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Influence of inhomogeneity on correlated single-electron tunneling in one-dimensional array of small tunnel junctions
PHYSICA
Physica B 194-196 (1994) 1309-1310 North-Holland
I n f l u e n c e o f i n h o m o g e n e i t y on correlated single-electron t u n n e l i n g in o n e - d i m e n s i o n a l array o f small tunnel j u n c t i o n s N. Yoshikawa, K. Fukushima and M. Sugahara Faculty of Engineering, Yokohama National University, Hodogaya, Yokohama 240, Japan Simulation calculations are made on the influence of inhomogeneity of junction parameters on correlated single-electron tunneling in a long one-dimensional array of small tunnel junctions. Current-voltage characteristics, static electric field dependence and frequency spectrum of voltage oscillations of the array are numerically calculated using the randomly scattered junction parameters. It is found that correlation of singleelectron transfer is not seriously disturbed by inhomogeneity of the junction parameters. 1. I N T R O D U C T I O N Recent theoretical and experimental studies have verified that a voltage-biased long one-dimensional array of small tunnel junctions exhibits a correlated single-electron transfer not only in space but also in time [1-4]. In the array, a spatially ordered singleelectron soliton lattice propagates along the array causing coherent oscillation of the junction voltage. Experimental results show that there exists substantial coherency in the single-electron oscillation in spite of inevitable inhomogeneity of junction parameters in the array. In this paper we investigate the influence of inhomogeneity of the array on the correlated single-electron transfer.
Using eq.(1) and the free energy of the system, the electron transport properties can be calculated using the Monte Carlo method [3,4]. In this calculation, we assume inhomogeneous arrays with randomly scattered junction parameters xi (xi=Ci, Gi, C01, and ni which is the number of fixed charges on the ith electrode), xi is scattered around average value x using the Gaussian distribution function with the standard deviation a'x~-Xax. 3, R E S U L T S Figure 1 shows the frequency spectrum of the voltage oscillations Sv(tO) = ~-
2. M O D E L F O R C A L C U L A T I O N The investigated model of the array is the same as that studied by Bakhvalov et al. [3]. The array consists of N small tunnel junctions with a junction conductance Gi and a capacitance Ci. A capacitance C0i between a ground and the ith electrode is added to the model circuit. The array is driven at one end by a voltage source V, and at both ends by V0, which corresponds to the transport voltage and the average potential (gate voltage) of the array, respectively. The electron tunneling rate in the ith junction per unit time is given by AFi ti-_ Gi e2 1 - exp(-eAFJkT) , (1) where AFi is the free energy difference between before and after the tunneling event in the ith junction.
[(V(t)V(t+x))" (V(t))2] c°s t°~ dx
,f0 °
of the first junction in a 21-junction array at T=0 for C0/C=0.1 and V/(e/C)=2.36 where G and n are scattered parameters. It can be seen in Figl.(a) that the scattering of G does not seriously increase the spectral linewidth even for t~c=0.5. Similar dependence of the linewidth on the standard deviation is also observed for C and Co. In contrast with this, the scattering of fixed charge number n considerably increases the linewidth, as shown in Fig.l(b), and completely smears coherency of the oscillation at ~a=0.2. Figure 2 shows the influence of the parameter scattering on space correlation. In the homogeneous case (Fig.2(a)), dependence of the average array current becomes a periodic function of the array potential V0 with periodicity Vo/(e/C)=10, which corresponds
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0921-4526(93)E1235-E
(2)
1310
to the single electron charging on each electrode. For ¢ro=0.5 (Fig.2(b)), the -V0 relation does not change considerably. The scattering of C causes a complex -Vo relation, but the periodicity is roughly maintained (Fig.2(c)). On the other hand, the potential distribution caused by the fixed charge strongly affects the periodicity of the -V0 relation
0.01 V/(e/C)=2.75
(Fig.2(d)).
