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increasing disorder is known to suppress superconductivity in low - dimensional systems such as thin films and narrow wires@xcite . this occurs because the disorder causes electrons to move diffusively rather than ballistically , making them less efficient at screening the coulomb repulsion between electrons . the increased coulomb repulsion decreases both the electron - electron attraction needed for superconductivity , and the density of states of electrons available for pairing at the fermi surface@xcite . typical types of experimental data are : ( i ) @xmath4 , the transition temperature as a function of normal state resistance per square@xcite ; ( ii ) @xmath5 , the order parameter at zero temperature , as a function of normal state resistance per square@xcite ; ( iii ) @xmath6 , the upper critical field as a function of temperature and normal state resistance per square@xcite ; ( iv ) @xmath7 , transition temperature as a function of resistance per square and spin - flip scattering rate in films with magnetic impurities.@xcite it is found experimentally that @xmath4 curves from a wide variety of materials fit a universal curve with a single fitting parameter , whilst the few experimental measurements of @xmath8 , seem to have @xmath9 roughly constant@xcite . this fitting to a single curve , whilst pleasing in showing that the basic ingredients of our theories are correct , does not allow detailed analysis of the theory . data of types ( iii ) and ( iv ) are more promising because there is an additional parameter to vary  the magnetic field in ( iii ) , and spin - flip scattering rate in ( iv ) . to the best of our knowledge , only one experiment of type ( iv ) has been performed@xcite , and we discuss it elsewhere.@xcite several experiments of type ( iii ) have been performed@xcite ; some seem to show a positive curvature in @xmath0 at low temperature as disorder is increased . moreover this effect is predicted by theory@xcite , and this seems to be another confirmation of the basic theoretical model . however , we need to be careful : positive curvature in @xmath0 is a ubiquitous feature of exotic superconductors@xcite , and occurs in many systems where localization is not believed to be the cause . indeed any pair - breaking mechanism that varies as a function of magnetic field can lead to such anomalous behaviour in @xmath2 . this means that it is often difficult to distinguish between the various mechanisms that might be present . it is therefore particularly important to be sure of our theory , and in this light we re - examine the predictions of localization theory . one of the main problems of the localization theory is that even first - order perturbation theory results are hard to obtain correctly . the first - order results are capable of explaining experimental data in the weak disorder regime , but for stronger disorder it is clear that we need something else . as an example consider the prediction for @xmath10 suppression@xcite , @xmath11 where @xmath12 is transition temperature for clean system , @xmath13 , and @xmath14 is the elastic scattering time . this yields an exponential curve for @xmath4 , which behaves like a straight line for small @xmath1 . it is clear that @xmath10 deduced from this equation can never go to zero for finite @xmath1 , as happens in experiment . a very simple ad hoc way of going beyond simple perturbation theory is to replace @xmath12 on the right hand side by @xmath10 , pleading perhaps to self - consistency . if we define @xmath15 , @xmath16 , and @xmath17 , the new equation has the cubic form @xmath18 and can easily be solved . however a new problem emerges because there are two positive roots for every value of @xmath1 . at first we can take the larger of the roots , on physical principles , because it is this root which tends to @xmath19 at @xmath20 . however we eventually come to a re - entrance point beyond which no solutions exist . it is clear that this re - entrance is unphysical , an artefact of our ad hoc extension of perturbation theory . in the case of @xmath4 the story has a happy ending in that perturbation theory