- Email: [email protected]
Effects of nanocrystal shape on the surface charge density of ionic colloidal nanoparticles
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 272–276 (2004) 1668–1669
Effects of nanocrystal shape on the surface charge density of ionic colloidal nanoparticles Fanyao Qua,*, R.H. Oliveiraa, P.C. Moraisb a
# # Faculdade de F!ısica, Universidade Federal de Uberlandia, Campus Santa Monica LNMIS, Uberlandia-MG 38400-902, Brazil b ! Universidade de Bras!ılia, Instituto de F!ısica, Nucleo de F!ısica Aplicada, Bras!ılia-DF 70919-970, Brazil
Abstract Both size and shape of nanocrystals play an important role in determination of the energy spectrum, the surface charge density, and the stability of semiconductor-based ionic colloids. When the nanocrystal size decreases, an improvement in colloidal stability is expected due to the increase of the surface charge density. A relatively small deviation from spherical-towards ellipsoidal shape may improve even further colloid stability, due to quantum confinement effects. r 2003 Elsevier B.V. All rights reserved. PACS: 75.50.Tt; 68.66.Hb; 02.70.Bf Keywords: Ionic colloid; Charge density; Shape-effects; Colloidal stability
Ionic magnetic fluid (MFs) are one kind of ionic colloids, which are based on semiconductor nanocrystals dispersed in low- or high-pH aqueous solution [1]. In recent years, this kind of semiconductor nanoparticles (SNPs) has attracted more attention, because they show atomic-like discrete energy levels due to quantum confinement [2]. They are often called ‘‘artificial atoms’’ or zero-dimensional quantum dots’’. A challenging aspect of nanocrystals dispersed in aqueous solution is the colloidal stability. Colloidal stability in ionic MFs depends primarily upon the nanocrystal surface charge density [3]. Among other parameters, the surface charge density of SNPs is determined by its chemical nature, size, and shape [4]. Thus, experimental and theoretical investigation in the effects of geometrical form of SNP on electronic structure becomes a crucial issue. The understanding of the structure-dependence of both the electronic states and the surface charge distribution would provide a powerful tool to exploit the flexibility of the synthesizing processes to generate different SNP shapes and compositions for, e.g., device optimization, and improvement of colloidal stability in ionic MFs. Calculation of the surface charge density in spherical *Corresponding author. Tel.: +55-34-32394190; fax: +5534-32394106. E-mail address: [email protected] (F. Qu).
SNPs dispersed in aqueous solution as a function of both nanocrystal size and pH, has been successfully used to explain the stability of ionic MFs at the usual lower/ higher pH values [5,6]. Nevertheless, a theoretical study of the effects of SNP-shape on the surface charge density has not yet been systematically carried out. The effects of the SNP-shape on the surface charge density as well as colloidal stability of a typical ionic colloid (containing ZnO nanocrystals) were investigated, using a truly three dimensional model, in which both spherical and nonspherical SNPs, such as ellipsoidal hemispherical, lens-shaped, quantum rings, and two or more aligned and interacting nanocrystals, could be easily treated. While, in this article, the authors mainly focus their attention on spherical and ellipsoidal SNPs. To clearly show the geometric effects, particular ellipsoids with semi-axes a ¼ c are selected for calculation. In addition, the ratio b=a ¼ 0:24 holds for all ellipsoidal SNPs. Thus, b ¼ 0:62R and a ¼ 1:61R; where R is sphere radius, a; b; and c represent three principal ellipsoid semi-axes, respectively. The corresponding effective-mass equation for the carrier envelope wave function, Cn;M ðr; zÞ; in cylindrical coordinates is [3]: 1 1 r r þ V ðr; zÞ Cn;M ðr; zÞ ¼ En;M Cn;M ðr; zÞ; 2 mðrÞ
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.1076
ARTICLE IN PRESS F. Qu et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 1668–1669
1 m2 : 2mðrÞ r2
ð3Þ
SSNP, which are possessed of equal-volume, is illustrated in inset of Fig. 1. As the size of nanoparticle decreases, the difference ðPESNP PSSNP Þ of carrier probability first increases, then reaches its maximum value, and finally drops down towards zero. The difference of electron ground-state energy between ESNP and SSNP was also computed. It presents a similar behavior to the carrier probability (data not shown). These results can be well explained by the variation of quantum confinement effect. Actually, in the limit of R-N; the quantum confinement effects for both SSNP and ESNP is negligible. Thus, the sub-band energies and their spacing become very small. As a result, a very small difference in properties between SSNP and ESNP is expected. As the SNP-size decreases, however, the quantum confinement effect increases. Because of b ¼ 0:62R; confinement effects should be stronger in ESNP than in SSNP, leading to an increased PESNP PSSNP : As the SNP-size decreases further, however, the ground-state energy approaches to the top of the well ðV0 Þ: Consequently, the wave function starts to leak into the barrier. Because this penetration of wave function is more pronounced for ESNPs than for SSNPs, the PESNP-PSSNP tends to decrease. It is important to emphasize that for the nanoparticles with very large value of V0 ; the different behaviour of PESNP PSSNP is expected. In this case, the penetration of electron wave function is prohibited due to very high barrier wall. Thus, PESNP PSSNP increases monotonically with decrease of nanoparticles-volume. In summary, the shape effects of SNPs dispersed in aqueous medium on the energy spectrum and surface charge density were investigated in the frame of FEM. It was found that both size and shape of nanocrystals play an important role in the determination of the colloidal stability. From the colloidal stability point of view, ellipsoidal-shaped nanocrystals may produce colloidal suspension with higher colloidal stability in comparison to spherical-shaped nanocrystals.
