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Optical response in a laser-driven quantum pseudodot system
Author’s Accepted Manuscript Optical response in a laser-driven quantum pseudodot system D. Gul Kilic, S. Sakiroglu, F. Ungan, U. Yesilgul, E. Kasapoglu, H. Sari, I. Sokmen www.elsevier.com/locate/physb
PII: DOI: Reference:
S0921-4526(16)30617-2 http://dx.doi.org/10.1016/j.physb.2016.12.031 PHYSB309772
To appear in: Physica B: Physics of Condensed Matter Received date: 19 December 2016 Accepted date: 29 December 2016 Cite this article as: D. Gul Kilic, S. Sakiroglu, F. Ungan, U. Yesilgul, E. Kasapoglu, H. Sari and I. Sokmen, Optical response in a laser-driven quantum pseudodot system, Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2016.12.031 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.
Optical response in a laser-driven quantum pseudodot system D. Gul Kilica , S. Sakiroglub,∗, F. Unganc , U. Yesilgulc , E. Kasapoglud , H. Sarie , I. Sokmenb a
Physics Department, Graduate School of Natural and Applied Sciences, Dokuz Eyl¨ ul University, 35390 Izmir, Turkey b Physics Department, Faculty of Science, Dokuz Eyl¨ ul University, 35390 Izmir, Turkey c Department of Optical Engineering, Faculty of Technology, Cumhuriyet University, 58140 Sivas, Turkey d Physics Department, Faculty of Science, Cumhuriyet University, 58140 Sivas, Turkey e Department of Primary Education, Faculty of Education, Cumhuriyet University, 58140 Sivas, Turkey
Abstract We investigate theoretically the intense laser-induced optical absorption coefficients and refractive index changes in a two-dimensional quantum pseudodot system under an uniform magnetic field. The effects of non-resonant, monochromatic intense laser field upon the system are treated within the framework of high-frequency Floquet approach in which the system is supposed to be governed by a laser-dressed potential. Linear and nonlinear absorption coefficients and relative changes in the refractive index are obtained by means of the compact-density matrix approach and iterative method. The results of numerical calculations for a typical GaAs quantum dot reveal that the optical response depends strongly on the magnitude of external magnetic field and characteristic parameters of the confinement potential. Moreover, we have demonstrated that the intense laser field modifies the confinement and thereby causes remarkable changes in the linear and nonlinear optical properties of the system. Keywords: quantum dot, intense laser field, nonlinear optics
1. Introduction Recent progress in nanofabrication technology allows the possibility of realizing quantum dots (QDs) with controllable size, shape and composition [1–3]. As a consequence of quantum confinement effect, formation of discrete energy levels in QDs results in particular electronic and optical properties different than their bulk counterparts. Feasibility of tailoring the energy spectrum to ∗
Corresponding Author. Email address: [email protected] (S. Sakiroglu)
Preprint submitted to Physica B
December 29, 2016
produce enviable optical transitions is important from the aspect of optoelectronic device technology [4]. Hence, many experimental and theoretical researches have been conducted to investigate the size-dependent optoelectronic properties of quantum dots. A great deal of work has been devoted on the linear and nonlinear optical absorption coefficients and relative changes in refractive index owing to small energy separation between the subbands and enhanced electric dipole matrix elements corresponding to transitions between the subbands in low-dimensional systems [5–11]. The intersubband optical absorption coefficients and refractive index changes in a quantum box ¨ with finite confining potential have been calculated by Unlu et al. [12]. The linear and nonlinear absorption coefficients and refractive index changes in parabolic quantum dots in the presence of electric and magnetic fields have been theoretically examined by Zhang et al. [13]. Rezaei and coworkers have considered a two-dimensional elliptic quantum dot with infinite confining potential height and have studied the effects of electromagnetic field intensity, size and shape of dot on the optical properties due to the intersubband transitions [14]. Analysis of the optical absorption coefficients of impurity doped quantum dots under the influence of noise, magnetic and electric fields has been conducted by Mandal et al. [15]. It is well-known that external perturbations give rise to remarkable changes in the electronic and optical response of quantum dots [16, 17]. In this respect, understanding of interaction of intense laser fields (ILFs) with carriers in low-dimensional semiconductor structures can offer the manipulation and control of physical properties of these systems [18]. A large number of studies on matter-field interaction reported in the literature put forth that ILFs can modify confinement potential, reduce energy gaps, distort optical absorption edges, shift critical temperatures of magnetic solids, and etc. [2, 19–22]. Optical absorptions and refractive index changes in parabolic quantum dot with hydrogenic impurity under laser radiation have been studied by Xie [23]. Prasad and Silotia have investigated the optical characteristics of disk shaped quantum dots with simultaneous influence of laser and magnetic field [24]. Linear and nonlinear optical absorption and relative changes in refractive index of spherical quantum dot placed at the center of cylindrical nanowire have been analyzed in Ref. [20]. Recently, intense laser-induced electronic and optical properties of double quantum dots under applied electric field have been reported by Bejan and Niculescu [25]. Though remarkable interest has been shown on the research of electronic and optical properties in the spherical, parabolic, cylindrical and rectangular quantum dots [26–28], the nonlinear optical 2
properties of a two-dimensional quantum pseudodot system under ILF has not been investigated so far. In this paper, we focus on addressing the effects of high-frequency intense laser field on the absorption coefficients and relative changes in the refractive index in two-dimensional quantum dot subjected to an uniform external magnetic field and represented by a pseudoharmonic potential. Effects of an intense laser field have been treated within the framework of laser-dressed potential approach and electronic structure of the system has been calculated by using finite element method [29]. Linear and third-order nonlinear optical absorption coefficients and refractive index changes are obtained from the compact-density matrix approach and iterative procedure. The remaining part of the paper is organized as follows: In Section 2, the theoretical framework is briefly given. Numerical results presented in Section 3 are followed by the conclusions given in Section 4. 2. Theoretical framework The two-dimensional quantum dot system is described by the confining potential that includes both the dot and antidot harmonic potential [30]: r0 2 r − , V (r) = V0 r0 r
(1)
where V0 is the chemical potential of the two-dimensional electron gas and r0 is the zero-point of the pseudoharmonic potential. We assume that the system is jointly exposed to a high-frequency laser field and static magnetic field along z−direction defined by the vector potential A(r) = (0, Br/2, 0) . The non-resonant, monochromatic, intense laser beam of frequency Ω is circularly polarized whose x sin Ωt+ yˆ cos Ωt), where x ˆ and yˆ define the unit vectors orthogonal vector potential is A(t) = A0 (−ˆ to the direction of propagation [31]. In the high-frequency intense laser field, the electron motion in quantum dot is determined by the time-averaged laser-dressed potential which is obtained as follows [32]: 1 Vd (r, α0 ) = T
T V (r + α(t))dt ,
(2)
0
where T = 2π/Ω is the period of the radiation. α(t) is related to the vector potential of ILF as t α(t) = A(t )dt and can be interpreted as a vector corresponding to the classical displacement along the polarization direction [32, 33]. Accordingly x cos Ωt + yˆ sin Ωt) , α(t) = α0 (ˆ 3
(3)
where the quiver amplitude of the electron in ILF is defined as the laser-dressing intensity parameter, α0 =
eA0 m∗ cΩ .
