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A general analytic approach to the study of the electrodynamic properties of d.c. SQUIDs
26 February 2001
Physics Letters A 280 (2001) 209–214 www.elsevier.nl/locate/pla
A general analytic approach to the study of the electrodynamic properties of d.c. SQUIDs R. De Luca DIIMA, Università degli Studi di Salerno, I-84084 Fisciano (SA), Italy Received 12 September 2000; received in revised form 18 January 2001; accepted 23 January 2001 Communicated by A.P. Fordy
Abstract A general analytic approach to the study of the dynamical properties of Josephson junctions in d.c. SQUIDs is developed. The periodic properties of the observable electrodynamic quantities are rigorously derived by means of elementary linear algebra concepts and the case of d.c. SQUIDs containing one or two π-junctions is discussed. 2001 Elsevier Science B.V. All rights reserved. PACS: 74.50.+r; 85.25.Dq
1. Introduction SQUIDs (superconducting quantum interference devices) have been thoroughly studied in the literature [1,2]. These systems find application in a large variety of fields. Their still limited use as non-invasive diagnostic instruments [3–5] or as non-destructive monitoring devices [6] is to be attributed mainly to the need of keeping their superconducting elements at low working temperatures by means of cryogenic liquids. However, the high field sensitivity attainable by SQUIDs makes these systems the only future candidates for detailed neuromagnetic measurements [7]. After the discovery of high-Tc superconductivity [8] these devices have been fabricated using thin films of the novel superconductors. More recently, due to the peculiar structure of high-Tc superconductors and to their physical properties, a d-wave symmetry in the
superconducting order parameter has been hypothesized [9]. Therefore, by following the schematic description of d-wave superconducting strips given by Copetti et al. [10], one can envision a simple circuital model for non-standard d.c. SQUIDs. Fabrication of a d.c. SQUID containing one 0-junction and one π -junction has recently been realized by Schulz et al. [11] an the device has been referred to as “π SQUID”. We shall here use the same diction and shall refer to standard (0–0) or π –π d.c. SQUIDs simply as “0-SQUIDs”. The dynamical properties of Josephson junctions (Jjs), which are responsible for the appearance of the macroscopic quantum effects in this class of instruments, have been studied by standard numerical and analytical methods. In the present Letter we shall develop a novel analytic approach to the problem, presenting the equations of the motion for the phase differences across the two small Jjs in a vectorial form. The relative phase shift in the Jjs (which may be, for
E-mail address: [email protected] (R. De Luca). 0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 0 6 2 - 7
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R. De Luca / Physics Letters A 280 (2001) 209–214
instance, either 0 or π in these devices) is taken into account in the Bohm–Aharonov relation, so that the periodic behavior of the superconducting phase differences in the running state can be investigated, for the two cases, by means of the invariance properties of a single vectorial equation of the motion. Moreover, by exploiting elementary concepts of matrix spaces, the periodic properties of the time-averaged electrodynamic observable quantities with respect to the external forcing terms (the applied field and the bias current) are seen to be independent from the type of device considered. The Letter is thus organized as follows. In the following section the dynamical equations for the gaugeinvariant phase differences across the junctions are derived in a general way for 0- and π -SQUIDs. In the third section a general study of the resulting instantaneous voltages and currents and of the corresponding observable time-averaged quantities is made. The magnetic field dependence of the maximum Josephson current and the V –I characteristics of π -SQUIDs are derived in the fourth section. Conclusions are drawn in the last section.
