document: with the advent of high temperature superconductivity in the cuprates and the possibility of exotic gap symmetry including nodal behavior , a renewed effort to find novel experimental probes of order parameter symmetry has ensued . 
   one result of this effort was the proposal by sauls and co - workers@xcite to examine the nonlinear current response of d - wave superconductors . 
   they showed that a nonanalyticity in the current - velocity relation at temperature @xmath1 is introduced by the presence of nodes in the order parameter . 
   one prediction was that an anisotropy should exist in the nonlinear current as a function of the direction of the superfluid velocity relative to the position of the node . 
   this would be reflected in an anisotropy of a term in the inverse penetration depth which is linear in the magnetic field @xmath2 . 
   early experimental work did not verify these predictions@xcite and it was suggested that impurity scattering@xcite or nonlocal effects@xcite may be responsible . 
   however , a more recent reanalysis of experiment has claimed to confirm the predictions@xcite . 
   an alternative proposal was given by dahm and scalapino@xcite who examined the quadratic term in the magnetic response of the penetration depth , which shows a @xmath3 dependence at low @xmath4 as first discussed by xu et al.@xcite . 
   dahm and scalapino demonstrated that this upturn would provide a clear and unique signature of the nodes in the d - wave gap and that this feature could be measured directly via microwave intermodulation effects . 
   indeed , experimental verification of this has been obtained@xcite confirming that nonlinear microwave current response can be used as a sensitive probe of issues associated with the order parameter symmetry . 
   thus , we are led to consider further cases of gap anisotropy and turn our attention to the two - band superconductor mgb@xmath0 which is already under scrutiny for possible applications , including passive microwave filter technology@xcite . 
   mgb@xmath0 was discovered in 2001@xcite and since this time an enormous scientific effort has focused on this material . on the basis of the evidence that is available , it is now thought that this material may be our best candidate for a classic two - band electron - phonon superconductor , with s - wave pairing in each channel@xcite . 
   a heightened interest in two - band superconductivity has led to claims of possible two - band effects in many other materials , both old@xcite and new@xcite . 
   our goal is to compare in detail the differences between one - band and two - band s - wave superconductors in terms of their nonlinear response , that would be measured in the coefficients defined by xu et al.@xcite and dahm and scalapino@xcite . 
   this leads us to reconsider the one - band s - wave case , where we study issues of dimensionality , impurities , and strong electron - phonon coupling . 
   we find new effects due to strong - coupling at both high and low @xmath4 . 
   we then examine the situation for two - band superconductors , starting from a case of highly decoupled bands . here 
   , we are looking for signatures of the low energy scale due to the smaller gap , the effect of integration of the bands , and the response to inter- and intraband impurities . 
   unusual behavior exists distinctly different from the one - band case and not necessarily understood as a superposition of two separate superconductors . finally , we return to the case of mgb@xmath0 which was studied previously via a more approximate approach@xcite . in the current work , 
   we are able to use the complete microscopic theory with the parameters and the electron - phonon spectral functions taken from band structure@xcite . in this way 
   , we provide more detailed predictions for the nonlinear coefficient of mgb@xmath0 .    in section  ii , we briefly summarize the necessary theory for calculating the gap and renormalization function in two - band superconductors , from which the current as a function of the superfluid velocity @xmath5 is then derived . in section  iii 
   , we explain our procedure for extracting the temperature - dependent nonlinear term from the current and we examine the characteristic features for one - band superconductors in light of issues of dimensionality , impurity scattering and strong coupling . 
   section  iv presents the results of two - band superconductors and simple formulas are given for limiting cases which aid in illuminating the effects of anisotropy . the case of mgb@xmath0 
   is also discussed . 
   we form our conclusions in section  v. 
   the superfluid current has been considered theoretically in the past by many authors for s - wave@xcite and for other order parameters , such as d - wave and f - wave@xcite . 
   most recently , the case of two - band superconductivity has been examined@xcite with good agreement obtained between theory and experiment for the temperature dependence of the critical current@xcite .    in this work , 
   we wish to calculate the superfluid current as a function of superfluid velocity @xmath5 or momentum @xmath6 and extract from this the nonlinear term . to do this , we choose to evaluate the expression for the superfluid current density @xmath7 that is written on the imaginary axis in terms of matsubara quantities.@xcite this naturally allows for the inclusion of impurity scattering and strong electron - phonon coupling in a numerically efficient manner . 
