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Resonance modes in one-dimensional parallel arrays of Josephson junctions
PHYSICA]
Physica B 194-196 (1994) 1779-1780 North-Holland
Resonance Modes in One-Dimensional Parallel Arrays of Josephson Junctions * H.S.J. van der Zant, ~ K.A. Delin, b R.D. Bock," D. Berman," J.R. Phillips, ~ and T.P. Orlando"
Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology , Cambridge , MA 02139, U.S.A. b Massachusetts Institute o] Technology, Lincoln Laboratory Lexington, MA 02173, U.S.A. We investigate both experimentally and numerically the dynamics of discrete one-dimensional parallel arrays of underdamped Josephson junctions. In a magnetic field, measurements show steps in the current-voltage characteristics which are the discrete analogs of Fiske steps in a long Josephson junction. From the position of the steps, one can construct a plot of the dispersion relation w(k). We observe a Sine-dependence in the dispersion relation due to the discrete nature of our arrays. We also observe an additional, smaller gap at a k-value determined by the periodicity of the vortex lattice. Our measurements are supported by numerical simulations of the full dynamics. The Fiske steps provide an experimental method to measure the self-inductance of 1D parallel arrays.
i I. I n t r o d u c t i o n
50
Resonance modes have extensively been investigated in continuous, long Josephson junctions [1]. In a magnetic field, Fiske [2] steps occur as current steps in the current-voltage characteristics at equally spaced voltage values. A onedimensional (1D) array of junctions connected in parallel by superconducting wires can be viewed as a discrete version of a long Josephson junction [3] with the ratio of the dosephson inductance Lj to the loop self-inductance L as the discreteness parameter. Recently, studies have been devoted to resonances in such arrays [4,5]. In this paper, we show :measurements of Fiske modes in discrete 1D arrays as illustrated in Fig. I. The most important difference from continuous long Josephson junctions is that the higher the step, the smaller the voltage difference between adjacent steps becomes. This is reminiscent of the discrete nature of our arrays and is connected to a sine-dependence in the dispersion relation w(k). A simple 1D discrete transmission line model ex*Part of this work was conducted under auspices of the Consortitun for Superconducting Electronics with partim support by the Defense Advanced Research Projects Agency (Contract No. MDA972-90-C-0021). The work at MIT was supported by the NSF through grant number DMR*910874B.
0921-4526/94/$07.00 © 1994 SSDI 0921-4526(93) 1521-M
-
T=7.65 K f=0.42
345 < 0
-50 I
-0.5
0
0.5
V (mY) Figure 1: Current-voltage characteristic of a 1D array ( N = 9) at T = 7.65 K and f=0.42.
plains some of the basic features of our data. We have also solved the full dynamics of 1D arrays with inclusion of the mutual inductances between all pairs of cells. The simulations show deviations from the transmission line model and agree with our experimental observations. II. M e a s u r e m e n t s We have fabricated 1D arrays with a selectiveniobium-anodization process (SNAP). The NbA12Ox-Nb junctions are underdamped and have defined areas of 4x4 or 2x2/~m 2. Measurements on single junctions show nearly ideal tunnel june-
Elsevier Science B.V. All rights reserved
1780
tion behavior. Our 1D arrays consist of 9 junctions ( g = 9) with a lattice parameter (p) of 12 #m. Current is fed into the array at the junction in the middle of the array. Measurements are performed in a He-4 probe, in which a magnetic field can be applied perpendicular to the junctions and the current loops. The magnetic field is given in units of the frustration parameter f, which is the applied flux per loop normalized to the flux q u a n t u m ¢o. In Fig. 1 we show the current-voltage characteristic (I-V) of an array with 2x2 p m 2 junctions measured at 7.65 K. The m a x i m u m voltage in this plot is well within the gap voltage. The junction normal-state resistance of this device is 58.5 fL One clearly sees the current steps in the 1-V of Fig. 1 and for this particular value of the magnetic field we see 6 steps in both the positive branch and negative branch. The first and second steps become visible for frustrations larger than 0.15. Only at high values of f , 6 and sometimes even 8 steps can be found at the same time. Steps occur for temperatures around 7.5 K. III. Transmission Line Model As a first intuitive approach to understand the step positions, we consider the dispersion relation of a 1D transmission line of inductors L and capacitors C. From current conservation at each node, one can calculate the dispersion curve for plane wave solutions,
w=2Wo]sin(?)]
