Response of thin-film SQUIDs to applied fields and vortex fields: Linear SQUIDs John R. Clem Ames Laboratory - DOE and Department of Physics and Astronomy, Iowa State University, Ames Iowa 50011 Ernst Helmut Brandt arXiv:cond-mat/0507524 v2 25 Sep 2005 Max-Planck-Institut für Metallforschung, D-70506 Stuttgart, Germany (Dated: September 27, 2005) In this paper we analyze the properties of a dc SQUID when the London penetration depth λ is larger than the superconducting film thickness d. We present equations that govern the static behavior for arbitrary values of Λ = λ2 /d relative to the linear dimensions of the SQUID. The SQUID’s critical current Ic depends upon the effective flux Φ, the magnetic flux through a contour surrounding the central hole plus a term proportional to the line integral of the current density around this contour. While it is well known that the SQUID inductance depends upon Λ, we show here that the focusing of magnetic flux from applied fields and vortex-generated fields into the central hole of the SQUID also depends upon Λ. We apply this formalism to the simplest case of a linear SQUID of width 2w, consisting of a coplanar pair of long superconducting strips of separation 2a, connected by two small Josephson junctions to a superconducting current-input lead at one end and by a superconducting lead at the other end. The central region of this SQUID shares many properties with a superconducting coplanar stripline. We calculate magnetic-field and current-density profiles, the inductance (including both geometric and kinetic inductances), magnetic moments, and the effective area as a function of Λ/w and a/w. PACS numbers: 74.78.-w I. INTRODUCTION The research in this paper has been motivated by several important recent developments in superconductivity: (a) the fabrication of thin-film SQUIDs (superconducting quantum interference devices) made of high-Tc superconductors,1 (b) the study of noise generated by vortices in active and passive superconducting devices,1,2,3,4,5,6,7,8,9,10 and (c) line-width reduction in superconducting devices to eliminate noise due to vortices trapped during cooldown in the earth’s magnetic field.11,12,13,14,15,16 The fact that the London penetration depth λ increases as T increases and diverges at Tc is an important consideration for high-Tc SQUIDs operated at liquidnitrogen temperature. When λ is larger than the film thickness d, the physical length that enters the equations governing the spatial variation of currents and fields is the Pearl length17 Λ = λ2 /d. Accordingly, the equations governing the behavior of active or passive thin-film superconducting devices depend upon the ratio of Λ to the linear dimensions of the device. In particular, the equations governing SQUIDs involve not just the magnetic flux up through a contour within the SQUID, but the effective flux Φ, which is the sum of the magnetic flux and a term proportional to the line integral of the current density around the same contour. While the effective flux Φ is similar to London’s fluxoid,18 which is quantized in multiples of the superconducting flux quantum φ0 = h/2e, we show in the next section that Φ is not quantized. We also give in Sec. II the basic equations, valid for any value of Λ, that govern the behavior of a dc SQUID. A vortex trapped in the body of the SQUID during cooldown through the superconducting transition temperature Tc in an ambient magnetic field generates a magnetic field and a screening current that together make a sizable vortex-position-dependent contribution Φv to the effective flux Φ. If such a vortex remains fixed in position and the temperature remains constant, this simply produces a harmless bias in Φ. On the other hand, both vortex motion due to thermal agitation and temperature fluctuations generate corresponding fluctuations in Φv and noise in the SQUID output. In Sec. II we show that to calculate Φv , it is not necessary to calculate the spatial dependence of the vortex-generated fields and currents. Instead, one may determine Φv with the help of the sheet-current distribution of a circulating current in the absence of the vortex. In Sec. III we apply the basic equations of Sec. II to calculate the properties of a model linear SQUID, which has the basic topology of a SQUID but is greatly stretched along one axis, such that the central portion resembles a coplanar stripline. The advantage of using such a model is that simple analytical results can be derived that closely approximate the exact numerically calculated quantities in the appropriate limits. In addition to calculating the field and current distributions for several values of Λ, we calculate the total inductance, geometric inductance, kinetic inductance, and magnetic moment when the SQUID carries a circulating current. We calculate the field and current distributions, magnetic moment, and effective area Aeff = Φf /Ba when a perpendicular magnetic induction Ba is applied and the effective flux Φf is focused into the SQUID. Finally, we calculate the field and current distributions and the magnetic moment for the zero-fluxoid state when the junctions are 2 short-circuited and the sample remains in the state with Φ = 0 when a perpendicular magnetic induction Ba is applied. In Sec. IV, we present a brief summary of our results. II. I BASIC EQUATIONS y Our purpose in this section is to derive general equations that govern the behavior of a dc SQUID consisting of thin superconducting films of thickness d less than the weak-field London penetration depth λ, such that the fields and currents are governed by the two-dimensional screening length or Pearl length17 Λ = λ2 /d. We calculate both the current I = I1 + I2 through the SQUID [see Fig. 1] and the circulating current19 Id = (I2 − I1 )/2 and describe how to calculate the critical current Ic of the SQUID for arbitrary values of the SQUID’s inductance L. For this case, the contributions from line integrals of the current density to the effective flux in the hole cannot be neglected, and the kinetic inductance makes a significant contribution to L. When a perpendicular magnetic induction Ba is applied, we calculate how much magnetic flux is focused into the SQUID’s hole; this flux also can be expressed in terms of the effective area20 of the hole. We also show how to calculate how much magnetic flux generated by a vortex in the main body of the SQUID is focused into the hole. Consider a dc SQUID in the xy plane, as sketched in Fig. 1. We suppose that the SQUID is symmetric about the y axis, which lies along the centerline. The maximum Josephson critical current is I0 for each of the Josephson junctions, shown as small black squares. The currents up through the left and right sides of the SQUID can be written as I1 = I/2 − Id and I2 = I/2 + Id . When the magnitude of I, the total current through the SQUID, is less than the critical current Ic , the equations that determine I = I1 + I2 and the circulating current Id = (I2 − I1 )/2 can be derived using a method similar to that used in Ref. 21. We begin by writing the local current density j in the superconductors (i.e., the main body of the SQUID and the counterelectrode) as18 j = −(1/µ0 λ2 )[A + (φ0 /2π)∇γ], (1) where A is the vector potential, φ0 = h/2e is the superconducting flux quantum, and γ is the phase of the order parameter. The quantity inside the brackets, which is gauge-invariant, can be thought of as the superfluid velocity expressed in units of vector potential. From the point of view of the Ginzburg-Landau theory, implicit in the use of this London-equation approach is the assumption that the applied fields and currents are so low that the magnitude of the order parameter is not significantly reduced from its equilibrium value in the absence of fields and currents. To obtain the SQUID equations, we integrate the vector potential around a contour C that passes in a counterclockwise direction through both junctions, the main I1 I2 x C b a I FIG. 1: I enters the main body of the SQUID from the counterelectrode below through two Josephson junctions (small black squares, labeled a and b) and divides into the currents I1 = I/2 − Id and I2 = I/2 + Id as shown. body of the SQUID, and the counterelectrode as shown in Fig. 1 and write the result in two ways. Since B = ∇×A, this integral yields, on the one hand, the magnetic flux in the z direction Z Bz (x, y)dS, (2) S where S is the area surrounded by the contour C and Bz (x, y) is the z component of the net magnetic induction in the plane of the SQUID produced by the sum of a perpendicular applied field Ba and the self-field Bsz (x, y) generated via the Biot-Savart law by the supercurrent density j(x, y). On the other hand, for those portions of the contour lying in the superconductors, we eliminate the line integrals of A in favor of line integrals of j using Eq. (1). We then express the line integrals of ∇γ in terms of the values of γ at the junctions. Equating the two expressions for the line integral of A, we find that the effective flux Φ in the z direction through the SQUID is given by Φ = (φ0 /2π)(φ1 − φ2 ), (3) where Φ= Z S Bz (x, y)dS + µ0 λ2 Z C j · dl, (4) with the integration contour C now passing through both superconductors but excluding the junction barriers. These equations are equivalent to Eq. (8.67) in 3 Ref. 21. The gauge-invariant phase differences across the junctions b and a are, respectively,22 φ1 = γbc − γbs − (2π/φ0 ) Z bs A · dl, Zbcas φ2 = γac − γas − (2π/φ0 ) A · dl, (5) (6) ac where bc labels a point on the counterelectrode side of the junction b and bs labels the point directly across the insulator in the SQUID washer, and ac and as label corresponding points for junction a. According to the Josephson equations,22 the junction supercurrents are I1 = I0 sin φ1 and I2 = I0 sin φ2 . In the above derivation we have assumed that the linear dimensions of the Josephson junctions are much less than the Josephson penetration depth λJ ,22 and that the applied fields are sufficiently small that the Josephson current densities and gauge-invariant phase differences are very nearly constant across the junction areas. The magnetic moment m = mẑ generated by the currents in the SQUID is23 Z 1 m= r × jd3 r. (7) 2 It can be shown with the help of the London fluxoid quantization condition,18 Z Z 2 Bz (x, y)dS + µ0 λ j · dl = nφ0 , (8) S′ C′ where n is an integer and C ′ is a closed contour that surrounds an area S ′ within the body of the SQUID, that if there are no vortices present (i.e., when n = 0), the expression for Φ in Eq. (4) is independent of the choice of contour C. Any convenient path can be chosen for C, provided only that the path remains in the superconducting material in the body of the SQUID and the counterelectrode. On the other hand, when there are vortices in the main body of the SQUID, the quantity Φ increases by φ0 each time the contour C is moved from a path inside the vortex axis to one enclosing the vortex axis. Thus, without specifying the precise contour C, Φ is determined only modulo φ0 . However, this is of no physical consequence, because the gauge-invariant phases φ1 and φ2 , which also enter Eq. (3), are also determined only modulo 2π. The final equations determining the currents I and Id are independent of the choice of contour C and remain valid even when vortices are present in the main body of the SQUID. When the thickness d of the SQUID is much larger than λ, the contours C and C ′ can be chosen to be at the midpoint of the thickness, where j is exponentially small, such that the line integrals of j can be neglected. The resulting equations are then the familiar ones found in many reference books, such as Refs. 21,24,25,26,27,28,29. However, we are interested here in the case for which d < λ, such that the fields and currents are governed by the two-dimensional screening length or Pearl length17 Λ = λ2 /d. The term in Eq. (4) involving j then must be carefully accounted for. For this case, j is very nearly constant over the thickness and it is more convenient to deal with the sheet-current density J(x, y) = jd, such that Eqs. (4) and (8) take the form30 Z Z J · dl (9) Bz (x, y)dS + µ0 Λ Φ= C S and Z Bz (x, y)dS + µ0 Λ Z C′ S′ J · dl = nφ0 . (10) For the general case when the SQUID is subject to a perpendicular applied magnetic induction Ba , carries a current I unequally divided between the two arms, I1 = I/2−Id and I2 = I/2+Id, where the circulating current19 is Id = (I2 − I1 )/2, and contains a vortex at the position rv in the body of the SQUID, the effective flux Φ in the z direction can be written as the sum of four independent contributions: Φ = Φ I + Φd + Φf + Φv , (11) The first term on the right-hand side of Eq. (11) is that which would be produced by equal currents I/2 in the y direction on the left and right sides of the SQUID shown in Fig. 1: Z Z JI · dl, (12) BI (x, y)dS + µ0 Λ ΦI = S C where BI (x, y) is the z component of the self-field generated via the Biot-Savart law by the sheet-current density JI (x, y), subject to the condition that the same current I/2 flows through the two contacts a and b. For a symmetric SQUID, JI (x, y), the y component of JI (x, y), is then an even function of x, and JI (x, y) and BI (x, y) are odd functions of x. As a result, both terms on the righthand side of Eq. (12) vanish by symmetry, and ΦI = 0. Since ∇ · JI = 0 except at the contacts a and b, we may write JI = −(I/2)∇ × GI , where GI = ẑGI , such that JI (x, y) = (I/2)ẑ×∇GI (x, y). The contours of the scalar stream function GI (x, y) = const correspond to streamlines of JI (x, y), and we may choose GI = 0 for points ri = (xi , yi ) all along the inner edges of the superconductors and GI = 1 for points ro = (xo , yo ) all along the outer right edges and GI = −1 for points ro = (xo , yo ) all along the outer left edges. The second term on the right-hand side of Eq. (11) is due to the circulating current19 Id = (I2 − I1 )/2 in the counterclockwise direction when unequal currents flow in the two sides of the SQUID shown in Fig. 1: Z Z Jd · dl, (13) Bd (x, y)dS + µ0 Λ Φd = S C where Bd (x, y) is the z component of the self-field generated via the Biot-Savart law by the circulating sheetcurrent density Jd (x, y) when a current Id flows through 4 contact a from the counterelectrode into the body of the SQUID, passes around the central hole, and flows through contact b back into the counterelectrode. The magnetic moment md generated by the circulating current is proportional to Id , as can be seen from Eq. (7). Since ∇ · Jd = 0 except at the contacts a and b, we may write Jd = −Id ∇ × Gd , where Gd = ẑGd , such that Jd (x, y) = Id ẑ × ∇Gd (x, y). The contours of the scalar stream function Gd (x, y) = const correspond to streamlines of Jd (x, y), and we may choose Gd = 0 for points ri = (xi , yi ) all along the inner edges of the superconductors and Gd = 1 for points ro = (xo , yo ) all along the outer edges. Once a numerical result for Φd is found, the result can be used to determine the inductance L of the SQUID via L = Φd /Id , as was done for a circular ring in Ref. 31. The resulting expression for L is the sum of the geometric and kinetic inductances. The third term on the right-hand side is a flux-focusing term due to the applied field: Z Z Jf · dl, (14) Bf (x, y)dS + µ0 Λ Φf = S C where Bf (x, y) is the z component of the net magnetic induction in the plane of the SQUID produced by the sum of a perpendicular applied field Ba and the z component of the self-field Bsf (x, y) generated via the Biot-Savart law by the sheet-current density Jf (x, y) induced in response to Ba , subject to the condition that no current flows through the junctions a and b. In other words, the desired fields are those that would appear in response to Ba if the junctions a and b were open-circuited. Since ∇·Jf = 0, we may write Jf = −∇×Gf , where Gf = ẑGf , such that Jf (x, y) = ẑ × ∇Gf (x, y). The contours of the scalar stream function Gf (x, y) = const correspond to streamlines of Jf (x, y), and we may chose Gf = 0 for all points (x, y) along the inner and outer edges of the superconductor. Once a numerical result for Φf is found, the result can be used to determine the effective area20 of the SQUID’s central hole, Aeff = Φf /Ba , as was done for a circular ring in Ref. 31. To prove that the