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magnetic tunnel junctions ( mtj s ) consisting of two ferromagnetic layers separated by the insulating spacer exhibit the tunneling magnetoresistance effect ( tmr ) when they are switched by applying a magnetic field from the antiparallel to parallel alignment of magnetizations in magnetic layers . the tmr effect was first observed by m.  julliere in 1975 @xcite . julliere found that the tunneling conductance of the trilayer structure fe / ge / co depended on the angle between magnetizations in the fe and co layers . the measured amplitude of the tmr in these experiments was 14% at 4.2k . only 20 years later , the large values of tmr at room temperature were obtained in magnetic junctions comprising the al@xmath2o@xmath3 barrier @xcite . since that time , there has been a renewed technological and fundamental interest to the tunneling phenomenon and during the last decade a lot of experimental and theoretical papers were published on this topic ( see reviews [ 5,6,7,8 ] ) . experimentally it was observed @xcite that the tmr depends critically on the material of the insulating barrier and on the conditions of its preparation , in particular on the imperfections of the interfaces between the metal and the insulating layer @xcite . on the other hand , the first theory suggested by m.  julliere expresses the tmr ratio in terms of the effective spin polarizations @xmath4 and @xmath5 of two magnetic layers via the expression @xmath6 , and thus predicts no dependence of the tmr on the parameters of the barrier . in spite of julliere s formula is widely used for the interpretation of the experimental data @xcite , it does not have rigorous theoretical foundation @xcite . the dependence of the tmr magnitude on the parameters of the tunnel barrier and the metal / insulator interface was taken into account in the subsequent theories . slonczewski @xcite considered a quantum mechanical problem of tunneling of a free electron through a simple spatially uniform barrier and showed that the tmr ratio depends on the height of the potential barrier and on the effective mass of the tunneling electron inside the insulator . later on , the influence of spin - flip scattering at the interfaces on temperature and bias - voltage dependences of the tmr was investigated by zhang _ et al . _ it was shown that mixing of spin - up and spin - down tunnel channels leads to a decrease of the tmr .    due to a permanent progress in the development of different methods of ab - inito calculations of the electronic properties of solids , in the last four years the transport properties of tunnel junctions were investigated using the realistic band structure of the ferromagnetic layers and the insulator ( refs . [ 1420 ] ) . the systems where the conductance was calculated were ideal epitaxial fe / znse / fe(001 ) @xcite and fe / mgo / fe(001 ) tunnel junctions @xcite . for these structures the electron transport is assumed to be ballistic , i.e.  the electron momentum @xmath7 parallel to the ferromagnet / insulator interface is conserved . experimentally it was also demonstrated recently by heinrich _ et al . _ @xcite that fe / mgo / fe(001 ) junctions can be indeed grown by depositing the mgo epitaxially onto a fe whisker and then depositing another fe electrode epitaxially on a top of the mgo . the results of ab - initio calculations showed that tunneling has more complicated behavior than the predictions of the simple barrier model proposed by slonczewski @xcite . the main conclusions are as follows @xcite . ( i )  tunneling conductance depends strongly on the symmetry of the electron states in the ferromagnetic electrodes and in the insulating layer . ( ii )  the decay rates of evanescent states in the barrier are different for the states with different symmetry . the slowest decay rates have the evanescent states which are compatible with @xmath0 symmetry . the bloch states in the metal couple more efficiently through the interface with the decaying states of the same symmetry in the barrier . therefore , mostly free - like @xmath8-electrons from the bands with @xmath0 character in the ferromagnet give the essential contribution to the tunneling conductance . ( iii ) for thin insulating layers the tunneling current in the minority channel is dominated by the interfacial resonance states that gives rise to `` hot spots '' for the @xmath7-resolved conductance in the surface brillouin zone . the contribution from the resonance states is substantially suppressed for the thicker barriers since the bloch states at `` hot spots '' have no @xmath0 character @xcite . moreover , as it follows from recent discussions @xcite , the surface resonance states are very sensitive to the details of the interface . the asymmetry of the potential barrier and the interfacial roughness considerably reduce the contribution from the surface states to a total conductance .    nowadays the ab - initio calculations of the tmr are possible only for ballistic junctions like above mentioned fe / znse / fe and fe / mgo / fe structures which are characterized by a small lattice mismatch between the metal and the insulator , and by the well - defined band structure of the oxide or the semiconductor . in realistic junctions the electron transport has a diffuse character , i.e.  when the electron crosses the ferromagnet / insulator interface its @xmath7 momentum is not conserved . for f / o / f tunnel junctions the most successful material until now has been alumina , al@xmath2o@xmath3@xcite . the al@xmath2o@xmath3 tunnel barriers are usually fabricated by the natural or plasma oxidation of the al layer @xcite . the subsequent structural analysis , e.g.  with the use of the x - ray photoelectron spectroscopy , shows that alumina is amorphous and the obtained alo@xmath9 tunnel barrier deviates from the ideal al@xmath2o@xmath3 structure @xcite . for a uniform coverage the al film thickness is usually ranged from about 7 to 18   , depending on a type of the ferromagnetic electrode @xcite . there is a small range of al thicknesses that yields to the best tmr ratio for a given oxidation condition . when the al layer is thin , the ferromagnet surface becomes oxidized leading to the formation of coo and co@xmath3o@xmath10 oxides @xcite or to the fe@xmath3o@xmath10 oxide @xcite . on the other hand , with too thick al film , an excess of al metal is left unoxidized . the amorphous barriers , the roughness of the interface and its structural inhomogeneity make the rigorous ab - initio calculations of the tmr virtually impossible and therefore more simplified models are required to treat the diffuse electron transport in mtj s .    in our previous paper @xcite , we attempted to investigate the influence of scattering processes of the electrons ( with and without spin - flip ) at the interfaces on the tmr using a simple two - band ( spin - up and spin - down ) free - electron model . it was shown that for this simplified model spin - conserving scattering may or may not lead to a decrease of the tmr depending on the amplitude of the scattering potential . in this paper , we proceed to study the diffuse electron transport in magnetic tunnel junctions of the from f / o / f with f s being 3@xmath1 transition metal electrodes and o being the insulating barrier ( al@xmath2o@xmath3 ) . we use the results of ab - initio calculations to model the band structure of ferromagnetic electrodes , namely , the most important feature of these calculations that at least two groups of electrons form the total band structure : the almost free - like spin - up and spin - down bands and the narrow strongly exchange split bands . we will call the first group @xmath0 and the second group @xmath1 electrons and will consider that a periodic part of the @xmath0-@xmath1 hybridization between bands is taken into account that results in the nonequivalence of spin - up and spin - down @xmath0-bands . the parameters of the adopted model can be adjusted to reproduce a value of the tmr observed in the experiments . we assume that for the case of al@xmath2o@xmath3 barrier the exchange split @xmath0-like quasi - free electrons give the main contribution to the tmr effect . in the framework of this simplified model we will be able to investigate in a proper way the influence of electron scattering at the interfaces on the tunneling conductance .    