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Nonlinear optical rectification of a coupled semiconductor quantum dot – Metallic nanosphere system under a strong electromagnetic field
Accepted Manuscript Nonlinear Optical Rectification of a Coupled Semiconductor Quantum Dot – Metallic Nanosphere System under a Strong Electromagnetic Field
Sofia Evangelou PII:
S0921-4526(18)30825-1
DOI:
10.1016/j.physb.2018.12.030
Reference:
PHYSB 311239
To appear in:
Physica B: Physics of Condensed Matter
Received Date:
17 September 2018
Accepted Date:
18 December 2018
Please cite this article as: Sofia Evangelou, Nonlinear Optical Rectification of a Coupled Semiconductor Quantum Dot – Metallic Nanosphere System under a Strong Electromagnetic Field, Physica B: Physics of Condensed Matter (2018), doi: 10.1016/j.physb.2018.12.030
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.
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Nonlinear Optical Rectification of a Coupled Semiconductor Quantum Dot – Metallic Nanosphere System under a Strong Electromagnetic Field Sofia Evangelou Department of Physics, School of Natural Sciences, University of Patras, Patras 265 04, Greece
Abstract We present results for the nonlinear optical rectification of an asymmetric GaAs/AlGaAs quantum dot structure coupled to a gold nanosphere under the influence of a strong electromagnetic field. We show that the nonlinear optical rectification coefficient depends strongly on the distance between the quantum dot and the metallic nanosphere and on the direction of the applied electromagnetic field.
Keywords: Semiconductor quantum dot, metallic nanosphere, coupled nanostructure, exciton, nonlinear optical rectification, applied electromagnetic field.
Corresponding Author: Sofia Evangelou Telephone: +30 6932 416713 E-mail: [email protected]
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1. Introduction A relatively new field of intense research activity deals with the study of optical properties of quantum systems (atoms/molecules and semiconductor quantum dots) coupled with plasmonic (mainly metallic) nanostructures [1]. In such nanosystems, strong fields and intense light confinement, which are related to plasmonic resonances, lead to strong interaction between electromagnetic fields and quantum systems near plasmonic nanostructures. In addition, using quantum systems, external control of the optical properties of the coupled nanophotonic structures can be achieved. In particular, in the last decade very intense and increasing research activity exists in the optical properties of a semiconductor quantum dot system, which is described by a twolevel quantum system, coupled with a metallic nanoparticle, which is described by a metallic sphere of nanometric dimensions, see e.g. refs. [2-15]. In the majority of the existent studies, quantum dots are assumed to be described by symmetric confinement potentials and so they do not have permanent electric dipoles. Still, in a few cases, the optical response of asymmetric quantum dots coupled to metallic nanoparticles have been studied [15-18]. Phenomena that have been studied in quantum dots with asymmetric confinement potentials that are coupled to metallic nanostructures are absorption and resonance fluorescence [13], second harmonic generation [16,17] and difference frequency generation [18]. The goal of this work is to study a basic second-order nonlinear optical effect, the nonlinear optical rectification, which, in our case, appears due to excitation of excitons in quantum dots coupled to a spherical metallic nanostructure. The nonlinear optical rectification phenomenon creates a static electric field (DC field) from an electromagnetic field in a material with second-order nonlinearity [19] and has been extensively studied for the case of (isolated) asymmetric semiconductor quantum dots, see, for example, refs. [20-30]. To the best of our knowledge, there is no other work where the metallic nanoparticle effect on nonlinear optical rectification from excitonic excitations in a quantum dot under strong excitation has been studied. The only directly related work that we know is that of Thanopulos et al. [31], where the effect of a complex plasmonic nanostructure (a twodimensional periodic metal-dielectric nanospheres array) on the nonlinear optical 2
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rectification from organic molecules has been studied. In that work [31] emphasis has been given on the effect of the significant modification of a molecule’s spontaneous emission rate, due to the presence of a plasmonic nanostructure, on nonlinear optical rectification. Given the fact that in ref. [31] a very weak electromagnetic field was considered, the effect of the intensity of light on the nonlinear optical rectification, which is the main issue in our work, was not analyzed. Furthermore, the present work is quite different from ref. [31], given that for the frequencies under consideration here the metallic nanosphere does not modify the quantum dot’s spontaneous emission rate. The main study here concerns the interaction of an asymmetric quantum dot with a strong electromagnetic field, which is strongly influenced by the metallic nanoparticle, pointing out the effect of the intensity of the electromagnetic field on the nonlinear optical rectification effect from a coupled quantum dot – metal nanoparticle structure. This work is organized as follows: in the next section we present the theory for the interaction of the quantum dot with the electromagnetic field in the presence of the metallic nanosphere, show the main formula for the nonlinear optical rectification coefficient, and outline the theory for the quantum dot electronic structure. In section 3 we present the main results of the paper and show the effect of the applied electromagnetic field’s polarization, as well as the distance of the quantum dot from the metallic nanosphere on the nonlinear optical rectification coefficient. Finally, in section 4 we give a summary of our findings. 