Download Blocking Oscillation Due to Combination of Discrete and Continuous

PIERS Proceedings, Kuala Lumpur, MALAYSIA, March 27–30, 2012
768
Blocking Oscillation Due to Combination of Discrete and
Continuous Charge/Flux Transfer in Systems Including
Nano/Josephson Tunnel Junctions
Yoshinao Mizugaki
The University of Electro-Communications (UEC Tokyo), Japan
Abstract— Nanotechnology enables us to manipulate a single electron in solid-state circuits.
Single-electron tunneling is known to transfer electric charge in units of the fundamental charge e
through tiny (nano-scale) tunnel junctions. So far, many single-electron devices, such as a singleelectron transistor, a single-electron turnstile, and a single-electron pump, have been proposed
and demonstrated. The author proposes a blocking oscillator composed of tiny tunnel junctions
and a dissipative element (resistor). The key is the combination of the discrete and continuous
charge transfer. Monte-Carlo simulation demonstrates blocking oscillation of an island charge.
Similar blocking oscillation occurs in a superconducting loop including Josephson junctions and a
resistor. Electromagnetic duality between single-electron devices and superconducting Josephson
devices are observed in these results. In this paper, the author numerically demonstrates these
blocking oscillation phenomena and discuss the possibility of experimental observation.
1. INTRODUCTION
It is commonly understood that electric current in a metal wire is carried by electrons. Because
the number of electrons is huge, electric current is considered to be continuous. It is valid if the
system is macroscopic. However, if the system is microscopic or mesoscopic, we sometimes need to
handle not continuous electric current but discrete electrons.
In fact, recent progress of nanofabrication techniques has enabled us to manipulate a single
electron in solid-state circuits [1]. Single-electron tunneling based on the Coulomb blockade is
realized in tiny (nano-scale) tunnel junctions when the following conditions are satisfied: e2 /2C À
kB T and Rt À RQ ≡ h/e2 , where e, C, kB , T , Rt , RQ , and h are the fundamental charge, tunnel
capacitance, Boltzmann constant, absolute temperature, tunnel resistance, quantum resistance, and
Planck constant, respectively. A single-electron transistor composed of two tiny tunnel junctions
and a capacitive gate is known as a ultra-sensitive charge sensor [2]. Systems including more than
three tiny tunnel junctions can transfer a single-electron one by one, known as a single-electron
turnstile [3] and a single-electron pump [4].
On the other hand, a single flux quantum in a superconducting loop can be used as a bit
carrier in superconducting Josephson electronics [5]. Interestingly, electromagnetic duality can be
found between single-electron devices and single-flux-quantum devices [2]. Although the duality
is phenomenological and imperfect, it sometimes helps us to develop a new device. Examples are
an RF single-electron transistor [6] dual to an RF superconducting quantum interference device
(RF-SQUID), a single-flux-quantum pump [7] dual to a single-electron pump, and a single-electron
current mirror [8] dual to a single-flux-quantum voltage mirror.
In this paper, the author describes numerical results of a single-electron blocking oscillator [9]
and a single-flux-quantum blocking oscillator [10], both of which have been proposed recently. The
key is the combination of the discrete and continuous charge/flux transfer.
2. CONFIGURATION OF BLOCKING OSCILLATORS
2.1. Single-electron Blocking Oscillator
The author has proposed a blocking oscillator composed of tiny tunnel junctions and a dissipative
element (resistor) [9], of which configuration is shown in Figure 1(a). Because of charge leakage through the resistive path, the island charge Q0 becomes continuous, whereas single-electron
tunneling events through the junctions carry fundamental charges into/from the island electrode
discretely. Combination of continuous and discrete charge transfer eventually shifts the operation
point into the Coulomb blockade, resulting in blocking oscillation.
