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Response of thin-film SQUIDs to applied fields and vortex fields: Linear SQUIDs

In this paper we analyze the properties of a dc SQUID when the London penetration depth \lambda is larger than the superconducting film thickness d. We present equations that govern the static behavior for arbitrary values of \Lambda = \lambda^2/d relative to the linear dimensions of the SQUID. The SQUID's critical current I_c depends upon the effective flux \Phi, 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 \Lambda, 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 \Lambda. 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 \Lambda/w and a/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-T c 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 T c is an important consideration for high-T c 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 length 17 Λ = λ 2 /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 φ 0 = 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 T c in an ambient magnetic field generates a magnetic field and a screening current that together make a sizable vortex-position-dependent contribution Φ v 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 Φ v and noise in the SQUID output. In Sec. II we show that to calculate Φ v , it is not necessary to calculate the spatial dependence of the vortex-generated fields and currents. Instead, one may determine Φ v 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 A eff = Φ f /B a when a perpendicular magnetic induction B a is applied and the effective flux Φ f 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 short-circuited and the sample remains in the state with Φ = 0 when a perpendicular magnetic induction B a is applied.
In Sec. IV, we present a brief summary of our results.

II. BASIC EQUATIONS
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 length 17 Λ = λ 2 /d. We calculate both the current I = I 1 + I 2 through the SQUID [see Fig. 1] and the circulating current 19 I d = (I 2 − I 1 )/2 and describe how to calculate the critical current I c 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 B a 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 area 20 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 I 0 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 I 1 = I/2 − I d and I 2 = I/2 + I d .
When the magnitude of I, the total current through the SQUID, is less than the critical current I c , the equations that determine I = I 1 + I 2 and the circulating current I d = (I 2 − I 1 )/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) as 18 where A is the vector potential, φ 0 = 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 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 where S is the area surrounded by the contour C and B z (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 B a and the self-field B sz (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 where with the integration contour C now passing through both superconductors but excluding the junction barriers. These equations are equivalent to Eq. (8.67) in Ref. 21. The gauge-invariant phase differences across the junctions b and a are, respectively, 22 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 I 1 = I 0 sin φ 1 and I 2 = I 0 sin φ 2 . In the above derivation we have assumed that the linear dimensions of the Josephson junctions are much less than the Josephson penetration depth λ J , 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 is 23 It can be shown with the help of the London fluxoid quantization condition, 18 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 φ 0 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 φ 0 . However, this is of no physical consequence, because the gauge-invariant phases φ 1 and φ 2 , which also enter Eq. (3), are also determined only modulo 2π. The final equations determining the currents I and I d 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 length 17 Λ = λ 2 /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 form 30 and For the general case when the SQUID is subject to a perpendicular applied magnetic induction B a , carries a current I unequally divided between the two arms, I 1 = I/2−I d and I 2 = I/2+I d , where the circulating current 19 is I d = (I 2 − I 1 )/2, and contains a vortex at the position r v in the body of the SQUID, the effective flux Φ in the z direction can be written as the sum of four independent contributions: 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: where B I (x, y) is the z component of the self-field generated via the Biot-Savart law by the sheet-current density J I (x, y), subject to the condition that the same current I/2 flows through the two contacts a and b. For a symmetric SQUID, J I (x, y), the y component of J I (x, y), is then an even function of x, and J I (x, y) and B I (x, y) are odd functions of x. As a result, both terms on the righthand side of Eq. The second term on the right-hand side of Eq. (11) is due to the circulating current 19 I d = (I 2 − I 1 )/2 in the counterclockwise direction when unequal currents flow in the two sides of the SQUID shown in Fig. 1: where B d (x, y) is the z component of the self-field generated via the Biot-Savart law by the circulating sheetcurrent density J d (x, y) when a current I d flows through 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 m d generated by the circulating current is proportional to I d , as can be seen from Eq. (7). Since ∇ · J d = 0 except at the contacts a and b, we may . The contours of the scalar stream function G d (x, y) = const correspond to streamlines of J d (x, y), and we may choose G d = 0 for points r i = (x i , y i ) all along the inner edges of the superconductors and G d = 1 for points r o = (x o , y o ) all along the outer edges. Once a numerical result for Φ d is found, the result can be used to determine the inductance L of the SQUID via L = Φ d /I d , 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: where B f (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 B a and the z component of the self-field B sf (x, y) generated via the Biot-Savart law by the sheet-current density J f (x, y) induced in response to B a , 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 B a if the junctions a and b were open-circuited. Since . The contours of the scalar stream function G f (x, y) = const correspond to streamlines of J f (x, y), and we may chose G f = 0 for all points (x, y) along the inner and outer edges of the superconductor. Once a numerical result for Φ f is found, the result can be used to determine the effective area 20 of the SQUID's central hole, A eff = Φ f /B a , as was done for a circular ring in Ref. 31.