Then the Hamiltonian of electron confined in a two-dimensional quantum dot with a laserdressed pseudoharmonic potential and subjected to external static magnetic field is defined by the Hamiltonian H=
1 (p + eA)2 + Vd (r, α0 ) , 2m∗
(4)
where m∗ is the electron effective mass, e is the absolute value of the elementary charge, p is the momentum operator and A(r) is the vector potential corresponding to the magnetic field. By the substitution of Eq. (1) into Eq. (2) the analytical expression for the laser-dressed confinement potential is obtained as
Vd (r, α0 ) = V0
r02 r 2 α20 + − 2 + r02 r02 |r 2 − α20 |
.
(5)
It should be noted that, our calculations we merely performed in the interval α0 < r < ∞ due to the singularities of the potential at r = α0 . The electronic states of an electron can be obtained by the numerical solution of corresponding Schr¨ odinger equation expressed in terms of cylindrical coordinates: ∂ 1 ∂2 ωc ∂Ψ 2 1 ∂ r + 2 2 Ψ−i − ∗ 2m r ∂r ∂r r ∂ϕ 2 ∂ϕ ∗ 2 2 m ωc r Ψ + Vd (r, α0 )Ψ = EΨ , + 8
(6)
where ωc = eB/m∗ is the cyclotron frequency. Choosing the magnetic quantum number m, we can set the wave function Ψ as
eimϕ U (r) √ . Ψ(r, ϕ) = √ 2π r
(7)
∗ = 2 /m∗ a∗ 2 as the length Adopting the effective Bohr radius a∗0 and effective Hartree energy EH 0
and energy scales, respectively yields to
2 (m − 1/4) mγ 1 d2 U (r) + + − 2 2 dr 2r 2 2 2 1 γ ∗ r 2 + Vd (r, α0 )/EH U (r) = EU (r) , + 2 2
(8)
∗ . Numerical calcuwhere magnetic field is defined via the dimensionless parameter γ = ωc /EH
lations for eigenenergies and corresponding wave functions have been carried out by using finite element method. 4
Calculation of the optical absorption coefficients is based on the compact density matrix approach and iterative scheme, for which the total absorption coefficient is given by α(ω, I) = α(1) (ω) + α(3) (ω, I) , where the linear and third-order nonlinear contributions are as follows, respectively [34]:
μ σs |M10 |2 Γ0 (1) α (ω) = ω εr (E10 − ω)2 + (Γ0 )2
σs Γ0 |M10 |2 μ I (3) α (ω, I) = −ω εr 2nr ε0 c [(E10 − ω)2 + (Γ0 )2 ]2 2 × 4|M10 | − |M11 − M00 |2
(9)
(10)
(11)
2 − 4E ω + 2 (ω 2 − Γ2 ) 3E10 10 0 × . 2 + (Γ )2 E10 0
Here μ is the permeability of the system, I is the intensity of the incident probe electromagnetic field, nr is the refractive index of medium, c is the speed of the light, ρs is the electronic density, Eij = Ei − Ej (i,j = 0,1) is the energy difference between these states, Mij = |ψi |er|ψj | are the off-diagonal matrix elements of the dipole moment and Γ0 = 1/T0 is damping term associated with the lifetime of the electrons due to intersubband scattering [35]. The linear and the third-order nonlinear refractive index changes are obtained as, respectively [36]:
and
1 E10 − ω Δn(1) (ω) 2 = 2 |M10 | σs nr 2nr ε0 (E10 − ω)2 + (Γ0 )2 Δn(3) (ω, I) nr
= −
(12)
μc σs I |M10 |2 3 4nr ε0 [(E10 − ω)2 + (Γ0 )2 ]2
× 4(E10 − ω)|M10 |2
−
(M11 − M00 )2 2 + (Γ )2 {(E10 − ω) E10 0
× E10 (E10 − ω) − (Γ0 )2 − (Γ0 )2 (2E10 − ω) . 5
(13)
5
5 r0 = 6 nm Black r0 = 8 nm Red r0 = 10 nm Blue α 0 = 0 nm Solid α 0 = 5 nm Dashed α 0 = 10 nm Dotted V0 = 225 meV
4
3
2
1
1
5
10
15
20
0 0
25
(b)
3
2
0 0
V0 = 175 meV Black V0 = 225 meV Red V0 = 275 meV Blue α 0 = 0 nm Solid α 0 = 5 nm Dashed α 0 = 10 nm Dotted r0 = 6 nm
4
V (eV )
V (eV )
(a)
r (nm)
5
10
15
20
25
r (nm)
Figure 1: (color online) (a) The confinement potential versus r for different values of the laser-dressing parameter and r0 with V0 = 225 meV. (b) Variation of the confinement potential as a function of r for different values of α0 and chemical potential V0 with r0 = 6 nm.
