Axial interface optical phonon modes in a double-nanoshell system

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Axial interface optical phonon modes in a double-nanoshell system

This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2008 Nanotechnology 19 285713 (http://iopscience.iop.org/0957-4484/19/28/285713) View the table of contents for this issue, or go to the journal homepage for more

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IOP PUBLISHING

NANOTECHNOLOGY

Nanotechnology 19 (2008) 285713 (8pp)

doi:10.1088/0957-4484/19/28/285713

Axial interface optical phonon modes in a double-nanoshell system C Kanyinda-Malu1,2, F J Clares1 and R M de la Cruz1,3 1

Departamento de F´ısica, Universidad Carlos III de Madrid, EPS Avenida de la Universidad 30, 28911 Legan´es (Madrid), Spain 2 Departamento de Econom´ıa Financiera y Contabilidad II, Universidad Rey Juan Carlos, FCJS, Paseo de los Artilleros s/n, 28032 Madrid, Spain E-mail: [email protected] and [email protected]

Received 7 April 2008, in final form 12 May 2008 Published 3 June 2008 Online at stacks.iop.org/Nano/19/285713 Abstract Within the framework of the dielectric continuum (DC) model, we analyze the axial interface optical phonon modes in a double system of nanoshells. This system is constituted by two identical equidistant nanoshells which are embedded in an insulating medium. To illustrate our results, typical II–VI semiconductors are used as constitutive polar materials of the nanoshells. Resolution of Laplace’s equation in bispherical coordinates for the potentials derived from the interface vibration modes is made. By imposing the usual electrostatic boundary conditions at the surfaces of the two-nanoshell system, recursion relations for the coefficients appearing in the potentials are obtained, which entails infinite matrices. The problem of deriving the interface frequencies is reduced to the eigenvalue problem on infinite matrices. A truncating method for these matrices is used to obtain the interface phonon branches. Dependences of the interface frequencies on the ratio of inter-nanoshell separation to core size are obtained for different systems with several values of nanoshell interdistance. Effects due to the change of shell and embedding materials are also investigated in interface phonon modes. (Some figures in this article are in colour only in the electronic version)

(CQDs) [22] can be fabricated. Along with devices fabrication, the works on phonon modes were extended to these new systems. On the other hand, intense research in semiconductor QDs was carried out for potential applications in optoelectronic devices and high-density memories. Recently, two groups demonstrated simultaneously that semiconductor QDs could be made water soluble and could be conjugated with biological molecules [23, 24]. Moreover, since the QD photoluminescence emission maximum can be manipulated by changing the particle size, its use as a fluorescent label for biological macromolecules has attracted considerable attention in medical imaging compared with conventional organic dyes. This new feature makes semiconductor QDs the object of intensive investigations in fundamental as well as applied aspects relative to their surfaces in order to make more feasible the conjugation with biological molecules [25–27]. Thus, the fluorescence quenching of the QDs can be used to measure the concentration of glucose in aqueous solution, which is based on the transfer of electrons from QDs to enzymes.

1. Introduction Since the pioneering works of Fuch and Kliewer [1] and Licari and Evrad [2] on the polar vibration modes in coupling quantum well (CQW) systems, the study of electron– phonon interactions in low-dimensional quantum structures have attracted much interest [3–12]. Electron–phonon interactions play a key role in many physical processes, such as transport or electron relaxation processes in confined systems. Of special interest are phonon confinements in quantum dots (QDs) and nanocrystals [13–15], which manifest themselves via additional peaks in the low-frequency range, blue shift of confined acoustic phonon peaks with decreasing nanoparticle size, and red shift and asymmetric broadening of optical phonons [16, 17]. Due to the rapid progress in semiconductor nanotechnology, sophisticated systems such as multi-layer planar CQWs, multi-layer coupling quantum well– wire (CQWW) [18–21] and multi-shell coupling quantum dots 3 Author to whom any correspondence should be addressed.