effective area is also given by Aeff = md /Id ,32 we Rconsider the electromagnetic energy cross term Efd = (Bf · Bd /µ0 + µ0 λ2 jf · jd )d3 r, where the integral extends over all space. Here, Bf (r) = Ba (r) + Bsf (r) = ∇ × Af (r), where ja (r) = ∇ × Ba (r)/µ0 is the current density in the distant coil that produces a nearly uniform field Ba in the vicinity of the SQUID, jf = ∇ × Bsf /µ0 is the induced current density in the SQUID, and Bsf is the corresponding self-field under the conditions of flux focusing, i.e., when jf = 0 through the junctions. Also, Bd = ∇ × Ad is the dipole-like field distribution generated by the circulating current Id with density jd in the SQUID; at large distances from the SQUID23 Ad = µ0 md × r/4πr3 . We evaluate Efd in two ways, making use of the vector identities ∇ · (A × B) = B · (∇ × A) − A · (∇ × B) and ∇ · (γj) = γ∇ · j + ∇γ · j, and applying the divergence theorem, first with A = Ad , B = Bf , γ = γd , and j = jf , from which we obtain Efd = Ba md , and then with A = Af , B = Bd , γ = γf , and j = jd , from which we obtain Efd = Φf Id with the help of Eq. (3). Since Φf = Ba Aeff , the effective area obeys Aeff = md /Id . The fourth term on the right-hand side of Eq. (11) is due to a vortex at position rv = x̂xv + ŷyv in the body of the SQUID: Z Z Jv · dl, (15) Bv (x, y)dS + µ0 Λ Φv (rv ) = C S where Bv (x, y) is the z component of the self-field generated by the vortex’s sheet-current density Jv (x, y) via the Biot-Savart law when no current flows through the junctions a and b. The desired fields are those that would appear in response to the vortex if the junctions a and b were open-circuited. Since ∇ · Jv = 0, it is possible to express Jv (x, y) in terms of a scalar stream function, as we did for Jf (x, y) and Jd (x, y). However, as shown below, it is possible to use energy arguments to express Φv (x, y) in terms of the stream function Gd (x, y).33 To obtain Φv (r) when a vortex is at the position r = x̂x + ŷy, imagine disconnecting the counterelectrode in Fig. 1 and attaching leads from a power supply to the contacts a and b. The power supply provides a constant current Id in the counterclockwise direction, and the sheet-current distribution through the body of the SQUID is given by Jd (x, y) = Id ẑ × ∇Gd (x, y), as discussed above. We also imagine attaching leads from a high-impedance voltmeter to the contacts a and b. If the vortex moves, the effective flux Φv changes with time, and the voltage read by the voltmeter will be34 Vab = dΦv /dt. The power delivered by the power supply can be expressed in terms of the Lorentz force on the vortex, Jd × ẑφ0 = Id φ0 ∇Gd ; i.e., the rate at which work is done on the moving vortex is Id φ0 ∇Gd · dr/dt. Equating this to the power P = Id dΦv /dt = Id ∇Φv · dr/dt delivered by the power supply to maintain constant current, we obtain the equation ∇Φv (r) = φ0 ∇Gd (r). Thus Φv (r) = φ0 Gd (r)+const, where the constant can have one of two possible values depending upon whether the integration contour C is chosen to run inside or outside the vortex axis at r = x̂x+ ŷy. Choosing C to run around the outer boundary of the SQUID, we obtain Φv (r) = φ0 Gd (r). (16) Since Gd (ro ) = 1 for points r = ro on the outer edges of the SQUID and Gd (ri ) = 0 for points r = ri on the inner edges (at the perimeter of the central hole or along the edges of the slit), we have Φv (ro ) = φ0 and Φv (ri ) = 0. The derivation of Eq. (16 ) implicitly assumes that the vortex-core radius is much smaller than the linear dimensions of the SQUID. We now return to the problem of how to find the currents I and Id in the SQUID, as well as the critical current Ic . As discussed above, we have ΦI = 0 for a symmetric SQUID. For simplicity, we assume first that there are no vortices in the body of the SQUID, such that 5 y 1 0.8 0.6 Ic 2I0 -w -a a w x 0.4 0.2 0 0.1 0.2 0.3 Ff Φ0 0.4 0.5 FIG. 2: Ic /2I0 vs Φf /φ0 , calculated from Eqs. (17) and (18), for πLI0 /φ0 = 0, 1, 2, 3, 4, and 5 (bottom to top). Φv = 0 and Φ = Φf + Φd in Eq. (3), where Φd = LId . From the sum and the difference of I1 and I2 we obtain πΦ πLId f I = 2I0 cos sin φ̄, (17) + φ0 φ0 πΦ πLId f Id = −I0 sin + cos φ̄, (18) φ0 φ0 where φ̄ = (φ1 + φ2 )/2 is determined experimentally by how much current is applied to the SQUID. When φ̄ = 0, the current I is zero. As φ̄ increases, the magnitude of I increases and reaches its maximum value Ic at a value of φ̄ that must be determined by numerically solving Eqs. (17) and (18). A simple solution is obtained for arbitrary Φf only in the limit πLI0 /φ0 → 0, for which I = Ic = 2I0 | cos(πΦf /φ0 )| and Id = 0 at the critical current. For values of πLI0 /φ0 of order unity, as is the case for practical SQUIDs, one may obtain Ic for any value of Φf by solving Eq. (18) self-consistently for Id for a series of values of φ̄ and by substituting the results into Eq. (17) to determine which value of φ̄ maximizes I. Equations (17) and (18) have been solved numerically by de Bruyn Ouboter and de Waele,24 (some of their results are also shown by Orlando and Delin21 ), who showed that at Ic Ic (Φf ) Id (Φf ) I1 (Φf ) I2 (Φf ) = = = = Ic (Φf + nφ0 ) = Ic (−Φf ), Id (Φf + nφ0 ) = −Id (−Φf ), I1 (Φf + nφ0 ) = I2 (−Φf ), I2 (Φf + nφ0 ) = I1 (−Φf ), (19) (20) (21) (22) where n is an integer. Hence all the physics is revealed by displaying Ic (Φf ) over the interval 0 ≤ Φf ≤ φ0 /2, as shown in Fig. 2. When a vortex is present, Eqs. (17) and (18) still hold, except that Φf in these equations is replaced by the sum Φf + Φv . Thermally agitated motion of vortices in the body of the SQUID can produce flux noise via the term Φv (rv ) and the time dependence of the vortex position FIG. 3: Sketch of central portion of the long SQUID considered in Sec. III. rv . From Eq. (16) we see that the sensitivity of Ic to vortex-position noise is proportional to the magnitude of ∇Φv (r) = φ0 ∇Gd = Jd × ẑφ0 /Id . Thus Ic is most sensitive to vortex-position noise when the vortices are close to the inner or outer edges of the SQUID, where the magnitude of Jd is largest. These equations provide more accurate results for the vortex-position sensitivity than the approximations given in Refs. 2 and 35. So far, we have investigated how the general equations governing the behavior of a dc SQUID are altered when the contributions arising from line integrals of the current density are included. As we have shown in Ref. 31, these additional contributions are important when the Pearl length Λ is an appreciable fraction of the linear dimensions of the SQUID. We have found that the basic SQUID equations, Eqs. (17) and (18), are unaltered, except that the magnetic flux (sometimes called Φext 21 ) generated in the SQUID’s central hole by the externally applied field in the absence of a vortex is replaced by the effective flux Φf , given in Eq. (14). Similarly, we have shown that the total inductance L of the SQUID has contributions both from the magnetic induction (geometric inductance) and the associated supercurrent (kinetic inductance). We also have shown in principle how to calculate the effect of the return flux from a vortex at position rv in the body of the SQUID, and we have found that the effective flux arising from the vortex is Φv (rv ), given in Eqs. (15) and (16). To demonstrate that all the above quantities can be calculated numerically for arbitrary values of Λ, we next examine the behavior of a model SQUID as decribed in Sec. III. III. LONG SQUID IN A PERPENDICULAR MAGNETIC FIELD Here we consider a long SQUID whose thickness d is less than the London penetration depth λ and whose topology is like that of Fig. 1 but which is stretched 6 to a large length l in the y direction, as sketched in Fig. 3. SQUIDs of similar geometry have been investigated experimentally in Refs. 14,36,37,38,39. We treat here the case for which the length l is much larger than the width 2w of the body of the SQUID, and we focus on the current and field distributions in and near the left (−w < x < −a) and right (a < x < w) arms and near the center of the SQUID, where to a good approximation the current density jy is uniform across the thickness and depends only upon