fi fj r dr dz;
ð4Þ
The authors acknowledge the financial support of the Brazilian agencies CNPq and FINATEC.
Radius (Å) 20
40
60
80
Probability (10-4)
4
3
5 4
250
3 2 200
1 0 0.0
0.5
1.0
1.5
2.0
150
Volume (106Å3)
2
Energy(meV)
PESNP- PSSNP(10-5)
300
100 1
PESNP
50
PSSNP
0 0.0
0
0.5
1.0
1.5
2.0
6 3
Volume (10 Å )
Fig. 1. Maximum probability as a function of volume of nanoparticles for SSNP (solid line) and ESNP (dashed line). Dotted line illustrates energy (right-hand side scale) versus radius (upper sale) of SSNP. Inset shows volume-dependent difference of probability between SSNP and ESNP.
where M ¼ 0; 71; 72; y is the quantum number of the angular momentum ðLÞ projection onto the z-axis, n is the main quantum number, mðrÞ is the effective mass, V ðr; zÞ is quantum dot potential with V ¼ 0 ðV0 Þ inside the SNP (outside the SNP). Triangular element meshes were used to discretize the wave equation in the frame of finite element method (FEM) by means of Galerkin’s weighted residue method [7]. Cn;M ðr; zÞ is approximated by an expansion over the FEM basis function, fi ðz; rÞ: Then a generalized eigenproblem for the coefficients cðkÞ and eigenvalues ln;M is obtained through H c ¼ ln;M D c; where Z Z 1 ½rfi rfj þ 2mðrÞ HijðkÞ ¼ 2mðrÞ ðkÞ Veff ðz; rÞ fi fj r dr dz; Veff ¼ V ðr; zÞ þ DðkÞ ij ¼
Z Z ðkÞ
1669
ð2Þ
and the vector cðkÞ contains the unknown expansion coefficients of wave function for the kth element. The size-dependence of carrier probability ðrjCn;M ðz; rÞj2 Þ for spherical (solid line) and ellipsoidal (dashed line) semiconductor nanocrystals is shown in Fig. 1. It is found that the carrier probability decreases with increasing the size (volume) of SNP, for both spherical (SSNP) and ellipsoidal (ESNP) semiconductor nanocrystals. To understand the mechanism of nanoparticle-shape effects, the carrier energy (right-hand side scale) as a function of radius (upper sale) of SSNPs is also shown in Fig. 1 (dotted line). Moreover, the difference of carrier probability between ESNP and
References [1] [2] [3] [4]
R. Massart, IEEE Trans. Magn. 17 (1981) 1247. Fanyao Qu, P.C. Morais, J. Chem. Phys. 111 (1999) 8588. N. Buske, Progr. Colloid Polym. Sci. 95 (1994) 175. Z. Hens, D. Vanmaekelbergh, Phys. Rev. Lett. 88 (2002) 236803. [5] P.C. Morais, Fanyao Qu, J. Magn. Magn. Mater. 252 (2002) 117. [6] Fanyao Qu, P.C. Morais, J. Phys. Chem. B 104 (2000) 5232. [7] J.E. Pask, B.M. Klein, P.A. Sterne, C.Y. Fong, Comp. Phys. Commun. 135 (2001) 1.