0957-4484/08/285713+08$30.00

1

© 2008 IOP Publishing Ltd Printed in the UK

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C Kanyinda-Malu et al

Actually, artificial molecules consisting of a few nanocrystals bound by organic backbone molecules are used to define a desired symmetry and degree of electronic coupling between QDs. In this way, two-dimensional (2D) or three-dimensional (3D) arrays of QDs are build into assembled blocks by covalent, hydrogen bonding or van der Waals interaction [28, 29] giving rise to QD molecules. These QD molecules exhibit, in some cases, excellent optical properties up to room temperature, making them potential candidates as functional building blocks for quantum computing and quantum communication applications in the solid state [30, 31]. In spite of extensive literature on coupled QDs, little is known about the phonon modes in a double system of nanoshells. Many interesting works dealing with eccentric or coupled two-sphere systems have been devoted to electrostatic force or electrophoresis mechanisms within the DC models [32]. One method for developing an exact solution is the use of bispherical coordinates. Hence, in the present work, we will deduce the interface optical phonon modes in a double-nanoshell system within the framework of the dielectric continuum model. To emphasize the effect of interdistance and size of the QD nanoshell, we compare the interface frequency (IF) modes in the nearly touching nanoshells (small interdistance) with the roughly decoupled ones (large interdistance) using the bispherical coordinates. The paper is organized as follows. In section 2, we describe briefly the application of the DC model to the two-nanoshell system in mathematical form. Subsequently, solutions for the electrostatic potentials and boundary conditions (BCs) are introduced, giving rise to the eigenvalue problem on infinite matrices. The results on II–VI doublenanoshell systems evaluating the effect of shell and insulating materials on the IF phonon characteristics are discussed in section 3. Finally, the conclusions are given in section 4.

Figure 1. Scheme of the double coupled nanoshell system and the bispherical polar coordinates employed in the analysis, with their respective parameters.

the nanoshells is denoted by 2 D , with the origin located at z = 0. The respective radii of the nanoshells are R1 and R2 , and the foci of the bispherical system are referred to as a± . Figure 1 shows the scheme of our system defined by bispherical coordinates. In this system, we assume that the focus is located inside the core region of the nanoshells and both nanoshells are centered at z = ±D . For the coupled spherical systems, it is convenient to use the bispherical polar coordinates (η, α, φ) which are related to the rectangular coordinate system by

2. DC model in the two-nanoshell system The dielectric continuum (DC) model is extensively applied to study phonon modes in polar semiconductor nanostructures [33]. In this model, the surface modes governing equation is given by (ω)∇ 2  = 0 (1)

x = a sin α cos φ/(cosh η − cos α),

(2a )

y = a sin α sin φ/(cosh η − cos α),

(2b )

z = a sinh η/(cosh η − cos α), (2c)  where a = 12 D −1 D 4 − 2 D 2 (R12 + R22 ) + (R12 − R22 )2 [34], and it represents a geometric focus distance, located in the inner-spherical core (see figure 1). The surfaces of the two concentric spheres are defined by η = ±η1,2 with D = Ri cosh ηi , Ri = a/| sinh ηi |, (i = 1, 2). Typical solutions of Laplace’s equation in bispherical coordinates are e±(n+1/2)η Pnm (cos α) or sinh((n + 1/2)η)Pnm (cos α), and their linear combinations. So far, the general solution of Laplace’s equation in these coordinates, which is regular on the axis, assumes the form

where  is the electrostatic potential and (ω) the frequency2 dependent dielectric function defined by (ω) = ∞ (ωLO 2 2 2 − ω )/(ωTO − ω ), with ωLO and ωTO the longitudinal and transversal optical frequencies of the bulk material. Since (ω) = 0 gives rise to the bulk-like LO optical mode, we will restrict our treatment to the case (ω) = 0, where the solution of Laplace’s equation along with the BCs lead to surface or interface modes. The system under study consists of two identical spherical equidistant nanoshells embedded in an insulating medium. We denote by 1 + (1−), 2 + (2−) and 3 the upper (lower) core, shell and surrounded regions with their respective dielectric functions 1 (ω), 2 (ω) and a frequency-independent dielectric constant 3 . The center-to-center distance between

 = (cosh η − cos α)1/2

∞  n  

   Mn exp n + 12 η

n=0 m=−n

    + Nn exp − n + 12 η × Pnm (cos α)eimφ 2

(3)

Nanotechnology 19 (2008) 285713

C Kanyinda-Malu et al 1    2 (ω) n Bn−1 e−(n− 2 )η2 + sinh η2 − (2n + 1) cosh η2

where Pnm (cos α) are associated Legendre functions of first kind. Because of the axial symmetry of the system, functions having different m are not coupled and we can solve separately for each m . We will only solve for m = 0, which is the axial mode. For m = 0 we choose potentials which are symmetric with respect to reflections through the x y plane. Then, we build the potentials inside the upper and lower nanoshell and in the insulating medium given by ∞  1 1 = (cosh η − cos α)1/2 An e−(n+ 2 )η Pn (cos α) (4a )