2. Theory We consider the interaction of the asymmetric quantum dot with an external electromagnetic field. The quantum dot is described within the two-level system approximation, which is usual in such quantum structures [1-15]. We denote as ħ𝜔1, ħ𝜔2, respectively, the energies of the ground and single exciton states of the quantum dot as well as, ħ𝜔21 = ħ(𝜔2 ― 𝜔1) the transition energy of the single exciton state. We also denote as 𝜇12 the induced electric dipole matrix element of the quantum dot and 𝜇11, 𝜇22, respectively, the permanent electric dipole matrix elements for the ground and the single exciton state. The permanent dipoles are non-zero for an asymmetric quantum dot structure. 3
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The quantum dot interacts with an electric field of the form 𝛦(𝑡) = 𝐸0𝑒 ―𝑖𝜔𝑡 + 𝛦0∗ 𝑒𝑖𝜔𝑡,
(1)
with amplitude 𝐸0 and angular frequency 𝜔. The nonlinear optical rectification coefficient for a strong electromagnetic field is given by [32]: 𝜒(2) 0 =
2𝛮(𝜇22 ― 𝜇11)𝜇212𝛵1𝛵2 𝜀0ħ2
1+
1 𝛵22(𝜔
2
― 𝜔21) +
2 4𝜇2 12|𝛦0| 𝛵1𝛵2 ħ2
,
(2)
where 𝑇1, 𝑇2 are, respectively, the population decay time (longitudinal relaxation time) and the dephasing time (transverse relaxation time) of the quantum dot, and 𝜀0 is the vacuum permittivity. The equation above gives the dependence of the nonlinear optical rectification coefficient on the frequency and the amplitude of the electric field, which is related to the intensity of the light through the equation 𝐼 = 2𝑛𝜀0𝑐|𝐸0|2, where 𝑛 is the quantum dot’s index of refraction. We note that eq. (2) that we presented above, has been extracted without considering permanent electric dipole effects. This methodology has been proposed by Zaluzny [32] and it is based on the usual implementation of the rotating wave approximation. This methodology has been updated by Paspalakis et al. [26,33] using a form of the rotating wave approximation appropriate for two-level systems with permanent dipoles. In that method [26,33] the density matrix equations contain Bessel functions of the form 𝐽𝑛(𝑦), where 𝑦 =
|𝜇22 ― 𝜇11||𝛦0| ħ𝜔
. These terms are essential in the low frequency regime and in
addition for high light intensities. For higher frequencies, where excitonic transitions take place, that we are interested in, e.g. typically for ħ𝜔~1 ― 3 𝑒𝑉, and for light intensities 𝑊
up to 1012𝑚2, that are actually in the destruction limit of the quantum dots and given that the |𝜇22 ― 𝜇11| term’s values are of the order of some e nm approximately, then y takes very low values (the maximum is about 0.01), so only the zero order Bessel function contributes to the dynamics, which for low values of y gives 𝐽0(𝑦)~1. So, in this case, the contribution of the Bessel terms is ignorable, and these terms can be safely neglected. Consequently, the methodology proposed by Zaluzny [32] is adequate for the study of this problem, and so is eq. (2).
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In the case when the quantum dot is placed nearby a metallic nanosphere (see fig. 1), which is the case that we study in the present work, then the electric field that the quantum dot feels is altered due to the presence of the metallic nanoparticle. Initially, we assume that the electric field is polarized parallel to the interparticle axis, which is taken as the z-axis. The electric field can be given according to the following equation in the quasi-static limit, where retardation effects of the electromagnetic field can be neglected [6,34]:
[
]
2𝛾(𝜔)𝑎3
𝐸0 = 1 + (𝑑 + 𝑎)3 𝐸0,
(3)
where 𝐸0 is the electric field of the incident radiation to the coupled nanostructure, 𝐸0 is the field that the quantum dot feels, a is the metallic nanoparticle radius and d is the distance 𝜀𝑚(𝜔) ― 𝜀𝑑
of the quantum dot from the surface of the metallic nanoparticle. Here, 𝛾(𝜔) = 𝜀𝑚(𝜔) + 2𝜀𝑑 , where 𝜀𝑚(𝜔) is the dielectric function of the metallic nanoparticle and 𝜀𝑑 is the dielectric constant of the environment, that in our study is the same as the dielectric constant of the quantum dot. Actually, a more accurate description of the electric field interacting with the quantum dot in the quasi-static limit contains both the field 𝐸0 of Eq. (3) and the self-interaction field 𝐸𝑑𝑑 from the dipole-dipole exciton-plasmon coupling [2-4,7-15]. This field, without 1 𝛾(𝜔)𝑎3
including multipole effects, is given, in the case studied here, by [2-4,7-15] 𝐸𝑑𝑑 = 𝜋𝜀𝑑(𝑑 + 𝑎)6 𝜇12𝜎, with 𝜎 being the off-diagonal density matrix element. As |𝜎| has maximum value 1 𝛾(𝜔)𝑎3
1/2, the value of 𝐸𝑑𝑑 is determined by |𝜋𝜀𝑑(𝑑 + 𝑎)6𝜇12|, which, in our study for the parameters of interest, takes values that are about five to six orders of magnitude lower than |𝐸0|. Therefore, its contribution can be safely omitted in this study. Before closing this section, we will describe the quantum dot structure. We consider a quasi-one-dimensional quantum dot with semi-parabolic confinement potential, which has been studied in several works, see e.g. refs. [20,21,30,35-38]. The Hamiltonian for an exciton in the quantum dot structure in the effective mass approximation is given by ħ2 ∂2
ħ2 ∂2
𝑒2
