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in anisotropic superconductors , local rotations of the lattice modulate the order parameter . the most striking manifestation of this effect is the influence of grain boundaries in high - t@xmath2 superconductors@xcite . less pronounced effects are also to be expected from lattice defects such as dislocations , as they twist the lattice axes in their vicinity . transversely polarized phonons can , in principle , also couple to the superconducting order parameter through the local rotations that they induce . if , along a given direction , there are strains which change the orientation of the lattice axes , the equilibrium order parameter must follow that distortion . for instance , in a two dimensional square lattice , a parameter with @xmath3 symmetry , @xmath4 , must change into @xmath5 after a lattice rotation by an angle @xmath6 . we express the local order parameter in terms of its components in a fixed frame of reference which is independent of the orientation of the lattice . we expand the local order parameter in terms of the functions @xmath7 , defined in this external frame . in principle , the symmetry of the square lattice allows for mixing of a pure @xmath8 function with higher spherical harmonics . under a rotation , each of these harmonics behaves differently , and requires a specific description in the external frame of reference mentioned earlier . for simplicity , we will first ignore these higher order components . their inclusion will be discussed later , as well as extensions to more complex order parameters . we will now set up an effective ginzburg landau description of the situation described above . let us first consider a model superconductor with very small coherence length . at each position , the order parameter will match perfectly the orientation of the lattice axes . there is no cost in free energy due to misorientations in the order parameter . we can ignore gradient terms , and use an expression which contains only quartic and quadratic terms , provided that we insert in them the value of order parameter with respect to the local axes . the coupling between neighboring regions is described by gradient terms . there is a free energy cost associated to the inhomegeneities of the order parameter . in the present case , the degree of inhomegeneity must be referred to the fixed , external frame of reference . there is an increase in the free energy of the system when different areas of the lattice are misoriented , even if the local order parameter is aligned with the respective axes . a particular example of this situation is a grain boundary , as described in the introduction . the difference between the two ways of describing the order parameter , with respect to an external frame or to the local one , is determined by the degree of rotation of the lattice from one region to another . let us define the rotation which takes the axes from one point to another by the angle @xmath6 . as mentioned previously , an order parameter which follows the lattice will look , in the external frame , like @xmath4 in one point and @xmath9 in the other . the transformation which takes one expression into the other can be accomplished by modifying the two component vector @xmath7 by means of an operator of the type @xmath10 , such that : @xmath11 and : @xmath12 the integral is to be taken along the path in real space along which the lattice is rotated . as @xmath6 is the total rotation along the path considered , we can write , for the example considered here : @xmath13 where @xmath14 is the gradient operator in real space , @xmath15 specifies the local rotation of the lattice axes , and @xmath16 is a pauli matrix . the rotation @xmath15 can be written as@xcite : @xmath17 where @xmath18 are the local deviations of the lattice node at @xmath19 from equilibrium . ( [ rotation ] ) is valid for small deviations ( strains ) . the expressions above allow us to relate the changes in the order parameter as seen in the two frames , and the rotations of the lattice , in terms of the local strains . the gradient term in the ginzburg landau description , which should be expressed with respect to the external frame , looks , in terms of the order parameter with respect to the local axes : @xmath20 where @xmath21 is the free energy . @xmath22 is a number with dimensions of temperature , and of order @xmath23 . finally , @xmath24 are proportionality constants which play no role in the analysis presented below . we take the critical temperatures of the @xmath25 order parameters as different , as appropiate for a square lattice . equation ( [ gl ] ) and the definitions ( [ gauge ] ) and ( [ rotation ] ) suffice to study the phenomenology of layered superconductors whose order parameter is well approximated by @xmath26 $ ] . it is easy to see that , when @xmath4 and @xmath27 