Physica C 433 (2006) 195–204 www.elsevier.com/locate/physc
Shielding current effects on diffraction patterns in the critical current of superconducting quantum interference devices R. De Luca b
a,*
, F. Romeo
b
a CNR-INFM and DIIMA, Universita` degli Studi di Salerno, I-84084 Fisciano (SA), Italy Dipartimento di Fisica, E. R. Caianiello, Universita` degli Studi di Salerno, I-84081 Baronissi (SA), Italy
Received 21 September 2005; accepted 27 October 2005 Available online 15 December 2005
Abstract By a simple schematization of the current distribution in a two-finite-size-junction interferometer, we study the shielding current effects on the diffraction envelope of the interference figure given by the critical current as a function of the applied magnetic field. By this scheme we are able to reproduce the known results in the limit of negligible shielding currents for both conventional interferometers and for those containing one p-junction. For non-negligible values of the circulating current contribution to the superconducting phase in the junctions, we find that the superimposed quantum interference and diffraction patterns suffer a contraction with respect to the horizontal field axis. Ó 2005 Elsevier B.V. All rights reserved. PACS: 74.50.+r; 85.25.Dq Keywords: Josephson junctions; Quantum interference; 0-SQUID; p-SQUID
1. Introduction The properties of superconducting quantum interference devices (SQUIDÕs) have been thoroughly studied in the past [1–4]. The application *
Corresponding author. Tel.: +39 089 964263; fax: +39 089 964191. E-mail address:
[email protected] (R.D. Luca).
of these devices as ultra-sensitive magnetic sensors has spread knowledge of these systems among the entire scientific community. In particular, the electrodynamic response of dc SQUIDÕs has been analyzed by means of the two-junction interferometer model [2–4]. For negligibly small values of the characteristic parameter b, the two-junction interferometer model reduces to a single non-linear ordinary differential equation [2] which is similar
0921-4534/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2005.10.014
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to a single-junction equation. In fact, it is possible to study the properties of a dc SQUID starting from the dynamical equation of the superconducting phase difference across the electrodes of a single overdamped Josephson junction. In order to do so, one only needs to take the effective maximum Josephson current IJ to be modulated by the externally applied normalized flux Wex ¼ l0UHS0 0 , where H is the applied magnetic field, S0 the loop area and U0 the elementary flux quantum, according to the following relation: I J ¼ 2I J0 j cosðpWex Þj;
IB
I1
I I I+I L(x)
I-I R(x)
I+IL(x)
I-IR(x) I
ð1Þ
where IJ0 is the maximum Josephson current of the two Josephson junctions (JJÕs) in the device, when assumed to be identical. In this context, the junctions are generally assumed to be point-like. For non-negligibly small values of the parameter b, even retaining these simplifying assumptions, the resulting coupled differential equations, one for each JJ, are nonetheless not exactly solvable. Recently, the authors have proposed a reduced two-junction interferometer model, by means of which, in the small b limit, the systemÕs dynamical equations can be reduced to a single junction nonlinear ordinary differential equation characterized by an unconventional effective current-phase relation (CPR) with a second harmonic contribution [5]. In the present work, we remove the hypothesis that the junctions in the device are small and analyze, by means of the schematic model represented in Fig. 1, the case in which the junction area is comparable with the cross section of the normal inner region. In the literature, the problem of the finiteness of the size of the JJÕs in a SQUID has been naturally compared with the analogous problem of a single long Josephson junction with nonuniform critical current density [2], which gives Fraunhofer-like diffraction, superimposed to an interference pattern, in the critical current vs. applied flux curves. Furthermore, much of the analysis on the self-field generated by circulating currents has been devoted, in the early days of superconductivity [6,7], to the case of single long Josephson junctions. To our knowledge, however, no attempt has yet been made to describe in details the dc SQUID electrodynamics in the presence
I2
I1’
I
I2’ IB
Fig. 1. Schematic representation of the current distribution in a two-junction interferometer containing extended Josephson junctions. The bias current IB splits into two branch currents (I1 and I2) which flow in a superconducting region, close to external boundary, having extension comparable to the penetration length of the material. In the upper and in the lower portion of the superconducting matrix of the system, the same shielding current I flows, while the tunnel current absorbs the currents I1 and I2 over the entire junction length, giving rise to the currents IL and IR, which depend on the distance from one end of the left and right junction, respectively.
