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Fractal clusters in Eu doped polymer fiber studied by near-field scanning optical microscopy
1 July 2002
Optics Communications 208 (2002) 97–101 www.elsevier.com/locate/optcom
Fractal clusters in Eu doped polymer fiber studied by near-field scanning optical microscopy Sun Xiaohong a,*, Ming Hai a, Xie Aifang b, Lu Yonghua a, Xu Xingsheng a, Xie Jianping a, Zhang Qijin c, Hu Jun c, Zhang Zebo b, Gu Benyuan b a
Laboratory of Structure Center, Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China b Laboratory of Optical Physics, Institute of Physics, Chinese Academy of Sciences, Bejing 100080, People’s Republic of China c Department of Material Science and Engineering, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China Received 24 February 2002; received in revised form 25 March 2002; accepted 15 May 2002
Abstract Using near-field scanning optical microscopy (NSOM), we explore the chromatic dispersion and absorption characteristics of fractal clusters in Eu doped polymer fiber. We present the near-field transmitted light intensity of fiber slice. The near-field light intensity of dots with clusters and without clusters is analyzed and compared and the agreement is obtained between experiment and theory which is based on the effective-medium approximation (EMA) and the differential effective-medium approximation (DEMA) of fractal clusters, respectively. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: EMA; DEMA; Chromatic dispersion; NSOM; Eu doped polymer fiber
1. Introduction Fractal clusters exist in the process of doping rare-earth ions into polymer optical fiber (POF) like in many growth processes resulting from irreversible kinetic aggregation [1,2], which will introduce a harmful effect to fiber laser and amplifier [3–5]. Eu doped polymer fiber is a new kind of material of fiber laser and amplifier. To obtain the
*
Corresponding author. Tel.: +86-0551-3603504; fax: +860551-3601073. E-mail address: [email protected] (S. Xiaohong).
appropriate doped concentration and better control its fabricating process, it is very important to study the optical properties of clusters. In this paper we lay emphasis on the chromatic dispersion and absorption of fractal clusters in Eu doped polymer fiber, and explore the change of dielectric function and relative transmission rate with the exciting wavelength. NSOM is chosen, because it can give out the topography of the measured surface and transmitted light intensity of sample simultaneously. In this way, we can eliminate the effect produced by the wave of surface, on the other hand, we can compare the intensity of the dots with clusters and
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 5 7 4 - 2
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S. Xiaohong et al. / Optics Communications 208 (2002) 97–101
without clusters of the same sample and give out the relative transmission rate. Doped polymer fiber is a kind of random inhomogeneous material. One of the most successful methods of treating the optical properties of such material is a self-consistent or effective-medium approximation (EMA) approach due originally to Bruggeman [6], studied quantitatively by Landauer [7] and developed by Stroud [8]. In this paper EMA is used to analyze the dielectric function of the area without clusters. Fractal clusters have been reported in colloidal gold suspensions by Weitz and Oliveria [9]. A fractal cluster is a system of interacting material particles called monomers. Fractal clusters were analyzed by Hui and Stroud [10] by differential EMA (DEMA). The complex dielectric function for the clusters in this approach is determined by a cubic equation. DEMA is used to give out the dielectric function of clusters.
c
eA ðxÞ ec ðxÞ eB ec ðxÞ þ ð1 cÞ ¼ 0; eA ðxÞ þ 2ec ðxÞ eB þ 2ec ðxÞ
ð2Þ
where c is the volume of metal particles, chosen as 800 ppm. Substituting the above given parameters into Eq. (2) we get ec ðxÞ. Its 100 times imaginary part is shown as solid line in Fig. 1. Its real part indicates the chromatic dispersion and the change with the wavelength is shown in Fig. 2 as a solid line. The fractal cluster dielectric function at radius R, eR ðxÞ is described by differential EMA method as the following: deR ðxÞ 3f 0 ðRÞ eB eR ðxÞ ¼ eR ðxÞ dR f ðRÞ eB þ 2eR ðxÞ
ð3Þ
2. Theoretical basis Eu doped polymer fiber is a composite made of Eu doped poly-methylmethacrylate (PMMA). For simplicity, we take the dielectric constant of the host PMMA to be unity, that is eB ¼ 1. The metallic dielectric function is described by a Drude dielectric function ð1Þ
Fig. 1. The 100 times imaginary part of ec ðxÞ without clusters (solid line) and the imaginary part of eR ðxÞ with fractal clusters (dashed line) change with the wavelength.
