The effect of metal nano particle on optical absorption coefficient of multi-layer spherical quantum dot

Physica B 490 (2016) 57–62

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Physica B journal homepage: www.elsevier.com/locate/physb

The effect of metal nano particle on optical absorption coefficient of multi-layer spherical quantum dot N. Zamani a, A. Keshavarz b,n, H. Nadgaran a a b

Department of Physics, College of Science, Shiraz University, Shiraz 71454, Iran Department of Physics, Shiraz University of Technology, Shiraz 71555-313, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 17 March 2015 Received in revised form 1 March 2016 Accepted 4 March 2016 Available online 7 March 2016

In this paper, we investigate the optical absorption coefficient of hybrid structure consisting of metal nano particle (MNP) coupled to multi-layer spherical quantum dot (MSQD). Energy eigenvalues and eigenfunctions of Schrödinger equation in this structure are obtained by using numerical solution (by the fourth-order Runge–Kutta method). The effect of MNP in the vicinity of MSQD is calculated by considering local field theory. Then the variation of optical absorption coefficient hybrid structure is calculated. The results show that the presence of MNP near MSQD enhances the optical absorption coefficient. Also, by changing the distance between MNP and MSQD and radius of MNP, variation of optical absorption coefficient and refractive index changes are introduced. & 2016 Elsevier B.V. All rights reserved.

Keywords: Metal nano particle Multi-layered spherical quantum dot Absorption coefficient Local field theory

1. Introduction In the past decades, significant attention has been paid to the investigation of optical properties of nano-structure systems such as quantum dots (QDs), quantum wires, and quantum wells [1–4]. Because of the occurrence of transitions with large oscillator strengths and narrow bandwidth [5] these low dimensional materials are a good candidate for designing and fabricating optoelectronic and photonic devices [6–8]. Today with advances in nano material fabrication, through many procedures such as chemical lithography, molecular beam epitaxy (MBE), etching method and Sol-gel, it has become possible to fabricate and confine the charge carriers in quantum dots with well-controlled shapes and sizes like sphere, pyramid, ellipsoid, cone-like [9–12]. QDs are ideal fluorophores for biological imaging and are widely used in biomedical science as a fluorescence marker as well as in chemistry. Their applications are in quantum computing [13], devices for biological application [14], and devices for emission and absorption of light in optoelectronic applications [15]. So far, the electronic and optical properties of QDs have been widely studied [16,17]. In this field, inter-subband transition related to optics has become one of the most important research subjects in QDs with different structures during the past two decades both from fundamental and technological points of view [18–21]. n

Corresponding author. E-mail addresses: [email protected] (N. Zamani), [email protected] (A. Keshavarz), [email protected] (H. Nadgaran). http://dx.doi.org/10.1016/j.physb.2016.03.010 0921-4526/& 2016 Elsevier B.V. All rights reserved.

Among the various types of QDs, multi-layer spherical quantum dot (MSQD) shows an interesting behaviour and has been studied theoretically by many researchers [22]. One of the outstanding properties of MSQD is the changes in energy spectrum which occur when some kind of modification is applied to the system. These modifications can be done by changing the size or the geometry of the system. The binding energy of a hydrogenic impurity in a GaAs-AlGaAs MSQD has been studied as a function of barrier and the inner dot thickness for different potentials [23]. Karimi et al. [24] investigated the effects of geometrical size, hydrogenic impurity, hydrostatic pressure and temperature on linear and nonlinear optical properties of multi-layered spherical quantum dots. Boz et al. [25] explored the effects of geometry on energy states of a hydrogenic impurity in MSQD by using a fourth order Runge–Kutta method. Also, this group reported the effects of magnetic field on multi-layered spherical quantum dot in ref [26]. All previous works show that the optical properties, as well as other characteristics, depend on the shape and size of the confining potential [27–29]. In recent years, the interaction between QDs and metal nano particles (MNP) has attracted considerable interest [30–32]. It is axiomatic that, when the plasmon resonances in MNP match the excitonic transitions of QDs, many diverse optical effects can appear. Furthermore, by changing the radius of the QD, the energies of the MNP plasmon and the QD exciton can match with each other so that they are strongly coupled, which will significantly enhance the optical effects. Hybrid QD-MNP systems have been used in the investigation of DNA sensors [33,34] laser systems without cavities [35], manipulation