to describe the nonideal tunnel junction we assume that defects and impurity centers ( the al or o ions , or other artificially embedded ions ) are randomly distributed within few monolayers near the f / o interface . within the interface , an electron undergoes scattering when it comes to the defect or impurity center . we take into account these processes assuming that the hybridization between @xmath0 and @xmath1 bands changes randomly on the interface because of this parameter is the most important in the adopted model among the other ones , characterizing the scattering potential . it yields to the possibility of @xmath0-like electrons to scatter into the @xmath1 sub - band ( and vice versa ) and thus strongly affects tunneling . to treat the electron scattering on the random potential we use the coherent - potential approximation ( cpa ) @xcite that allows to consider the case of strong scattering . we apply the generalized kubo formalism and the green s functions method to calculate the conductance of the system , and find the vertex corrections to the conductivity with the use of `` ladder '' approximation @xcite combined with the cpa . it is shown that the so - called ward identity is satisfied in the framework of this approximation that provides the necessary condition for a conservation of a tunneling current . note , that if it is not the case , the conclusions may be completely misleading . as a result , we show that , due to substantial difference in majority and minority @xmath1 density of states at the fermi energy for 3@xmath1 ferromagnetic metals , the inter - band @xmath0-@xmath1 scattering on the interface can strongly reduce the tmr even down to zero value . in accordance with results of zhang _ et al . @xcite , it is also shown that the spin - flip scattering of electrons on the surface magnons within the interface leads to a further decrease of the tmr ratio at finite temperature . the paper is organized as follows . in sec . ii we describe the model hamiltonian , the calculation of the tunneling conductance and vertex corrections . the discussion of the obtained results is presented in sec . conclusions are in sec . the proof of ward identity is given in appendix a. the details of the derivation of the cpa equations are described in appendix b. we will consider a trilayer tunnel junction of the form f@xmath11/o / f@xmath3 , where f@xmath11 and f@xmath3 are two semi - infinite ferromagnetic layers and o is a dielectric oxide spacer ( al@xmath2o@xmath3 ) . our arguments in behalf of the two - band @xmath0-@xmath1 model which was briefly described in the introduction are as follows . one of the conclusions of ab - initio calculations of the tmr for ballistic tunnel junctions @xcite is that the expected spin - dependence of the tunneling current can be deduced from the symmetry of the bloch states in the ferromagnet at the fermi energy . spin - polarized band structure for bcc fe , fcc ni , and fcc co can be found in ref .  [ 26 ] . the type and symmetry of the bloch states for different crystal faces with @xmath12 for fe , co and ni are presented in table  i ( in accordance with ref .  [ 18 ] ) . for example , in case of fe electrodes , the examination of the band structure shows that both the majority and minority bands with @xmath0 character in [ 110 ] and [ 111 ] directions ( @xmath13 and @xmath14 ) cross the fermi energy . for [ 100 ] direction the band with @xmath0 symmetry ( @xmath15 ) crosses the fermi energy in case of the majority channel only . the similar analysis can also be performed in case of ni and co. thus one can assume that in the polycrystalline fe , co or ni - based films the states with @xmath0 character present for both spin directions and the bloch states will couple efficiently through the f / o interface with @xmath0 states in the insulator and will decay with the equal rates in the barrier region . we will call the electrons from these bands as @xmath0-like electrons and will describe them as free electrons with the effective mass @xmath16 ( where @xmath17 is a bare electron mass ) and with different fermi momenta @xmath18 and @xmath19 ( @xmath20 ) for up and down spins . the idea about a dominant contribution of the mostly itinerant electrons to tunneling was originally proposed by m.  stearns @xcite and explained the positive polarization of the spin - dependent current in the experiments on tunneling with the superconductors @xcite . according to estimations of stearns @xcite , in the case of fe , @xmath21  @xmath22 , @xmath23  @xmath22 .    . type and symmetry of the bloch states with * k*@xmath24 for fe , co and ni for three different crystal faces ( in accordance with ref . the symmetry of these bands is as follows : @xmath15 , @xmath14 , and @xmath13 ( @xmath0,@xmath25,@xmath1 ) ; @xmath26 , and @xmath27 ( @xmath25 and @xmath1 ) ; and @xmath28 , @xmath29 , @xmath30 , and @xmath31 ( @xmath1 ) . [ cols="^,^,^,^",options="header " , ]     other more localized bands ( compatible with @xmath1 symmetry ) crossing the fermi energy also will be described by two exchange split bands with the isotropic quadratic dispersion law but with larger effective mass @xmath32 . the fermi momenta @xmath33 and @xmath34 of @xmath1-like electrons can be chosen to reproduce the typical for 3@xmath1 transition metals ratio of the values of spin - up and spin - down @xmath1 density of states at the fermi energy @xmath35 , @xmath36 @xcite . in accordance with the band structure of fe , co and ni @xcite the narrow majority and minority @xmath1-bands are practically filled . therefore , the particles with a large effective mass , @xmath37 , must be regarded as holes . the values of the fermi momenta define the positions of band bottoms ( fig . 1 ) @xmath38 , @xmath39 ( @xmath40 ; @xmath41 ) with respect to the fermi energy @xmath35 . we note , that the aim of this work is to calculate the relative change of the tmr due to scattering . therefore the proposed model is rather adequate for this purpose since the scattering rate depends mostly on the density of states and on the matrix elements of the scattering potential . the al@xmath2o@xmath3 tunnel barriers obtained by oxidation of the al film are amorphous @xcite . concerning the @xmath42-al@xmath2o@xmath3 crystals , it is known from the band structure calculations @xcite that the gap ( which is not direct ) between the upper valence band and the conduction band is of the width of @xmath43  ev . the dispersion law of the lowest conduction band is not isotropic , and the effective electron masses along the different directions in the brillouin zone vary from @xmath44 to @xmath45 with an average value of about @xmath46 @xcite . this lowest conduction band is formed by a mixture of al 3@xmath0 , o 2@xmath0 and o 3@xmath0 orbitals . in the top of the upper valence band the dispersion curves are very flat , i.e.  the effective masses of holes are large as compared with the mass of the conduction electrons . at the @xmath47 point , the top of the valence band consists of hybridized o 2@xmath25 and al 3@xmath25 orbitals . in a view of that , one can describe the conduction and valence bands of the amorphous al@xmath2o@xmath3 by isotropic quadratic laws with effective masses @xmath48 and @xmath49 , @xmath50 .    next , the @xmath0-like electrons from the ferromagnet can easily penetrate into the oxide since the lead bands with @xmath0 character can couple efficiently with @xmath0 states in the barrier , and the tunneling conductance , caused by the specular transmission of the bloch waves through the interface , will decay for both spin - up and spin - down @xmath0-channels with the same law as   @xmath51 $ ] , where @xmath52 is the width of a barrier , @xmath53 is a bottom of the conduction band and @xmath54 is the electron momentum parallel to the f / o interface . following maclaren _ et al._@xcite we can suppose that @xmath1-like electrons from those bands in the ferromagnet without @xmath0 character can not couple efficiently with @xmath0 states in the oxide . nevertheless , these @xmath1-bands have an admixture of @xmath25 symmetry and therefore can couple with the valence bands of the al@xmath2o@xmath3 . however , in the @xmath1 channel the tunneling conductance due to specular transmission will decay very rapidly with the thickness @xmath52 of the barrier , as@xmath55 $ ] , because of a large mass @xmath49 of holes in the alumina ( here @xmath56 is the top of the upper valence band ) . the model potentials describing the propagation of the electron through a tunnel junction are presented in fig .  