can be correctly extended by a renormalization group ( rg ) treatment based on finkelstein s interacting non - linear sigma model@xcite . this leads to the result @xmath21 where @xmath22 . this equation reduces to the first - order result for small @xmath23 and now @xmath10 goes smoothly to zero at @xmath24 . the three curves are plotted for comparison in fig . the reason for discussing @xmath4 in detail above is that the same problem occurs for @xmath6 . the standard theory in this case@xcite , due to maekawa , ebisawa and fukuyama ( mef ) is the equivalent of the ad hoc extension discussed above , and has the form @xmath25\right)-r_{hf}-r_v \nonumber\\ r_{hf}&=&\displaystyle-{1\over 2}t\ln^2{\left({1\over 2\pi t_c\tau}\right ) } -t\ln{\left({1\over 2\pi t_c\tau}\right ) } \left[\psi\left({1\over 2}\right ) -\psi\left({1\over 2}+{1\over 2\pi t_c\tau_h}\right)\right]\nonumber\\ r_v&=&\displaystyle-{1\over 3}t\ln^3{\left({1\over 2\pi t_c\tau}\right ) } -t\ln^2{\left({1\over 2\pi t_c\tau}\right ) } \left[\psi\left({1\over 2}\right ) -\psi\left({1\over 2}+{1\over 2\pi t_c\tau_h}\right)\right],\end{aligned}\ ] ] where @xmath26 . this equation suffers similar re - entrance problems at finite @xmath27 to those found at @xmath28 , when it is just the @xmath10 equation . indeed the @xmath0 curves can only be plotted down to the value of @xmath3 at which re - entrance occurs , and at this point the curves appear to have infinite slope . this leads to us asking the question of whether the positive curvature in @xmath0 is also an artefact of the ad hoc approximation used . what we need to answer this question is the finite magnetic field analogue of the rg result discussed above . however the rg is very difficult , and the answer is not forthcoming from this source . fortunately oreg and finkelstein@xcite have recently shown that the rg result can be obtained from a diagrammatic resummation technique . this method has the great advantage of being analytically tractable and easy to use . in this paper we extend this approach to finite magnetic field to see whether we really do expect positive curvature in @xmath0 . indeed this paper is intended as somewhat of a showcase for this resummation method , to demonstrate the ease of its extension to a wide variety of problems . oreg and finkelstein@xcite take the coulomb interaction to be featureless , and we shall follow them . we should therefore explain why it is legitimate to use a featureless interaction rather than the correct screened coulomb interaction . the screened coulomb interaction has a singularity at low momentum , and one might naively think that this would lead to a strong enhancement of the suppression of transition temperature . however a cancellation occurs between diagrams 14 and diagram 5 of fig . 2 , which effectively removes the low momentum singularity@xcite . the coulomb interaction is then effectively featureless , all diagrammatic sums being dominated by large frequency and momentum . it turns out that one gets the same result from doing the perturbation theory correctly with the screened coulomb interaction as from using a constant interaction of strength @xmath29 and keeping only diagrams 14 . this is what we shall do in this paper . we also note that the re - summation method of oreg and finkelstein@xcite keeps only diagrams 3 and 4 , which is again legitimate as their contribution is greater than that from diagrams 1 and 2 . however it is not difficult to include these terms in the re - summation , as we will show in section . the cancellation of low momentum singularities is expected to be a general feature , enforced by gauge invariance , so we are free to ignore them and use a featureless interaction as long as we include the diagrams that give the dominant contribution at large frequency and momentum . the outline of the rest of the paper is as follows . in section we review the first - order perturbation calculation for @xmath0 assuming a featureless coulomb interaction . we derive the analytical mef formula by making an asymptotic approximation to the matsubara frequency sums which arise in perturbation theory . we make plots of @xmath0 both by solving