To prove that the effective area is also given by A eff = m d /I d , 32 we consider the electromagnetic energy cross where j a (r) = ∇ × B a (r)/µ 0 is the current density in the distant coil that produces a nearly uniform field B a in the vicinity of the SQUID, j f = ∇ × B sf /µ 0 is the induced current density in the SQUID, and B sf is the corresponding self-field under the conditions of flux focusing, i.e., when j f = 0 through the junctions. Also, B d = ∇ × A d is the dipole-like field distribution generated by the circulating current I d with density j d in the SQUID; at large distances from the SQUID 23 A d = µ 0 m d × r/4πr 3 . We evaluate E fd 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 = A d , B = B f , γ = γ d , and j = j f , from which we obtain The fourth term on the right-hand side of Eq. (11) is due to a vortex at position r v =xx v +ŷy v in the body of the SQUID: where B v (x, y) is the z component of the self-field generated by the vortex's sheet-current density J v (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 ∇ · J v = 0, it is possible to express J v (x, y) in terms of a scalar stream function, as we did for J f (x, y) and J d (x, y). However, as shown below, it is possible to use energy arguments to express Φ v (x, y) in terms of the stream function G d (x, y). 33 To obtain Φ v (r) when a vortex is at the position r =xx +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 I d in the counterclockwise direction, and the sheet-current distribution through the body of the SQUID is given by J d (x, y) = I dẑ × ∇G d (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 Φ v changes with time, and the voltage read by the voltmeter will be 34 V ab = dΦ v /dt. The power delivered by the power supply can be expressed in terms of the Lorentz force on the vortex, J d ×ẑφ 0 = I d φ 0 ∇G d ; i.e., the rate at which work is done on the moving vortex is I d φ 0 ∇G d · dr/dt. Equating this to the power P = I d dΦ v /dt = I d ∇Φ v · dr/dt delivered by the power supply to maintain constant current, we obtain the equation 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 =xx+ŷy. Choosing C to run around the outer boundary of the SQUID, we obtain Since G d (r o ) = 1 for points r = r o on the outer edges of the SQUID and G d (r i ) = 0 for points r = r i on the inner edges (at the perimeter of the central hole or along the edges of the slit), we have Φ v (r o ) = φ 0 and Φ v (r i ) = 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 I d in the SQUID, as well as the critical current I c . As discussed above, we have Φ I = 0 for a symmetric SQUID. For simplicity, we assume first that there are no vortices in the body of the SQUID, such that (17) and (18), for πLI0/φ0 = 0, 1, 2, 3, 4, and 5 (bottom to top).
From the sum and the difference of I 1 and I 2 we obtain whereφ = (φ 1 + φ 2 )/2 is determined experimentally by how much current is applied to the SQUID. Whenφ = 0, the current I is zero. Asφ increases, the magnitude of I increases and reaches its maximum value I c at a value ofφ that must be determined by numerically solving Eqs. (17) and (18). A simple solution is obtained for arbitrary Φ f only in the limit πLI 0 /φ 0 → 0, for which I = I c = 2I 0 | cos(πΦ f /φ 0 )| and I d = 0 at the critical current. For values of πLI 0 /φ 0 of order unity, as is the case for practical SQUIDs, one may obtain I c for any value of Φ f by solving Eq. (18) self-consistently for I d for a series of values ofφ and by substituting the results into Eq. (17) to determine which value ofφ 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 Delin 21 ), who showed that at I c where n is an integer. Hence all the physics is revealed When a vortex is present, Eqs. (17) and (18) still hold, except that Φ f in these equations is replaced by the sum Φ f + Φ v . Thermally agitated motion of vortices in the body of the SQUID can produce flux noise via the term Φ v (r v ) and the time dependence of the vortex position  17) and (18), are unaltered, except that the magnetic flux (sometimes called Φ ext 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 Φ f , 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 r v in the body of the SQUID, and we have found that the effective flux arising from the vortex is Φ v (r v ), 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 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 j y 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 J y (x) = j y (x)d.