1

× Bn e−(n+ 2 )η2 + (n + 1)Bn+1 e−(n+ 2 )η2     + sinh η2 cosh n + 12 η2 + (2n + 1) cosh η2       × sinh n + 12 η2 Cn − nCn−1 sinh n − 12 η2    − (n + 1)Cn+1 sinh n + 32 η2     = 3 sinh η2 cosh n + 12 η2 + (2n + 1) cosh η2       × sinh n + 12 η2 Dn − n Dn−1 sinh n − 12 η2   − (n + 1)Dn+1 sinh n + 32 .

n=0

2 = (cosh η − cos α)

∞   1/2

1 Bn e−(n+ 2 )η

A− n,

   + Cn cosh n + 12 η Pn (cos α) ∞  1 (n+ 2 )η 1/2 − A− Pn (cos α) ne 1 = (cosh η − cos α) 1/2 − 2 = (cosh η − cos α)

(4b ) (4c)

1

Bn− e(n+ 2 )η

n=0

   + cosh n + 12 η Pn (cos α) ∞     3 = (cosh η − cos α)1/2 Dn cosh n + 12 η Cn−

(4d )

n=0

× Pn (cos α)

(4e)

with the indices representing the respective regions of the space. The equations for unknown coefficients can be obtained with the help of the usual electrostatic BCs (i.e., continuity of electrostatic potential and continuity of normal component of displacement electric field). Unlike the 11 separable system coordinates, the factor (cosh η − cos α)1/2 in the R-separable bispherical eigenfunctions causes trouble when we come to fix the BCs. So, to overcome this trouble, we multiply both terms by 2(cosh η − cos α)1/2 in this case and apply the recurrence relation of Legendre polynomials (5)

to expand all functions of α in terms of Legendre polynomials Pn (cos α). Exploiting the orthogonality of the Pn s and equating their coefficients, the BCs yield the following expressions: 1 1    An e−(n+ 2 )η1 = Bn e−(n+ 2 )η1 + Cn cosh n + 12 η1 , (6a ) 1       Bn e−(n+ 2 )η2 + Cn cosh n + 12 η2 = Dn cosh n + 12 η2 1    1 (ω) n An−1 e−(n− 2 )η1 + sinh η1 − (2n + 1) cosh η1 1 3 × An e−(n+ 2 )η1 + (n + 1)An+1 e−(n+ 2 )η1 1   = 2 (ω) sinh η1 − (2n + 1) cosh η1 Bn e−(n+ 2 )η1 1

(7b )

Cn−

The lower nanoshell equations are obtained similarly with the above-mentioned substitutions. To eliminate the coefficient Dn in the BC equations we substitute (6b) in equation (7b), and reordering the Bn and Cn ’s coefficients we obtain the following expressions: 1     n e−(n− 2 )η2 2 (ω) + 3 tanh n − 12 η2 Bn−1    + sinh η2 − (2n + 1) cosh η2 2 (ω) − sinh η2 1    + (2n + 1) cosh η2 tanh n + 12 η2 3 Bn e−(n+ 2 )η2 3     + (n + 1) 2 (ω) + 3 tanh n + 32 η2 Bn+1 e−(n+ 2 )η2        − n 2 (ω) − 3 sinh n − 12 η2 Cn−1 + 2 (ω) − 3     × sinh η2 cosh n + 12 η2 + (2n + 1) cosh η2      × sinh n + 12 η2 Cn − (n + 1) 2 (ω) − 3    (9) × sinh n + 32 η2 Cn+1 = 0.

(2n + 1) cos α Pn (cos α) = (n + 1)Pn+1 (cos α) + n Pn−1 (cos α)

Bn− ,

The equations for the coefficients and can be obtained from equations (6a) and (6b), (7a) and (7b) by the − − substitutions An → A− n , Bn → Bn , Cn → Cn , η1 → η1− and η2 → η2− . From equations (6a) and (7a) and reordering the Bn and Cn ’s coefficients for the upper nanoshell we obtain 1    n e−(n− 2 )η1 1 (ω) − 2 (ω) Bn−1 + sinh η1 1   − (2n + 1) cosh η1 1 (ω) − 2 (ω) e−(n+ 2 )η1 Bn 3   + (n + 1) 1 (ω) − 2 (ω) e−(n+ 2 )η1 Bn+1        + n 1 (ω) cosh n − 12 η1 + 2 (ω) sinh n − 12 η1    × Cn−1 + 1 (ω) sinh η1 − (2n + 1) cosh η1        × cosh n + 12 η1 − 2 (ω) sinh η1 cosh n + 12 η1    + (2n + 1) cosh η1 sinh n + 12 η1 Cn + (n + 1)     × 1 (ω) cosh n + 32 η1 + 2 (ω)    (8) × sinh n + 32 η1 Cn+1 = 0.

