Phase diagram of geometric d-wave superconductor Josephson junctions
aa r X i v : . [ c ond - m a t . s up r- c on ] F e b Phase diagram of geometri d -wave super ondu tor Josephson jun tionsA. Gumann and N. S hopohlInstitut für Theoretis he Physik and Center for Colle tive Quantum Phenomena,Universität Tübingen, Auf der Morgenstelle 14, D-72076 Tübingen, Germany(Dated: February 24, 2009)We show that a onstri tion-type Josephson jun tion realized by an epita ti thin (cid:28)lm of a d -wavesuper ondu tor with an appropriate boundary geometry exhibits intrinsi phase di(cid:27)eren es between and π depending on geometri parameters and temperature. Based on mi ros opi Eilenbergertheory, we provide a general derivation of the relation between the hange of the free energy of thejun tion and the urrent-phase relation. From the hange of the free energy, we al ulate phasediagrams and dis uss transitions driven by geometri parameters and temperature.PACS numbers: 74.50.+r, 85.25.Cp, 74.20.RpI. INTRODUCTIONThe hange of the free energy of a Josephson jun tion(JJ) evoked by the variation of the phase determines theintrinsi phase di(cid:27)eren e in the unbiased ground state.Usually, the oupling energy between the ele trodes of aJJ is positive and the urrent-phase relation is sinusoidal, orresponding to a vanishing intrinsi phase di(cid:27)eren e.For the pe uliar ase of negative oupling, however, in-trinsi phase di(cid:27)eren es of π are possible (see Refs. [1,2℄and referen es therein).In the rossover regime between positive and nega-tive oupling, higher harmoni s dominate the urrent-phase relation. This behavior has been studied fordi(cid:27)erent types of Josephson devi es: tunneling andin parti ular grain boundary JJs involving d -wave su-per ondu tors3,4,5, ontrollable super ondu tor-normalmetal-super ondu tor JJs (SNS)6, super ondu tor-ferromagnet-super ondu tor JJs7,8, periodi ally alter-nating - π JJs9,10 and grain boundary JJs in non en-trosymmetri super ondu tors11. The dominating higherharmoni s lose to the - π rossover lead to additional ze-ros of the urrent-phase relation. Whether or not theseadditional zeros are related to stable energeti al minima,and a ordingly to intrinsi phase di(cid:27)eren es of neither nor π , an only be de ided by onsideration of the freeenergy of the JJ.In the present work, we show that intrinsi phase dif-feren es in the full range ≤ γ ≤ π o ur a ross d -wave super ondu tor mi robridges. The geometry un-der onsideration onsists of a stripe of a ˆ c -axis orientedepita ti thin (cid:28)lm of a d -wave super ondu tor, whi h isnarrowed down from one side by a wedge-shaped in i-sion (Fig. 1). A ordingly, the Josephson e(cid:27)e t followssolely from the lateral onstri tion of the thin (cid:28)lm andemerges if the width w of the bridge is of the order ofthe oheren e length ξ of the super ondu ting materialor below. Su h a mi robridge on(cid:28)guration is learlydistin t from a grain boundary, super ondu tor-isolator-super ondu tor or SNS tunneling JJ12,13. It should beemphasized that in the geometry under onsideration,Fig. 1, there exists no grain boundary. A ording to theterminology introdu ed in Ref. [14℄, this mi robridge on- FIG. 1: (Color online) Geometry de(cid:28)ning the mi robridgeJosephson jun tion based on a ˆ c -axis oriented epita ti thin(cid:28)lm of a d -wave super ondu tor. A stripe of the super on-du ting material is narrowed down from one side by a wedge-shaped in ision with the opening angle β . The width of theresulting mi robridge-like jun tion is given by w . Two typ-i al quasiparti le traje tories a ross the jun tion: 1-withoutre(cid:29)e tion, 2-with re(cid:29)e tion.