2. The equations Let us consider a planar symmetric d.c. SQUID in the presence of an external magnetic field H = H zˆ applied in the direction zˆ orthogonal to the plane containing the system. By defining a column vector ϕ, whose components are the gauge-invariant phase differences ϕ1 and ϕ2 across the Josephson junctions (Jjs) on the two corresponding SQUID branches, we may symbolically write the dynamical equations for the Jjs as follows: OJ (ϕ) = I,
(1)
where I is the column vector whose components are the currents I1 and I2 flowing in the two SQUID branches, and where OJ is a non-linear operator acting on a single component ϕi as follows: OJ (ϕi ) =
Φ0 dϕi + IJ sin ϕi , 2πR dt
(2)
Φ0 being the elementary flux quantum, and R and IJ the Josephson junction parameters. Fluxoid quantiza-
tion implies Φ , (3) Φ0 where Φ is the total flux linked to the SQUID loop and where the integer k is equal to 2n for a 0-SQUID and to 2n − 1 for a π -SQUID, n being integer too. We notice that in the above equation the intrinsic phase difference in π -junctions in the case of non-standard SQUIDs has been taken into account. The total flux is expressed in terms of the externally applied flux Φex = µ0 H S0 , S0 being the area enclosed in the loop, and of the currents flowing in the two branches as follows:
ϕ2 − ϕ1 + kπ = 2π
Φ = Φex + L(I1 − I2 ),
(4)
where L is the self inductance of one branch of the SQUID. By expressing the currents in terms of the phase differences and of the forcing terms, here represented by Φex and by the bias current IB = I1 + I2 , we have 1 dϕ1 + sin ϕ1 + (ϕ1 − ϕ2 ) dτ 4πβ 1 Ψex − k/2 , = iB − 2 β dϕ2 1 + sin ϕ2 − (ϕ1 − ϕ2 ) dτ 4πβ 1 Ψex − k/2 = iB + (5) , 2 β where τ = 2πRIJ t/Φ0 is the normalized time, Ψex = Φex /Φ0 is the applied flux number, β = LIJ /Φ0 is the usual SQUID parameter and iB = IB /IJ . In terms of the phase difference vector ϕ Eqs. (5) can be written as follows: dϕ (6) + Aϕ + sin ϕ = f, dτ where we have introduced the forcing vector f , having as components the right-hand sides of Eqs. (5), and the matrix A, defined as 1 +1 −1 . A= (7) 4πβ −1 +1 3. Observable electrodynamic quantities We would like to investigate the general properties of the electrodynamic SQUID response by studying
R. De Luca / Physics Letters A 280 (2001) 209–214
the time periodicity of the instantaneous voltage and of the instantaneous currents and the periodicity of the corresponding time-averaged quantities with respect to the applied flux number Ψex . Let us introduce the normalized voltage vector v, whose components are the voltages across the SQUID branches divided by RIJ , so that di dϕ + 2πβ , (8) dτ dτ where i = I /IJ . The normalized current vector i, on its turn, can be expressed in terms of the phase differences and of the forcing term as follows: v(τ ) =
i = f − Aϕ.
(9)
The normalized voltage vector, because of Eq. (9), can now be written as follows: dϕ 1 1 1 dϕ = , v(τ ) = (1 − 2πβA) (10) dτ 2 1 1 dτ where 1 is the 2 × 2 identity matrix. One trivial consequence of Eq. (10) is, as expected, the following: 1 d (ϕ1 + ϕ2 ). 2 dτ The normalized voltage is seen to be invariant with respect to the following general transformation: 1 ϕ (τ ) = ϕ(τ + mT ) = ϕ(τ ) + 2πm (11) , 1
v1 = v2 =
where m is an integer. For m = 0 we have the static solution ϕ = ϕ, which can be realized, for any τ , only if the forcing term f is such to satisfy the stationary equation related to Eq. (6). For m = 0, on the other hand, both junctions are in the running state. Therefore, T is the period of the function v(τ ). One can easily argue that the current vector i has same period T . Indeed, from Eq. (1) we can write dϕ + sin ϕ. i(τ ) = (12) dτ By noticing that both terms in the right-hand side of Eq. (12) are invariant with respect to the time translation given in Eq. (11), the proof is complete. The observable physical quantities, however, are v
and i , where the symbol stands for time-average. By recalling Eq. (10), v can be readily evaluated: v =
1 2T
1 1
1 1
T 0
2π 1 dϕ dτ = . dτ T 1
(13)
211
Similarly, by Eq. (9), we may write i = f − A ϕ .
(14)
Let us now investigate the periodic behavior of v
and i with respect to the forcing terms Ψex and iB . In general, if a period F in the v vs. f graph exists, an increase of the forcing term from f = f ∗ to f = f ∗ + F leaves the average voltage invariant, so that ϕ = ϕ + ϕ0 ,
(15)
where ϕ is the solution for f = f ∗ , ϕ is the one relative to f = f ∗ + F , and ϕ0 is a constant vector. By taking the time average of both sides of Eq. (6) for f = f ∗ and for f = f ∗ + F , respectively, we may write dϕ + A ϕ + sin ϕ = f ∗ , dτ dϕ + A ϕ + Aϕ0 + sin(ϕ + ϕ0 ) = f ∗ + F. dτ (16) By inspection we see that the constant vector ϕ0 may only take the following form: m1 ϕ0 = 2π (17) , m2 where m1 and m2 are integers. In this way, we only need to require that Aϕ0 = F.