the implicit mef equation , and its equivalent where the matsubara sums are performed exactly . surprisingly we find no positive curvature in the latter case . in section we derive @xmath0 using the resummation technique both in the form used by oreg and finkelstein , and also in an extended form which includes self - energy diagrams missed in their formalism . again we find no positive curvature in @xmath0 . in section we discuss the experimental situation and draw conclusions . in this section we will carefully review the first - order perturbation theory calculation@xcite of @xmath0 . we do this in detail because we will show that accurately performing sums rather than making an analytical approximation to them removes the upward curvature in @xmath0 . we identify @xmath10 as the temperature at which the magnetic field dependent pair propagator diverges . to find the latter we calculate the correction to the momentum - dependent pair propagator , @xmath30 , and then make the usual substitution @xmath31 , where @xmath26 . the zeroth - order ( mean field ) pair propagator is given by @xmath32\ ] ] so that upon substitution @xmath31 we get the usual abrikosov - gorkov result@xcite for the pair - breaking effect of the magnetic field @xmath33 we will calculate the corrections to the pair polarization bubble , @xmath34 , which will lead to a change in @xmath10 given by @xmath35 since we will assume a featureless interaction , we need to evaluate diagrams 1 to 4 of fig . diagrams 1 and 3 involve the summation of three terms to form a hikami box@xcite , the general form for which is @xmath36 where @xmath37 and @xmath38 , the @xmath39 being the incoming momenta , and the @xmath40 the matsubara frequencies on the electron green functions in the box . using standard diagrammatic rules , and the above result for the hikami box , we obtain the 4 contributions to the pair polarization bubble @xmath41 we note that the relative signs of @xmath42 and @xmath43 are irrelevant in the sum , since the summand depends only upon @xmath44 and @xmath45 , and that the featurelessness of the potential allows us to shift the momentum sum in terms 2 and 4 . we find that terms 1 and 2 partially cancel , whilst terms 3 and 4 reinforce , to yield the sum @xmath46[\epsilon_l+\epsilon_n]}+ { 1\over[\epsilon_l+1/\tau_h][\epsilon_n+1/\tau_h ] } \left[\ln{\left({1\over[\epsilon_l+\epsilon_n]\tau}\right ) } + { 1\over[\epsilon_l+\epsilon_n]\tau_h}\right]\right\}\ ] ] where @xmath47 and we have performed the @xmath48-sum and set @xmath31 .    we first reproduce mef s analytic approximation to the sums over matsubara frequencies . to do this it turns out to be easier to make a choice of relative sign of matsubara frequencies , @xmath49 , and to set @xmath50 . if @xmath51 and @xmath52 , the sum becomes @xmath53m}+ { 1\over[l+1/2+\alpha][m - l-1/2+\alpha ] } \left[\ln{\left({m\over m}\right)}+{\alpha\over m}\right ] \right\}\nonumber\\ & = & \displaystyle-{g\over 4\pi^2d}\sum_{m=1}^{m } \left[\psi\left({1\over 2}+m+\alpha\right ) -\psi\left({1\over 2}+\alpha\right)\right ] \left\{{1\over m}+{2\over ( m+2\alpha)}\left[\ln{\left({m\over m}\right ) } + { \alpha\over m}\right]\right\}\end{aligned}\ ] ] where @xmath54 , @xmath55 . if we first evaluate the sum for @xmath56 , we see that it will be dominated by large @xmath57 , so we replace the difference of digamma functions by @xmath58 , and the sum over @xmath57 by an integral to get @xmath59.\ ] ] to evaluate the result for finite @xmath60 we subtract off the @xmath56 result and try to analytically approximate the difference . again we expect the sum to be dominated by large @xmath57 , so we can ignore the @xmath60 in @xmath61 and @xmath62 , and ignore the term proportional to @xmath60 as it is less divergent . the difference then has the form @xmath63 \left\{{1\over m}+ { 2\over m}\ln{\left({m\over m}\right)}\right\}\nonumber\\ & = & \displaystyle-{g\over 4\pi^2d } \left[\psi\left({1\over 2}\right)-\psi\left({1\over 2}+\alpha\right)\right ] \left\{\ln{\left({1\over 2\pi t_c\tau}\right)}+ \ln^2{\left({1\over 2\pi t_c\tau}\right)}\right\}\end{aligned}\ ] ] from our knowledge of the calculation which includes the full screened coulomb interaction we know that we should set @xmath64 , from which it follows that @xmath65 . putting the eqns . ( [ mefpapprox ] ) and ( [ mfpapprox ] ) into eqn . ( [ tch ] ) yields the mef formula cited in eqn . ( [ mef ] ) . it turns out that this approximation , although apparently reasonable , is not justified . if we plot the values of @xmath66 calculated by performing the sums directly , we find that they do not agree with eqn . ( [ mef ] ) . consequently the @xmath0 curves predicted by the analytic approximation and exact sum are also different .    