1 . we suppose that the ferromagnet / insulator interfaces are rough and contaminated by impurity ions . the contamination of a few metal monolayers close to the f / o interface always presents after the oxidation of the al film . other impurities ( e.g.  cr , ru @xcite or cu ions @xcite ) can be artificially inserted into the interface . we will characterize these structural defects by the random potentials , which may be divided into the spin - conserving and the spin - flip parts . let @xmath57 and @xmath58 be the positions of interfaces , the @xmath59-plane be the plane parallel to the interface , and the @xmath60-axis be the axis perpendicular to the barrier ( see fig .  1 ) . the hamiltonian of the system is written as : @xmath61 with @xmath62 \psi^s_{\mu}({\bf r})\phantom { , } \nonumber \\   { } & \displaystyle   + \sum_{\mu}\int d{\bf r}\ , \psi^{d+}_{\mu}({\bf r } )   \left [ -\frac{\hbar^2}{2m_d(z)}\nabla^2 + u^d_{\mu}(z )   \right ] \psi^d_{\mu}({\bf r } ) , \nonumber\end{aligned}\ ] ] @xmath63 , \nonumber\end{aligned}\ ] ] @xmath64 . \nonumber\end{aligned}\ ] ] here @xmath65 is a spin suffix ; @xmath66 , @xmath67 and @xmath68 , @xmath69 are field operators of the creation and annihilation of @xmath0 and @xmath1-type electrons with the spin @xmath70 at the point  @xmath71 ; @xmath72 and @xmath73 are the effective masses of electrons and holes in the ferromagnetic layers ( @xmath74 and @xmath37 ) or in the barrier ( @xmath48 and @xmath49 ) . @xmath75 , @xmath76 are the spin - dependent step - like potentials shown in fig .  1 : @xmath77 @xmath78 where @xmath38 , @xmath39 ( @xmath41 ) are the majority ( @xmath79 ) and minority ( @xmath80 ) band bottoms in the ferromagnet , @xmath53 and @xmath56 are the bottom of the conduction band and the top of the upper valence band in the insulator . the positions of `` steps '' for the potential @xmath76 are shifted by a value @xmath81 with respect to the points @xmath82 and @xmath83 in order to describe the finite thickness ( @xmath84 ) of the interface . the explanation , why it is done in this way , is given below in the text . @xmath85 is the spin - conserving part of the hamiltonian , @xmath86 are the interface numbers . to describe the defect structure of the nonideal f / o interface we consider that the impurity ions and the ions of a ferromagnetic metal constitute the more or less random alloy of a type a@xmath9b@xmath87 where a denotes the ions of the ferromagnet ( fe , co or ni ) and b denotes the impurities . therefore , we suppose that each site @xmath88 on the interface @xmath42 is characterized by the random parameter @xmath89 of @xmath0-@xmath1 hybridization taking two different values @xmath90 and @xmath91 with the probabilities @xmath92 and @xmath93 , respectively . we also suppose that a periodic part of @xmath0-@xmath1 hybridization is taken into account leading to the nonequivalence of majority and minority @xmath0-bands in the ferromagnet .    for the simple two - band tight - binding model of the binary alloys @xcite one assumes that : 1 ) parameters @xmath94 and @xmath95 describing the positions of @xmath0 and @xmath1 bands on the energy scale are different for the alloy s components ; 2 ) also the parameter @xmath96 of the hybridization between @xmath0 and @xmath1 bands depends on the type of an ion ( a or b ) . in our particular case , one can not take into account the former effect since the well - defined two types of the electrons will as before exist in the vicinity of the interface and the adopted model of tunneling will not change significantly . however , the random hybridization makes possible the processes of scattering of quasi - free @xmath0-electrons into the localized @xmath1 sub - band and vice versa and , therefore , can strongly influence on tunneling . @xmath97 is a part of the hamiltonian describing the spin - flip scattering . we take into account only the spin - flip processes for @xmath0-like electrons since these electrons are itinerant and give the most essential contribution to the tunneling current . operators @xmath98 , @xmath99 are defined as @xmath100 , \nonumber \\ \hat{s}_{-}(\rho_n^{\alpha } ) = \frac{1}{\sqrt{2sn } }   \sum_{\bf q}e^{iq\rho_n^{\alpha } } [ b^{+}_{\bf q } + \dots\ ] . \nonumber\end{aligned}\ ] ] here @xmath101 and @xmath102 denote the creation and annihilation operators of the surface magnons , @xmath103 is a number of the lattice sites on the interface , @xmath104 is a spin value . we used the well - known representation of the spin operators in terms of @xmath102 and @xmath101 and left the first terms of the series . @xmath105 is a random exchange integral which also takes the values @xmath106 and @xmath107 with probabilities @xmath92 and @xmath93 . let us now turn back to the step - like potential for @xmath1-holes . as it will be clear from the following consideration , the amplitude of the effective scattering potential on the interface ( @xmath82 ) for @xmath0-like electrons ( i.e.   @xmath108 , where @xmath109 is the self - energy ) is determined by the value @xmath110 , here @xmath111 is the imaginary part of the @xmath1-electron green s function at the point @xmath112 ( density of states ) . if one put @xmath113 in eq . ( [ u_d ] ) , then the vertex contribution to the conductivity ( that is the contribution due to tunneling assisted by interfacial scattering ) will be rather small as compared with the contribution due to direct tunneling . this result is not enough accurate and is the sequence of a continual type of the model when one neglects the existence of the atomic lattice and for this reason the self - energy has a @xmath114like behavior on the interface . however , the width @xmath81 of the interfacial layer is about the distance between atomic planes or even larger . for the case of bcc lattice , @xmath115 for [ 100 ] direction ( here @xmath116 is a lattice constant ) . one can show that for the present model with @xmath113 the imaginary part @xmath117 of the unperturbed @xmath1-electron green s function ( with @xmath118 ) , i.e.  density of states of @xmath1-electrons ( holes ) , has small value at the point of interface @xmath112 and increases inside the f - layer up to the distance of the order of @xmath119 , and then it oscillates near the average value which is approximately ten times larger than @xmath120 . the period of oscillations is determined by @xmath33 or @xmath34 , depending on the electron spin and on the orientation of magnetization in the f - layer . such a behavior is easily understood , if to take into account that @xmath1-electrons ( holes ) are almost completely reflected on the metal / insulator interface . thus , if one shifts the positions of `` steps '' for @xmath76 with respect to @xmath75 as it is given by ( [ u_d ] ) , one can expect the more effective mechanism of scattering due to the larger value of @xmath121 , and therefore the model becomes more realistic .      to calculate the non - local conductivity we apply the kubo formula of the linear response @xcite ( it is valid under the small applied voltage which is much less than the value of the energy gap in the insulator ) : @xmath122 where a matrix operator @xmath123 is defined as @xmath124 and @xmath125 is the asymmetric gradient operator , @xmath126 and @xmath127 are the retarded and advanced @xmath128matrix green s functions ( with components @xmath129 , @xmath130 , @xmath131 and @xmath132 ) , @xmath133 are the spin suffixes , brackets @xmath134 denote the averaging over the configurations and magnon degrees of freedom , the trace ( sp ) goes over @xmath0 and @xmath1 indices of the bands . below , for convenience , it is assumed that @xmath135 . we will recall about @xmath136 in the final expressions for the conductance .    