𝐻 = ― 2𝑚 ∗ ∂𝑧2 ― 2𝑚 ∗ ∂𝑧2 + 𝑉𝑒0(𝑧𝑒) + 𝑉ℎ0(𝑧ℎ) ― 𝜀𝑑|𝑧𝑒 ― 𝑧ℎ|. 𝑒
𝑒
ℎ
ℎ
(4)
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Here, e refers to the electron and h to the hole. Also, 𝑚𝑒∗ ,𝑚ℎ∗ are the effective masses of the electron and the hole respectively, and 𝑉𝑒0, 𝑉ℎ0 are the confining potentials of the electron and the hole. We consider a semi-parabolic confining potentials of the form [20,21,30,351
1
38] 𝑉𝑒0(𝑧𝑒) = 2𝑚𝑒∗ 𝜔20𝑧2𝑒 , if 𝑧𝑒 ≥ 0 and 𝑉𝑒0→∞ if 𝑧𝑒 < 0, and 𝑉ℎ0(𝑧ℎ) = 2𝑚ℎ∗ 𝜔20𝑧2ℎ, if 𝑧ℎ ≥ 0 and 𝑉ℎ0→∞ if 𝑧ℎ < 0, where 𝜔0 is the characteristic oscillator angular frequency. In the strong confinement regime, the frequency difference between the ground and first excited states is given by [21] 𝜔21 =
7 2ħ𝜔0
3
- 2ħ𝜔0 ħ
= 2𝜔0 and the corresponding permanent and
induced electric matrix elements, needed for the calculation of the nonlinear optical rectification coefficient, are given by 𝜇22 - 𝜇11 = 𝑒
ħ 𝜋 𝜇𝜔0
1
and 𝜇12 = 𝑒
ħ 2 3𝜋 𝜇𝜔0
, where
∗ ∗ 𝜇 = 𝑚𝑒 𝑚ℎ (𝑚𝑒∗ + 𝑚ℎ∗ ) is the reduced mass.
3. Results and Discussion We will study a GaAs/AlGaAs quantum dot and will take the effective masses of the electron and hole, respectively, as 𝑚𝑒∗ = 0.067𝑚0, 𝑚ℎ∗ = 0.09𝑚0 (where 𝑚0 is the free electron mass), 𝜀𝑑 = 12.5, 𝑇1 = 1 ps, 𝑇2 = 0.5 ps and 𝑁 = 5 𝑥 1024 𝑚 ―3. For the semiparabolic confinement potential, choosing ħ𝜔0 = 0.773 eV, we obtain ħ𝜔21 = 1.546 eV, and using the above values for the effective masses we also take 𝜇12 = 0.738 e nm and 𝜇22 ― 𝜇11 = 0.904 e nm. Also, the metallic nanoparticle will be described by the Drude 𝜔2𝑝
dielectric function, with 𝜀𝑚(𝜔) = 1 ― 𝜔2 + 𝑖𝜔𝛤. We will assume parameters ħ𝜔𝑝 = 3.71 eV and ħ𝛤 = 0.1855 eV that represent properly the dielectric function of gold up to 2.7 eV [7]. Before analyzing the effect of the metallic nanosphere on the nonlinear optical rectification, we will study the effect of different values of the incident electromagnetic field’s intensities on the nonlinear optical rectification of a quantum dot without the metallic nanosphere. Specifically, in fig. 2(a) we present the nonlinear optical rectification coefficient as a function of the photon’s energy in a semi-parabolic GaAs/AlGaAs quantum dot (without 6
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the metallic nanosphere) for three different light intensities. We observe that as the light intensity increases the nonlinear optical rectification significantly decreases. Namely, we observe the well-known saturation of the nonlinear optical rectification when the light intensity increases [26,29,30,32,33]. We extract the same result from fig. 2(b) which shows the dependence of the nonlinear optical rectification coefficient’s maximum as a function of the intensity of light. Subsequently, we will study the coupled system of a quantum dot with a gold metallic nanosphere. The radius of the nanosphere is taken a = 10 nm. We initially consider the electric field being z-polarized (along the interparticle axis, see fig. 1). Fig. 3(a) depicts the nonlinear optical rectification coefficient of the quantum dot as a function of the photon energy, for different distances of the quantum dot from the gold nanoparticle, for the same value of light intensity. We find that depending on the distance between the quantum dot and the metallic nanoparticle, the curve of the coefficient of the nonlinear optical rectification is significantly affected, relative to the photon energy. Specifically, we see that for d = 80 nm the curve (solid curve) is low-heighted and of large width, while inversely for d = 5 nm the curve (dot-dashed curve) is of great height and with small width. In the distances between (dotted and dashed curves) the height of the curves increases, and their width decreases as the distance between the quantum dot and the metallic nanoparticle decreases. This shows that saturation occurs as the quantum dot moves away from the metallic nanosphere. Therefore, the nonlinear optical rectification coefficient increases as the quantum dot and the metallic nanoparticle come closer for the same value of light intensity (compare, for example, the dashed and dot-dashed curves of fig. 3(a) with the dashed curve of fig. 2(a)). Moreover, fig. 3(b) presents the maximum value of the nonlinear optical rectification coefficient as a function of the distance d from the surface of the gold nanoparticle. In this case, even if there is an extra dependence of the nonlinear optical rectification to the frequency of light 𝜔 in the last term of the denominator of Eq. (2), as can be seen from fig. 3(a) the maximum of the coefficient is still found at resonance (for 𝜔 = 𝜔21). This happens as 𝛾(𝜔), which carries the frequency dependence, varies insignificantly and practically remains constant in the region of frequencies very close to resonance. As we can see the 7
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maximum nonlinear optical rectification coefficient initially reduces as the distance between the quantum dot and the metallic nanoparticle increases, while later for larger distances the maximum of the nonlinear optical rectification coefficient becomes essentially constant. The latter happens because when the metallic nanoparticle is far away from the quantum dot, the metallic nanoparticle does not practically affect the electric field felt by the quantum dot. Additionally, we present results for the case that the electric field is polarized along the xaxis (and, of course, the quantum dot is grown along the x-axis). In this case, in the presence of the metallic nanosphere, the field that interacts with the quantum dot is [6,34]:
[
𝛾(𝜔)𝑎3
]
𝛦0 = 1 ― (𝑑 + 𝑎)3 𝛦0.
