Tunable ±φ , φ 0 and φ 0 ±φ Josephson junction
aa r X i v : . [ c ond - m a t . s up r- c on ] J un Tunable ± ϕ , ϕ and ϕ ± ϕ Josephson junction
E. Goldobin, D. Koelle, and R. Kleiner Physikalisches Institut and Center for Collective Quantum Phenomena in LISA + ,Universit¨at T¨ubingen, Auf der Morgenstelle 14, D-72076 T¨ubingen, Germany (Dated: May 6, 2018)We study a 0- π dc superconducting quantum interference device (SQUID) with asymmetric in-ductances and critical currents of the two Josephson junctions (JJs). By considering such a dcSQUID as a black box with two terminals, we calculate its effective current-phase relation I s ( ψ )and the Josephson energy U ( ψ ), where ψ is the Josephson phase across the terminals. We show thatthere is a domain of parameters where the black box has the properties of a ϕ JJ with degenerateground state phases ψ = ± ϕ . The ϕ domain is rather large, so one can easily construct a ϕ JJexperimentally. We derive the current phase relation and show that it can be tuned in situ byapplying an external magnetic flux resulting in a continuous transition between the systems withstatic solutions ψ = ± ϕ , ψ = ϕ ( ϕ = 0 , π ) and even ψ = ϕ ± ϕ . The dependence of ϕ on appliedmagnetic flux is not 2 π (one flux quantum) periodic. PACS numbers: 74.50.+r, 85.25.CpKeywords: current-phase relation
I. INTRODUCTION
One of the most important characteristics of a Joseph-son junction (JJ) is its current-phase relation (CPR), i.e. ,the relation between the supercurrent I s flowing throughthe junction and the Josephson phase φ across it. TheCPR, aka the first Josephson relation, plays a fundamen-tal role and is responsible for almost all properties of theJJ. The CPR usually depends on the microscopic physicsof the Josephson barrier. The CPR is usually a 2 π peri-odic function of φ and, in the most simple case, is givenby I s = I c sin φ . A review of different types of CPRs canbe found elsewhere .The quantity directly related to the CPR is the Joseph-son energy profile U ( φ ). It is defined so that I s ( φ ) = ∂ U ( φ ) /∂φ . For a sinusoidal CPR U ( φ ) = E J (1 − cos φ ),where E J = Φ I c / π is the Josephson energy, andΦ ≈ .
068 fWb is the magnetic flux quantum. U ( φ )is also a 2 π periodic function of φ .The most interesting JJs both in terms of applicationsand fundamental studies are those having a non-trivialJosephson energy profile U ( φ ). For example, π JJs thatreceived a lot of attention have negative critical cur-rent, which results in the sign change of U ( φ ). As aresult, the energy minimum (and the ground state) cor-responds to φ = π , where the usual JJ (called 0 JJ inthis context) has a maximum. Further, there was a theo-retical predictions that one can obtain the so-called ϕ JJ, i.e. , the one having a single U ( φ ) minimum (witheach 2 π period) situated at φ = ϕ = 0 , π . More recently, ϕ JJs having a periodic double-well potential U ( φ ) and,therefore, a degenerate ground state (two U ( φ ) minima)with the phases φ = ± ϕ were investigated . Uponapplication of magnetic field one can also obtain a U ( φ )without reflection symmetry.Theoretically, it was predicted that a JJ made of d -wave superconductors in a specific range of parame-ters (orientation angle, temperature, etc. ) can posses a ϕ ground state. A degenerate ground state was obtainedfrom measurements of the CPR of d -wave based nanoJJs . Later on, some other indications of a ϕ state, suchas an anomalous temperature dependence of the criticalcurrent, were observed also on nano JJs. The facetingalong longer grain-boundary JJs based on d -wave super-conductors results in an effective (facet-averaged) ϕ JJ .In the latter case one expects non-quantized splinteredvortices , which were observed using superconduct-ing quantum interference device (SQUID) microscopy.However, in all cases no state manipulation or readoutwere demonstrated, probably because of high dampingand poor control over JJ properties.Recently, we have suggested and successfullydemonstrated a ϕ JJ based on conventional low- T c su-perconductors with tailored ferromagnetic barrier .This junction has a degenerate ground state phase ± ϕ ,i.e. its Josephson energy profile looks like a 2 π -periodicdouble well potential (in the absence of bias current).The two ground states can be used to store information .The unusual physics of ϕ JJs was discussed in severalworks . However, the