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a usual josephson tunnel junction , the current @xmath3 passing through the junction is related to the phase difference @xmath4 between the two superconductors by @xcite @xmath5 the equilibrium phase difference is zero in the absence of current ( current is also zero when @xmath6 , but this corresponds to the maximum of the josephson energy rather than a minimum ) . the simple relation ( [ i1 ] ) should be replaced with a more complicated one @xcite , @xcite in the case of constriction junctions , but the equilibrium phase difference will still be at @xmath7 .    a conventional dc - squid consists of a superconducting ring with two ordinary josephson junctions on its opposite arms .
the total current passing through the squid is the sum of the currents crossing each of the josephson junctions . as a result of interference between the two currents , the total current - phase relation depends on the flux @xmath8 threading the ring .
in particular , the critical current ( the maximum current allowed before going to non - stationary state with finite voltage drop across the junction ) is given by @xcite @xmath9 where @xmath10 is related to the external flux @xmath8 threading the loop through @xmath11 , with @xmath12 being the flux quantum , @xmath13 the planck constant , and @xmath14 the charge of electron . here
we neglect the self - inductance of the loop .
notice that @xmath10 in eq .
( [ ic0 ] ) has period @xmath15 which corresponds to one flux quantum . at small @xmath10 , the relation ( [ ic0 ] )
is almost flat ( @xmath16 ) ; consequently , the sensitivity of the measurement of small fluxes is low .
moreover , the critical current does not depend on the direction of the magnetic field . to avoid this problem
, one can apply a biasing flux to move the working point of the squid away from the flat region .
however , this biasing flux increases the noise as well as the complexity of the device and sometimes can have unwanted influence on the measured system .
therefore , it is desirable to shift the working point of the squid away from the flat region without applying an external magnetic field .
this is possible by replacing one of the josephson junctions with a junction with non - zero equilibrium phase difference . a dc - squid with a @xmath2-phase difference between the equilibrium phases of the two josephson junctions
has already been studied @xcite . here
we propose another type of squid ( we call it a @xmath1-squid ) with the equilibrium phase difference of @xmath1 between the junctions .
the frustration caused by the two junctions produces the desired shift of the working point . in the next section
we introduce such a device using high @xmath0 superconducting grain boundary junctions . in section
iii we study the @xmath1-squid using a simple model .
section iv is devoted to the quasiclassical calculation of the current in the @xmath17-wave structure and demonstrating agreement with the simple model of section ii .
finally , we summarize our results in section v.
in high @xmath0 superconductors , because of the @xmath17-wave symmetry of the order parameter , the current - phase relations are nontrivial and generally depend on the orientations of the order parameters on both sides of the junction @xcite , @xcite .
if we keep only the first two harmonics , we can write the current - phase relation as @xmath18 when @xmath19 the equilibrium phase will occur at @xmath20 , where @xmath21 is neither 0 nor @xmath2 .
in particular , when the first harmonic vanishes , the equilibrium phase difference happens at @xmath22 .
these types of junctions , known as @xmath1-junctions , can be realized at the grain boundary junction between two @xmath17-wave superconductors with crystalline orientations of @xmath23 and @xmath24 with respect to the grain boundary .
at such a junction ( known as an asymmetric junction ) , there is an exact cancellation of the first harmonic and the second harmonic dominates the current @xcite , @xcite .
these junctions have been realized and tested experimentally @xcite .
a symmetric junction can also be made with @xmath24 misorientation angle by choosing -22.5@xmath25 and @xmath26 crystalline orientations at the two sides .
this junction is very different from the asymmetric one ; specifically , the exact cancellation of the first harmonic does not occur in this case @xcite . in a realistic junction with non - ideal transparency and roughness at the boundary , the second harmonic gets suppressed by the boundary imperfections more strongly than the first harmonic and , as a result , the latter dominates the current .
this leads to a conventional junction ( 0-junction ) with the usual current - phase relation of eq .
( [ i1 ] ) .
these junctions have also been studied experimentally @xcite .    by combining symmetric and asymmetric junctions
, we can have a squid with @xmath1 equilibrium phase difference between the junctions .
1 illustrates such an structure .
first we study this system using a simple model .
the more realistic numerical simulation of the system using a quasiclassical model will come after that .
consider a dc - squid , consisting of a conventional and a @xmath1 josephson junction , with current - phase relations @xmath27 respectively . to simplify the calculations
, we will assume @xmath28 .
the more general case of arbitrary @xmath29 can also be treated as the solutions are similar .
we also neglect the self inductance of the ring in our calculations . when there is a flux @xmath8 threading the squid ring , the phases at two junctions are related by @xmath30 .
the total current passing through the squid is then given by @xmath31 .
\label{cpr}\ ] ] the critical current is the maximum current in this current - phase relation . to find the critical current , one has to find the solutions to the equation @xmath32 .
these solutions are @xmath33 and @xmath34 , with @xmath35 being an integer number . substituting into eq .
( [ cpr ] ) we find the maximum current to be    @xmath36    here `` max '' means the maximum of the three cases with @xmath37 .