(5)
For this case, as it shown in fig. 4(a) we observe that the nonlinear optical rectification coefficient follows the opposite response from the one found in fig. 3(a), since here as the distance between the quantum dot and the metallic nanoparticle decreases, the nonlinear optical rectification coefficient decreases too. Therefore, saturation effects occur as the quantum dot and the metal nanoparticle come closer. We also see that the curve of the nonlinear optical rectification coefficient as a function of the photon energy is less affected when the electric field is polarized along the x-axis compared to the case of a z-polarized field of the same intensity for the different values of the interparticle distance. Additionally, fig. 4(b) shows the maximum value that the nonlinear optical rectification coefficient follows as a function of the distance d from the surface of the gold nanoparticle, for a specific value of the light intensity. In this case too, where the electric field is polarized along the x-axis, we observe that the nonlinear optical rectification coefficient increases initially as the distance between the quantum dot and the metallic nanosphere increases until it takes a practically constant value for very big interparticle distances. The behavior of the results for the two different polarizations of light can be explained from the dependence of the electric field felt by the quantum dot on distance d. Using eqs. (3) and (5) we show in fig. 5 the dependence of the normalized field intensity (intensity enhancement or suppression) for the resonance frequency for a gold nanoparticle as a function of distance d. We find that for a z-polarized field its value decreases as the quantum dot and the metallic nanoparticle come closer, while the opposite happens for an 8
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x-polarized field. This is caused due to destructive interference between the incident field and the field that is created from the metallic nanoparticle when the field is polarized along the z-axis and constructive interference between the two fields when the field is polarized along the x-axis. By using this in eq. (2) we can understand the increase or the decrease of the nonlinear optical rectification coefficient as a function of distance for the two different polarizations of light. 4. Summary In the present work we studied the optical properties of an asymmetric quantum dot that is coupled to a spherical metallic nanoparticle. Specifically, we presented results for the nonlinear optical rectification coefficient for a GaAs/AlGaAs quantum dot near a gold nanosphere for different values of the distance between the quantum dot and the metallic nanoparticle. We found that when the electric field was polarized along the z-axis (the axis that connects the quantum dot with the metallic nanoparticle) then as the distance between the quantum dot and the metallic nanoparticle decreases, the value of the nonlinear optical rectification increases. The opposite happens when the electric field is polarized along the x axis, i.e. as the distance between the quantum dot and the metallic nanoparticle decreases, the value of the nonlinear optical rectification decreases too. Acknowledgements: The author is grateful for financial support by Greece and the European Union - European Regional Development Fund via the “Supporting Postdoctoral Researchers” project of the Greek State Scholarships Foundation (IKY). She also thanks Professor Andreas Terzis of the Physics Department of the University of Patras for useful discussions.
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Fig. 1: The schematic of the system considered. A quantum dot in the presence of a metallic nanosphere.
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Fig. 2: (a) The nonlinear optical rectification coefficient for the exciton transition as a function of applied field energy for different values of the applied electromagnetic field intensity. Solid curve: I = 109 W/m2, dotted curve I = 1010 W/m2, and dashed curve I = 5×1010 W/m2. (b) The maximum nonlinear optical rectification coefficient as a function of the applied field intensity for 1.546 eV excitation energy. These calculations are in the absence of a metallic nanosphere.
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Fig. 3: (a) The nonlinear optical rectification coefficient for the exciton transition as a function of applied field energy for different values of the distance from the surface of the gold nanoparticle. The applied electromagnetic field is z-polarized and has intensity I = 5×1010 W/m2. The solid curve is for d = 80 nm, the dotted curve is for d = 10 nm, the dashed curve is for d = 7.5 nm and the dot-dashed curve is for d = 5 nm. (b) The maximum nonlinear optical rectification coefficient as a function of the distance from the surface of the gold nanoparticle for 1.546 eV excitation energy and intensity I = 5×1010 W/m2.
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Fig. 4: The same as in fig. 3 when the applied electromagnetic field is x-polarized.