0.25truein    on the left side of fig . 3 we display @xmath0 predicted by the mef result of eqn . ( [ mef ] ) for several values of @xmath1 in the case where @xmath67 . we clearly see the positive curvature at low temperature , and the result that the curves all terminate at finite @xmath3 due to re - entrance problems . on the right side of fig . 3 we plot @xmath0 deduced directly from the first - order perturbation theory result of eqn . ( [ passum ] ) . we note that since the answers we get depend upon the logarithm of the upper cut - off , if we treat the upper cut - off differently , we will get slightly different answers . however , this difference will always be lost in fitting to experiment since we determine the unknown upper cut - off parameter from the initial slope of the data . the reason we mention this is that the exact sums can be performed in two slightly different ways : we can have the sum over @xmath42 and @xmath68 , or the sum over @xmath42 and @xmath43 , the cut - offs in each case being taken at @xmath69 . the results are slightly different due to different treatment of the upper cut - off . however in both cases we find no positive curvature in @xmath0 . there is still the problem that the curves finish at finite @xmath3 because of re - entrance , and this tells us that the ad hoc prescription being used to go beyond first - order perturbation theory is still inadequate . but again we stress the key point of this section : even within the ad hoc extension of first - order perturbation theory , we do not get positive curvature in @xmath0 if we do the matsubara sums exactly . in this section we answer the question of what is the correct way to go beyond first - order perturbation theory . to do this , we extend a resummation technique recently developed by oreg and finkelstein@xcite ( of ) . this involves calculating the pair scattering amplitude , @xmath70 , and identifying @xmath10 as the temperature at which it first diverges . the ladder summation involved is shown diagrammatically in fig . 4 this is actually an extended version of the of approach because as well as considering diagrams 3 and 4 of fig .  2 , which correspond to an effective coulomb pseudopotential , and end up in the block @xmath71 , we also consider diagrams 1 and 2 of fig .  2 , which correspond to an effective self - energy , and end up in the block @xmath72 which renormalises the cooperon impurity ladder . in fact we will consider both versions of the summation technique : the simpler one involves using the bare cooperon @xmath73 rather than the dressed one @xmath74 . of demonstrate that the matrix equation they obtain in 2d can be approximated by a parquet - like differential equation that turns out to be identical to that obtained from renormalization group analysis . instead of generating a continuous approximation to the matrix equation , we simply solve the equation and change the temperature ( which affects the upper cut - off @xmath55 ) until we find the first zero eigenvalue of the matrix , at which point it has become singular , and we have reached @xmath10 . numerically this involves diagonalizing matrices of rank less than 2000 or so . let us now proceed to the calculation of the pair amplitude matrix @xmath75 . if we evaluate the diagrams of fig .  