ϕ JJs constructed so far arerather large. Making them smaller (shorter) requires veryexact control on parameters such as critical current den-sities and the lengths of the 0 and π regions.In this paper, we propose an effective ϕ JJ based onan asymmetric 0- π dc SQUID, i.e. , a dc SQUID with one0 and one π JJ, with finite inductance and asymmetriccritical currents of the 0 and π JJ. This not only hasadvantages over the previous proposals in termsof geometrical dimensions, margins, and the size of the ϕ domain in parameter space, but also shows other uniquefeatures. For example, it can be operated not only as a ϕ JJ, but also as a ϕ JJ or as a combination of both, i.e. , as a ϕ ± ϕ JJ. This goes much beyond the trivial useof a dc SQUID as the substitution for a single JJ withmagnetic-field-tunable critical current.The paper is organized as follows. In sec. II we intro-duce the considered model system and the equations de-scribing it. In the main sec. III we present our numericalresults. Sec. IV summarizes our findings. In appendix Awe consider the vicinity of the 0 ↔ ϕ transition in param-eter space and derive many results analytically similar tothe Ginzburg-Landau approach. II. MODEL
The SQUID circuit is shown in the inset of Fig. 1. Theapplied bias current I splits into two branches with induc-tances L and L and Josephson junctions with criticalcurrents I c and I c .For the sake of simplicity, we will derive everything innormalized units. The current will be normalized to thelargest (by absolute value) critical current of the two JJs.Without loss of generality we assume that I c > | I c | ≤ I c . Then we define α = I c /I c as the normalizedcritical current of the second JJ, with | α | ≤
1. Thus, α > α < π SQUID. Although, our main focus is on a 0- π SQUID( α <
0) the results for a 0-0 SQUID will be includedautomatically as well. Further, we introduce normalizedinductances β = 2 πI c L Φ , β = 2 πI c L Φ . (1)Note that the definition of β uses I c , so that by chang-ing β and β one can see the effect of inductances only,while changing α one can see the effect of critical cur-rent asymmetry only. These definitions are related to theconventional β L = 2 I c L Σ / Φ , as πβ L = β + β = β Σ ,where L Σ = L + L .The total phase ψ across the SQUID, see the inset ofFig. 1, can be expressed in two ways ψ = φ + β sin φ + r φ e ; (2a) ψ = φ + αβ sin φ − r φ e , (2b)where r + r = 1, and r and r are the ratios, in whichthe externally applied normalized flux/phase φ e = 2 πf =2 π Φ e / Φ ( f is the normalized flux, aka frustration) is di-vided between the two branches. If the external magneticfield is applied by a coil, r , φ e = 2 πM , I / Φ , where I is the current in the coil creating the magnetic field and M , is the mutual inductance between this coil and theleft and the right arms of the SQUID, respectively. Forthe sake of simplicity, we neglect the mutual inductancebetween L and L .The sum of the currents in both branches is given by γ = sin φ + α sin φ , (3)where γ = I/I c is the normalized bias current.Since we are interested in a geometrically small systemwhich should not have too many internal states, we focuson the case of small, but finite, inductances, i.e., 0 ≤ β ≤ ≤ β ≤ -1.0 -0.5 0.0 0.5 1.0-0.50.00.51.01.52.02.53.03.5 L L I e n e r gy U () +0.9 FIG. 1. (Color online) The energy U ( ψ ) of the system givenby Eq. (5) for different values of the asymmetry parameter α and for β = β = 0 . α c ≈ − . III. RESULTS
The easiest way to solve Eqs. (2) and (3) is to cal-culate φ and φ from Eqs. (2) for given ψ . Note thatfor β , β , | α | ≤ unique solu-tion φ or φ for given ψ . It is convenient to define auniversal function φ ( v, p ), which is a solution φ of theequation v = φ + p sin( φ ) for given argument v and pa-rameter | p | ≤
1. Then φ ( ψ ) = φ ( ψ − r φ e , β ) and φ ( ψ ) = φ ( ψ + r φ e , αβ ). The function φ ( v, p ) has anobvious, but useful, property: φ ( v + πn, p ) = πn + φ [ v, ( − n p ] , (4)where n is an integer. A. Current-phase relation and Josephson energy
