3479 Noise effects and thermally induced switching errors in Josephson junctions Andrey L. Pankratov 1 and Bernardo Spagnolo 2 1 Institute for Physics of Microstructures of RAS, Nizhny Novgorod, Russia 2 INFM and Dipartimento di Fisica e Tecnologie Relative, Group of Interdisciplinary Physics, Università di Palermo, Viale delle Scienze, I-90128 Palermo, Italy We present an analytical and numerical analysis of influence of fluctuations and periodic driving on temporal characteristics of an overdamped Josephson junction. We obtain the mean switching time and its standard deviation in the presence of a dichotomous driving force for arbitrary noise intensity and in the frequency range of practical interest. The presented analysis allows to choose the proper range of parameters to minimize noise-induced errors in RSFQ devices. INTRODUCTION The influence of thermal fluctuations on time characteristics of Josephson junction has been subject of earlier theoretical and experimental investigations [1 3]. Recently an increasing attention is devoted to the noise induced switching errors in short overdamped Josephson junctions, due to their broad applications in logic devices, as rapid single flux quantum (RSFQ) for example [4]. The reproduction of quantum pulses due to spasmodic changing by 2π of the phase difference of overdamped Josephson junctions is the main mechanism in RSFQ devices [5]. In addition the operating temperatures offered by high-t c superconductors lead to higher noise levels and increases the occurrence of thermally induced errors [6]. As a consequence the major restriction in the development of a working RSFQ logic circuits is given by influence of fluctuations on pulse jitter and timing errors in RSFQ circuits [7]-[10]. Three different types of digital errors may be identified. (i) Storage errors, due to the passive storage of data in quantizing loops of RSFQ cells [11]. (ii) Switching errors, produced in two junction comparator of RSFQ circuits, due to a finite probability that the wrong junction flips [12]. (iii) Timing errors due to thermal-induced fluctuations of the time interval between input and output pulses of an RSFQ circuit operating at high speed. The effect of fluctuations on timing errors has been studied in the linearized overdamped JJ model [7]. However, the considered system is significantly nonlinear, and such interesting nonlinear phenomenon as noise-induced increase of the switching time has been observed in the same system, when the input pulse has the rectangular form [11]. In a nonlinear system with a metastable state, the resonant activation [13]-[15] and noise enhanced stability phenomena [16]-[19] may be observed. These noise-induced effects through the nonlinearity of the system may play both positive and negative role in accumulation of fluctuational errors in RSFQ logic devices. A positive role is played by the resonant activation phenomenon, while negative effects are due to the noise-induced increase of the switching time, or noise enhanced stability phenomenon [16]-[19]. These effects however were not observed in refs. [7],[12], due to the approach based on linearized models. Moreover with the technical realization of the 16-bit
3480 20GHz RSFQ microprocessor prototype, there remains unsolved questions about limiting frequencies of RSFQ devices and possible optimizations in order to increase working frequencies and reduce pulse jitter and timing errors in RSFQ circuits. The aim of the present paper is to analyze noise-induced effects in order to minimize timing errors and to help experimentalists and designers to construct RSFQ circuits optimized for high frequency operation. An analytical and numerical analysis of influence of fluctuations and dichotomous signal on temporal characteristics of the overdamped Josephson junction is carried out. In particular, the analytical expression of standard deviation of switching time works in the frequency range of practical interest and for arbitrary noise intensity. THE MODEL Processes going on in a single JJ of a small size under a current I in the presence of fluctuations are well described by Langevin equation. Let us restrict our consideration to JJs with high damping β 1, which are widely used in logic elements with high-speed switching. Here β =2eI c RN 2 C/ h is the McCamber Stewart parameter, I c is the critical current, R 1 N = G N is the normal conductivity of a JJ, C is the capacitance, e is the electron charge and h is the Planck constant. In this case the Langevin equation takes the following form where ωc 1 dϕ(t) = du(ϕ) dt dϕ i F (t), (1) u(ϕ) =1 cos ϕ iϕ, i = i 0 + f(t). (2) is the dimensionless potential profile, ϕ is the difference in the phases of the order parameter on opposite sides of the junction, i = I/I c, i F (t) =I F /I c, I F is the random component of the current and ω c = 2eR N I c / h is the characteristic frequency of the JJ. When only thermal fluctuations are taken into account [4], the random current may be represented by white Gaussian noise i F (t) =0, i F (t)i F (t + τ) = 2γ δ(τ), ω c where γ =2ekT/( hi c )=I T /I c is the dimensionless intensity of fluctuations, T is the temperature and k is the Boltzmann constant. For the description of our system, i. e. a single overdamped JJ with noise, we will use the Fokker-Planck equation for the probability density W (ϕ, t) that corresponds to Langevin equation (1) W(ϕ, t) t { } G(ϕ, t) du(ϕ) W(ϕ, t) = = ω c W (ϕ, t)+γ. (3) ϕ ϕ dϕ ϕ