4 , but ignore the magnetic field and self - energy correction to the cooperon , we obtain the same equation as of , @xmath76 { 1\over |\epsilon_m|}\gamma(\epsilon_m,\epsilon_l)\ ] ] except that we have explicitly kept both positive and negative matsubara frequencies because the expression for @xmath77 will turn out to depend upon whether its two matsubara frequencies have the same or opposite sign . this is a reflection of the breaking of time - reversal invariance caused by the magnetic field . let us first see how to solve the above equation , and then later put in the magnetic field , and the correction to the cooperon . as a matrix equation , it takes the form @xmath78{\hat\epsilon}^{-1}{\hat\gamma}\ ] ] where @xmath79 , @xmath80 , @xmath81 and @xmath82 . this has the solution @xmath83 where @xmath84{\hat\epsilon}^{-1/2}\ ] ] and @xmath85 is the identity matrix . it follows that when the matrix @xmath86 has an eigenvalue equal to @xmath87 , the pair amplitude diverges , and we have found @xmath10 . our approach is therefore to start at the mean field transition temperature , @xmath12 , and decrease temperature until one of the eigenvalues of @xmath88 changes sign . the matrix @xmath89 depends upon @xmath3 both through the dependence of @xmath90 upon @xmath3 and through its rank @xmath91 , where @xmath92 . we start at the mean - field value of @xmath93 , which we will call @xmath94 , and decrease the temperature by increasing @xmath93 successively by one . we diagonalize the matrix @xmath95 for each value of @xmath93 until an eigenvalue changes sign . at this point we have found @xmath10 for the given problem , and @xmath96 . we can then change a parameter such as @xmath23 or @xmath60 and repeat the procedure . this removes the need for any perturbative expansion . let us now put back in the magnetic field and cooperon self - energy corrections . the @xmath97 denominator in the eqn . ( [ of ] ) comes from the cooperon @xmath98 and in the presence of a magnetic field , @xmath97 is replaced by @xmath99 , and hence @xmath100 , where @xmath101 . the contributions to @xmath71 from the diagrams of fig .  4 which correspond to diagrams 3 and 4 of fig . 2 are given by @xmath102}\theta(-\epsilon_n\epsilon_m)\nonumber\\ t\lambda_4(\epsilon_n,\epsilon_m)&=&\displaystyle { g\over\pi n(0)}\sum_{q'}{1\over [ dq'^2+|\epsilon_n|+|\epsilon_m| ] } \theta(\epsilon_n\epsilon_m)\end{aligned}\ ] ] and so performing the @xmath48-sum and setting @xmath31 gives @xmath103 } & \quad$\epsilon_n\epsilon_m>0$\cr \displaystyle\ln{\left[{1\over ( |\epsilon_n|+|\epsilon_m|)\tau}\right ] } + { 2\over ( |\epsilon_n|+|\epsilon_m|)\tau_h}&\quad $ \epsilon_n\epsilon_m<0$\cr}\ ] ] it follows that the matrix elements of @xmath90 are @xmath104    to include the self - energy correction into the cooperon , we note that the corrected cooperon is given by @xmath105^{-1}= { 1\over 2\pi n(0)\tau^2}\displaystyle \left[dq^2 + 2|\epsilon_m|-{1\over 2\pi n(0)\tau^2}\sigma\right]^{-1}\ ] ] so that we need to absorb a factor @xmath106 into @xmath72 . the contributions from the diagrams of fig .  4 which correspond to diagrams 1 and 2 of fig .  2 are @xmath107\theta(-\epsilon_n\epsilon_m ) \nonumber\\ \sigma_2(\epsilon_n)&=&\displaystyle-{2g\over n(0)}t\sum_{\epsilon_m}\sum_{q ' } { 1\over ( dq'^2+|\epsilon_n|+|\epsilon_m|)}\theta(\epsilon_n\epsilon_m)\end{aligned}\ ] ] which partially cancel to give the result @xmath108\ ] ] the weak - localization contribution to the cooperon self - energy is given by @xmath109\ ] ] incorporating @xmath72 into the cooperon means that @xmath110 has elements @xmath111 -2\alpha t\ln{\left({m\over n+1/2}\right)}\right\}\delta_{nm}\ ] ] using the new formulas for @xmath110 and @xmath90 , we now plot @xmath0 for various values of @xmath17 . the results are shown in fig .  5 : the plot on the left does not include self - energy corrections ; the plot on the right does . the results are very similar and show no sign of upward curvature in @xmath0 . we note that there are no re - entrance problems in our calculation of @xmath10 : we can plot the curves down to as low a temperature as we like if we are prepared to diagonalize large enough matrices .    