Progress In Electromagnetics Research Symposium Proceedings, KL, MALAYSIA, March 27–30, 2012 769
Island charge Q 0
J1
(Cj , R j )
Ib
J2
(Cj , R j )
L lp
Vb
JJ 1
(lc , Rn, C jj )
Cg
Rg
R lp
JJ 2
(lc , Rn, C jj )
(a)
(b)
Figure 1: Blocking oscillators utilizing combination of discrete and continuous chrage/flux transfer. (a)
Single-electron blocking oscillator comprising series array of two tiny tunnel junctions with capacitive and
resistive ground paths from its island electrode. Cj and Rj are the junction capacitance and junction resistance, respectively. (b) Single-flux-quantum blocking oscillator comprising parallel array of two Josephson
junctions connected via a superconducting inductor and a resistor. Ic , Rn , and Cjj are the superconducting
critical current, junction resistance, and junction capacitance, respectively.
Vb = 27.0 mV
-19
Island charge Q0 (× 10 C)
-19
Island charge Q0 (× 10 C)
Vb = 26.7 mV
1.5
1
0.5
0
-0.5
-1
2
3
4
5
Time (ns)
6
7
1.5
1
0.5
0
-0.5
-1
1.5
2
2.5
3
3.5
Time (ns)
4
4.5
5
(a)
(b)
Figure 2: Numerical waveforms of the island charge. Cj , Rj , Cg , and Rg are set at 1 aF, 100 kΩ, 1 aF, and
20 MΩ, respectively. Absolute zero temperature and no cotunneling processes are assumed. Vb is set at (a)
26.7 mV and (b) 27.0 mV.
2.2. Single-flux-quantum Blocking Oscillator
A single-flux-quantum blocking oscillator [10] dual to the single-electron blocking oscillator is presented in Figure 1(b). The flux in the JJ1 -Llp -Rlp -JJ2 loop is continuously changing toward zero
due to the resistive element Rlp , whereas switching events of the Josephson junctions (JJ1 and
JJ2 ) transfer flux quanta into/from the loop discretely. Combination of continuous and discrete
flux transfer eventually moves the operation point into the superconducting state, resulting in
blocking oscillation.
3. NUMERICAL RESULTS
3.1. Waveforms of Island Charge in Single-electron Blocking Oscillator
Two examples of the island charge evolution in the single-electron blocking oscillator are shown
in Figure 2. Simulation was carried out using a simulator for single-electron devices and circuits
(SIMON) on the basis of the Monte Carlo method [11].
Blocking oscillation can be confirmed in both results, whereas sporadic single-electron tunneling
events are seen for the lower bias condition in Figure 2(a), at the time durations from 3 ns to 5 ns
and from 7 ns to 7.5 ns. Since single-electron tunneling events are stochastic, intervals among events
are not constant in both Figures 2(a) and 2(b). Especially at low bias conditions, time intervals
vary widely, resulting in sporadic events.
3.2. Waveforms of Junction Voltages in Single-flux-quantum Blocking Oscillator
Waveforms of the junction voltages in the single-flux-quantum blocking oscillator are shown in
Figure 3. For discussion on the voltages across the junctions below, not the magnetic flux in
the loop but the junction voltages are presented. Simulation was carried out using a Josephson
integrated circuit simulator (JSIM) [12]. The voltage V across a Josephson junction is expressed
PIERS Proceedings, Kuala Lumpur, MALAYSIA, March 27–30, 2012
770
as V = (Φ0 /2π)(∂φ/∂t), where Φ0 and φ are the flux quantum and the quantum phase difference
across the junction, respectively.
The numbers of junction switching in JJ1 and JJ2 in one period of blocking oscillation are
rather small in these conditions: the numbers of switching in one period are 2 in JJ1 and 1 in JJ2
for Figure 3(a), and those are 3 in JJ1 and 2 in JJ2 for Figure 3(b). The switching number in JJ1 is
one greater than that in JJ2 , which is a feature of the single-flux-quantum blocking oscillator [10].
Differently from the results in Figure 2, the blocking oscillation is deterministic and periodic,
which demonstrates imperfect duality between single-electron devices and single-flux-quantum devices.
4. EVIDENCE OF BLOCKING OSCILLATION OBTAINED FROM STATIC
CHARACTERISTICS
It can be seen in Figures 2 and 3 that the signals from the blocking oscillators are rather small and
fast. The RF-SET technique [6] and the Josephson sampling technique [13, 14] would be applied
to detect such signals, although each of them is a challenging task.