(1)
where wo is the characteristic freqency k/1/(LC). By imposing the standing wave condition and writing frequencies in terms of voltages (V = ¢bow/(27r)), we find that resonances should occur at voltages V., -
~o
7r~
sin (mTr
\ 2N ]
(2)
where rn is an integer, m = 1,2, 3,...., N - 1. In Fig. 2, we plot the positions of the steps as shown in Fig. 1 versus kp. The drawn line is a fit to a continuous form of Eq. (2) and clearly shows the sine-dependence. Our measurements at f =
i
13 O " zx
300
i
i
data f-0.42 simulations f~0.46 data f=0.46 ~
T=7.65 K
200 100 i kp=nf
0 0
m 1
I
,
2
kp
, 3
Figure 2: Dispersion relations V(k) of discrete 1D arrays. The drawn line is a fit to Eq. (2) in a
continuous form. Open symbols give experimental results. 0.42 are in very good agreement with this model. From the fit we obtain a value of 1.9x10 -12 s for the ~ product. We estimate our capacitance to be 290 fF, so that L = 12.3 pH ~ 0.8#op and L j / L = 2.1 at this temperature. The open triangulars show the dispersion relation for steps measured at f = 0.46. There is a clear difference from the drawn line with possibly an additional gap at kp = 7rf indicated by the dashed line. At this k-value one expects to see a gap at the first Brillouin zone introduced by the periodicity (= p/f) of the vortex lattice. First results on simulations of the full dynamics show similar behavior and more research is underway to m a p out the dispersion relations for different inductance ratios and magnetic fields [6]. References Likharev, Dynamics of Josephson Junctions and Circuits, (Gordon and Breach,
[1] K.K.
1986). [2] M.D. Fiske, Rev. Mod. Phys. 36, 221 (1964). [3] K. Nakajima and Y. Ondera, J. Appl. Phys. 49, 2958 (1978). [4] H.S.J. van der Zant, E.H. Visscher, D.R. Curd, T.P. Orlando, and K.A. Delin, IEEE Trans Appl. Supercd., to be published. [5] A.V. Ustinov, M. Cirillo, and B.A. Malomed, Phys. Rev. B 47, 8357 (1993). [6] R.D. Bock, thesis, M I T (1993), unpublished.
Physica B 194-196 (1994) 1779-1780 North-Holland
Resonance Modes in One-Dimensional Parallel Arrays of Josephson Junctions * H.S.J. van der Zant, ~ K.A. Delin, b R.D. Bock," D. Berman," J.R. Phillips, ~ and T.P. Orlando"
Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology , Cambridge , MA 02139, U.S.A. b Massachusetts Institute o] Technology, Lincoln Laboratory Lexington, MA 02173, U.S.A. We investigate both experimentally and numerically the dynamics of discrete one-dimensional parallel arrays of underdamped Josephson junctions. In a magnetic field, measurements show steps in the current-voltage characteristics which are the discrete analogs of Fiske steps in a long Josephson junction. From the position of the steps, one can construct a plot of the dispersion relation w(k). We observe a Sine-dependence in the dispersion relation due to the discrete nature of our arrays. We also observe an additional, smaller gap at a k-value determined by the periodicity of the vortex lattice. Our measurements are supported by numerical simulations of the full dynamics. The Fiske steps provide an experimental method to measure the self-inductance of 1D parallel arrays.
i I. I n t r o d u c t i o n
50
Resonance modes have extensively been investigated in continuous, long Josephson junctions [1]. In a magnetic field, Fiske [2] steps occur as current steps in the current-voltage characteristics at equally spaced voltage values. A onedimensional (1D) array of junctions connected in parallel by superconducting wires can be viewed as a discrete version of a long Josephson junction [3] with the ratio of the dosephson inductance Lj to the loop self-inductance L as the discreteness parameter. Recently, studies have been devoted to resonances in such arrays [4,5]. In this paper, we show :measurements of Fiske modes in discrete 1D arrays as illustrated in Fig. I. The most important difference from continuous long Josephson junctions is that the higher the step, the smaller the voltage difference between adjacent steps becomes. This is reminiscent of the discrete nature of our arrays and is connected to a sine-dependence in the dispersion relation w(k). A simple 1D discrete transmission line model ex*Part of this work was conducted under auspices of the Consortitun for Superconducting Electronics with partim support by the Defense Advanced Research Projects Agency (Contract No. MDA972-90-C-0021). The work at MIT was supported by the NSF through grant number DMR*910874B.