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Fig. 5: The normalized field intensity felt by the quantum dot due to the presence of the gold nanoparticle as a function of the distance from the surface of the gold nanoparticle. For (a) the electric field is z-polarized and for (b) it is x-polarized.
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Sofia Evangelou PII:
S0921-4526(18)30825-1
DOI:
10.1016/j.physb.2018.12.030
Reference:
PHYSB 311239
To appear in:
Physica B: Physics of Condensed Matter
Received Date:
17 September 2018
Accepted Date:
18 December 2018
Please cite this article as: Sofia Evangelou, Nonlinear Optical Rectification of a Coupled Semiconductor Quantum Dot – Metallic Nanosphere System under a Strong Electromagnetic Field, Physica B: Physics of Condensed Matter (2018), doi: 10.1016/j.physb.2018.12.030
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.
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Nonlinear Optical Rectification of a Coupled Semiconductor Quantum Dot – Metallic Nanosphere System under a Strong Electromagnetic Field Sofia Evangelou Department of Physics, School of Natural Sciences, University of Patras, Patras 265 04, Greece
Abstract We present results for the nonlinear optical rectification of an asymmetric GaAs/AlGaAs quantum dot structure coupled to a gold nanosphere under the influence of a strong electromagnetic field. We show that the nonlinear optical rectification coefficient depends strongly on the distance between the quantum dot and the metallic nanosphere and on the direction of the applied electromagnetic field.
Keywords: Semiconductor quantum dot, metallic nanosphere, coupled nanostructure, exciton, nonlinear optical rectification, applied electromagnetic field.
Corresponding Author: Sofia Evangelou Telephone: +30 6932 416713 E-mail: [email protected]
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1. Introduction A relatively new field of intense research activity deals with the study of optical properties of quantum systems (atoms/molecules and semiconductor quantum dots) coupled with plasmonic (mainly metallic) nanostructures [1]. In such nanosystems, strong fields and intense light confinement, which are related to plasmonic resonances, lead to strong interaction between electromagnetic fields and quantum systems near plasmonic nanostructures. In addition, using quantum systems, external control of the optical properties of the coupled nanophotonic structures can be achieved. In particular, in the last decade very intense and increasing research activity exists in the optical properties of a semiconductor quantum dot system, which is described by a twolevel quantum system, coupled with a metallic nanoparticle, which is described by a metallic sphere of nanometric dimensions, see e.g. refs. [2-15]. In the majority of the existent studies, quantum dots are assumed to be described by symmetric confinement potentials and so they do not have permanent electric dipoles. Still, in a few cases, the optical response of asymmetric quantum dots coupled to metallic nanoparticles have been studied [15-18]. Phenomena that have been studied in quantum dots with asymmetric confinement potentials that are coupled to metallic nanostructures are absorption and resonance fluorescence [13], second harmonic generation [16,17] and difference frequency generation [18]. The goal of this work is to study a basic second-order nonlinear optical effect, the nonlinear optical rectification, which, in our case, appears due to excitation of excitons in quantum dots coupled to a spherical metallic nanostructure. The nonlinear optical rectification phenomenon creates a static electric field (DC field) from an electromagnetic field in a material with second-order nonlinearity [19] and has been extensively studied for the case of (isolated) asymmetric semiconductor quantum dots, see, for example, refs. [20-30]. To the best of our knowledge, there is no other work where the metallic nanoparticle effect on nonlinear optical rectification from excitonic excitations in a quantum dot under strong excitation has been studied. The only directly related work that we know is that of Thanopulos et al. [31], where the effect of a complex plasmonic nanostructure (a twodimensional periodic metal-dielectric nanospheres array) on the nonlinear optical 2
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rectification from organic molecules has been studied. In that work [31] emphasis has been given on the effect of the significant modification of a molecule’s spontaneous emission rate, due to the presence of a plasmonic nanostructure, on nonlinear optical rectification. Given the fact that in ref. [31] a very weak electromagnetic field was considered, the effect of the intensity of light on the nonlinear optical rectification, which is the main issue in our work, was not analyzed. Furthermore, the present work is quite different from ref. [31], given that for the frequencies under consideration here the metallic nanosphere does not modify the quantum dot’s spontaneous emission rate. The main study here concerns the interaction of an asymmetric quantum dot with a strong electromagnetic field, which is strongly influenced by the metallic nanoparticle, pointing out the effect of the intensity of the electromagnetic field on the nonlinear optical rectification effect from a coupled quantum dot – metal nanoparticle structure. This work is organized as follows: in the next section we present the theory for the interaction of the quantum dot with the electromagnetic field in the presence of the metallic nanosphere, show the main formula for the nonlinear optical rectification coefficient, and outline the theory for the quantum dot electronic structure. In section 3 we present the main results of the paper and show the effect of the applied electromagnetic field’s polarization, as well as the distance of the quantum dot from the metallic nanosphere on the nonlinear optical rectification coefficient. Finally, in section 4 we give a summary of our findings. 