Stability of dynamic states in Josephson junctions

Physica 23D (1986) 68-71 North-Holland, Amsterdam

STABILITY OF DYNAMIC STATES IN J O S E P H S O N JUNCTIONS Peter L. CHRISTIANSEN Laboratory of Applied Mathematical Physics, The Technical University of Denmark, DK-2800 Lyngby, Denmark

Instabilities of dynamic states in linear Josephson junctions modelled by the perturbed sine-Gordon equation are investigated experimentally,computationally,and by stabilitytheory. The narrow line-widthof the electromagneticradiation from the circular Josephsonoscillatoris determinedcomputationaUyand by perturbation theory.

1. Introduction The present paper describes recent results concerning instability of dynamic states in the linear overlap Josephson junction obtained in ref. 1 and line width of the electromagnetic radiation from the circular Josephson oscillator obtained in refs. 2-3. In both cases the Josephson oscillator is modelled by the perturbed sine-Gordon equation [4] ¢~, - %-

s i n ¢ = a ¢ , - B C x . , - "Y

(1)

in normalized units. Here, ~(x, t) is the usual Josephson phase variable, x is the distance along the junction, and t is time. The term in a represents shunt loss due to quasiparticies crossing the junction, the term in 13 represents series loss due to surface resistance of the superconducting films, and 7 is the uniform bias current. For the linear oscillator we apply inhomogeneous Neumann boundary conditions

eG(O,t)=eG(l,t)=~,

(2a)

where 1 is the length of the junction, and ~ is the external magnetic field. In the circular oscillator periodic boundary conditions

dpx(0 , t) =

x(l, t),

t) =,,(t, t),

(2b)

1 being the circumference of the oscillator, are used. The detailed choice of initial conditions in the numerical solutions of (1)-(2) is described in [1-2].

2. The linear Josephson oscillator Fig. 1 shows a portion of an I - V characteristic obtained as the relationship between bias current, 7', and resulting average voltage on the junction, (~t(0, t)), by computational integration of eqs. (1) and (2a). For intermediate-length (1 < 5) Josephson junction we use an extension of the multimode theory developed by Enpuku et al. [5], which amounts to an expansion of the solutions of the perturbed sine-Gordon equation in a truncated series of time-dependent Fourier spatial components. This approach provides very accurate resuits at a lower computational cost than direct numerical solution of the mathematical model [6]. In fig. 1 we observe an instability for 3'- < ~/< "/+ or ~0_< (~t) < ~0+. In this region the dynamic state without spatial structure, ~ = ~ ( t ) , which corresponds to the McCumber branch (MCB) in the I - V characteristic, becomes unstable (to small perturbations). As a result a switching occurs to a dynamic state with a spatial one-soliton structure which corresponds to the first zero-field step (ZFS 1) in the I - V characteristic. The transition which may involve creation of breather-modes [7]

0167-2789/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

P.L. Christiansen/ Stabilityof dynamicstatesin Josephsonjunctions ZFS 1 0.6

I lrn/

MCB

I I

69

/o / I- /

- - - -,~.

>. I

I I

E, 0.4 I.IJ c¢" Or" SD U

/

I

i/

/ / oo/./

0.2

r--~--77t

J

of

J

o.j--~_ O - -

1

0

2

4-

50 p.v _

~, 1

~_ 1

1

1

4 6 8 10 AVERAGE VOLTAGE - < e t >

(a)

t-

i

12

Fig. 1. I-V characteristic calculation from eqs. (1) and (2a) using a=0.05, fl=0.02, l=2, and *1=0, showing the McCumber background curve (MCB) and the first zero-field step (ZFS 1). Inset showsin detail how ZFS 1 joins MCB.