(cid:28)guration belongs to the weak link type JJs sin e theele trodes are not ele tri ally separated by a tunnelingbarrier.In order to hara terize the d Josephson e(cid:27)e t, we al ulate urrent-phase relations based on mi ros opi Eilenberger theory. For the derivation of the intrinsi phase di(cid:27)eren e, we make use of the relation betweenthe hange of the free energy of the jun tion E ( γ ) − E (0) and the urrent-phase relation I ( γ ) . We give a very gen-eral derivation of this relation whi h is valid for arbitrarystru tures exhibiting a urrent-phase relation in the fulltemperature range < T < T c as well as in the pres-en e of an external magneti (cid:28)eld. The intrinsi phasedi(cid:27)eren e of the mi robridge will be dis ussed in termsof phase diagrams, justi(cid:28)ed by the thoroughly derived hange of the free energy.II. BASIC MECHANISMThe intrinsi phase shift of the devi e is a dire t on-sequen e of the d -wave symmetry. If the width of thejun tion w is large, quasiparti le traje tories without andwith a re(cid:29)e tion at the straight edge opposite to thewedge ontribute to the total urrent a ross the jun -tion (traje tories of type 1 and 2 in Fig. 1). If the onstri tion is narrow enough, however, the dominant ontribution to the total urrent stems from traje to-ries whi h get re(cid:29)e ted (type 2). If the orientation ofthe d -wave is for example α = π/ (nodal surfa e), allre(cid:29)e ted traje tories su(cid:27)er a sign hange of the pairingpotential whi h leads to the formation of pronoun ed zeroenergy Andreev bound states at the surfa e15,16,17. For < α < π/ , still a fra tion of all traje tories ontributesto the formation of zero energy Andreev bound states,whi h engender anomalous ounter(cid:29)owing quasiparti lesurfa e urrents18,19 and also intrinsi phase shifts20.III. THEORYA. Current-Phase RelationIn order to al ulate urrent-phase relations of the JJ,we employ mi ros opi Eilenberger theory21. The self- onsisten y equation whi h has to be solved for the pair-ing potential ∆( r , k F ) reads ∆( r , k F ) = Z F S d k ′ F (2 π ) [ V pair ] k F , k ′ F | ~ v ′ F | πk B T X ε n ′ > a a b (1) Here, F S is the Fermi surfa e, v ′ F = v F ( k ′ F ) is the Fermivelo ity, [ V pair ] k F , k ′ F is the pairing intera tion matrix, ε n = (2 n + 1) πk B T are fermioni Matsubara frequen ies,and a = a ( r , k ′ F , ε n ′ ) and b = b ( r , k ′ F , ε n ′ ) are the Ri - ati amplitudes22,23. The self onsisten y equation for the urrent density j ( r ) is given by j ( r ) = 2 e Z F S d k ′ F (2 π ) ( − i )2 πk B T | ~ v ′ F | X ε n ′ > v ′ F − a b a b (2)The self onsisten y equations (1), (2) allow for the mi- ros opi al ulation of the urrent-phase relation20,24.Self onsistent solutions guarantee urrent onservation( ∇ · j ( r ) = 0 ), but an in general only be found numeri- ally. B. Free EnergyIn order to derive the relation between the hangeof the free energy of the jun tion E ( γ ) − E (0) and the urrent-phase relation I ( γ ) , we start from the Eilen-berger fun tional21 for the free energy E (∆ , ∆ † , A ; a, b ) ,parametrized by the pairing potentials ∆ , ∆ † , the ve torpotential A ( r ) and the Ri ati amplitudes a , b : E (∆ , ∆ † , A ; a, b ) = Z d r ′ µ rot ′ A · rot ′ A − µ B ext · rot ′ A + R F S d k ′ F R F S d k ′′ F ∆ † ( r ′ , k ′ F )[ V pair ] − k ′ F , k ′′ F ∆( r ′ , k ′′ F ) − R F S d k ′ F (2 π ) πk B T | ~ v ′ F | P ε n ′ > a b (cid:20) ∆ † ( r ′ , k ′ F ) a + ∆( r ′ , k ′ F ) b + (1 − a b ) (cid:0) ε n ′ − i v ′ F · e A + ~ v ′ F · ∇ ′ ln ab (cid:1) (cid:21) (3)Here, A = A ( r ′ ) is the total magneti ve tor poten-tial, B ext = B ext ( r ′ ) is the external magneti (cid:28)eld, and a = a ( r ′ , k ′ F , ε n ′ ) and b = b ( r ′ , k ′ F , ε n ′ ) are the Ri atiamplitudes.Consider a general variation of this fun tional: d E = ∂ E ∂ ∆ d ∆ + ∂ E ∂ ∆ † d ∆ † + ∂ E ∂ A d A + ∂ E ∂a da + ∂ E ∂b db The variation with respe t to ∆ † yields the self onsis-ten y equation for ∆( r , k F ) , Eq. (1), and the variationwith respe t to ∆ a orresponding self onsisten y equa-tion for ∆ † ( r , k F ) : ∆ † ( r , k F ) = Z F S d k ′ F (2 π ) [ V pair ] k F , k ′ F | ~ v ′ F | πk B T X ε n ′ > b a b (4) The variation with respe t to b yields the Ri ati di(cid:27)er-ential equation for a and vi e versa22,23: ~ v F · ∇ a + 2( ε n − i v F · e A ) a + ∆ † a − ∆ = 0 (5) ~ v F · ∇ b − ε n − i v F · e A ) b − ∆ b + ∆ † = 0 (6)A ordingly, all variations vanish in the ase of a self- onsistent solution, i.e. at the stationary point of thefun tional, independent of the gauge.After making use of the self onsisten y equation for the urrents, Eq. (2), and identi(cid:28) ation of the external ur-rents via rot B ext = µ j ext , we (cid:28)nd for ∂ E /∂ A : d E = Z d r ′ (cid:20)(cid:16) j ( r ′ ) + j ext ( r ′ ) (cid:17) − µ rot rot A ( r ′ ) (cid:21) d A If one integrates over all spa e, the variation of E with respe t to A vanishes due to Maxwell's equation,rot rot A = µ ( j + j ext ) . Restri ting the integration toa (cid:28)nite volume V en losing the jun tion, the external urrents j ext drop out. Now we arry out the gaugetransformation { ∆ → ∆ e iφ , ∆ † → ∆ † e − iφ , a → ae iφ , b → be − iφ , A → A + ~ e ∇ φ } , leading to d E = ~ e Z V d r ′ h j ( r ′ ) − µ rot rot A ( r ′ ) i d ( ∇ ˜ φ ) (7)with the gauge-invariant phase ˜ φ = φ + e ~ R r ∞ d l · A .Considering the volume V a ording to Fig. 2 and usingbasi ve tor al ulus, it an be shown that the ontri-bution of the ve tor potential vanishes. Integration byparts, exploitation of urrent onservation and appli a-tion of Gauss's theorem then results in d E = ~ e Z S = ∂V dσ ′ n · j ( r ′ ) d ˜ φ (8)Only the parts of the surfa e S = ∂V where the urrententers into or leaves the volume V ontribute. Sin e rossse tions of the super ondu tor far from the jun tion areplanes of onstant gauge-invariant phase, d E = ~ e h I d ˜ φ R − I d ˜ φ L i = ~ e I ( γ ) dγ (9)with the total urrent I and the gauge-invariant phasedi(cid:27)eren e γ = φ R − φ L − e ~ R r R r L d l · A and (cid:28)nally E ( γ ) − E (0) = ~ e Z γ dγ ′ I ( γ ′ ) (10)Be ause of urrent onservation, the total urrent for the urrent-phase relation I ( γ ) an be taken at any rossse tion of the super ondu tor S sc : I ( γ ) = Z S sc dσ ′ n · j ( r ′ ) (11) = Z S sc dσ ′ n · e Z F S d k ′ F (2 π ) ( − i )2 πk B T | ~ v ′ F | X ε n ′ > v ′ F − ab ab Quasiparti le bound states as well as the super urrent ontributions are in luded via the mi ros opi Ri atiamplitudes a and b .For the derivation of the result (10), self onsisten y hasbeen assumed. However, even if the urrent-phase rela-tion used to evaluate Eq. (10) has not been al ulatedself onsistently, an upper bound for the hange of the L R I φ JJ V φ∼∼ FIG. 2: The volume V for the integration of the free energyen loses the Josephson jun tion (JJ) through whi h a total urrent I (cid:29)ows. Cross se tions of the super ondu tor far fromthe jun tion are planes of onstant gauge-invariant phase. FIG. 3: (Color online) Three exemplary urrent-phase rela-tions (upper panel) and the orresponding variations of thefree energy (lower panel). A: I = I c sin γ , a normal or -JJ with γ = 0 ; B: I = − I c sin γ , a π -JJ with γ = π ; C: I = − I c sin(2 γ ) , a so- alled ϕ -JJ with < γ < π .free energy follows. Eq. (10) is valid at arbitrary temper-ature as well as in the presen e of an