of circulating currents. Furthermore, p-SQUIDÕs properties are being recently investigated [8] and shielding current effects might be of some importance in interpreting the critical current vs. applied flux curves of these devices. The paper is thus organized as follows. In the next section, we give the details of the electrodynamic model adopted. In the third section, by simplifying the analysis to negligibly small contribution to the magnetic flux in the system by the circulating currents, we reproduce, by a rather different approach, the already known expression for the critical current of traditional and p-SQUIDÕs in this approximation. In the fourth section, we remove the assumption of negligible inductance of the current paths and give the equation governing the space variation of the superconducting phases in the junctions. By this equation, considering only shielding current contributions, we
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calculate the critical current vs. applied magnetic flux curves in this case. Conclusions are drawn in the last section and future developments of the present work are discussed.
L W d
H dx
2. Schematic electrodynamic model Consider the two-junction interferometer, represented in Fig. 1, consisting of two junctions with identical parameters, both of finite length L, with maximum Josephson current IJ, connected in parallel and separated by a normal circular region of cross-section S0 = pa2. One of the two JJÕs might be a p-junction. In this figure the bias current IB is applied to the two ends of the system and splits into two identical branches, I1 and I2, flowing in a superconducting region of width comparable to the penetration length of the material. In Fig. 1, we represent the shielding current I as separated from the branch currents, so that, we may adopt symmetry arguments for the branch currents when deriving the electrodynamic properties of the system. By taking the tunnelling current density in the JJÕs to the left and to the right are denoted as JL and JR, respectively, by KirchhoffÕs law, we can set: dI R;L ¼ WJ R;L ; dx
ð2Þ
where the x-axis are coincident with the longitudinal axes of the two JJÕs (taking different reference points and orientations for the two junctions, for convenience), where IL and IR are the residual parts of the branch currents flowing along the xdirection, and W is the width of the junction (see Fig. 2). We express the current densities in the JJÕs by means of the usual Josephson current-phase relation, as follows: J R;L ðxÞ ¼ J 0 sin uR;L ðxÞ;
ð3Þ
where J0 is the maximum superconducting current density allowed in the two JJÕs, and uR and uL are the gauge-invariant superconducting phase differences across the right and left JJ, respectively.
Fig. 2. Integration path (white full line) of the magnetic vector potential over which the magnetic flux per unit length in Eq. (4) is calculated. Notice that the effectively flux penetrated area of the junction is given by the following: SJ = Ld.