n=0

n=0 ∞  

3

(6b )

Equations (8) and (9) constitute a set of homogeneous difference equations whose coefficients are functions of the materials’ constitutive parameters. Let us introduce the vector notations for the matching coefficients; i.e., B = (B0 , B1 , . . . , Bn , . . .) and C = (C0 , C1 , . . . , Cn , . . .). Thus, the equations (8) and (9) can be reduced to matrix problems

3

+ n Bn−1 e−(n− 2 )η1 + (n + 1)Bn+1 e−(n+ 2 )η1     + sinh η1 cosh n + 12 η1 + (2n + 1) cosh η1       × sinh n + 12 η1 × Cn − nCn−1 sinh n − 12 η1    (7a ) × (n + 1)Cn+1 sinh n + 32 η1 ,

[M] B + [N]C = [0] 3

(10)

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C Kanyinda-Malu et al

[P] B + [Q]C = [0]

Table 1. Optical parameters and materials used in this work.

(11)

where [M], [N], [P] and [Q] represent the tridiagonal infinite matrices which relate the geometrical parameters, the vibrational frequencies and materials data for the upper nanoshell (see the appendix). Similar calculations applied to the lower nanoshell system provide another set of homogeneous difference equations, i.e.,

∞

ωLO (cm−1 )

ωTO (cm−1 )

CdS HgS ZnS Polyethylene Glass

5.5 11.36 5.2 2.3 4.64

300.0 250.0 349.0 — —

233.0 197.5 269.0 — —

[R] B − + [S]C − = [0]

(12)

3. Results and discussion

[T ] B − + [U ]C − = [0]

(13)

To illustrate our results, we apply the theoretical model to the two commonly used II–VI semiconductor heterostructures; i.e., CdS/ZnS and CdS/HgS embedded in two insulating media: polyethylene and glass. The interest in CdS and ZnS nanostructures has grown due their biological applications, especially for their use in biological labels (see [27] and references therein). Recently, capping core nanocrystals with ZnS has been shown to increase the stability and performance, producing QDs with improved luminescence, higher photochemical stability and higher quantum yields at room temperature. However, it is known that ZnS capping is not sufficient, particularly in biological solutions. The modification of the product to make it biologically compatible with polyethyleneglycol increases the stability and reduces non-specific adsorption [38]. Therefore, the determination of the interface phonon modes in four systems of double nanoshells, i.e., CdS/ZnS/polyethylene, CdS/ZnS/glass, CdS/HgS/polyethylene and CdS/HgS/glass, can help in understanding some optical responses of these molecule-like QDs in different environments. The effects of the substitution of the shell and embedding materials will be investigated on the interface phonon branches. Core-like and shell-like phonon modes are expected to appear. Also, all the interface frequencies are expected to satisfy the restrahlen condition: ωTO  ωIF  ωLO . Our results are shown as a function of the geometrical parameter D/R1 for two selected constant values of D/R2 . Hence, the value of D/R2 = 1.127 corresponds to the case of nearly touching nanoshells, while the value of D/R2 = 2.151 corresponds to nearly decoupled nanoshells. The derivation of the interface frequencies is reduced to the eigenvalue problem on infinite matrices. This treatment is similar to that reported in the literature for surface modes of two spheres and van der Waals energy between voids in dielectrics [39, 40]. In this work, we truncate the infinite matrices up to the 4 × 4 treatment, showing mainly our results for the 3 × 3 truncating case. Values of ωTO , ωLO and ∞ of the constitutive materials are given in table 1 [41, 42]. By truncating the derived infinite matrices by the 3 × 3 truncating method, we obtain a symmetrical 18th-order polynomial in terms of ω, where the numerical resolution leads to nine real roots, which can be clustered in five bands of branches. Finally, we treat the 4 × 4 truncating method and obtain a 24th-order polynomial in terms of ω. In the light of the above results, we would expect for higher-order truncation treatment a higher number of branches clustered to five bands of phonons. The IF values of these bands would