0.01
OG=0.5
0
2.5 N=21 T=0K
I
(a)
0
A v
~
0.01
(~rG=
[
1.5
E 0.01 0.5
0 0
0 0.005
0.01
0.015
(~/2~)/(G/C) 2.5
N=21 2
5
10
Vo/(e/C)
[
(b)
T~
T=OK
Figure 2. Dependence of the array current on the gate voltage Vo for several kinds of junction parameter scattering. (a) ~=0, (b) ~c=0.5, (c) ~c=0.5, (d) or,=0.2. (N=21, Co/C=0.1, V/(e/C)=2.75, and T=0) 4. CONCLUSIONS
~
We investigated the effect of inhomogeneity on the correlated transfer of single electrons in onedimensional arrays of small tunnel junctions. Numerical calculation shows that correlation is not seriously disturbed by inhomogeneity of the junction parameters such as G, C, and Co, whereas the existence of a fixed charge strongly affects it.
1.5
%
0.5
On=0.2 o
0.005
o
o
•
•
• •
•
• •
• qJ
I
0.01
I
l
t
O
!
I
References
0.015
((d2n) / (CVC)
Figure 1. Frequency spectrum of the voltage oscillation of a junction in a 21-junction array, where (a) G and (b) n are randomly scattered with the standard deviations Go and o . , respectively. For all curves, T=0, C0/C=0.1, V/(e/C)=2.36, /(eG/C) =0.01, and theoretical frequency of spectrum maximum is (co/2~z)/(G/C)=0.01. Note that each data point is offset by increments of 0.5 for clarity.
1. A. V. Averin and K. K. Likharev, J. Low Temp. Phys. 62 (1986) 345. 2. P. Delsing, K. K. Likharev, L. S. Kuzmin and T. Claeson, Phys. Rev. Lett. 63 (1989) 1861. 3. N. S. Bakhalov, G. S. Kazacha, K. K. Likharev and S. I. Serdyukova, Sov. Phys. JETP 68 (1989) 581. 4. M. Amman, E. Ben-Jacob and K. Mullen, Phys. Lett. A 142 (1989) 431.
Physica B 194-196 (1994) 1309-1310 North-Holland
I n f l u e n c e o f i n h o m o g e n e i t y on correlated single-electron t u n n e l i n g in o n e - d i m e n s i o n a l array o f small tunnel j u n c t i o n s N. Yoshikawa, K. Fukushima and M. Sugahara Faculty of Engineering, Yokohama National University, Hodogaya, Yokohama 240, Japan Simulation calculations are made on the influence of inhomogeneity of junction parameters on correlated single-electron tunneling in a long one-dimensional array of small tunnel junctions. Current-voltage characteristics, static electric field dependence and frequency spectrum of voltage oscillations of the array are numerically calculated using the randomly scattered junction parameters. It is found that correlation of singleelectron transfer is not seriously disturbed by inhomogeneity of the junction parameters. 1. I N T R O D U C T I O N Recent theoretical and experimental studies have verified that a voltage-biased long one-dimensional array of small tunnel junctions exhibits a correlated single-electron transfer not only in space but also in time [1-4]. In the array, a spatially ordered singleelectron soliton lattice propagates along the array causing coherent oscillation of the junction voltage. Experimental results show that there exists substantial coherency in the single-electron oscillation in spite of inevitable inhomogeneity of junction parameters in the array. In this paper we investigate the influence of inhomogeneity of the array on the correlated single-electron transfer.