By evaluating the circuitation of the magnetic vector potential across an infinitesimal path of length dx in the x-direction, as shown in Fig. 2, either in the left, either in the right JJ, by means of Bohm–Aharonov condition, we can write [2]: duR;L ðxÞ 2p e ¼ U ð4Þ R;L ðxÞ; U0 dx e R;L ðxÞ is the flux, calculated per unit length where U in the direction x, linked to the path depicted in Fig. 2 and where the minus sign pertains to the right JJ (RJJ) and the plus sign to the left JJ (LJJ). This flux consists of three different terms: e ex ¼ l0 Hd due to the externally applied (a) U field, where d is the thickness of the flux penetrated region of either junctions, (b) the flux per unit length due to the shielding current I, (c) the flux per unit length due to the residual current IL or IR, depending on the JJ we are considering. Taking into account all contributions, we write: e R ðxÞ ¼ 2dlðI I R ðxÞÞ þ l0 Hd; U e L ðxÞ ¼ 2dlðI þ I L ðxÞÞ þ l0 Hd; U
ð5aÞ ð5bÞ
l L
where dl ¼ is the inductance per unit length pertaining to the current paths and where the factor of 2 is given by the contributions of the upper and lower superconducting portions of the system. Notice now that the shielding current I can be obtained by the condition of null flux (U = 0) in the superconducting regions, so that U ¼ plI þ l0 HS SUP ¼ 0;
ð6Þ
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with p 2 þ p þ 2p La, provides the following expression for I: I ¼ al0 Hd;
ð7Þ
where the geometric factor a can be set equal to 2pLðLþ2aÞ 1 . It is here important to notice that this ð2þpÞLþ2pa d geometric factor is null for L = 0. By Eq. (5a,b) and by Eq. (7), we can now obtain the fluxes e R;L ðxÞ U e R;L ðxÞ ¼ l0 Hdð1 þ aÞ 2l I R;L ðxÞ. U L
ð8Þ
By substituting Eq. (8) into Eq. (4), the gradient of the superconducting phases uR,L(x) can be found to be: duR;L ðxÞ 4pl ¼ k a I R;L ðxÞ; U0 L dx
ð9Þ
where k a ¼ 2plU00Hd ð1 þ aÞ. Now, by noticing that the injected current must be equal to the total current tunnelling through the JJÕs, calculated over the junctions length L as follows: Z L ðsin uR ðxÞ þ sin uL ðxÞÞ dx; ð10Þ I B ¼ WJ 0 0
we can obtain the critical current of the system by maximizing IB with respect to the indetermined phase parameter coming from evaluation of the superconducting phases at one of the fixed origins, as we shall see in the following sections.
3. Critical current in the case of negligible inductance In this section, we shall show that the above schematic approach correctly gives the Fraunhofer-like pattern superimposed to the interference pattern already known from the literature [2]. We thus proceed by neglecting the flux contribution due to all circulating currents. In this approximae R;L ðxÞ ¼ U e ex and the x-dependence of the tion, U superconducting phase difference uR and uL is readily found by Eq. (4) as follows: uL ðxÞ ¼ u0 þ kx þ ep; uR ðxÞ ¼ u1 kx;
ð11Þ
where e = 0, 1 for a dc SQUID containing a conventional or a p-junction on the left branch, respectively, k ¼ k a¼0 ¼ 2plU00Hd , and where we notice the appearance of two indeterminate superconducting phase values u0 and u1, which can be related by means of an appropriate integration of Eq. (4). Indeed, by extending the integration path of Fig. 2 over the two junctions, including the inner normal region, we get: S0 u1 u0 ¼ kL 2 þ ð12Þ Ld and finally Z L sinðu0 þ kx þ epÞ I B ¼ WJ 0 0 S0 þ sin u0 þ kL 2 þ kx dx. Ld
ð13Þ
By performing the definite integrals in the expression of the bias current IB found in Eq. (13), we obtain: S kL p 0 I B ¼ Im 2WJ 0 Leiu0 ei 2 2þLd þe2 kL S0 p sin kL2 1þ cos ; ð14Þ e kL 2 2 Ld 2 so that the maximum value Ic (the critical current of the device) with respect to u0 can be obtained by taking the modulus of the pre-factor of eiu0 in parentheses in Eq. (14). By doing so, we get: kL S0 p sin kL2 1þ I c ¼ 2I 0 cos e kL ; ð15Þ 2 2 Ld 2 where I0 = J0WL is the maximum Josephson current of each JJ. Notice that, for L ! 0, Eq. (15) correctly corresponds to the well known expression given in Eq. (1). The expression in Eq. (15), moreover, is the simple superimposition of two terms, as one could simply deduce by looking at the two-slit diffraction grating expression, in which one term gives the interference pattern and the second the diffraction due to the finite amplitude of the slit. As remarked in Ref. [2], Eq. (15), carries on an additional information due to the coherence of superconducting systems; namely, when the flux penetrated area SJ = Ld in the junction cannot be
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sinðrpWex Þ I c ¼ 2I 0 j sinðpð1 þ rÞWex Þj rpWex
neglected with respect to the cross-sectional area S0 of the inner normal region, the information about the length of the JJ enters the interference term. In order to make Eq. (15) more clearly recognizable in terms of the quantities appearing in Eq. (1), we rewrite the former as follows:
ðp SQUIDÞ; ð16bÞ where r ¼ SS0J ¼ Ld , so that we may define the apS0 plied flux Wex as a reference flux, in terms of which we shall plot the critical current vs. applied field curves. In Fig. 3a and b, the quantum interference pattern for the normalized critical current ic ¼ II 0c as a function of the normalized externally applied flux Wex is reported for a 0-SQUID with r = 1.1.
