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recently , magnetic - field - induced superconductivity ( fisc ) has been observed in organic superconductors @xmath4   and @xmath3  @xcite , where bets is bis(ethylenedithio)tetraselenafulvalene . in these salts , localized spins on fe@xmath5 exhibit antiferromagnetic long - range order at ambient pressure at low temperatures . the fisc in these compounds is considered to be due to the jaccarino - peter mechanism  @xcite , where the localized spins are aligned uniformly at high fields . et al . _ have observed in the compound @xmath3  that superconductivity coexists with the antiferromagnetic long - range order in a low - field region around the zero field  @xcite . they have fitted the experimental phase diagrams by fisher s theory  @xcite based on the jaccarino - peter mechanism . the resultant upper critical field of low - field superconductivity is much smaller than their experimental data . they have suggested that the reason for the discrepancy is that the jaccarino - peter - fisher theory does not take into account the antiferromagnetic long - range order at low fields  @xcite .    in recent works , one of the authors has extended the jaccarino - peter mechanism to antiferromagnetic superconductors with canted spin structures in magnetic fields  @xcite . the canted spin structure generates the ferromagnetic moments that create exchange fields acting on the conduction electrons through kondo interactions . if the kondo interactions are antiferromagnetic , the exchange fields partly cancel the zeeman energy . as a result , the pauli paramagnetic pair - breaking effect can be largely reduced , and the upper critical field can exceed the pauli paramagnetic limit ( chandrasekhar and clongston limit )  @xcite . this mechanism occurs even in the presence of the orbital pair - breaking effect  @xcite . we call this mechanism an extended jaccarino - peter mechanism in this paper . since the canted antiferromagnetic phase occurs in the compound @xmath3   for @xmath6  @xcite , we apply the mechanism to this compound .    in the compound @xmath3 , the fisc has been observed both for @xmath7 and @xmath8  @xcite . the phase diagrams for @xmath6 and @xmath9 are rather different , and it is attributed to the anisotropy of the fermi surface and the kondo interactions between the localized spins and the conduction electrons . we take into account the fermi surface anisotropy by effective masses . the effective - mass model was introduced in ginzburg - landau equations by ginzburg  @xcite . hohenberg and werthamer  @xcite pointed out that detailed structures of the fermi surface affect the upper critical field . rieck and scharnberg  @xcite and langmann  @xcite obtained general equations for arbitrary fermi surfaces . et al . _ @xcite and kim _ et al . _ @xcite calculated the upper critical fields of mixed _ d_-wave and _ s_-wave superconductors with effective - mass anisotropy . recently , kita and arai  @xcite have formulated an equation for the upper critical field , taking into account the fermi surface anisotropy and the gap anisotropy on the basis of the rieck and scharnberg theory  @xcite . they have performed the quantitative calculations of the upper critical fields for type - ii superconductors nb , nbse@xmath10 , and mgb@xmath10 using fermi surfaces obtained by first - principles calculations  @xcite . a theory of the upper critical field for layered superconductors has been proposed by lebed and yamaji  @xcite , and developed by lebed and hayashi  @xcite . they have found that when the layer spacing is large the upper critical field exhibits a reentrant transition or an enhancement at low temperatures in the quantum region , due to an effect of dimensional crossover induced by the magnetic field  @xcite . in the compounds @xmath3 , however , since the upper critical field of the low - field superconductivity did not exhibit either a reentrant transition or an enhancement in the experimental phase diagrams , the dimensional crossover does not seem to take place . therefore , from a phenomenological consideration , we use the effective - mass model as an approximation instead of strict equations in refs . @xcite and @xcite . the effective - mass model is adequate in the ginzburg - landau region for layered superconductors .    in this paper , first , we derive critical field equations for _ s_-wave , @xmath1-wave , and @xmath2-wave superconductors with effective - mass anisotropy in three directions , taking into account both orbital and paramagnetic pair - breaking effects . secondly , we take into account the extended jaccarino - peter mechanism . lastly , we reproduce the phase diagrams of @xmath3   including both the fisc and low - field superconductivity . we also examine the possibility of the fflo state . the fflo state has extensively been studied  @xcite since pioneering works by fulde and ferrell , and larkin and ovchinnikov  @xcite . the state is taken into account by an extension of the bcs mean - field theory to include the possibility of finite center - of - mass momenta @xmath11 s . in this