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Conductance resonance of coupled supported metal clusters
__ __ . ii!!!!
21 August 1995
&
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 204 (1995) 291-294
Conductance resonance of coupled supported metal clusters Xiaoshuang Department
of Physics
Chen, Jijun Zhao, Fengqi Lui, Qing Sun, Guanghou Wang and National Laboratory
Center for Advanced
Received 20 February
of Solid State Microstructures,
Studies in Science
and Technology
Nanjing University, Nanjing 210093,
of Microstructures,
Nanjing 210093,
* China
China
received 6 April 1995; accepted for publication 16 June 1995 Communicated by J. Flouquet
1995; revised manuscript
Abstract The conductance resonance of a tunneling structure with a few metal clusters, deposited on an insulating film, is studied by the generalized Breit-Wigner formula in a tight-binding approximation. We find that in the conductance resonance the multiple peak structure comes from the interaction between supported metal clusters on an insulating film and that the different arrangements of metal clusters can cause a difference of the conductance resonance peaks. Therefore it is possible to predict the effect of the interaction between metal clusters on the conductance resonance and develop some new microelectronic devices by artificially arranging metal clusters on the surface of the insulating film. PACS:
36.40. +
d; 73.40.Gk: 73.40.R~
In solid state physics as a unique transitional
metal clusters are considered state between a single atom
and a bulk [l]. Considerable effort has been devoted to supported metal clusters [2,3], in which the discrete electron energy levels of metal clusters have been observed, and metal Coulomb islands [4] because they represent the first stage of the metal film growth [5-71 and the development of some new microelectronic devices [s]. Therefore the interaction between supported metal clusters is of great importance for determining the properties of cluster-based materials and developing the microelectronic devices by artificially arranging metal Coulomb islands. Recently, conductance studies of electron transport
* Corresponding 0375.9601/95/$09.50
author.
through a single Coulomb island or supported metal cluster have attracted much attention [2-4,9,10]. For example, Lin et al. [2,3] have observed the discrete electron energy levels on an individual nanometersize supported Au cluster. Van Bentum et al. [9] have observed the incremental charging effect of a single isolated metal cluster by carefully positioning the tip of a STM above a metal cluster and found steplike structures in the I-V characteristic emerging from the quantization of the charge on metal clusters. Barner and Ruggiero [ll] have also observed conductance resonance in a structure consisting of a granular layer of Ag clusters embedded in the oxide layer between two continuous Ag or Au films. Very recently, we have investigated the influence of structural and size effects for a single metal cluster on conductance resonance [12]. In these studies only the single metal cluster is taken into account and the
0 1995 Elsevier Science B.V. All rights reserved
SSDI 0375-9601(95)00452-l
292
X. Chen et al. / Physics Letters A 204 (1995) -791-294
effect of the interaction between metal clusters on the conductance data is neglected. On the other hand, Nagamune et al. [13] have found Coulomb oscillations with an unexplained multiple peak structure in small quantum dots. They suggest that these unknown peaks may come from electronic correlations or potential fluctuations resulting from doped impurities which cause a breakup of the quantum dot in a few smaller dots. Therefore it is necessary to study the electronic transport properties of a resonant tunneling structure with a few supported metal clusters. In this Letter we consider the resonant tunneling structure with a few supported metal clusters or Coulomb islands on an insulating film surface, where the tip of the STM and the metal substrate are chosen as two electrodes. Only one metal cluster of the structure is coupled with the tip of the STM and all metal clusters are coupled with the metal substrate below the insulating film. In the structure under consideration, the Fermi level of the metal electrode can be shifted by changing the applied voltage. When the Fermi energy in the metal electrode matches the energy of a localized quantum state in the system, the tunneling probability increases sharply, producing a peak in the conductance data. We consider that a resonant level E,, of the metal cluster is close to the Fermi level E of the metal electrode, which can be realized by changing the voltage between the two electrodes to shift the Fermi level of the metal electrode, and that the value 1E - E, 1 is small compared to the characteristic electron energy in metal clusters. To obtain the different conductance, using the multi-dimensional Landauer approach [14], one can find the conductance of this structure by means of the generalized Breit-Winger formula in the tight-binding approximation [12,1.51 G = sTr(
r,Rr2Rt)
where the trace means summing the tunneling structure and
over all clusters in
ff=
E,
6,,
6
E,
f’
...