The total magnitude of the relative change in the refractive index Δn(ω, I)/nr is then the summation of the contributions given in Eqs. (12) and (13). 3. Results and discussion In this section, we will discuss the linear and third-order nonlinear optical absorption coefficients and relative changes in refractive index for the (1-2) transition in GaAs quantum dot with two-dimensional pseudoharmonic potential under ILF and static magnetic field. The parameters adapted in our calculations are as follows: m∗ = 0.067me (where me is the free-electron mass), εr = 13.18, μ = 4π × 10−7 Hm−1 , nr = 3.2, σs = 5 × 1024 m−3 , I = 4 × 109 W/m2 , and Γ0 = 5 ps−1 . Before discussing the optical response of the system under debate, it’s important to identify the variation of confinement potential deduced from ILF-quantum dot interaction. In Fig. 1 we show the confinement potential profile for different values of laser-dressing parameter α0 considering a zero-magnetic field case. It’s clearly seen from Fig. 1(a) that by applying an intense laser field, the bottom of potential shifts to higher energy values while the zero-point moves to higher values of r. Moreover, en6
α(1) Dashed α(3) Dotted α(tot) Solid
20
1.6
(b)
1.2
α0 = 0 nm Black α0 = 5 nm Red α0 = 10 nm Blue r0 = 6 nm V0 = 225 meV B = 10 T
15
10
5
0
−5 200
Δn(1) /nr Dashed Δn(3) /nr Dotted Δn(tot)/nr Solid
(a) Ref ractive index changes
Absorption Coef f icients (106 m−1 )
25
α0 = 0 nm Black α0 = 5 nm Red α0 = 10 nm Blue
0.8
r0 = 6 nm V0 = 225 meV B = 10 T
0.4 0 −0.4 −0.8 −1.2
250
300
350
400
450
−1.6 200
500
250
300
350
400
450
500
P hoton energy (meV )
P hoton energy (meV )
Figure 2: (color online) The linear, nonlinear and total (a) optical absorption coefficients and (b) refractive index changes as a function of the energy of incident photon for three values of α0 . We use V0 = 225 meV, B = 10 T and r0 = 6 nm.
hancement in the confinement potential for increasing (decreasing) chemical potential (zero-point parameter) becomes evident from Fig. 1(b). The linear (dashed), third-order nonlinear (dotted) and the total (solid) optical absorption coefficients and relative changes in refractive index as a function of incident photon energy are depicted in Fig. 2(a) and (b), respectively for varying values of α0 . Black curves stand for zero-ILF case whereas red and blue lines represent the cases for laser-dressing parameters of 5 and 10 nm, respectively where a pseudodot system with V0 = 225 meV and r0 = 6 nm is considered to be subjected to magnetic field of 10T. The main observation is the obvious blue-shift of resonant peak positions of optical absorption coefficients with increasing values of laser-dressing parameter α0 . This can be explained by the additive confining enhancement due to the intense laser which yields to the increment in the energy difference between the first lower-lying levels. Another important observation is that despite the diminishing peak magnitudes both in the linear and nonlinear contributions of absorption coefficients, peak magnitudes of the total absorption coefficient remains almostly unchanged with strengthening in the laser field. It should be noticed that, although the changes in the absorption 7
25
1.2
α (1) Dashed α (3) Dotted α (tot) Solid B = 0 T Black B = 10 T Red B = 15 T Blue V0 = 225 meV r0 = 6 nm α 0 = 5 nm
Δn(1) /nr Dashed Δn(3) /nr Dotted Δn(tot) /nr Solid B = 0 T Black B = 10 T Red B = 15 T Blue V0 = 225 meV r0 = 6 nm α 0 = 5 nm
20
15
0.8
Ref ractive index changes
Absorption Coef f icients (106 m−1 )
(a)
10
5
0
−5 290
0.4
(b)
0
−0.4
−0.8
295
300
305
310
315
−1.2 290
320
295
300
305
310
315
320
P hoton energy (meV )
P hoton energy (meV )
Figure 3: (color online) The linear, nonlinear and the total (a) optical absorption coefficients and (b) refractive index changes versus the photon energy for different values of magnetic field. The system is defined by V0 = 225 meV and r0 = 6 nm where the laser-dressing parameter is set to α0 = 5 nm .
coefficient and refractive index changes are predominantly determined by the linear terms, the nonlinear contributions of opposite sign have also significant effect on the total optical property. On the other hand, reduction in the peak of the total refractive index changes accompanied by a blue-shift for stronger ILF-cases is clearly seen from Fig. 2(b). In order to expose the effect of external magnetic field on the optical characteristics of pseudodot, in Fig. 3(a)-(b) we present the absorption coefficients and refractive index changes as a function of incident photon energy for three values of magnetic field by considering a system having a chemical potential V0 = 225 meV and zero-point r0 = 6 nm under a laser field defined by α0 = 5 nm. In both subfigures, the linear, third-order nonlinear and the total optical property is represented by dashed, dotted and solid line whereas magnetic fields of 0 T, 10 T and 15 T are shown with black, red and blue lines, respectively. The common feature of both figures is the shift of the resonant peaks toward the higher energy region for increasing strengths of magnetic field. The physical origin for this shift is separation of the energy levels as a consequence of the enhanced confinement effect due to magnetic field. Moreover, a slight variation imperceptible to eye is 8
25
1.5
α (1) Dashed α (3) Dotted α (tot) Solid r0 = 6 nm Black r0 = 6.4 nm Red r0 = 6.8 nm Blue V0 = 225 meV B = 10 T α 0 = 5 nm
Δn(1) /nr Dashed Δn(3) /nr Dotted Δn(tot) /nr Solid
20
15
10
5
0
−5 230
(b)
r0 = 6 nm Black r0 = 6.4 nm Red r0 = 6.8 nm Blue V0 = 225 meV B = 10 T α 0 = 5 nm
1
Ref ractive index changes
Absorption Coef f icients (106 m−1 )
(a)
0.5
0
−0.5
−1
260
290
320
−1.5 230
350
P hoton energy (meV )
260
290
320
350
P hoton energy (meV )
Figure 4: (color online) Variation of the linear, third-order nonlinear and the total (a) optical absorption coefficients and (b) refractive index changes as a function of the photon energy with varying values of r0 . We set V0 = 225 meV, B = 10 T and α0 = 5 nm.