ZFS 1

............ / "

Fig. 2 illustrates the corresponding experimental findings both at the first and second zero-field steps (ZFS 1 and ZFS 2). In the two cases one and two solitons respectively are travelling back and forth on the oscillator in different configurations

(3)

Here % ( 0 is the McCumber solution in the power-balance approximation [9]

% ( t ) = 2am [t/k; k],

I

(3b)

E ( k ) being the complete elliptic integral of second kind. The small perturbation ~(x, t) is given by ~ ( x , t) = y ( t ) exp (i bx)

(b) Fig. 2. (a) Detail of the I- V characteristic of Nb-NbxOv_Pb overlap junction (experimentalsample $6-7/4 with l = 4.4 and a = 0.006 at 4.2 K) measured at a temperature slightlybelow the transition temperature of the lead counter electrode and in zero magnetic field. Arrows indicateswitchingto gap state. (b) Same characteristic with 10 x -magnifiedcurrent scale. Dotted lines indicate switching from higher-voltage to lower-voltage states.

where overdots denote differentiation with respect to t. For small k ' s the stability boundaries for the average voltage, to, and to_, are determined approximately as solutions to the equations 1 to------7 8 )

(3c)

with b = n~r/l, n = 0,1, 2 . . . . . Insertion into eqs. (1) and (2a) yields the damped Hill's equation

Y + ( ~ + f l b 2 ) p + (b+c°Se~o(t)}y=O,

V

(3a)

where am is the Jacobian elliptic function of modulus k, and k satisfies

Y = 4aE(k)/~rk,

........

50 t~V

[81. A stability analysis is carried out for ,/---0 using

ZFS 2

1oopA

will be an interesting object to study by means of spectral methods.

~ ( x , t) = ~o(t) + ~ ( x , t).

v

W÷

(4)

4_ 1

_

1(1-8@)28¢o2

+

Bb2)]l/2 (5)

P.L. Christiansen / Stability of dynamic states in Josephson junctions

70 <¢

>

3. T h e circular Josephson oscillator

3

o

1

-0.5

I

0.0

I

0.5

1.0

Fig. 3. Stability boundaries for ZFS 1 in average voltage (q,t> as a function of magnetic field ~ measured experimentally (circles) and calculated from approximative equations for stability boundaries (solid curves). Parameter values: a = 0.026, l = 3.16 and 0 < fl < 0.07.

For small values of 7/ (4=0) the stability boundaries have been determined approximately by a generalization of the method leading to more complex equations than (5). Parmentier [10] has reduced the damped Hill's equation (4) to a Lam6 equation for which exact stability boundaries are given. Fig. 3 shows a comparison between the experimentally determined stability boundaries (circles) in a magnetic field associated with ZFS 1 and those obtained from approximative equations for stability boundaries (eq. (5) and its generalization for 7/4: 0), shown as solid curves. The effect of varying fl is indicated by the slight thickening of the curves. The instability region becomes smaller as/3 and ~/ are increased. The agreement between experimental and theoretical values for to+ is reasonable. The discrepancy for the to_ branch may be due to the fact that the approximations used in the theoretical expression are poorer for low values of the average voltage. In fig. 1 the numerical procedure predicts T + = 0.1712 and 7_=0.1401, while eq. (5) leads to T+ = 0.1711 and and T- = 0.1404. Thus, the agreement between the computational results and the results obtained by stability theory is very good.

So far, we have focussed on the instabilities of the dynamic states on the linear oscillator. In this section we shall demonstrate the extraordinary stability of the dynamic state in which one soliton rotates on the circular junction. Thermal noise or external microwave radiation is unable to perturb the soliton velocity much from the power-balance velocity predicted by perturbation theory [11]. As a result the electromagnetic radiation emitted by the oscillator has a very well-defined frequency. Experimentally, a line-width of less than 5 kHz at a resonance frequency of 10 GHz has been found. A relative accuracy of about 10 -8 is therefore required for computational line-width determinations. We have performed simulations of eqs. (1) and (2b) with this degree of accuracy on a CRAY1-S vector processor. In the case of microwave radiation, T in eq. (1) was replaced by "Y ~--- )tdc q- ~ a c

sin (I2t),

(la)

where )tdc is a constant, and Tac and I2 are amplitude and frequency of the microwave. In fig. 4 the solid curve is the resulting computational determination of the line-width as the standard deviation of the electromagnetic radiation frequency, or, versus microwave frequency, D. A perturbation theory using q~(x, t) = dpS(x, t) + ~ ( t ) ,

(6)

where q,S(x, t) is the travelling soliton and ~(t) is a small background, leads to the ordinary differential equation for ~(t) - q~- sin ~ = ctq) + Tdc + ")tae sin (~2t)

(7)

as well as a determination of the soliton velocity u(t). The soliton revolution time, T. (and the corresponding frequency, f . = 1/T.) has first been determined from the equation

2~rl ft t+ r"u(t) dt = 2~r,

(8a)

P.L. Christiansen/ Stability of dynamic states in Josephsonjunctions Standard deviation - Of

?