Journal of Magnetism and Magnetic Materials 272–276 (2004) 1668–1669
Effects of nanocrystal shape on the surface charge density of ionic colloidal nanoparticles Fanyao Qua,*, R.H. Oliveiraa, P.C. Moraisb a
# # Faculdade de F!ısica, Universidade Federal de Uberlandia, Campus Santa Monica LNMIS, Uberlandia-MG 38400-902, Brazil b ! Universidade de Bras!ılia, Instituto de F!ısica, Nucleo de F!ısica Aplicada, Bras!ılia-DF 70919-970, Brazil
Abstract Both size and shape of nanocrystals play an important role in determination of the energy spectrum, the surface charge density, and the stability of semiconductor-based ionic colloids. When the nanocrystal size decreases, an improvement in colloidal stability is expected due to the increase of the surface charge density. A relatively small deviation from spherical-towards ellipsoidal shape may improve even further colloid stability, due to quantum confinement effects. r 2003 Elsevier B.V. All rights reserved. PACS: 75.50.Tt; 68.66.Hb; 02.70.Bf Keywords: Ionic colloid; Charge density; Shape-effects; Colloidal stability
Ionic magnetic fluid (MFs) are one kind of ionic colloids, which are based on semiconductor nanocrystals dispersed in low- or high-pH aqueous solution [1]. In recent years, this kind of semiconductor nanoparticles (SNPs) has attracted more attention, because they show atomic-like discrete energy levels due to quantum confinement [2]. They are often called ‘‘artificial atoms’’ or zero-dimensional quantum dots’’. A challenging aspect of nanocrystals dispersed in aqueous solution is the colloidal stability. Colloidal stability in ionic MFs depends primarily upon the nanocrystal surface charge density [3]. Among other parameters, the surface charge density of SNPs is determined by its chemical nature, size, and shape [4]. Thus, experimental and theoretical investigation in the effects of geometrical form of SNP on electronic structure becomes a crucial issue. The understanding of the structure-dependence of both the electronic states and the surface charge distribution would provide a powerful tool to exploit the flexibility of the synthesizing processes to generate different SNP shapes and compositions for, e.g., device optimization, and improvement of colloidal stability in ionic MFs. Calculation of the surface charge density in spherical *Corresponding author. Tel.: +55-34-32394190; fax: +5534-32394106. E-mail address: [email protected] (F. Qu).
SNPs dispersed in aqueous solution as a function of both nanocrystal size and pH, has been successfully used to explain the stability of ionic MFs at the usual lower/ higher pH values [5,6]. Nevertheless, a theoretical study of the effects of SNP-shape on the surface charge density has not yet been systematically carried out. The effects of the SNP-shape on the surface charge density as well as colloidal stability of a typical ionic colloid (containing ZnO nanocrystals) were investigated, using a truly three dimensional model, in which both spherical and nonspherical SNPs, such as ellipsoidal hemispherical, lens-shaped, quantum rings, and two or more aligned and interacting nanocrystals, could be easily treated. While, in this article, the authors mainly focus their attention on spherical and ellipsoidal SNPs. To clearly show the geometric effects, particular ellipsoids with semi-axes a ¼ c are selected for calculation. In addition, the ratio b=a ¼ 0:24 holds for all ellipsoidal SNPs. Thus, b ¼ 0:62R and a ¼ 1:61R; where R is sphere radius, a; b; and c represent three principal ellipsoid semi-axes, respectively. The corresponding effective-mass equation for the carrier envelope wave function, Cn;M ðr; zÞ; in cylindrical coordinates is [3]: 1 1 r r þ V ðr; zÞ Cn;M ðr; zÞ ¼ En;M Cn;M ðr; zÞ; 2 mðrÞ
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.1076
ARTICLE IN PRESS F. Qu et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 1668–1669
1 m2 : 2mðrÞ r2
ð3Þ
SSNP, which are possessed of equal-volume, is illustrated in inset of Fig. 1. As the size of nanoparticle decreases, the difference ðPESNP PSSNP Þ of carrier probability first increases, then reaches its maximum value, and finally drops down towards zero. The difference of electron ground-state energy between ESNP and SSNP was also computed. It presents a similar behavior to the carrier probability (data not shown). These results can be well explained by the variation of quantum confinement effect. Actually, in the limit of R-N; the quantum confinement effects for both SSNP and ESNP is negligible. Thus, the sub-band energies and their spacing become very small. As a result, a very small difference in properties between SSNP and ESNP is expected. As the SNP-size decreases, however, the quantum confinement effect increases. Because of b ¼ 0:62R; confinement effects should be stronger in ESNP than in SSNP, leading to an increased PESNP PSSNP : As the SNP-size decreases further, however, the ground-state energy approaches to the top of the well ðV0 Þ: Consequently, the wave function starts to leak into the barrier. Because this penetration of wave function is more pronounced for ESNPs than for SSNPs, the PESNP-PSSNP tends to decrease. It is important to emphasize that for the nanoparticles with very large value of V0 ; the different behaviour of PESNP PSSNP is expected. In this case, the penetration of electron wave function is prohibited due to very high barrier wall. Thus, PESNP PSSNP increases monotonically with decrease of nanoparticles-volume. In summary, the shape effects of SNPs dispersed in aqueous medium on the energy spectrum and surface charge density were investigated in the frame of FEM. It was found that both size and shape of nanocrystals play an important role in the determination of the colloidal stability. From the colloidal stability point of view, ellipsoidal-shaped nanocrystals may produce colloidal suspension with higher colloidal stability in comparison to spherical-shaped nanocrystals.