   written in general for two - bands having a current @xmath8 and @xmath9 , for the first and second band , respectively , we have : @xmath10 where @xmath11 is the electric charge , @xmath12 is the electron mass , @xmath4 is the temperature , @xmath13 , @xmath14 is the electron density and @xmath15 is the fermi velocity of the @xmath16th band ( @xmath17,2 ) . 
   the @xmath18 represents an integration for the @xmath16th band which is given as @xmath19 for a 3d band and @xmath20 for a 2d band , with @xmath21 in the 2d case . 
   also , in the expression for the current , the 3 should be changed to a 2 for 2d . 
   this is done within a mean - field treatment and ignoring critical fluctuations near @xmath22 . here 
   , we have taken the approximation of a spherical fermi surface in 3d and a cylindrical one in 2d as we will see further on that the differences between 2d and 3d are not significant to more than a overall numerical factor and so providing more precise fermi surface averages will not changes the results in a meaningful way . to evaluate this expression , we require the solution of the standard s - wave eliashberg equations for the renormalized gaps and frequencies @xmath23 and @xmath24 , respectively . these have been generalized to two bands and must also include the effect of the current through @xmath6 . with further details given in refs . 
   @xcite , we merely state them here : @xmath25\nonumber\\ & \times&\biggl\langle \frac{\tilde\delta_j(m)}{\sqrt{(\tilde\omega_j(m)-is_jz)^2+\tilde\delta_j^2(m)}}\biggr\rangle_j\nonumber\\ & + & \pi\sum_jt^+_{lj}\biggl\langle \frac{\tilde\delta_j(n ) } { \sqrt{(\tilde\omega_j(n)-is_jz)^2+\tilde\delta_j^2(n)}}\biggr\rangle_j\label{eq : del}\end{aligned}\ ] ] and @xmath26 where @xmath27 sums over the number of bands and the sum over @xmath12 is from @xmath28 to @xmath29 . here , @xmath30 is the ordinary impurity scattering rate and @xmath31 indexes the @xmath31th matsubara frequency @xmath32 , with @xmath33 , where @xmath34 
   . the @xmath35 are coulomb repulsions , which require a high energy cutoff @xmath36 , taken to be about six to ten times the maximum phonon frequency , and the electron - phonon interaction enters through @xmath37 with @xmath38 the electron - phonon spectral functions and @xmath39 the phonon energy . 
   note that the dimensionality does not change the gap equations when there is no current . for finite @xmath6 
   , it does and we will see later the result of this effect . likewise , an essential ingredient is that the current enters the eliashberg equations and provides the bulk of the nonlinear effect for temperatures above @xmath40 . 
   indeed , at @xmath22 all of the nonlinearity arises from the gap . 
   we now proceed to the case of one - band superconductors , to illustrate the generic features of the superfluid current and demonstrate how we extract the nonlinear term . in the section following , we will return to the two - band case . 
   ( 250,200 )    in fig .  [ fig1 ] , we illustrate that these equations reproduce the standard results for @xmath7 versus @xmath6 for a one - band superconductor in the weak coupling bcs limit . equations ( [ eq : js])-([eq : z ] ) were solved for both the 2d and 3d cases at @xmath41 and 0.95 . 
   the @xmath1 result of past literature@xcite is recovered in the case of 3d . 
   one sees for @xmath42 at low @xmath6 , the curve is essentially linear , reflecting the relationship of @xmath43 , with @xmath44 the superfluid density . for strong coupling , the slope would be reduced by approximately @xmath45 as the superfluid condensate is also reduced by this factor . 
   likewise the reduction in the slope with temperature would reflect the temperature dependence of the superfluid density . indeed , to provide these curves using the eliashberg equations 
   , we used the @xmath46 spectrum of al and made the corrections for the @xmath45 factor . 
   al is a classic bcs weak coupling superconductor , that agrees with bcs in every way and is generally used for bcs tests of the eliashberg equations . 