x, and the magnetic induction B = ∇ × A depends only upon x and z. In the equations that follow, we deal with the sheet-current density, whose component in the y direction is Jy (x) = jy (x)d. The self-field magnetic induction generated by Jy (x) is BJ (x, z) = ∇ × AJ (x, z), where AJ (x, z), the y component of the vector potential obtained from Ampere’s law, is Z µ0 C AJ (x, z) = Jy (x′ ) ln p dx′ . (23) 2π (x − x′ )2 + z 2 The integration here and in the following equations is carried out only over the strips, and C is a constant with dimensions of length remaining to be determined. In the presence of a perpendicular applied field Ba = ẑBa = ∇× Af , the total vector potential is A = AJ + Af , where Af = ŷBa x. A. Formal solutions We now use the approach of Ref. 40 to calculate the inplane magnetic-induction and sheet-current distributions appearing in Eqs. (11)-(14) in Sec. II. For all of these contributions we shall take into account the in-plane (z = 0) self-field contribution AJ (x) = AJ (x, 0) to the y component of the vector potential, where Z C µ0 dx′ . (24) Jy (x′ ) ln AJ (x) = 2π |x − x′ | We first examine the equal-current case and consider the contributions BI (x), the z component of BI (x), and JI (x), the y component of JI (x), due to equal currents I/2 in the left and right sides of the SQUID. Since JI (−x) = JI (x), the corresponding y component of the vector potential is also a symmetric function of x: AI (−x) = AI (x), where the subcripts I refer to the equal-current case. There are no flux quanta between the strips (ΦI = 0), and the second term in the brackets on the right-hand side of Eq. (1) vanishes; (φ0 /2π)∇γ = 0. However, the constant C must be chosen such that JI (x) = −AI (x)/µ0 Λ in the superconductor. Combining this equation with Eq. (24), making use ofR the symmetry JI (−x) = JI (x), and noting that w I = 2 a JI (x)dx, we obtain Z wh i I b b2 1 ′ ′ ′ ln = ln 2 ′ 2 + Λδ(x−x ) JI (x )dx 2π C 2π |x −x | a (25) for a < x < w. Here b can be chosen to be any convenient length, such as the length l or w, but not C. We now define the inverse integral kernel K sy (x, x′ ) for the symmetric-current case via Z w i h 1 b2 ′′ ′ ′′ ln ′′ 2 K sy (x, x′′ ) ′ 2 + Λδ(x − x ) dx 2π |x − x | a = δ(x − x′ ). (26) Applying this kernel to Eq. (25), we obtain b Z w I JI (x) = ln K sy (x, x′ )dx′ . 2π C a Rw Since I = 2 a JI (x)dx, we find b .Z wZ w ln =π K sy (x, x′ )dxdx′ , C a a (27) (28) such that JI (x) = I 2 Z a w .Z wZ w K sy (x, x′ )dx′ K sy (x′ , x′′ )dx′ dx′′ . a a For a ≤ x ≤ w, the stream function is Z 2 x GI (x) = JI (x′ )dx′ , I a (29) (30) and for −w ≤ x ≤ −a, GI (−x) = −GI (x). The corresponding z component of the magnetic induction BI (x) can be obtained from the Biot-Savart law or, since BI (x) = dAI (x)/dx, from Eq. (24). Note that BI (−x) = −BI (x). Although the kernel K sy (x, x′ ) depends upon b, we find numerically that JI (x) and BI (x) are independent of b. We next examine the circulating-current case and consider the contributions Bd (x), the z component of Bd (x), and Jd (x), the y component of Jd (x), due to a circulating current Id ; the current in the y direction on the right side of the SQUID is I2 = Id and that on the left side is I1 = −Id . The vector potential is still given by Eq. (24), except that we add subscripts d. However, since Jd (−x) = −Jd (x), the vector potential is now an antisymmetric function of x; Ad (−x) = −Ad (x). The circulating current is generated by the fluxoid Φd [see Eq. (13)] associated with a nonvanishing gradient of the phase γ around the loop. The second term inside the bracket on the right-hand side of Eq. (1), (φ0 /2π)∇γ, is −ŷΦd /2l for a < x < w and ŷΦd /2l for −w < x < −a. Thus, Jd (x) = −[Ad (x) − Φd /2l]/µ0Λ for a < x < w. Combining this equation with that for Ad (x), noting that the inductance of the SQUID is L = Φd /Id , and making use of the symmetry Jd (−x) = −Jd (x), we obtain Z wh x + x′ i LId 1 = µ0 ln +Λδ(x−x′ ) Jd (x′ )dx′ (31) ′ 2l 2π x−x a for a < x < w. We now define the inverse integral kernel K as (x, x′ ) for the asymmetric-current case via Z w x′′ + x′ i h 1 ′′ ′ + Λδ(x − x ) dx′′ ln ′′ K as (x, x′′ ) 2π x − x′ a = δ(x − x′ ). (32) 7 Applying this kernel to Eq. (31) and noting that Id = Rw J (x)dx, we obtain d a Jd (x) = αId Z w K as (x, x′ )dx′ (33) a and L = 2αµ0 l, (34) where α=1 .Z wZ a w K as (x, x′ )dxdx′ (35) a is a dimensionless function of a, b, w, and Λ, which we calculate numerically in the next section. For a ≤ x ≤ w, the stream function is Z 1 x Gd (x) = Jd (x′ )dx′ , (36) Id a and for −w ≤ x ≤ −a, Gd (−x) = Gd (x). In this formulation, as in Ref. 31, L = Lm + Lk is the total inductance. The geometricR inductance contriw bution Lm = 2Em /Id2 , where Em = l a Jd (x)Ad (x)dx is the stored magnetic energy, andRthe kinetic contribution w Lk = 2Ek /Id2 , where Ek = µ0 Λl a Jd2 (x)dx is the kinetic energy of the supercurrent, can be calculated using Eq. (33) from31,44,45 Z Z µ0 l w w x + x′ Lm = ln Jd (x)Jd (x′ )dxdx′ , πId2 a a x − x′ (37) Z w 2 2µ0 lΛ 2µ0 lΛ < Jd > Lk = , (38) J 2 (x)dx = Id2 a d (w − a) < Jd >2 where the brackets (<>) denote averages over the film width. We can show that Lm + Lk = L with the help of Eqs. (32) and (35). The z component of the magnetic induction Bd (x) generated by Jd (x) can be obtained from the Biot-Savart law, or, since Bd (x) = dAd (x)/dx, from Eq. (24). Note that Bd (−x) = Bd (x). When l ≫ w, the magnetic moment in the z direction generated by the circulating current Id is (to lowest order in w/l) Z w md = 2l xJd (x)dx, (39) a where the factor 2 accounts for the fact41 that the currents along the y direction and those along the x direction at the ends (U-turn) give exactly the same contribution to md , even in the limit l → ∞. To next higher order in w/l one has to replace l in Eq. (39) by l − (w − a)q, where (w − a)q/2 is the distance of the center of gravity of the x-component of the currents near each end from this end. For a single strip of width w−a, one has, e.g., q = 1/3 for the Bean critical state (with rectangular current stream lines) and q = 0.47 for ideal screening (Λ ≪ w).42 We next examine flux focusing. As discussed in Sec. II, to calculate the effective area of the slot, we need to calculate the fields produced in response to a perpendicular applied magnetic induction Ba , subject to the condition that no current flows through either junction. Since this is equivalent to having both junctions open-circuited, the problem reduces to finding the fields produced in the vicinity of a pair of long superconducting strips connected by a superconducting link at only one end, i.e., when the slot of width 2a between the two strips is open at one end. However, the desired fields may be regarded as the superposition of the solutions of two separate problems when the slot has closed ends: (a) the fields generated in response to Ba , when Φ = 0 (the zero-fluxoid case) and a clockwise screening current flows around the slot [second term on the right-hand side of Eq. (41) below], and (b) the fields generated in the absence of Ba , when flux quanta in the amount of Φf are in the slot and a counterclockwise screening current flows around the slot [first term on the right-hand side of Eq. (41)]. The desired flux-focusing solution is obtained by setting the net circulating current equal to zero. The equations describing the fields in the flux-focusing case are derived as follows. The z component of the net magnetic induction Bf (x) is the sum of Ba and the self-field Bsf (x) generated by Jf (x). The vector potential Ay (x) is the sum of Ba x, which describes the applied magnetic induction, and the self-field contribution given by Eq. (24) but with subscripts f. Since Jf (−x) = −Jf (x), the vector potential is again an antisymmetric function of x; Af (−x) = −Af (x). The fluxoid Φf [see Eq. (14)] contributes a nonvanishing gradient of the phase γ around the loop, such that the second term inside the brackets on the right-hand side of Eq. (1), (φ0 /2π)∇γ, is −ŷΦf /2l for a < x < w and ŷΦf /2l for −w < x < −a. Equation (1) yields Jf (x) = −[Ba x + Af (x) − Φf /2l]/µ0Λ for a < x < w. Combining this equation with