to calculate the conductivity ( [ cond ] ) of the system one has to find the green s function of the hamiltonian ( [ hamilt ] ) , which can be found by solving the following system of differential equations in the mixed @xmath137 representation @xcite : @xmath138g_{\mu\kappa}^{\gamma\beta}(z , z ' )   = \delta_{\alpha\beta}\delta(z - z ' ) , \label{green_eq}\ ] ] @xmath139 where @xmath54 is a projection of the electron momentum on the @xmath59plane ( parallel to the f / o interface ) , the greek indices @xmath42 , @xmath140 and @xmath141 assume values @xmath0 and @xmath1 , @xmath70 denotes the electron spin . @xmath142 is the @xmath128matrix linear differential operator where @xmath143 and @xmath144 ( @xmath145 ; @xmath65 ) denote the coherent potentials for spin - up and spin - down electrons which take into account the scattering of the electron by random spin - conserving and spin - flip potentials on the interfaces . they were found with the use of the coherent potential approximation ( cpa ) @xcite , the details of these calculations are presented in the subsequent section  iic and in appendix b. the operator @xmath146 represents the effective single - particle hamiltonian of the system which , however , is non - hermitian since coherent potentials are imaginary quantities .    in order to solve eq . ( [ green_eq ] ) for the green s functions , we will follow the procedure described below . first , let s solve the shrdinger equation with the hamiltonian @xmath146 : @xmath147 \psi_{\beta}(z ) = 0 . \label{schrod}\ ] ] the solutions of this equation can be easily found since the potentials @xmath148 have a step - like form . let us put @xmath149 , where @xmath35 is the fermi energy , and introduce the following notations : @xmath150 @xmath151 are fermi momenta in f@xmath11 and f@xmath3 ferromagnetic layers for electrons with the spin @xmath70 , and @xmath152 let also @xmath153 and @xmath154 be the components of electron momentum with spin @xmath70 along @xmath60-axis in f@xmath11 and f@xmath3 layers , respectively ( @xmath155 is the in - plane component of the momentum , @xmath42 is a band index ) , and let @xmath156 be the imaginary electron momentum in the insulating layer . further , for convenience , we will omit indices @xmath70 and @xmath155 in the notation of some functions . equation ( [ schrod ] ) has four linear - independent solutions which we denote as @xmath157 and @xmath158 we choose these independent solutions so that two functions @xmath159 and @xmath160 would describe two waves corresponding to the @xmath0-like electrons , and functions @xmath161 and @xmath162 would correspond to the @xmath1-like electrons . namely , in a layer f@xmath11 @xmath163 the solutions @xmath164 have the form @xmath165 \\ 0 \end{array } \right ) \quad z < a , \label{phi}\ ] ] @xmath166 \end{array } \right ) \quad z < a,\ ] ] and in a layer f@xmath3 @xmath167 the solutions @xmath168 are @xmath169 \\ 0 \end{array } \right ) \quad z > b , \label{psi}\ ] ] @xmath170 \end{array } \right ) \quad z > b.\ ] ] since @xmath149 , then @xmath171 ( @xmath172 ; @xmath173 ) . thus these solutions satisfy the following boundary conditions : @xmath174 starting from expressions ( [ phi ] ) and ( [ psi ] ) , the solutions @xmath164 , @xmath168 can be easily extended in two other layers . let us introduce the matrices @xmath175 @xmath176 the wronskian of the system ( [ schrod ] ) is @xmath177 where the matrix operator @xmath178 is defined similar to eq . ( [ d_r ] ) , and the subscript @xmath179 denotes the transposition operation . it is known from the theory of differential equations that the matrix @xmath180 is a constant since it satisfies the equation @xmath181 . taking into account that @xmath149 and @xmath182 , the solution of eq . ( [ green_eq ] ) for the retarded green s function can be written in the matrix form as @xmath183^{-1}\psi^t(z ' ) ,   \quad \mbox{if}\quad z < z ' , \label{green_r } \\ g^r(z , z ' ) & = & \psi(z)\delta^{-1}\phi^t(z ' ) ,   \qquad \mbox{if}\quad z > z ' . \nonumber\end{aligned}\ ] ] to find the advanced green s function , one has to put @xmath184 in eq . ( [ green_eq ] ) and assume that @xmath185 . then we obtain @xmath186^{*}. \label{green_a}\ ] ] in the expressions ( [ green_r ] ) and ( [ green_a ] ) the green s functions depend on the in - plane momentum @xmath155 and on the spin @xmath70 of the electron because solutions @xmath168 and @xmath164 also depend on @xmath70 and @xmath155 . next , one has to find the two - point conductivity ( [ cond ] ) of the system using the green s functions ( [ green_r ] ) and ( [ green_a ] ) . for that , we introduce the current matrices @xmath187 and @xmath188 ( @xmath65 ) constructed with the use of solutions @xmath168 and @xmath164 , respectively : @xmath189 the total conductance of the system @xmath190 may be presented in the usual form as a sum of the `` bubble '' part and the vertex corrections ( see fig . 2 ) @xcite : @xmath191    the `` bubble '' contribution to the conductance describes direct electron tunneling from the electrode to another one through the barrier when electron momentum parallel to the f / o interface @xmath54 is conserved ( the specular transmission ) . the scattering on the interfaces affects direct tunneling so that the effective height of the potential barrier increases because the electron also has to pass through @xmath114-like potentials at the points @xmath112 and @xmath192 formed by the self - energies @xmath193 and @xmath194 . the vertex corrections to the conductance describe the tunneling assisted by interfacial roughness  that is the processes when the electron with momentum @xmath155 comes to the impurity center on the interface , undergoes scattering ( @xmath195 ) in another channel with @xmath196 , and then goes away to the electrode or to the barrier . it also can be shown that the contribution to the tunneling conductance is negligibly small ( @xmath197 ) from the diagram containing both vertex parts @xmath198 and @xmath199 compared with other contributions that are of the order of @xmath200 . the diagram with vertices @xmath198 and @xmath199 corresponds to the interference of waves scattered from both interfaces . therefore , we can neglect this interference term .     as a sum of `` bubble '' and `` vertex '' parts . here the full lines correspond to the green s functions @xmath201 and @xmath202 , and wavy lines denote the asymmetric gradient operator @xmath203 of velocity at the points @xmath60 and @xmath204 . the shaded square designates the vertex part @xmath205 at the interface ; _ h.c . _  denotes the complex conjugate terms . ] substituting obtained expressions ( [ green_r ] ) , ( [ green_a ] ) for the green s functions and using the definition ( [ j_current ] ) for the current matrices , we come to the following results . the analytical expression for the `` bubble '' conductance is given by @xmath206^t\frac{1}{\delta_{\mu}^{\dagger } } j_{\mu}^{\psi}(z')\right\ } , \label{s_0}\ ] ] here it is assumed that @xmath207 , @xmath208 is the wronskian ( [ wronsc ] ) , @xmath209 denotes the junction area .    