The self-field magnetic induction generated by , the y component of the vector potential obtained from Ampere's law, is 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 B a =ẑB a = ∇× A f , the total vector potential is

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 A J (x) = A J (x, 0) to the y component of the vector potential, where We first examine the equal-current case and consider the contributions B I (x), the z component of B I (x), and J I (x), the y component of J I (x), due to equal currents I/2 in the left and right sides of the SQUID. Since J I (−x) = J I (x), the corresponding y component of the vector potential is also a symmetric function of x: A I (−x) = A I (x), where the subcripts I refer to the equal-current case. There are no flux quanta between the strips (Φ I = 0), and the second term in the brackets on the right-hand side of Eq. (1) vanishes; (φ 0 /2π)∇γ = 0. However, the constant C must be chosen such that J I (x) = −A I (x)/µ 0 Λ in the superconductor. Combining this equation with Eq. (24), making use of the symmetry J I (−x) = J I (x), and noting that 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 Applying this kernel to Eq. (25), we obtain Since such that and Combining this equation with that for A d (x), noting that the inductance of the SQUID is L = Φ d /I d , and making use of the symmetry J d (−x) = −J d (x), we obtain for a < x < w. We now define the inverse integral kernel K as (x, x ′ ) for the asymmetric-current case via w a K as (x, x ′′ ) 1 2π ln Applying this kernel to Eq. (31) and noting that I d = w a J d (x)dx, we obtain and where 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 and In this formulation, as in Ref. 31, L = L m + L k is the total inductance. The geometric inductance contribution dx is the stored magnetic energy, and the kinetic contribution where the brackets (<>) denote averages over the film width. We can show that L m + L k = L with the help of Eqs. (32) and (35). The z component of the magnetic induction B d (x) generated by J d (x) can be obtained from the Biot-Savart law, or, since When l ≫ w, the magnetic moment in the z direction generated by the circulating current I d is (to lowest order in w/l) where the factor 2 accounts for the fact 41 that the currents along the y direction and those along the x direction at the ends (U-turn) give exactly the same contribution to m d , 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 B a , 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 B a , 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) 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), (φ 0 /2π)∇γ, is −ŷΦ f /2l for a < x < w andŷΦ f /2l for −w < x < −a. Equation (1) yields for a < x < w. We again use the inverse integral kernel K as (x, x ′ ) for the asymmetric-current case [Eq. (32)] to obtain The effective area of the SQUID A eff is found from the condition that the net current around the loop is zero [ w a J f (x)dx = 0], which yields For a ≤ x ≤ w, the stream function is and for −w ≤ x ≤ −a, G f (−x) = G f (x). The spatial distribution of the resulting z component of the in-plane magnetic induction is given by where B sf (x) can be obtained from the Biot-Savart law or by substituting J f (x) into Eq. (24) and making use of The resulting magnetic moment m f in the z direction can be calculated by replacing J d by J f 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, Φ v = 0, and the effective flux is zero: Φ = Φ d +Φ f = 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 B a . 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.

B. Numerical solutions
In the previous section we have presented formal solutions for the sheet-current density J y (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 x i (i = 1, 2, . . . , N ) spanning only the strip (but avoiding the edges, where the integrand may have infinities), a < |x i | < w, such that for any function Here w i are the weights, approximately equal to the local spacing of the x i ; the weights obey N i=1 w i = w − a. We have chosen the grid such that the weights w i are narrower and the grid points x i more closely spaced near the edges a and w, where J y (x) varies more rapidly. We have accomplished this by choosing some appropriate continuous function x(u) and an auxiliary discrete variable u i ∝ i − 1 2 , such that w i = x ′ (u i )(u 2 − u 1 ). 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 where δ ij = 0 for i = j, δ ii = 1, and The optimum choice of the diagonal term Q sy ii (i.e., with |x 2 i − x 2 j | replaced by x i w i /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.
and applying it to Eq. (44), we obtain such that The superscript (as) is a reminder that this is for an Defining applying it to Eq. (51), and noting that and where The stream function G d (x) can be evaluated as 43 Shown in Fig. 6 are plots of J d (x), G d (x), and B d (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 critical current I c is proportional to the magnitude of 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 L m and L k can be calculated separately from Eqs. (37) and (38) using Eqs. (54) and (56). We can show that L m + L k = 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 L m and L k vs a/w. For Λ = 0, when L k = 0, L = L m 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 L m and L k are those 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 = L m + L k 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 L k .
The solid curves in Fig. 9(b) show L m and L k vs Λ/w. The geometric inductance L m depends upon Λ but only weakly, varying slowly between its Λ = 0 asymptote  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 L m except for small values of Λ/w and a/w. The dotted curves in Fig. 9(a) are obtained from L = L m + L k 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 L k . In Eq. the circulating current is, from Eq. (39), 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 J d (x) ∝ 1/x, Eq. (A5). The contribution of these small x to m d , Eq. (60), stays finite due to the factor x, but the total current I d to which m d is normalized, diverges when a/w → 0, thus suppressing the plotted ratio m d /I d . Interestingly, the curves in Fig. 10 coincide with those in Fig. 13; see below.
