of electric and magnetic fields.
II.
METHOD
The system Hamiltonian containing impurity (H0 ) is a combination of four terms and can be given by H0 = H00 + Vimp + |e|F (x + y) + Vnoise ,
(1)
where H00 , Vimp , |e|F (x + y) and Vnoise represent the impurity-free Hamiltonian, the impurity potential, the applied static electric field (along x and y axes) of strength F and the noise term, respectively. |e| is the absolute value of electron charge. The noise term follows a Gaussian distribution and is endowed with zero average and spatial δ-correlation conditions. Moreover, depending upon the mode of application, noise may be either additive or multiplicative. Subject to the lateral parabolic confinement in the x−y plane and an orthogonal magnetic field, the effective mass approximation gives H00 =
1 h e i2 1 ∗ 2 2 −i~∇ + A + m ω0 (x + y 2 ). 2m∗ c 2
(2)
m∗ and ω0 are the effective mass of the electron and the harmonic confinement frequency, respectively. A denotes the vector potential and in Landau gauge it reads A = (By, 0, 0), where B is the magnetic field strength. In this gauge H00 can alternatively be given by 2 ~2 ∂ ∂2 1 1 ∂ 0 H0 = − ∗ + 2 + m∗ ω02 x2 + m∗ Ω2 y 2 − i~ωc y . (3) 2 2m ∂x ∂y 2 2 ∂x p Ω(= ω02 + ωc2 ) indicates the effective confinement frequency in the y-direction and ωc (= eB ) m∗ c
represents the cyclotron frequency which visibly depends on magnetic field strength.
In the present study the impurity (dopant) potential (Vimp ) is represented by a Gaussian 2 2 function and is given by V = V e−γ [(x−x0 ) +(y−y0 ) ] . The important parameters pertaining imp
0
to this dopant potential are (x0 , y0 ), V0 and γ −1/2 . They stand for the location of dopant inclusion, the strength of the dopant potential, and the spatial domain over which the impurity potential is disseminated, respectively. γ can be viewed as γ = kε, where k is a constant 4
and ε is the dielectric constant of the medium. We have invoked Box-Muller algorithm to generate the characteristic noise term having strength ζ. We have constructed the Hamiltonian matrix (H0 ) in the direct product basis of the harmonic oscillator eigenfunctions. The said matrix is then diagonalized to obtain the energy eigenvalues and the eigenfunctions. In this context the required convergence test has been carried out. Polarizability (αp ) and dipole moment (µ) are given by: αp = −
e [hψ|x + y|ψiF 6=0 − hψ|x + y|ψiF =0 ] , F
(4)
where ψ is the wave function of the system and µ = αp F.
(5)
However, during the calculation of dipole moment we consider the dimensionless quantity µ/µ0 , where µ0 = ea0 and a0 is the effective Bohr radius. And SPE of impurity doped QD, on the other hand, can be given by [50] P = −hψ| (x − x0 ) |ψi + hψ 0 | (x − x0 ) |ψ 0 i − hψ| (y − y0 ) |ψi + hψ 0 | (y − y0 ) |ψ 0 i, e
(6)
where ψ is the wave function describing the system and ψ 0 is the wave function in absence of confinement effects. BE (EB ) for the ground state is defined as the difference between the ground state energies in absence and in presence of impurity and is given by EB = E0 − E,
(7)
where E0 is the ground state energy in absence of impurity and E is the same in presence of impurity.
III.
RESULTS AND DISCUSSION
The calculations are performed using ε = 12.4, m∗ = 0.067m0 , where m0 is the free electron mass. The values of the other parameters are: ~ω0 = 250.0 meV, F = 100 KV/cm, B = 20.0 T, V0 = 280.0 meV, r0 = 0.0 nm, and ζ = 1.0 × 10−2 . The parameters are suitable for GaAs QDs.
5
A.
Polarizability (αp ):
Fig. 1 evinces the change of αp with BE when noise is absent [fig. 1(i)] and when noise is introduced via additive [fig. 1(ii)] and multiplicative [fig. 1(iii)] pathways, respectively. In absence of noise αp decreases steadily with increase in BE [31, 41] although the rate of (i) (ii) (iii)
1.0 0.8
5
αP x 10 Å
3
(i)
0.6 0.4
(ii)
0.2 (iii)
0.0 0
50
100
150
200
250
BE (meV)
FIG. 1: Plots of αp vs BE: (i) without noise, (ii) noise applied in additive mode and (iii) noise applied in multiplicative mode.
decrease falls at large BE (> 150 meV). However, in presence of noise, initially αp decreases with increase in BE up to BE ∼ 100 meV beyond which polarizability does not change any further with increase in BE. The behavior divulges progressive compression in the spatial spread of wave function with increase in BE. In absence of noise, the extent of compression falls a bit at high value of BE. However, in presence of noise, the compression terminates after a typical value of BE. Fig. 1 also reveals that the presence of noise causes drastic reduction in the αp value from that of noise-free condition. In other words, application of noise causes significant shrinkage of spatial extension of wave function. The role of mode of application of noise becomes also evident from the above plot. This is because of the fact that the extent of drop in the αp value has been found to be more with multiplicative noise than its additive analogue. This is because of the fact that, multiplicative noise - by virtue of its mode of application - undergoes more close connection with the system than its additive counterpart leading to greater suppression of spatial spread of wave function.