0921-4526/94/$07.00 © 1994 SSDI 0921-4526(93) 1521-M
-
T=7.65 K f=0.42
345 < 0
-50 I
-0.5
0
0.5
V (mY) Figure 1: Current-voltage characteristic of a 1D array ( N = 9) at T = 7.65 K and f=0.42.
plains some of the basic features of our data. We have also solved the full dynamics of 1D arrays with inclusion of the mutual inductances between all pairs of cells. The simulations show deviations from the transmission line model and agree with our experimental observations. II. M e a s u r e m e n t s We have fabricated 1D arrays with a selectiveniobium-anodization process (SNAP). The NbA12Ox-Nb junctions are underdamped and have defined areas of 4x4 or 2x2/~m 2. Measurements on single junctions show nearly ideal tunnel june-
Elsevier Science B.V. All rights reserved
1780
tion behavior. Our 1D arrays consist of 9 junctions ( g = 9) with a lattice parameter (p) of 12 #m. Current is fed into the array at the junction in the middle of the array. Measurements are performed in a He-4 probe, in which a magnetic field can be applied perpendicular to the junctions and the current loops. The magnetic field is given in units of the frustration parameter f, which is the applied flux per loop normalized to the flux q u a n t u m ¢o. In Fig. 1 we show the current-voltage characteristic (I-V) of an array with 2x2 p m 2 junctions measured at 7.65 K. The m a x i m u m voltage in this plot is well within the gap voltage. The junction normal-state resistance of this device is 58.5 fL One clearly sees the current steps in the 1-V of Fig. 1 and for this particular value of the magnetic field we see 6 steps in both the positive branch and negative branch. The first and second steps become visible for frustrations larger than 0.15. Only at high values of f , 6 and sometimes even 8 steps can be found at the same time. Steps occur for temperatures around 7.5 K. III. Transmission Line Model As a first intuitive approach to understand the step positions, we consider the dispersion relation of a 1D transmission line of inductors L and capacitors C. From current conservation at each node, one can calculate the dispersion curve for plane wave solutions,
w=2Wo]sin(?)]
(1)
where wo is the characteristic freqency k/1/(LC). By imposing the standing wave condition and writing frequencies in terms of voltages (V = ¢bow/(27r)), we find that resonances should occur at voltages V., -
~o
7r~
sin (mTr
\ 2N ]
(2)
where rn is an integer, m = 1,2, 3,...., N - 1. In Fig. 2, we plot the positions of the steps as shown in Fig. 1 versus kp. The drawn line is a fit to a continuous form of Eq. (2) and clearly shows the sine-dependence. Our measurements at f =
i
13 O " zx
300
i
i
data f-0.42 simulations f~0.46 data f=0.46 ~
T=7.65 K
200 100 i kp=nf
0 0
m 1
I
,
2
kp
, 3
Figure 2: Dispersion relations V(k) of discrete 1D arrays. The drawn line is a fit to Eq. (2) in a
continuous form. Open symbols give experimental results. 0.42 are in very good agreement with this model. From the fit we obtain a value of 1.9x10 -12 s for the ~ product. We estimate our capacitance to be 290 fF, so that L = 12.3 pH ~ 0.8#op and L j / L = 2.1 at this temperature. The open triangulars show the dispersion relation for steps measured at f = 0.46. There is a clear difference from the drawn line with possibly an additional gap at kp = 7rf indicated by the dashed line. At this k-value one expects to see a gap at the first Brillouin zone introduced by the periodicity (= p/f) of the vortex lattice. First results on simulations of the full dynamics show similar behavior and more research is underway to m a p out the dispersion relations for different inductance ratios and magnetic fields [6]. References Likharev, Dynamics of Josephson Junctions and Circuits, (Gordon and Breach,
[1] K.K.
1986). [2] M.D. Fiske, Rev. Mod. Phys. 36, 221 (1964). [3] K. Nakajima and Y. Ondera, J. Appl. Phys. 49, 2958 (1978). [4] H.S.J. van der Zant, E.H. Visscher, D.R. Curd, T.P. Orlando, and K.A. Delin, IEEE Trans Appl. Supercd., to be published. [5] A.V. Ustinov, M. Cirillo, and B.A. Malomed, Phys. Rev. B 47, 8357 (1993). [6] R.D. Bock, thesis, M I T (1993), unpublished.