To determine the CPR γ ( ψ ) we calculate φ ( ψ ) and φ ( ψ ) as mentioned above and then use Eq. (3) to obtain γ . We do not show γ ( ψ ) plots here, but show Josephsonenergy U ( ψ ) plots instead.The total energy of the system is given by U ( ψ ) = U J ( ψ ) + U L ( ψ ) , (5)where U J ( ψ ) = [1 − cos φ ( ψ )] + α [1 − cos φ ( ψ )] , (6) . . . . . . . . . . . . - . - . - . - . . JJ a sy mm e t r y FIG. 2. (Color online) The domain of the ϕ state (pink/gray)as a function of parameters α , β and β . is the Josephson energy of both JJs, and U L ( ψ ) = β φ ( ψ ) + β α sin φ ( ψ ) , (7)is the magnetic field energy stored in the inductors. Bydirect substitution, one can see that U ′ ( ψ ) ≡ γ ( ψ ) (hereand below the prime denotes ∂/∂ψ by default), like forany JJ.Consider the case of zero applied magnetic flux φ e = 0.Several examples of U ( ψ ) are presented in Fig. 1. Atpositive α (conventional 0-0 SQUID) the energy profileresembles the usual U conv ( ψ ) = 1 − cos( ψ ) profile of aconventional single JJ with the ground state at ψ = 0.Deviations from U conv ( ψ ) are due to finite inductancesand make U ( ψ ) sharper than U conv ( ψ ) near the maximaand more shallow than U conv ( ψ ) near the minima. As α decreases down to 0, the height of U ( ψ ) decreases as2(1 + α ) while the shape of U ( ψ ) almost does not change.As α becomes negative (0- π SQUID) the U ( ψ ) becomesmore flat near ψ = 0. For α below some critical value α c the energy profile develops two minima, as can beseen in Fig. 1. The system has a degenerate ground state ψ = ± ϕ , i.e. , forms a ϕ JJ .The transition to the degenerate ground state takesplace at the value of α = α c , for which U ′′ (0) = γ ′ (0) = 0.This allows to calculate α c = −
11 + β + β = −
11 + β Σ = −
11 + πβ L . (8)This is one of the central results of the paper.The domain of the ϕ state is shown in Fig. 2. From thesides it is limited by our choice of parameters 0 < β , < α > −
1. From the top it islimited by the surface α c ( β , β ) given by Eq. (8). The ϕ domain has the maximum size (height) along the α axisequal to − ≤ α ≤ − / β , = 1. For β , → ϕ domain along the α axis vanishes linearly ∝ β Σ . We also note that in the case of β Σ = 0, usedby many authors because it is solvable analytically, one never obtains a ϕ JJ.An important practical difference between the 0- π dc SQUID considered here and a 0- π JJ of finitelength is that our system has a rather large ϕ do-main where the theory presented here works. Even in thecase of β Σ → α = − . . . α c , correspond-ing to the ϕ domain, shrinks as ∆ α = α c − ( −
1) = β Σ , i.e. , linearly. Instead, for 0- π JJs the ϕ domain shrinksas δ ≈ L , where δ = ( L − L π ) / π facet length from the average length L = ( L + L π ) /
2. This imposes additional requirementson fabrication accuracy or one should move to a region ofthe phase diagram where the theory works only qualita-tively, as in the first experiments . In other works some alternative techniques to enlarge ϕ domains weresuggested. However, the relative volume of the ϕ do-mains in parameter space is still much smaller than inthe present work.Let us now consider U ( ψ ) at finite applied magneticflux φ e = 0. First, we point out that Eqs. (2) and (3)have several important symmetry properties, when theapplied flux changes by a half-integer number of flux-quanta Φ φ new e = φ e + nπ ; (9a) ψ new = r nπ ; (9b) φ new1 = φ ; (9c) φ new2 = φ + nπ ; (9d) α new = ( − n α ; (9e)(9f)Upon such transformation the current does not change, i.e. , γ ( ψ, α ) = γ ( ψ new , α new ) , (10)while the Josephson energy, which is still 2 π periodic in ψ , can be expressed in terms of new variables as U ( ψ, α ) = U ( ψ new , α new ) + α new [( − n − . (11)Thus, by applying a half-integer f we can turn a 0-0SQUID with α > | α c | (effective 0 JJ) into a 0- π SQUID(effective ϕ JJ) with α new = − α < α c and vice versa.An even more interesting point is that ψ shifts by anamount r nπ , which is, in general, not a multiple of 2 π .Since after transformation (9) the value of γ does notchange (it depends only on ( φ , φ ) mod 2 π ), the CPR γ ( ψ ) shifts along the ψ axis by the amount r nπ . Thisis another central result of the paper.Examples of U ( ψ ) calculated for different values of ap-plied magnetic flux are shown in Fig. 3. For α > | α c | ,see Fig. 3(a), U ( ψ ) at f = 0 has a single minimum at ψ = 0. At f = 0 . α = − . -0.50.00.51.01.52.02.53.03.5 -1.0 -0.5 0.0 0.5 1.0-0.50.00.51.01.52.02.53.03.5 e n e r gy U () (a)(b) e n e r gy U () FIG. 3. (Color online) Josephson energy U ( ψ ) for β = 0 . β = 0 . r = 0 . f = 0 . . . α = 0 . > | α c | = 0 .
5; (b) α = 0 . < | α c | =0 . and f = 0, but shifted by (centered at) ψ = r π = 0 . π and lifted by 2 α = − α new = 1 .