study , we adopt a model in which @xmath12 is assumed following gruenberg and gunther  @xcite , since we consider the situation in which substantial orbital pair - breaking effect is present . in the organic compounds @xmath4 , the possibility of the fflo state in the fisc has been pointed out by uji _ et al . _ @xcite and balicas _ et al . _ @xcite , and also examined theoretically  @xcite . the shape of the phase boundary of the fisc is well reproduced by taking into account the fflo state  @xcite . _ have also argued that the fflo state may occur in @xmath13 from their experimental data  @xcite . recently , the quasi - two - dimensional heavy - fermion superconductor cecoin@xmath14 has been believed to exhibit the fflo state  @xcite . adachi and ikeda  @xcite and won _ et al . _ @xcite have calculated the critical fields for @xmath1-wave pairing taking into account the fflo state in connection with cecoin@xmath14 . this paper is constructed as follows . in the next section , we extend the theory of the critical field to the systems with anisotropic fermi surfaces . s_-wave , @xmath1-wave , and @xmath2-wave pairing superconductors are examined . in sec . [ sec : fisc ] , we take into account the extended jaccarino - peter mechanism  @xcite . in sec .  [ sec : res ] , we apply the present theory to the organic superconductor @xmath3 . we compare experimental and theoretical phase diagrams . the final section is devoted to the summary and a discussion . in this section , we derive critical field equations for _ s_-wave , @xmath1-wave , and @xmath2-wave superconductors with an effective - mass anisotropy . we consider the electron system with the energy dispersion @xmath15 and the pairing interaction @xmath16 where we have defined @xmath17 and introduced anisotropic effective masses @xmath18 , @xmath19 , and @xmath20 . in the magnetic field @xmath21 , the hamiltonian is written as @xmath22 where @xmath23 ^ 2   \psi_\sigma({\mathbf{r } } ) \ ; , \\ { \mathcal{h}}_m & =    -\int { d^3}{\mathbf{r}}\ ; \mu_{\text{e}}\mathbf{h } \cdot \left [   \sum_{\sigma\sigma'}\psi_\sigma^\dag({\mathbf{r } } )    \boldsymbol{\sigma}_{\sigma\sigma ' }    \psi_{\sigma'}({\mathbf{r } } ) \right ] \;. \end{split}\ ] ] here , @xmath24 denotes the field operator which annihilates an electron of spin @xmath25 at a point @xmath26 . we have defined the electronic magnetic moment @xmath27 with the bohr magneton @xmath28 and the _ g _ factor of the conduction electrons @xmath29 . we use the units such that @xmath30 unless it is explicitly expressed . we define the @xmath31 axis in the spin space along the direction of @xmath32 . we should note that the @xmath31 axis does not necessarily coincide with the @xmath33 axis of the electron coordinate depending on the direction of the @xmath32 . therefore , we have @xmath34 where @xmath35 with @xmath36 , and @xmath37 and @xmath38 for up and down spin states , respectively . we apply the bcs mean - field approximation to the interaction hamiltonian  ( [ eq : inth ] ) as @xmath39 \ ; , \end{split}\ ] ] defining the order parameter @xmath40 we define the center - of - mass coordinate @xmath41 and the relative coordinate @xmath42 , and redefine the gap function as @xmath43 . by following the procedure of refs . @xcite and @xcite , we obtain the linearized gap equation @xmath44 near the second - order phase - transition point . we could examine both orbital and paramagnetic pair - breaking effects with eq . ( [ eq : gap1 ] ) . in the derivation of eq . ( [ eq : gap1 ] ) , it is assumed that the spatial variation of vector potential @xmath45 is sufficiently slow . therefore , the order parameter becomes a slowly varying function of the center - of - mass coordinate . we have defined @xmath46 the matsubara frequencies @xmath47 with integer @xmath48 , and the free - electron green s function @xmath49    we introduce @xmath50 with @xmath51 so that the dispersion relation becomes isotropic as @xmath52 in the @xmath53 space , where @xmath54 . we also introduce @xmath55 with @xmath56 and @xmath57 , so that @xmath58 . we also define the operator @xmath59 with @xmath60 where @xmath61 and @xmath62 . we will calculate the explicit form of @xmath59 afterward . we replace @xmath63 and @xmath64 in @xmath65 , @xmath66 , and @xmath67 with the fermi momentum @xmath68 since electrons with momenta @xmath53 and @xmath69 near the fermi surface mainly contribute to the gap equation . therefore , we write @xmath65 and @xmath66 as @xmath70 and @xmath71 , respectively , with unit vectors @xmath72 and @xmath73 . after simple algebra using eq . ( [ eq : freeg ] ) , we obtain @xmath74 t } \\ & \qquad\qquad \times    \exp\left[\frac{t}{2i}\text{sgn}(\omega_n )   \tilde{\mathbf{v}}_{\text{f}}(\hat{{\tilde{\mathbf{p } } } } ' )   \cdot\tilde{{\boldsymbol{\pi } } } \right ]   \delta_{-\sigma}({\tilde{\mathbf{r}}},\hat{{\tilde{\mathbf{p } } } } ' ) \ ; , \end{split}\ ] ] where we have defined @xmath75 and the density of states at the fermi energy @xmath76  @xcite . since eq . ( [ eq : gap2 ] ) does not depend on the spin value @xmath25 , we omit the spin suffix from now on . by