where xp ¼ 2pc=kp is the plasmon resonance frequency and kp correspond to the absorption peaks of Eu doped materials which are 394, 466 and 534 nm in the visible light range, and s is a characteristic relaxation time which is chosen as s ¼ 1014 , and x ¼ 2pc=k, c and k are the light speed and the incident light wavelength, respectively. Next, in the wavelength range from 300 to 600 nm, based on EMA and DEMA theories, the dielectric functions ec ðxÞ and eR ðxÞ of POF under homogeneously (without clusters) and inhomogeneously (with fractal clusters) doped circumstances are discussed. According to the EMA, the dielectric function ec ðxÞ is described by Stroud [8] as
Fig. 2. The real part of ec ðxÞ without clusters and eR ðxÞ with fractal clusters changes with the wavelength.
x2p ; eA ðxÞ ¼ 1 xðx þ i=sÞ
S. Xiaohong et al. / Optics Communications 208 (2002) 97–101
99
where f ðRÞ is the metal concentration in the clusters and f 0 ðRÞ is its differential form. Eq. (3) can be integrated to give 3 3ð3df Þ eR ðxÞ eB eA ðxÞ R ¼ ð4Þ eA ðxÞ eB eR ðxÞ a which is a cubic equation for eR ðxÞ, where a is the radius of the metal particles, and df is the fractal dimension. Eq. (4) then can be solved for the cluster dielectric function in terms of R, df and wavelength k. The imaginary part and the real part are shown in Figs. 1 and 2 as the dashed line, respectively. In the figure, we choose df ¼ 2:5, R ¼ 10a (because according to different values the shape of curve are basically the same). From the diagram, we see that the absorption and the chromatic dispersion of fractal clusters is much greater than that of POF without clusters. Up to now, the dielectric function ec ðxÞ and eR ðxÞ of POF without clusters and with fractal clusters, have been extracted. Next we calculate the relative transmission rate of the area with clusters relative to that without clusters of the same sample. As is well known, the dielectric function is relevant to refractive index of the material. Considering multi-reflection between the surface of the sample, because the extinction coefficient of the host is small and the sample is enough thin, for the normal incidence light, as reported in the paper [11], we can derive the transmission rate depending on the reflection coefficient as follows: T ¼
I 1R ; ¼ I0 1þR
Fig. 3. The relative transmission rate of the location with clusters relative to the one without clusters changes with the wavelength.
The relative transmission rate of the regions with clusters, relative to those without clusters is, shown in Fig. 3.
3. Experimental results and discussion In this work, we used a commercial scanning near-field optical microscopy (NSOM) from RHK Technology (USA) and its block diagram is shown in Fig. 4. NSOM has the ability to achieve a high
2
R¼
ðn 1Þ þ j2 2
ðn þ 1Þ þ j2
;
ð5Þ
where n and j represent the phase refractive index and the extinction coefficient, which can be solved numerically by the following set of equations: e 1 ¼ n 2 j2 ;
e2 ¼ 2nj;
ð6Þ
where e1 ; e2 are the real and imaginary part of the dielectric function, respectively. From Eqs. (5) and (6), it is easy to deduce the evolvement of the transmission rate with k, that is h i1=2 2 2 1=2 2e þ 2ðe þ e Þ 1 1 2 2n ¼ T ¼ 2 : ð7Þ 2 2 1=2 n þ j2 þ 1 ðe1 þ e2 Þ þ 1
Fig. 4. Block diagram of NSOM. FC, fiber coupler; APD, avalanche photodiode; PD, position detector; LD, laser diode.