that for Af (x) [Eq. (24)], making use of the symmetry Jf (−x) = −Jf (x), and introducing the effective area via Φf = Ba Aeff [see Sec. II], we obtain Ba (Aeff − 2lx)/2l = Z wh x + x′ i 1 ′ + Λδ(x − x ) Jf (x′ )dx′ µ0 ln 2π x − x′ a (40) for a < x < w. We again use the inverse integral kernel K as (x, x′ ) for the asymmetric-current case [Eq. (32)] to obtain Z Ba w Aeff Jf (x) = (41) − x′ K as (x, x′ )dx′ . µ0 a 2l The effective area of the SQUID Aeff is found from the condition that the net current around the loop is zero Rw [ a Jf (x)dx = 0], which yields Z wZ w Aeff = 2αl x′ K as (x, x′ )dxdx′ . (42) a a 8 For a ≤ x ≤ w, the stream function is Gf (x) = Z 4 x Jf (x′ )dx′ , Ba= 0, I > 0 I1 = I2 = I / 2 3 4G a J, B, 4G 2 and for −w ≤ x ≤ −a, Gf (−x) = Gf (x). The spatial distribution of the resulting z component of the in-plane magnetic induction is given by Bf (x) = Ba + Bsf (x), where Bsf (x) can be obtained from the Biot-Savart law or by substituting Jf (x) into Eq. (24) and making use of Bsf (x) = dAf (x)/dx. Note that Bf (−x) = Bf (x). The resulting magnetic moment mf in the z direction can be calculated by replacing Jd by Jf in Eq. (39). In the next section we also present numerical results for field and current distributions in the zero-fluxoid case, in which I = 0, Φv = 0, and the effective flux is zero: Φ = Φd +Φf = 0. Such a case could be achieved by shortcircuiting the Josephson junctions in Fig. 1, cooling the device in zero field such that initially Φ = 0, and then applying a small perpendicular magnetic induction Ba . A circulating current J(x), given by the second term on the right-hand side of Eq. (41), would spontaneously arise in order to keep Φ = 0, as in the Meissner state. 1 equal currents (43) J 0 1 J 1 0 B 0 1 Λ/w = 0 0.03 0.1 0.3 1 −1 −2 0 0.2 0.4 0.6 0 B 0.8 B 1 1.2 x/w FIG. 4: Profiles of the sheet current JI (x), Eq. (49), stream function GI (x), Eq. (50), and magnetic induction BI (x) for the equal-current case (Ba = 0, I1 = I2 = I/2 > 0). Shown are the examples a/w = 0.3 with Λ/w = 0 (solid lines with dots), 0.03 (dot-dashed lines), 0.1 (dashed lines), 0.3 (dotted lines), and 1 (solid lines). Here B/µ0 and J are in units I2 /w and G in units I2 . where δij = 0 for i 6= j, δii = 1, and B. Qsy ij = Numerical solutions In the previous section we have presented formal solutions for the sheet-current density Jy (x) in Eqs. (29), (33), and (41), which are expressed as integrals involving the geometry-dependent inverse kernels K sy (x, x′ ) and K as (x, x′ ). As in Ref. 31 for thin rings, these integrals are evaluated on a grid xi (i = 1, 2, . . . , N ) spanning only the strip (but avoiding the edges, where the integrand may have infinities), a < |xi | < w, such that for any function Rw PN f (x) one has a f (x)dx = i=1 wi f (xi ). Here wi are the weights, approximately equal to the local spacing of the P xi ; the weights obey N i=1 wi = w − a. We have chosen the grid such that the weights wi are narrower and the grid points xi more closely spaced near the edges a and w, where Jy (x) varies more rapidly. We have accomplished this by choosing some appropriate continuous function x(u) and an auxiliary discrete variable ui ∝ i − 21 , such that wi = x′ (ui )(u2 − u1 ). We can choose x(u) such that its derivative x′ (u) vanishes (or is reduced) at the strip edges to give a denser grid there. By choosing an appropriate substitution function x(u) one can make the numerical error of this integration method arbitrarily small, decreasing rapidly with any desired negative power of the grid number N , e.g., N −2 or N −3 . For the equal-current case, Eq. (25) becomes Qsy ii = b2 1 , i 6= j , ln 2 2π |xi − x2j | 1 πb2 . ln 2π xi wi (45) The optimum choice of the diagonal term Qsy ii (i.e., with |x2i − x2j | replaced by xi wi /π for i = j, which reduces the numerical error from order N −1 to N −2 or higher depending on the grid) is discussed in Eq. (3.12) of Ref. 41 for strips and in Eq. (18) of Ref. 31 for disks and rings. The superscript (sy) is a reminder that this is for a symmetric current distribution [JI (−x) = JI (x)]. sy −1 Defining Kij = (wj Qsy , such that ij + Λδij ) N X sy Kik (wj Qsy kj + Λδkj ) = δij , (46) k=1 and applying it to Eq. (44), we obtain JI (xi ) = Since I = 2 PN i=1 ln N b X I K sy . ln 2π C j=1 ij (47) wi JI (xi ), we find b C =π N N X .X sy wi Kij , (48) i=1 j=1 such that b I ln = 2π C N X j=1 N (wj Qsy ij + Λδij )JI (xj ) (44) JI (xi ) = N N I X sy . X X sy K wk Kkl . 2 j=1 ij k=1 l=1 (49) 9 Λ/w = 0 4 Ba= 0, I = 0 − I1 = I2 > 0 3 circulating currents J, B, 4G 1 0 2 Λ/w = 0.01 4G 1 1 J B 0 0 1 0 Λ/w = 0 0.03 0.1 0.3 1 −1 Λ/w = 0.1 −2 0 0.2 0.4 0.6 1 0 B 0.8 B 1 1.2 x/w FIG. 6: Profiles of Jd (x), Eq. (54), Gd (x), Eq. (57), and magnetic induction Bd (x) for the circulating-current case (Ba = 0, −I1 = I2 > 0). Shown are the examples a/w = 0.3 with Λ/w = 0 (solid lines with dots), 0.03 (dot-dashed lines), 0.1 (dashed lines), 0.3 (dotted lines), and 1 (solid lines). B/µ0 and J are in units I2 /w and G in units I2 . Λ/w = 50 as −1 Defining Kij = (wj Qas , such that ij + Λδij ) FIG. 5: Magnetic field lines in the equal-current case for a/w = 0.1 and Λ/w = 0, 0.01, 0.1, and 50 (or ∞). N X as Kik (wj Qas kj + Λδkj ) = δij , (53) k=1 It is remarkable that although the parameter b appears in Eq. (45), the final result for JI (xi ) in Eq. (49) does not depend upon b. The stream function GI (x) can be evaluated as43 GI (xi ) = i X N X N X N .X sy wj Kjk j=1 k=1 applying it to Eq. (51), and noting that Id PN i=1 wi Jd (xi ), we obtain Jd (xi ) = αId sy wl Klm . LId (wj Qas = µ0 ij + Λδij )Jd (xj ) 2l j=1 (54) and Shown in Fig. 4 are plots of JI (x), GI (x), and the corresponding magnetic induction BI (x) vs x for a/w = 0.3 and various values of Λ/w = 0, 0.03, 0.1, 0.3, 1. The curves for Λ = 0 exactly coincide with the analytic expressions of Appendix A. The magnetic field lines for this case are depicted in Fig. 5. For the circulating-current case, Eq. (31) becomes N X as Kij j=1 (50) l=1 m=1 N X = L = 2αµ0 l, (55) N X N .X (56) where α=1 as wi Kij . i=1 j=1 The stream function Gd (x) can be evaluated as43 (51) Gd (xi ) = α i X N X sy wj Kjk . (57) j=1 k=1 where 1 xi + xj ln , i 6= j , 2π |xi − xj | 4πxi 1 , ln = 2π wi Qas ij = Qas ii (52) The superscript (as) is a reminder that this is for an asymmetric current distribution [Jd (−x) = −Jd (x)]. Shown in Fig. 6 are plots of Jd (x), Gd (x), and Bd (x) vs x for a/w = 0.3 and various values of Λ/w. Note that these curves look similar to those in Fig. 4, but they all have opposite parity, as can be seen from the different profiles B(x) near x = 0. The magnetic field lines for this case are shown in Fig. 7. As discussed in Sec. II, when a vortex is present in the region a < |x| < w, the sensitivity of the SQUID’s 10 Λ/w = 0 3 (a) Λ/w=1 0.3 2.5 0.1 L / ( µ0 l ) 0.03 Λ/w = 0.01 2 1.5 Λ/w = 0 1 0.5 Λ/w = 0.1 0 0 FIG. 7: Magnetic field lines in the circulating-current case for a/w = 0.1 and Λ/w = 0, 0.01, 0.1, and 50 (or ∞). Lm / ( µ0 l ) Λ/w = 50 2.5 Lk / ( µ0 l ), 3 1.5 0.2 0.4 (b) a/w 0.6 0.8 Lk 1 L k Λ/w=1 0.3 0.1 2 L m 1 0.5 1, 0.1, 0 Λ / w = 0.03 0 0 critical current Ic is proportional to the magnitude of dΦv /dx = φ0 dGd /dx = φ0 Jd (x)/Id . From Fig. 6 we see that when Λ ≪ w, this sensitivity is greatly enhanced when the vortex is close to the edges a and w but that when Λ ≥ w, the sensitivity is nearly independent of position. Shown as the solid curves in Fig. 8(a) are plots of the inductance L vs a/w for various values of Λ/w = 0, 0.03, 0.1, 0.3, and 1. The solid curves in Fig. 9(a) show the same L vs Λ/w (range 0.0045 to 2.2) for several values of a/w = 0.01, 0.1, 0.4, 0.8, 0.95, and 0.99. The geometric and kinetic contributions Lm and Lk can be calculated separately from Eqs. (37) and (38) Lm = N N 2µ0 l X X wi wj Qas ij Jd (xi )Jd (xj ) , Id2 i=1 j=1 N Lk = (58) N 2µ0 lΛ X X wi Jd2 (xi ) , Id2 i=1 j=1 (59) using Eqs. (54) and (56). We can show that Lm + Lk = L using the property of inverse matrices that M · M −1 = M −1 · M = I, where I is the identity matrix. Shown as solid curves in Fig. 8(b) are Lm and Lk vs a/w. For Λ = 0, when Lk = 0, L = Lm exactly coincides