6
B.
Dipole moment (DPM):
Fig. 2 exhibits the analogous profile for DPM consisting of similar features. It can (i) (ii) (iii)
7.0
µ/µ0
5.6
(i)
4.2 2.8
(ii)
1.4 (iii)
0.0 0
50
100
150
200
250
BE (meV)
FIG. 2: Plots of µ/µ0 vs BE: (i) without noise, (ii) noise applied in additive mode and (iii) noise applied in multiplicative mode.
therefore be inferred that, in absence of noise, the dot-impurity separation undergoes steady decline with increase in BE. The extent of decline, slows down a little bit at high BE value (> 150 meV). In presence of noise the said decline ceases beyond BE ∼ 100 meV. Moreover, presence of noise happens to cause a drastic reduction in the dot-impurity separation and DPM falls severely from that of noise-free value. And the reduction in DPM (from noise-free state) becomes more severe in presence of multiplicative noise than its additive relative.
C.
Self-polarization effect (SPE):
Fig. 3 evinces the change of SPE with BE when noise is absent [fig. 3(i)] and when noise is introduced via additive [fig. 3(ii)] and multiplicative [fig. 3(iii)] pathways, respectively. Both in absence and presence of noise SPE decreases moderately in a steady way with increase in BE [49]. The behavior divulges progressive compression in the spatial elongation of wave function with increase in BE under all conditions. However, the magnitude of SPE is much reduced in presence of noise than under noise-free condition. In other words, it is again revealed that the incorporation of noise induces considerable shrinkage of spatial extension of wave function. As found earlier, also in the present case multiplicative noise causes greater diminish of SPE than its additive analogue (from noise-free situation). 7
80.0 (i)
P/e (Å)
60.0 (ii)
40.0
20.0
(iii)
(i) (ii) (iii)
0.0 0
50
100
150
200
250
BE (meV)
FIG. 3: Plots of SPE vs BE: (i) without noise, (ii) noise applied in additive mode and (iii) noise applied in multiplicative mode. IV.
CONCLUSION
The profiles of polarizability, dipole moment and self-polarization effect of doped GaAs QD have been deciphered under the aegis of noise with special emphasis on the role played by binding energy. In absence of noise αp decreases steadily with increase in BE although the rate of decrease falls at large BE (> 150 meV). However, in presence of noise, initially αp decreases with increase in BE up to BE ∼ 100 meV beyond which polarizability does not change any further. Moreover, presence of noise causes drastic reduction in the αp value from that of noise-free condition. Also, the extent of drop in the αp value has been found to be more with multiplicative noise than its additive analogue. DPM reveals similar features as that of αp . SPE, on the other hand, decreases moderately in a steady way with increase in BE both in absence and presence of noise. Furthermore, multiplicative noise causes greater fall of SPE than its additive analogue. Morales et al. have found quite similar observation since polarizability follows the BE variation very closely, i.e., a smallest BE ensures that the system is in its maximum deformation leading to the highest value of polarizability [31]. The observations of Zounoubi et al. also appear to be in agreement as they have reported that an increase in the magnetic field strength increases BE and strongly reduces the polarizability [41]. In case of SPE, Erdogan et al. have reported that SPE decreases linearly as BE increases [49]. In the present study we also find similar linear drop in SPE as BE increases. However, the drop becomes slightly nonlinear at high BE value. On the whole, present work reveals that a variation of BE, in effect, controls the effective confinement of the system. 8
The effective confinement governs the spatial elongation of wave function and the extent of dot-impurity separations which are reflected through the profiles of polarizability, dipole moment and SPE. Application of noise chiefly modifies the magnitude of polarizability, dipole moment and SPE from that of noise-free situation. However, the nature of such modification manifestly depends on the mode of introduction of noise. The results seem to carry ample significance in the related field of research.
V.
ACKNOWLEDGEMENTS
The authors A. G., A. B. 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.
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11
1.0
(i) noise-free (ii) additive noise (iii) multiplicative noise
0.8
5
P x 10 Å
3
(i)
0.6 0.4
(ii)
0.2 (iii)
0.0 0
50
100
150
BE (meV)
200
250
Research Highlights
Dipole moment, polarizability and self-polarization effect (SPE) of doped quantum dot are studied.
The dot is subjected to Gaussian white noise.
Special emphasis is put on role of binding energy (BE).
Interplay between noise and BE prominently affects above properties.