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Enhanced quantum interference effects in normal and superconducting arrays
Superlattices
and Microstructures,
159
Vol. 17, No. 2. 1992
ENHANCED QUANTUM INTERFERENCE
EFFECTS IN NORMAL AND SUPERCONDUCTING
J. Huang, J.H. Miller.Jr.,
ARRAYS
T.D. Gelding, and G.H. Gunaratne
Department of Physics, Space Vacuum Epitaxy Center. and Texas Center for Superconductivity University of Houston-University Park 3800 Calhoun Road IHoustrm, TX 77203-5507 U.S.A. (Received 19 May 1991)
We have generalized the Aharanov-Bohm effect from a single loop to a twodimensional rectangular network containing MxN loops. each of equal area. For a given geometry, as the total number of loops increases, the periodic principal maxima m the t’lux-dependent conductance exhihit a spacing ol h/e per loop while becoming increasingly narrow and pronounced, and we observe M-2 subsidiary maxima between consecutive principal maxima. We predict a similar enhancement and narrowing of the principal maxima in the flux-dependent crttical current of a multi-junction SQUID array, although in this case the llux periodicity is N2e.
1. Introduction The Aharonov-Bohm (AB) effect has been the subject of research since its conception in 1959 [ 11. Both h/e and h/2e periodicitv in the magnetoconductance has been observed in normal metal loops, and the superconducting auantum interference device (SQUID) is a well know manifestation of the AB effect invol&g josephson junctions or weak links. Both of these systems can be considered to be analogous to the optical two slit interference effect. Recently the possibility that quantum interference effects can result from multiple paths of the electron wavefunction has been proposed [2] as an interpretation of the experimentally observed universal conductance fluctuations in disordered mesoscopic systems [3]. While universal conductance fluctuations result from interference between random paths, the advent of nanolithography will allow the possibility of controlled fabrication of devices exploiting quantum interference effects in ordered systems. For this reason, we have conducted a theoretical study of quantum interference effects in normal and superconducting ordered arrays. 2. AB effect in two-dimensional metal arrays
M
normal
Without losing generality, we simplify the geometry of an Aharonov-Bohm ring into 2 parallel conductor channels as showed in Fig. 1. Following conventional
analysis, and choosing the gauge A as indicated, we relate the wavefunction at 1, to the wavefunction at 2, given by
where ‘+’ and ‘-’ refer to top and bottom channels respectively, and where 6=2x0/00, @o=e/hc and O=BS, with S the area of the ring.
0749-6036/92/020159+04$02.00/0
The electron transmission magnitude is calculated as a function of applied magnetic flux for a two terminal two dimensional (NxM) network configured as shown in figure l(c). We account for all possible paths throughout the network, and account for multiple reflections with the phase shift given along each path according to (l), assuming that the electron starts at one side of network and terminates at the other side.
c)
It2
d)
lEl3 N
Figure 1: (a) Aharonov-Bohm rmg; (b) the simplified model used in calculation; (c) Schematic of two terminal (NxM) network; (d) the notation for the transmission and reflection coefficients for the various junction geometries.
0 1992 Academic
Press Limited
Superlattices
160
and
Microstructures,
Vol.
17, No.
2, 7 992
I
81 L
3x3
!
r:
6.
0
0.2
0.4
0.6
0.3
OL
0
1
0
t=0.72
+
t-0.52
0.2
c
0.8
0.6
0.8
1.0
P
5x5
200 -
0.6
0.4
P
5’
t=0.68
+
t=0.48
0
t=0.28
0.4
1.0
0.6
0.8
@,a 1
Figure 2: Transmission magnitude for (2x2), (3x3). (4x4), and (5x5) networks showing dependence of interference on transmission coefficient of constituent junction
We have computed the transmission magnitude for various values of N and M, but restricted our analysis to networks with M,N ~10. In addition, the transmission magnitude has been computed for all transmission and reflection coefficients subject to the normalization constraint CT+R=~. In figure 2 we show the magnetic flux-dependent transmission magnitude for (2x2), (3x3), (4x4). and (5x5) networks for various junction transmission and reflection
of the transmtssion and reflection coefficients. It can be seen from figure 3 that, as the size of the network increases, the peak transmission magnitude becomes increasingly sensitive to the particular values of the transmission and reflection coefficients. We have also calculated the effect of complex transmission and reflection coefficients,in the form t’=ltleib and r’=-lrleih. Figure 4 shows the intensity of the
coefficients. For this analysis we have taken &O. According to the results observed in figure 2, for a given geometry, as the total number of loops increases, the periodic principal maxima in the flux-dependent conductance exhibit a spacing of h/e per loop while becoming increasingly narrow and pronounced, and we observe M-2 subsidiary maxima between consecutive principal maxima. We also observe that the magnetic flux-dependent conductance is strongly affected by the transmission coefficients of the constituent junctions in the network. This effect is clearly observed in figure 3 where we have plotted the magnitude of each principle maximum, obtained from its respective flux-dependent transmission graph, as a function
find that while the intensity
wavefunction
for a (3x3) network
the periodicity
Multiple
of 6. We
of the peaks are dependent
of the peaks position
3.