(18)
In the case we keep iB constant and vary Ψex , we may look for the period "Ψex by writing: "Ψex −1 m1 , F = 2πA (19) = m2 1 2β so that "Ψex = m2 − m1 . Being the quantity (m2 − m1 ) an integer, we argue that the periodicity of vi
(i = 1, 2) with respect to Ψex is "Ψex = 1. We may also notice that the presence of a period in the v − Ψex curve implies a periodic behavior of T with Ψex , since 2π 1 v = . T 1 In Fig. 1 we show vi vs. Ψex curves of 0- and π SQUIDs for β = 0.05 and for iB = 2.5, 3.2. In the same way, we can argue that i presents the same periodicity with respect to Ψex . Indeed,
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Fig. 1. vi (i = 1, 2) vs. Ψex curves of 0-SQUIDs (triangles) and of π -SQUIDs (boxes) for β = 0.05. The lower curves are obtained for iB = 2.5, while the upper curves are obtained for iB = 3.2.
by recalling Eq. (9), we may write, analogously to Eqs. (16):
4. Quantum interference and V –I characteristics
i = f ∗ − A ϕ ,
In the present section we shall investigate the ic vs. Ψex curves and the V –I characteristics of π SQUIDs. As usual, we define the critical current ic as the maximum value of iB for which the system is in the zero-voltage state at a fixed value of Ψex . In the zero-voltage state i = sin ϕ, so that iB = i1 + i2 = sin ϕ1 + sin ϕ2 . On the other hand, by Eq. (3), ϕ2 = ϕ1 + 2πΨ − (2n − 1)π so that the function to be maximized with respect to the variable ϕ1 is the following:
i = f ∗ + F − A ϕ + ϕ0 ,
(20)
F
so that, in general, = F , and the same periodicity with respect to the external magnetic flux is expected. We may further notice that, while it is possible to have periodic behavior of v and i with respect to Ψex , by keeping iB constant, the same is not true for v –iB and i –iB curves for constant Ψex . Indeed, if we keep Ψex constant, the equation defining the possible periodicity "iB in the bias current is the following: "iB 1 m1 F = 2πA (21) = , m2 1 2 and it cannot be satisfied by any non-null choice of "iB . This is not surprising, since, for example, the v – iB characteristics of standard SQUIDs are monotonic functions for v greater than zero. We can finally notice that the above results do not depend on the choice of the integer k in Eq. (3), so that the same period in the external field is expected for the two types of d.c. SQUIDs, while no periodicity with respect to the bias current is detectable in both cases.
iB = sin ϕ1 − sin(ϕ1 + 2πΨ ).
(22)
Notice that Ψ can be expressed in terms of ϕ1 , by means of Eq. (4), as follows:
Ψ = Ψex + β sin ϕ1 − sin(ϕ1 + 2πΨ ) . (23) This equation implicitly defines Ψ in terms of ϕ1 for a fixed Ψex . For very small values of β, however, the analysis may be simplified by setting Ψ Ψex . In this latter case one obtains ic = 2 sin(πΨex ) , (24) while a numerical algorithm must be used to solve the problem for finite β values. Notice that the above result could be obtained by simply shifting of π/2
R. De Luca / Physics Letters A 280 (2001) 209–214
213
Fig. 2. vi (i = 1, 2) vs. iB curves of π -SQUIDs for β = 0.1. The upper curve (triangles) is obtained for Ψex = 0, while the lower curve (boxes) is obtained for Ψex = 0.5.