Using eq.(1) and the free energy of the system, the electron transport properties can be calculated using the Monte Carlo method [3,4]. In this calculation, we assume inhomogeneous arrays with randomly scattered junction parameters xi (xi=Ci, Gi, C01, and ni which is the number of fixed charges on the ith electrode), xi is scattered around average value x using the Gaussian distribution function with the standard deviation a'x~-Xax. 3, R E S U L T S Figure 1 shows the frequency spectrum of the voltage oscillations Sv(tO) = ~-
2. M O D E L F O R C A L C U L A T I O N The investigated model of the array is the same as that studied by Bakhvalov et al. [3]. The array consists of N small tunnel junctions with a junction conductance Gi and a capacitance Ci. A capacitance C0i between a ground and the ith electrode is added to the model circuit. The array is driven at one end by a voltage source V, and at both ends by V0, which corresponds to the transport voltage and the average potential (gate voltage) of the array, respectively. The electron tunneling rate in the ith junction per unit time is given by AFi ti-_ Gi e2 1 - exp(-eAFJkT) , (1) where AFi is the free energy difference between before and after the tunneling event in the ith junction.
[(V(t)V(t+x))" (V(t))2] c°s t°~ dx
,f0 °
of the first junction in a 21-junction array at T=0 for C0/C=0.1 and V/(e/C)=2.36 where G and n are scattered parameters. It can be seen in Figl.(a) that the scattering of G does not seriously increase the spectral linewidth even for t~c=0.5. Similar dependence of the linewidth on the standard deviation is also observed for C and Co. In contrast with this, the scattering of fixed charge number n considerably increases the linewidth, as shown in Fig.l(b), and completely smears coherency of the oscillation at ~a=0.2. Figure 2 shows the influence of the parameter scattering on space correlation. In the homogeneous case (Fig.2(a)), dependence of the average array current becomes a periodic function of the array potential V0 with periodicity Vo/(e/C)=10, which corresponds
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0921-4526(93)E1235-E
(2)
1310
to the single electron charging on each electrode. For ¢ro=0.5 (Fig.2(b)), the -V0 relation does not change considerably. The scattering of C causes a complex -Vo relation, but the periodicity is roughly maintained (Fig.2(c)). On the other hand, the potential distribution caused by the fixed charge strongly affects the periodicity of the -V0 relation
0.01 V/(e/C)=2.75
(Fig.2(d)).
0.01
OG=0.5
0
2.5 N=21 T=0K
I
(a)
0
A v
~
0.01
(~rG=
[
1.5
E 0.01 0.5
0 0
0 0.005
0.01
0.015
(~/2~)/(G/C) 2.5
N=21 2
5
10
Vo/(e/C)
[
(b)
T~
T=OK
Figure 2. Dependence of the array current on the gate voltage Vo for several kinds of junction parameter scattering. (a) ~=0, (b) ~c=0.5, (c) ~c=0.5, (d) or,=0.2. (N=21, Co/C=0.1, V/(e/C)=2.75, and T=0) 4. CONCLUSIONS
~
We investigated the effect of inhomogeneity on the correlated transfer of single electrons in onedimensional arrays of small tunnel junctions. Numerical calculation shows that correlation is not seriously disturbed by inhomogeneity of the junction parameters such as G, C, and Co, whereas the existence of a fixed charge strongly affects it.
1.5
%
0.5
On=0.2 o
0.005
o
o
•
•
• •
•
• •
• qJ
I
0.01
I
l
t
O
!
I
References
0.015
((d2n) / (CVC)
Figure 1. Frequency spectrum of the voltage oscillation of a junction in a 21-junction array, where (a) G and (b) n are randomly scattered with the standard deviations Go and o . , respectively. For all curves, T=0, C0/C=0.1, V/(e/C)=2.36, /(eG/C) =0.01, and theoretical frequency of spectrum maximum is (co/2~z)/(G/C)=0.01. Note that each data point is offset by increments of 0.5 for clarity.
1. A. V. Averin and K. K. Likharev, J. Low Temp. Phys. 62 (1986) 345. 2. P. Delsing, K. K. Likharev, L. S. Kuzmin and T. Claeson, Phys. Rev. Lett. 63 (1989) 1861. 3. N. S. Bakhalov, G. S. Kazacha, K. K. Likharev and S. I. Serdyukova, Sov. Phys. JETP 68 (1989) 581. 4. M. Amman, E. Ben-Jacob and K. Mullen, Phys. Lett. A 142 (1989) 431.