for @xmath210 , @xmath211 , the vertex corrections from the left and from the right interfaces can be written as @xmath212 where summation is also performed over repeating indices @xmath213 . here @xmath214 and @xmath215 are the vertex parts on the interfaces @xmath216 and @xmath217 . @xmath218 are @xmath128matrices , their components are defined by @xmath219^{\beta_1\beta_2 } \right\ } , \label{lambda } \\ \lambda^{\gamma_1\gamma_2}_{\mu\kappa}(a , z ' ) = { \rm sp } \left\ { \frac{1}{\delta_{\mu } } \left[\rho^{\varphi t}_{\mu}(a)\right]^{\gamma_1\gamma_2 } \frac{1}{\delta_{\mu}^{\dagger } } j^{\psi}(z ' ) \right\ } , \nonumber\end{aligned}\ ] ] where @xmath220^{\beta_1\beta_2}$ ] and @xmath221^{\gamma_1\gamma_2}$ ] are density matrices with the components : @xmath222^{\beta_1\beta_2}_{ik } & = & \psi^{\beta_1*}_i(a)\psi^{\beta_2}_k(a)\quad ( i , k = s , d ) , \label{rho } \\ \left[\rho^{\varphi}_{\mu}(a)\right]^{\gamma_1\gamma_2}_{ik } & = & \varphi^{\gamma_1*}_i(a)\varphi^{\gamma_2}_k(a)\quad ( i , k = s , d ) . \nonumber\end{aligned}\ ] ] the expressions similar to expressions ( [ lambda ] ) may be also written for the matrices @xmath223 , @xmath224 and for other position of the points @xmath60 , @xmath204 with respect to @xmath216 and @xmath217 . the vertex parts @xmath225 and @xmath226 are calculated in the `` ladder '' approximation @xcite . the derivation of the equation for @xmath205 is presented below in the subsection iid . we have to note , that coherent potentials @xmath193 , @xmath194 calculated in the framework of the cpa and vertices @xmath225 , @xmath226 calculated in the `` ladder '' approximation satisfy the so - called ward identity which in our case , for example , for the interface @xmath82 can be written as follows ( for the details , see appendix a ) : @xmath227 here the summation is also performed over repeating indices @xmath228 and @xmath229 . the fulfillment of ( [ ward ] ) provides the necessary condition of the nondivergence of the current through the tunnel junction : @xmath230 = 0 . \label{current0}\ ] ]    according to eq . ( [ current0 ] ) , the total conductance of the system is a constant value . in view of that , we will derive the exact expression for @xmath190 evaluating the conductance at points @xmath231 and @xmath232 , i.e.  at the left and at the right sides from the interface . the wronskian matrix @xmath208 and the current matrices @xmath233 and @xmath234 are expressed in terms of matrices @xmath235 and @xmath236  these matrices , as follows from eq . ( [ green_r ] ) , determine the green s function . the straightforward evaluation of the conductance according to formulae ( [ s_0][lambda ] ) leads to the result that @xmath237 is expressed in terms of the retarded green s functions at the points of interfaces @xmath238 and @xmath239 constructed according to eq . ( [ green_r ] ) . the explicit form of @xmath240 in the @xmath137 representation is given by the expression :    @xmath241    - \sigma_{\mu}^{dd}(a ) \end{array}\right)^{-1 } , \label{g_int}\ ] ]    where @xmath242 the similar expression can be written for the green s function @xmath243 on the right interface . for that , one has to make the substitutions @xmath244 , @xmath245 , @xmath246 in the expression for @xmath240 . here @xmath247 ( @xmath173 ) and @xmath248 are the effective electron ( hole ) masses in the ferromagnetic and insulating layers ; @xmath249 and @xmath250 ( @xmath251 , @xmath173 ) are the functions on @xmath155 introduced above in the text after eq . ( [ schrod ] ) . let us define the `` transport '' density of states as follows : @xmath252 where @xmath253    expression ( [ s_0 ] ) for the `` bubble '' conductance then reads @xmath254 where @xmath255 @xmath256 @xmath257 and @xmath258 is the `` width '' of @xmath1barrier .    for the vertex correction from the left interface ( @xmath82 ) we obtain @xmath259 in the similar way , the vertex correction from the right interface ( @xmath83 ) reads @xmath260 expressions ( [ s_0_ab][s_vertex_b ] ) determine the total conductance of the system . the scattering of the electron by random potentials within the interface ( terms @xmath261 and @xmath262 in the hamiltonian ) is taken into account by introducing the self - energy operators @xmath143 and @xmath144 into the effective single - particle hamiltionian ( [ h_eff ] ) . to calculate the self - energies we apply the coherent - potential approximation ( cpa ) @xcite . let s denote kets latexmath:[$|\gamma,\mu,\rho_n^{\alpha}\rangle =    the wannier states of the electron on the interface @xmath42 ( @xmath264 ) at the given site @xmath88 in the @xmath265-plane , where @xmath141 refers to the state @xmath0 or @xmath1 and @xmath70 is the spin . the symbol @xmath266 denotes the direct product . the problem of finding of the single - particle green s function @xmath267 of the many - body hamiltonian  ( [ hamilt ] ) is reduced to the related single electron problem of the propagation of the electron in a random interfacial potential @xmath268 where @xmath269 and the summation goes over the interface number @xmath42 and over the sites @xmath270 . here @xmath271 is the random potential of @xmath272 hybridization associated with the site @xmath270 and @xmath273 is the exchange - like interaction with the surface magnons . the random quantities @xmath89 and @xmath274 used here were introduced in sec . now one can formulate the cpa by the ordinary way and the only difference with respect to the usual situation of the bulk scattering is that the initial green s functions have to be calculated for the trilayer system . we assume that the averaged propagator of the system @xmath275 differs from the initial green s function , corresponding to the hamiltonian @xmath276 [ see eq . ( [ hamilt ] ) ] , by the self - energy correction in eqs . ( [ green_eq ] ) , ( [ h_eff ] ) . this means that the system behaves as if coherent potentials @xmath277 had been assigned to each site of the interface @xmath216 and @xmath217 . after the introducing of the effective medium @xmath278 , each site @xmath279 becomes a source of the random potential @xmath280 . the single - site @xmath281-matrix associated with potential @xmath282 is given by @xmath283 where @xmath284 or @xmath217 , @xmath285 is the averaged green s function at the interface @xmath286 which is expressed via @xmath278 in accordance with eq . ( [ g_int ] ) . the upper limit @xmath287 is a cut - off of the in - plane momentum which originates from the finite size of the brillouin zone . for that , one has to substitute the brillouin zone s projection onto @xmath288-plane by a circle of the same square with a radius @xmath287 . for the bcc lattice , @xmath289 , where @xmath116 is a lattice constant . the single - site @xmath281-matrix  ( [ t_matrix ] ) is obviously different for different sites . at the same time it is supposed that scattering by the random potential is taken into account in the averaged propagator @xmath290 by the self - energy operator @xmath291 . therefore , we require that the ensemble average of the single - site @xmath281-matrix vanishes , i.e. , @xmath292 here @xmath293 and @xmath294 are the values of the single - site @xmath281-matrix associated with a given site @xmath270 which is occupied by the a - type ion ( ferromagnet s ion ) or by the b - type ion ( impurity ) , respectively . brackets @xmath295 denote the averaging over magnon degrees of freedom . the equation ( [ t_av ] ) is the well - known self - consistent coherent potential approximation ( cpa ) @xcite that implicitly determines the self - energy operator @xmath278 . the cpa equations ( [ t_av ] ) are formulated for the particular case of electron scattering within the f / o interface . the straightforward calculations show that the single - site @xmath281-matrix @xmath296 can be represented in the form : @xmath297 with respect to spin - up and spin - down subspaces . here @xmath298 , @xmath299 , and the blocks @xmath300 , @xmath301 are @xmath302-matrices , functionally depending on @xmath303 and @xmath304 , with the components designated by indices of bands @xmath0 and @xmath1 .    to satisfy the condition  ( [ t_av ] ) , one has to consider only the spin - conserving part of this equation as long as @xmath305 since the expression to be averaged contains unequal number of the creation and annihilation operators of the magnons . in order to calculate @xmath306 we adopted the further approximation and assumed that @xmath307 here @xmath308 is the average number of magnons at the given temperature . in other words , we substituted the operators @xmath309 by its averaged values . the function @xmath310 is given by the familiar expression @xmath311 where @xmath312 is the spectrum of surface magnons . the approximation we made to take into account the spin - flip processes is the static approximation for magnons . the inelasticity of the electron - magnon processes may be taken into account in the energy conservation rule . however , the characteristic magnon energy @xmath313 is much less than the fermi energy @xmath35 , and in the first approximation one can neglect this energy @xmath313  it becomes important in the case of finite voltages when the process of emitting of a magnon by a hot electron influences on the form of @xmath314 dependence . we , therefore , restricted our calculation to the case of small voltage bias with the voltage less than @xmath313 . within this approximation , for the system of cpa equations we get @xmath315 as far as matrices @xmath316 are @xmath128blocks , thus for the general case eq . ( [ cpa_system ] ) represents the system of two matrix equations for 8 unknown quantities @xmath317 . the system ( [ cpa_system ] ) is one of the possible forms of the cpa equations . but actually we used another representation of the cpa which was more convenient for the numerical implementation . for that we exploited the augmented - space formalism by mookerjee @xcite , and the details are presented in appendix b.      to find the vertex corrections we used the `` ladder '' approximation combined with the cpa , that was originally proposed by velick  @xcite . the diagrammatic representation of the `` ladder '' approximation is given in fig .  