as a function
Junction
is independent
on 6,
of 6.
SQUID’s
A similar enhancement of quantum interference effects is predicted for suitably designed Josephson junction (JJ) arrays. Previous studies of JJ arrays [4] show a rather complicated flux-dependence of the critical currents. However, sharply-defined maxima in the flux-dependent critical current can be obtained by eliminating the JJ’s from the horizontal interconnects.
Superlattices
and Microstructures,
161
Vol. 17, No. 2, 1992
r
i
L
8X8 2E+3
r
Figure 3: Graphs of principle maximum for (2x2) (4x4), and (8x8) networks, as a function of the (6x6), transmission and reflection coefficients.
Consider
the N-loop
SQUID
illustrated
in figure
5,
with N+l junctions. In the figure, & represents the phase difference between the complex superconducting order parameters on opposite sides of the Josephson junction, or weak link, on the left-hand side of the parallel array. In the
-2
Figure
4. Intensity
UI
36
of the wavefunction
for W@O =O.O, 0.1 and 0.5 respectively
38
1
as a function
low inductance
limit, the flux @i through
to the externally
applied
current,
IcT,
6 ml
current
I
of 6
restricted generality.
each loop is equal
flux 0,. In this case. the total critical
is obtained
is maximized. to the range For a given &
by adjusting
Lin such that the total
The phase difference -TC 5 60
5 K
&
without
, the total current I is
can be loss of
given by
162
Superlattices
Figure
5.
(N+l junction interferometer
'T
and
Microstructures,
Vol.
7 I, No.
2, 1992
principal maxima in the flux-dependent critical current, as compared which that of a conventional two-junction SQUID, which is similar to the enhanced flux-dependent aansmission magnitude predicted for the normal metal array. We have predicted similar results when the effects of finite self- and mutual inductances and nonuniformity of the junction critical currents are taken into account. A current-biased multiSQUID is expected to exhibit sharp minima in the voltageflux characteristics- a feature which would enable this device to measure well-spaced, precisely-defined, quantized values of magnetic field. Furthermore, the multi-junction SQUID may offer significant advantages for high temperature operation, where the inductance of a conventional dc SQUlD is required to be low. 4. Conclusion
k&a
In this paper we have predicted a significant enhancement of quantum interference effects in suitably designed normal metal and JJ artays. For the case of normal metal arrays, we find that the transmission magnitude is strongly dependent on the transmission coefficients of the constituent junctions in the network. This, we believe, should be an important consideration when such networks arc fabricated in an attempt to experimentally observe enhanced quantum interference effects. Figure 6: Critical currents of the 12-junction SQUID
N
I
=
c
f,.sin(&-5)
,
n=o
where Ic,, represents the critical current of the & junction. The results of this calculation for a 12-junction SQUID with uniform critical currents are plotted in figures 6. In the figure, we see an enhancement and narrowing of the
References 1. Y. Aharonov and D. Bohm Phys. Rev. Let?. 115 485 (1959). 2. P.A. Lee and A.D. Stone Phys. Rev. Len. 55 1622 (1985). 3. C. P. Umbach, S. Washburn, R. B. Laibowitz, and R. R. A. A. Webb, Phys. Rev. B 30, 4048 (1984); Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, Phys. Rev. Lerr. 54, 2696 (1985); S. Washburn, C. P. Umbach, R. B. Laibowitz, and R. A. Webb, Phys. Rev. B 32, 4789 (1985). 4. S. P. Benz, M. S. Rzchowski, M. Tinkham, and C. J. Lobb, Phys. Rev. B 42, 6165 (1990).