   the @xmath47 for al is 0.43 . 
   while at low @xmath4 the curves show little deviation from linearity at low @xmath6 , and thus the nonlinear correction will be essentially zero ( exponentially so with temperature in bcs theory ) , at @xmath4 near @xmath22 , one sees that there is more curvature for @xmath48 and hence a larger nonlinear term is expected . 
   however , while the 2d and 3d curves differ in behavior near the maximum in @xmath7 , one finds that the behavior at low @xmath6 is very similar . 
   indeed , the nonlinearity is a very small effect on these plots and hard to discern , however , it will be borne out in our paper that the nonlinear current does not show significant differences in the @xmath4-dependence between 2d and 3d . 
   nevertheless , we will still include both the 2d and 3d calculation in our two - band calculations as mgb@xmath0 has a 2d @xmath49-band and a 3d @xmath50-band , and there is a overall factor of 2/3 between the two in the nonlinear term due to dimensionality .    to obtain the nonlinear current as @xmath48 , the general expression for the current 
   can be expanded to second lowest order in powers of @xmath6 leading to the general formula @xmath51 , \label{eq : jexpand}\ ] ] where only first and third order terms arise . here by choice @xmath52 and the variable for the expansion 
   was taken as @xmath53 , where @xmath54 . 
   @xmath55 and @xmath56 are temperature - dependent coefficients which follow when solutions of the eliashberg equations ( [ eq : del ] ) and ( [ eq : z ] ) are substituted in the expression ( [ eq : js ] ) for the current . in practice , it is complicated to expand eqs . 
   ( [ eq : js])-([eq : z ] ) to obtain an explicit form @xmath56 and so we chose to extract @xmath55 and @xmath56 numerically by solving our full set of equations with no approximations for @xmath7 versus @xmath6 . from this numerical data , we find the intercept and slope of @xmath57 versus @xmath58 for @xmath48 from which we obtain the @xmath55 and @xmath56 , respectively . 
   ( 250,200 )    the results for @xmath59 and @xmath60 as a function of temperature are shown in fig . 
   one sees , in the inset , @xmath59 which , in the one - band case , is just the superfluid density @xmath44 normalized to the clean bcs value at @xmath1 . there is no difference between 2d and 3d bcs . also , shown is the @xmath59 extracted for the strong electron - phonon coupling superconductor pb with no impurities and with impurity scattering of @xmath61 . 
   one sees that strong coupling pushes the temperature dependence of the curve higher , even slightly so at @xmath1 , and this is a well - documented effect@xcite . with impurities , 
   the superfluid density is reduced in accordance with standard theory . 
   these curves were obtained from our @xmath7 calculations and agree exactly with bcs and eliashberg calculations done with the standard penetration depth formulas@xcite , confirming that our numerical procedure is accurate . the second term in eq . 
   ( [ eq : jexpand ] ) gives the nonlinear current and the coefficient @xmath60 , which is a measure of this , is also shown for the four cases . 
   [ note that @xmath60 is the same as the @xmath62 of ref . 
   @xcite to within a constant of proportionality . ] here one does find a difference between the 2d and 3d bcs curves showing that dimensionality can affect the nonlinear current . in the case of strong 
   coupling one finds an increase in the nonlinear piece near @xmath22 and also a finite contribution at low @xmath4 which is unexpected in the usual bcs scenario . 
   impurities have the effect of further increasing the low @xmath4 contribution and reducing the curve near @xmath22 . 
   near @xmath4 equal to @xmath22 ( @xmath63 ) in bcs , it can be shown analytically that @xmath64 and for 3d @xmath65 as obtained in our previous paper , ref . 
   the value of @xmath56 for 2d is increased by a factor of @xmath66 . 
   these numbers agree with the numerical calculations in fig . 
   [ fig2 ] , where the 3d bcs curve goes to 0.21 for 2d and 0.14 in 3d . 
   there are two definitions in the literature for the nonlinear coefficient : one is denoted as @xmath67 due to xu et al.@xcite and the other , @xmath68 , used by dahm and scalapino@xcite , is the one that is related to the intermodulation power in microstrip resonators . rewriting eq . 