observed in the peak magnitudes of the total absorption coefficients and refractive index changes. Besides, we should note that for zero-ILF case the magnitudes of both optical properties show the similar tendency which is in accordance with the results reported in Ref. [36]. The optical response of quantum systems is strongly dependent on the geometrical size of the structure. Hence forth, Fig. 4 is dedicated to the effect of zero-point parameter of pseudodot system onto the optical absorption coefficients and refractive index changes. We set the chemical potential to V0 = 225 meV, and consider the magnetic field of 10 T and laser-dressing parameter of α0 = 5 nm. As seen from Fig. 4(a) and (b), both optical properties are strongly size-dependent and their peak positions exhibit red-shift due to the reduced energy difference of the electronic states for increasing zero-point values. We should point out that despite the magnitudes of the linear and third-order nonlinear absorption contributions becomes smaller for decreasing r0 values, the the magnitudes of the total absorption coefficients increase slightly. On the other hand, increasing value of structure parameter r0 leads to increment in the peak value of the total changes in refractive index. 9
25
1.5
α(1) Dashed α(3) Dotted α(tot) Solid V0 = 175 meV Black V0 = 225 meV Red V0 = 275 meV Blue B = 10 T r0 = 6 nm α0 = 5 nm
Δn(1) /nr Dashed Δn(3) /nr Dotted Δn(tot)/nr Solid V0 = 175 meV Black V0 = 225 meV Red V0 = 275 meV Blue B = 10 T r0 = 6 nm α0 = 5 nm
20
15
(b)
1 Refractive index changes
Absorption Coefficients (106 m−1 )
(a)
10
5
0
−5 235
0.5
0
−0.5
−1
260
285 310 335 P hoton energy (meV )
360
385
−1.5 235
265
295 325 355 P hoton energy (meV )
385
Figure 5: (color online) The linear, nonlinear and total (a) optical absorption coefficients and (b) refractive index changes as a function of incident photon energy for different values of chemical potential. We consider a system with r0 = 6 nm, B = 10 T and α0 = 5 nm.
It’s also noteworthy to demonstrate the influence of the structure parameter V0 on the the linear, third-order nonlinear and the total optical absorption coefficients and relative changes in the refractive index in two-dimensional pseudodot system. In the Fig. 5, the chemical potential of 175 meV, 225 meV and 275 meV is represented by black, red and blue lines, respectively where the representation of the optical properties is the same as the other figures. Pseudodot with zero-point of 6 nm is considered to be exposed by magnetic field of 10 T and ILF with α0 = 5 nm. Remarkable variation in the position of the peaks of absorption coefficients and refractive index changes toward the higher energy regions for increasing V0 values is evident. The underlying fact for this shift is the increase of the quantum confinement which leads to an enlargement in the energy interval. Besides, it’s readily seen that as V0 increases, the peak values of absorption coefficient (refractive index changes) increase (decrease). We note that dipole moment matrix elements and the energy interval between different electronic states determine the behavior of the optical property. Competing effects between these factors assigns the variation in the magnitude of the optical absorption coefficient and the refractive index changes. As a result, one may conclude that optical absorption coefficients and relative changes in the 10
refractive index are rather sensitive to the non-resonant intense laser field and external magnetic field as well as to the structure parameters defining the confinement of the system. 4. Conclusion We have theoretically studied the influence of a non-resonant intense laser field and uniform magnetic field on the optical absorption coefficients and relative changes in the refractive index in a two-dimensional quantum pseudodot system. Circularly polarized laser field has been considered within the context of high-frequency Floquet theory and analytical expressions for the optical properties have been inferred from the compact-density matrix approach. Our results reveal that the position of resonant peak of the linear, third-order nonlinear and total absorption coefficients and refractive index changes as well as its magnitude shows a considerable sensitivity to the strength of intense laser field, magnetic field and structure parameters of pseudodot. External fields and an increase of the chemical potential cause a blue-shift in the peak positions of the optical properties whereas these optical quantities experience a red-shift for an increment in zero-parameter values. Besides, variations in the magnitudes of the resonant peaks of the absorption coefficients and refractive index changes differ for each external perturbation and structure parameter. Results of our work may make a contribution for utilization of quantum-size effects of quantum dots in THz optoelectronic devices. Acknowledgements This research is produced from MSc thesis of D. Gul Kilic completed at Dokuz Eyl¨ ul University, The Graduate School of Natural and Applied Sciences, Department of Physics. [1] L. Jacak, P. Hawrylak, A. W´ ojs, Quantum Dots, 1st ed., Springer, Berlin, 1998. [2] L. Lu, W. Xie, H. Hassanabadi, The effects of intense laser on nonlinear properties of shallow donor impurities in quantum dots with the Woods-Saxon potential, J. Luminesc. 131 (2011) 2538-2543. [3] R. Khordad, Confinement of an exciton in a quantum dot: effect of modified Kratzer potential, Indian J. Phys. 87 (2013) 623-628. [4] T. Chen, W. Xie, S. Liang, Optical and electronic properties of a two-dimensional quantum dot with an impurity, J. Luminesc. 139 (2013) 64-68. [5] G. Rezaei, B. Vaseghi, R. Khordad, H.A. Kenary, Optical rectification coefficient of a two-dimensional quantum pseudodot system, Physica E 43 (2011) 1853-1856.