the Ψex term in the argument of the cosine term in the homologous expression for a standard d.c. SQUID [1]. The different modulation of the maximum Josephson current with respect to the externally applied magnetic flux in a π -SQUID already gives a hint on the shape of the vi vs. iB curves (V –I characteristics) of these systems. As before, in the case of finite β values one needs to resort to some numerical procedure in order to solve the problem. We here illustrate one possible procedure, which we shall adopt to find the vi
vs. iB curves numerically. We first solve the differential equation for the phase differences ϕ1 and ϕ2 by means of Mathematica 3.0, letting the bias current iB increase after a sufficiently high number of representative (τ , ϕi ) points are collected. In this way, v(τ ) can be calculated by means of Eqs. (10) and (6). Indeed, by combining these two equations, we get v1 (τ ) = v2 (τ ) =
1
iB − (sin ϕ1 + sin ϕ2 ) , 2
(25)
so that 1
iB − sin ϕ1 + sin ϕ2 , (26) 2 where the average values of the sine terms are evaluated by means of the list of collected (τ , ϕi ) points. V –I characteristics of π -SQUIDs for β = 0.1 and for Ψex = 0, 0.5 are shown in Fig. 2. We might here notice v1 = v2 =
that the curve for β 0 can be obtained analytically as in the case of symmetric 0-SQUIDs. By carrying out the calculations in the same way as in a standard d.c. SQUID, indeed, we have 0, for 0 iB 2| sin(πΨex )|, vi = (27) iB2 − 4 sin2 (πΨex ), for iB 2| sin(πΨex )|. Again, we notice that the above result can be derived by a π/2 shift in Ψex in the homologous formula obtained for a 0-SQUID.
5. Conclusions We have presented a general analytic approach to the study of the electrodynamic response of 0- and π -SQUIDs. By writing the dynamical equations for the phase differences across the Josephson junctions in vectorial form and by applying to these equations elementary linear algebra concepts, the periodic properties of the instantaneous voltage and current vectors are derived. A formal proof that the corresponding time averaged quantities v and i show periodic behavior with respect to the applied flux number Ψex is given. A similar reasoning applied to the second forcing term iB shows that no periodic pattern of v
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R. De Luca / Physics Letters A 280 (2001) 209–214
and i with respect to the bias current exists. The time evolution equations for the superconducting phase differences are seen to be similar in the two types of devices, the only difference being an effective relative shift of the Ψex forcing term of 1/2. In this way, the observable time-averaged voltages and currents, when plotted against Ψex , though preserving the same type of periodicity ("Ψex = 1), appear to be shifted of exactly 1/2 on the Ψex -axis. This implies that the V –I characteristics for a π -SQUID are such that the curve corresponding to the highest I -value for a fixed voltage is at Ψex = (2k + 1)/2, where k is an integer, while the curve corresponding to lowest I -value is obtained for Ψex = kπ . This behavior is just opposite to the well known features of 0-SQUIDs. This type of analysis is not confined to the d.c. SQUID problem only. Indeed, a candidate model for a three-dimensional (3D) magnetic field sensor has been recently developed [12]. In this 3D model system, consisting of twelve junctions and inductors, the Jj–inductor pairs being located at the edges of a current-biased cubic superconducting network, the dynamical equations for the superconducting phase differences take on the same form as in the case of a d.c. SQUID [13]. In the 3D case, however, a twelvecomponent ϕ-vector arises, as opposed to the simplest case of a two-component vector characterizing the d.c. SQUID problem. Therefore, in a future work the same type of approach will be applied to this 3D system, whose properties might open new perspectives in the field of detailed neuromagnetic measurements.
Acknowledgements The author would like to thank R. Cavaliere for helpful discussions.