sinðrpWex Þ I c ¼ 2I 0 j cosðpð1 þ rÞWex Þj rpWex ð0 SQUIDÞ;
199
ð16aÞ
2
ic
1.5
1
0.5
0 −4
−2
(a)
0
2
4
Ψex
2
ic
1.5
1
0.5
0 −1 (b)
−0.5
0
Ψex
0.5
1
Fig. 3. Quantum interference pattern (full line) given by the normalized critical current ic versus the normalized applied flux Wex for a 0-SQUID with r ¼ SS0J ¼ 1:1. The Fraunhofer-like diffraction envelope is represented by a dashed line. (a) Extended view of the ic vs. Wex curves in the interval [5, +5]. (b) Detailed view of the ic vs. Wex curves in the interval [1, +1].
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In Fig. 3a, in particular, we let Wex range between the values 5 and 5, while in Fig. 3b, we restrict the range of variation of Wex to the interval [1, +1]. In Fig. 4a and b, on the other hand, ic vs. Wex curves are reported for a p-SQUID with r = 1.1. In Fig. 4a and in Fig. 4b we let Wex range in the intervals [5, +5] and [1, +1] respectively. Notice that in Figs. 3a and b and 4a and b, the junction area is slightly greater than the cross-section S0 (r = 1.1). On the contrary, if the junction area is small with respect to S0 (r 1), we have patterns which slightly differ from those given in Eq. (1). In order to detect the difference in this
case, let us consider the results shown in Fig. 5a and b, where the ic vs. Wex curves for r = 0.1 are reported for a p-SQUID. In Fig. 5a the curve coming from Eq. (16b) in the text is represented as a full line and the diffraction envelope is shown as a dashed line. In this figure it is evident that, for the applied flux range represented, the maxima of the oscillating curve in Eq. (16b) tend to decrease as the field increases. Consider next the ic vs. Wex curves shown in Fig. 5b. In this diagram the curve given by Eq. (16b) is again plotted as a full line, while the graph of the curve coming from Eq. (1) is plotted as a dashed line. From this last figure
2
ic
1.5
1
0.5
0 −4
−2
(a)
0
2
4
Ψex
2
ic
1.5
1
0.5
0
(b)
−1
−0.5
0
0.5
1
Ψex
Fig. 4. Quantum interference pattern (full line) given by the normalized critical current ic versus the normalized applied flux Wex for a p-SQUID with r ¼ SS0J ¼ 1:1. The Fraunhofer-like diffraction envelope is represented by a dashed line. (a) Extended view of the ic vs. Wex curves in the interval [5, +5]. (b) Detailed view of the ic vs. Wex curves in the interval [1, +1].
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201
2
ic
1.5
1
0.5
0 −3
−2
−1
0
1
2
3
0.5
1
1.5
Ψex
(a)
2
ic
1.5
1
0.5
0 −1.5
−1
−0.5
(b)
0
Ψex
Fig. 5. Quantum interference pattern (full line) given by the normalized critical current ic versus the normalized applied flux Wex for a p-SQUID with r ¼ SS0J ¼ 0:1. (a) Extended view of the ic vs. Wex curves in the interval [3, +3]. The dashed curve represents the Fraunhofer-like diffraction envelope. (b) Detailed view of the ic vs. Wex curves in the interval [1.5, +1.5]. The dashed curve represents the interference pattern for a conventional dc SQUID with unitary periodicity (DWex = 1).
two different periodicities (or pseudo-periodicities, in the case of Eq. (16b)) are evident. The latter feature can be experimentally difficult to detect, unless one can unambiguously determine the junction area Ld, assuming that the junction area S0 can be precisely measured, in such a way that a comparison between the two curves reported in Fig. 5b can effectively be made. In the following section, by taking account of shielding current effects, we shall consider this question in more details. In particular, we shall see how circulating
currents in the system further affect the ic vs. Wex curves.