redefining @xmath77 and summing up the matsubara frequencies , the gap equation ( [ eq : gap2 ] ) acquires the form @xmath78   \delta({\tilde{\mathbf{r}}},\hat{{\tilde{\mathbf{p } } } } ' ) \;. \end{split}\ ] ] it is obvious that if we rewrite the integral variable @xmath79 as @xmath80 , the linearized gap equation  ( [ eq : gap3 ] ) is similar to those obtained by many authors  @xcite . the differences between eq . ( [ eq : gap3 ] ) and the equations obtained so far for isotropic systems are that the vector potential @xmath45 is scaled as @xmath81 in @xmath82 , and that the pairing interaction @xmath70 is deformed in the @xmath79-space anisotropic superconductivity . the former scale transformation in @xmath45 has been studied in ginzburg - landau theory  @xcite for _ s_-wave pairing . for _ s_-wave pairing , since @xmath83 , eq .  ( [ eq : gap3 ] ) is exactly reduced to the equations for the system with the isotropic fermi surface , except the scale of @xmath84 , which results in the scaling @xmath85 . therefore , the only difference due to the fermi surface anisotropy is that the critical field in the @xmath86 direction is enhanced or reduced by the factor @xmath87 . the latter deformation of @xmath70 occurs because @xmath88 for anisotropic superconductivity . we take into account the deformation of the pairing interaction in the @xmath53 space . these changes affect the mass anisotropy dependence of the critical field . now , we derive a more explicit form of the upper critical field equation . we take the _ z _ axis in the direction where the effective mass is the largest . the compounds with layered structures with large layer spacing are approximately described by the models with @xmath89 . for example , in the application to the organic superconductor @xmath3 , we take the @xmath90 and @xmath91 axes along @xmath92 and @xmath93 axes of the compounds , respectively . we study a quasi - two - dimensional superconductor under magnetic fields parallel to the layers ( @xmath94 or @xmath95 ) hereafter . we derive the critical field equations only for @xmath94 , because those for @xmath95 are obtained by exchanging @xmath96 and @xmath97 . we assume the uniform magnetic field @xmath32 in the @xmath98 direction parallel to the layers , which is expressed by a vector potential @xmath99 . then , we obtain the explicit form of @xmath82 as @xmath100 from eqs . ( [ eq : pi ] ) and ( [ eq : tpi ] ) . we define the differential operators @xmath101 which satisfy the bosonic commutation relations , where @xmath102 the factor @xmath103 is due to the effective - mass anisotropy . we consider pairing interactions of the form @xmath104 where @xmath105 denotes the symmetry function of symmetry @xmath106 and @xmath107 , @xmath1 , @xmath2 and so on . the symmetry function @xmath108 is normalized so that @xmath109 ^ 2\rangle = 1 $ ] , where the average on the fermi surface @xmath110 is defined by @xmath111 , where @xmath112 denotes the angle - dependent density of states on the fermi surface . the gap function is proportional to the symmetry function @xmath108 and expanded by the abrikosov functions @xmath113 of the landau - level indexes @xmath114 , as @xmath115 the abrikosov functions @xmath113 are expressed as @xmath116 where @xmath117 denotes the solution of @xmath118 . in the gap function  ( [ eq : gf1 ] ) , we have taken into account the possibility of nonzero @xmath119 for the fflo state , where @xmath120 . the linearized gap equation  ( [ eq : gap3 ] ) can be written as @xmath121 with @xmath122 where we have defined @xmath123 ^ 2    \frac{1-e^{2\tilde{p } \omega_{\text{d}}/{\tilde{v}_{\text{f } } } } }        { \tilde{p}^2 \sinh(2 \pi t \tilde{p } / { \tilde{v}_{\text{f } } } ) } \ ; , \\ d_{nn'}^\alpha   & = -\frac{t}{2{\tilde{v}_{\text{f } } } }    \int { d^3}{\tilde{\mathbf{p}}}[\gamma_\alpha(\hat{{\mathbf{p}}})]^2    \frac{1-e^{2\tilde{p } \omega_{\text{d}}/{\tilde{v}_{\text{f } } } } }        { \tilde{p}^2 \sinh(2 \pi t \tilde{p } / { \tilde{v}_{\text{f } } } ) }   e^{i(n'-n)\tilde{\varphi } } \\ & \qquad\qquad\times   \bigg [   \sum_{j=1}^\infty    d_{nn'}^{(j)}(\tilde{p},{\tilde{\theta } } )   \left(\delta_{n'-n,2j } + \delta_{n'-n,-2j}\right )   \\ & \qquad\qquad\qquad\qquad   + d_{nn'}^{(0)}(\tilde{p},{\tilde{\theta}})\delta_{nn ' }    \bigg ] \end{split}\ ] ] and @xmath124 \\ & \times   \sum_{k=0}^{\min(n , n ' ) }   \left [ -\frac{\kappa_x}{2 } \tilde{p}^2\sin^2{\tilde{\theta}}\right]^{k+j } \\ & \qquad \times   \frac{\sqrt{n!n'!}}{k!(k+2j)![\min(n , n')-k ] ! }   \\ & \qquad \times   \cos\left[\frac{2\tilde{p}}{{\tilde{v}_{\text{f } } } }   ( h+\frac{\tilde{q}{\tilde{v}_{\text{f}}}}{2}\cos{\tilde{\theta}})\right ]   -\delta_{nn'}\delta_{j,0 } \;. \end{split}\ ] ] more explicit expressions of the matrix elements @xmath125 , @xmath126 , and @xmath127 are given below . defining the zero - field transition temperature @xmath128 , the linearized gap equation  ( [ eq : mgeq1 ] ) is written as @xmath129 the transition temperature and the critical field are given by the condition that eq . ( [ eq : mgeq1 ] ) has a nontrivial solution of @xmath130 for the first time when the magnetic field and temperature decrease , respectively , where @xmath119 is optimized .      