References [1] A. Barone, G. Paternó, Physics and Applications of the Josephson Effect, Wiley, New York, 1982. [2] K.K. Likharev, Dynamics of Josephson Junctions and Circuits, Gordon and Breach, Amsterdam, 1986. [3] S. Tanaka, H. Itozaki, H. Toyoda, N. Harada, A. Adachi, K. Okajima, H. Kado, T. Nagaishi, Appl. Phys. Lett. 54 (1994) 514. [4] A.P. Rijpma, Y. Seppenwoolde, H.J.M. ter Brake, M.J. Peters, H. Rogalla, Inst. Phys. Conf. Ser. 158 (1997) 771. [5] R. Weidl, S. Brabetz, F. Schmidl, F. Klemm, S. Wunderlich, P. Seidel, Supercond. Sci. Technol. 10 (1997) 95. [6] H.-J. Krause, Y. Zhang, R. Hohmann, M. Gruneklee, M.I. Faley, D. Lomparski, M. Maus, H. Bousack, A.I. Braginski, Inst. Phys. Conf. Ser. 158 (1997) 775. [7] J. Clarke, in: H. Weinstock, R.W. Ralston (Eds.), The New Superconducting Electronics, Kluwer Academic, 1993. [8] J.G. Bednorz, K.A. Muller, Z. Phys. B 64 (1986) 189. [9] D.J. Van Harlingen, Rev. Mod. Phys. 67 (1995) 515. [10] C.A. Copetti, F. Rüders, B. Oelze, Ch. Buchal, B. Kabius, J.W. Seo, Physica C 253 (1995) 63. [11] R.R. Schulz, B. Chesca, B. Goetz, C.W. Schneider, A. Schmehl, H. Bielefeldt, H. Hilgenkamp, J. Mannhart, C.C. Tsuei, Appl. Phys. Lett. 76 (2000) 912. [12] T. Di Matteo, R. De Luca, J. Paasi, A. Tuohimaa, IEEE Trans. Appl. Superc. 9 (1999) 3515. [13] R. De Luca, T. Di Matteo, A. Tuohimaa, J. Paasi, Philos. Mag. B 80 (2000) 897.
Physics Letters A 280 (2001) 209–214 www.elsevier.nl/locate/pla
A general analytic approach to the study of the electrodynamic properties of d.c. SQUIDs R. De Luca DIIMA, Università degli Studi di Salerno, I-84084 Fisciano (SA), Italy Received 12 September 2000; received in revised form 18 January 2001; accepted 23 January 2001 Communicated by A.P. Fordy
Abstract A general analytic approach to the study of the dynamical properties of Josephson junctions in d.c. SQUIDs is developed. The periodic properties of the observable electrodynamic quantities are rigorously derived by means of elementary linear algebra concepts and the case of d.c. SQUIDs containing one or two π-junctions is discussed. 2001 Elsevier Science B.V. All rights reserved. PACS: 74.50.+r; 85.25.Dq
1. Introduction SQUIDs (superconducting quantum interference devices) have been thoroughly studied in the literature [1,2]. These systems find application in a large variety of fields. Their still limited use as non-invasive diagnostic instruments [3–5] or as non-destructive monitoring devices [6] is to be attributed mainly to the need of keeping their superconducting elements at low working temperatures by means of cryogenic liquids. However, the high field sensitivity attainable by SQUIDs makes these systems the only future candidates for detailed neuromagnetic measurements [7]. After the discovery of high-Tc superconductivity [8] these devices have been fabricated using thin films of the novel superconductors. More recently, due to the peculiar structure of high-Tc superconductors and to their physical properties, a d-wave symmetry in the
superconducting order parameter has been hypothesized [9]. Therefore, by following the schematic description of d-wave superconducting strips given by Copetti et al. [10], one can envision a simple circuital model for non-standard d.c. SQUIDs. Fabrication of a d.c. SQUID containing one 0-junction and one π -junction has recently been realized by Schulz et al. [11] an the device has been referred to as “π SQUID”. We shall here use the same diction and shall refer to standard (0–0) or π –π d.c. SQUIDs simply as “0-SQUIDs”. The dynamical properties of Josephson junctions (Jjs), which are responsible for the appearance of the macroscopic quantum effects in this class of instruments, have been studied by standard numerical and analytical methods. In the present Letter we shall develop a novel analytic approach to the problem, presenting the equations of the motion for the phase differences across the two small Jjs in a vectorial form. The relative phase shift in the Jjs (which may be, for
E-mail address: [email protected] (R. De Luca). 0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 0 6 2 - 7
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R. De Luca / Physics Letters A 280 (2001) 209–214
instance, either 0 or π in these devices) is taken into account in the Bohm–Aharonov relation, so that the periodic behavior of the superconducting phase differences in the running state can be investigated, for the two cases, by means of the invariance properties of a single vectorial equation of the motion. Moreover, by exploiting elementary concepts of matrix spaces, the periodic properties of the time-averaged electrodynamic observable quantities with respect to the external forcing terms (the applied field and the bias current) are seen to be independent from the type of device considered. The Letter is thus organized as follows. In the following section the dynamical equations for the gaugeinvariant phase differences across the junctions are derived in a general way for 0- and π -SQUIDs. In the third section a general study of the resulting instantaneous voltages and currents and of the corresponding observable time-averaged quantities is made. The magnetic field dependence of the maximum Josephson current and the V –I characteristics of π -SQUIDs are derived in the fourth section. Conclusions are drawn in the last section.