4, see Eqs. (9) and (11).In essence, at this f one obtains a ϕ ± ϕ JJ. For f = 1the energy U ( ψ ) transforms again to the same profile asat f = 0, but shifted by (centered at) ψ = r π = 0 . π ,see Eqs. (9) and (11). For 0 < α < | α c | , see Fig. 3(b),we again start at f = 0 from having a single minimumof U ( ψ ) at ψ = 0. At f = 0 . U ( ψ )profile as the 0- π SQUID with α new = − α , but at f = 0,shifted by ψ = r π . However, in this case it is not adouble-well as − α < α c . At f = 1 we obtain the originalprofile (as for f = 0), but shifted by r π , see Eqs. (9)and (11). B. Ground state phase
The ground state ( f = 0, γ = 0) phase ϕ in the ϕ domain α < α c can be calculated numerically from γ ( ϕ ) = 0.To simplify and accelerate this computation we use thefollowing procedure. Since for γ = 0 the phases φ and φ (but not ψ ) depend only on β Σ , for calculation of φ and φ , without loosing generality, we assume that β = β Σ and β = 0. Then Eqs. (2) collapse to φ = φ + β Σ sin( φ ) . (12)Substituting this into Eq. (3) with γ = 0 we arrive at a -1.0 -0.8 -0.6 -0.4 -0.2 0.8 1.0-0.50.00.5 g r ound s t a t e pha s e JJ asymmetry = = 0.7 c FIG. 4. The ground state phase ± ϕ ( α ). The horizontaldashed line shows the unstable static solution ψ = 0 in theregion α < α c . Thin line shows the approximation given byEq. (A11). rather simple transcendental equation,sin( φ ) + α sin[ φ + β Σ sin( φ )] = 0 , which we solve to find φ . Then the value of φ , if neces-sary, can be calculated from Eq. (12). The ground statephase ϕ is obtained using one of the Eqs. (2), with β and β corresponding to the real circuit.An example of ϕ ( α ) at fixed β , is shown in Fig. 4.In essence this is ϕ along a vertical line crossing a ϕ domain in Fig. 2. The ground state phase is zero as α decreases from 1 down to the bifurcation point α c . Afterthe bifurcation point ( α < α c ) the zero solution becomesunstable, as indicated by the dashed line. Instead twodegenerate stable solutions appear.The phase ϕ corresponding to the degenerate state de-parts from zero as described by Eq. (A11). At α → − ϕ max ,which is given by ϕ max = π y β − β β + β , (13)where y is a solution of the equation2 y = β Σ cos( y ) . (14)The possible range of y is from y = 0 for β = β = 0to y = y ⋆ , where y ⋆ ≈ .
739 is a solution of the equation y ⋆ = cos( y ⋆ ). The corresponding range of ψ max is thenfrom π/ − y ⋆ (reached for β → β →
1) to π/ y ⋆ (reached for β → β → y is a function of β Σ only, it follows from (13) that thephase ϕ max is antisymmetric with respect to the diagonaldirection β = β . In particular, ϕ max = π at β = β (symmetric system). It is interesting that even for a π JJ,which is weaker than the 0 JJ, one can obtain a groundstate phase | ϕ | > π — a situation, which is not possiblein a continuous 0- π JJ studied earlier . state L1 L state state L1 L state FIG. 5. The phase diagram of 0- π dc SQUID (ground state phase ϕ ( β , β ) for (a) α = − . α = − .
8. The dashedline shows the boundary between trivial ψ = 0 and ψ = ± ϕ ground states given by expression (15). Continuous lines are thelines of the constant ground state phase ϕ . Its value is given next to each line. It is also interesting to plot the ground state phaseon the ( β , β ) plane for fixed α , i.e. , in essence, in thehorizontal plane crossing the ϕ domain in Fig. 2 at fixed α . In such a plane, with the help of Eq. (8), the ϕ domainis given by β > − α − − β for α < . (15)Thus, the boundary is just a straight line, see Fig. 5.Below this line the ground state phase is ψ = 0, whilein the filled area above this line the ground state phaseis ψ = ± ϕ . Note, that the boundary given by Eq. (15)shifts towards the origin as α decreases. At α = − α < − ϕ = π domain willappear close to the origin and will grow as α decreasesfurther. C. Persistent current
If the ground state of the system ( f = γ = 0) is the ± ϕ state, one has a persistent current circulating clockwiseor counterclockwise around the SQUID. From Eq. (3), itsvalue is given by I circ = sin[ φ ( ± ϕ )] = α sin[ φ ( ± ϕ )] . (16)Since in the ground state the phases φ and φ dependonly on β Σ , so does I circ . This means that the value ϕ of the ground state phase, which depends on β − β and β Σ , can be chosen independently from the value of thepersistent current I circ , which depends only on β Σ . -1.0 -0.8 -0.6 -0.4 -0.2 0.8 1.0-1.0-0.50.00.51.0 maxcirc c i r c u l a t i ng pe r s i s t en t c u rr en t I c i r c () JJ asymmetry = 1.0 c FIG. 6. Circulating current as a function of α . Tilted dashedline shows the value of the critical current of the α -junction.The horizontal dashed-dotted line shows the maximum valueof persistent current reached at α = − α maxcirc calculated using Eq. (18). For fixed β Σ the value of persistent current grows ∝√ α c − α near α c and reaches the maximum value at α = −