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of heat generation in MNPs [36,37], nanothermometers [38], etc. So far, a few works have focused on optical properties enhanced by coupling between QD and MNP. In this field, the third-order nonlinear optical susceptibility for third-harmonic generation (THG) of a QD in the vicinity of a MNP has been calculated. The aim of the studies of [39–41] was to find the influence of MNP on the optical nonlinearity and the THG of the QD. In the present paper, we have a theoretical computation about optical absorption coefficient for MSQD near MNP. The purpose of this paper is to find the influence of MNP on the optical absorption coefficient of MSQD by means of effective-mass approximation as well as local field theory [42]. The outline of the paper is as follows: in Section 2, using compact density matrix approach and iterative method, the eigenstates and eigenenergies as well as the analytical expression for the linear and nonlinear susceptibility influenced by MNP are described. The numerical results and the conclusion of the paper are presented in Sections 3 and 4, respectively.

0 ≤ r ≤ R1 R1 < r < R2 R2 ≤ r ≤ R3 R3 < r < ∞

(2)

Here V0 = Q c ΔEg , where Q c ( = 0.67) is the conduction band offset parameter, and ΔEg = 1.247x eV (x corresponding mole fractional of Al) [44]. Due to the fact that the potential is spherically symmetrical, we are going to use spherical coordinates. In three dimensional spherical coordinates, the eigenfunctions of the Hamiltonian system, Eq. (1), are as below:

ψn, l, m (r , θ , ϕ) = Rn, l (r ) Yl, m (θ , ϕ)

(3)

where Rn, l (r ) is the radial wave function of the electron and Yl, m (θ , ϕ) is the spherical harmonic. n, l, m are the radial, angular and azimuthal quantum numbers, respectively. From Eqs. (1) and (3)), the radial part of the Schrödinger equation in the spherical coordinate can be expressed as:

2 d l (l + 1) ⎤ ℏ2 ⎡ d2 − ⎢ 2 + ⎥ Rn, l (r ) + [E − V (r )] Rn, l (r ) = 0, ⁎ ⎣ 2m (r ) dr r dr r2 ⎦

2. Theory 2.1. Eigenvalues and eigenstates Fig. 1 shows a schematic representation of a hybrid structure. In this figure, Wl = R1, Wr = R3 − R2 and Wb = R2 − R1 are defined as the left (inner) well, right (outer) well and barrier widths, respectively. The MNP with radius Rm, and the dielectric constant of MNP and MSQD are, ϵm and ϵs, respectively. The center-to-center distance of MSQD and MNP is d, and the background dielectric constant is ϵb. Effective mass approximation is employed in order to investigate the system wave functions, energy levels, and dipole transition matrix elements. Therefore, within the framework of effective mass approximation, the Hamiltonian of the system is given by [43]:

H=

⎧ 0, ⎪ ⎪ V0, V (r ) = ⎨ ⎪ 0, ⎪ ⎩ V0,

⎞ l (l + 1)ℏ2 −ℏ2 ⎛ 1 ∇r ⎜ ⁎ ∇r ⎟ + + V (r ) ⎝ m (r ) ⎠ 2 2m⁎ (r ) r 2

(1)

h

where ℏ = 2π (h is the Planck constant) and m⁎ (r ) is the positiondependent effective mass of electron, r is the radial position of the electron, and l is the angular momentum quantum number. V(r) is the confinement potential profile of the MSQD which can be seen in Fig. 1 and defined as:

m

d

ε (ω)