for _ s_-wave pairing , we insert @xmath131 into eq . ( [ eq : mgeq2 ] ) . it is easily verified by integrating over @xmath132 that @xmath133 for @xmath134 . therefore , the matrix elements @xmath135 in the linearized gap equation  ( [ eq : mgeqg ] ) are expressed as @xmath136      we consider @xmath1-wave pairing interaction given by eq . ( [ eq : gpi ] ) with @xmath137 , where @xmath138 . the function @xmath139 is rewritten as @xmath140 \ ; , \end{split}\ ] ] where we have defined the polar coordinates ( @xmath141 ) taking the @xmath142 axis as the polar axis , so that @xmath143 is measured from the @xmath142 axis , and @xmath132 is measured from the @xmath144 axis . therefore , in the linearized gap equation  ( [ eq : mgeqg ] ) , the matrix elements @xmath126 are written as @xmath145 where @xmath146 is defined by @xmath147 with @xmath148   \left[\frac{c({\tilde{\theta}})-1}{c({\tilde{\theta}})+1 } \right]^j \\ & \times   \bigg{\ { } \delta_{j,0}-2c({\tilde{\theta } } )   + 4j[c({\tilde{\theta}})]^2 \\ & \qquad   + 2[c({\tilde{\theta}})]^3\bigg{\ } } \ ; , \end{split}\ ] ] and @xmath149 . the matrix equation  ( [ eq : mgeqg ] ) is decoupled two sets of equations for @xmath150 of even and odd @xmath151 s . next , we consider @xmath2-wave pairing interaction with @xmath152 the matrix elements @xmath127 take the form @xmath153 where @xmath154 is defined by @xmath155 with @xmath156 \\ & \times   \bigg{\ { } 2c({\tilde{\theta}})\left\{1-[c({\tilde{\theta}})]^2\right\ } \delta_{j,0 } \\ & \qquad\quad   -\frac{4j}{[c({\tilde{\theta}})]^{2j-1 } }   \left[\frac{c({\tilde{\theta}})-1}{c({\tilde{\theta}})+1 } \right]^j   \bigg{\ } } \;. \end{split}\ ] ] the order parameter components @xmath157 and @xmath158 couple only with @xmath159 and @xmath160 , respectively , where @xmath161 . therefore the matrix equation  ( [ eq : mgeqg ] ) for @xmath2-wave pairing is decoupled into two sets of equations for @xmath162 of even and odd @xmath151 s . in this section , we review an extended jaccarino - peter mechanism in antiferromagnetic superconductors  @xcite . we have formulated the upper critical field equation of the conduction electron system in the previous section . now , we shall introduce a localized spin system that is coupled with the conduction electron system through the kondo interaction @xmath163 . we consider a situation in which the antiferromagnetic long - range order has canted spin structures in magnetic fields . then , ferromagnetic moments are induced in the localized spins and create exchange fields on the conduction electrons . the exchange fields partly or completely cancel the magnetic fields in the zeeman energy terms of the conduction electrons when @xmath164 . the exchange fields are not real magnetic fields in the sense that they only modify the zeeman energy terms but not the vector potential . on the other hand , the ferromagnetic moments also create real internal magnetic fields . however , such internal magnetic fields are weak when the localized spins are spatially separated from the conduction electrons . therefore , we neglect them in this study . we examine the case in which the antiferromagnetic transition occurs at a temperature much higher than the superconducting transition temperature and thus the magnetic ordered state is rigid . therefore , we neglect modification of the magnetic structure by occurrence of superconductivity . strictly speaking , the localized spin state is modified by the mobile electrons , and the state of the total system should be determined self - consistently  @xcite . however , we consider a model in which such modification has already been included and the canted spin structure occurs as a result . we modify the extended jaccarino - peter mechanism  @xcite for the system with anisotropic kondo interactions , introducing kondo coupling constants @xmath165 between the localized spins and the conduction electrons of the @xmath166 component in spin space . when the magnetic field is oriented to the @xmath167 direction , where @xmath168 , we replace @xmath169 in eq . ( [ eq : allh ] ) with the effective hamiltonian @xmath170 where @xmath171 denotes the effective zeeman energy . therefore , in the critical field equations derived in the previous section , the zeeman energy @xmath172 is replaced with @xmath171 . for @xmath173 , @xmath171 is given by @xmath174 with @xmath175 . ( [ eq : thlf ] ) , @xmath33 , @xmath176 , @xmath177 , and @xmath178 denote number of antiferromagnetic bonds for a given site , number of lattice sites that participate in the kondo interaction with the conduction electrons at @xmath179 , coupling constant of the exchange interaction of the localized spins , and the _ g _ factor of the localized spins , respectively . here , @xmath180 denotes the critical field of the antiferromagnetic phase which is given by @xmath181 , where @xmath182 denotes the magnitude of the localized spins . therefore , when @xmath183 , the pauli paramagnetic pair - breaking effect is reduced in antiferromagnetic superconductors .    