6 2N >
6’NI
fsN2
r l(2) 0
.‘.
iN 0
0
1
G,,, =
6 IN
0
,y
. 0
0
..
r
I(31,
N
Here I is the unit matrix and r is the matrix of energy level width. The symmetric matrix H is the tight-binding energy matrix and the non-diagonal elements of the energy matrix, aikr are the overlap integrals between the eigenstates of clusters i and k. c’ and c2 are the partial decay width from the state in cluster i into the tip of the STM and the metal substrate, respectively, which can be expressed as rjlc2’ a exp( - 2yd!(“), where y is the absolute value of the wave number in the barrier region and di”’ is the distance between cluster i and the tip of the STM (the metal substrate). In our study, as an example, we consider the tunneling structure with a few Au clusters, deposited on an insulating film, where the discrete electron energy levels have been observed [2,3] $or a diameter of the supported Au cluster of d = 10 A. For convenience we also take the diameter of the supported Au clusters as d = 10 A in our calculation. According to Ref. [9], we take the thickness of the insulating layer to be d: = 10 A and the potential height V, of insulating barrier to be 2.0 eV [16], relative to the Fermi level of the metal substrate. We assume that the distance between the tip of the STM and the Au cluster coupled with the tip of the STM is df = 10 A. The potential between the tip of the STM and the Au clusters is assumed to be flat in order to simplify the calculations, and is taken as V, = 1.0 eV. For the generalized Breit-Wigner formula in the tight-binding approximation, we consider the tunneling structure with one, two and four metal clusters. In Fig. 1, the structures with two and four metal clusters are shown in (a) and (b), respectively. Based on the above assumption, the diagonal terms are chosen to
X. Chen et al. /Physics
Letters A 204 (1995) 291-294
293
Fig. 1. Distribution of supported metal clusters on the surface of insulating film. 1 means that only one supported metal cluster of the structure is coupled with the the tip of the STM. All metal clusters are coupled with the metal substrate below the insulating film. The supported metal cluster number N: (a) two and (b) four.
be the resonant level E,, where we take E, = - 3.0 eV [2,3], and the hopping integral parameters, Sik, are taken to be a non-zero constant t, only in the case of nearest neighbor interaction. Dimensional arguments imply that the hopping matrix element between two Au clusters of size d at unit transmission is t N h’/m *d, where m* is electronic effective mass. c’ and c” can easily be obtained by using the above-mentioned formula. Numerical results are shown in Fig. 2, where one can see the splittings of the conductance resonance peak. For instance, one resonant peak appears for the structure containing only a single metal cluster. However, two and three resonant peaks in the vicinity of E, appear for structures with two and four metal clusters, respectively. The splitting of the resonant peak is
1.0
l-
CJ0.5
0.0 -: 1.1
-2.95
-
-I
-2.85
E(ev) Fig. 2. Conductance G in units of e’/nh as a function of the Fermi level E of the metal substrate. Short dashed, long dashed and solid lines correspond with one, two and four supported metal clusters, respectively.
Fig. 3. Different formations of the tunneling structure consisting of four metal clusters are shown in (a), (b), tc) and cd). Notations as in Fig. 1.
caused by the interaction between the metal clusters. Therefore, the multiple peak structure of the conductance resonance observed by Nagamune et al. [13] can be explained by generalizing our model. Furthermore, according to the conductance data, one can easily determine the strength of the interaction between the supported metal clusters by analyzing the conductance data. Next we consider a tunneling structure with four supported Au clusters but different arrangements, shown in Fig. 3, which is based on the experimental observation of a fractal structure in the coalescence process of metal clusters deposited on insulating substrates. Random packing formation of metal cluster-based film are observed by transmission electron microscopy and STM by our group [17,18]. In these arrangements, conductance resonance peaks are numerically obtained, see Fig. 4. The calculated number and positions of conductance resonance peaks are different for different arrangements of Au clusters. The two resonant peaks for the arrangement in Fig. 3a and three resonant peaks for other arrangements in the vicinity of E, are shown in Fig. 4a and Figs. 4b-4d, respectively. In Fig. 4a, the positions of the two splitting peaks are - 3.066 eV and - 2.934 eV, respectively. Though the number of splitting peaks for other arrangements is the same, the positions are different. For example, in Fig. 4b, the positions of three splitting peaks are at -3.064 eV, - 3.021 eV and - 2.916 eV for the arrangement in Fig. 3b, but the positions of the splitting peaks in
294
X. Chen et al. /Physics
Letters A 204 (1995) 291-294
This work is supported by the National Science Foundation in China and a Tianma Fellowship financed by Tianma Microelectronics Co. Ltd. in Shenzhen.