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PII: DOI: Reference:
S0921-4526(16)30617-2 http://dx.doi.org/10.1016/j.physb.2016.12.031 PHYSB309772
To appear in: Physica B: Physics of Condensed Matter Received date: 19 December 2016 Accepted date: 29 December 2016 Cite this article as: D. Gul Kilic, S. Sakiroglu, F. Ungan, U. Yesilgul, E. Kasapoglu, H. Sari and I. Sokmen, Optical response in a laser-driven quantum pseudodot system, Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2016.12.031 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.
Optical response in a laser-driven quantum pseudodot system D. Gul Kilica , S. Sakiroglub,∗, F. Unganc , U. Yesilgulc , E. Kasapoglud , H. Sarie , I. Sokmenb a
Physics Department, Graduate School of Natural and Applied Sciences, Dokuz Eyl¨ ul University, 35390 Izmir, Turkey b Physics Department, Faculty of Science, Dokuz Eyl¨ ul University, 35390 Izmir, Turkey c Department of Optical Engineering, Faculty of Technology, Cumhuriyet University, 58140 Sivas, Turkey d Physics Department, Faculty of Science, Cumhuriyet University, 58140 Sivas, Turkey e Department of Primary Education, Faculty of Education, Cumhuriyet University, 58140 Sivas, Turkey
Abstract We investigate theoretically the intense laser-induced optical absorption coefficients and refractive index changes in a two-dimensional quantum pseudodot system under an uniform magnetic field. The effects of non-resonant, monochromatic intense laser field upon the system are treated within the framework of high-frequency Floquet approach in which the system is supposed to be governed by a laser-dressed potential. Linear and nonlinear absorption coefficients and relative changes in the refractive index are obtained by means of the compact-density matrix approach and iterative method. The results of numerical calculations for a typical GaAs quantum dot reveal that the optical response depends strongly on the magnitude of external magnetic field and characteristic parameters of the confinement potential. Moreover, we have demonstrated that the intense laser field modifies the confinement and thereby causes remarkable changes in the linear and nonlinear optical properties of the system. Keywords: quantum dot, intense laser field, nonlinear optics
1. Introduction Recent progress in nanofabrication technology allows the possibility of realizing quantum dots (QDs) with controllable size, shape and composition [1–3]. As a consequence of quantum confinement effect, formation of discrete energy levels in QDs results in particular electronic and optical properties different than their bulk counterparts. Feasibility of tailoring the energy spectrum to ∗
Corresponding Author. Email address: [email protected] (S. Sakiroglu)
Preprint submitted to Physica B
December 29, 2016
produce enviable optical transitions is important from the aspect of optoelectronic device technology [4]. Hence, many experimental and theoretical researches have been conducted to investigate the size-dependent optoelectronic properties of quantum dots. A great deal of work has been devoted on the linear and nonlinear optical absorption coefficients and relative changes in refractive index owing to small energy separation between the subbands and enhanced electric dipole matrix elements corresponding to transitions between the subbands in low-dimensional systems [5–11]. The intersubband optical absorption coefficients and refractive index changes in a quantum box ¨ with finite confining potential have been calculated by Unlu et al. [12]. The linear and nonlinear absorption coefficients and refractive index changes in parabolic quantum dots in the presence of electric and magnetic fields have been theoretically examined by Zhang et al. [13]. Rezaei and coworkers have considered a two-dimensional elliptic quantum dot with infinite confining potential height and have studied the effects of electromagnetic field intensity, size and shape of dot on the optical properties due to the intersubband transitions [14]. Analysis of the optical absorption coefficients of impurity doped quantum dots under the influence of noise, magnetic and electric fields has been conducted by Mandal et al. [15]. It is well-known that external perturbations give rise to remarkable changes in the electronic and optical response of quantum dots [16, 17]. In this respect, understanding of interaction of intense laser fields (ILFs) with carriers in low-dimensional semiconductor structures can offer the manipulation and control of physical properties of these systems [18]. A large number of studies on matter-field interaction reported in the literature put forth that ILFs can modify confinement potential, reduce energy gaps, distort optical absorption edges, shift critical temperatures of magnetic solids, and etc. [2, 19–22]. Optical absorptions and refractive index changes in parabolic quantum dot with hydrogenic impurity under laser radiation have been studied by Xie [23]. Prasad and Silotia have investigated the optical characteristics of disk shaped quantum dots with simultaneous influence of laser and magnetic field [24]. Linear and nonlinear optical absorption and relative changes in refractive index of spherical quantum dot placed at the center of cylindrical nanowire have been analyzed in Ref. [20]. Recently, intense laser-induced electronic and optical properties of double quantum dots under applied electric field have been reported by Bejan and Niculescu [25]. Though remarkable interest has been shown on the research of electronic and optical properties in the spherical, parabolic, cylindrical and rectangular quantum dots [26–28], the nonlinear optical 2
properties of a two-dimensional quantum pseudodot system under ILF has not been investigated so far. In this paper, we focus on addressing the effects of high-frequency intense laser field on the absorption coefficients and relative changes in the refractive index in two-dimensional quantum dot subjected to an uniform external magnetic field and represented by a pseudoharmonic potential. Effects of an intense laser field have been treated within the framework of laser-dressed potential approach and electronic structure of the system has been calculated by using finite element method [29]. Linear and third-order nonlinear optical absorption coefficients and refractive index changes are obtained from the compact-density matrix approach and iterative procedure. The remaining part of the paper is organized as follows: In Section 2, the theoretical framework is briefly given. Numerical results presented in Section 3 are followed by the conclusions given in Section 4. 2. Theoretical framework The two-dimensional quantum dot system is described by the confining potential that includes both the dot and antidot harmonic potential [30]: r0 2 r − , V (r) = V0 r0 r
(1)
where V0 is the chemical potential of the two-dimensional electron gas and r0 is the zero-point of the pseudoharmonic potential. We assume that the system is jointly exposed to a high-frequency laser field and static magnetic field along z−direction defined by the vector potential A(r) = (0, Br/2, 0) . The non-resonant, monochromatic, intense laser beam of frequency Ω is circularly polarized whose x sin Ωt+ yˆ cos Ωt), where x ˆ and yˆ define the unit vectors orthogonal vector potential is A(t) = A0 (−ˆ to the direction of propagation [31]. In the high-frequency intense laser field, the electron motion in quantum dot is determined by the time-averaged laser-dressed potential which is obtained as follows [32]: 1 Vd (r, α0 ) = T
T V (r + α(t))dt ,
(2)
0
where T = 2π/Ω is the period of the radiation. α(t) is related to the vector potential of ILF as t α(t) = A(t )dt and can be interpreted as a vector corresponding to the classical displacement along the polarization direction [32, 33]. Accordingly x cos Ωt + yˆ sin Ωt) , α(t) = α0 (ˆ 3
(3)
where the quiver amplitude of the electron in ILF is defined as the laser-dressing intensity parameter, α0 =
eA0 m∗ cΩ .