   ( [ eq : jexpand ] ) in the form @xmath69 , \label{eq : jexpand2}\ ] ] dahm and scalapino define@xcite @xmath70 xu , yip and sauls@xcite keep the form of eq . 
   ( [ eq : jexpand ] ) but define a variable @xmath71 , where @xmath72 is the temperature dependent gap equal to @xmath73 . with this 
   they identify the coefficient @xmath74 in this work , we always take @xmath75 to be the usual bcs temperature dependence of the gap function . 
   ( 250,200 )    in fig . 
   [ fig3 ] , we show the calculations for the @xmath67 coefficient of xu et al .. here , we have made a number of points 
   . first , the 2d bcs curve derived from our procedure agrees with that shown by xu et al.@xcite , once again validating our numerical work for extracting the very tiny nonlinear coefficient . 
   second , for bcs one sees a difference between 2d and 3d in the nonlinear coefficient . 
   the 2d curve goes to 1 at @xmath22 and to 2/3 for the 3d case . 
   the question arises as to whether the difference between 2d and 3d is simply a numerical factor and so with the dotted curve , we show the 3d case scaled up by 3/2 . 
   we do note that there is a small difference in the temperature variations at an intermediate range of @xmath4 , but the major difference between 2d and 3d is the overall numerical factor of 2/3 . 
   third , one might question the necessity of including the effect of the current on the gap itself and to answer this , we show the long - dashed curve where the @xmath6 dependence was omitted in the eliashberg equations ( [ eq : del ] ) and ( [ eq : z ] ) . 
   one finds that the nonlinear coefficient is reduced substantially at temperatures above @xmath76 and disappears at @xmath22 . 
   thus , without the @xmath6 dependence in the gap , the true nonlinear effects will not be obtained for high temperatures as the gap provides the major contribution to the nonlinearity .    in the lower frame of fig . 
   [ fig3 ] , we examine the case of pb to illustrate strong electron - phonon coupling and impurity effects . 
   it is seen that the strong coupling increases the value at @xmath22 and also gives a finite value at low @xmath4 . 
   the behavior at low @xmath4 is surprising in light of the bcs result@xcite , but is related to the inelastic electron - phonon scattering which appears to increase the nonlinear coefficient at small @xmath4 in a similar way to what is already known about the effect of impurities in bcs@xcite . 
   the strong coupling behavior near @xmath22 is similar to that seen for other quantities such as the specific heat@xcite , where the downward bcs curvature is now turned concave upward to higher values at @xmath22 . 
   impurities have the effect of reducing the nonlinearity near @xmath22 and increasing it at low @xmath4 .    once again , in bcs we can provide some analytic results near and at @xmath22 for @xmath77 which provide a useful check on our numerical work . for three dimensions near @xmath22 : @xmath78 and , upon substituting for @xmath75 , @xmath79 
   doing the same algebra for the two - dimensional case corrects these expressions by a factor of 3/2 and gives 1 instead of 2/3 for @xmath80 .    to characterize the strong - coupling effects seen in the figure for pb , we can develop a strong - coupling correction formula for @xmath81 and @xmath80 . 
   these formulas have been provided in the past for many quantities and form a useful tool for experimentalists and others to estimate the strong coupling corrections.@xcite this was done by evaluating this quantity for ten superconductors using their known @xmath46 spectra and their @xmath22 values . 
   we used al , v , sn , in , nb , v@xmath82ga , nb@xmath82ge , pb , pb@xmath83bi@xmath84 , and pb@xmath85bi@xmath86 . 
   these materials were chosen to span the range of typical s - wave superconductors with strong coupling parameter @xmath87 ranging from 0.004 to 0.2 . 
   the details of these materials and references for the spectra may be found in the review by carbotte@xcite . 
   the parameter @xmath88 is defined as : @xmath89.\ ] ] by fitting to these materials , we arrived at the following strong coupling correction formulas for three dimensions : @xmath90\ ] ] and @xmath91 note that , even though @xmath92 is the usual form of the strong coupling correction , in this last equation , we have found no advantage in fitting with the additional parameter offered by the log factor . 
   these formula should be seen as approximate tools to give the trend for @xmath87 for values restricted to the range of 0 to 0.2 . 
   pb has @xmath87 value of 0.128 and is intermediate to this range , and al is a weak coupling superconductor with a value of 0.004 . 