Expressions for m d in the limits Λ/w → 0 and Λ/w → ∞ are given in Eqs. (A13) and (C7). For the flux-focusing case, Eq. (40) becomes Applying Eq. (53), we obtain  where, since The stream function G f (x) can be evaluated as 43 Shown in Fig. 11  The ratio B f (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 B f (0)/Ba ≈ (w/a)/ ln(4w/a) good for a/w ≤ 0.3. 46 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, I 1 = I 2 = 0. The corresponding magnetic field lines are depicted in Fig. 12. Shown in Figs. 13 and 14 are plots of the effective area A eff (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, A eff is given by Eq. (A19), and when Λ/w → ∞, A eff = l(w + a). Note in Fig. 14 that A eff 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 A eff of the gap tends to a constant, except in the limit Λ → 0, where it vanishes very slowly, A eff /2wl ≈ (π/2)/ ln(4w/a) [Eq. (A19)]. When a/w → 0, the enhancement factor A eff /2al → ∞ and thus diverges even for Λ = 0. In the limit a/w ≪ 1, A eff (0, Λ/w) tends to  Fig. 11], plotted versus a/w and Λ/w, respectively. The ratio B f (0)/B a ≥ 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 B f (0)/B a = wE(k ′ )/aK(k ′ ) obtained from Eq. (A17). For a/w ≪ 1 this yields B f (0)/B a ≈ (w/a)/ ln(4w/a), which is a good approximation for 0 < a/w ≤ 0.3. 46 The magnetic moment m f for the flux-focusing case, calculated from Eq. (60) but  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 = λ 2 /d, where the London penetration depth λ is larger than d, the film thickness. The SQUID's critical current I c 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. circulating-current term Φ d , a vortex-field term Φ v , and a flux-focusing term Φ f , 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 Φ d can be expressed in terms of the SQUID inductance L and the circulating current I d via Φ d = LI d . The SQUID inductance has two contributions, L = L m + L k , 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, L m depends only weakly upon Λ, because for the same circulating current I d , 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, L k 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 Φ v = φ 0 G d , where G d is a dimensionless stream function describing the circulating sheet-current density J d . Roughly speaking, when Λ is small, I c is most strongly dependent upon the vortex position when the vortex is close to the edges of the film, but when Λ is large, I c is equally sensitive to the vortex position wherever the vortex is. Recent experiments 49 have used the relationship Φ v = φ 0 G d to determine the vortex-free sheet-current density J d (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 J d (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 Φ f = B a A eff , where B a is the applied magnetic induction and A eff is the effective area of the central hole of the SQUID. Although A eff 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, L m , and L k , the effective area A eff , 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. neglecting corrections proportional to a 2 /w 2 , and for small (w − a)/w by L m = (µ 0 l/π) ln[16/(1 − a 2 /w 2 )], neglecting corrections proportional to 1 − a 2 /w 2 . In the limit that Λ = 0, the kinetic inductance vanishes (L k = 0), and the inductance in Eq. (A10) becomes the total inductance: L = L m . The magnetic moment [see Eq.
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 is 47 and the z component of the magnetic induction in the plane z = 0 of the strips is 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 Φ f = πB a lw/K(k ′ ), (A18) and the effective area A eff of the slot is Note that A eff = m d /I d . The magnetic moment generated by J f (x) is m f = −πl[w 2 + a 2 − 2w 2 E(k ′ )/K(k ′ )]B a /µ 0 . (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: The corresponding z component of the magnetic induction is 47 B(x) = B a x 2 −[1−E(k)/K(k)]w 2 [(x 2 − a 2 )(x 2 − w 2 )] 1/2 , |x| > w, (A22) = 0, a < |x| < w, L k = 2µ 0 l Λ w (w 2 + δ 2 ) (w 2 − a 2 + 2δ 2 ) (f a + f w ) [F (λ a , q) − F (λ w , 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 L k 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.
When Λ ≫ w, the total inductance L is dominated by the kinetic inductance (L k ≫ L m ), such that L ≈ L k . Since J d is uniform, the magnetic moment is easily found from Eq. (39) to be m d = l(w + a)I d .
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 J f (x) = B a (w + a − 2x)/2µ 0 Λ for a < x < w, the effective area is A eff = l(w + a) = m d /I d , and the z component of the magnetic induction in the plane of the strips is B f (x) = B a + B sf (x), where from the Biot-Savart law The magnetic moment generated by J f (x) in this limit is 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) ≈ B a . The y component of the vector potential is approximately given by A(x) = B a x, and the y component of the induced sheetcurrent density, obtained from Eq. (1) with γ = 0, is J(x) = −(B a /µ 0 Λ)x. To the next order of approximation, B(x) = B a + B s (x), where the self-field B s is found from the Biot-Savart law The magnetic moment generated by J(x) in this limit is m = −[2l(w 3 − a 3 )/3Λ](B a /µ 0 ). (C11)