for @xmath184 , since the spins are aligned uniformly , the effective zeeman energy @xmath171 is given by @xmath185 because the second term is a negative constant when @xmath186 , the present mechanism is reduced to the conventional jaccarino - peter mechanism . the zeeman energy is completely compensated at @xmath187 . it is noted that fisc occurs only when @xmath188  @xcite . theoretical phase diagram for _ s_-wave pairing in the magnetic fields parallel to the _ c _ axis . here , sc stands for superconductivity . the solid and dotted curves show boundaries of superconductivity in the presence and the absence of the fflo state , respectively . the closed circles show the experimental data of the superconducting transition points for @xmath6 in @xmath3   by konoike _ et al_.  ( ref . the dot - dashed line shows the critical field of the antiferromagnetic phase.,width=245 ]     theoretical phase diagram for _ s_-wave pairing in the magnetic fields parallel to the _ a _ axis . the notations are the same as those in fig . [ fig : sc ] . since the canted spin structure does not occur for @xmath9 at low field , the upper critical field of low - field superconductivity has not been calculated.,width=245 ]    in this section , we apply the present theory to the antiferromagnetic organic superconductor @xmath3   taking the _ x _ and _ y _ axes along the crystallographic _ c _ and _ a _ axes , respectively . therefore , we write the effective masses as @xmath189 , @xmath190 , and @xmath191 , and the anisotropic kondo interactions as @xmath192 and @xmath193 . in @xmath3 , the canted spin structure was observed when @xmath6 , and we can apply the extended jaccarino - peter theory . however , for the magnetic field parallel to the magnetic easy axis , i.e. , @xmath9 , the canted spin structure has not been observed and the details of the metamagnetic transition have not been revealed . therefore , we do not apply the present theory to the low - field region for @xmath9 . it is convenient to introduce a constant @xmath194 with which @xmath195 and @xmath196 are expressed as @xmath197 ^ 2   \ ; , \\ \kappa_y & = a_m h    \left ( \frac{m_a}{m_c } \right)^{1/4 }    \left [ \frac{2\pi t_{\text{c}}^{(0)}}{{\tilde{v}_{\text{f } } } } \right]^2 \;. \end{split}\ ] ] we also define the strength ratio of the paramagnetic and orbital pair - breaking effects @xmath198    we explain the procedure to analyze the experimental phase diagram with the present theory . our critical field equations contain five microscopic parameters : @xmath199 , @xmath200 , @xmath201 , @xmath202 , and @xmath203 . first , we fit the experimental data of the critical field of the fisc for @xmath6 with the parameters @xmath199 , @xmath200 , @xmath201 , and @xmath202 . secondly , using the parameter values of @xmath199 , @xmath200 and @xmath201 determined in the first step , we fit the data of the fisc for @xmath9 with a single parameter @xmath203 . it is noted that all five parameters are determined only from the curves of the fisc for @xmath6 and @xmath9 . crudely speaking , the curve of the fisc can be characterized by three real numbers , that is , the magnetic field where the transition temperature is maximum , the maximum transition temperature , and the width of the magnetic fields at @xmath204 . therefore , we obtain six real numbers from the two experimental data of the fisc , which sufficiently determine all five microscopic parameters . lastly , using those five parameters determined in the first and second steps , we calculate the upper critical field of low - field superconductivity for @xmath6 , where the canted spin structure coexists . in this last step , we do not use any additional fitting parameter . theoretical phase diagram for @xmath1-wave in the magnetic fields parallel to the _ c _ axis . the notation is the same as those in fig . [ fig : sc ] . the inset shows the theoretical phase diagram below 0.8  k and between 2 and 2.6  t.,width=245 ]     theoretical phase diagram for @xmath1-wave in the magnetic fields parallel to the _ a _ axis . the notation is the same as those in fig . [ fig : sc].,width=245 ]    we carry out this procedure for _ s_-wave , @xmath1-wave , and @xmath2-wave pairings . in figs . [ fig : sc][fig : ffq ] , we set @xmath205  k and @xmath206  t , which have been observed in the compounds @xmath3 . for all pairing symmetries examined , the theoretical curves agree well with the experimental data , as shown in figs .  [ fig : sc][fig : d2a ] . in particular , the agreements at low field for @xmath6 are quantitative , because all parameters are determined only from the data of the fisc at high fields . in what follows , we explain the differences depending on the pairing symmetries . the results for _ s_-wave pairing are shown in figs . [ fig : sc ] and [ fig : sa ] . we obtain the effective - mass ratio @xmath207 , which is reasonable in comparison with the value estimated from the shubnikov  de haas  ( sdh ) oscillations  @xcite . it is confirmed that the fflo state is not realized in the fisc . in contrast , it is shown that low - field superconductivity exhibits the fflo state at low temperatures . the results for @xmath1-wave pairing are shown in figs . [ fig : d1c ] and [ fig : d1a ] . we obtain @xmath208 , which is too small in comparison with the value estimated from the sdh oscillations  @xcite . it is confirmed that the fflo state is not realized in the fisc . there are bends in the theoretical curves of the low fields , where the landau level indexes @xmath151 s of the optimum solution alternate . the solution near @xmath204 is expressed by a linear combination of the abrikosov functions with odd @xmath151 s . in fig . [ fig : d1c ] , it is found that there are two tricritical points in the phase diagram for @xmath6 . one at @xmath209  k is the critical point of the normal state , the bcs state with even @xmath151 s , and the fflo state with even @xmath151 s , and another at @xmath210  k is that of the normal state , the fflo states with even @xmath151 s and odd @xmath151 s . the results for @xmath2-wave pairing are shown in figs . [ fig : d2c ] and [ fig : d2a ] . we obtain @xmath211 , which is reasonable in comparison with the value estimated from the sdh oscillations  @xcite . in this case , the area of the fflo state is very limited . it is found that the fflo state is not realized in the fisc and low - field superconductivity near @xmath204 . there is a bend in the theoretical curve , where the parity of the landau - level indexes @xmath151 s alternates . the solution of the odd @xmath151 s gives the highest critical field near @xmath204 . theoretical phase diagram for @xmath2-wave in the magnetic fields parallel to the _ c _ axis . the notation is the same as those in fig . [ fig : sc ] . the inset shows the theoretical phase diagram below 0.8  k and between 2 and 2.6  t.,width=245 ]     theoretical phase diagram for @xmath2-wave in the magnetic fields parallel to the _ a _ axis . the notation is the same as those in fig . [ fig : sc].,width=245 ]    in fig . [ fig : ffq ] , we show temperature dependences of @xmath212 along the upper critical field curves of low - field superconductivity for _ s_-wave , @xmath1-wave , and @xmath2-wave pairings when @xmath6 . for @xmath1-wave pairing , it is found that the @xmath119 jumps at a temperature where the landau - level indexes @xmath151 of the optimum solution alternate . for @xmath2-wave pairing , it is found that the @xmath119 vanishes for @xmath213 , where the fflo state with odd @xmath151 is unstable .    in a brief summary of this section , experimental results could be reproduced for all _ s_-wave , @xmath1-wave , and @xmath2-wave parings . for @xmath214-wave pairing , the upper critical fields of low - field superconductivity are smaller than the experimental data , but they are improved by taking into account the fflo state . the in - plane mass anisotropies are estimated as @xmath215 , 0.468 , and 0.865 for _ s_-wave , @xmath1-wave , and @xmath2-wave pairings , respectively . the results for _ s_-wave and @xmath2-wave pairings agree with the value obtained from the sdh oscillations  @xcite , whereas the result for @xmath1-wave pairing largely deviates from it . we found that the fflo state does not occur in the fisc , while it occurs in low - field superconductivity . temperature dependence of the magnitudes of the total pair momentum along the upper critical field curves for ( a ) _ s_-wave , ( b ) @xmath1-wave , and ( c ) @xmath2-wave pairings in the magnetic fields parallel to the _ c _ axis . the sets of parameters for ( a ) , ( b ) , and ( c ) are the same as in figs . [ fig : sc ] , [ fig : d1c ] , and [ fig : d2c ] , respectively.,width=245 ] we have studied the upper critical fields of antiferromagnetic _ s_-wave , @xmath1-wave , and @xmath2-wave superconductors with fermi surface anisotropy . we have derived the linearized gap equations when the magnetic field is applied parallel to layers , taking into account the effective - mass anisotropy , the fflo state , and the extended jaccarino - peter mechanism . we have applied the theory to the organic superconductor @xmath3 , and obtained good agreement between the theoretical and experimental results . while the agreements have been obtained for all cases of _ s_-wave , @xmath1-wave , and @xmath2-wave pairings , the resultant in - plane mass ratios @xmath216 are different from one another . the mass ratio for @xmath1-wave pairing seems too small , although those for _ s_-wave and @xmath2-wave pairings are close to the value estimated from the sdh oscillations  @xcite . the mass anisotropy dependence of the pure orbital limits of the upper critical fields is shown in fig . [ fig : mae ] . the upper critical fields for _ s_-wave , @xmath1-wave , and @xmath2-wave pairings become larger when the ratio @xmath217 decreases . it is found that the upper critical field is the most influenced for @xmath2-wave pairing by a change in the ratio @xmath217 , while it is less influenced for @xmath1-wave pairing . this is why the resultant ratio @xmath217 to fit the experimental data becomes much smaller for @xmath1-wave pairing than for @xmath2-wave pairing .     