with their corresponding tridiagonal infinite matrices [R], [S], [T ] and [U ]. Checking again the BC equations for the upper and lower nanoshells, we have 1    Bn e−(n+ 2 )η2 + Cn cosh n + 12 η2 1   = Bn− e(n+ 2 )η2 + Cn− cosh n + 12 ) . (14) This relation can be transformed to matrix form using the identical rules applied to the former BC equations. Then we obtain [F] B − + [G]C − = [H ] B + [ J ]C (15) with four tridiagonal infinite matrices [F], [G], [H ] and [ J ] for the vectors B − , C − , B and C . In summary, joining the continuity and linking the BC relations for the two nanoshells, we have a system of matricial homogeneous equations with four unknown quantities. Thus, equation (15) appears to be a constraint condition which connects the upper and lower nanoshells throughout the medium; i.e.,

−[H ] B [M] B

Material

−[ J ]C +[N]C

+[F] B −

+[G]C −

=0 =0 [T ] B − +[U ]C − = 0   [P] B +[Q]C = 0. To determine the interface frequencies, we have to resolve the above system of equations. For this, we impose that the determinant of the resulting matrix generated from the ten tridiagonal infinite matrices must be zero. An exact solution requires an infinite number of terms, so the series must be truncated at some point. For numerical implementation, we truncate the different matrices at n = 4 (also called 4 × 4). We will largely discuss the 3 × 3 truncating method since we do not observe a big deviation between (3 × 3) and (4 × 4) results. The number of terms in our numerical treatment was limited by the PC Mathematica package [35] and memory requirements. Some authors have demonstrated that, in some case, a small number of terms (e.g. n = 5 or n = 10) is enough [36] to get an insight into the physical phenomena. A similar truncating treatment was also applied in other research fields such as the modal methods of diffraction gratings [37]. The convergence of the IF frequencies towards the real values is discussed in the next section. 4

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C Kanyinda-Malu et al 360

340

Interface frequency (cm-1)

Interface frequency (cm-1)

360

(a)

320 300 CdS/ZnS/polyethylene

280

D/R2= 1 .127

260 240 220

340 320

(a)

300

CdS/ZnS/polyethylene D/R2= 2 .151

280 260 240 220

D/R1

D/R1

360

320

Interface frequency (cm-1)

Interface frequency (cm-1)

360 340 (b)

300 280 260

CdS/ZnS/glass D/R2= 1 .127

240

340 320

(b)

300

CdS/ZnS/glass

280

D/R2= 2 .151

260 240

220

220 D/R1

D/R1

Figure 2. Interface frequencies of CdS/ZnS/polyethylene (a) and CdS/ZnS/glass (b) as a function of D/R1 and for a fixed value of D/R2 = 1.127.

Figure 3. Interface frequencies of CdS/ZnS/polyethylene (a) and CdS/ZnS/glass (b) as a function of D/R1 and for a fixed value of D/R2 = 2.151.

be approximately around average values of IF frequencies corresponding to that obtained in the 1 × 1 truncating treatment. As an example, for the CdS/ZnS/islanding material systems when D/R2 = 1.127, we would have values around 349, 343, 305, 250, and 237 cm−1 . The above values are arithmetical median values obtained in the geometrical parameter D/R1 range investigated. Because, in all investigated treatments, the number of branches can be clustered in five bands of branches, we can conclude that the number of IF branches of two double-nanoshell systems is five. Let us remember that for one islanding quantum dot there are two IF branches, which corresponds to 1 separation barrier +1 [43]. For one unique nanoshell, there are three IF branches [27, 33], which corresponds to 2 separation barriers +1. Consequently, it can be argued that, for double-nanoshell systems, there would be five branches: i.e., 4 separation barriers +1, as is obtained in this work. Finally, we will comment that the existence of a band of phonons is a feature appearing in other lowdimensional heterostructures such as superlattices [44]. Therefore, as an example, the results of the four investigated systems are shown for the 3 × 3 truncating treatment. Figure 2 shows the interface frequencies for CdS/ZnS embedded in two different insulating media, polyethylene and glass, as a function of D/R1 and for a value of D/R2 = 1.127. For both cases, there are nine branches, which can be clustered into five bands of frequencies around values of 340 cm−1 , 320 cm−1 , 290 cm−1 , 240 cm−1 and 233 cm−1 , respectively. Two of these branches (340 and 320 cm−1 ) are of shell type, while the branches around 240 cm−1 and 233 cm−1