have the same critical temperature , and @xmath28 , the effects of the field can be made to vanish by performing a local rotation of the order parameter@xcite . in this case , the system has isotropic superconducting properties . hence , rotations of the underlying crystal lattice leave the ( degenerate ) order parameter unaffected . the formalism described here can be expressed in terms of a connexion , which depends on the lattice orientation . the definition of parallel transport of a vector , in our case @xmath29 , needs to be modified by the rotation of the lattice . the usual derivative is changed into a covariant derivative@xcite . this technique lies at the basis of the extensive work done in topological phases@xcite , and has found use in many fields in condensed matter physics@xcite . in addition to the effects described by ( [ gl ] ) , there may be an explicit coupling of the order parameter to the lattice strains . the simplest coupling is proportional to @xmath30 , where @xmath31 defines the local compression , or expansion , in terms of the symmetric part of the strain tensor . this term can also exist in a @xmath32 wave superonductor . a coupling of this type is not negligible in the high - t@xmath2 compounds , due to the strong dependence of t@xmath2 on pressure@xcite . the previous analysis can be generalized to three dimensional superconductors with arbitrary order parameters . local rotations of the lattice can be expressed in terms of the parameter : @xmath33 where @xmath34 is the fully antisymmetric tensor . in order to determine the transformations induced by these rotations in the order parameter , we must first decompose it into spherical harmonics with well defined transformation properties . then , a gauge field which performs rotations in order parameter space is added to each component . finally , as in ( [ gl ] ) , all terms allowed by the lattice symmetry are included in the free energy . the procedure is cumbersome , but straightforward . for instance , we can add to ( [ gl ] ) higher harmonics which transform like @xmath35 , \sin [ ( 4 n + 2 ) \theta ] $ ] . the gauge field associated to these components is of the type @xmath36 , where @xmath37 transforms one of these parameters into the other . the gauge field induced by the higher harmonics of the order parameter are also proportional to the gradient of the lattice rotations , but with a larger constant of proportionality . the term with @xmath38 in the free energy induces a reduction of these components of the order parameter . this reduction will be stronger for higher harmonics than for lower ones . we now consider specific lattice distortions within the layers . the simplest static defect is a dislocation . the strains in the vicinity of a dislocation can be calculated from the theory of elasticity . if the burgers vector is @xmath39 , then@xcite , @xmath40 where we are neglecting a possible anisotropy in the elastic constants of the lattice . we now assume that the @xmath27 component is negligible , which is justified if , for instance , the parameter @xmath41 in ( [ gauge ] ) is less than zero . then , the coupling of the lattice rotations to @xmath4 is due to terms quadratic in the gauge potential , and we obtain : @xmath42 and latexmath:[$| { \bf \vec{a } } |^2 = \frac {     term is to reduce the effective critical temperature near the dislocation . it suppresses a possible local enhancement of t@xmath2 due to the explicit dependence of the critical temperature on pressure@xcite . in the presence of a magnetic field , the normal cores of vortices will be attracted towards the dislocations , to take adventage of the supression of t@xmath2 , leading to pinning . near the core of the dislocation , @xmath44 , we can estimate the reduction of t@xmath2 ( see below ) as @xmath45 . this effect can be significant in a material with a short coherence length , such as the copper oxides . a grain boundary can be modelled as an array of dislocations . the gauge fields induced by each of them can be added , to obtain the effect of the whole boundary . after some algebra , we find that , if the rotation of the axes across the boundary is not large : @xmath46 ^ 2 + \sin^2 \left ( \frac{\pi y}{\bar{a } } \right)}{\left [ \cosh \left ( \frac{\pi x}{\bar{a } } \right ) - \cos \left ( \frac{\pi y}{\bar{a } } \right ) \right]^4 } \nonumber \\ & \sim & \frac{\pi^2 \sin^4 ( \theta ) } { 4 a^2 } e^{- \frac { \pi x}{\bar{a } } } , \ , \ , \ , \ , \ , x \rightarrow \infty \label{grainb}\end{aligned}\ ] ] where @xmath47 is the lattice constant , @xmath6 is the angle between the axes at both sides of the boundary , and @xmath48 . from ( [ grainb ] ) and ( [ gl ] ) , we can estimate the reduction of the critical temperature at the boundary . we assume that @xmath24 , and @xmath49 . then , @xmath50 , where @xmath51 . this calculation is rigorously valid