3481 The initial and boundary conditions of the probability density W (ϕ, t) and of the probability current G(ϕ, t) for the potential profile (2) are: W (ϕ, 0) = δ(ϕ ), W (+,t)=0,g(,t)=0. Let us consider the following setup of the problem: let, initially, the JJ is biased by the current smaller than the critical one, that is i 0 < 1, and the junction is in the superconductive state. The current pulse f(t), such that i(t) =i 0 + f(t) > 1, switches the junction into the resistive state. However, an output pulse will appear not immediately, but at the later time. Such a time is called the turn-on delay time or the switching time between the input, i. e. a current pulse leading to i>1, and the output voltage pulse [3, 4]. If noise exists in the system, the switching time is a random quantity, and may be characterized by its mean and standard deviation. As an example of a driving with sharp fronts we will consider the meander-like (dichotomous) signal f(t) = Asign(sin(ωt)). THE MAIN RESULTS Our aim is to investigate the following temporal characteristics: the mean switching time (MST) and its standard deviation (SD). These quantities may be introduced as characteristic scales of the evolution ϕ 2 of the probability P (t) = W (ϕ, t)dϕ, to find the phase within one period of the potential profile of Eq. (2). We choose therefore ϕ 2 = π, = π and we put the initial distribution on the bottom of a potential well: = arcsin(i 0 ). A widely used definition of such characteristic time scales is the integral relaxation time, see the review [15] and references therein. The mean switching time τ = t may be introduced in the form τ = t = 0 tw(t)dt = where the probability density of switching time w(t is: [P (t) P ( )]dt 0 [P (0) P ( )], (4) and the SD of the switching time is w(t) = 1 [P ( ) P (0)] P(t), (5) t σ = t 2 t 2, t 2 = 0 t 2 w(t)dt. (6) The results of computer simulations are shown in Fig. 1. Both MST and its SD does not depend on the driving frequency below a certain cut-off frequency, above which the characteristics degrade. In the frequency range from 0 to 0.2ω c therefore we can describe the effect of dichotomous driving by time characteristics in a constant potential. The exact analytical expression, as well as asymptotic representation of the MST τ c ( ) in a time-constant potential has been obtained in [11]. For an arbitrary γ we have
3482 τ(ω), σ(ω) 6.0 τ(ω), γ=0.2 τ(ω), γ=0.02 4.0 σ(ω), γ=0.2 2.0 σ(ω), γ=0.02 0.0 0.01 0.1 1 ω FIG. 1: Semilog plot of the mean switching time τ (solid line) and standard deviation σ(diamonds and circles) as a function of the frequency ω. The dashed line represents the Eq.(12). and for γ 1 τ c ( )= t = 1 γω c ϕ 2 x e u(x)/γ e u(ϕ)/γ dϕdx+ ϕ 2 e u(ϕ)/γ dϕ ϕ 2 e u(ϕ)/γ dϕ, (7) τ c ( )= 1 ω c {f 1 (ϕ 2 ) f 1 ( )+γ [f 2 (ϕ 2 )+f 2 ( )] +...}, (8) where ( 2 i tan(x/2) 1 f 1 (x) = i2 1 arctan i2 1 ),f 2 (x) = 1 2(i sin x) 2. (9) Using the approach of ref. [15], the exact expression for second moment of the switching time τ 2c ( )= t 2 in a time-constant potential, which corresponds also to a single unit-step pulse, may be derived as τ 2c ( )=τ 2 c ( ) 2 (γω c ) 2 ϕ 2 ϕ 2 e u(x)/γ e u(v)/γ e u(y)/γ e u(x)/γ dx x v y ϕ 2 e u(v)/γ e u(y)/γ v y e u(z)/γ dzdydvdx+ (10) e u(z)/γ dzdydv, (11)
3483 3.0 σ(γ) 2.0 1.0 0.0 0.001 0.01 0.1 1 γ FIG. 2: The asymptotic standard deviation (Eq.(12)) as a function of the noise intensity for two values of the current in the time-constant case (i >1). Theoretical results are compared with numerical simulations: (a) i=1.5, solid line (theory), circles (simulation); (b) i=1.2, dashed line (theory), diamonds (simulation). where τ c ( ) is given by (7). The asymptotic expression of σ = τ 2c τc 2 in the small noise limit γ 1 is σ( )= 1 ω c 2γ [F (ϕ2, )+f 3 (ϕ 2, )] +..., (12) and scales as the square root of noise intensity. Here and F (ϕ 2, )=f 1 (ϕ 2 )f 2 (ϕ 2 ) 2f 1 ( )f 2 ( )+f 1 ( )f 2 ( )+ f 1(ϕ 2 ) f 1 ( ) (i sin( )) 2, (13) f 3 (ϕ 2, )= ϕ 2 [ ] cos(x)f1 (x) (sin(x) i) 3 3 2(sin(x) i) 3 dx. (14) The comparison of the approximate expression of σ (Eq.(12)) with the results of computer simulation, is presented in Fig. 2 for i =1.2 andi =1.5. We can see that formula (12) works rather well up to γ = 0.05. Not only low temperature devices (γ 0.001, see [7]), but also high temperature devices therefore may be described by formulas (8) and (12). It is important to mention that, since the largest