   ( 250,200 )    in fig . 
   [ fig4 ] , we show the coefficient used by dahm and scalapino@xcite for the same cases as previously considered . with this coefficient 
   one finds qualitatively similar curves . 
   the 2d and 3d bcs curves go to zero rapidly at low temperature , but once again the strong coupling effects in pb give a finite value for @xmath68 at low @xmath4 . with impurities the tail at low temperature is raised significantly . due to the divergence in @xmath68 near @xmath22 because of the division by three powers of the superfluid density which is going to zero at @xmath22 , we prefer to work with a new quantity @xmath93 , which removes this divergence . 
   thus , we define @xmath94 and this is shown in the inset in fig . 
   it has the advantage of illustrating the detailed differences between the curves more clearly and providing finite values at @xmath22 which can be evaluated analytically in bcs theory . in this instance , we obtain @xmath95 for three dimensions in agreement with what we obtain from our numerical work , shown in the fig .  [ fig4 ] . 
   once again we can develop strong coupling formulas for this quantity and they are given as : @xmath96\ ] ] and @xmath97 for three dimensions . once again , there was no extra advantage to fitting @xmath98 with the usual form that includes the log factor . 
   this last quantity @xmath99 is related to the intermodulation power in microstrip resonators and hence can be measured directly . 
   having identified the features of one - band superconductors , we now turn to the two - band case where signatures of the two - band nature may occur in these nonlinear coefficients . 
   the generalization of eq . 
   ( [ eq : jexpand2 ] ) to the two - band case proceeds as follows . 
   the total current @xmath7 is the sum of the two partial currents @xmath100 , @xmath101 with @xmath102 for the two - dimensional @xmath49- and three - dimensional @xmath50-band , respectively . for our numerical work 
   , we do take into account the different dimensionality of the bands but , for simplicity in our analytic work below , we take them both to be three dimensional . 
   a decision needs to be taken about the normalization of the current @xmath7 in the second term . 
   dahm and scalapino have used @xmath103 . here instead , we prefer to use the more symmetric form @xmath104 which reduces properly to the one - band case when our two bands are taken to be identical , with @xmath105 , where @xmath31 is the total electron density per unit volume . 
   for the combined system , eqs . 
   ( [ eq : dougb ] ) and ( [ eq : saulsa ] ) still hold with @xmath55 and @xmath56 modified as follows : @xmath106 and @xmath107 with these definitions eq . 
   ( [ eq : jexpand ] ) also holds with @xmath108 replacing @xmath109 and the xu , yip , and sauls variable , @xmath71 , of the one - band case is replaced by @xmath110 , with @xmath75 the usual temperature profile of the bcs gap . 
   other choices could be made . 
   the superfluid density @xmath44 is proportional to @xmath55 for the combined system , specifically @xmath111 is given by eq . 
   ( [ eq : atb ] ) with the first two factors omitted . 
   ( 250,200 )    in fig . 
   [ fig5 ] , we show both the @xmath60 and the @xmath68 for a model which uses truncated lorentzians for the @xmath112 spectra . this same model was used in our previous work@xcite and so we refer the reader to that paper for details . also , in ref . 
   @xcite may be found the curves for the @xmath113 , the penetration depth , and other quantities for the same parameters used here . 
   the essential parameters of this model are @xmath114 , @xmath115 and the interband electron - phonon coupling is varied from @xmath116 ( nearly decoupled case ) to 0.1 ( more integrated case ) . 
   in addition , the @xmath117 , @xmath118 and @xmath119 . in the nearly decoupled case of @xmath116 , it can be seen that the solid curve looks like a superposition of two separate superconductors , one with a @xmath22 which is about 0.33 of the bulk @xmath22 . 
   the lower temperature part of this curve is primarily due to the @xmath50-band ( or band 2 ) which is three dimensional , and indeed , when examined in detail , it has the characteristic behavior of the 3d example studied in the one - band case . 
   the part of the curve at higher temperatures above about @xmath120 is due to the @xmath49-band ( or band 1 ) which is taken to be 2d and indeed , in the case of @xmath60 it shows a dependence approaching @xmath22 that expected for 2d strong - coupling with some interband anisotropy effects . 
   the relative scale of the two sections of the curve is set by the value of the gap anisotropy @xmath121 and the ratio @xmath122 . 
   the overall scale on the y - axis for @xmath68 differs from that of fig . 