3 . since the scattering potentials on the different interfaces ( at the points @xmath216 and @xmath217 ) do not correlate with each other , the vertex parts @xmath318 can be found independently for each interface and we will omit interface suffix in the subsequent expressions . let @xmath319 be the average of the product of two random @xmath281-matrices over configurations and magnon distribution , where as before @xmath320 are spin indices and @xmath321 are orbital indices . on the diagram in fig.3 , it is presented by the dashed line . the analytical expression for the correlator @xmath322 can be found in accordance with the adopted approximate scheme of averaging over the magnon degrees of freedom . taking into account eq . ( [ t_form ] ) we obtain : @xmath323 where @xmath324 and @xmath325 are the components of the single - site @xmath281-matrix , eq . ( [ t_form ] ) . we also define the operator @xmath326 \nonumber\\ & & -\ g_{\mu}^{*\beta_1\gamma_1}(z_{\alpha})g_{\mu}^{\beta_2\gamma_2}(z_{\alpha } ) \label{d}\end{aligned}\ ] ] denoting the propagator of the pair of electrons in the `` ladder '' equation at the interface @xmath327 . its definition follows from the fact that in the diagram representation of the cpa the multiple scattering on the given site is assumed to be incorporated into the single - site @xmath281-matrix @xmath328 , corresponding to the single vertex of any diagram . due to that , the subsequent sites in the `` ladder '' diagrammatic equation ( fig.3 ) must not reproduce each other . therefore , the necessary correction is subtracted in eq . ( [ d ] ) . the summation over @xmath155 goes up to @xmath287 similar to eq . ( [ g_kappa ] ) . after that definitions the analytical equation for the vertex part reads as @xmath329 eq . ( [ gamma_eq ] ) is the ordinary system of linear equations . it can be proved that the vertex parts found with the use of the `` ladder '' approximation and self - energies found with the use of the cpa satisfy the ward identity ( [ ward ] ) , and hence the total tunneling current is a constant value @xmath330 , i.e.  it does not depend on @xmath60 . the proof of the ward identity is presented in appendix  a. we considered the case of fe / al@xmath2o@xmath3/fe tunnel junction . the following parameters were chosen to describe the system . according to estimations of stearns @xcite , for the itinerant @xmath0like electrons in fe we set @xmath21  @xmath22 , @xmath23  @xmath22 , and @xmath331 ( here @xmath17 is bare electron mass ) . for the more localized @xmath1electrons ( holes ) , we put @xmath332 . the @xmath1 density of states at the fermi energy is larger for the minority spin band @xcite , therefore , @xmath333 , and we put @xmath334  @xmath22 , @xmath335  @xmath22 . the values of the fermi momenta define the positions of band bottoms @xmath336 , @xmath337 ( @xmath173 , @xmath338 ) with respect to @xmath35 . for @xmath1-electrons the fermi momenta were chosen such a way that the interface density of states @xmath339 ( where @xmath340 are the positions of interfaces ) comply with the following ratio @xmath341 , which is the typical situation for the case of @xmath342 transition metals ( see , for a example , the calculations of tsymbal and pettifor @xcite ) . we also put the width of the barrier @xmath343   , and @xmath115 where @xmath344    is a lattice constant for bcc fe ( @xmath81 is a parameter describing the thickness of the interface ) . the main features of the band structure of @xmath42-al@xmath2o@xmath3 crystals were briefly presented in sec . iia . in a view of that discussion , the following parameters of the model were taken to describe the amorphous al@xmath2o@xmath3 barrier : the effective masses of electrons and holes are @xmath345 , @xmath346 , the positions of the conduction band bottom ( @xmath53 ) and the top of a valence band ( @xmath56 ) are @xmath347  ev , i.e.  the width of the band gap is @xmath348  ev , and @xmath35 is assumed to be a zero of energy .    in order to illustrate the general formalism presented in the previous sections , let us first consider the case of only @xmath272 impurity scattering ( i.e.  @xmath349 and there are no spin - flip processes ) when concentration of impurity ions on the interface is @xmath350 . in this case all formulae have a simple analytical form . as it was mentioned previously in sec . iia , in the two - band tight - binding model description of a binary alloy the random variables are diagonal matrix elements of the hamiltonian @xmath351 , @xmath352 and @xmath353 depending on whether the site @xmath270 is occupied by an a or b ion . to be more precise , one must consider the concentration dependence of these matrix elements . it reflects the fact that their values are modified by the existence of a charge transfer in the alloy . since in the framework of our model we are focusing on the hybridization effects ( see sec . iia for justification of this approach ) , we have @xmath354 , @xmath355 where @xmath356 and @xmath357 are hybridizations of the pure components . let us assume for simplicity that in eq . ( [ operators ] ) @xmath358 . then for @xmath359 we get @xmath360 , and thus @xmath361 is the single parameter characterizing the amplitude of @xmath0-@xmath1 scattering on the interface . the nonzero value of @xmath362 will lead to a slight modification of the @xmath0 and @xmath1-like eigenstates describing @xmath0 and @xmath1-like electrons in the vicinity of the interface , and thus will not affect the qualitative results presented below . we will omit spin suffixes since we consider the spin - conserving scattering now . at @xmath359 only diagonal elements of the self - energy matrix @xmath363 in the @xmath130-space have non - zero values . then the green s functions , e.g.  at the point @xmath216 , become [ see eq . ( [ g_int ] ) ] @xmath364   \nonumber \\ & & -\ \sigma_{\mu}^{dd}(a ) \biggr\}^{-1}\frac{\kappa d\kappa}{2\pi } ,   \nonumber\end{aligned}\ ] ] and cpa equations are written as follows @xmath365 which must be solved self - consistently by means of converging iterative procedure . as far as green functions are diagonal , one can denote @xmath366 and can introduce the similar notations for quantities @xmath179 and @xmath367 , defined in section iid . then one gets @xmath368 @xmath369 @xmath370 @xmath371 since the mass of holes @xmath49 in the insulator is much larger than the electron mass @xmath48 , the exponential factor @xmath372 in formulae ( [ s_0_ab ]  [ s_vertex_b ] ) for tunneling conductance is negligibly small as compared with one for @xmath0like electrons . therefore , one may neglect the contribution from @xmath1holes to the tunneling current . then for the tunneling conductance we obtain @xmath373 where @xmath374 is the conductance corresponding to the `` bubble '' diagram ( here @xmath375 ) , @xmath376 \nonumber \\ & + & \frac{e^2}{2\pi\hbar}\left(\int_0^{\kappa_{max } } a^{ss}_{\kappa\mu}(a ) \frac{\kappa d\kappa}{2\pi}\right)\,\gamma^{(a)\mu\mu}_{\,\,ss } \nonumber \\ & \times &    \left [ \int_0^{\kappa_{max } } { \left| g_{\kappa'\mu}^{ss}(a ) \right|}^2 { \left(\frac{q_2^s}{m^0_s}\right)}^2 a^{ss}_{\kappa'\mu}(b)e^{-2 q_2^s w}\frac{\kappa ' d\kappa'}{2\pi}\right ] \nonumber\end{aligned}\ ] ] is the `` vertex '' contribution to the conductance on the left interface , and @xmath377 \nonumber \\ & + & \frac{e^2}{2\pi\hbar}\left(\int_0^{\kappa_{max } } a^{ss}_{\kappa\mu}(b ) \frac{\kappa d\kappa}{2\pi}\right)\,\gamma^{(b)\mu\mu}_{\,\,ss } \nonumber \\ & \times &   \left[\int_0^{\kappa_{max } } { \left| g_{\kappa'\mu}^{ss}(b ) \right|}^2 { \left(\frac{q_2^s}{m^0_s}\right)}^2 a^{ss}_{\kappa'\mu}(a)e^{-2 q_2^s w}\frac{\kappa ' d\kappa'}{2\pi}\right ] \nonumber\end{aligned}\ ] ] is the `` vertex '' contribution to the conductance on the right interface . here @xmath378 are the transport densities of states which for the present case on the left interface are given by the expressions @xmath379/2m_d - \sigma^{dd}_{\mu}(a)\right|}^2}. \quad \nonumber\end{aligned}\ ] ] and by the analogous expressions in the case of right interface . we remind , that physically the `` bubble '' term in the total conductance is a contribution to the current due to direct tunneling when the electron momentum @xmath54 in the plane of the interface is conserved . scattering processes renormalize the `` bubble '' term with respect to the case of ballistic transport such that the effective `` height '' of the potential barrier seen by electron increases due to self - energy corrections @xmath380 arising on the interfaces . the vertex contributions to the conductance describe the tunneling assisted by interfacial scattering   that is the processes of tunneling with scattering on the f / o interface when the in - plane momentum is not conserved , i.e.  @xmath381 for the scattered electron . thus , both momenta @xmath155 and @xmath382 in eq . ( [ s_vertex ] ) determine the vertex corrections . note also , that both vertex corrections @xmath383 and @xmath384 ( [ sa_diff ] ) , ( [ sb_diff ] ) consist of two terms with vertices @xmath385 and @xmath386 . the terms with vertex part @xmath386 are contribution to the tunneling conductance from @xmath0-like electrons only . the scattering in the @xmath0 channel described by @xmath109 ( [ cpa_50 ] ) is caused initially by @xmath0-@xmath1 scattering . the terms with @xmath385 describe either the process of diffuse scattering of @xmath1-like electron to the @xmath0 state in the f / o interface and then tunneling of @xmath0-like electron in the barrier , or the process of tunneling of @xmath0-electron and then its scattering to @xmath1 state in the o / f interface and leaving into the electrode . we have calculated the tunneling conductance and the tmr ratio defined as @xmath387 , where @xmath388 and @xmath389 are the total conductances for the parallel ( p ) and antiparallel ( ap ) alignment of magnetic moments in the f - layers . the cpa equations defining the self - energies were solved numerically . the validity of the ward identity , eq . ( [ ward ] ) , was checked after the vertex parts @xmath47 had been computed at every step for a given value of the parameter @xmath141 .     per unit square 1@xmath390 ) for the parallel ( p ) alignment of magnetic moments in the ferromagnetic layers as a function of the scattering parameter @xmath141 on the interface in the absence of spin - flip processes . the parameters of the model are : @xmath0-like electrons   @xmath391@xmath22 , @xmath392@xmath22 , @xmath393 , @xmath394 ; @xmath1-like holes   @xmath395@xmath22 , @xmath396@xmath22 , @xmath397 , the height of the potential barrier @xmath347  ev , the width of the barrier @xmath343   , the concentration of fe ions on the interface @xmath398 . ] the results are presented in figs . 47 . in fig . 4 the `` bubble '' ( [ s_ballistic ] ) and the `` vertex '' ( [ sa_diff ] ) , ( [ sb_diff ] ) contributions to the conductance for spin - up and spin - down channels are shown for the parallel ( p ) alignment of magnetic moments as a function of the scattering parameter @xmath141 on the interface . ( we remind that up to now only the spin - conserving scattering is considered . ) as it was mentioned previously , scattering suppresses the `` bubble '' conductance , and hence @xmath399 and @xmath400 are decreasing functions on @xmath141 . the contribution from the `` bubble '' part is larger for the majority spin @xmath401 channel since @xmath402 . on the contrary , the contribution from the vertex corrections dominates for the minority spin @xmath403 channel  that behavior can be understood as follows . first , at small @xmath404 the imaginary part of the self - energy @xmath109 , which describes the scattering of @xmath0-electrons , behaves as @xmath405 [ see eq . ( [ cpa_50 ] ) ] , where @xmath406 are spin - up ( down ) @xmath1 density of states on the interface . as far as @xmath407 is of the order of magnitude greater than @xmath408 , the scattering for @xmath0-like itinerant electrons from minority spin band ( @xmath80 ) is more effective than for the majority spin ( @xmath79 ) electrons . the more effective scattering of spin - down @xmath0-electons leads to the predominance of the minority spin contribution with vertex part @xmath409 over the majority spin contribution with vertex @xmath410 , since at small @xmath141 , @xmath411 and @xmath412 , therefore @xmath413 . second , the transport density of states @xmath414 [ eq . ( [ a_transport ] ) ] which determines the partial conductance due to mixing of @xmath0 and @xmath1 channels in the interface ( terms with @xmath385 ) is also larger for minority spin channel . thus , the combination of these factors results in domination of the vertex contribution to the conductance from the minority spin channel . the results of calculations also showed that the @xmath129 contribution to the vertex corrections @xmath415 ( terms with vertices @xmath386 ) was much less than the @xmath130 contribution @xmath416 ( terms with @xmath385 ) .     per unit square 1@xmath390 ) for the antiparallel ( ap ) alignment of magnetic moments as a function of the scattering parameter @xmath141 on the interface in the absence of spin - flip processes . parameters of the model are the same as in fig .  4 . ]    the `` bubble '' and vertex contributions to the conductance ( which are the same for spin - up and spin - down channels ) for the antiparallel ( ap ) alignment of moments as a function of @xmath141 are presented in fig . the total conductances for p and ap configurations are shown in fig . 6 . finally , in fig . 7 the tmr ratio @xmath417 as a function of scattering parameter @xmath141 is shown . when the amplitude of scattering is negligibly small ( @xmath418 ) , we have positive value of the tmr  @xmath419  that is the result of slonczewski s theory @xcite under the chosen parameters for @xmath0like electrons . with increasing of @xmath141 the tmr amplitude is monotonically decreasing and can become even negative if @xmath420  ev .    to understand qualitatively the obtained results , conductance and magnetoresistance can be rewritten in the general approximate form : @xmath421 where we took into account that @xmath422 , and for high and thick enough barrier the main contribution to the tunneling conductance is due to electrons with momentum almost perpendicular to the barrier ( factors @xmath423 ) . consequently , @xmath424 are the quasi - one - dimensional @xmath0 density of states near the interface , and @xmath425 are the corresponding three - dimensional @xmath1 density of states . the @xmath426 are renormalized vertex corrections . in expression ( [ dsigma ] ) for @xmath427 , the first term due to direct tunneling is always positive . concerning the second term , @xmath428 since @xmath429 , but @xmath430 and , therefore , this term is negative . this contribution decreases the magnetoresistance when scattering parameter @xmath141 becomes larger . at @xmath431 the @xmath130 contribution overlaps the contribution to @xmath427 from the @xmath129 channel , and we have @xmath432 . thus , the inverse tmr ratio arisen in our model is caused by extremely strong scattering of negatively polarized @xmath1-like electrons ( which give the indirect contribution to the tunneling current ) to the @xmath0-band on the interface .     per unit square 1@xmath390 ) for the parallel ( p ) and antiparallel ( ap ) alignment of magnetic moments as the function of the scattering parameter @xmath141 on the interface in the absence of spin - flip processes . the parameters are the same as in fig .  