Then the Hamiltonian of electron confined in a two-dimensional quantum dot with a laserdressed pseudoharmonic potential and subjected to external static magnetic field is defined by the Hamiltonian H=
1 (p + eA)2 + Vd (r, α0 ) , 2m∗
(4)
where m∗ is the electron effective mass, e is the absolute value of the elementary charge, p is the momentum operator and A(r) is the vector potential corresponding to the magnetic field. By the substitution of Eq. (1) into Eq. (2) the analytical expression for the laser-dressed confinement potential is obtained as
Vd (r, α0 ) = V0
r02 r 2 α20 + − 2 + r02 r02 |r 2 − α20 |
.
(5)
It should be noted that, our calculations we merely performed in the interval α0 < r < ∞ due to the singularities of the potential at r = α0 . The electronic states of an electron can be obtained by the numerical solution of corresponding Schr¨ odinger equation expressed in terms of cylindrical coordinates: ∂ 1 ∂2 ωc ∂Ψ 2 1 ∂ r + 2 2 Ψ−i − ∗ 2m r ∂r ∂r r ∂ϕ 2 ∂ϕ ∗ 2 2 m ωc r Ψ + Vd (r, α0 )Ψ = EΨ , + 8
(6)
where ωc = eB/m∗ is the cyclotron frequency. Choosing the magnetic quantum number m, we can set the wave function Ψ as
eimϕ U (r) √ . Ψ(r, ϕ) = √ 2π r
(7)
∗ = 2 /m∗ a∗ 2 as the length Adopting the effective Bohr radius a∗0 and effective Hartree energy EH 0
and energy scales, respectively yields to
2 (m − 1/4) mγ 1 d2 U (r) + + − 2 2 dr 2r 2 2 2 1 γ ∗ r 2 + Vd (r, α0 )/EH U (r) = EU (r) , + 2 2
(8)
∗ . Numerical calcuwhere magnetic field is defined via the dimensionless parameter γ = ωc /EH
lations for eigenenergies and corresponding wave functions have been carried out by using finite element method. 4
Calculation of the optical absorption coefficients is based on the compact density matrix approach and iterative scheme, for which the total absorption coefficient is given by α(ω, I) = α(1) (ω) + α(3) (ω, I) , where the linear and third-order nonlinear contributions are as follows, respectively [34]:
μ σs |M10 |2 Γ0 (1) α (ω) = ω εr (E10 − ω)2 + (Γ0 )2
σs Γ0 |M10 |2 μ I (3) α (ω, I) = −ω εr 2nr ε0 c [(E10 − ω)2 + (Γ0 )2 ]2 2 × 4|M10 | − |M11 − M00 |2
(9)
(10)
(11)
2 − 4E ω + 2 (ω 2 − Γ2 ) 3E10 10 0 × . 2 + (Γ )2 E10 0
Here μ is the permeability of the system, I is the intensity of the incident probe electromagnetic field, nr is the refractive index of medium, c is the speed of the light, ρs is the electronic density, Eij = Ei − Ej (i,j = 0,1) is the energy difference between these states, Mij = |ψi |er|ψj | are the off-diagonal matrix elements of the dipole moment and Γ0 = 1/T0 is damping term associated with the lifetime of the electrons due to intersubband scattering [35]. The linear and the third-order nonlinear refractive index changes are obtained as, respectively [36]:
and
1 E10 − ω Δn(1) (ω) 2 = 2 |M10 | σs nr 2nr ε0 (E10 − ω)2 + (Γ0 )2 Δn(3) (ω, I) nr
= −
(12)
μc σs I |M10 |2 3 4nr ε0 [(E10 − ω)2 + (Γ0 )2 ]2
× 4(E10 − ω)|M10 |2
−
(M11 − M00 )2 2 + (Γ )2 {(E10 − ω) E10 0
× E10 (E10 − ω) − (Γ0 )2 − (Γ0 )2 (2E10 − ω) . 5
(13)
5
5 r0 = 6 nm Black r0 = 8 nm Red r0 = 10 nm Blue α 0 = 0 nm Solid α 0 = 5 nm Dashed α 0 = 10 nm Dotted V0 = 225 meV
4
3
2
1
1
5
10
15
20
0 0
25
(b)
3
2
0 0
V0 = 175 meV Black V0 = 225 meV Red V0 = 275 meV Blue α 0 = 0 nm Solid α 0 = 5 nm Dashed α 0 = 10 nm Dotted r0 = 6 nm
4
V (eV )
V (eV )
(a)
r (nm)
5
10
15
20
25
r (nm)
Figure 1: (color online) (a) The confinement potential versus r for different values of the laser-dressing parameter and r0 with V0 = 225 meV. (b) Variation of the confinement potential as a function of r for different values of α0 and chemical potential V0 with r0 = 6 nm.