I0-2

the sine-Gordon model is an excellent testing ground for nonlinear phenomena in the sense that computational results and theoretical predictions can be verified by comparison with experimental measurements. In this paper we have only considered soliton dynamic states and their instabilities. Chaotic phenomena also occur on the Josephson junction. To predict these theoretically, MelnikovArnold techniques which apply to low-dimensional systems (see e.g. [12]) must be generalized to many-dimensional systems.

t

' t '" %,

1 0 -3

•

ii

lO-Z,

",°.. •.

l'~k \.

10-5 05

'.

\"x.."

f

"x "x '

1/ i I

•""""

:/

10 Driving frequency-~

71

I

15

Fig. 4. Standard deviation of electromagnetic radiation frequency, or, versus microwave frequency 12. Par~aaete~s in eqs. (1), (la) and (2b): a = 0.01, fl = 0, "/de= 0.02, `/ac= 0.01, and l = 8. Solid curve: computational result. Dashed-dotted curve: perturbative result from eq. (8a). Dotted curve: perturbative result from eq. (8b) using linearized version of eq. (7). Dashed curve: perturbative result from eq. (8b) using full eq. (7). Dotted and dashed curves overlap away from resonance region.

which yields the dashed-dotted curve in fig. 4 in poor agreement with the solid curve. Realizing that the main contribution to the line-width stems from the background radiation, ~(t), and from perturbations of the soliton velocity, we replace eq. (8a) by

not u(t),

2-~--~ftt+~u(t)dt+~(t+ T,)-~(t)=2~r

(8b)

and find the dotted curve, when a linearized version of eq. (7) is used. The hysteresis phenomenon is recovered (dashed curve in fig. 4), when the full eq. (7) is used. The level of the line-width is now predicted correctly by the perturbation theory while the location of the maximum is still predicted at a slightly too high frequency.

4. Conclusion The results presented in this paper demonstrate that the Josephson junction in combination with

Acknowledgements The financial support of the Danish Council for Scientific and Industrial Research and of the European Research Office of the United States Army through contract No. DAJA-45-85-C-0042 is acknowledged.

References [1] S. Pagano, M.P. Soerensen, R.D. Parmentier, P.L. Christiansen, O. Skovgaard, N.F. Pedersen, J. Mygind and M.R. Samuelsen, Phys. Rev. B 33 (1986) 174. [2] F. If, P.L. Christiansen, R.D. Parmentier, O. Skovgaard and M.P. Soerensen, Phys. Rev. B 32 (1985) 1512. [3] M. Fordsmand, P.L. Christiansen and F. If, Phys. Lett. A 116 (1986) 71. [4] A.C. Scott, F.Y.F. Chu and S.A. Reible, J. Appl. Phys. 47 (1976) 3272. [5] K. Enpuku, K. Yoshida and F. Irie, J. Appl. Phys. 52 (1981) 344. [6] M.P. Soereusen, R.D. Parmentier, P.L. Christiansen, O. Skovgaard, B. Dueholm, E. Joergensen, V.P. Koshelets, O.A. Levring, R. Monaco, J. Mygind, N.F. Pedersen and M.R. Samuelsen, Phys. Rev. B 30 (1984) 2640. [7] D.W. McLaughlin and N. Ercolani, private communication. [8] P.S. Lomdahl, O.H. Soerensen and P.L. Christiansen, Phys. Rev. B 25 (1982) 5737. [9] R.D. Parmentier, in Solitons in Action, K. Lonngren and A.C. Scott, eds. (Academic, New York, 1978) p. 173. [10] R.D. Parmentier, private communication. [11] D.W. McLaughlin and A.C. Scott, Phys. Rev. A 18 (1978) 1652. [12] M. Bartuccelli, P.L. Christiansen, N.F. Pedersen and M.P. Soerensen, Phys. Rev. B 33 (1986) 4686.