when @xmath52 , that is , when strains change slowly . a significant reduction of t@xmath2 can be expected near a twin boundary , which will act in a way similar to a weak link . it would be interesting to check this experimentally@xcite . the orientation of the lattice can also be modified dynamically , by transversely polarized phonons . a longitudinal phonon induces local compressions and dilatations of the lattice . a transverse phonon does not change the volume of the unit cell , but induces a shear deformation which changes from position to position . this deformation can be viewed as a local rotation of the axes . in a material with a complex unit cell , there are also optical phonons which rotate locally the axes . the rotations can be analyzed by means of the formalism discussed before . the only difference with a static distortion is that the gauge field which accounts for the lattice rotations depends on dynamic displacements . the time scales associated to the superconducting response are of the order of the inverse plasma frequency . this frequency is much higher than typical phonon frequencies , so that the changes in t@xmath53 induced by phonons can be studied as if the lattice distortions were static . transversely polarized acoustic phonons give rise to strains proportional to the gradient of their amplitudes . hence , we expect that the fourier components of @xmath15 are given by : @xmath54 where @xmath55 is the amplitude of a phonon of momentum @xmath56 , and @xmath57 is the component of @xmath56 perpendicular to @xmath55 . for transversely polarized phonons , @xmath56 is always perpendicular to @xmath55 , and we do not need to take this restriction into account . the gauge potential in ( [ gauge ] ) has a finite average , related to the thermal ( or quantum ) average value of the phonon amplitudes . it is given by : @xmath58    we are analyzing materials which show strong differences in the superconducting properties along the in plane and out of plane directions . phonons , however , need not be so anisotropic . we assume that they are described by an isotropic sound velocity , @xmath59 , and a debye frequency @xmath60 , where @xmath47 is the lattice constant , which we take to be of the same order of magnitude in all directions . the three dimensional integral implicit in ( [ gaugeph ] ) is extended to those modes such that @xmath61 and @xmath62 and @xmath63 . note that the ginzburg - landau description used here is valid at length scales greater than @xmath64@xcite . for simplicity , we take the bose occupancy of a mode of frequency @xmath65 to be @xmath66 . none of these approximations will alter the order of magnitude of the effect . we can distinguish three regimes : i ) if @xmath67 , the upper cutoff in ( [ gaugeph ] ) is due to the bose factor . ii ) when @xmath68 , the upper cutoff is @xmath69 and @xmath70 and iii ) if @xmath71 , the maximum value of @xmath72 is @xmath73 . we can now estimate the reduction in the ( average ) value of the critical temperature in the same way as in the previous section . we find : @xmath74 where @xmath75 is the density . this reduction of the critical temperature should not be negligible in the high - t@xmath2 compounds , as @xmath64 is not very different from @xmath47 , the lattice constant . in addition to a reduction in the average value of t@xmath2 , we can also analyze the fluctuations in this value . we are treating phonons as a source of static disorder , which reduces t@xmath2 . the value of @xmath76 fluctuates , changing the local values of the critical temperature . if these fluctuations take place on scales larger than @xmath64 , a distribution of t@xmath2 s will be observed . as the changes in t@xmath2 go as : @xmath77 we find that : @xmath78 and : @xmath79 by comparing ( [ tcphonons ] ) and this expression , one can see that fluctuations can not be neglected . finally , the typical length scale for fluctuations inside the planes is @xmath80 where @xmath81 is the electronic bandwidth . hence , we expect large fluctuations in small scale domains . transport measurements will be unable to resolve this structure , and the average @xmath82 will be observed . transverse optical phonons of frequency @xmath83 . give an additional reduction of t@xmath2 , @xmath84 where @xmath85 denotes the mean square displacement of the phonon . it is , typically , a fraction , @xmath86 of @xmath87 . as in the previous case , we assume that the superconducting order parameter has a short coherence length in the out of plane direction . there is a contribution like ( [ phononopt ] ) for each optical phonon transversely polarized in the plane . the frustration of the order parameter in anisotropic superconductors by dynamical vibrations implies that @xmath88 can not be much larger than the frequency of the phonons responsible of the effect . the ginzburg landau expression ( [ gl ] ) contains terms which are linear in @xmath76 , which couple one component of the order parameter and the spatial derivatives of the other ( note that @xmath76 is an operator which rotates one component into the other ) . these derivatives are different from zero in the presence of a supercurrent , because of the variation in the phase . in the case described by ( [ gl ] ) , however , these terms do not contribute , as @xmath89 . terms of this type play a role in materials where the superconducting properties differ significantly along the lattice axes . let us consider an orthorhombic superconductor , where the order parameter is @xmath1 . in addition to the terms in ( [ gl ] ) we also have : @xmath90 where @xmath91 is the @xmath32 component of the order parameter . for the sake of simplicity , we ignore terms induced by the orthorhombic symmetry which are quadratic in the gradients@xcite . these terms , needed for a complete description of the anisotropy in the penetration depth , do not modify qualitatively the picture discussed in the previous sections .    in the absence of internal lattice rotations , @xmath92 . the first term in ( [ ortho ] ) leads to a difference in the penetration depth along the two lattice axes . local rotations of the lattice can be incorporated by adding the gauge field ( [ gauge ] ) to the derivatives of the @xmath0 components of the order parameter in ( [ ortho ] ) . then , assuming that the equilibrium order parameter has both @xmath4 and @xmath91 components , we obtain terms of the type : @xmath93 this expression gives a contribution if @xmath94 , and , in particular , in the presence of a current . the changes in a current carrying state induced by lattice rotations are best studied within london s framework , which is valid if the magnitude of the order parameter does not change . we define the london tensor , with the same symmetry properties of the perfect lattice , such that @xmath95 , where @xmath96 is the electromagnetic vector potential . for the study of currents within the layers , we have : @xmath97 where @xmath98 are the penetration depths along the two axes , and @xmath59 is the velocity of light . london s expression for the free energy is@xcite : @xmath99 the deviation of tetragonal symmetry is given by @xmath100 . taking into account that the gauge field related to the lattice distortions , @xmath101 , is itself a derivative , we can integrate by parts in eq.([orthoder ] ) , to obtain : @xmath102 this equation can be interpreted as a rotation of the london tensor , ( [ londont ] ) , induced by the distortion of the lattice . the currents adjust to the directions of the local principal axes . at a grain boundary , for instance , @xmath96 must be continuous . the component of the current parallel to the boundary must be discontinous , leading to a kink in the direction of the total current , as the london tensor which gives the current at both sides is discontinuous@xcite . if , on the other hand , the boundary conditions prevent the current from acquiring a transverse component , the vector potential shows a kink , and a perpendicular magnetic field is generated at the boundary , as sketched in fig([twin]@xcite . this picture , derived from london s theory , is valid at distances comparable to the penetration depth . the discontinuity at the boundary is rounded off at shorter scales . the present analysis implies that currents can also be deflected by other types of static defects , and by phonons . the currents flowing near a dislocation , and the instantaneous current in the presence of phonons , are sketched in fig.([currents ] ) . magnetic fields will be induced . it is interesting to note that phonons can modulate the currents and give rise to electromagnetic fields . so far , we have only considered the role of in plane distortions in layered superconductors . if we include rotations around in plane axes , we must allow for components like @xmath103 in the order parameter . because of the strong anisotropy of these compounds , components which transform like higher powers of @xmath104 are also to be expected . in analogy with ( [ london ] ) , these components lead to contributions of the type @xmath105 , where the direction @xmath106 lies in the planes . note that this effect should also be present in layered superconductors with @xmath32 symmetry within the layers . the magnitude of the @xmath107 coupling is proportional to @xmath108 .    a particular defect which generates rotations in the out of the plane direction is a screw dislocation , which seems to be common in sputtered films@xcite . in cylindrical coordinates , we have that only @xmath109 is different from zero@xcite . hence , the free energy of the superconductor has a term like @xmath110 . a current flowing around the dislocation induces a current parallel to the axis of the dislocation . this situation may arise if a vortex is pinned by a screw dislocation . a similar effect can be realized in a monocrystalline whisker subject to shear strains . then , @xmath111 , leading again to a @xmath110 coupling . a current flowing along the axis of the whisker induces a tangential component , giving rise to a magnetic field in its interior . a schematic picture of both situations is sketched in fig.