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S. Xiaohong et al. / Optics Communications 208 (2002) 97–101
spatial resolution optical contrast and simultaneously measure the sample topography. The light from Ar ion laser is coupled into a very small aluminum-coated quartz single-mode glass fiber tip with a 50 nm aperture. The tip illustrates the sample and scans near the sample surface of interest. The transmitted light of sample is detected through the objective by a photon counting avalanche photodiode (SPCM-AQR-15, EG&G, USA). The counting unit of photon counting avalanche photodiode is 1 Hz ¼ 1 photon/s. On the other hand, the light intensity is relevant to the dwell time of tip at each scanning point. Typical pixel dwell time in our images was 10 ms. Images were acquired and processed by using the software SPM32 5.06 version (RHK, USA). In the experiment, the constant separation between the tip and the sample surface of interest is controlled by an atom force microscopy (AFM) operation in non-contact mode with a laser beam deflection, so that the movement of the optical fiber tip is recorded by a four-phase detector to determine the vertical position of the tip. The fourphase detector transforms light signals reflected by the tip to voltage signals and give out the topography of surface. The transverse resolution is decided by the aperture of tip, 50 nm. In the experiment, the different wavelength light of Ar ion laser is used to illustrate the sample and different transmitted light intensity is obtained. The experimental results are shown in Fig. 5. Fig. 5(a) is the topography of the sample surface, decided by the voltage difference of four-phase detector under the constant separation mode of NSOM. And Fig. 5(b) is the transmitted light intensity of the sample scanned by the tip, given by a photon counter whose counting unit is 1 Hz ¼ 1 photon/s. To give out a more intuitive and quantitative result, we use NSOM software to mark a light line in the corresponding area of each diagram of Fig. 5 and obtain the average surface height and the corresponding transmitted intensity as a function of distance shown in Figs. 6(a) and (b). From top to bottom of Figs. 5 and 6, the incident light wavelengths are 457, 488 and 514 nm in sequence. From Fig. 6(b), we can see the maximum values keep consistent basically in each diagram and the other values different from each
Fig. 5. (a) The near-field topography obtained by four-phase detector and (b) the transmitted light intensity recorded by photon counter when the sample is illuminated with Ar ions laser at k ¼ 457, 488, 514 nm (top to bottom).
Fig. 6. (a) The average surface height and (b) the light intensity of the marked location in light line as a function of the distance.
other, while the difference of surface height is very small in Fig. 6(a). This indicates that the intensity difference is caused by clusters inside materials. To
S. Xiaohong et al. / Optics Communications 208 (2002) 97–101
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cussed in Eu doped polymer fiber by using NSOM. The near-field relative transmission rate of the area with clusters relative to that without clusters is compared between theory and experiment. The experimental results are in agreement with theory based on EMA and DEMA.
Acknowledgements
Fig. 7. The relative transmitted rate of the location with clusters relative to that without clusters in the right as a function of the distance at k ¼ 457 nm (solid line), k ¼ 488 nm (dahsed line) and k ¼ 514 nm (dot line).
facilitate the comparison among different wavelengths, the relative transmitted rates to the maximum intensity in the light line are shown in Fig. 7. The location difference of the maximum is caused by the opposite beginning location of tip while scanning at different wavelengths. Without consideration of the difference, the relative transmission rates increase basically with the increasing wavelength. This is primarily in agreement with the theoretical curve in Fig. 3.
4. Conclusions In this paper, the chromatic dispersion and absorption characteristic of fractal clusters is dis-
This subject was supported by National Science Fund of China, Nos. 19974042 and 50025309 and the Research Fund for the Doctoral Program of Higher Education, No. 1999035822 and the Chuangxin Project Fund of Chinese Academy, KGCX220202.
References [1] F. Family, D.P. Landau, Kinetics of Aggregation and Gelation, North-Holland, Amsterdam, 1984. [2] M.V. Berry, I.C. Percival, Opt. Acta 33 (1986) 577. [3] L.A. Riseberg, M.J. Weber, in: E. Wolf (Ed.), Prog. Opt. 14 (1976) 91. [4] X.L. Wang, R.J. Wei, Phys. Rev. Lett. 78 (1997) 2744. [5] Yang Bao, Ming Hai, Zhang Qi-jin, Xie Jian-ping, Chin. Phys. Lett. 15 (1998) 272. [6] D.A.G. Bruggeman, Ann. Phys. (Leipz.) 24 (1935) 636. [7] R. Landauer, J. Appl. Phys. 23 (1952) 779. [8] D. Stroud, Phys. Rev. B. 12 (8) (1975) 3369. [9] D.A. Weitz, M. Oliveria, Phys. Rev. Lett. 52 (1984) 1433. [10] P.M. Hui, D. Stroud, Phys. Rev. B 33 (4) (1986) 2163. [11] R. Bowling Barnes, M. Czerrny, Phys. Rev. 38 (1931) 339.