2. The equations Let us consider a planar symmetric d.c. SQUID in the presence of an external magnetic field H = H zˆ applied in the direction zˆ orthogonal to the plane containing the system. By defining a column vector ϕ, whose components are the gauge-invariant phase differences ϕ1 and ϕ2 across the Josephson junctions (Jjs) on the two corresponding SQUID branches, we may symbolically write the dynamical equations for the Jjs as follows: OJ (ϕ) = I,
(1)
where I is the column vector whose components are the currents I1 and I2 flowing in the two SQUID branches, and where OJ is a non-linear operator acting on a single component ϕi as follows: OJ (ϕi ) =
Φ0 dϕi + IJ sin ϕi , 2πR dt
(2)
Φ0 being the elementary flux quantum, and R and IJ the Josephson junction parameters. Fluxoid quantiza-
tion implies Φ , (3) Φ0 where Φ is the total flux linked to the SQUID loop and where the integer k is equal to 2n for a 0-SQUID and to 2n − 1 for a π -SQUID, n being integer too. We notice that in the above equation the intrinsic phase difference in π -junctions in the case of non-standard SQUIDs has been taken into account. The total flux is expressed in terms of the externally applied flux Φex = µ0 H S0 , S0 being the area enclosed in the loop, and of the currents flowing in the two branches as follows:
ϕ2 − ϕ1 + kπ = 2π
Φ = Φex + L(I1 − I2 ),
(4)
where L is the self inductance of one branch of the SQUID. By expressing the currents in terms of the phase differences and of the forcing terms, here represented by Φex and by the bias current IB = I1 + I2 , we have 1 dϕ1 + sin ϕ1 + (ϕ1 − ϕ2 ) dτ 4πβ 1 Ψex − k/2 , = iB − 2 β dϕ2 1 + sin ϕ2 − (ϕ1 − ϕ2 ) dτ 4πβ 1 Ψex − k/2 = iB + (5) , 2 β where τ = 2πRIJ t/Φ0 is the normalized time, Ψex = Φex /Φ0 is the applied flux number, β = LIJ /Φ0 is the usual SQUID parameter and iB = IB /IJ . In terms of the phase difference vector ϕ Eqs. (5) can be written as follows: dϕ (6) + Aϕ + sin ϕ = f, dτ where we have introduced the forcing vector f , having as components the right-hand sides of Eqs. (5), and the matrix A, defined as 1 +1 −1 . A= (7) 4πβ −1 +1 3. Observable electrodynamic quantities We would like to investigate the general properties of the electrodynamic SQUID response by studying
R. De Luca / Physics Letters A 280 (2001) 209–214
the time periodicity of the instantaneous voltage and of the instantaneous currents and the periodicity of the corresponding time-averaged quantities with respect to the applied flux number Ψex . Let us introduce the normalized voltage vector v, whose components are the voltages across the SQUID branches divided by RIJ , so that di dϕ + 2πβ , (8) dτ dτ where i = I /IJ . The normalized current vector i, on its turn, can be expressed in terms of the phase differences and of the forcing term as follows: v(τ ) =
i = f − Aϕ.
(9)
The normalized voltage vector, because of Eq. (9), can now be written as follows: dϕ 1 1 1 dϕ = , v(τ ) = (1 − 2πβA) (10) dτ 2 1 1 dτ where 1 is the 2 × 2 identity matrix. One trivial consequence of Eq. (10) is, as expected, the following: 1 d (ϕ1 + ϕ2 ). 2 dτ The normalized voltage is seen to be invariant with respect to the following general transformation: 1 ϕ (τ ) = ϕ(τ + mT ) = ϕ(τ ) + 2πm (11) , 1
v1 = v2 =
where m is an integer. For m = 0 we have the static solution ϕ = ϕ, which can be realized, for any τ , only if the forcing term f is such to satisfy the stationary equation related to Eq. (6). For m = 0, on the other hand, both junctions are in the running state. Therefore, T is the period of the function v(τ ). One can easily argue that the current vector i has same period T . Indeed, from Eq. (1) we can write dϕ + sin ϕ. i(τ ) = (12) dτ By noticing that both terms in the right-hand side of Eq. (12) are invariant with respect to the time translation given in Eq. (11), the proof is complete. The observable physical quantities, however, are v
and i , where the symbol stands for time-average. By recalling Eq. (10), v can be readily evaluated: v =
1 2T
1 1
1 1
T 0
2π 1 dϕ dτ = . dτ T 1
(13)
211
Similarly, by Eq. (9), we may write i = f − A ϕ .