0.25truein the central message of the paper is that the theory of localization and interaction does not predict an anomalous positive curvature in the upper critical field @xmath0 of thin film superconductors . a subsidiary message is that the resummation method developed by oreg and finkelstein@xcite is a very powerful and adaptable tool for going beyond perturbation theory in disordered superconductors . we suspected that the positive curvature in @xmath0 was an artefact related to the re - entrance problem found in @xmath4 , and that it would not survive a systematic non - perturbative treatment . the latter is true , but the cause of the positive curvature artefact turns out to be an incorrect analytic approximation to sums of matsubara frequencies . we will now consider the experimental situation by attempting to fit experimental data to both the mef formula of eqn . ( [ mef ] ) , and the exact first - order perturbation theory result of eqn . ( [ passum ] ) . the data which fits best to the localization theory is that of graybeal and beasley@xcite on amorphous films of mo - ge , because values of @xmath10 for all films can be obtained by choosing a single value of the parameter @xmath16 . the fits in fig . 6 used the parameter value @xmath112 for the mef curves and @xmath113 for the exact sum curves . values of @xmath114 and @xmath115koe were taken directly from the data . the experimental data appears to fit better to the exact sum curves than the mef curves , and there seems little sign of an upward curvature in @xmath0 . when the same procedure is applied to the data of okuma et al@xcite on zn films , we find that we need to use different values of @xmath116 for the two films to get the correct @xmath10 for given @xmath1 . the plots in fig .  7 use values @xmath117 for the @xmath118 film and @xmath119 for the @xmath120 film for the mef formula ; @xmath121 for the @xmath118 film and @xmath122 for the @xmath120 film for the exact sum formula . both films have the same thickness of @xmath123 , but different resistances , and hence different diffusion constants @xmath124 . the latter were calculated using the material parameters for zn to get values of @xmath125 for zero - resistance films of the same composition equal to 5.08koe for the @xmath118 film and 7.62koe for @xmath120 film . @xmath12 was taken to be @xmath126 . again the data seems to fit better to the exact sum curves than the mef formula . to fit the data of hebard and paalanen@xcite on amorphous in - ino@xmath127 films again requires a different value of @xmath116 for each film . the plots in fig . 8 use @xmath128 for the @xmath129 film and @xmath130 for the @xmath131 and @xmath132 films for the mef curves ; @xmath133 for @xmath129 film and @xmath134 for the @xmath131 and @xmath132 films for the exact sum curves . the value of @xmath135 quoted in the paper was used , and a value of @xmath136 was estimated from the experimental plots . there is a clear upturn in @xmath0 at low @xmath3 , but it is not explained by the mef curves which are effectively all upturn at these parameter values . again the fit to exact sum curves seems better .    in conclusion it seems that the experimental curves fit better to the exact sum curves than the mef curves . any low temperature upturns in @xmath0 are not explained by the mef formula , and presumably correspond to different physics . we thank i. aleiner , a. clerk , a.m. finkelstein , i.v . lerner and y. oreg for helpful discussions . ras acknowledges the support of a nuffield foundation award to newly appointed lecturers in science and mathematics . bsh acknowledges the support of a uk epsrc graduate studentship . va is supported by the us national science foundation under grant dmr-9805613 .
we present calculations of the upper critical field @xmath0 in superconducting films as a function of increasing disorder ( as measured by the normal state resistance per square @xmath1 ) . in contradiction to previous work , we find that there is no anomalous low temperature positive curvature in @xmath2 vs @xmath3 as disorder is increased . 
 we show that the previous prediction of this effect is due to an unjustified analytical approximation of sums occuring in the perturbative calculation . 
 our treatment includes both a careful analysis of first - order perturbation theory , and a non - perturbative resummation technique . 
 no anomalous curvature is found in either case . 
 we present our results in graphical form . 
 psfig # 1#2