Optics Communications 208 (2002) 97–101 www.elsevier.com/locate/optcom
Fractal clusters in Eu doped polymer fiber studied by near-field scanning optical microscopy Sun Xiaohong a,*, Ming Hai a, Xie Aifang b, Lu Yonghua a, Xu Xingsheng a, Xie Jianping a, Zhang Qijin c, Hu Jun c, Zhang Zebo b, Gu Benyuan b a
Laboratory of Structure Center, Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China b Laboratory of Optical Physics, Institute of Physics, Chinese Academy of Sciences, Bejing 100080, People’s Republic of China c Department of Material Science and Engineering, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China Received 24 February 2002; received in revised form 25 March 2002; accepted 15 May 2002
Abstract Using near-field scanning optical microscopy (NSOM), we explore the chromatic dispersion and absorption characteristics of fractal clusters in Eu doped polymer fiber. We present the near-field transmitted light intensity of fiber slice. The near-field light intensity of dots with clusters and without clusters is analyzed and compared and the agreement is obtained between experiment and theory which is based on the effective-medium approximation (EMA) and the differential effective-medium approximation (DEMA) of fractal clusters, respectively. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: EMA; DEMA; Chromatic dispersion; NSOM; Eu doped polymer fiber
1. Introduction Fractal clusters exist in the process of doping rare-earth ions into polymer optical fiber (POF) like in many growth processes resulting from irreversible kinetic aggregation [1,2], which will introduce a harmful effect to fiber laser and amplifier [3–5]. Eu doped polymer fiber is a new kind of material of fiber laser and amplifier. To obtain the
*
Corresponding author. Tel.: +86-0551-3603504; fax: +860551-3601073. E-mail address: [email protected] (S. Xiaohong).
appropriate doped concentration and better control its fabricating process, it is very important to study the optical properties of clusters. In this paper we lay emphasis on the chromatic dispersion and absorption of fractal clusters in Eu doped polymer fiber, and explore the change of dielectric function and relative transmission rate with the exciting wavelength. NSOM is chosen, because it can give out the topography of the measured surface and transmitted light intensity of sample simultaneously. In this way, we can eliminate the effect produced by the wave of surface, on the other hand, we can compare the intensity of the dots with clusters and
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 5 7 4 - 2
98
S. Xiaohong et al. / Optics Communications 208 (2002) 97–101
without clusters of the same sample and give out the relative transmission rate. Doped polymer fiber is a kind of random inhomogeneous material. One of the most successful methods of treating the optical properties of such material is a self-consistent or effective-medium approximation (EMA) approach due originally to Bruggeman [6], studied quantitatively by Landauer [7] and developed by Stroud [8]. In this paper EMA is used to analyze the dielectric function of the area without clusters. Fractal clusters have been reported in colloidal gold suspensions by Weitz and Oliveria [9]. A fractal cluster is a system of interacting material particles called monomers. Fractal clusters were analyzed by Hui and Stroud [10] by differential EMA (DEMA). The complex dielectric function for the clusters in this approach is determined by a cubic equation. DEMA is used to give out the dielectric function of clusters.
c
eA ðxÞ ec ðxÞ eB ec ðxÞ þ ð1 cÞ ¼ 0; eA ðxÞ þ 2ec ðxÞ eB þ 2ec ðxÞ
ð2Þ
where c is the volume of metal particles, chosen as 800 ppm. Substituting the above given parameters into Eq. (2) we get ec ðxÞ. Its 100 times imaginary part is shown as solid line in Fig. 1. Its real part indicates the chromatic dispersion and the change with the wavelength is shown in Fig. 2 as a solid line. The fractal cluster dielectric function at radius R, eR ðxÞ is described by differential EMA method as the following: deR ðxÞ 3f 0 ðRÞ eB eR ðxÞ ¼ eR ðxÞ dR f ðRÞ eB þ 2eR ðxÞ
ð3Þ
2. Theoretical basis Eu doped polymer fiber is a composite made of Eu doped poly-methylmethacrylate (PMMA). For simplicity, we take the dielectric constant of the host PMMA to be unity, that is eB ¼ 1. The metallic dielectric function is described by a Drude dielectric function ð1Þ
Fig. 1. The 100 times imaginary part of ec ðxÞ without clusters (solid line) and the imaginary part of eR ðxÞ with fractal clusters (dashed line) change with the wavelength.