external magneti (cid:28)eld.To the best of our knowledge, the derivation of Eq. (10)given in the present work is the (cid:28)rst mi ros opi deriva-tion with general validity. Previous derivations were ei-ther based on thermodynami reasoning and the appli- ation of the (se ond) Josephson relation dγ/dt = 2 eV / ~ with the voltage V a ross the jun tion25 or were re-stri ted to tunneling jun tions26. The derivation givenin the present work does not depend on the a tual re-alization of the JJ but is valid for arbitrary stru turesexhibiting a urrent-phase relation.By examination of Eq. (10), one (cid:28)nds that a zero of the urrent-phase relation I ( γ ) = 0 with dI ( γ ) /dγ | γ = γ > orresponds to a stable lo al minimum of the free energy,and thus yields the intrinsi phase di(cid:27)eren e γ . If thereexists no more than one nontrivial zero with < γ < π ,four ases an be distinguished: (1) γ = 0 orresponds toa normal JJ; (2) γ = π orresponds to a π -JJ. Finally, < γ < π and I c > ( I c < ) orresponds to a so- alled ϕ -JJ10,27 with a positive (negative) riti al urrent,where the riti al urrent I c is de(cid:28)ned as the absolutemaximum of the urrent-phase relation.In Fig. 3, we sket h three exemplary urrent-phase re-lations and the orresponding variations of the free en-ergy. In the ase of a normal JJ with γ = 0 , the urva-ture of the variation of the free energy at γ = 0 is positive, d E ( γ ) /dγ | γ =0 > (see urves A in Fig. 3). For a π JJwith γ = π , however, the urvature of the variation ofthe free energy at γ = 0 is negative, d E ( γ ) /dγ | γ =0 < (see urves B in Fig. 3). In the rossover regime with in-termediate intrinsi phase di(cid:27)eren es < γ < π , higherharmoni s dominate, but still the urvature of the varia-tion of the free energy at γ = 0 is negative (see urves C inFig. 3). A ordingly, in the ase of a normal or -JJ, thefree energy of the jun tion (cid:28)rstly in reases with in reas-ing phase di(cid:27)eren e γ > , whereas it (cid:28)rstly de reases inthe ase of π and ϕ JJs.IV. SELFCONSISTENT SOLUTIONSIn this se tion, we present full two-dimensional self on-sistent solutions for the mi robridge geometry depi tedin Fig. 1. Therefore, we numeri ally al ulate a self on-sistent solution of the self onsisten y equation for thepairing potential ∆( r , k F ) , Eq. (1). Based on this solu-tion for the pairing potential, we numeri ally solve theequation for the urrent density j ( r ) , Eq. (2).For the self onsistent al ulations, we assume a ylindri- al Fermi surfa e with the ylinder axis aligned perpen-di ular to the (cid:28)lm plane. The geometry used for the al- ulations spreads over an area of about . × . ξ withthe oheren e length ξ = ~ v F / ( π ∆ ∞ ( T = 0)) . Spe u-lar boundary onditions lead to ˆ n · j = 0 with the sur-fa e normal ˆ n at all surfa es of the geometry. For theleft and right end of the geometry depi ted in Fig. 1,periodi boundary onditions have been used. Self on-sisten y automati ally guarantees urrent onservation, ∇ · j = 0 . Details of the self onsistent al ulations havebeen published in a previous work on s -wave super on-du ting mi robridges24.In Fig. 4, we present self onsistent on(cid:28)gurations forthe amplitude and the phase of the pairing potential aswell as for the orresponding urrent density. For this(cid:28)gure, an orientation angle of the d -wave of α = π/ hasbeen used be ause, in this ase, the e(cid:27)e t of the d -wavesymmetry is most pronoun ed. The opening angle of thewedge has been hosen to be β = 0 and the width of themi robridge w = 3 . ξ . An intermediate temperatureof T = 0 . T c has been used.For the orientation angle α = π/ , the d -wave symme-try leads to a suppression of the amplitude of the pairingpotential at all surfa es of the re tangular geometry (seeFig. 