1, see Fig. 6. This maximum value is equal to sin φ ,where φ is a solution of the following transcendentalequation 2 φ − β Σ sin φ = π. (17)In the limit β Σ →
0, from Eq. (17) φ → π/ → | α | — the critical current of the weaker JJ? It turns outthat for fixed β Σ this happens for α = α maxcirc , which is asolution of the following transcendental equation αβ Σ + arcsin( α ) + π , (18)see also Fig. 6. To proof this we take the state with φ = π/
2, then sin φ = 1. From Eq. (3) sin φ = − α >
0, and therefore φ = arcsin( − α ) (exactly this root!).By substituting this into Eqs. (2) and, without loosinggenerality assuming β = 0, β = β Σ , we arrive at (18). D. Self-generated flux
The self-generated flux in the loop (not including ex-ternal flux φ e ) is given by2 π ΦΦ = β sin φ ( ψ ) − αβ sin φ ( ψ ) = φ ( ψ ) − φ ( ψ ) − φ e . (19)Similar to the circulating current, the value of sponta-neous flux in the absence of the bias ( γ = 0) dependsonly on β Σ , which is not obvious at all from the firstpart of Eq. (19), but apparent from its last part. Thisallows to write2 π ΦΦ = β Σ sin φ = − αβ Σ sin φ = β Σ I circ . (20)Therefore, spontaneous flux behaves similar to the circu-lating current. The maximum value of the flux is alsoreached at α → − ( ) s pon t aneou s f l u x () C i r c u l a t i ng c u rr en t I c i r c () total normalized inductance | | = 0.7 1.136 I circ ( ) FIG. 7. The amplitudes of the spontaneous circulating cur-rent I circ ( β Σ ) and spontaneous flux Φ( β Σ ) / Φ for fixed α = − . It is instructive to replot I circ and Φ as a function of β Σ at fixed α , see Fig. 7. One can see that at smallinductance β Σ < (1 + α ) / ( − α ) [inverted Eq. (8)] thesystem is in the zero state. At larger inductances thesystem enters into ± ϕ state and the spontaneous flux andcurrent increase. However, Φ( β Σ ) grows monotonously,reaching ∼ Φ / β Σ = 2, while the I circ ( β Σ ) exhibit a maximum, where I circ = | α | — the maximum possiblevalue in our SQUID. It happens at β Σ = (2 arcsin( α ) + π ) / ( − α ), which was obtained by inverting Eq. (18). E. Critical currents
In general, our system has four critical currents in the ϕ domain and two critical currents outside of it. Thesecritical currents correspond to the escape of the phasefrom the left ( − ϕ ) or the right (+ ϕ ) wells of the double-well potential, see Fig. 1, for different directions of thebias current γ . The critical current corresponds to themaximum of the CPR γ ( ψ ). -1.0 -0.5 0.0 0.5 1.00.00.51.01.52.0 -1.0 -0.5 0.0 0.5 1.00.00.51.01.52.0 c r i t i c a l c u rr en t s c asymmetry parameter (a) L1 = L2 = 0.7 c r i t i c a l c u rr en t s c asymmetry parameter (b) L1 = 0.1, L2 = 0.9 FIG. 8. Numerically calculated dependences γ c ± ( α ) for (a) β = β = 0 . β = 0 . β = 0 . The numerically calculated dependence of γ c ( α ) forgiven β and β is shown in Fig. 8. One can see severalkey points on the γ c ± ( α ) dependence. First, γ c + (0) ≡ β = β , γ c − ( −
1) = γ c + ( − i.e. , the twodependences converge at α = −
1, see Fig. 8(a). In thecase β < β , see Fig. 8(b), the dependences cross atsome − < α x < α c . F. Critical current as a function of applied flux
The presence of the degenerate ground state and twocritical currents also manifests itself in the dependence ofthe maximum supercurrent on magnetic field γ c ( f ). Inthe general case this dependence can be calculated onlynumerically. Several examples of γ c ( f ) dependences arepresented in Fig. 9. For the parameters in the ϕ domain( α < α c ) one sees the domains corresponding to the dif-ferent flux states overlapping in the vicinity of integer f . In this overlapping region one observes in total fourcritical currents corresponding to the escape of the phasefrom different energy minima (wells) of the Josephsonpotential in two different directions. Note the strikingsimilarity of these curves with those for ϕ JJs based oncontinuous 0- π JJs .As α increases and approaches α c , see Fig. 9, thebistability region is shrinking. Exactly at α = α c theoverlap near integer f disappears, which corresponds tothe disappearance of two distinct ground states ψ = ± ϕ . However, for a small range of α > α c , one ob-serves a small triangular-like bistability region where thebranches meet, see Fig. 9(d). Further investigation of thissmall domain is outside the scope of this paper. We notethat the intersecting domains, similar to those shown inFig. 