   [ fig4 ] due to our choice of @xmath108 for the normalization in the nonlinear term . 
   indeed , for nearly decoupled bands ( solid curve ) , the value of the nonlinear coefficient @xmath68 is small at reduced temperature @xmath123 just above the sharp peak due to band 2 . 
   specifically , it is of order 0.5 . 
   if it had been referred to @xmath124 instead of @xmath108 , it would be smaller still by a factor of 1.7 and comparable to the single band 2d bcs result at the same reduced temperature ( fig .  [ fig4 ] bottom frame , solid curve ) . 
   however , as the non - diagonal electron - phonon couplings @xmath125 and @xmath126 are increased and a better integration of two bands proceeds , @xmath68 at @xmath123 can increase by an order of magnitude as , for example , in the dashed curve . the actual scale in this region is set by the details of the electron - phonon coupling ( see later the specific case of mgb@xmath0 ) .    with more integration between the bands , 
   one finds that the sharp peak at lower @xmath4 is reduced and rounded with a tail reaching to @xmath22 . 
   when @xmath127 , the feature characteristic of the @xmath50-band @xmath22 is almost gone in @xmath60 and absent entirely in @xmath68 , even for modest interband coupling . 
   the same conclusion holds for the effects of interband scattering ( shown in fig . 
   [ fig5 ] for a value of @xmath128 for the nearly decoupled case ) which also integrates the bands and eliminates the lower energy scale . 
   however , while the structure at the lower @xmath22 is now reduced to the point of giving a monotonic curve for @xmath68 , there still remains a large nonlinear contribution well above that for the one - band s - wave case , which marks the presence of the second band . 
   we can have further insight into these results and check our work by developing some simple analytic results in renormalized bcs theory ( rbcs ) . for a summary of the approximations of rbcs and a comparison with full numerical solution for various properties including @xmath7 
   , we refer the reader to our previous work@xcite . for simplicity 
   , we take both bands to be three dimensional in the following . near @xmath129 , eqs . 
   ( [ eq : rbcsa ] ) and ( [ eq : rbcsb ] ) are modified for each band to : @xmath130 and @xmath131 where the functions @xmath132 and @xmath133 have been derived in ref . 
   @xcite and @xmath134 is independent of @xmath135 , where @xmath136 . 
   the @xmath137 s depend on the microscopic parameters of the theory . in rbcs 
   , they are @xmath138 , @xmath139 , @xmath140 , and @xmath141 , from which @xmath22 and @xmath142 follow . while the expressions obtained for the @xmath132 and @xmath133 are lengthy , and hence we do not repeat them here , they are explicit algebraic forms . 
   it is useful in this work to consider several simplifying limits . for decoupled bands @xmath143 and @xmath144 . 
   as the band 2 does not contribute near @xmath22 , @xmath145 and @xmath146 take on the form of the single band case ( eqs . 
   ( [ eq : rbcsa ] ) and ( [ eq : rbcsb ] ) ) . 
   another limiting case is the separable anisotropy model.@xcite in this model , there are only two gap values with a ratio of @xmath147 , with @xmath148 an anisotropy parameter often assumed small . in this case , 
   @xmath149 , @xmath150 and @xmath151 , where @xmath152 and @xmath153 . as a result @xmath154 and @xmath155 in this model , taking in addition that @xmath118 and @xmath156 leads to the one - band case and this can be used as a check of our algebra .    to see the consequences of this algebra for our nonlinear coefficient @xmath68 , we begin with the decoupled band case near @xmath129 for which @xmath144 and @xmath145 and @xmath146 reduce to their single band value . in this limit of @xmath157 
   , @xmath158 ^ 2 , \label{eq : dougdc}\ ] ] where @xmath159 is the gap anisotropy parameter @xmath121 , and @xmath142 is the gap at @xmath1 . 