are core type. On the other hand, a non-well-defined branch around the value of 290 cm−1 (see figure 2) appears between the core-like and shell-like modes, that can be ascribed to one mixed mode. By comparing with the case of one nanoshell [27], we obtain similar behavior of the branches with the geometrical parameter. In fact, two additional branches bands around 320 and 233 cm−1 are found. This feature could be interpreted as a consequence of the double system, where at least five regions of space are observed. The effect of the insulating material is to enlarge the gap between the two lowest shell-type branches in CdS/ZnS/glass, where the values of the dielectric constants of the core, shell and insulating material are nearly similar (see table 1). This effect was also observed in the case of one nanoshell [27]. If the interdistance between nanoshells is increased, the mixed branch of 290 cm−1 becomes a nearly constant branch with value 269 cm−1 , and the branch around 233 cm−1 also becomes constant (see figure 3). The effect of the insulating material is similar to that obtained for D/R2 = 1.127; that is, an enlargement of the gap between the lowest shell-type and the mixed branch. Figure 4 shows the interface frequencies of CdS/HgS/polyethylene and CdS/HgS/glass as a function of D/R1 for a value of D/R2 = 1.127. A change in the shell material (substitution of ZnS by HgS) makes the IF modes to have two clearly defined shell-like around 210 and 220 cm−1 , one core-like mode around 280 cm−1 , and two mixed bands around 245 and 240 cm−1 . The effect of the insulating material is similar to that obtained in the previous cases. If D/R2 increases, the dependence of the IF branches with the geometri5

Nanotechnology 19 (2008) 285713

C Kanyinda-Malu et al

280

260

Interface frequency (cm-1)

Interface frequency (cm-1)

280 (a)

CdS/HgS/polyethylene D/R2= 1 .127

240

220

(a)

CdS/HgS/polyethylene D/R2= 2 .151

260 240 220 200

200 D/R1

D/R1

280 (b)

260

Interface frequency (cm-1)

Interface frequency (cm-1)

280 CdS/HgS/glass D/R2= 1 .127

240

220

(b)

CdS/HgS/glass D/R2= 2 .151

260 240 220 200

200 D/R1

D/R1

Figure 4. Interface frequencies of CdS/HgS/polyethylene (a) and CdS/HgS/glass (b) as a function of D/R1 and for a fixed value of D/R2 = 1.127.

Figure 5. Interface frequencies of CdS/HgS/polyethylene (a) and CdS/HgS/glass (b) as a function of D/R1 and for a fixed value of D/R2 = 2.151.

cal parameter is slighter than for the value D/R2 = 1.127 (see figure 5). Again the change in embedding material modifies the gap separation between the lowest core and highest branches. In order to prove the convergence of IF frequencies towards the real ones obtained from the infinite tridiagonal matrices, we truncate the above matrices until 4 × 4 order, obtaining five band branches of phonons, similar to that obtained for a smaller order of truncating method. Figure 6 shows the interface frequencies of CdS/ZnS/insulating media as a function of D/R1 and for a value of D/R2 = 1.127. Although not shown here, a similar behavior for 4 × 4 truncating order is obtained for the other systems of nanoshells. On the other hand, it is interesting to investigate the case when η2 tends to ∞, which would correspond to the case of one isolated nanoshell. In fact, figure 7 shows the interface frequencies for CdS/ZnS/polyethylene as a function of D/R1 and for a value of D/R2 = 5.71 × 1025 . We obtain three individual branches; two are shell type and one is core type, typical of one isolated nanoshell [27, 33]. Although not shown here, the behavior of IF branches with different geometrical parameters is similar for the other systems of nanoshells. In all these investigations, we assume that D/R1 is greater than D/R2 , that makes it easy to analyze the effect of interdistance between the external spheres.

360

Interface frequency (cm-1)

340 320 CdS/ZnS/polyethylene D/R2= 1 .127

300 280 260 240 220

1,0

1,5

2,0

2,5

3,0 D/R1

3,5

4,0

4,5

5,0

Figure 6. Interface frequencies of CdS/ZnS/polyethylene as a function of D/R1 and for a fixed value of D/R2 = 5.71 × 1025 .

of the dielectric continuum model, we have determined the interface frequencies in a II–VI semiconductor doublenanoshell system. Resolution of Laplace’s equation in bispherical coordinates is made by considering the usual electrostatic boundary conditions at the surfaces of the twonanoshell system. Consequently, recursion relations for the coefficients appearing in the potentials are obtained, which entails infinite matrices. We have treated these matrices by the truncating method up to 4 × 4 order, where the convergence of IF modes is confirmed. In fact, the solution for IF branches consists of five bands of phonon modes which correspond to