with Eq. (A10), which may be approximated by Eq. (A11) for a/w < 0.7 [open circles in Fig. 8(a)] and by Eq. (A12) for a/w > 0.7 [open squares in Fig. 8(a)]. The dotted curves in Fig. 8(b) for Lm and Lk are those 0.2 0.4 a/w 0.6 0.8 1 FIG. 8: (a) Solid curves show the inductance L = Lm + Lk [Eq. (55)] vs a/w calculated for Λ/w = 0, 0.03, 0.1, 0.3, and 1. See the text for descriptions of analytic approximations shown by the open circles, open squares, dots, and dashes. (b) Solid curves show the geometric inductance Lm [Eq. (58)] vs a/w for Λ/w = 0, 0.1, and 1 and the kinetic inductance Lk [Eq. (59)] vs a/w for Λ/w = 0.03, 0.1, 0.3, and 1. See the text for descriptions of analytic approximations shown by the dotted and dashed curves. of Eqs. (C3) and (C6) in the limit Λ/w → ∞, when the circulating current density is uniform. The dotted curves in Fig. 8(a) are obtained from L = Lm + Lk using the approximations of Eqs. (C3) and (C6); they are an excellent approximation to L for Λ/w ≥ 0.03 except for small values of a/w. Improved agreement for small values of Λ/w and a/w is shown by the dashed curves in Fig. 8(a), which show the approximation of Eq. (B7) for L, and in Fig. 8(b), which show the approximation of Eq. (B12) for Lk . The solid curves in Fig. 9(b) show Lm and Lk vs Λ/w. The geometric inductance Lm depends upon Λ but only weakly, varying slowly between its Λ = 0 asymptote [Eq. (A10), horizontal dot-dashed line] and its Λ = ∞ asymptote [Eq. (C3), horizontal dotted line]. For larger values of of a/w, Lm is nearly independent of Λ. On the other hand, the kinetic inductance Lm , is approxi- 11 1 1 10 (a) 0.9 a / w = 0.99 0.95 magnetic moment for the circulating − current case 0.8 0.7 L / ( µ0 l ) 0.8 0.4 0.6 0.1 m 0.01 10 0.5 Λ / w = 10 1 0.3 0.1 0.03 0.01 0.003 0 0.4 0 10 0 0.3 0.2 0.1 −2 −1 10 10 Lk (b) a/w= 0 0 0 10 Λ/w Lk Lm / ( µ0 l ) 0.2 0.3 0.8 10 Lm 0.4 0.1 0.6 0.7 0.8 0.9 1 FIG. 10: The magnetic moment md for the circulating current case (Ba = 0, −I1 = I2 = Id > 0) plotted versus a/w for various values of Λ/w in units 2wlId . These curves coincide with those in Fig. 13 below. Lm 0.01 4 −1 10 B −1 10 Λ/w 0 10 FIG. 9: (a) Solid curves show the inductance L = Lm + Lk [Eq. (55)] vs Λ/w for a/w = 0.01, 0.1, 0.4, 0.8, 0.95, and 0.99. See the text for descriptions of analytic approximations shown by the dotted and dashed curves. (b) Solid curves show both the geometric inductance Lm [Eq. (58)] and the kinetic inductance Lk [Eq. (59)] vs Λ/w for a/w = 0.01, 0.1, 0.4, 0.8, 0.95, and 0.99. See the text for descriptions of analytic approximations shown by the dotted, dashed, and dot-dashed curves. mately proportional to Λ/w. The straight dotted lines in Fig. 9(b), calculated from the large-Λ approximation given in Eq. (C6), are a good approximation to Lm except for small values of Λ/w and a/w. The dotted curves in Fig. 9(a) are obtained from L = Lm + Lk using the approximations of Eqs. (C3) and (C6). Improved agreement for small values of Λ/w and a/w is shown by the dashed curves in Fig. 9(a), which show the approximation of Eq. (B7) for L, and in Fig. 9(b), which show the approximation of Eq. (B12) for Lk . In Eq. (3) of Ref. 44, Yoshida et al. derived an approximate expression for the kinetic inductance when Λ/w ≪ 1. We have found that their expression for Lk is not an accurate approximation to our exact numerical results. To eliminate the logarithmic divergences due to the inverse square-root dependence of the current density near the edges, Yoshida et al. followed an J, B, −2G −2 0.5 0.99 0 10 0.4 a/w 0.95 Lk / ( µ0 l ), 0.1 B >0 a I = I =0 J Λ/w = 0 3 0.03 0.1 0.3 2 1 1 B B 2 flux focusing 0 0 1 1 0 1 −1 0 −2G J −2 0 0.2 0.4 0.6 x/w 0.8 1 1.2 FIG. 11: Profiles Jf (x), Eq. (62), Gf (x), Eq. (64), and magnetic induction Bf (x) for the flux-focusing case (Ba > 0, I1 = I2 = 0). Shown are the examples a/w = 0.3 with Λ/w = 0 (solid lines with dots), 0.03 (dot-dashed lines), 0.1 (dashed lines), 0.3 (dotted lines), and 1 (solid lines). B and µ0 J are in units Ba , and G in units wBa /µ0 . approach used by Meservey and Tedrow,45 and chose a cutoff length of the order of d, the film thickness. When d < λ, however, this approach cannot be correct, because the equations describing the fields and currents in superconducting strips contain only the two-dimensional screening length Λ = λ2 /d. The cutoff length therefore must instead be chosen to be of the order of Λ, as we have done in Appendix B. The magnetic moment in the z direction generated by 12 1 Λ/w = 0 0.9 0.8 Aeff / 2wl 0.7 Λ/w = 0.01 Λ / w = 10 0.6 Λ / w >> 1 10 0.3 0.1 0.03 0.01 0.003 0.001 0.5 0.4 0.1 0.01 0 0.3 0.2 Λ/w=0 0 0.1 Λ/w = 0.1 0 0 0.1 0.2 0.3 0.4 a/w, 0.5 0.6 0.7 0.8 0.9 1 20*a / w + 0.3 FIG. 13: The effective area Aeff , Eq. (63), plotted versus the gap half width a/w for several values of Λ/w = 0, 0.001, 0.003, 0.01, 0.03, 0.1, 0.3, and 10. The lower-right curves show the same data shifted and stretched along a/w. The dots show the exact result (A19) in the limit Λ/w → 0. For Λ/a ≥ 10 one has Aeff /2wl ≈ (1 + a/w)/2, Appendix C. These curves coincide with Fig. 10 since Aeff = md /Id . Λ/w = 0.5 0.6 FIG. 12: Magnetic field lines in the flux-focusing case for a/w = 0.1 and Λ/w = 0, 0.01, 0.1, and 0.5. 0.55 a / w = 0.2 0.5 0.45 md = 2 N X wi xi Jd (xi ). Aeff / 2wl the circulating current is, from Eq. (39), (60) i=1 0.1 0.4 0.03 0.35 0.3 0.01 As shown in Fig. 10, this magnetic moment vanishes very slowly when the gap width and Λ go to zero, a/w → 0 and Λ/w → 0. This can be explained by the fact that for Λ = 0 and a < x ≪ w one has Jd (x) ∝ 1/x, Eq. (A5). The contribution of these small x to md , Eq. (60), stays finite due to the factor x, but the total current Id to which md is normalized, diverges when a/w → 0, thus suppressing the plotted ratio md /Id . Interestingly, the curves in Fig. 10 coincide with those in Fig. 13; see below. Expressions for md in the limits Λ/w → 0 and Λ/w → ∞ are given in Eqs. (A13) and (C7). For the flux-focusing case, Eq. (40) becomes 0.25 0.003 0.001 0.2 −5 10 −4 10 −3 Ba (Aeff /2l − xi ) = µ0 (wj Qas ij + Λδij )Jf (xj ). 10 −1 Λ/w 10 0 10 1 10 FIG. 14: The effective area Aeff , Eq. (63), plotted versus Λ/w (range 7 · 10−6 to 14) for several values of a/w = 0, 0.001, 0.003, 0.01, 0.03, 0.1, and 0.2. Same data as in Fig. 13. where, since PN i=1 wi Jf (xi ) = 0, the effective area is Aeff = 2αl N X −2 10 N X N X as wi Kij xj . (63) i=1 j=1 (61) j=1 Applying Eq. (53), we obtain The stream function Gf (x) can be evaluated as43 Gf (xi ) = i X wj Jf (xj ). (64) j=1 Jf (xi ) = N N Ba Aeff X as X as Kij xj , Kij − µ0 2l j=1 j=1 (62) Shown in Fig. 11 are plots of the flux-focusing Jf (x), Gf (x), and Bf (x) vs x for a/w = 0.3 and Λ/w = 0, 0.03, 13 5 0.3 B (0)/B ≈ (w/a) / ln(4w/a) f a 3 0.3 10 1 0 1 0.2 a/w a / w = 0.001 0.05 0.9 enhancement of B (x=0) 0.8 by flux focusing f 0 Λ/w=0 0.001 0.003 0.7 0.5 0.01 0.4 0.03 Λ/w=0 0.1 0.3 2× 10 0.1 0.2 0.3 0.4 a / w, 0.5 0.6 0.7 0.8 −2 −1 10 10 Λ/w 0 1 10 10 1 5*a / w + 0.35 FIG. 15: The minimum of the magnetic induction in the fluxfocusing case, Bf (0) = Bf (x = 0), referred to the applied field Ba and plotted versus the half gap width a/w. Top: The ratio Bf (x = 0)/Ba , tending to unity for a/w → 1 and for Λ/w ≫ 1, and diverging for a/w → 0 when Λ = 0. Bottom: The same ratio multiplied by a/w to avoid this divergence and fit all data into one plot. The lower right plot depicts the small-gap data two times enlarged along the ordinate, and shifted and five times stretched along the abscissa. The circles show the approximation Bf (0)/Ba ≈ (w/a)/ ln(4w/a) good for a/w ≤ 0.3.46 magnetic moment for the flux − focusing case Λ/w=0 0.003 0.01 0.03 0.1 0.3 1 10 2 0.9 −3 10 2.5 0.3 10 0.2 −4 10 FIG. 16: The minimum field Bf (0) = Bf (x = 0) for the fluxfocusing case as in Fig. 15 but plotted versus Λ/w (range 7 · 10−6 to 14) for several values of a/w = 0, 0.001, 0.003, 0.01, 0.03, 0.1, and 0.2. 0.6 0.1 −5 10 1.5 −m f 0.01 0.003 0.3 a [B (0) / B ] (a / w) 0.03 0.1 0.1 1 0 0 0.2 0.15 0.1 2 0.1 0.25 f 4 0.2 for Λ=0, a/w < − 0.3 : [Bf(0) / Ba] (a / w) a 6 B (0) / B 0.35 Λ/w=0 0.003 0.01 0.03 0.01 0.3 1 10 7 1 0.1 0.5 0.3 1 10 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a/w FIG. 17: The magnetic moment mf for the flux-focusing case plotted versus a/w for various values of Λ/w in units w2 lBa /µ0 . 