References [ll P.R. Andres, R.S. Averback, W.L. Brown, L.E. Brus, W.A.
0.0 -3.15
A, -3.05
-2.95
-2.85
E(eV) Fig. 4. Conductance G in units of e’,/nfi as a function of the Fermi level E of the metal substrate, where (a), (b), cc) and (d) are as in Fig. 3.
Fig. 4c are at -3.057 eV, -2.988 eV and -2.917 eV for the arrangement in Fig. 3c. This is because the structures with different arrangements of Au clusters possess different interactive forms and produce different splitting energy levels in the vicinity of E,. Therefore we predict that the formation of metal clusters deposited on an insulating film has influence on the conductance resonance. In summary, we have studied conductance resonance of a tunneling structure with a few metal clusters deposited on an insulating film. We have found that in conductance resonance multiple peak structures arise from the interaction between the supported metal clusters, which can be used to explain the multiple peak structures observed by Nagamune et al. [13] by generalizing our model to a system of a few quantum dots. We have also obtained that the conductance resonant peaks are different for different arrangements of the metal clusters on the insulating film. Recent advances in nanotechnology and in experimental observation of conductance resonance provide the possibility of studying the interaction between supported metal clusters and the arrangements of metal clusters deposited on the insulating film. It is also possible to develop some new microelectronic devices by artificially arranging metal clusters on the surface of an insulating film.
Goddard, A. Kaldor, S.G. Louie, M. Moscovits. P.S. Peercy. S.J. Riley, R.W. Siegel. F. Spaepen and Y. Wang, J. Mater. Res. 4 (1989) 84. [‘I M.E. Lin, R.P. Andres and R. Reifenberger. Phys. Rev. Lett. 67 (1991) 447. [31 M.E. Lin, R. Reifenberger and P.R. Andre% Phys. Rev. B 46 (1992) 15490. [41 U. Meirav, M.A. Kastner and S.J. Wind, Phys. Rev. Lett. 65 (1990) 771; M.A. Kastner, Rev. Mod. Phys. 64 (1992) 849; P.L. McEuen et al., Phys. Rev. Lett. 68 (1991) 1936. El R.N. Siegel, Phys. Today 64 (October 1993); Nanostructured Mater. 3 (1993) 1. 161K.H. Bennemann, Surf. Sci. 156 (1985) 1040. [71 D. Schonauer, H. Lauer and U. Kreibig, Z. Phys. D 20 (1991) 301. [81 H. van Houten and C.W. Devort, eds.. Single charge tunneling (Plenum, New York, 1989). 191 P.J.M. van Bentum, R.T.M. Somkers and H. van Kempen, Phys. Rev. Lett. 60 (1988) 2543: J.G.A. Dubors, E.N.G. Verheijen, J.W. Gerritsen and H. van Kempen, Phys. Rev. B 48 (1993) 11260. [lOI C.W.J. Beenakker and H. van Houten, in: Single charge tunne!ing, eds. H. Grabert. J.M. Martinis and M.H. Devoret (Plenum. New York, 1991). [Ill J.B. Barner and ST. Ruggiero, Phys. Rev. L&t. 59 (1987) 807. [l’l Xiaoshuang Chen, Jijun Zhao and Guanghou Wang, Appl. Phys. Lett. 65 (1994) 2419. L.C. Mur. iI31 Y. Nagamune, H. Sakaki. L.P. Kouwenhoven, C.P.M. Harmans. J.M. Tohisa and H. Noge, Appl. Phys. Lett. 64 (1994) 2379. [I41 M. Buttiker, Y. Imry, R. Landauer and S. Pinhas. Phys. Rev. B 31 (1985) 6207: A.D. Stone and A. Szater, IBM J. Rcs. Rev. 32 (1988) 384. Ml M.Yu. Sumetskii, JETP Lett. 44 (1986) 369; J. Phys. Condens. Matter 3 (1991) 2651. [161 V. Kalmeyer and R.B. Laughlin. Phys. Rev. B 35 (1987) 9805. [171 J.X. Ma. M. Han, H.Q. Zhang, Y.C. Gong and G.H. Wang. Appl. Phys. Lett. 65 (1994) 1315. Ml G.H. Wang. H.Q. Zhang, M. Han. J.X. Ma and Q. Wang, Phys. Lctt. A 189 (1994) 218.