fi fj r dr dz;
ð4Þ
The authors acknowledge the financial support of the Brazilian agencies CNPq and FINATEC.
Radius (Å) 20
40
60
80
Probability (10-4)
4
3
5 4
250
3 2 200
1 0 0.0
0.5
1.0
1.5
2.0
150
Volume (106Å3)
2
Energy(meV)
PESNP- PSSNP(10-5)
300
100 1
PESNP
50
PSSNP
0 0.0
0
0.5
1.0
1.5
2.0
6 3
Volume (10 Å )
Fig. 1. Maximum probability as a function of volume of nanoparticles for SSNP (solid line) and ESNP (dashed line). Dotted line illustrates energy (right-hand side scale) versus radius (upper sale) of SSNP. Inset shows volume-dependent difference of probability between SSNP and ESNP.
where M ¼ 0; 71; 72; y is the quantum number of the angular momentum ðLÞ projection onto the z-axis, n is the main quantum number, mðrÞ is the effective mass, V ðr; zÞ is quantum dot potential with V ¼ 0 ðV0 Þ inside the SNP (outside the SNP). Triangular element meshes were used to discretize the wave equation in the frame of finite element method (FEM) by means of Galerkin’s weighted residue method [7]. Cn;M ðr; zÞ is approximated by an expansion over the FEM basis function, fi ðz; rÞ: Then a generalized eigenproblem for the coefficients cðkÞ and eigenvalues ln;M is obtained through H c ¼ ln;M D c; where Z Z 1 ½rfi rfj þ 2mðrÞ HijðkÞ ¼ 2mðrÞ ðkÞ Veff ðz; rÞ fi fj r dr dz; Veff ¼ V ðr; zÞ þ DðkÞ ij ¼
Z Z ðkÞ
1669
ð2Þ
and the vector cðkÞ contains the unknown expansion coefficients of wave function for the kth element. The size-dependence of carrier probability ðrjCn;M ðz; rÞj2 Þ for spherical (solid line) and ellipsoidal (dashed line) semiconductor nanocrystals is shown in Fig. 1. It is found that the carrier probability decreases with increasing the size (volume) of SNP, for both spherical (SSNP) and ellipsoidal (ESNP) semiconductor nanocrystals. To understand the mechanism of nanoparticle-shape effects, the carrier energy (right-hand side scale) as a function of radius (upper sale) of SSNPs is also shown in Fig. 1 (dotted line). Moreover, the difference of carrier probability between ESNP and
References [1] [2] [3] [4]
R. Massart, IEEE Trans. Magn. 17 (1981) 1247. Fanyao Qu, P.C. Morais, J. Chem. Phys. 111 (1999) 8588. N. Buske, Progr. Colloid Polym. Sci. 95 (1994) 175. Z. Hens, D. Vanmaekelbergh, Phys. Rev. Lett. 88 (2002) 236803. [5] P.C. Morais, Fanyao Qu, J. Magn. Magn. Mater. 252 (2002) 117. [6] Fanyao Qu, P.C. Morais, J. Phys. Chem. B 104 (2000) 5232. [7] J.E. Pask, B.M. Klein, P.A. Sterne, C.Y. Fong, Comp. Phys. Commun. 135 (2001) 1.