2. Theory We consider the interaction of the asymmetric quantum dot with an external electromagnetic field. The quantum dot is described within the two-level system approximation, which is usual in such quantum structures [1-15]. We denote as ħ𝜔1, ħ𝜔2, respectively, the energies of the ground and single exciton states of the quantum dot as well as, ħ𝜔21 = ħ(𝜔2 ― 𝜔1) the transition energy of the single exciton state. We also denote as 𝜇12 the induced electric dipole matrix element of the quantum dot and 𝜇11, 𝜇22, respectively, the permanent electric dipole matrix elements for the ground and the single exciton state. The permanent dipoles are non-zero for an asymmetric quantum dot structure. 3
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The quantum dot interacts with an electric field of the form 𝛦(𝑡) = 𝐸0𝑒 ―𝑖𝜔𝑡 + 𝛦0∗ 𝑒𝑖𝜔𝑡,
(1)
with amplitude 𝐸0 and angular frequency 𝜔. The nonlinear optical rectification coefficient for a strong electromagnetic field is given by [32]: 𝜒(2) 0 =
2𝛮(𝜇22 ― 𝜇11)𝜇212𝛵1𝛵2 𝜀0ħ2
1+
1 𝛵22(𝜔
2
― 𝜔21) +
2 4𝜇2 12|𝛦0| 𝛵1𝛵2 ħ2
,
(2)
where 𝑇1, 𝑇2 are, respectively, the population decay time (longitudinal relaxation time) and the dephasing time (transverse relaxation time) of the quantum dot, and 𝜀0 is the vacuum permittivity. The equation above gives the dependence of the nonlinear optical rectification coefficient on the frequency and the amplitude of the electric field, which is related to the intensity of the light through the equation 𝐼 = 2𝑛𝜀0𝑐|𝐸0|2, where 𝑛 is the quantum dot’s index of refraction. We note that eq. (2) that we presented above, has been extracted without considering permanent electric dipole effects. This methodology has been proposed by Zaluzny [32] and it is based on the usual implementation of the rotating wave approximation. This methodology has been updated by Paspalakis et al. [26,33] using a form of the rotating wave approximation appropriate for two-level systems with permanent dipoles. In that method [26,33] the density matrix equations contain Bessel functions of the form 𝐽𝑛(𝑦), where 𝑦 =
|𝜇22 ― 𝜇11||𝛦0| ħ𝜔
. These terms are essential in the low frequency regime and in
addition for high light intensities. For higher frequencies, where excitonic transitions take place, that we are interested in, e.g. typically for ħ𝜔~1 ― 3 𝑒𝑉, and for light intensities 𝑊
up to 1012𝑚2, that are actually in the destruction limit of the quantum dots and given that the |𝜇22 ― 𝜇11| term’s values are of the order of some e nm approximately, then y takes very low values (the maximum is about 0.01), so only the zero order Bessel function contributes to the dynamics, which for low values of y gives 𝐽0(𝑦)~1. So, in this case, the contribution of the Bessel terms is ignorable, and these terms can be safely neglected. Consequently, the methodology proposed by Zaluzny [32] is adequate for the study of this problem, and so is eq. (2).
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In the case when the quantum dot is placed nearby a metallic nanosphere (see fig. 1), which is the case that we study in the present work, then the electric field that the quantum dot feels is altered due to the presence of the metallic nanoparticle. Initially, we assume that the electric field is polarized parallel to the interparticle axis, which is taken as the z-axis. The electric field can be given according to the following equation in the quasi-static limit, where retardation effects of the electromagnetic field can be neglected [6,34]:
[
]
2𝛾(𝜔)𝑎3
𝐸0 = 1 + (𝑑 + 𝑎)3 𝐸0,
(3)
where 𝐸0 is the electric field of the incident radiation to the coupled nanostructure, 𝐸0 is the field that the quantum dot feels, a is the metallic nanoparticle radius and d is the distance 𝜀𝑚(𝜔) ― 𝜀𝑑
of the quantum dot from the surface of the metallic nanoparticle. Here, 𝛾(𝜔) = 𝜀𝑚(𝜔) + 2𝜀𝑑 , where 𝜀𝑚(𝜔) is the dielectric function of the metallic nanoparticle and 𝜀𝑑 is the dielectric constant of the environment, that in our study is the same as the dielectric constant of the quantum dot. Actually, a more accurate description of the electric field interacting with the quantum dot in the quasi-static limit contains both the field 𝐸0 of Eq. (3) and the self-interaction field 𝐸𝑑𝑑 from the dipole-dipole exciton-plasmon coupling [2-4,7-15]. This field, without 1 𝛾(𝜔)𝑎3
including multipole effects, is given, in the case studied here, by [2-4,7-15] 𝐸𝑑𝑑 = 𝜋𝜀𝑑(𝑑 + 𝑎)6 𝜇12𝜎, with 𝜎 being the off-diagonal density matrix element. As |𝜎| has maximum value 1 𝛾(𝜔)𝑎3
1/2, the value of 𝐸𝑑𝑑 is determined by |𝜋𝜀𝑑(𝑑 + 𝑎)6𝜇12|, which, in our study for the parameters of interest, takes values that are about five to six orders of magnitude lower than |𝐸0|. Therefore, its contribution can be safely omitted in this study. Before closing this section, we will describe the quantum dot structure. We consider a quasi-one-dimensional quantum dot with semi-parabolic confinement potential, which has been studied in several works, see e.g. refs. [20,21,30,35-38]. The Hamiltonian for an exciton in the quantum dot structure in the effective mass approximation is given by ħ2 ∂2
ħ2 ∂2
𝑒2