4 (a)). The phase di(cid:27)eren e γ a ross the mi robridgewhi h has been used for this (cid:28)gure orresponds to the riti al urrent. At T = 0 . T c , the phase of the pairingpotential monotoni ally in reases from − γ/ at the leftboundary of the geometry to + γ/ at the right boundary(see Fig. 4 (b)). The orresponding urrent distributionin Fig. 4 ( ) exhibits ontributions whi h (cid:29)ow along thegradient of the phase (from left to right, in positive dire -tion) as well as ba k(cid:29)owing surfa e urrents18,19. Theseba k(cid:29)owing surfa e urrents are dire tly related to the d -wave symmetry and they are arried by Andreev boundstates whi h exist at surfa es of d -wave super ondu torswith orientation angles α = 0 (c)(b)(a) FIG. 4: (Color online) Amplitude (a) and phase (b) of thepairing potential as well as the orresponding urrent density( ) at the riti al urrent for a d -wave orientation angle of α = π/ , an opening angle of the wedge of β = 0 and a widthof w = 3 . ξ at a temperature of T = 0 . T c . The boundarygeometry is indi ated by the thi k (red) lines. In ( ), theshading is proportional to the urrent density.sponding to those in Fig. 4, but for a lower temperatureof T = 0 . T c . From Fig. 5 (a), it is obvious that the am-plitude of the pairing potential hardly hanges between T = 0 . T c and T = 0 . T c . However, the on(cid:28)guration ofthe phase of the pairing potential hanges ompletely (seeFig. 5 (b)). With de reasing temperature, the in(cid:29)uen e (c)(b)(a) FIG. 5: (Color online) Amplitude (a) and phase (b) of thepairing potential as well as the orresponding urrent density( ) at a temperature of T = 0 . T c . Other parameters andpresentation orresponding to Fig. 5.of surfa e Andreev bound states strongly in reases, whi hleads to dominating ba k(cid:29)owing surfa e urrents (Fig. 5( )). A ordingly, the phase exhibits a nonmonotoni variation with a phase shift (Fig. 5 (b)). At T = 0 . T c ,integrating the urrent density over a ross se tion of thegeometry yields a negative total urrent.From the self onsistent on(cid:28)gurations presented inFigs. 4 and 5, it follows that the behavior of the mi- robridge strongly depends on the temperature T . With de reasing temperature, the in(cid:29)uen e of surfa e Andreevbound states in reases and, a ordingly, the relativeweight of the ba k(cid:29)owing surfa e urrents. An in reas-ing importan e of surfa e Andreev bound states with de- reasing temperature has also been found in a previouswork on the in(cid:29)uen e of the surfa e Andreev bound stateson the Bean-Livingston barrier28. The ba k(cid:29)owing sur-fa e urrents dominate at low temperatures T as well asfor small widths w of the jun tion. This leads to negativevalues of the urrent-phase relations29 and even to neg-ative riti al urrents20. As follows from Eq. (10) whi hlinks the urrent-phase relation and the variation of thefree energy, these negative urrents are related to (cid:28)niteintrinsi phase di(cid:27)eren es < γ ≤ π . The intrinsi phase di(cid:27)eren e γ determines the state of the mi ro-bridge JJ, whi h an be either a normal or -JJ, a ϕ -JJor a π -JJ.The two-dimensional self onsistent solutions shown inFigs. 