0.1, 0.3, and 1. The first term in Eq. (62) equals the circulating-current sheet-current density, Eq. (54), with appropriate weight factor such that the total circulating current vanishes, I1 = I2 = 0. The corresponding magnetic field lines are depicted in Fig. 12. Shown in Figs. 13 and 14 are plots of the effective area Aeff (a/w, Λ/w) versus a/w and Λ/w, respectively, in units of the maximum possible area 2wl. In the limit Λ/w → 0, Aeff is given by Eq. (A19), and when Λ/w → ∞, Aeff = l(w + a). Note in Fig. 14 that Aeff increases with increasing Λ, particularly for small gap widths 2a. Flux focusing is reflected by the fact that for small a/w → 0 the effective area Aeff of the gap tends to a constant, except in the limit Λ → 0, where it vanishes very slowly, Aeff /2wl ≈ (π/2)/ ln(4w/a) [Eq. (A19)]. When a/w → 0, the enhancement factor Aeff /2al → ∞ and thus diverges even for Λ = 0. In the limit a/w ≪ 1, Aeff (0, Λ/w) tends to a universal function [see Fig. 14]. Interestingly, Figs. 13 and 10 show identical curves; this is because the identity Aeff = md /Id holds for all values of a/w and Λ/w, as proved in general in Sec. II. Figures 15 and 16 show the minimum of the magnetic induction in the flux-focusing case, Bf (0) = Bf (x = 0) [see Fig. 11], plotted versus a/w and Λ/w, respectively. The ratio Bf (0)/Ba ≥ 1 tends to unity for a/w → 1 and for Λ/w ≫ 1, and it diverges for a/w → 0 when Λ = 0. The curve for Λ = 0 exactly coincides with the analytic expression Bf (0)/Ba = wE(k ′ )/aK(k ′ ) obtained from Eq. (A17). For a/w ≪ 1 this yields Bf (0)/Ba ≈ (w/a)/ ln(4w/a), which is a good approximation for 0 < a/w ≤ 0.3.46 The magnetic moment mf for the flux-focusing case, calculated from Eq. (60) but 14 5 4 −J zero−fluxoid case 0.03 0 3 − J, B Λ/w = 0 B >0 a I1 = − I2 > 0 Λ/w = 1 0.3 2 0.1 0.03 0 1 B B 0.1 0.3 −J 0 Λ/w = 0.01 −J 1 1 0 Λ/w = 0.1 B 0.2 0.4 0.6 x/w 0.8 1 1.2 FIG. 18: Profiles of the sheet-current density J(x) [second term in Eq. (62)] and magnetic induction B(x) generated when a perpendicular magnetic induction Ba is applied in the zero-fluxoid case when Φ = 0, I1 = −I2 > 0, and I = 0. Shown are the examples a/w = 0.3 with Λ/w = 0 (solid lines with dots), 0.03 (dot-dashed lines), 0.1 (dashed lines), 0.3 (dotted lines), and 1 (solid lines). B and µ0 J are in units Ba . with Jd (xi ) replaced by Jf (xi ), is shown in Fig. 17. Expressions for mf in the limits Λ/w → 0 and Λ/w → ∞ are given in Eqs. (A20) and (C9) Shown in Fig. 18 are profiles for the zero-fluxoid case with plots of J(x) and the corresponding B(x) generated by an applied magnetic induction Ba > 0 when the junctions are short-circuited such that Φ = 0 and I1 = −I2 > 0; for comparison see analogous profiles in Sec. 2.5 of Ref. 47 for two parallel strips and in Sec. IV of Ref. 48 and Sec. 4 of Ref. 31 for rings. That the current density J(x) in the zero-fluxoid case is given by the second term on the right-hand sides of Eqs. (41) and (62), can be seen by setting Φf = Ba Aeff = 0 in Eqs. (40), (41), (61), and (62). Depicted in Fig. 18 are the examples a/w = 0.3 with Λ/w = 0, 0.03, 0.1, 0.3, and 1. Figure 19 shows the magnetic field lines for this case and Fig. 20 the magnetic moment m. Expressions for J, B, and m for the zero-fluxoid case in the limits Λ/w → 0 and Λ/w → ∞ are given in Appendixes A and C. IV. SUMMARY In Sec. II of this paper we have presented general equations governing the static behavior of a thin-film dc SQUID for all values of the Pearl length Λ = λ2 /d, where the London penetration depth λ is larger than d, the film thickness. The SQUID’s critical current Ic depends upon the effective flux Φ, which is the sum of the magnetic flux up through a contour surrounding the central hole and a term proportional to the line integral of the current density around this contour. For a symmetric SQUID there are three important contributions to Φ: a Λ/w = 0.5 FIG. 19: Magnetic field lines in the zero-fluxoid case Φ = 0, Ba > 0, and I = 0 for a/w = 0.1 and Λ/w = 0, 0.01, 0.1, and 0.5. 3 2.5 magnetic moment for the zero−fluxoid case B > 0 a 2 −m −1 0 Λ/w=0 0.003 0.01 0.03 0.1 1.5 0.3 1 0.5 1 10 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a/w FIG. 20: The magnetic moment m for the zero-fluxoid case Φ = 0, Ba > 0, and I = 0 plotted versus a/w for various values of Λ/w in units w2 lBa /µ0 . At a = Λ = 0 one has −m = πw2 lBa /µ0 . circulating-current term Φd , a vortex-field term Φv , and a flux-focusing term Φf , all of which depend upon Λ. Since Λ is a function of temperature, an important consequence is that all of the contributions to Φ are temperaturedependent. The circulating-current term Φd can be expressed in 15 terms of the SQUID inductance L and the circulating current Id via Φd = LId . The SQUID inductance has two contributions, L = Lm + Lk , where the first term is the geometric inductance (associated with the energy stored in the magnetic field) and the second is the kinetic inductance (associated with the kinetic energy of the circulating supercurrent). Both contributions are functions of Λ, since they both depend on the spatial distribution of the current density. However, Lm depends only weakly upon Λ, because for the same circulating current Id , the energy stored in the magnetic field does not vary greatly as Λ ranges from zero to infinity. On the other hand, because the kinetic energy density is proportional to Λ, Lk is also nearly proportional to Λ, with deviations from linearity occurring only for small values of Λ/w. The vortex-field term can be written as Φv = φ0 Gd , where Gd is a dimensionless stream function describing the circulating sheet-current density Jd . Roughly speaking, when Λ is small, Ic is most strongly dependent upon the vortex position when the vortex is close to the edges of the film, but when Λ is large, Ic is equally sensitive to the vortex position wherever the vortex is. Recent experiments49 have used the relationship Φv = φ0 Gd to determine the vortex-free sheet-current density Jd (x, y) from vortex images obtained via low-temperature scanning electron microscopy.6,9 The experimental data obtained in magnetic fields up to 40 µT are in excellent agreement with numerical calculations of Jd (x, y), confirming the validity of the above relationship, even in the presence of many (up to 200) vortices in the SQUID washer. The flux-focusing term can be expressed as Φf = Ba Aeff , where Ba is the applied magnetic induction and Aeff is the effective area of the central hole of the SQUID. Although Aeff is primarily determined by the dimensions of the SQUID, it also depends upon the value of Λ. To illustrate the Λ dependence of the above quantities, in Sec. III of this paper we analyzed in detail the behavior of a long SQUID whose central region resembles a coplanar stripline. We numerically calculated the profiles of the sheet-current density, stream function, and magnetic induction in the equal-current, circulating-current, flux-focusing, and zero-fluxoid cases for various representative values of Λ. We presented plots of the inductances L, Lm , and Lk , the effective area Aeff , and the magnetic moments for these cases. Useful analytic approximations are provided for the Λ/w → 0 limit in Appendix A, for small Λ/w and a/w in Appendix B, and for the Λ/w → ∞ limit in Appendix C. We are in the process of applying the above theory to square and circular SQUIDs, using the numerical method of Ref. 50. Acknowledgments We thank D. Koelle for stimulating discussions. This work was supported in part by Iowa State University of Science and Technology under Contract No. W-7405ENG-82 with the U.S. Department of Energy and in part by the German Israeli Research Grant Agreement (GIF) No G-705-50.14/01. APPENDIX A: THE LIMIT Λ/w = 0 In the ideal-screening limit Λ/w = 0, the y component of the sheet-current density in the strips (a < |x| < w) for the equal-current case is47 JI (x) = |x| I , 2 2 π [(x − a )(w2 − x2 )]1/2 (A1) and the z component of the magnetic induction