the mass anisotropy dependence of the pure orbital limits for @xmath6 . the open circles , open diamonds , and open triangles show the results for _ s_-wave , @xmath1-wave , and @xmath2-wave pairings , respectively . the solid , dotted , and dashed curves are guides to the eye . here , @xmath218 denotes the orbital limit @xmath219 of the isotropic case ( @xmath220).,width=245 ]    this result is physically interpreted as follows . for example , we consider the situation that @xmath221 . the origin of the orbital pair - breaking effect is the lorentz force that acts on electrons . when the magnetic fields are applied along the _ c _ axis , the lorentz force is the strongest for electrons near the @xmath222 axis , where the fermi velocity is the largest , in the two - dimensional momentum space . when the mass ratio @xmath217 decreases with the fixed electron number , i.e. , when the effective mass @xmath223 increases , the fermi velocity in the _ a _ direction decreases . therefore , the lorentz force acting on the electrons on the fermi surface near the @xmath222 axis becomes stronger and superconductivity is more suppressed when @xmath217 decreases . the order parameter of @xmath2-wave pairing has nodes on the @xmath222 axis , while that of @xmath1-wave pairing has the largest amplitude there . therefore , @xmath1-wave pairing is less affected than @xmath2-wave pairing by a change in the ratio @xmath217 . it has been found from fig . [ fig : d2c ] that the fflo state does not occur for @xmath2-wave pairing in low - field superconductivity near @xmath204 . this result is physically explained as follows . since the effective density of states for @xmath106-wave pairing is proportional to @xmath224 ^ 2 $ ] in the gap equation , areas of the fermi surface near the peak of @xmath224 ^ 2 $ ] have a greater effect on the nesting condition for the fflo state , but those near the nodes of @xmath224 ^ 2 $ ] have less effect  @xcite . hence , in the absence of the orbital pair - breaking effect , the vector @xmath11 tends to points in the areas of the fermi surfaces where @xmath224 ^ 2 $ ] exhibits a peak . in this study , however , we adopt the model in which @xmath12 is assumed due to the orbital pair - breaking effect . therefore , for @xmath2-wave pairing , the fermi surface nesting is less effective when @xmath225 or @xmath226 , because the order parameter has nodes on the @xmath227 and @xmath228 axes . in contrast , for @xmath1-wave pairing , because the order parameter has peaks on the @xmath227 and @xmath228 axes , the fflo state occurs in low - field superconductivity , when @xmath225 or @xmath226 . theoretical predictions of the phase boundary of superconductivity for various values of @xmath229 . the other parameters are common to fig . [ fig : sc ] . the thin dotted and thin two - dot - dashed curves show the pauli paramagnetic and orbital limits , respectively . in this figure , the fflo state is taken into account , but modifies the curves slightly at low temperatures . at the upper critical fields of the fisc for @xmath230 , the fflo state does not occur.,width=245 ]    in the application to @xmath3 , it is found that the fflo state is not present in the fisc in contrast to @xmath4 , while it is present in low - field superconductivity . in @xmath4 , the phase boundary of the fisc can be well reproduced if the possibility of the fflo state is taken into account  @xcite . for the fflo state to occur , the orbital pair - breaking effect needs to be sufficiently suppressed . in @xmath4 , this condition seems to be satisfied even at high field due to low dimensionality . in contrast , it was suggested that @xmath3   has a larger interlayer electron hopping energy than @xmath4  @xcite . therefore , in @xmath3 , the orbital effect is substantial at high fields , as verified by comparison of the zero - field transition temperature and the maximum transition temperature of the fisc . the upper critical fields of low - field superconductivity agree with the experimental data , as shown in figs . [ fig : sc ] , [ fig : d1c ] , and [ fig : d2c ] . further , for @xmath214-wave pairing , the agreement is improved if the fflo state is taken into account . however , they are slightly smaller than the experimental data . this discrepancy may be removed by taking into account a mixing effect in the presence of the weak triplet pairing interaction , as in the compounds @xmath4  @xcite . a slight discrepancy also occurs in the lower critical fields of the fisc , where the fflo state does not occur as explained above . if the mixing effect occurs , this discrepancy may also be removed .    for @xmath1-wave and @xmath2-wave pairings in low - field superconductivity , we found an internal transition between the vortex states with different landau level indexes @xmath151 , which is analogous to internal transitions in a two - dimensional system in tilted magnetic fields  @xcite . both the fflo state and the vortex state with higher @xmath151 s originate from the pauli paramagnetic effect as discussed in ref . @xcite , because in the absence of zeeman energy only the vortex state with lower @xmath151 s occurs . due to the transitions to the fflo state and the vortex states with higher @xmath151 s , the upper critical fields of low field superconductivity exhibit downward convex curves for all pairing symmetries examined , which agrees with the experimental data .    