9 were calculated long ago . At that time, how-ever, π junctions were unknown, so only a 0-0 SQUID,where the domains intersect near half-integer f , was con-sidered. However some key results can be adopted to ourcase easily.In particular, we can find the positions of key points ± A and ± X where ∂γ c ( f ) /∂f = 0. We note thatthe phases φ = ± π/ φ = ± π/ γ ′ = 0 for any asymmetryand applied flux. It turns out that each of these fourcombinations corresponds to an extremum of the γ ( f ) de-pendence, i.e. , to points ± A and ± X in Fig. 9. To findthe values of the external flux 2 πf ≡ φ e , correspond-ing to these points, we substitute the above phases intoEq. (2) and obtain φ e = ± h π β i ∓ h π αβ i . (21)The corresponding values of the critical current are ob-tained from Eq. (3) γ extr c = ± ± α. (22)In Eqs. (21) and (22) both ± signs are independent,providing four combinations in total. The trivial con-sequence of Eq. (22) is that for a SQUID with symmetriccritical currents ( | α | = 1) the points ± X are situated atthe horizontal axis ( γ c = 0).The bistability region around integer f can be used tostore one bit of information in the ± ϕ states as demon-strated recently . In some sense it is similar to the ear-lier proposals to use n = 0 and n = 1 states of theSQUID biased to f ≈ /
2. However in our case, the flux -2-1012-2-1012-2-1012 -1.0 -0.5 0.0 0.5 1.0-2-1012 X + XA c r i t i c a l c u rr en t I c (a) + A (b) c r i t i c a l c u rr en t I c (c) c r i t i c a l c u rr en t I c (d) frustration f c r i t i c a l c u rr en t I c FIG. 9. The dependence of the critical current γ c of the deviceon the normalized applied magnetic field (frustration) f for β = β = 0 . πβ L = 1 . α c ≈ − . α . bias is not needed. Asymmetry also provides differentcritical currents at f = 0, which simplifies readout.Finally, the practical question is: can one detect ± ϕ , ϕ and ϕ ± ϕ states in experiment by measuring the γ c ( f ) dependence? For the ± ϕ state the answer is givenin Fig. 9. One should observe intersection of branchesand four critical currents. For the ϕ ± ϕ state the sit-uation is similar. However, since this state appears athalf-integer f the whole γ c ( f ) curve is shifted so thatthe bistability regions are situated around half-integer f .Finally, in the ϕ state, which in our system appears atfinite field only, one has only two critical currents andthe junction looks just like a conventional one, althoughthe γ c ( f ) dependence is unusual (periodic), but nevermulti-state. To prove the ϕ state one has to do a phasesensitive experiment, e.g. , putting our black box in a su-perconducting loop. IV. SUMMARY
We have shown that an asymmetric 0- π SQUID canbe used as an effective ϕ JJ with magnetic field tunablecurrent-phase relation γ ( ψ ) and, accordingly, a Joseph-son energy U ( ψ ). The critical value α c of the criticalcurrent asymmetry parameter α required to obtain thedegenerate ± ϕ ground state depends on the sum L + L of inductances in two branches of the SQUID. Upon ap-plying an integer number of flux quanta f = n , the phase ψ across the structure advances by an amount r · πn ,where r is the fraction of external flux induced in the leftbranch of the SQUID. Since, in general, r is an arbitrarynumber depending on design, the phase shift r · πn isnot a multiple of 2 π . By applying a half-integer numberof flux quanta f = n + 1 / α < | α c | ,one can turn the effective 0-JJ into a ϕ = r πn JJ. If α > | α c | , then one can turn the effective 0-JJ into a JJwith ground states ϕ ± ϕ . The dependence of the criticalcurrent on magnetic flux clearly shows bistability regionstypical for ϕ JJ .In terms of designing a practical device (bistable ϕ JJ)the target parameters can be, e.g. , β = β = 0 . . . . . α ≈ − . . . . − . ϕ domainin Fig. 2. This will provide very large operation margins.Note that a finite inductance is essential to obtain a ϕ domain of finite size. In the limit β Σ → ϕ -domainshrinks to a point. ACKNOWLEDGMENTS
We thank M. Fistul, V. Ryazanov, A. Ustinov, M. Wei-des for useful discussions. This work was funded by theDeutsche Forschungsgemeischaft (DFG) via projects No.GO-1106/5 and No. SFB/TRR 21 A5 and by the EU-FP6-COST Action MP1201.