4 and 5 ea h are representative on(cid:28)gurations forone set of the parameters T , α , β and w and for a (cid:28)xedvalue of the phase di(cid:27)eren e γ . Fig. 4 for T = 0 . T c orresponds to a normal or -JJ whereas Fig. 5 orre-sponds to a π -JJ. In order to obtain urrent-phase re-lations and, a ordingly, intrinsi phase di(cid:27)eren es fordiverse ombinations of the relevant parameters, the fulltwo-dimensional al ulations have to be repeated manytimes.Finally, it should be noted that the self onsistent on-(cid:28)gurations shown in Figs. 4 and 5 do not hange sub-stantially if the orientation of the d -wave deviates from α = π/ . Small deviations do not lead to an abruptdisappearan e of the ba k(cid:29)owing surfa e urrents. Sim-ilarly, opening angles of the wedge other than β = 0 donot lead to an abrupt disappearan e of the ba k(cid:29)owingsurfa e urrents and do not substantially hange the ur-rent on(cid:28)gurations as presented in Figs. 4 and 5.V. STEP MODELIn order to al ulate omplete phase diagrams of themi robridge, the self onsistent al ulations presented inthe last se tion would have to be repeated for arbitrary ombinations of the relevant parameters: temperature T ,width of the jun tion w , orientation angle of the d -wave α , opening angle of the wedge β and, (cid:28)nally, in order toobtain urrent-phase relations, phase di(cid:27)eren es a rossthe jun tion in the range ≤ γ ≤ π . Unfortunately,the numeri al osts of the full two-dimensional self on-sistent al ulations inhibit the self onsistent al ulationof omplete phase diagrams. Therefore, in the following,we employ a non-self onsistent step model for the pairingpotential ∆( r , k F ) in order to al ulate urrent-phase re-lations and the a ording variations of the free energy forthe mi robridge. Nevertheless, based on the mi ros opi derivation of the relation between the urrent-phase re-lation and the variation of the free energy, Eq. (10), weknow that a non-self onsistent al ulation provides anFIG. 6: (Color online) Boundary geometry for the step model,Eq. (12), together with an exemplary quasiparti le traje tory r ( s ) and the Ri ati amplitudes a ( s ) , b ( s ) orresponding toEqs. (13).upper bound for the variation of the free energy. A de-tailed omparison of the full two-dimensional self onsis-tent al ulations and the step model will be dis ussedsubsequently.We assume a ylindri al Fermi surfa e with the ylin-der axis aligned perpendi ular to the (cid:28)lm plane. A - ordingly, v F = v F (ˆ x cos θ + ˆ y cos θ ) with ˆ x , ˆ y beingunit ve tors in the (cid:28)lm plane and θ being the polarangle. Thus, in the ase of d -wave pairing, V k F , k ′ F = V cos(2 θ − α ) cos(2 θ ′ − α ) . The step model orrespondsto an opening angle of the wedge of β = 0 and assumes astep-like variation of the phase of the pairing potential,whereas its amplitude is taken to be onstant: ∆ L,R ( r , k F ) = ∆ ∞ ( T ) cos(2 θ − α ) e ∓ iγ/ (12)Here, the indi es L, R label the left and right side ofthe jun tion and ∆ ∞ ( T ) is the temperature-dependentamplitude of the pairing potential in the bulk. Thestep model (12) has to be solved taking into a ountthe boundary geometry de(cid:28)ning the mi robridge JJ (seeFig. 6).In order to (cid:28)nd the urrent density (11) at the ross se -tion of the onstri tion, we solve the Ri ati equationsalong traje tories r ( s ) = ( x = 0 , y ) + s (cos θ, sin θ ) , seeFig. 6. Introdu ing Ω( θ ) = p ε n + | ∆ ∞ ( T ) cos(2 θ − α ) | a L,R ( θ ) = b † L,R ( θ ) = ∆ L,R ( θ ) ε n + Ω( θ ) as well as η ( θ ) = 2Ω( θ ) / ( ~ v F ) and l = | y / sin θ | , we (cid:28)ndfor < θ < π/ : a ( s = 0) = a L ( θ ) (13) b ( s = 0) = b R ( θ ) + 1 e η ( θ ) l b R (2 π − θ ) − b R ( θ ) + ∆ R ( θ )2Ω( θ ) ( e η ( θ ) l − Results for π/ < θ < π follow a ordingly.Based on Eqs. (13), the total urrent (11) an be al- ulated by integrating the urrent density over the rossse tion of the mi robridge JJ, i.e. along the negative y -axis of the geometry depi ted in Fig. 6. The integral overthe ross se tion has to be taken from y = 0 to the