   this expression shows explicitly the corrections introduced by the two - band nature of the system over the pure one - band case . 
   note that @xmath68 is always increased by the presence of the correction term . in ( [ eq : dougdc ] ) 
   , @xmath159 can never be taken to be one since we have assumed band 2 is weaker than band 1 . before leaving the decoupled case , 
   it is worth noting that @xmath68 will show a change at the band 2 critical temperature @xmath160 . for @xmath4 below @xmath160 , @xmath161 and @xmath162 
   will be finite while above this temperature they are both zero . when the coupling @xmath125 and @xmath126 is switched on but still small 
   , we expect that these quantities will acquire small tails and that they vanish only at @xmath22 . 
   this is the hallmark of nearly decoupled bands .    for the anisotropic @xmath163 model near @xmath22 @xmath164 ^ 3},\ ] ] where the average gap @xmath165 is related to @xmath22 by @xmath166 $ ] . for @xmath167 
   this expression reduces properly to the one - band limit . 
   therefore , it is seen that anisotropy increases @xmath68 for @xmath4 near @xmath22 . 
   another interesting limiting case is to assume both bands are the same , i.e. isotropic gap case , but that the fermi velocities differ in the two bands . 
   near @xmath22 , we obtain @xmath168 ^ 3 } \frac{1}{8}\frac{(v_{f1}+v_{f2})^2}{(v_{f1}v_{f2})^2}(v_{f1}^2+v_{f2}^2).\ ] ] in this case , the fermi velocity anisotropy changes the nonlinear coefficient , but when @xmath118 the expression reduces properly to the one - band result . 
   we find that the fermi velocity anisotropy increases @xmath68 near @xmath22 , a result that is seen in one of our calculations for mgb@xmath0 shown in fig . 
   [ fig6 ] . 
   ( 250,200 )    with fig . 
   [ fig6 ] , we now turn to the specific case of mgb@xmath0 , where we have used the parameters and @xmath112 given by band structure calculations , and as a result , there are , in principle , no free parameters other than varying the impurity scattering rate . 
   the basic parameters are @xmath169 , @xmath170 , @xmath171 , @xmath172 , @xmath173 , @xmath174 , @xmath175 , @xmath176 , with a @xmath177  k and a gap anisotropy of @xmath178 . 
   the ratio of the two density of states is @xmath179 and of the fermi velocities is @xmath180 . 
   we have found excellent agreement between theory and experiment for these parameters , as have other authors@xcite . as we have found in our previous work , mgb@xmath0 is quite integrated between the bands . 
   it is also an intermediate strong coupler with @xmath181 and thus there is competition between the strong coupling effects and the anisotropy@xcite . in fig .  6 
   , the solid curve gives the prediction for mgb@xmath0 for @xmath60 and @xmath68 . 
   a strong nonmonotonic feature around the lower band energy scale is observed in @xmath60 , but the @xmath68 is monotonically increasing with temperature . to see the second band effects in @xmath68 , 
   it is better to plot @xmath182 ( the inset ) which accentuates the subtle variations found at the lower energy scale associated with the @xmath50 band . 
   also shown in the inset for the upper frame is @xmath59 , which gives the temperature dependence of the superfluid density . 
   the solid curve agrees with our previous calculation by other means@xcite . 
   the variation in @xmath59 appears to be sufficient to remove the bump in @xmath60 when divided by three factors of @xmath59 to obtain the definition of @xmath68 . also , shown are the effects of intraband scattering with @xmath183 for the dashed curve and @xmath184 for the dot - dashed . 
   scattering in the @xmath50-band reduces its contribution and provides an impurity tail at low @xmath4 , as found for the one - band case . 
   however , scattering in the @xmath49-band , while lowering @xmath60 near @xmath22 as expected , does not appear to add weight at low @xmath4 . 
   this is because the parameters for mgb@xmath0 heavily weight the @xmath50-band and the @xmath49-band is a small component . 