𝐻 = ― 2𝑚 ∗ ∂𝑧2 ― 2𝑚 ∗ ∂𝑧2 + 𝑉𝑒0(𝑧𝑒) + 𝑉ℎ0(𝑧ℎ) ― 𝜀𝑑|𝑧𝑒 ― 𝑧ℎ|. 𝑒
𝑒
ℎ
ℎ
(4)
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Here, e refers to the electron and h to the hole. Also, 𝑚𝑒∗ ,𝑚ℎ∗ are the effective masses of the electron and the hole respectively, and 𝑉𝑒0, 𝑉ℎ0 are the confining potentials of the electron and the hole. We consider a semi-parabolic confining potentials of the form [20,21,30,351
1
38] 𝑉𝑒0(𝑧𝑒) = 2𝑚𝑒∗ 𝜔20𝑧2𝑒 , if 𝑧𝑒 ≥ 0 and 𝑉𝑒0→∞ if 𝑧𝑒 < 0, and 𝑉ℎ0(𝑧ℎ) = 2𝑚ℎ∗ 𝜔20𝑧2ℎ, if 𝑧ℎ ≥ 0 and 𝑉ℎ0→∞ if 𝑧ℎ < 0, where 𝜔0 is the characteristic oscillator angular frequency. In the strong confinement regime, the frequency difference between the ground and first excited states is given by [21] 𝜔21 =
7 2ħ𝜔0
3
- 2ħ𝜔0 ħ
= 2𝜔0 and the corresponding permanent and
induced electric matrix elements, needed for the calculation of the nonlinear optical rectification coefficient, are given by 𝜇22 - 𝜇11 = 𝑒
ħ 𝜋 𝜇𝜔0
1
and 𝜇12 = 𝑒
ħ 2 3𝜋 𝜇𝜔0
, where
∗ ∗ 𝜇 = 𝑚𝑒 𝑚ℎ (𝑚𝑒∗ + 𝑚ℎ∗ ) is the reduced mass.
3. Results and Discussion We will study a GaAs/AlGaAs quantum dot and will take the effective masses of the electron and hole, respectively, as 𝑚𝑒∗ = 0.067𝑚0, 𝑚ℎ∗ = 0.09𝑚0 (where 𝑚0 is the free electron mass), 𝜀𝑑 = 12.5, 𝑇1 = 1 ps, 𝑇2 = 0.5 ps and 𝑁 = 5 𝑥 1024 𝑚 ―3. For the semiparabolic confinement potential, choosing ħ𝜔0 = 0.773 eV, we obtain ħ𝜔21 = 1.546 eV, and using the above values for the effective masses we also take 𝜇12 = 0.738 e nm and 𝜇22 ― 𝜇11 = 0.904 e nm. Also, the metallic nanoparticle will be described by the Drude 𝜔2𝑝
dielectric function, with 𝜀𝑚(𝜔) = 1 ― 𝜔2 + 𝑖𝜔𝛤. We will assume parameters ħ𝜔𝑝 = 3.71 eV and ħ𝛤 = 0.1855 eV that represent properly the dielectric function of gold up to 2.7 eV [7]. Before analyzing the effect of the metallic nanosphere on the nonlinear optical rectification, we will study the effect of different values of the incident electromagnetic field’s intensities on the nonlinear optical rectification of a quantum dot without the metallic nanosphere. Specifically, in fig. 2(a) we present the nonlinear optical rectification coefficient as a function of the photon’s energy in a semi-parabolic GaAs/AlGaAs quantum dot (without 6
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the metallic nanosphere) for three different light intensities. We observe that as the light intensity increases the nonlinear optical rectification significantly decreases. Namely, we observe the well-known saturation of the nonlinear optical rectification when the light intensity increases [26,29,30,32,33]. We extract the same result from fig. 2(b) which shows the dependence of the nonlinear optical rectification coefficient’s maximum as a function of the intensity of light. Subsequently, we will study the coupled system of a quantum dot with a gold metallic nanosphere. The radius of the nanosphere is taken a = 10 nm. We initially consider the electric field being z-polarized (along the interparticle axis, see fig. 1). Fig. 3(a) depicts the nonlinear optical rectification coefficient of the quantum dot as a function of the photon energy, for different distances of the quantum dot from the gold nanoparticle, for the same value of light intensity. We find that depending on the distance between the quantum dot and the metallic nanoparticle, the curve of the coefficient of the nonlinear optical rectification is significantly affected, relative to the photon energy. Specifically, we see that for d = 80 nm the curve (solid curve) is low-heighted and of large width, while inversely for d = 5 nm the curve (dot-dashed curve) is of great height and with small width. In the distances between (dotted and dashed curves) the height of the curves increases, and their width decreases as the distance between the quantum dot and the metallic nanoparticle decreases. This shows that saturation occurs as the quantum dot moves away from the metallic nanosphere. Therefore, the nonlinear optical rectification coefficient increases as the quantum dot and the metallic nanoparticle come closer for the same value of light intensity (compare, for example, the dashed and dot-dashed curves of fig. 3(a) with the dashed curve of fig. 2(a)). Moreover, fig. 3(b) presents the maximum value of the nonlinear optical rectification coefficient as a function of the distance d from the surface of the gold nanoparticle. In this case, even if there is an extra dependence of the nonlinear optical rectification to the frequency of light 𝜔 in the last term of the denominator of Eq. (2), as can be seen from fig. 3(a) the maximum of the coefficient is still found at resonance (for 𝜔 = 𝜔21). This happens as 𝛾(𝜔), which carries the frequency dependence, varies insignificantly and practically remains constant in the region of frequencies very close to resonance. As we can see the 7
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maximum nonlinear optical rectification coefficient initially reduces as the distance between the quantum dot and the metallic nanoparticle increases, while later for larger distances the maximum of the nonlinear optical rectification coefficient becomes essentially constant. The latter happens because when the metallic nanoparticle is far away from the quantum dot, the metallic nanoparticle does not practically affect the electric field felt by the quantum dot. Additionally, we present results for the case that the electric field is polarized along the xaxis (and, of course, the quantum dot is grown along the x-axis). In this case, in the presence of the metallic nanosphere, the field that interacts with the quantum dot is [6,34]:
[
𝛾(𝜔)𝑎3
]
𝛦0 = 1 ― (𝑑 + 𝑎)3 𝛦0.