εs

z

ε

D

2.2. Optical properties In this section, we will derive the optical properties of the hybrid system using the compact density matrix method and the iterative procedure. We consider that such a system interacts with a laser field E (t ) = ϵE0 cos (ωt ) where ϵ is the polarization unit vector, E0 is the electric field amplitude, and ω is the angular frequency of the applied field. We assume that only the first two energy levels of the MSQD contribute to the dynamics of the system, and other transitions are considered to be out of resonance with the field frequency. The total electric field inside a MSQD comes from three parts: EMSQD = (E1 + E2 + E3 ) e−iωt . E1e−iωt is the electric field induced by E0 inside a MSQD in the absence of a MNP [40]:

E1 =

3ϵb 1 E0 = E0 ϵ s + 2ϵb ϵeff

(5)

here ϵeff is the effective dielectric constant of the MSQD. E2 e−iωt is the local field induced only from surface charges of the MNP and is given by:

MSQD

R

It should be noted that we have taken into consideration the lowest two-level energies which are S (n = 0, l = 0) and P (n = 1, l = 1). The fourth-order Runge–Kutta method was employed to obtain energy eigenvalues and eigenfunctions of the system due to complicated analytical solution of MSQD [45,46].

E2 = sα

3 γ1R m E0, ϵeff d3

(6)

b

m

where γ1 = [ϵm (ω) − ϵb ] /[ϵm (ω) + 2ϵb ] and sα = 2 ( − 1). sα is 2 when the electric field polarization E0 is parallel to z axis and it is
1 when the electric field polarization is parallel to x,y axes. E3 e−iωt is the field induced by the dipole (n = 1) and multidipole (n > 1)> interaction between MNP and MSQD and

MNP

W

V(r) Wl

Wb

r

∞

V

0

E3 =

∑ n= 1

2n + 1 sn γn R m

ϵb ϵ2eff d2n + 4

p

(7) 1)2

R

1

R

2

R

3

R4

r

Fig. 1. Schematic representation of a multi-layered spherical quantum dot (MSQD) with its potential profile near MNP in radial direction.

The parameter sn = (n + when E0 is parallel to the z axis. And sn = Pn′ (1) when E0 is vertical to the z axis. Here Pn is the Legendre function and Pn′ (1) is the differential of Legendre function of argument of 1. γn = [ϵm (ω) − ϵb ] /[ϵm (ω) + (1 + 1/n)ϵb ] and p is the polarization. In order to calculate the energy damping and the significance of

N. Zamani et al. / Physica B 490 (2016) 57–62

59

˚, Fig. 2. The confinement potential profile of the MSQD (blue colour) and the probability distribution of the ground state S and the first excited state P versus r (a) Wl = 30 A ˚ , (c) Wl = 50 A ˚ . (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.) (b) Wl = 40 A

E3, we use a two-level system to describe the optical process. The Hamiltonian of the system in the rotating-wave approximation can be written as [47,48]: † † H = ℏω12 a^ a^ − μ12 EMSQD (a^ + a^ )

(8)

where a^ and a^ are the creation and annihilation operators of i . μ12 is the transition matrix element and ω12 is the transition frequency when the electron transits from state 1 to 2 . In the rotating wave and steady state approximation, by solving the optical Bloch function, we can obtain [49]: †

2T1 Im (μ12 Eρ21 ), ℏ

ρ22 =

ρ21 =

−ΩΔ (ω − ω12 + GΔ + i/T20 )

(9)

ϵ 0 κ (1) (ω) =

σv |μ21|2 ′ − ω − iγ12 ′) ℏ(ω12

(13)

where σv is the carrier density. The susceptibility χ (ω) is related to the change in refractive index and optical absorption coefficient which is satisfied in [50]:

⎡ χ (ω) ⎤ Δn (ω) = Re ⎢ ⎥, nr ⎣ 2nr2 ⎦ α (ω) = ω

μ Im [ϵ 0 χ (ω)], ϵR

(14)

where nr is the refractive index, μ is the permeability of the system, and ϵR is the real part of the permittivity.

in which

Ω=

μ12 ·(E1 + E2 ) μ f = 12 2 , ℏ ℏ

∞

G=

∑ n= 1

2n + 1 μ12 sn γn R m

ϵb ϵ2eff d2n + 4ℏ

≡ GR + iGI

3. Numerical results and discussion

(10)