(14)
Let us now investigate the periodic behavior of v
and i with respect to the forcing terms Ψex and iB . In general, if a period F in the v vs. f graph exists, an increase of the forcing term from f = f ∗ to f = f ∗ + F leaves the average voltage invariant, so that ϕ = ϕ + ϕ0 ,
(15)
where ϕ is the solution for f = f ∗ , ϕ is the one relative to f = f ∗ + F , and ϕ0 is a constant vector. By taking the time average of both sides of Eq. (6) for f = f ∗ and for f = f ∗ + F , respectively, we may write dϕ + A ϕ + sin ϕ = f ∗ , dτ dϕ + A ϕ + Aϕ0 + sin(ϕ + ϕ0 ) = f ∗ + F. dτ (16) By inspection we see that the constant vector ϕ0 may only take the following form: m1 ϕ0 = 2π (17) , m2 where m1 and m2 are integers. In this way, we only need to require that Aϕ0 = F.
(18)
In the case we keep iB constant and vary Ψex , we may look for the period "Ψex by writing: "Ψex −1 m1 , F = 2πA (19) = m2 1 2β so that "Ψex = m2 − m1 . Being the quantity (m2 − m1 ) an integer, we argue that the periodicity of vi
(i = 1, 2) with respect to Ψex is "Ψex = 1. We may also notice that the presence of a period in the v − Ψex curve implies a periodic behavior of T with Ψex , since 2π 1 v = . T 1 In Fig. 1 we show vi vs. Ψex curves of 0- and π SQUIDs for β = 0.05 and for iB = 2.5, 3.2. In the same way, we can argue that i presents the same periodicity with respect to Ψex . Indeed,
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Fig. 1. vi (i = 1, 2) vs. Ψex curves of 0-SQUIDs (triangles) and of π -SQUIDs (boxes) for β = 0.05. The lower curves are obtained for iB = 2.5, while the upper curves are obtained for iB = 3.2.
by recalling Eq. (9), we may write, analogously to Eqs. (16):
4. Quantum interference and V –I characteristics
i = f ∗ − A ϕ ,
In the present section we shall investigate the ic vs. Ψex curves and the V –I characteristics of π SQUIDs. As usual, we define the critical current ic as the maximum value of iB for which the system is in the zero-voltage state at a fixed value of Ψex . In the zero-voltage state i = sin ϕ, so that iB = i1 + i2 = sin ϕ1 + sin ϕ2 . On the other hand, by Eq. (3), ϕ2 = ϕ1 + 2πΨ − (2n − 1)π so that the function to be maximized with respect to the variable ϕ1 is the following:
i = f ∗ + F − A ϕ + ϕ0 ,
(20)
F
so that, in general, = F , and the same periodicity with respect to the external magnetic flux is expected. We may further notice that, while it is possible to have periodic behavior of v and i with respect to Ψex , by keeping iB constant, the same is not true for v –iB and i –iB curves for constant Ψex . Indeed, if we keep Ψex constant, the equation defining the possible periodicity "iB in the bias current is the following: "iB 1 m1 F = 2πA (21) = , m2 1 2 and it cannot be satisfied by any non-null choice of "iB . This is not surprising, since, for example, the v – iB characteristics of standard SQUIDs are monotonic functions for v greater than zero. We can finally notice that the above results do not depend on the choice of the integer k in Eq. (3), so that the same period in the external field is expected for the two types of d.c. SQUIDs, while no periodicity with respect to the bias current is detectable in both cases.
iB = sin ϕ1 − sin(ϕ1 + 2πΨ ).