we have determined the microscopic parameters only from the curves of the fisc , and calculated the upper critical field of low - field superconductivity with those parameter values without any additional fitting parameters . from the accordance between the theoretical and experimental results on low - field superconductivity , the extended jaccarino - peter mechanism  @xcite seems to be realized in @xmath3 . applying the extended jaccarino - peter mechanism to the same models with different parameter values , we obtain some unusual phase diagrams as shown in fig . [ fig : sj ] . when @xmath231 , the transition temperature curve has double peaks . when @xmath232 , superconductivity occupies large area in the phase diagram . similar phase diagrams have been obtained in two- and three - dimensional systems  @xcite . in this study , we confirmed that the phase diagrams as obtained above can be realized in a more realistic model of organic superconductors . if we control the value of @xmath163 by material design , such critical field curves would be observed .    in the present theory , the effective mass @xmath233 is included in the parameter @xmath199 . since we have obtained the value of @xmath199 , we can estimate a value of @xmath233 , if we have the value of @xmath234 appropriate for @xmath3 . the upper critical field in low fields for @xmath9 could not be reproduced by the present theory . it has been found that the magnetic easy axis of antiferromagnetic long - range order in low fields is along the _ a _ axis  @xcite , and that the metamagnetic transition occurs at the antiferromagnetic phase boundary close to the upper critical field  @xcite . in order to reproduce the upper critical field in low fields for @xmath9 , we need to reproduce the localized spin structure . this remains for future study .    in conclusion , the experimental phase diagrams of the antiferromagnetic superconductor @xmath3  are theoretically reproduced by the present theory . in particular , the low - field phase for @xmath6 is well reproduced by the model parameters that are determined from the critical fields of the fisc . therefore , the extended jaccarino - peter mechanism seems to be realized in the present compound at low fields . due to this mechanism , some unusual phase diagrams may occur in compounds with appropriate energy parameters , in the future . we would like to thank s.  uji and t.  konoike for discussions and for providing us with the experimental data . we wish to acknowledge useful discussions with k.  doi . this work was partially supported by the ministry of education , science , sports and culture of japan , grant - in - aid for scientific research  ( c ) , no .  16540320 , 2005 . t.  konoike , s.  uji , t.  terashima , m.  nishimura , s.  yasuzuka , k.  enomoto , h.  fujiwara , b.  zhang , e.  fujiwara , and h.  kobayashi , phys . b * 72 * , 094517 ( 2005 ) ; t.  konoike , h.  fujiwara , b.  zhang , h.  kobayashi , m.  nishimura , s.  yasuzuka , k.  enomoto , and s.  uji , physica c * 412 - 414 * , 107 ( 2004 ) . l.  balicas , j.s . brooks , k.  storr , d.  graf , s.  uji , h.  shinagawa , e.  ojima , h.  fujiwara , h.  kobayashi , a.  kobayashi , and m.  tokumoto , solid state commun . * 116 * , 557 ( 2000 ) ; s.  uji , h.  shinagawa , y.  terai , t.  yakabe , t.  terashima , l.  balicas , j.s . brooks , e.  ojima , h.  fujiwara , h.  kobayashi , a.  kobayashi , and m.  tokumoto physica b * 298 * , 557 ( 2001 ) .
we derive critical field @xmath0 equations for antiferromagnetic _ s_-wave , @xmath1-wave , and @xmath2-wave superconductors with effective mass anisotropy in three dimensions , where we take into account ( i )  the jaccarino - peter mechanism of magnetic - field - induced superconductivity ( fisc ) at high fields , ( ii )  an extended jaccarino - peter mechanism that reduces the pauli paramagnetic pair - breaking effect at low fields where superconductivity and an antiferromagnetic long - range order with a canted spin structure coexist , and ( iii )  the fulde - ferrell - larkin - ovchinnikov ( fflo or loff ) state . as an example , experimental phase diagrams observed in organic superconductor @xmath3 are theoretically reproduced . in particular , the upper critical field of low - field superconductivity 
 is well reproduced without any additional fitting parameter other than those determined from the critical field curves of the fisc at high fields . 
 therefore , the extended jaccarino - peter mechanism seems to occur actually in the present compound . 
 it is predicted that the fflo state does not occur in the fisc at high fields in contrast to the compound @xmath4 , but it may occur in low - field superconductivity for _ s_-wave and @xmath1-wave pairings . 
 we also briefly discuss a possibility of compounds that exhibit unconventional behaviors of upper critical fields .