in the plane z = 0 of the strips is µ0 I x , |x| > w, (A2) 2π [(x2 − a2 )(x2 − w2 )]1/2 = 0, a < |x| < w, (A3) x µ0 I = , |x| < a, (A4) 2π [(a2 − x2 )(w2 − x2 )]1/2 BI (x) = − and the √ constant C in Eqs. (24), (25), (27), and (28) is C = w2 − a2 /2. For the circulating-current case, the y component of the sheet-current density in the strips (a < |x| < w) in the limit Λ/w = 0 is47 Jd (x) = w2 2B0 x , µ0 |x| [(x2 − a2 )(w2 − x2 )]1/2 (A5) and the z component of the magnetic induction in the plane z = 0 of the strips is w2 , |x| > w, (A6) [(x2 − a2 )(x2 − w2 )]1/2 = 0, a < |x| < w, (A7) 2 w , |x| < a, (A8) = B0 2 [(a − x2 )(w2 − x2 )]1/2 Bd (x) = −B0 where the parameter B0 , the magnetic flux Φd in the z direction in the slot, the circulating current Id , and the geometric inductance Lm are related by Φd = Lm Id = 2B0 lwK(k) (A9) Lm = µ0 lK(k)/K(k ′), (A10) and where K(k) is the complete elliptic integral of the first kind √ of modulus k = a/w and complementary modulus ′ k = 1 − k 2 . The geometric inductance is well approximated for small a/w by Lm = (πµ0 l/2)/ ln(4w/a), (A11) 16 neglecting corrections proportional to a2 /w2 , and for small (w − a)/w by Lm = (µ0 l/π) ln[16/(1 − a2 /w2 )], 2 (A12) 2 neglecting corrections proportional to 1 − a /w . In the limit that Λ = 0, the kinetic inductance vanishes (Lk = 0), and the inductance in Eq. (A10) becomes the total inductance: L = Lm . The magnetic moment [see Eq. (39)] can be obtained from Eqs. (13)-(16) of Ref. 47: md = [πlw/K(k ′ )]Id . (A13) For the flux-focusing case, the y component of the sheet-current density in the strips (a < |x| < w) in the limit Λ/w = 0 is47 Jf (x) = 2Ba x E(k ′ )w2 − 2K(k ′ )x2 , µ0 K(k ′ ) |x| [(x2 − a2 )(w2 − x2 )]1/2 (A14) and the z component of the magnetic induction in the plane z = 0 of the strips is Ba E(k ′ )w2 − 2K(k ′ )x2 , K(k ′ ) [(x2 − a2 )(x2 − w2 )]1/2 |x| > w, (A15) = 0, a < |x| < w, (A16) Ba E(k ′ )w2 − 2K(k ′ )x2 = , K(k ′ ) [(a2 − x2 )(w2 − x2 )]1/2 |x| < a, (A17) Bf (x) = − where E(k ′ ) is the complete elliptic integral √ of the second kind of complementary modulus k ′ = 1 − k 2 and modulus k = a/w. The magnetic flux in the z direction in the slot is Φf = πBa lw/K(k ′ ), (A18) and the effective area Aeff of the slot is Aeff = Φf /Ba = πlw/K(k ′ ). (A19) Note that Aeff = md /Id . The magnetic moment generated by Jf (x) is mf = −πl[w2 + a2 − 2w2 E(k ′ )/K(k ′ )]Ba /µ0 . (A20) For the zero-fluxoid case, the y component of the sheetcurrent density in the strips can be obtained from Sec. 2.5 of Ref. 47: 2Ba x x2 −[1−E(k)/K(k)]w2 J(x) = − . (A21) µ0 |x| [(x2 − a2 )(w2 − x2 )]1/2 The corresponding z component of the magnetic induction is47 x2 −[1−E(k)/K(k)]w2 B(x) = Ba , |x| > w, (A22) [(x2 − a2 )(x2 − w2 )]1/2 = 0, a < |x| < w, (A23) [1−E(k)/K(k)]w2 −x2 , |x| < a. (A24) = Ba [(a2 − x2 )(w2 − x2 )]1/2 APPENDIX B: BEHAVIOR FOR SMALL Λ AND SMALL a In this section we present some expressions for L, Lk , and Lm that follow from approximating the circulatingcurrent distribution for small values of Λ and a. When the slot is very narrow (a/w ≪ 1), we approximate the sheet-current density in the region a < x < w generated by the fluxoid Φd via w Jd (x) = I0 p , (B1) 2 2 2 (x − a + δ )(w2 − x2 + δ 2 ) where I0 = 2Φd /πµ0 l and δ is a quantity of order Λ = λ2 /d determined as follows. When l → ∞ and then a → 0 and w → ∞, an exact calculation yields for x > 0 Z ∞ −xt/Λ e dt 2Φd . (B2) Jy (x) = 2 πµ0 lΛ 0 t +1 Rb We find 0 Jy (x)dx = (2Φd /πµ0 l) ln(γb/2Λ) when b ≫ Λ, where γ = eC = 1.781..., and C = 0.577... is EuRb ler’s constant. From Eq. (B1) we find 0 Jd (x)dx = (2Φd /πµ0 l) ln(2b/δ) when a = 0 and δ ≪ b ≪ w. Comparing these two integrals we obtain δ = (4/γ)Λ = 2.246Λ. Integrating Eq. (B1) from a to w to obtain Id , we find w Id = I0 √ [F (λa , q) − F (λw , q)], (B3) w2 + δ 2 where F (φ, k) is the elliptic integral of the first kind and r w 2 − a2 + δ 2 , (B4) λa = arcsin w2 − a2 + 2δ 2 δ , (B5) λw = arcsin √ 2 w − a2 + 2δ 2 r w2 − a2 + 2δ 2 q = . (B6) w2 + δ 2 Expanding Eq. (B3) for a ≪ w and δ ≪ w, using I0 = 2Φd /πµ0 l, and neglecting terms of order a2 /w2 and δ 2 /w2 , we obtain L = Φd /Id = (πµ0 l/2)/{ln[4w/(a + δ)] − δ/w}, (B7) where δ = 2.246Λ. Note that Eq. (B7) reduces to Eq. (A11) when Λ = 0. From Eq. (B1) we obtain the approximation Z w w (fa + fw ), (B8) Jd2 (x)dx = I02 2 (w − a2 + δ 2 ) a where fa The magnetic moment generated by J(x) is m = −πl[2w2 E(k)/K(k) − w2 + a2 ]Ba /µ0 . (A25) fw √ 2 2 w −1 (w − a) δ − a tan = √ , 2 2 2 a(w − a) + δ δ −a a < δ, (B9) √ 2 2 (w − a) a − δ w tanh−1 , = √ 2 2 a(w − a) + δ 2 a −δ a > δ, (B10) √ 2 2 (w − a) w + δ w . (B11) tanh−1 = √ 2 2 w(w − a) + δ 2 w +δ 17 Using Eqs. (B3) and (B8), we obtain from Eq. (38) (w2 + δ 2 ) (fa + fw ) Λ , 2 2 2 w (w − a + 2δ ) [F (λa , q) − F (λw , q)]2 (B12) where δ = 2.246Λ. Although our intention in using the ansatz of Eq. (B1) initially was to obtain an improved approximation to Lk for small values of Λ and a, we see from Figs. 8(b) and 9(b) that Eq. (B12) provides a reasonably good approximation for all values of Λ and a. neglecting corrections proportional to (1 − a/w)2 . From Eq. (38) we obtain the kinetic inductance Lk = 2µ0 l Lk = 2µ0 lΛ/(w − a). (C6) When Λ ≫ w, the total inductance L is dominated by the kinetic inductance (Lk ≫ Lm ), such that L ≈ Lk . Since Jd is uniform, the magnetic moment is easily found from Eq. (39) to be md = l(w + a)Id . (C7) APPENDIX C: THE LIMIT Λ/w → ∞ In the weak-screening limit Λ/w → ∞, K sy (x, x′ ) = Λ−1 δ(x − x′ ), the y component of the sheet-current density in the strips (a < |x| < w) in the equal-current case is uniform, JI = I/2(w−a), the z component of the magnetic induction in the plane of the strips obtained from the Biot-Savart law is (x − w)(x + a) µ0 I BI (x) = ln (C1) , 4π(w − a) (x + w)(x − a) and the constant C in Eqs. (24), (25), (27), and (28) is C = w exp[−πΛ/(w − a)]. For the circulating-current case in the limit Λ/w → ∞, K as (x, x′ ) = Λ−1 δ(x − x′ ), α = Λ/(w − a), the y component of the sheet-current density in the strips is again uniform, Jd = Id /(w − a) for a < x < w, and the z component of the magnetic induction in the plane of the strips obtained from the Biot-Savart law is Bd (x) = x2 − w2 µ0 Id ln 2 . 2π(w − a) x − a2 (C2) The geometric inductance is, from Eq. (37) Lm = h 4w2 µ0 l 2 w ln π(w − a)2 w 2 − a2 4a2 i w + a + a2 ln −2aw ln , (C3) w−a w 2 − a2 which is independent of Λ. Equation (C3) is well approximated for small a/w by Lm = (µ0 l/π)(1 + 2a/w) ln 4, 1 i 2 3 1 + − (1 − a/w) , 1 − a/w 2 2 x2 − w2 Ba h (w + a) ln 2 4πΛ x − a2 (x + w)(x − a) i + 2x ln − 4(w − a) . (C8) (x − w)(x + a) Bsf (x) = The magnetic moment generated by Jf (x) in this limit is mf = −[l(w − a)3 /6Λ](Ba /µ0 ). (C9) For the zero-fluxoid case in the limit Λ/w → ∞, the applied field is only weakly screened, and the z component of the magnetic flux density is nearly equal to the applied magnetic induction, B(x) ≈ Ba . The y component of the vector potential is approximately given by A(x) = Ba x, and the y component of the induced sheetcurrent density, obtained from Eq. (1) with γ = 0, is J(x) = −(Ba /µ0 Λ)x. To the next order of approximation, B(x) = Ba + Bs (x), where the self-field Bs is found from the Biot-Savart law Bs (x) = (C4) neglecting corrections proportional to a2 /w2 , and for small (w − a)/w by h Lm = (µ0 l/π) ln For the flux-focusing case in the limit Λ/w → ∞, K as (x, x′ ) = Λ−1 δ(x − x′ ), the y component of the sheetcurrent density is Jf (x) = Ba (w + a − 2x)/2µ0 Λ for a < x < w, the effective area is Aeff = l(w + a) = md /Id , and the z component of the magnetic induction in the plane of the strips is Bf (x) = Ba + Bsf (x), where from the Biot-Savart law (x + w)(x − a) i Ba h x ln − 2(w − a) . (C10) 2πΛ (x − w)(x + a) The magnetic moment generated by J(x) in this limit is m = −[2l(w3 − a3 )/3Λ](Ba /µ0 ). (C11) (C5) See the review by D. Koelle, R. Kleiner, F. Ludwig, E. Dantsker, and J. Clarke, Rev. Mod. Phys. 71, 631 (1999), and references therein. 2 3 M. J. Ferrari, J. J. Kingston, F. C. Wellstood, and J. Clarke, Appl. Phys. Lett. 58, 1106 (1991). A. H. Miklich, D. 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