(22)
Notice that Ψ can be expressed in terms of ϕ1 , by means of Eq. (4), as follows:
Ψ = Ψex + β sin ϕ1 − sin(ϕ1 + 2πΨ ) . (23) This equation implicitly defines Ψ in terms of ϕ1 for a fixed Ψex . For very small values of β, however, the analysis may be simplified by setting Ψ Ψex . In this latter case one obtains ic = 2 sin(πΨex ) , (24) while a numerical algorithm must be used to solve the problem for finite β values. Notice that the above result could be obtained by simply shifting of π/2
R. De Luca / Physics Letters A 280 (2001) 209–214
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Fig. 2. vi (i = 1, 2) vs. iB curves of π -SQUIDs for β = 0.1. The upper curve (triangles) is obtained for Ψex = 0, while the lower curve (boxes) is obtained for Ψex = 0.5.
the Ψex term in the argument of the cosine term in the homologous expression for a standard d.c. SQUID [1]. The different modulation of the maximum Josephson current with respect to the externally applied magnetic flux in a π -SQUID already gives a hint on the shape of the vi vs. iB curves (V –I characteristics) of these systems. As before, in the case of finite β values one needs to resort to some numerical procedure in order to solve the problem. We here illustrate one possible procedure, which we shall adopt to find the vi
vs. iB curves numerically. We first solve the differential equation for the phase differences ϕ1 and ϕ2 by means of Mathematica 3.0, letting the bias current iB increase after a sufficiently high number of representative (τ , ϕi ) points are collected. In this way, v(τ ) can be calculated by means of Eqs. (10) and (6). Indeed, by combining these two equations, we get v1 (τ ) = v2 (τ ) =
1
iB − (sin ϕ1 + sin ϕ2 ) , 2
(25)
so that 1
iB − sin ϕ1 + sin ϕ2 , (26) 2 where the average values of the sine terms are evaluated by means of the list of collected (τ , ϕi ) points. V –I characteristics of π -SQUIDs for β = 0.1 and for Ψex = 0, 0.5 are shown in Fig. 2. We might here notice v1 = v2 =
that the curve for β 0 can be obtained analytically as in the case of symmetric 0-SQUIDs. By carrying out the calculations in the same way as in a standard d.c. SQUID, indeed, we have 0, for 0 iB 2| sin(πΨex )|, vi = (27) iB2 − 4 sin2 (πΨex ), for iB 2| sin(πΨex )|. Again, we notice that the above result can be derived by a π/2 shift in Ψex in the homologous formula obtained for a 0-SQUID.
5. Conclusions We have presented a general analytic approach to the study of the electrodynamic response of 0- and π -SQUIDs. By writing the dynamical equations for the phase differences across the Josephson junctions in vectorial form and by applying to these equations elementary linear algebra concepts, the periodic properties of the instantaneous voltage and current vectors are derived. A formal proof that the corresponding time averaged quantities v and i show periodic behavior with respect to the applied flux number Ψex is given. A similar reasoning applied to the second forcing term iB shows that no periodic pattern of v
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and i with respect to the bias current exists. The time evolution equations for the superconducting phase differences are seen to be similar in the two types of devices, the only difference being an effective relative shift of the Ψex forcing term of 1/2. In this way, the observable time-averaged voltages and currents, when plotted against Ψex , though preserving the same type of periodicity ("Ψex = 1), appear to be shifted of exactly 1/2 on the Ψex -axis. This implies that the V –I characteristics for a π -SQUID are such that the curve corresponding to the highest I -value for a fixed voltage is at Ψex = (2k + 1)/2, where k is an integer, while the curve corresponding to lowest I -value is obtained for Ψex = kπ . This behavior is just opposite to the well known features of 0-SQUIDs. This type of analysis is not confined to the d.c. SQUID problem only. Indeed, a candidate model for a three-dimensional (3D) magnetic field sensor has been recently developed [12]. In this 3D model system, consisting of twelve junctions and inductors, the Jj–inductor pairs being located at the edges of a current-biased cubic superconducting network, the dynamical equations for the superconducting phase differences take on the same form as in the case of a d.c. SQUID [13]. In the 3D case, however, a twelvecomponent ϕ-vector arises, as opposed to the simplest case of a two-component vector characterizing the d.c. SQUID problem. Therefore, in a future work the same type of approach will be applied to this 3D system, whose properties might open new perspectives in the field of detailed neuromagnetic measurements.
Acknowledgements The author would like to thank R. Cavaliere for helpful discussions.
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