4. Critical current in the presence of shielding currents In this section, we shall consider the effects due to the shielding currents in the superconducting regions. Let us however start by considering the most general case, in which not only the shielding
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currents but also the residual branch currents IR(x) and IL(x) are taken into account, and let us derive the equations governing space variation of the superconducting phase differences uR,L(x). By differentiating both sides of Eqs. (9) and by Eqs. (2,3), we get the static sine-Gordon equation d2 uR;L 1 ¼ 2 sin uR;L ðxÞ; dx2 k0
ð17Þ
where the effective penetration length k0 is rescaled qffiffiffiffiffiffiffiffiffiffiffiffi U0 ¼ with respect toqthe usual expression k J ffiffiffiffiffiffiffiffiffiffiffi 2pl0 J 0 d l Ld by the factor 2l1 0W , which can be thought to be given by the ratio of the inductance of a thin hollow cylinder of effective section Ld and height W and the combined inductance 2l of the current paths along the junctions. Eq. (17) may be solved, in terms of Jacobian elliptic functions [7], with the following boundary conditions duR;L ð0Þ 4pl ¼ k a I 2;1 ; U0 L dx
ð18Þ
given by evaluating Eq. (9) at x = 0, where the minus sign and the index 2 pertains to the RJJ and the plus sign and the index 1 to the LJJ. For a 0-SQUID, for example, by symmetry reasons we can take I 1 ¼ I 2 ¼ I2B , so that the boundary conditions (18) can be expressed in terms of IB. Therefore, when trying to maximize IB, in order to obtain the value of the critical current of the device, we need to consider a self-consistent solution of Eq. (10), given that the bias current is also contained on the right hand side in the expressions for the phase differences, because of Eq. (18). This is a rather involved problem, which will be treated in future work. Here, however, let us consider the contribution due solely to shielding currents. By neglecting the contribution due to the branch currents, we replace k ! ka in Eq. (15), so that the expressions for the critical current of the device become, in this case: I c ¼ 2I 0 j cosðpð1 þ rÞð1 þ aÞWex Þj sinðrð1 þ aÞpWex Þ ð0 SQUIDÞ; ð19aÞ rð1 þ aÞpWex I c ¼ 2I 0 j sinðpð1 þ rÞð1 þ aÞWex Þj sinðrð1 þ aÞpWex Þ ðp SQUIDÞ; ð19bÞ rð1 þ aÞpWex
The expressions given in Eqs. (19a) and (19b) are only slightly different from those in Eqs. (16a) and (16b), respectively. In order to detect the salient differences between each pair of relations, we report, in Fig. 6a and in b, the normalized value of the critical current as a function of the normalized flux Wex for a 0-SQUID and a p-SQUID, respectively. For each curve we take r = 0.5; in obtaining the full-line and the dashed-line curves we have set a = 0.25 and a = 0, respectively. Even thought the full-line curves clearly show deviations from the a = 0 case, the interference-diffraction pattern in the presence of shielding currents is seen to be only ‘‘compressed’’ in a smaller interval of variation. Indeed, if we consider the zeroes of the diffraction factor for a 0-SQUID, for example, we may write: W ex ¼
2n 1 1 ; 2 ð1 þ aÞð1 þ rÞ
ð20Þ
where n is an integer. The zeroes of the interference cosine factor, on the other hand, are Wex ¼
k ; ð1 þ aÞr
ð21Þ
k being an integer. If we attempt to take the ratio of the first zeroes (n = k = 1), we get W rþ1 ex ; ¼ 2 r Wex
ð22Þ
which does not depend on a since the zeroes in Eqs. (20,21) are both rescaled by a factor (1 + a). Naturally, the relative position of these zeroes would retain a-dependence and, for n = k = 1, we have DW0ex ¼ W ex Wex ¼
1 1 rþ2 . 2 1 þ a rðr þ 1Þ
ð23Þ