([dislocation ] ) . we have presented a formalism to study the influence of lattice rotations on anisotropic superconductors . the analysis has been particularized to d - wave layered systems , but it is general enough to be applicable to more complex lattices and order parameters . we expect the scheme outlined here to be also useful for the study of heavy fermions and other materials which show an anisotropic order parameter .    for the systems considered here , we find that    \i ) lattice distortions couple directly to an anisotropic order parameter , irrespective of other mechanisms , like the dependence of t@xmath2 on pressure . \ii ) higher harmonics of the order parameter are more strongly frustrated by local rotations@xcite . \iii ) lattice defects , such as dislocations , reduce locally t@xmath53 . in the presence of a magnetic field , the vortex cores will be attracted to these regions , giving rise to pinning . \iv ) transversely polarized phonons also reduce t@xmath2 . the value of the critical temperature can not be much larger than the typical frequency of these phonons . \v ) local rotations deflect supercurrents in superconductors with inequivalent lattice axes . in particular , the current through a grain boundary is deflected . this work was done with support from cicyt ( spain ) through grant pb96 - 0875 . valuable discussions with k. maki , and hospitality at the istituto eduardo caianello ( vietri , italy ) , are acknowledged . c. c. tsuei , j. r. kirtley , c. c. chi , l. s. yu - jahnes , a. gupta , t. s. shaw , j. z. sun and m. b. ketchen , phys . lett . * 73 * , 595 ( 1994 ) . j. r. kirtley , c. c. tsuei , j. z. sun , c. c. hi , l. s. yu - jahnes , a. gupta , m. rupp and m. b. ketchen , nature ( london ) * 373 * , 225 ( 1995 ) . m. sigrist and t. m. rice , rev . phys . * 67 * , 503 ( 1995 ) . the curl of @xmath112 is non zero only in the presence of a disclination . such a defect induces a finite curvature in the two dimensional layer considered so far . its description requires the inclusion of other harmonics , related to out of plane rotations of the @xmath4 order parameter . m. nakahara , * geometry , topology and physics * , iop publishing , london , ( 1990 ) . a. shapere and f. wilczek , * geometric phases in physics * , world scientific , singapore ( 1988 ) . a. m. tsvelik , * quantum field theory in condensed matter physics * , cambridge u. p. , cambridge ( 1995 ) . l. d. landau and e. f. lifschitz , * theory of elasticity * pergamon press , london ( 1959 ) . f. guinea , europhys . * 7 * , 549 ( 1988 ) . a. b. kuklov , phys . b * 52 * , r7002 ( 1995 ) . the present scheme can not be used to analyze the pair breaking due of short wavelength phonons . it will be studied elsewhere . k. a. kuznetsov , a. g. sun , b. chen , a. s. katz , s. r. bahcall , john clarke , r. c. dynes , d. a. gajewsky , s. h. han , m. b. maple , j. giapintzakis , j .- kim and d. m. ginsberg , phys . 79 * , 3050 ( 1997 ) . p. g. de gennes , * superconductivity of metals and alloys * , addison - wesley , new york ( 1966 ) . f. guinea , phys . rev . b * 40 * , 9362 ( 1989 ) . f. guinea , phys . b * 41 * , 4733 ( 1990 ) . in addition to the deflection discussed here , currents can be induced at a boundary , when the quartic terms in the ginzburg - landau expression are taken into account . see d. l. feder , a. beardsall , a. j. berlinsky and c. kallin , preprint ( cond - mat/9705139 ) . these terms are discussed in detail in@xcite . currents flowing through a twin with a similar geometry have been studied by j. r. kirtley , k. a. moler and d. j. scalapino , phys . b * 54 * , 886 ( 1997 ) and d. f. agterberg and m. sigrist , preprint ( cond - mat/9711157 ) . g. gerber , d. anselmetti , j. g. bednorz , j. mannhart and d. g. schlom , nature ( london ) * 350 * , 279 ( 1991 ) . this effect may lead to the suppression of the kohn luttinger effect , which shows that an electron liquid is always unstable towards the formation of cooper pairs of high angular momenta . local rotations of the lattice will greatly reduce the ( already small ) estimated value of the critical temprature . w. kohn and j. m. luttinger , phys . * 15 * , 524 ( 1965 ) .
lattice distortions in which the axes are locally rotated provide an intrinsic source of frustration in anisotropic superconductors . a general framework to study this effect 
 is presented . 
 the influence of lattice defects and phonons in @xmath0 and @xmath1 layered superconductors is studied .