3 sα γ1Rm E0/d3) 1/ϵeff ,

where f2 = (1 + and GR and GI are the real and imaginary parts of G, respectively. Actually GR denotes the energy shift between states 1 and 2 , while GI denotes the shift of the dephasing rate. According to Ref. [42], the first-order density matrix can be written as: (1) ρmn =

Δμ mn E (ω) ℏ(ω mn − ω − iγmn )

σv = 5 × 1022 m−3, nr = 3.2, T12 = 0.14 ps where Γ12 =

(11)

By comparing Eqs. (9) and (11), we find that they have the same ′ = ω12 − ΔGR and form if we make the transformation ωmn γ′mn = 1/T20 + ΔGI . ⁎ Because p = μ12 (ρ21 + ρ21 ), we can rewrite Eq. (7) as E3 = f3 E0 , where ∞

f3 =

∑ n= 1

2 ⎤ 2n + 1 ⎡ f2 −μ12 sn γn R m ⎢ + c . c ⎥. 2 2 n 4 + ′ + iγ12 ′) ⎥⎦ ϵb ϵeff d ⎣⎢ ℏ(ω − ω12

Optical properties of MSQD are strongly affected by the geometrical size and any changes in the structure which lead to changes in the optical properties. Also the interaction of MSQD and MNP changes the optical properties. In this section, the influence of MNP near MSQD on variation of optical properties becomes evident. The unchanged parameters used in our calcula⁎ ⁎ tions are as follows: mAlGaAs = 0.088m0 , mGaAs = 0.067m0 ,

(12)

Therefore, the total electric field is EMSQD = ℓE0 and ℓ is the enhance factor: ℓ = f2 + f3. The MNP enhances linear susceptibilities for a two level quantum system which are given as χ (1) (ω) = ℓ(ω) κ (1) (ω) where

1 , T12

I ¼0.2

W cm2

.

We assume that the factor Δ = ρ11 − ρ22 = 1 − 2ρ22 ≃ 1 because the effect of light field is considerably weak. The MNP is made of gold, so for ϵm (ω) we use the experimental values for gold [51]. The background dielectric constant ϵb = 1.8ϵ0 while the semiconductor dielectric is ϵs = 9ϵ0 . The size of the gold nano particle is Rm ¼100 Å whereas the distance between gold and MSQD nano particles is D¼ 20 Å. Fig. 2 represents the confining potential and the probability distribution of ground and first states of MSQD as a function of the growth direction r with Wb ¼10 Å and Wr = 20 A˚ as fixed and Wl = 30, 40 and 50 Å as variable. It can be seen that confinement of ground state increases as inner well width increases. Figs. 3–5 display the optical absorption coefficients as a function of the photon energy ℏω for three different states. In Fig. 3, the first state is depicted where the optical absorption coefficients of MSQD are

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N. Zamani et al. / Physica B 490 (2016) 57–62

0.8

W

0.06

0.7

W1

0.04

1

W

0

Presence of MNP with E || to z

1

0.6

0

0.02

0.5 Δn/nr

α×105(1/m)

Absence of MNP Presence of MNP with E || to x(y)

0.4

0

x 10

−4

5

0.3 −0.02

0.2

0

−0.04

0.1

−5

0 60

65

70

75

80 85 90 Photon Energy(meV)

95

100

105

110

Fig. 3. Optical absorption coefficient for MSQD in the absence of MNP for the inner well width values Wl ¼ 30, 40 and 50 Å consecutively.