In general, however, experimental data do not possess knowledge about this compression a priori, so that the factor (1 + a), which takes account of the contribution due to the shielding currents in the system, can easily be masked by the effect of the coefficient r. For this reason it could be difficult to discriminate between the contribution due to the junction area and the one due to the shielding currents. As in the previous section, we conclude that, in order to discriminate between these
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203
2
1.5
ic
1
0.5
0 −2
−1
(a)
0
1
2
0
1
2
Ψex
ic
1.5
1
0.5
0 −2
−1
(b)
Ψex
Fig. 6. Normalized critical current ic versus the normalized applied flux Wex for a 0-SQUID (a) and a p-SQUID (b) in the presence of shielding currents for the following choice of parameters: r ¼ SS0J ¼ 0:5, a = 0.4 (full line). The dashed curve is obtained by setting a = 0 (no shielding current effect).
effects, an unambiguous measure of the effective area Ld of the JJ is of paramount importance.
5. Conclusions We have calculated the critical current Ic in terms of the external field H in a two-junction interferometer where the Josephson junctions are not necessarily considered as point-like objects. This analysis has been made possible by a schematic model for the current distribution in the system. This model is seen to reproduce already
known results in the case of negligibly small contributions of the circulating currents to the magnetic flux present in the region close to the junction barrier. In this case, indeed, the extended single junction model with non-uniform maximum Josephson current density [2] states that, in analogy with optics, the Ic vs. H curves are seen to be given by the product of the Fraunhofer-like diffraction pattern, due to the finiteness of the JJÕs, and the well known interference pattern observed in conventional dc SQUIDÕs. Differently from what happens in classical physics, however, a term proportional to the junction flux-penetrated area
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in the argument of the cosine function modulating the quantum interference pattern must be present. The appearance of this term is a consequence of the Bohm–Aharonov relation and can be qualitatively explained as a countersign of the coherence of the superconducting state [2]. The model is then applied to take account of the effect of the currents circulating in the system, and static sine-Gordon equations are seen to govern the field distribution in the two JJÕs. Solution of the problem in this general case has been postponed to future work and only shielding current effects are considered. In this simple case, the resulting interference and diffraction patterns for 0-SQUIDÕs and p-SQUIDÕs are seen to suffer a contraction with respect to the horizontal field axis. It is noted that this feature can be masked by the effect of the parameter r, which is the ratio of the effective junction area and the area of the
inner normal region. We therefore conclude that, in order to discriminate additional contribution due to shielding currents, a precise measurement of the effective junction area should be performed. References [1] R.C. Jaklevic, J. Lambe, A.H. Silver, J.E. Mercereau, Phys. Rev. Lett. 12 (1964) 159, and 274. [2] A. Barone, G. Paterno`, Physics and applications of the Josephson effect, Wiley, New York, 1982. [3] J. Clarke, in: H. Weinstock, R.W. Ralston (Eds.), The New Superconducting Electronics, Kluwer Academic, 1993. [4] H. Weinstock (Ed.), SQUID sensors: Fundamentals, fabrication and applications, Kluwer Academic, 1996. [5] F. Romeo, R. De Luca, Phys. Lett. A 328 (2004) 330. [6] R.A. Ferrel, R.E. Prange, Phys. Rev. Lett. 10 (1963) 479. [7] C.S. Owen, D.J. Scalapino, Phys. Rev. 164 (1967) 538. [8] C.W. Schneider, G. Hammerl, G. Logvenov, T. Kopp, J.R. Kirtley, P.J. Hirschfeld, J. Mannhart, Europhys. Lett. 68 (2004) 86.