and Microstructures,
159
Vol. 17, No. 2. 1992
ENHANCED QUANTUM INTERFERENCE
EFFECTS IN NORMAL AND SUPERCONDUCTING
J. Huang, J.H. Miller.Jr.,
ARRAYS
T.D. Gelding, and G.H. Gunaratne
Department of Physics, Space Vacuum Epitaxy Center. and Texas Center for Superconductivity University of Houston-University Park 3800 Calhoun Road IHoustrm, TX 77203-5507 U.S.A. (Received 19 May 1991)
We have generalized the Aharanov-Bohm effect from a single loop to a twodimensional rectangular network containing MxN loops. each of equal area. For a given geometry, as the total number of loops increases, the periodic principal maxima m the t’lux-dependent conductance exhihit a spacing ol h/e per loop while becoming increasingly narrow and pronounced, and we observe M-2 subsidiary maxima between consecutive principal maxima. We predict a similar enhancement and narrowing of the principal maxima in the flux-dependent crttical current of a multi-junction SQUID array, although in this case the llux periodicity is N2e.
1. Introduction The Aharonov-Bohm (AB) effect has been the subject of research since its conception in 1959 [ 11. Both h/e and h/2e periodicitv in the magnetoconductance has been observed in normal metal loops, and the superconducting auantum interference device (SQUID) is a well know manifestation of the AB effect invol&g josephson junctions or weak links. Both of these systems can be considered to be analogous to the optical two slit interference effect. Recently the possibility that quantum interference effects can result from multiple paths of the electron wavefunction has been proposed [2] as an interpretation of the experimentally observed universal conductance fluctuations in disordered mesoscopic systems [3]. While universal conductance fluctuations result from interference between random paths, the advent of nanolithography will allow the possibility of controlled fabrication of devices exploiting quantum interference effects in ordered systems. For this reason, we have conducted a theoretical study of quantum interference effects in normal and superconducting ordered arrays. 2. AB effect in two-dimensional metal arrays
M
normal
Without losing generality, we simplify the geometry of an Aharonov-Bohm ring into 2 parallel conductor channels as showed in Fig. 1. Following conventional
analysis, and choosing the gauge A as indicated, we relate the wavefunction at 1, to the wavefunction at 2, given by
where ‘+’ and ‘-’ refer to top and bottom channels respectively, and where 6=2x0/00, @o=e/hc and O=BS, with S the area of the ring.
0749-6036/92/020159+04$02.00/0
The electron transmission magnitude is calculated as a function of applied magnetic flux for a two terminal two dimensional (NxM) network configured as shown in figure l(c). We account for all possible paths throughout the network, and account for multiple reflections with the phase shift given along each path according to (l), assuming that the electron starts at one side of network and terminates at the other side.
c)
It2
d)
lEl3 N
Figure 1: (a) Aharonov-Bohm rmg; (b) the simplified model used in calculation; (c) Schematic of two terminal (NxM) network; (d) the notation for the transmission and reflection coefficients for the various junction geometries.
0 1992 Academic
Press Limited
Superlattices
160
and
Microstructures,
Vol.
17, No.
2, 7 992
I
81 L
3x3
!
r:
6.
0
0.2
0.4
0.6
0.3
OL
0
1
0
t=0.72
+
t-0.52
0.2
c
0.8
0.6
0.8
1.0
P
5x5
200 -
0.6
0.4
P
5’
t=0.68
+
t=0.48
0
t=0.28
0.4
1.0
0.6
0.8
@,a 1
Figure 2: Transmission magnitude for (2x2), (3x3). (4x4), and (5x5) networks showing dependence of interference on transmission coefficient of constituent junction
We have computed the transmission magnitude for various values of N and M, but restricted our analysis to networks with M,N ~10. In addition, the transmission magnitude has been computed for all transmission and reflection coefficients subject to the normalization constraint CT+R=~. In figure 2 we show the magnetic flux-dependent transmission magnitude for (2x2), (3x3), (4x4). and (5x5) networks for various junction transmission and reflection
of the transmtssion and reflection coefficients. It can be seen from figure 3 that, as the size of the network increases, the peak transmission magnitude becomes increasingly sensitive to the particular values of the transmission and reflection coefficients. We have also calculated the effect of complex transmission and reflection coefficients,in the form t’=ltleib and r’=-lrleih. Figure 4 shows the intensity of the
coefficients. For this analysis we have taken &O. According to the results observed in figure 2, for a given geometry, as the total number of loops increases, the periodic principal maxima in the flux-dependent conductance exhibit a spacing of h/e per loop while becoming increasingly narrow and pronounced, and we observe M-2 subsidiary maxima between consecutive principal maxima. We also observe that the magnetic flux-dependent conductance is strongly affected by the transmission coefficients of the constituent junctions in the network. This effect is clearly observed in figure 3 where we have plotted the magnitude of each principle maximum, obtained from its respective flux-dependent transmission graph, as a function
find that while the intensity
wavefunction
for a (3x3) network
the periodicity
Multiple
of 6. We
of the peaks are dependent
of the peaks position
3.