Appendix A: Solutions close to the bifurcation point α ≈ α c The ϕ JJ proposed here can also be used as a qubitat f = 0, when the barrier separating the − ϕ and + ϕ states is small. This is the case when α is only slightlysmaller than α c . In this limit the expressions for manyimportant quantities can be obtained analytically. Weare especially interested in the situation near the bottomof the energy profile U ( ψ ), i.e. , for small values of ψ sincethe important physics (formation of ± ϕ state, escape,macroscopic quantum tunneling) takes place there.For α = α c − ǫ (0 ≤ ǫ ≪
1) the Josephson energy ofthe system can be expanded like in the Ginzburg-Landau theory as U GL ( ψ ) ≈ aψ + bψ , | ψ | ≪ . (A1)where the coefficients a ( α ) and b ( α ) have to be deter-mined from our model given by Eqs. (2) and (3), namely,from U ′′ (0) = γ ′ (0) = 2 a ; (A2) U ′′′′ (0) = γ ′′′ (0) = 24 b, (A3)From Eq. (3) γ ′ ( ψ ) = cos( φ ) φ ′ + α cos( φ ) φ ′ . (A4)The derivatives φ ′ , can be calculated by differentiatingEqs. (2) with respect to ψ : φ ′ = 11 + β cos( φ ) , φ ′ = 11 + β α cos( φ ) . (A5)By substituting φ ′ , from Eq. (A5) into Eq. (A4) we ob-tain γ ′ ( ψ ) = cos( φ )1 + β cos( φ ) + α cos( φ )1 + β α cos( φ ) . (A6)According to Eqs. (2), ψ = 0 corresponds to φ = φ =0, so from Eqs. (A2) and (A6) we obtain the explicitexpression for a ( α ): a ( α ) = 12 (cid:20)
11 + β + α αβ (cid:21) , (A7)which is negative for α < α c . Near α c the leading termis a ( ǫ ) = −
12 1 α c (1 + β ) ǫ = a ǫ. (A8)Similarly, differentiating γ ′ ( ψ ) in Eq. (A6) two addi-tional times and using Eq. (A5) after each differentiation,we obtain at ψ = φ = φ = 0 from Eq. (A3) b ( α ) = − (cid:20) β ) + α (1 + αβ ) (cid:21) . (A9)Near α c the main term of b ( α ) is a constant b ( α c ) = 124 β Σ ( β + 3 β Σ + 3)(1 + β ) = 124 α c + 1( − α c ) (1 + β ) = b . (A10)Now we can readily calculate various quantities. Thevalue ϕ of the ground state phase is determined from γ GL ( ψ ) = U ′ GL ( ψ ) = 0, i.e. , ϕ GL = r − a b ≈ r − a b √ ǫ. (A11)The self-generated flux in the ground state is given byEq. (19) and for α → α c can be approximated by2 π Φ GL Φ ≈ β Σ β ϕ GL = β Σ β r − a b √ ǫ. (A12)where we took into account that for small ψ , accordingto Eq. (2), φ ≈ ψ/ (1 + β ) and φ ≈ ψ/ (1 + αβ ).The barrier between two wells is given by U GL (0) − U GL ( ϕ ) = a b ≈ a b ǫ . (A13)The depinning current γ c − can be found as an ex-tremum of γ ( ψ ). The extremum is reached at ψ = ψ dep satisfying the equation γ ′ ( ψ dep ) = 2 a + 12 bψ = 0.Thus, ψ dep = r − a b = ϕ GL √ ≈ r − a b √ ǫ. (A14)The value of γ c − = γ ( ψ dep ) is γ − = ψ dep a = 4 s − a b ǫ . (A15)The value of γ c + cannot be calculated in the frameworkof our GL-approximation as γ c + corresponds to a largedepinning phase.The eigenfrequency in each of the ± ϕ wells can becalculated as ω = U ′′ ( ϕ ) = γ ′ ( ϕ ) = − a ≈ − a ǫ. (A16)0 A. A. Golubov, M. Y. Kupriyanov, and E. Il’ichev, Rev.Mod. Phys. , 411 (2004). L. N. Bulaevski˘i, V. V. Kuzi˘i, and A. A. Sobyanin, JETPLett. , 290 (1977), [Pis’ma Zh. Eksp. Teor. Fiz. 25, 314(1977)]. A. I. Buzdin, L. N. Bulaevskii, and