   thus , upon comparison between @xmath49- and @xmath50-band scattering , @xmath68 could be lowered at @xmath185 , for example , by putting impurities in the @xmath50-band , but it would be raised if the impurity scattering is in the @xmath49-band . 
   the dotted curve in the figure is for pure mgb@xmath0 but where we have taken @xmath186 to mimic a case where transport may happen along the c - axis . in this instance , the bump in @xmath60 remains , but is gone in @xmath68 . 
   we see that @xmath68 is large due to the higher power of the fermi velocity ratio that enters the calculation , and , as a result , the nonlinearity is greatly increased . 
   as @xmath68 is a relevant quantity for microwave filter design , this study provides some insight into which factors may be used to optimize the material and reduce the nonlinear effects . 
   study of nonlinear current response is important for device applications and for providing fundamental signatures of order parameter symmetry , such as have been examined in the case of d - wave superconductivity . in this paper , we have considered the case of two - band superconductors . in so doing , we also reexamined the one - band case and discovered that there can exist extra nonlinearity at both low and high temperatures due to strong electron - phonon coupling , for which we have provided strong - coupling correction formulas , whereas the excess nonlinearity induced by impurity scattering occurs primarily at low temperatures . at @xmath22 
   , impurities will give an enhanced or decreased contribution depending on the particular nonlinear coefficient discussed . 
   in this paper , we have examined two nonlinear coefficients defined in the literature , one due to xu et al.@xcite and one defined by dahm and scalapino , with an emphasis on the latter as it is related to the intermodulation power in microstrip resonators@xcite . 
   we have also studied issues associated with dimensionality motivated by the two - band superconductor mgb@xmath0 , which has a two - dimensional @xmath49-band and a three - dimensional @xmath50-band . within our one - band calculation , aside from an overall factor of 2/3 , we find little difference in the temperature variation of the nonlinear coefficient in mean - field between 2d and 3d . 
   this is further reduced by strong coupling effects .    for two - band superconductors , 
   we show that for nearly decoupled bands a strong signature of the small gap @xmath50-band will appear in the nonlinear coefficients , but with increased interband coupling or interband scattering , such a signature will rapidly disappear . likewise 
   , intraband impurities in the @xmath50-band will wash out the temperature variation of the @xmath50-band , whereas the intraband impurities in the @xmath49-band largely effect the nonlinearity at higher temperatures above the energy scale of the @xmath50-band , for the parameters typical to mgb@xmath0 . 
   we provide a prediction for the nonlinear coefficient in mgb@xmath0 using the parameters set by band structure calculations . as the bands in mgb@xmath0 are quite integrated , we find that the nonlinear coefficient @xmath68 is monotonically increasing in contrast to a previous prediction , which was based on a number of approximations,@xcite and we find that the increased nonlinearity due to the @xmath50-band is best reduced at @xmath185 by adding impurities to the @xmath50-band . 
   should the supercurrent sample the c - axis direction , a larger anisotropy in the fermi velocity ratio between the bands would result and this effect is found to increase the nonlinearity . 
   finally , several simple formulas have been provided for near @xmath22 which aid in the understanding of the range of behavior observed in the numerical calculations . 
   we await experimental verification of our predictions . 
   we thank dr . 
   ove jepsen for supplying us with the mgb@xmath0 electron - phonon spectral functions . 
   djs would also like to acknowledge useful discussions with thomas dahm . 
   ejn acknowledges funding from nserc , the government of ontario ( prea ) , and the university of guelph . 
   jpc acknowledges support from nserc and the ciar . 
   djs acknowledges nsf support under grant no . 
   dmr02 - 11166 . 
   this research was supported in part by the national science foundation under grant no . 
   phy99 - 07949 and we thank the hospitality of the kitp , where this work was initiated . 
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summary: we have calculated the nonlinear current of a number of single band s - wave electron - phonon superconductors . among issues considered were those of dimensionality , strong electron - phonon coupling , impurities , and comparison with bcs . for the case of two bands , 
   particular attention is paid to the role of anisotropy , the integration effects of the off - diagonal electron - phonon interaction , as well as inter- and intraband impurities . for the specific case of mgb@xmath0 , 
   we present results based on the known microscopic parameters of band theory .