The total magnitude of the relative change in the refractive index Δn(ω, I)/nr is then the summation of the contributions given in Eqs. (12) and (13). 3. Results and discussion In this section, we will discuss the linear and third-order nonlinear optical absorption coefficients and relative changes in refractive index for the (1-2) transition in GaAs quantum dot with two-dimensional pseudoharmonic potential under ILF and static magnetic field. The parameters adapted in our calculations are as follows: m∗ = 0.067me (where me is the free-electron mass), εr = 13.18, μ = 4π × 10−7 Hm−1 , nr = 3.2, σs = 5 × 1024 m−3 , I = 4 × 109 W/m2 , and Γ0 = 5 ps−1 . Before discussing the optical response of the system under debate, it’s important to identify the variation of confinement potential deduced from ILF-quantum dot interaction. In Fig. 1 we show the confinement potential profile for different values of laser-dressing parameter α0 considering a zero-magnetic field case. It’s clearly seen from Fig. 1(a) that by applying an intense laser field, the bottom of potential shifts to higher energy values while the zero-point moves to higher values of r. Moreover, en6
α(1) Dashed α(3) Dotted α(tot) Solid
20
1.6
(b)
1.2
α0 = 0 nm Black α0 = 5 nm Red α0 = 10 nm Blue r0 = 6 nm V0 = 225 meV B = 10 T
15
10
5
0
−5 200
Δn(1) /nr Dashed Δn(3) /nr Dotted Δn(tot)/nr Solid
(a) Ref ractive index changes
Absorption Coef f icients (106 m−1 )
25
α0 = 0 nm Black α0 = 5 nm Red α0 = 10 nm Blue
0.8
r0 = 6 nm V0 = 225 meV B = 10 T
0.4 0 −0.4 −0.8 −1.2
250
300
350
400
450
−1.6 200
500
250
300
350
400
450
500
P hoton energy (meV )
P hoton energy (meV )
Figure 2: (color online) The linear, nonlinear and total (a) optical absorption coefficients and (b) refractive index changes as a function of the energy of incident photon for three values of α0 . We use V0 = 225 meV, B = 10 T and r0 = 6 nm.
hancement in the confinement potential for increasing (decreasing) chemical potential (zero-point parameter) becomes evident from Fig. 1(b). The linear (dashed), third-order nonlinear (dotted) and the total (solid) optical absorption coefficients and relative changes in refractive index as a function of incident photon energy are depicted in Fig. 2(a) and (b), respectively for varying values of α0 . Black curves stand for zero-ILF case whereas red and blue lines represent the cases for laser-dressing parameters of 5 and 10 nm, respectively where a pseudodot system with V0 = 225 meV and r0 = 6 nm is considered to be subjected to magnetic field of 10T. The main observation is the obvious blue-shift of resonant peak positions of optical absorption coefficients with increasing values of laser-dressing parameter α0 . This can be explained by the additive confining enhancement due to the intense laser which yields to the increment in the energy difference between the first lower-lying levels. Another important observation is that despite the diminishing peak magnitudes both in the linear and nonlinear contributions of absorption coefficients, peak magnitudes of the total absorption coefficient remains almostly unchanged with strengthening in the laser field. It should be noticed that, although the changes in the absorption 7
25
1.2
α (1) Dashed α (3) Dotted α (tot) Solid B = 0 T Black B = 10 T Red B = 15 T Blue V0 = 225 meV r0 = 6 nm α 0 = 5 nm
Δn(1) /nr Dashed Δn(3) /nr Dotted Δn(tot) /nr Solid B = 0 T Black B = 10 T Red B = 15 T Blue V0 = 225 meV r0 = 6 nm α 0 = 5 nm
20
15
0.8
Ref ractive index changes
Absorption Coef f icients (106 m−1 )
(a)
10
5
0
−5 290
0.4
(b)
0
−0.4
−0.8
295
300
305
310
315
−1.2 290
320
295
300
305
310
315
320
P hoton energy (meV )
P hoton energy (meV )
Figure 3: (color online) The linear, nonlinear and the total (a) optical absorption coefficients and (b) refractive index changes versus the photon energy for different values of magnetic field. The system is defined by V0 = 225 meV and r0 = 6 nm where the laser-dressing parameter is set to α0 = 5 nm .
coefficient and refractive index changes are predominantly determined by the linear terms, the nonlinear contributions of opposite sign have also significant effect on the total optical property. On the other hand, reduction in the peak of the total refractive index changes accompanied by a blue-shift for stronger ILF-cases is clearly seen from Fig. 2(b). In order to expose the effect of external magnetic field on the optical characteristics of pseudodot, in Fig. 3(a)-(b) we present the absorption coefficients and refractive index changes as a function of incident photon energy for three values of magnetic field by considering a system having a chemical potential V0 = 225 meV and zero-point r0 = 6 nm under a laser field defined by α0 = 5 nm. In both subfigures, the linear, third-order nonlinear and the total optical property is represented by dashed, dotted and solid line whereas magnetic fields of 0 T, 10 T and 15 T are shown with black, red and blue lines, respectively. The common feature of both figures is the shift of the resonant peaks toward the higher energy region for increasing strengths of magnetic field. The physical origin for this shift is separation of the energy levels as a consequence of the enhanced confinement effect due to magnetic field. Moreover, a slight variation imperceptible to eye is 8
25
1.5
α (1) Dashed α (3) Dotted α (tot) Solid r0 = 6 nm Black r0 = 6.4 nm Red r0 = 6.8 nm Blue V0 = 225 meV B = 10 T α 0 = 5 nm
Δn(1) /nr Dashed Δn(3) /nr Dotted Δn(tot) /nr Solid
20
15
10
5
0
−5 230
(b)
r0 = 6 nm Black r0 = 6.4 nm Red r0 = 6.8 nm Blue V0 = 225 meV B = 10 T α 0 = 5 nm
1
Ref ractive index changes
Absorption Coef f icients (106 m−1 )
(a)
0.5
0
−0.5
−1
260
290
320
−1.5 230
350
P hoton energy (meV )
260
290
320
350
P hoton energy (meV )
Figure 4: (color online) Variation of the linear, third-order nonlinear and the total (a) optical absorption coefficients and (b) refractive index changes as a function of the photon energy with varying values of r0 . We set V0 = 225 meV, B = 10 T and α0 = 5 nm.