−0.06 50 50

60

55

70

80

60

90

65

70 75 80 Photon Energy(meV)

85

90

95

Fig. 6. Refractive index changes for MSQD in the absence and presence of MNP with electric field polarization parallel (vertical) to z axis and the inner well width ˚. value Wl = 30 A

1.4 Presence of MNP with E0 || z

Absence of MNP 1.2

α×105(1/m)

1 0.8 0.6 0.4 0.2 0 50

55

60

65

70

75

80

85

90

95

Photon Energy(meV) Fig. 4. Optical absorption coefficient for MSQD in the presence of MNP for the ˚ with electric field polarization parallel to z axis. inner well width value Wl = 30 A

0.7

Presence of MNP with E vertical to z 0

0.6

Absence of MNP

5 α×10 (1/m)

0.5 0.4 0.3 0.2 0.1 0 60

65

70

75 80 85 Photon Energy(meV)

90

to (Fig. 4) or vertical on the z axis (Fig. 5). In Fig. 4 the peak is increasing while in Fig. 5 it is decreasing. In the second and third states, the resonance peak has a red shift due to GR. In order to see the effect of MNP on the refractive index changes, they are plotted as a function of photon energy in Fig. 6 for the MSQD in absence and presence of MNP with electric field polarization parallel(vertical) to z axis. It can be seen that the refractive index changes with MSQD near MNP increase for electric field polarization parallel to z axis while they decrease for electric field polarization vertical to z axis. And in both cases a red shift occurs. This behaviour is due to the reduction in the energy difference of two different electronic states, corresponding to the ground state (l ¼0) and the first excited state (l ¼1). Our aim is to increase the optical properties. From the above figures, we know that the enhancing optical properties occur for electric field polarization parallel to z axis. Therefore, for the following figures, we consider the case of E0 ∥ z axis. Fig. 7 shows the optical absorption coefficient as a function of photon energy and distance D between MSQD and MNP for electric field polarization parallel to z axis with Rm = 150 A˚ . It is observed that the α decreases gradually as the distance between MNP and MSQD increases. Also as the distance increases, the peak of α tends to blue shifts. Considering the fact that when the electric field polarization is parallel to z axis the optical absorption is enhanced, for E0 ∥ z we display the optical absorption coefficient as a function of photon energy and radius of MNP with D ¼30 Å in Fig. 8. It is observed that the α peak increases as the radius of MNP increases and it tends toward blue shifts.

95

Fig. 5. Optical absorption coefficient for MSQD in the presence of MNP for the ˚ with electric field polarization vertical to z axis. inner well width value Wl = 30 A

plotted as a function of the photon energy ℏω in the absence of MNP with Wb = 10 A˚ and Wr = 20 A˚ as fixed and Wl as variable. It is observed that the resonant peaks of optical absorption coefficients go toward blue shift with the increase of the radius of the inner well. In Figs. 4 and 5, the second and third states of the optical absorption coefficients of MSQD near MNP are plotted as a function of the photon energy ℏω for electric field polarization and are parallel with or vertical on z axis, respectively. Compare to Fig. 3 in which MNP is absent, Figs. 4 and 5 display the peaks of optical absorption coefficients in the presence of MNP where E0 is parallel

4. Conclusion In this paper, we studied the effect of MNP on optical absorption coefficient of MSQD. Accordingly, we acquired the eigenfunctions and energy eigenvalues through solving Schrödinger equation using a numerical method for MSQD. Then we utilized the calculated electron eigenfunctions and energies to evaluate the optical properties under study. The main focus of this work was to find the influence of gold metal nano particle on variation of optical absorption coefficients. For this purpose, the local field theory and effective mass approximation were employed to locate the effect of MNP on optical absorption coefficient of MSQD. The results showed that the MNP enhanced optical absorption coefficient in the hybrid system. It was found that by changing distance d, when the E0 ∥ z , α can be enhanced. In addition, by increasing the

N. Zamani et al. / Physica B 490 (2016) 57–62

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Fig. 7. Optical absorption coefficient for MSQD as a function of photon energy and distance between MSQD and MNP for electric field polarization parallel to z axis with the ˚. radius of gold nano particle is Rm = 100 A

Fig. 8. Optical absorption coefficient for MSQD as a function of photon energy and radius of MNP for electric field polarization parallel to z axis with the distance between MSQD and MNP is D ¼30 Å.

radius of the gold nano particle the optical absorption coefficient increased and tended to blue shift. We hope that this work can make a significant contribution both to the experimental work and also to the theoretical analysis in this area.

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