4. Conclusions In an attempt to study the optical properties of QDs molecules, we analyze the axial interface optical phonons modes in a system of double-nanoshell QDs. Within the framework 6

Nanotechnology 19 (2008) 285713

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Interface frequency (cm-1)

360



(2n + 3)η1 2

 (2n + 1)η1 + 2 (ω) = (n + 1) 1 (ω) cosh 2

 (2n + 1)η1 × sinh 2 = 1 (ω)[sinh η1 − (2n + 3) cosh η1 ]

 (2n + 3)η1 × cosh − 2 (ω) sinh η1 2

 (2n + 3)η1 × cosh + (2n + 3) cosh η1 2

 (2n + 3)η1 × sinh . 2

 c11 c12 Trid [P] = n 0 c21 c22 × sinh

340

b21

320 300 CdS/ZnS/polyethylene D/R2= 5 .71 x 1025

280

b22

260 240 6

7

8

9

10 11 D/R1

12

13 14 x 1025

15

Figure 7. Interface frequencies of CdS/ZnS/polyethylene (a) and CdS/ZnS/glass (b) as a function of D/R1 and for a fixed value of D/R2 = 5.71 × 1025 .

(A.3)

where

c11 = 2 (ω)[sinh η2 − (2n + 1) cosh η2 ] − 3 sinh η2



(2n+1)η2 (2n + 1)η1 + (2n + 1) cosh η2 tanh e− 2 2

 (2n+3)η2 (2n + 3)η2 c12 = (n + 1) 2 (ω) + 3 tanh e− 2 2

 (2n+1)η2 (2n + 1)η2 c21 = (n + 1) 2 (ω) + 3 tanh e− 2 2 c22 = 2 (ω)[sinh η2 − (2n + 3) cosh η2 ] − 3 sinh η2



(2n+3)η2 (2n + 3)η1 + (2n + 3) cosh η2 tanh e− 2 . 2

 d11 d12 Trid [Q] = n (A.4) 0 d21 d22

four separation barriers +1. This tendency is also obtained for simple QDs and one isolated nanoshell. Dependences of the interface frequencies on the ratio of inter-nanoshells separation to core size are obtained for different values of the nanoshell interdistance. These dependences are similar to that obtained in other low-dimensional heterostructures. By increasing the nanoshell interdistance, we would expect typical IF frequencies features of decoupled nanoshells. The effect of the embedding material on the dependence of IF frequencies consists in separating the intermediate branches.

Appendix As an example, we give the expressions for the infinite tridiagonal matrices related to the upper nanoshell; i.e.,

 a11 a12 Trid [M] = n0 (A.1) a21 a22

where

 (2n + 1)η2 d11 = 2 (ω) sinh η2 cosh 2

 (2n + 1)η2 + (2n + 1) cosh η2 sinh (−3 ) 2

 (2n + 3)η2 d12 = −(n + 1)[2 (ω) − 3 ] sinh 2

 (2n + 1)η2 d21 = −(n + 1)[2 (ω) − 3 ] sinh 2

 (2n + 3)η2 d22 = sinh η2 cosh 2

 (2n + 3)η2 + (2n + 3) cosh η2 sinh (2 (ω) − 3 ). 2

where

a11 = (sinh η1 − (2n + 1) cosh η1 )[1 (ω) − 2 (ω)]e− a12 = (n + 1)[1 (ω) − 2 (ω)]e−

(2n+3)η1 2

a21 = (n + 1)[1 (ω) − 2 (ω)]e−

(2n+1)η1 2

a22 = (sinh η1 − (2n + 3) cosh η1 )[1 (ω) − 2 (ω)]e−

 b11 b12 Trid [N] = n 0 b21 b22

(2n+1)η1 2

(2n+3)η1 2

.