widthof the jun tion at y = − w . (cid:144) T c w (cid:144) Ξ Πjj FIG. 7: (Color online) Phase diagram of the geometri Josephson jun tion shown in Fig. 1. The thi k (red) line sep-arates regions of positive and negative riti al urrent (aboveand below). For this (cid:28)gure, α = π/ .FIG. 8: (Color online) Intrinsi phase di(cid:27)eren e γ orre-sponding to the phase diagram in Fig. 7. For this (cid:28)gure, α = π/ . VI. RESULTS AND DISCUSSIONBased on the al ulation of urrent-phase relations and riti al urrents I c from the step model (12), we show inFig. 7 the phase diagram of the geometri mi robridge JJfrom Fig. 1 for a (cid:28)xed d -wave orientation angle α = π/ .From the phase diagram, it follows that if the width ofthe jun tion is smaller than a riti al value w c , the riti al urrent is negative. For T → , we (cid:28)nd a value of about w c ≈ . ξ . With in reasing temperature, the riti alwidth de reases to about w c ≈ . ξ near T c . Near T = T c , the urrent-phase relations assume the asymptoti forms I = I c sin γ for w > w c and I = − I c sin γ for w < w c , respe tively, and only the and the π stateo ur. With de reasing temperature, higher harmoni s ofthe urrent-phase relations be ome more important andthe ϕ state appears in the vi inity of the - π transition.At low temperatures, the ϕ state extends to widths w mu h larger than the riti al width w c whi h separates I c < from I c > .FIG. 9: (Color online) Phase diagram for the variation of the d -wave orientation angle α . The thi k (red) line separatesregions of positive and negative riti al urrent (above andbelow). For this (cid:28)gure, T = 0 . T c .In the limit w → , all traje tories traveling throughthe jun tion su(cid:27)er a re(cid:29)e tion. A ordingly, this situa-tion an be onsidered as a π point onta t, the omple-mentary on(cid:28)guration to a normal point onta t30. Thelimit w → implies l → and Eqs. (13) be ome parti -ularly simple: a ( s = 0) = a L ( θ ) b ( s = 0) = b R (2 π − θ ) (14)In this ase, the urrent-phase relations of the d -wavepoint onta t are being reprodu ed, but with an intrin-si phase shift of γ = π . From the phase diagram inFig. 7, one (cid:28)nds that the π point onta t exists at alltemperatures < T < T c .In Fig. 8, we show the intrinsi phase di(cid:27)eren e γ orresponding to the phase diagram in Fig. 7. The dis-appearan e of the ϕ state near T = T c be omes apparentas a dis ontinuous transition from γ = 0 to γ = π .At lower temperatures, however, a ontinuous transitionarises.In Fig. 9, we plot the phase diagram for a (cid:28)xed tem-perature of T = 0 . T c , fo using on the variation of theorientation of the d -wave α . Starting from the ideal ori-entation α = π/ for the o urren e of the π state, we(cid:28)nd that the riti al width w c de reases when the d -waveis being rotated. However, small deviations from α = π/ do not lead to an abrupt disappearan e of the π or the ϕ state whi h is important for the experimental realization.In Fig. 10, we show urrent-phase relations and the orresponding variations of the free energy for d -waveorientation angles α from α = 0 to α = π/ . The π stateis apparent for α lose to π/ sin e negative urrentso ur for all γ . A ordingly, lose to α = π/ , the freeenergy de reases with in reasing γ . With de reasing α ,a transition to the state o urs, with positive urrentsand an in reasing free energy for all γ . Close to thevery transition, a ϕ region with an additional zero of the urrent-phase relation o urs. Sin e the gradient of the urrent-phase relation at this additional zero is positive, FIG. 10: (Color online) Current-phase relations (upper panel)and the orresponding variations of the