21 August 1995
&
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 204 (1995) 291-294
Conductance resonance of coupled supported metal clusters Xiaoshuang Department
of Physics
Chen, Jijun Zhao, Fengqi Lui, Qing Sun, Guanghou Wang and National Laboratory
Center for Advanced
Received 20 February
of Solid State Microstructures,
Studies in Science
and Technology
Nanjing University, Nanjing 210093,
of Microstructures,
Nanjing 210093,
* China
China
received 6 April 1995; accepted for publication 16 June 1995 Communicated by J. Flouquet
1995; revised manuscript
Abstract The conductance resonance of a tunneling structure with a few metal clusters, deposited on an insulating film, is studied by the generalized Breit-Wigner formula in a tight-binding approximation. We find that in the conductance resonance the multiple peak structure comes from the interaction between supported metal clusters on an insulating film and that the different arrangements of metal clusters can cause a difference of the conductance resonance peaks. Therefore it is possible to predict the effect of the interaction between metal clusters on the conductance resonance and develop some new microelectronic devices by artificially arranging metal clusters on the surface of the insulating film. PACS:
36.40. +
d; 73.40.Gk: 73.40.R~
In solid state physics as a unique transitional
metal clusters are considered state between a single atom
and a bulk [l]. Considerable effort has been devoted to supported metal clusters [2,3], in which the discrete electron energy levels of metal clusters have been observed, and metal Coulomb islands [4] because they represent the first stage of the metal film growth [5-71 and the development of some new microelectronic devices [s]. Therefore the interaction between supported metal clusters is of great importance for determining the properties of cluster-based materials and developing the microelectronic devices by artificially arranging metal Coulomb islands. Recently, conductance studies of electron transport
* Corresponding 0375.9601/95/$09.50
author.
through a single Coulomb island or supported metal cluster have attracted much attention [2-4,9,10]. For example, Lin et al. [2,3] have observed the discrete electron energy levels on an individual nanometersize supported Au cluster. Van Bentum et al. [9] have observed the incremental charging effect of a single isolated metal cluster by carefully positioning the tip of a STM above a metal cluster and found steplike structures in the I-V characteristic emerging from the quantization of the charge on metal clusters. Barner and Ruggiero [ll] have also observed conductance resonance in a structure consisting of a granular layer of Ag clusters embedded in the oxide layer between two continuous Ag or Au films. Very recently, we have investigated the influence of structural and size effects for a single metal cluster on conductance resonance [12]. In these studies only the single metal cluster is taken into account and the
0 1995 Elsevier Science B.V. All rights reserved
SSDI 0375-9601(95)00452-l
292
X. Chen et al. / Physics Letters A 204 (1995) -791-294
effect of the interaction between metal clusters on the conductance data is neglected. On the other hand, Nagamune et al. [13] have found Coulomb oscillations with an unexplained multiple peak structure in small quantum dots. They suggest that these unknown peaks may come from electronic correlations or potential fluctuations resulting from doped impurities which cause a breakup of the quantum dot in a few smaller dots. Therefore it is necessary to study the electronic transport properties of a resonant tunneling structure with a few supported metal clusters. In this Letter we consider the resonant tunneling structure with a few supported metal clusters or Coulomb islands on an insulating film surface, where the tip of the STM and the metal substrate are chosen as two electrodes. Only one metal cluster of the structure is coupled with the tip of the STM and all metal clusters are coupled with the metal substrate below the insulating film. In the structure under consideration, the Fermi level of the metal electrode can be shifted by changing the applied voltage. When the Fermi energy in the metal electrode matches the energy of a localized quantum state in the system, the tunneling probability increases sharply, producing a peak in the conductance data. We consider that a resonant level E,, of the metal cluster is close to the Fermi level E of the metal electrode, which can be realized by changing the voltage between the two electrodes to shift the Fermi level of the metal electrode, and that the value 1E - E, 1 is small compared to the characteristic electron energy in metal clusters. To obtain the different conductance, using the multi-dimensional Landauer approach [14], one can find the conductance of this structure by means of the generalized Breit-Winger formula in the tight-binding approximation [12,1.51 G = sTr(
r,Rr2Rt)
where the trace means summing the tunneling structure and
over all clusters in
ff=
E,
6,,
6
E,
f’
...