(A.2)

where

 (2n + 1)η1 b11 = 1 (ω)[sinh η1 − (2n + 1) cosh η1 ] cosh 2

 (2n + 1)η1 − 2 (ω) sinh η1 cosh +(2n + 1) cosh η1 2

 (2n + 1)η1 × sinh 2

 (2n + 3)η1 b12 = (n + 1) 1 (ω) cosh + 2 (ω) 2

References [1] [2] [3] [4] [5]

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Fuchs R and Kliewer K L 1965 Phys. Rev. 140 A2076 Licari J and Evrard R 1977 Phys. Rev. B 15 2254 Shi J J and Pan S H 1996 J. Appl. Phys. 80 3863 Yan Z W and Liang X X 2001 Int. J. Mod. Phys. B 15 3539 Yu S and Kim K W 1997 J. Appl. Phys. 82 3363

Nanotechnology 19 (2008) 285713

C Kanyinda-Malu et al

[6] Kim K W, Stroscio M A, Bhatt A, Mickevicius R and Mitin V V 1991 J. Appl. Phys. 70 319 [7] Xie H J, Chen C Y and Ma B K 2000 J. Phys.: Condens. Matter 12 8623 [8] Zhang L and Xie H J 2003 J. Phys.: Condens. Matter 15 5881 [9] Klimin S N, Pokatilov E P and Fomin V M 1994 Phys. Status Solidi b 184 373 [10] Zhang L, Xie H J and Chen C Y 2002 Phys. Rev. B 66 205326 [11] Wang X F, da Cunha Silva I C and Lei X L 1998 Phys. Rev. B 58 12609 [12] Wang X F, da Cunha Silva I C, Troper A and Lei X L 1999 J. Appl. Phys. 85 6598 [13] Klein M C, Hache F, Ricard D and Flytzanis C 1990 Phys. Rev. B 42 11123 [14] Chamberlain M P, Trallero-Giner C and Cardona M 1995 Phys. Rev. B 51 1680 [15] Roca E, Trallero-Giner C and Cardona M 1994 Phys. Rev. B 49 6598 [16] Fujii M, Kanzawa Y, Hayashi S and Yamamoto K 1996 Phys. Rev. B 54 R8373 [17] Campbell I H and Fauchet P M 1986 Solid State Commun. 58 739 [18] Zhang L and Xie H J 2004 Phys. Scr. 70 66 [19] Suenaga K et al 1997 Science 278 653 [20] Zhang Y, Suenaga K, Colliex C and Iijima S 1998 Science 281 973 [21] Zeng Z Y, Xiang Y and Zhang L D 2000 Eur. Phys. J. B 17 699 [22] Eychm¨uler A, Mews A and Weller H 1993 Chem. Phys. Lett. 208 59 [23] Bruchez M Jr, Moronne M, Gin P, Weiss S and Alivisatos A P 1998 Science 281 2013 [24] Cahn W C W and Nie S 1998 Science 281 2016

[25] Wenhua L, Haiyan X, Zhixiong X, Zhexue L, Jainhong O, Xiangdong C and Ping S 2004 J. Biochem. Biophys. Methods 58 59 [26] Bailey R E, Smith A M and Nie S 2004 Physica E 25 1 [27] L´opez I, Kanyinda-Malu C and de la Cruz R M 2006 Phys. Status Solidi a 203 1370 [28] Wuister S F and Meijerink A 2003 J. Lumin. 105 35 [29] Meijerink A 2005 Opto-electrical properties of Quantum dot molecules and solids http://www.nwo.nl/projecten.nsf. [30] Lippen T V, N¨otzel R, Hamhuis G J and Wolter J H 2005 J. Cryst. Growth 278 88 [31] Kerr W E, Pancholi A and Stoleru V G 2006 Physica E 35 139 [32] Liu H and Bau H H 2004 Phys. Fluids 16 1217 [33] Comas F and Trallero-Giner C 2003 Phys. Rev. B 67 115301 [34] Ansorg M 2005 17 May 2005 http://arXiv:gr-pc/0505059v2 [35] Wolfram Mathematica*6 1988–2007 Wolfram Research, Inc. [36] Kuo H Y and Chen T 2007 Int. J. Eng. Sci. 45 980 [37] Li L 1999 J. Opt. A: Pure Appl. Opt. 1 531 [38] Jamieson T, Bkhshi R, Petrova D, Pocock R, Imani M and Seifalian A M 2007 Biomaterials 28 4717 [39] Lucas A A, Ronveaux A, Schmeits M and Delanaye F 1975 Phys. Rev. B 12 5372 [40] Ruppin R 1982 Phys. Rev. B 26 3440 [41] Brandrup J and Immergut E K (ed) 1989 Polymer Handbook (New York: Wiley) [42] Landolt-B¨ornstein 1982 Numerical Data and Functional Relationships in Science and Technology (Berlin: Springer) [43] Knipp P A and Reinecke T L 1992 Phys. Rev. B 46 7260 [44] Barnham K (ed) 2001 Low-Dimensional Semiconductor Structures: Fundamentals and Devices Applications (Cambridge: Cambridge University Press)

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