where xp ¼ 2pc=kp is the plasmon resonance frequency and kp correspond to the absorption peaks of Eu doped materials which are 394, 466 and 534 nm in the visible light range, and s is a characteristic relaxation time which is chosen as s ¼ 1014 , and x ¼ 2pc=k, c and k are the light speed and the incident light wavelength, respectively. Next, in the wavelength range from 300 to 600 nm, based on EMA and DEMA theories, the dielectric functions ec ðxÞ and eR ðxÞ of POF under homogeneously (without clusters) and inhomogeneously (with fractal clusters) doped circumstances are discussed. According to the EMA, the dielectric function ec ðxÞ is described by Stroud [8] as
Fig. 2. The real part of ec ðxÞ without clusters and eR ðxÞ with fractal clusters changes with the wavelength.
x2p ; eA ðxÞ ¼ 1 xðx þ i=sÞ
S. Xiaohong et al. / Optics Communications 208 (2002) 97–101
99
where f ðRÞ is the metal concentration in the clusters and f 0 ðRÞ is its differential form. Eq. (3) can be integrated to give 3 3ð3df Þ eR ðxÞ eB eA ðxÞ R ¼ ð4Þ eA ðxÞ eB eR ðxÞ a which is a cubic equation for eR ðxÞ, where a is the radius of the metal particles, and df is the fractal dimension. Eq. (4) then can be solved for the cluster dielectric function in terms of R, df and wavelength k. The imaginary part and the real part are shown in Figs. 1 and 2 as the dashed line, respectively. In the figure, we choose df ¼ 2:5, R ¼ 10a (because according to different values the shape of curve are basically the same). From the diagram, we see that the absorption and the chromatic dispersion of fractal clusters is much greater than that of POF without clusters. Up to now, the dielectric function ec ðxÞ and eR ðxÞ of POF without clusters and with fractal clusters, have been extracted. Next we calculate the relative transmission rate of the area with clusters relative to that without clusters of the same sample. As is well known, the dielectric function is relevant to refractive index of the material. Considering multi-reflection between the surface of the sample, because the extinction coefficient of the host is small and the sample is enough thin, for the normal incidence light, as reported in the paper [11], we can derive the transmission rate depending on the reflection coefficient as follows: T ¼
I 1R ; ¼ I0 1þR
Fig. 3. The relative transmission rate of the location with clusters relative to the one without clusters changes with the wavelength.
The relative transmission rate of the regions with clusters, relative to those without clusters is, shown in Fig. 3.
3. Experimental results and discussion In this work, we used a commercial scanning near-field optical microscopy (NSOM) from RHK Technology (USA) and its block diagram is shown in Fig. 4. NSOM has the ability to achieve a high
2
R¼
ðn 1Þ þ j2 2
ðn þ 1Þ þ j2
;
ð5Þ
where n and j represent the phase refractive index and the extinction coefficient, which can be solved numerically by the following set of equations: e 1 ¼ n 2 j2 ;
e2 ¼ 2nj;
ð6Þ
where e1 ; e2 are the real and imaginary part of the dielectric function, respectively. From Eqs. (5) and (6), it is easy to deduce the evolvement of the transmission rate with k, that is h i1=2 2 2 1=2 2e þ 2ðe þ e Þ 1 1 2 2n ¼ T ¼ 2 : ð7Þ 2 2 1=2 n þ j2 þ 1 ðe1 þ e2 Þ þ 1
Fig. 4. Block diagram of NSOM. FC, fiber coupler; APD, avalanche photodiode; PD, position detector; LD, laser diode.