(5)
For this case, as it shown in fig. 4(a) we observe that the nonlinear optical rectification coefficient follows the opposite response from the one found in fig. 3(a), since here as the distance between the quantum dot and the metallic nanoparticle decreases, the nonlinear optical rectification coefficient decreases too. Therefore, saturation effects occur as the quantum dot and the metal nanoparticle come closer. We also see that the curve of the nonlinear optical rectification coefficient as a function of the photon energy is less affected when the electric field is polarized along the x-axis compared to the case of a z-polarized field of the same intensity for the different values of the interparticle distance. Additionally, fig. 4(b) shows the maximum value that the nonlinear optical rectification coefficient follows as a function of the distance d from the surface of the gold nanoparticle, for a specific value of the light intensity. In this case too, where the electric field is polarized along the x-axis, we observe that the nonlinear optical rectification coefficient increases initially as the distance between the quantum dot and the metallic nanosphere increases until it takes a practically constant value for very big interparticle distances. The behavior of the results for the two different polarizations of light can be explained from the dependence of the electric field felt by the quantum dot on distance d. Using eqs. (3) and (5) we show in fig. 5 the dependence of the normalized field intensity (intensity enhancement or suppression) for the resonance frequency for a gold nanoparticle as a function of distance d. We find that for a z-polarized field its value decreases as the quantum dot and the metallic nanoparticle come closer, while the opposite happens for an 8
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x-polarized field. This is caused due to destructive interference between the incident field and the field that is created from the metallic nanoparticle when the field is polarized along the z-axis and constructive interference between the two fields when the field is polarized along the x-axis. By using this in eq. (2) we can understand the increase or the decrease of the nonlinear optical rectification coefficient as a function of distance for the two different polarizations of light. 4. Summary In the present work we studied the optical properties of an asymmetric quantum dot that is coupled to a spherical metallic nanoparticle. Specifically, we presented results for the nonlinear optical rectification coefficient for a GaAs/AlGaAs quantum dot near a gold nanosphere for different values of the distance between the quantum dot and the metallic nanoparticle. We found that when the electric field was polarized along the z-axis (the axis that connects the quantum dot with the metallic nanoparticle) then as the distance between the quantum dot and the metallic nanoparticle decreases, the value of the nonlinear optical rectification increases. The opposite happens when the electric field is polarized along the x axis, i.e. as the distance between the quantum dot and the metallic nanoparticle decreases, the value of the nonlinear optical rectification decreases too. Acknowledgements: The author is grateful for financial support by Greece and the European Union - European Regional Development Fund via the “Supporting Postdoctoral Researchers” project of the Greek State Scholarships Foundation (IKY). She also thanks Professor Andreas Terzis of the Physics Department of the University of Patras for useful discussions.
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Fig. 1: The schematic of the system considered. A quantum dot in the presence of a metallic nanosphere.
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Fig. 2: (a) The nonlinear optical rectification coefficient for the exciton transition as a function of applied field energy for different values of the applied electromagnetic field intensity. Solid curve: I = 109 W/m2, dotted curve I = 1010 W/m2, and dashed curve I = 5×1010 W/m2. (b) The maximum nonlinear optical rectification coefficient as a function of the applied field intensity for 1.546 eV excitation energy. These calculations are in the absence of a metallic nanosphere.
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Fig. 3: (a) The nonlinear optical rectification coefficient for the exciton transition as a function of applied field energy for different values of the distance from the surface of the gold nanoparticle. The applied electromagnetic field is z-polarized and has intensity I = 5×1010 W/m2. The solid curve is for d = 80 nm, the dotted curve is for d = 10 nm, the dashed curve is for d = 7.5 nm and the dot-dashed curve is for d = 5 nm. (b) The maximum nonlinear optical rectification coefficient as a function of the distance from the surface of the gold nanoparticle for 1.546 eV excitation energy and intensity I = 5×1010 W/m2.
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Fig. 4: The same as in fig. 3 when the applied electromagnetic field is x-polarized.
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Fig. 5: The normalized field intensity felt by the quantum dot due to the presence of the gold nanoparticle as a function of the distance from the surface of the gold nanoparticle. For (a) the electric field is z-polarized and for (b) it is x-polarized.
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