6 2N >
6’NI
fsN2
r l(2) 0
.‘.
iN 0
0
1
G,,, =
6 IN
0
,y
. 0
0
..
r
I(31,
N
Here I is the unit matrix and r is the matrix of energy level width. The symmetric matrix H is the tight-binding energy matrix and the non-diagonal elements of the energy matrix, aikr are the overlap integrals between the eigenstates of clusters i and k. c’ and c2 are the partial decay width from the state in cluster i into the tip of the STM and the metal substrate, respectively, which can be expressed as rjlc2’ a exp( - 2yd!(“), where y is the absolute value of the wave number in the barrier region and di”’ is the distance between cluster i and the tip of the STM (the metal substrate). In our study, as an example, we consider the tunneling structure with a few Au clusters, deposited on an insulating film, where the discrete electron energy levels have been observed [2,3] $or a diameter of the supported Au cluster of d = 10 A. For convenience we also take the diameter of the supported Au clusters as d = 10 A in our calculation. According to Ref. [9], we take the thickness of the insulating layer to be d: = 10 A and the potential height V, of insulating barrier to be 2.0 eV [16], relative to the Fermi level of the metal substrate. We assume that the distance between the tip of the STM and the Au cluster coupled with the tip of the STM is df = 10 A. The potential between the tip of the STM and the Au clusters is assumed to be flat in order to simplify the calculations, and is taken as V, = 1.0 eV. For the generalized Breit-Wigner formula in the tight-binding approximation, we consider the tunneling structure with one, two and four metal clusters. In Fig. 1, the structures with two and four metal clusters are shown in (a) and (b), respectively. Based on the above assumption, the diagonal terms are chosen to
X. Chen et al. /Physics
Letters A 204 (1995) 291-294
293
Fig. 1. Distribution of supported metal clusters on the surface of insulating film. 1 means that only one supported metal cluster of the structure is coupled with the the tip of the STM. All metal clusters are coupled with the metal substrate below the insulating film. The supported metal cluster number N: (a) two and (b) four.
be the resonant level E,, where we take E, = - 3.0 eV [2,3], and the hopping integral parameters, Sik, are taken to be a non-zero constant t, only in the case of nearest neighbor interaction. Dimensional arguments imply that the hopping matrix element between two Au clusters of size d at unit transmission is t N h’/m *d, where m* is electronic effective mass. c’ and c” can easily be obtained by using the above-mentioned formula. Numerical results are shown in Fig. 2, where one can see the splittings of the conductance resonance peak. For instance, one resonant peak appears for the structure containing only a single metal cluster. However, two and three resonant peaks in the vicinity of E, appear for structures with two and four metal clusters, respectively. The splitting of the resonant peak is
1.0
l-
CJ0.5
0.0 -: 1.1
-2.95
-
-I
-2.85
E(ev) Fig. 2. Conductance G in units of e’/nh as a function of the Fermi level E of the metal substrate. Short dashed, long dashed and solid lines correspond with one, two and four supported metal clusters, respectively.
Fig. 3. Different formations of the tunneling structure consisting of four metal clusters are shown in (a), (b), tc) and cd). Notations as in Fig. 1.
caused by the interaction between the metal clusters. Therefore, the multiple peak structure of the conductance resonance observed by Nagamune et al. [13] can be explained by generalizing our model. Furthermore, according to the conductance data, one can easily determine the strength of the interaction between the supported metal clusters by analyzing the conductance data. Next we consider a tunneling structure with four supported Au clusters but different arrangements, shown in Fig. 3, which is based on the experimental observation of a fractal structure in the coalescence process of metal clusters deposited on insulating substrates. Random packing formation of metal cluster-based film are observed by transmission electron microscopy and STM by our group [17,18]. In these arrangements, conductance resonance peaks are numerically obtained, see Fig. 4. The calculated number and positions of conductance resonance peaks are different for different arrangements of Au clusters. The two resonant peaks for the arrangement in Fig. 3a and three resonant peaks for other arrangements in the vicinity of E, are shown in Fig. 4a and Figs. 4b-4d, respectively. In Fig. 4a, the positions of the two splitting peaks are - 3.066 eV and - 2.934 eV, respectively. Though the number of splitting peaks for other arrangements is the same, the positions are different. For example, in Fig. 4b, the positions of three splitting peaks are at -3.064 eV, - 3.021 eV and - 2.916 eV for the arrangement in Fig. 3b, but the positions of the splitting peaks in
294
X. Chen et al. /Physics
Letters A 204 (1995) 291-294
This work is supported by the National Science Foundation in China and a Tianma Fellowship financed by Tianma Microelectronics Co. Ltd. in Shenzhen.