as a function
Junction
is independent
on 6,
of 6.
SQUID’s
A similar enhancement of quantum interference effects is predicted for suitably designed Josephson junction (JJ) arrays. Previous studies of JJ arrays [4] show a rather complicated flux-dependence of the critical currents. However, sharply-defined maxima in the flux-dependent critical current can be obtained by eliminating the JJ’s from the horizontal interconnects.
Superlattices
and Microstructures,
161
Vol. 17, No. 2, 1992
r
i
L
8X8 2E+3
r
Figure 3: Graphs of principle maximum for (2x2) (4x4), and (8x8) networks, as a function of the (6x6), transmission and reflection coefficients.
Consider
the N-loop
SQUID
illustrated
in figure
5,
with N+l junctions. In the figure, & represents the phase difference between the complex superconducting order parameters on opposite sides of the Josephson junction, or weak link, on the left-hand side of the parallel array. In the
-2
Figure
4. Intensity
UI
36
of the wavefunction
for W@O =O.O, 0.1 and 0.5 respectively
38
1
as a function
low inductance
limit, the flux @i through
to the externally
applied
current,
IcT,
6 ml
current
I
of 6
restricted generality.
each loop is equal
flux 0,. In this case. the total critical
is obtained
is maximized. to the range For a given &
by adjusting
Lin such that the total
The phase difference -TC 5 60
5 K
&
without
, the total current I is
can be loss of
given by
162
Superlattices
Figure
5.
(N+l junction interferometer
'T
and
Microstructures,
Vol.
7 I, No.
2, 1992
principal maxima in the flux-dependent critical current, as compared which that of a conventional two-junction SQUID, which is similar to the enhanced flux-dependent aansmission magnitude predicted for the normal metal array. We have predicted similar results when the effects of finite self- and mutual inductances and nonuniformity of the junction critical currents are taken into account. A current-biased multiSQUID is expected to exhibit sharp minima in the voltageflux characteristics- a feature which would enable this device to measure well-spaced, precisely-defined, quantized values of magnetic field. Furthermore, the multi-junction SQUID may offer significant advantages for high temperature operation, where the inductance of a conventional dc SQUlD is required to be low. 4. Conclusion
k&a
In this paper we have predicted a significant enhancement of quantum interference effects in suitably designed normal metal and JJ artays. For the case of normal metal arrays, we find that the transmission magnitude is strongly dependent on the transmission coefficients of the constituent junctions in the network. This, we believe, should be an important consideration when such networks arc fabricated in an attempt to experimentally observe enhanced quantum interference effects. Figure 6: Critical currents of the 12-junction SQUID
N
I
=
c
f,.sin(&-5)
,
n=o
where Ic,, represents the critical current of the & junction. The results of this calculation for a 12-junction SQUID with uniform critical currents are plotted in figures 6. In the figure, we see an enhancement and narrowing of the
References 1. Y. Aharonov and D. Bohm Phys. Rev. Let?. 115 485 (1959). 2. P.A. Lee and A.D. Stone Phys. Rev. Len. 55 1622 (1985). 3. C. P. Umbach, S. Washburn, R. B. Laibowitz, and R. R. A. A. Webb, Phys. Rev. B 30, 4048 (1984); Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, Phys. Rev. Lerr. 54, 2696 (1985); S. Washburn, C. P. Umbach, R. B. Laibowitz, and R. A. Webb, Phys. Rev. B 32, 4789 (1985). 4. S. P. Benz, M. S. Rzchowski, M. Tinkham, and C. J. Lobb, Phys. Rev. B 42, 6165 (1990).