100
S. Xiaohong et al. / Optics Communications 208 (2002) 97–101
spatial resolution optical contrast and simultaneously measure the sample topography. The light from Ar ion laser is coupled into a very small aluminum-coated quartz single-mode glass fiber tip with a 50 nm aperture. The tip illustrates the sample and scans near the sample surface of interest. The transmitted light of sample is detected through the objective by a photon counting avalanche photodiode (SPCM-AQR-15, EG&G, USA). The counting unit of photon counting avalanche photodiode is 1 Hz ¼ 1 photon/s. On the other hand, the light intensity is relevant to the dwell time of tip at each scanning point. Typical pixel dwell time in our images was 10 ms. Images were acquired and processed by using the software SPM32 5.06 version (RHK, USA). In the experiment, the constant separation between the tip and the sample surface of interest is controlled by an atom force microscopy (AFM) operation in non-contact mode with a laser beam deflection, so that the movement of the optical fiber tip is recorded by a four-phase detector to determine the vertical position of the tip. The fourphase detector transforms light signals reflected by the tip to voltage signals and give out the topography of surface. The transverse resolution is decided by the aperture of tip, 50 nm. In the experiment, the different wavelength light of Ar ion laser is used to illustrate the sample and different transmitted light intensity is obtained. The experimental results are shown in Fig. 5. Fig. 5(a) is the topography of the sample surface, decided by the voltage difference of four-phase detector under the constant separation mode of NSOM. And Fig. 5(b) is the transmitted light intensity of the sample scanned by the tip, given by a photon counter whose counting unit is 1 Hz ¼ 1 photon/s. To give out a more intuitive and quantitative result, we use NSOM software to mark a light line in the corresponding area of each diagram of Fig. 5 and obtain the average surface height and the corresponding transmitted intensity as a function of distance shown in Figs. 6(a) and (b). From top to bottom of Figs. 5 and 6, the incident light wavelengths are 457, 488 and 514 nm in sequence. From Fig. 6(b), we can see the maximum values keep consistent basically in each diagram and the other values different from each
Fig. 5. (a) The near-field topography obtained by four-phase detector and (b) the transmitted light intensity recorded by photon counter when the sample is illuminated with Ar ions laser at k ¼ 457, 488, 514 nm (top to bottom).
Fig. 6. (a) The average surface height and (b) the light intensity of the marked location in light line as a function of the distance.
other, while the difference of surface height is very small in Fig. 6(a). This indicates that the intensity difference is caused by clusters inside materials. To
S. Xiaohong et al. / Optics Communications 208 (2002) 97–101
101
cussed in Eu doped polymer fiber by using NSOM. The near-field relative transmission rate of the area with clusters relative to that without clusters is compared between theory and experiment. The experimental results are in agreement with theory based on EMA and DEMA.
Acknowledgements
Fig. 7. The relative transmitted rate of the location with clusters relative to that without clusters in the right as a function of the distance at k ¼ 457 nm (solid line), k ¼ 488 nm (dahsed line) and k ¼ 514 nm (dot line).
facilitate the comparison among different wavelengths, the relative transmitted rates to the maximum intensity in the light line are shown in Fig. 7. The location difference of the maximum is caused by the opposite beginning location of tip while scanning at different wavelengths. Without consideration of the difference, the relative transmission rates increase basically with the increasing wavelength. This is primarily in agreement with the theoretical curve in Fig. 3.
4. Conclusions In this paper, the chromatic dispersion and absorption characteristic of fractal clusters is dis-
This subject was supported by National Science Fund of China, Nos. 19974042 and 50025309 and the Research Fund for the Doctoral Program of Higher Education, No. 1999035822 and the Chuangxin Project Fund of Chinese Academy, KGCX220202.
References [1] F. Family, D.P. Landau, Kinetics of Aggregation and Gelation, North-Holland, Amsterdam, 1984. [2] M.V. Berry, I.C. Percival, Opt. Acta 33 (1986) 577. [3] L.A. Riseberg, M.J. Weber, in: E. Wolf (Ed.), Prog. Opt. 14 (1976) 91. [4] X.L. Wang, R.J. Wei, Phys. Rev. Lett. 78 (1997) 2744. [5] Yang Bao, Ming Hai, Zhang Qi-jin, Xie Jian-ping, Chin. Phys. Lett. 15 (1998) 272. [6] D.A.G. Bruggeman, Ann. Phys. (Leipz.) 24 (1935) 636. [7] R. Landauer, J. Appl. Phys. 23 (1952) 779. [8] D. Stroud, Phys. Rev. B. 12 (8) (1975) 3369. [9] D.A. Weitz, M. Oliveria, Phys. Rev. Lett. 52 (1984) 1433. [10] P.M. Hui, D. Stroud, Phys. Rev. B 33 (4) (1986) 2163. [11] R. Bowling Barnes, M. Czerrny, Phys. Rev. 38 (1931) 339.