free energy (lowerpanel) for angles α from α = 0 to α = π/ (as indi ated) insteps of π/ for w = 0 . ξ . The s ale for the urrents isgiven by I = πeN (0) v F k B T c ξ d . For this (cid:28)gure, T = 0 . T c .it orresponds to a stable energeti al minimum and anintermediate intrinsi phase di(cid:27)eren e < γ < π .In the present work, we use the step model (12) inorder to al ulate phase diagrams of the geometri mi- robridge Josephson jun tion depi ted in Fig. 1. Fromthe mi ros opi derivation of the relation between the urrent-phase relation and the variation of the free en-ergy, we know that an upper bound for the free energyfollows if a non-self onsistent model for the pairing po-tential is employed. Based on a detailed omparison of urrent-phase relations from the step model and fromfull two-dimensional self onsistent solutions29, we do notexpe t self onsisten y to qualitatively alter the generalproperties of the devi e as des ribed here. We rather(cid:28)nd that the riti al width w c whi h marks the transi-tion to negative riti al urrents is underestimated in thes ope of the non-self onsistent step model. It should benoted that, a ording to the self onsistent al ulations,the opening angle of the wedge β hardly in(cid:29)uen es the urrent-phase relations. However, small opening anglesof the wedge ould possibly lead to an in reased apa i-tive oupling of the ele trodes. As long as the width w isin the range of several ξ or below, the step model provesto be a useful approximation. Sin e mi ros opi sur-fa e roughness does not suppress surfa e Andreev boundstates18,31, the reported intrinsi phase di(cid:27)eren es areexpe ted to be robust features.FIG. 11: (Color online) D super ondu ting quantum inter-feren e devi e (d SQUID) geometry onsisting of two geo-metri mi robridge Josephson jun tion a ording to Fig. 1.For α = π/ , the geometri mi robridge Josephson jun tionlabeled 1 an be in the , ϕ or π state depending on its widthand on the temperature whereas the Josephson jun tion la-beled 2 always is in the normal or state33.VII. CONCLUSIONIn the present work, we provide a mi ros opi deriva-tion of the relation between the variation of the free en-ergy and the urrent-phase relation with very general va-lidity. Based on a omparison with full two-dimensionalself onsistent solutions, we use a step model for the pair-ing potential in order to al ulate urrent-phase relationsand intrinsi phase di(cid:27)eren es for the geometri mi ro-bridge Josephson jun tion depi ted in Fig. 1. The al- ulation of intrinsi phase di(cid:27)eren es is used to a ess phase diagrams of the Josephson jun tion, justi(cid:28)ed bythe relation to the variation of the free energy.From the phase diagrams, we on lude that the stru -ture sizes required for the experimental realization ofthe π and in parti ular of the ϕ state in uprate high-temperature super ondu tor mi robridges are withinrea h of modern fabri ation te hnology. Be ause of thelarger oheren e length, ele tron-doped materials are es-pe ially promising32. To test our predi tions we suggestan interferen e experiment with a d super ondu tingquantum interferen e devi e (d SQUID) onsisting oftwo mi robridge JJs with α = π/ and α = 0 , re-spe tively (see Fig. 11). Geometry- and temperature-dependent intrinsi phase di(cid:27)eren es a ording to thephase diagram will show up as shifts of the orrespond-ing (cid:29)ux-dependent interferen e pattern. The experimen-tal veri(cid:28) ation of the intrinsi phase di(cid:27)eren es would atthe same time imply a dire t on(cid:28)rmation of the anoma-lous ounter(cid:29)owing quasiparti le surfa e urrents whi hare a unique and intriguing (cid:28)ngerprint of dd