3484 τ(i), σ(i) 100.00 10.00 τ(i), γ=0.001 1.00 σ(i), γ=0.001 0.10 0.01 1 1.2 1.4 1.6 1.8 2 i FIG. 3: The mean switching time and the standard deviation as a function of the bias current for the time-constant case when γ = 0.001. Theoretical results given by Eqs. (8) and (12 (solid lines), versus computer simulations (diamonds and circles). contribution to the MST comes from the deterministic term τ c ( )= 1 ω c [f 1 (ϕ 2 ) f 1 ( )], the low noise limit formula (8) gives actually the same results as presented in the linear approach. However the formula (12) in some cases significantly deviates from the results of linearized calculations [7]. Considering the case γ =0.001, authors of ref. [7] have obtained σ =0.4ω 1 c but for larger current i =1.5 the discrepancy is larger: σ =0.06ω 1 c for i =1.2, while we get σ =0.436ω 1 c, in [7], and we get σ =0.14ω 1 c. The dependencies of the MST and its SD on the bias current are presented in Fig. 3 for γ =0.001. If the noise intensity is rather large, the phenomenon of noise enhanced stability may be observed in our system: MST increases with the noise intensity, as it may be easily seen from Eq. (8) [11]. Here we only note that it is very important to consider this effect in the design of large arrays of RSFQ elements, operating at high frequencies. To neglect the noise induced effects in such nonlinear devices it may lead to malfunctions due to the accumulation of errors. CONCLUSIONS In the present paper an analytical and numerical analysis of influence of fluctuations and periodic driving on temporal characteristics of the Josephson junction is performed. For dichotomous driving force we obtain analytically the standard deviation of switching time for arbitrary noise intensity. This expression works in the frequency range of practical interest. As a conclusion from our study we obtain that maximal clock frequencies may be reached together with minimal timing errors: clock frequencies
3485 may be maximized, if the bias current will be increased and kept close to the critical one, but keeping small storage errors as it has been estimated in ([7]). Storage errors are acceptably small up to i 0 =0.99 for γ =0.001. In this case the noise delayed switching effect is minimized and the standard deviation of switching time also decreases. Concerning applications in RSFQ elements, the presented analysis allows to choose proper range of parameters to minimize noise-induced error. This suppression of thermally induced errors should be very important in view of the possible use of RSFQ electronic devices in readout electronics for quantum computing [20, 21]. ACKNOWLEDGMENTS The work has been supported by INTAS Grants 2001-0450 and 2001-0367, by MIUR, by INFM, by Russian Foundation for Basic Research (projects 02-02-16775, 03-02-16533, 02-02-17517, and 03-02-06343), by the grant SS-1729.2003.2, and by the grant from BRHE and SOC PSNS NNSU. alp@ipm.sci-nnov.ru; spagnolo@unipa.it; http://gip.dft.unipa.it [1] Vinay Ambegaokar and B. I. Halperin, Phys. Rev. Lett., 22, 1364 (1969). [2] W. H. HenkelsandW. W. Webb, Phys. Rev. Lett., 26, 1164 (1971). [3] A. Barone and G. Paterno, Physics and Applications of the Josephson Effect, Wiley, 1982. [4] K. K. Likharev, Dynamics of Josephson Junctions and Circuits (Gordon and Breach, New York, 1986). [5] K. K. Likharev and V. K. Semenov, IEEE Trans. Appl. Supercond., 1, 3 (1991). [6] J. Satchell, IEEE Trans. Appl. Supercond., 9, 3841 (1999). [7] A. V. Rylyakov and K. K. Likharev, IEEE Trans. Appl. Supercond., 9, 3539 (1999). [8] Q. P. Herr, M. W. Johnson and M. I. Feldman, IEEE Trans. Appl. Supercond., 9, 3594 (1999). [9] V. Kaplunenko and V. Borzenets, IEEE Trans. Appl. Supercond., 11, 288 (2001). [10] T. Ortlepp, H. Toepfer and H. F. Uhlmann, IEEE Trans. Appl. Supercond., 11, 280 (2001). [11] A. N. Malakhov, and A.L. Pankratov, Physica C, 269, 46 (1996). [12] T. J. Walls, T. V. Filippov, and K. K. Likharev, Phys. Rev. Lett., 89, 217004 (2002). [13] A. L. Pankratov, and M. Salerno, Phys. Lett. A 273, 162 (2000). [14] R.N. Mantegna, and B. Spagnolo, Phys. Rev. Lett. 84, 3025 (2000). [15] A. N. Malakhov, and A. L. Pankratov, Adv. Chem. Phys. 121, 357 (2002). [16] R.N. Mantegna, and B. Spagnolo, Phys. Rev. Lett. 76, 563 (1996). [17] N. V. Agudov, and B. Spagnolo, Phys. Rev. E 64, 035102(R) (2001). [18] A. Fiasconaro, D. Valenti, and B. Spagnolo, Physica A 325, 136 (2003). [19] N. V. Agudov, A. A. Dubkov, and B. Spagnolo, Physica A 325, 144 (2003). [20] P. Rott and M. I. Feldman, IEEE Trans. Appl. Supercond., 11, 1010 (2001). [21] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, UK, 2000).