2 shows the flux dependence of the critical current using eq .
( [ ic ] ) .
notice that the periodicity of the function is @xmath2 ( half a flux quantum ) although the period of the argument of the sine - functions in eq .
( [ ic ] ) is actually @xmath38 .
the @xmath2-periodicity of eq .
( [ ic ] ) is a result of the symmetry @xmath39 in eq.([cpr ] ) . in other words , changing the applied flux by half a flux quantum shifts the current - phase relation by @xmath2 and therefore does not change the critical current .
this is evidently different from the ordinary squid for which the period is @xmath15 ( one flux quantum ) .
the important feature here is that the @xmath40 does not coincide with the maximum of the function as it does in usual the dc - squids .
therefore , an increase in sensitivity for measuring small fluxes is gained .
moreover , the direction of the flux is also detectable by measuring the critical current .
as was mentioned , the current - phase relations in @xmath17-wave grain boundary junctions does not usually have the simple forms of eq.([cp ] ) .
one has to find the realistic current - phase relation using a microscopic theory , taking into account the imperfections of the boundary .
here we use a quasiclassical model to study this system .
the equations we solve are the eilenberger equations @xcite    @xmath41=0 ,   \label{eqa1}\ ] ]    where @xmath42 @xmath43 is the superconducting order parameter and @xmath44 is the matrix green s function , which depends on the electron velocity on the fermi surface @xmath45 , the coordinate @xmath46 , and the matsubara frequency @xmath47 , with @xmath35 being an integer number and @xmath48 the temperature .
we also need to satisfy the normalization condition @xmath49 in general , @xmath43 depends on the direction of @xmath50 and is determined by the self - consistency equation    @xmath51    where @xmath52 is the interaction potential .
solution of matrix equation ( [ eqa1 ] ) together with ( [ gapeq ] ) determines the current density @xmath53 in the system    @xmath54    in two dimensions , @xmath55 is the 2d density of states and @xmath56 is the averaging over directions of the 2d vector @xmath57 .
the transparency of the boundary is described by the parameter @xmath58 @xcite , @xcite . to incorporate the roughness , we assume a narrow scattering region of width @xmath17 between the two superconductors @xcite , @xcite . the roughness is therefore parameterized by @xmath59 , where @xmath60 is the mean free path of the quasiparticles in this scattering region . in our calculations
, we use @xmath61 which are reasonable values for a realistic system .
different widths , @xmath62 and @xmath63 , are chosen for the two arms and the ratio of the widths is denoted by @xmath64 .
the details of our numerical method for solving these equations is given in @xcite .
3 displays the current - phase relation at different values of the external flux @xmath10 for a squid structure with @xmath65 . at @xmath40 ,
the graph has two local maxima , which is the signature of having a mixture of first and second harmonics .
as @xmath10 is increased , the two maxima become closer and at some value of @xmath10 , they will be equal . at this point ,
the global maximum jumps from one local maximum to another .
this actually explains the sharp kinks at the minimum positions in fig .  2 ( and also in fig .  4 )
4 shows the critical current as a function of @xmath10 for different values of @xmath66 .
one can immediately see the similarities between these graphs and the one obtained from our simple model in fig .  2 .
specifically , the maxima again occur with period of @xmath2 , although the @xmath2-periodicity is not exact here ( the real period is @xmath15 ) .
the maximum of the critical current is shifted from @xmath40 , as was desired .
one can also see from fig .  4 that the maximum range of variation of @xmath67 is obtained at @xmath65 .
this is actually the optimal ratio of the widths for squid design .
we proposed and studied a dc - squid which has @xmath1 difference between the equilibrium phases at its two junctions . the @xmath1 phase difference is easily obtained using the properties of high @xmath0 grain boundary symmetric and asymmetric junctions .
the frustration caused by this phase difference moves the working point of the squid to a point at which the maximum critical current is obtained at a non - zero external flux .
this will increase the sensitivity of the squid at small magnetic field , with a dependence on the direction of the flux , without any external biasing coil .
we introduced a simple model and compared its results with the ones obtained from a microscopic quasiclassical model .
the agreement between the two models is acceptable .
we found that the @xmath1-squid exhibits other new features , different from a conventional squid or a @xmath2-squid , among those is the @xmath68 ( instead of @xmath69 ) periodicity of the critical current .
we find the optimal ratio between the widths of the two arms to be @xmath65 .
the authors are grateful to a.m. zagoskin , e. ilichev , and j.p .
hilton for stimulating discussions
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available : http://xxx.lanl.gov/abs/cond-mat/0105486
a new dc - squid is proposed that exploits the properties of the grain boundary junctions in high @xmath0 superconductors . 
 the orientations of the grain boundaries are chosen in such a way to establish a @xmath1 ( rather than 0 or @xmath2 ) phase difference between the equilibrium phases of the two josephson junctions in the squid loop . 
 this property is exploited to increase the sensitivity and direction dependence of the squid for measuring magnetic flux without additional flux generating coils .    
 squid , high @xmath0 superconductivity , quasiclassical theory .