observed in the peak magnitudes of the total absorption coefficients and refractive index changes. Besides, we should note that for zero-ILF case the magnitudes of both optical properties show the similar tendency which is in accordance with the results reported in Ref. [36]. The optical response of quantum systems is strongly dependent on the geometrical size of the structure. Hence forth, Fig. 4 is dedicated to the effect of zero-point parameter of pseudodot system onto the optical absorption coefficients and refractive index changes. We set the chemical potential to V0 = 225 meV, and consider the magnetic field of 10 T and laser-dressing parameter of α0 = 5 nm. As seen from Fig. 4(a) and (b), both optical properties are strongly size-dependent and their peak positions exhibit red-shift due to the reduced energy difference of the electronic states for increasing zero-point values. We should point out that despite the magnitudes of the linear and third-order nonlinear absorption contributions becomes smaller for decreasing r0 values, the the magnitudes of the total absorption coefficients increase slightly. On the other hand, increasing value of structure parameter r0 leads to increment in the peak value of the total changes in refractive index. 9
25
1.5
α(1) Dashed α(3) Dotted α(tot) Solid V0 = 175 meV Black V0 = 225 meV Red V0 = 275 meV Blue B = 10 T r0 = 6 nm α0 = 5 nm
Δn(1) /nr Dashed Δn(3) /nr Dotted Δn(tot)/nr Solid V0 = 175 meV Black V0 = 225 meV Red V0 = 275 meV Blue B = 10 T r0 = 6 nm α0 = 5 nm
20
15
(b)
1 Refractive index changes
Absorption Coefficients (106 m−1 )
(a)
10
5
0
−5 235
0.5
0
−0.5
−1
260
285 310 335 P hoton energy (meV )
360
385
−1.5 235
265
295 325 355 P hoton energy (meV )
385
Figure 5: (color online) The linear, nonlinear and total (a) optical absorption coefficients and (b) refractive index changes as a function of incident photon energy for different values of chemical potential. We consider a system with r0 = 6 nm, B = 10 T and α0 = 5 nm.
It’s also noteworthy to demonstrate the influence of the structure parameter V0 on the the linear, third-order nonlinear and the total optical absorption coefficients and relative changes in the refractive index in two-dimensional pseudodot system. In the Fig. 5, the chemical potential of 175 meV, 225 meV and 275 meV is represented by black, red and blue lines, respectively where the representation of the optical properties is the same as the other figures. Pseudodot with zero-point of 6 nm is considered to be exposed by magnetic field of 10 T and ILF with α0 = 5 nm. Remarkable variation in the position of the peaks of absorption coefficients and refractive index changes toward the higher energy regions for increasing V0 values is evident. The underlying fact for this shift is the increase of the quantum confinement which leads to an enlargement in the energy interval. Besides, it’s readily seen that as V0 increases, the peak values of absorption coefficient (refractive index changes) increase (decrease). We note that dipole moment matrix elements and the energy interval between different electronic states determine the behavior of the optical property. Competing effects between these factors assigns the variation in the magnitude of the optical absorption coefficient and the refractive index changes. As a result, one may conclude that optical absorption coefficients and relative changes in the 10
refractive index are rather sensitive to the non-resonant intense laser field and external magnetic field as well as to the structure parameters defining the confinement of the system. 4. Conclusion We have theoretically studied the influence of a non-resonant intense laser field and uniform magnetic field on the optical absorption coefficients and relative changes in the refractive index in a two-dimensional quantum pseudodot system. Circularly polarized laser field has been considered within the context of high-frequency Floquet theory and analytical expressions for the optical properties have been inferred from the compact-density matrix approach. Our results reveal that the position of resonant peak of the linear, third-order nonlinear and total absorption coefficients and refractive index changes as well as its magnitude shows a considerable sensitivity to the strength of intense laser field, magnetic field and structure parameters of pseudodot. External fields and an increase of the chemical potential cause a blue-shift in the peak positions of the optical properties whereas these optical quantities experience a red-shift for an increment in zero-parameter values. Besides, variations in the magnitudes of the resonant peaks of the absorption coefficients and refractive index changes differ for each external perturbation and structure parameter. Results of our work may make a contribution for utilization of quantum-size effects of quantum dots in THz optoelectronic devices. Acknowledgements This research is produced from MSc thesis of D. Gul Kilic completed at Dokuz Eyl¨ ul University, The Graduate School of Natural and Applied Sciences, Department of Physics. [1] L. Jacak, P. Hawrylak, A. W´ ojs, Quantum Dots, 1st ed., Springer, Berlin, 1998. [2] L. Lu, W. Xie, H. Hassanabadi, The effects of intense laser on nonlinear properties of shallow donor impurities in quantum dots with the Woods-Saxon potential, J. Luminesc. 131 (2011) 2538-2543. [3] R. Khordad, Confinement of an exciton in a quantum dot: effect of modified Kratzer potential, Indian J. Phys. 87 (2013) 623-628. [4] T. Chen, W. Xie, S. Liang, Optical and electronic properties of a two-dimensional quantum dot with an impurity, J. Luminesc. 139 (2013) 64-68. [5] G. Rezaei, B. Vaseghi, R. Khordad, H.A. Kenary, Optical rectification coefficient of a two-dimensional quantum pseudodot system, Physica E 43 (2011) 1853-1856.
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