Here, the author proposes an indirect measurement method of blocking oscillation in the singleflux-quantum blocking oscillator. A voltage waveform across a Josephson junction is a pulse train
as shown in Figure 3. In experiments, however, pulses are smeared through conventional lowbandwidth equipments, and the average voltage expressed as hV i = Φ0 f is measured, where f is
the number of pulses per unit time (the pulse repetition frequency). In other words, the number of
pulses per unit time can be obtained by measuring the average voltage. Thus, the average voltage
across JJ1 in Figure 3(a) is twice as large as that across JJ2 . In Figure 3(b), the average voltage
ratio for JJ1 to JJ2 becomes 3/2 (= 1.5).
The current (Ib )-average voltage (hV i) characteristics, calculated using the device parameters
described in Figure 3, are shown in Figure 4(a). These characteristics are those obtained by means
of quasi-static (low-bandwidth) measurements. Several step structures can been seen, which will
be explained later.
lb = 0.185 mA
JJ1
Voltage (0.2 mV/div)
Voltage (0.2 mV/div)
lb = 0.170 mA
0
JJ2
0
200
250
300
350
400
Time (ps)
450
500
JJ1
0
JJ2
0
200
250
300
350
400
Time (ps)
450
500
0.25
0.2
JJ1
JJ2
0.15
0.1
0.05
0
-0.05
0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22
lb (mA)
Voltage ratio for JJ 1 to JJ 2
Average voltage <V> (mV)
(a)
(b)
Figure 3: Numerical waveforms of the junction voltages. Ic , Rn , Cjj , Llp , and Rlp are 0.10 mA, 3.8 Ω,
0.22 pF, 20 pH, and 0.38 Ω, respectively. Absolute zero temperature is assumed. Ib is set at (a) 0.170 mA
and (b) 0.185 mA.
2
2/1
3/2
1.5
4/3
5/4
1
0.16
0.17
0.18
0.19 0.2
lb (mA)
0.21
0.22
(a)
(b)
Figure 4: (a) Current (Ib )-average voltage (hV i) characteristics across JJ1 and JJ2 . (b) The voltage ratio
for JJ1 to JJ2 plotted as functions of Ib .
Progress In Electromagnetics Research Symposium Proceedings, KL, MALAYSIA, March 27–30, 2012 771
The voltage ratio for JJ1 to JJ2 is plotted in Figure 4(b) as functions of Ib . The constant
voltage ratios of 2/1, 3/2, 4/3, and 5/4 can be confirmed. These voltage ratios directly correspond
to the ratios of the pulse numbers per unit time. That is, the blocking oscillation in the mode of
“2 switching events in JJ1 and 1 switching event in JJ2 ” appears as the voltage ratio of 2/1, that
of “3 in JJ1 and 2 in JJ2 ” appears as the voltage ratio of 3/2, and so on. The step structures in
Figure 4(a) correspond to the boundaries between the oscillation modes.
According to these results, it is possible to obtain an evidence of blocking oscillation in the
single-flux-quantum blocking oscillator by means of quasi-static measurements.
5. CONCLUSION
In this paper, the author presented the single-electron blocking oscillator and the single-fluxquantum blocking oscillator. Numerical simulation demonstrated the waveforms of blocking oscillation in both. Some electromagnetic duality was confirmed between the two blocking oscillators,
although it was not perfect. Since the blocking oscillation in the single-flux-quantum blocking
oscillator was deterministic and periodic, it would be confirmed by measuring quasi-static currentvoltage characteristics using conventional low-bandwidth equipments.
ACKNOWLEDGMENT
The author thanks Prof. H. Shimada, Dr. M. Moriya, Mr. K. Kuroiwa, and Mr. Y. Sato for their
technical support and fruitful discussion. This work was partially supported by the Asahi Glass
Foundation, and also VLSI Design and Education Center (VDEC), the University of Tokyo, in
collaboration with Cadence Design Systems, Inc.
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