References [ll P.R. Andres, R.S. Averback, W.L. Brown, L.E. Brus, W.A.
0.0 -3.15
A, -3.05
-2.95
-2.85
E(eV) Fig. 4. Conductance G in units of e’,/nfi as a function of the Fermi level E of the metal substrate, where (a), (b), cc) and (d) are as in Fig. 3.
Fig. 4c are at -3.057 eV, -2.988 eV and -2.917 eV for the arrangement in Fig. 3c. This is because the structures with different arrangements of Au clusters possess different interactive forms and produce different splitting energy levels in the vicinity of E,. Therefore we predict that the formation of metal clusters deposited on an insulating film has influence on the conductance resonance. In summary, we have studied conductance resonance of a tunneling structure with a few metal clusters deposited on an insulating film. We have found that in conductance resonance multiple peak structures arise from the interaction between the supported metal clusters, which can be used to explain the multiple peak structures observed by Nagamune et al. [13] by generalizing our model to a system of a few quantum dots. We have also obtained that the conductance resonant peaks are different for different arrangements of the metal clusters on the insulating film. Recent advances in nanotechnology and in experimental observation of conductance resonance provide the possibility of studying the interaction between supported metal clusters and the arrangements of metal clusters deposited on the insulating film. It is also possible to develop some new microelectronic devices by artificially arranging metal clusters on the surface of an insulating film.
Goddard, A. Kaldor, S.G. Louie, M. Moscovits. P.S. Peercy. S.J. Riley, R.W. Siegel. F. Spaepen and Y. Wang, J. Mater. Res. 4 (1989) 84. [‘I M.E. Lin, R.P. Andres and R. Reifenberger. Phys. Rev. Lett. 67 (1991) 447. [31 M.E. Lin, R. Reifenberger and P.R. Andre% Phys. Rev. B 46 (1992) 15490. [41 U. Meirav, M.A. Kastner and S.J. Wind, Phys. Rev. Lett. 65 (1990) 771; M.A. Kastner, Rev. Mod. Phys. 64 (1992) 849; P.L. McEuen et al., Phys. Rev. Lett. 68 (1991) 1936. El R.N. Siegel, Phys. Today 64 (October 1993); Nanostructured Mater. 3 (1993) 1. 161K.H. Bennemann, Surf. Sci. 156 (1985) 1040. [71 D. Schonauer, H. Lauer and U. Kreibig, Z. Phys. D 20 (1991) 301. [81 H. van Houten and C.W. Devort, eds.. Single charge tunneling (Plenum, New York, 1989). 191 P.J.M. van Bentum, R.T.M. Somkers and H. van Kempen, Phys. Rev. Lett. 60 (1988) 2543: J.G.A. Dubors, E.N.G. Verheijen, J.W. Gerritsen and H. van Kempen, Phys. Rev. B 48 (1993) 11260. [lOI C.W.J. Beenakker and H. van Houten, in: Single charge tunne!ing, eds. H. Grabert. J.M. Martinis and M.H. Devoret (Plenum. New York, 1991). [Ill J.B. Barner and ST. Ruggiero, Phys. Rev. L&t. 59 (1987) 807. [l’l Xiaoshuang Chen, Jijun Zhao and Guanghou Wang, Appl. Phys. Lett. 65 (1994) 2419. L.C. Mur. iI31 Y. Nagamune, H. Sakaki. L.P. Kouwenhoven, C.P.M. Harmans. J.M. Tohisa and H. Noge, Appl. Phys. Lett. 64 (1994) 2379. [I41 M. Buttiker, Y. Imry, R. Landauer and S. Pinhas. Phys. Rev. B 31 (1985) 6207: A.D. Stone and A. Szater, IBM J. Rcs. Rev. 32 (1988) 384. Ml M.Yu. Sumetskii, JETP Lett. 44 (1986) 369; J. Phys. Condens. Matter 3 (1991) 2651. [161 V. Kalmeyer and R.B. Laughlin. Phys. Rev. B 35 (1987) 9805. [171 J.X. Ma. M. Han, H.Q. Zhang, Y.C. Gong and G.H. Wang. Appl. Phys. Lett. 65 (1994) 1315. Ml G.H. Wang. H.Q. Zhang, M. Han. J.X. Ma and Q. Wang, Phys. Lctt. A 189 (1994) 218.