S. V. Panyukov, JETPLett. , 178 (1982). J. J. A. Baselmans, A. F. Morpurgo, B. J. V. Wees, andT. M. Klapwijk, Nature (London) , 43 (1999). T. Kontos, M. Aprili, J. Lesueur, F. Genˆet, B. Stephanidis,and R. Boursier, Phys. Rev. Lett. , 137007 (2002). V. A. Oboznov, V. V. Bol’ginov, A. K. Feofanov, V. V.Ryazanov, and A. I. Buzdin, Phys. Rev. Lett. , 197003(2006). M. Weides, M. Kemmler, E. Goldobin, D. Koelle,R. Kleiner, H. Kohlstedt, and A. Buzdin, Appl. Phys.Lett. , 122511 (2006), arXiv:cond-mat/0604097. J. A. van Dam, Y. V. Nazarov, E. P. A. M. Bakkers,S. De Franceschi, and L. P. Kouwenhoven, Nature (Lon-don) , 667 (2006). H. Jorgensen, T. Novotny, K. Grove-Rasmussen, K. Flens-berg, and P. Lindelof, Nano Lett. , 2441 (2007). A. K. Feofanov, V. A. Oboznov, V. V. Bol’ginov, J. Lisen-feld, S. Poletto, V. V. Ryazanov, A. N. Rossolenko,M. Khabipov, D. Balashov, A. B. Zorin, P. N. Dmitriev,V. P. Koshelets, and A. V. Ustinov, Nat. Phys. , 593(2010). M. I. Khabipov, D. V. Balashov, F. Maibaum, A. B.Zorin, V. A. Oboznov, V. V. Bolginov, A. N. Rossolenko,and V. V. Ryazanov, Supercond. Sci. Technol. , 045032(2010). A. S. Vasenko, S. Kawabata, A. A. Golubov, M. Y.Kupriyanov, C. Lacroix, F. S. Bergeret, and F. W. J.Hekking, Phys. Rev. B , 024524 (2011). A. Buzdin, Phys. Rev. Lett. , 107005 (2008). M. Alidoust and J. Linder, Phys. Rev. B , 060503 (2013). E. Goldobin, D. Koelle, R. Kleiner, and R. G. Mints, Phys.Rev. Lett. , 227001 (2011), arXiv:1110.2326. H. Sickinger, A. Lipman, M. Weides, R. G. Mints,H. Kohlstedt, D. Koelle, R. Kleiner, and E. Goldobin,Phys. Rev. Lett. , 107002 (2012), arXiv:1207.3013[cond-mat.supr-con]. E. Goldobin, H. Sickinger, M. Weides, N. Ruppelt,H. Kohlstedt, R. Kleiner, and D. Koelle, Appl. Phys. Lett. , 242602 (2013). A. Lipman, R. G. Mints, R. Kleiner, D. Koelle,and E. Goldobin, Phys. Rev. B , 184502 (2014),arXiv:1208.4057. Y. Tanaka and S. Kashiwaya, Phys. Rev. B , R11957(1996). Y. Tanaka and S. Kashiwaya, Phys. Rev. B , 892 (1997). E. Il’ichev, M. Grajcar, R. Hlubina, R. P. J. IJssel-steijn, H. E. Hoenig, H.-G. Meyer, A. Golubov, M. H. S.Amin, A. M. Zagoskin, A. N. Omelyanchouk, and M. Y. Kupriyanov, Phys. Rev. Lett. , 5369 (2001). G. Testa, E. Sarnelli, A. Monaco, E. Esposito, M. Ejr-naes, D.-J. Kang, S. H. Mennema, E. J. Tarte, and M. G.Blamire, Phys. Rev. B , 134520 (2005). R. G. Mints, Phys. Rev. B , R3221 (1998). R. G. Mints and I. Papiashvili, Phys. Rev. B , 134501(2001). E. Goldobin, D. Koelle, R. Kleiner, and A. Buzdin, Phys.Rev. B , 224523 (2007), arXiv:0708.2624. R. G. Mints, I. Papiashvili, J. R. Kirtley, H. Hilgenkamp,G. Hammerl, and J. Mannhart, Phys. Rev. Lett. ,067004 (2002). M. Weides, H. Kohlstedt, R. Waser, M. Kemmler, J. Pfeif-fer, D. Koelle, R. Kleiner, and E. Goldobin, Appl. Phys.A , 613 (2007). M. Kemmler, M. Weides, M. Weiler, M. Opel, S. T. B.Goennenwein, A. S. Vasenko, A. A. Golubov, H. Kohlstedt,D. Koelle, R. Kleiner, and E. Goldobin, Phys. Rev. B ,054522 (2010). E. Goldobin, R. Kleiner, D. Koelle, and R. G. Mints, Phys.Rev. Lett. , 057004 (2013), arXiv:1307.8042 [cond-mat.supr-con]. J. Clarke and A. Braginski,
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