4 . ]    the parameter @xmath141 determining the amplitude of @xmath0-@xmath1 scattering on the interface is defined as @xmath361 . one can regard the difference between hybridizations , @xmath433 , as approximately a constant value for given constituents . however , @xmath93 is a concentration of impurity centers , and thus @xmath141 is a measure of the imperfection of the f / o interface . the proposed model , therefore , explains qualitatively the strong interface sensitivity of the tunneling magnetoresistance effect @xcite . according to heine s discussion of the hybridization in transition metals @xcite , the hybridization constants are from about 1.0 to 3.0 ev for different elements . one can assume that the difference @xmath434 is of the order of magnitude smaller and , therefore , the critical value of scattering parameter @xmath435  ev that we obtained is more or less realistic . the strong reduction of the tmr due to nonideal structure of the metal / insulator interfaces is a well - known observation . the oxidation of a thin al layer leads to the undesirable oxidation of few metal monolayers close to the f / o interface and thus to the formation of f - o oxides ( fe@xmath3o@xmath10 @xcite , coo and co@xmath3o@xmath10 @xcite )  that reduces the tmr @xcite . on the other hand , if the too thick al layer is not oxidized completely , the contamination of the interface by al ions also reduces the tmr @xcite . the dependence of the tmr _ vs._the thickness of the al overlayer has , therefore , a maximum and the best tmr values achieved by moodera s group lie in the range @xmath436    @xcite .     in the case of only elastic scattering on the interfaces . the parameters are the same as in fig .  4 . ]    the contamination of the f / o interface by oh ions in the early experiments by merservey and tedrow on tunneling with the superconductors @xcite led to the small measured values of a spin polarization @xmath437 for ni and gd . the contamination was due to oxidizing of the al films in the laboratory air . the improved technique of samples preparation in a pure oxygen increased the values of @xmath437 for ni and gd , and for some rare - earth metals @xcite .    in a recent work by leclair _ et al . _ @xcite the strong suppression of magnetoresistance was observed in co / al@xmath2o@xmath3/co tunnel junctions with a very thin cr interfacial layer . it was found that the tmr decayed exponentially on the cr interlayer thickness with a length scale  @xmath438  ( approximately 0.5 monolayers ) . with the addition of only 3  cr ( @xmath439 monolayers ) the reduced tmr was only 10% of the initial value for a control junction . et al . _ @xcite presented some qualitative arguments that the suppression of a spin polarization ( and , hence , the reduction of tmr ) was due to more strongly suppression of majority @xmath0-@xmath25 density of states compared with minority spins caused by the resonant scattering of majority spin @xmath8-electrons with the cr @xmath1 states . from the point of view of our model , we can explain the strong degradation of the tmr by the strong electron scattering within the interfacial co - cr alloy that is formed under the preparation of extremely thin ( @xmath438  monolayer ) cr interlayer .    . ]    we have also calculated the temperature dependence of the tmr taking into account spin - flip scattering in addition to @xmath272 impurity scattering as it is described in details in sections iic and iid . for that , the average magnon number @xmath310 ( [ n_t ] ) as a function of the temperature was found in analogy with debye s treatment of phonons using the similar approach that was proposed by s.  zhang _ et al _ @xcite . the magnon dispersion relation in eq . ( [ n_t ] ) was replaced by simple isotropic parabolic spectrum @xmath440 where @xmath441 is the equivalent radius of the two - dimensional brillouin zone ( see eq . [ g_kappa ] ) , and @xmath442 is related to curie temperature @xmath443 and in the mean - field approximation is given by @xmath444 . for the chosen model of the dispersion relation @xmath445 , one has to overcome the divergence on the lower limit of the integral in eq . ( [ n_t ] ) . therefore , one must introduce a lower wave - length cutoff @xmath446 @xcite . physically , it may represent a finite coherence length due to interfacial roughness . in our calculation we have taken the same parameters that were used by s.  zhang _ et al . _ @xcite  for the analysis of the zero - bias anomaly : @xmath447 , @xmath448  mev , and @xmath449  mev . then , for the temperature range within the room value , we have @xmath450 we have put @xmath451 , @xmath452  ev ( for fe atoms ) and @xmath453  ev , i.e.  it is supposed that spin - flip process is possible , if an electron scatters on the fe ion . the tmr _ vs. _  the scattering parameter @xmath141 at different temperatures @xmath454 and @xmath455  k are plotted in fig .  8 . in fig . 9 the temperature dependence of the resistance for the p and ap configuration of magnetizations is presented for the same parameters , but @xmath418 . the results show that the tmr ratio decreases with increasing of the temperature . moreover , the resistances @xmath456 and @xmath457 of the structure for both configurations , p and ap , are also decreasing when temperature increases  that is in the qualitative agreement with experimental data ( e.g. , see fig . 4b in ref . the physical mechanism of this effect is related with the excitation of spin - flip processes in the system . due to these processes , the new channels of electron scattering appear which are frozen at zero temperature . as the result , the conductance of the system increases for both p and ap configurations and , therefore , the resistance drops . the spin - flip processes mix the spin - up and spin - down channels . therefore , the relative difference of the resistances decreases at different configurations and the tmr decreases with increasing of temperature . in conclusion , we would like to outline the main results obtained in the present work . based on the analysis of the band structure of 3@xmath1 ferromagnetic metals and al@xmath2o@xmath3 crystals and on the results of ab - initio calculations of the magnetoresistance for epitaxial tunnel junctions , we built on an adaptation of the simplified two - band @xmath0-@xmath1 model to treat the diffuse electron transport in the non - ideal f / o / f magnetic tunnel junctions . we had modeled the rough f / o interface by the random binary alloy that is formed from the ions of the ferromagnet and impurities ( e.g.  the al or o ions ) , and assumed that the main mechanism of electron scattering on the interface which substantially affects tunneling is the @xmath0-@xmath1 scattering . we used the kubo formalism to calculate the tunneling conductance and found the vertex corrections to conductivity with the use of the `` ladder '' approximation combined with the cpa . the obtained results show that in the case of strong electron scattering within the interfacial alloy the vertex corrections give the essential contribution to the tunneling conductance . we proved that adopted approximations lead to the physically correct results , namely , the non - local conductivity tensor is a constant function and , therefore , a tunneling current is conserved . we showed that interfacial inter - band scattering substantially reduces a value of the tmr , which can become even negative in the case of extremely strong scattering . the reason of the suppression of the magnetoresistance is the indirect contribution of negatively polarized @xmath1-like electrons to the tunneling current due to strong scattering to the @xmath0 band on the interface . it is also shown that spin - flip electron scattering on the surface magnons within the interface leads to a further decrease of the tmr at finite temperature . thus , in the framework of the proposed model , we are able to explain qualitatively the strong interface sensitivity of the tunneling properties that is observed in the experiments . a.  vedyayev and d.  bagrets are grateful to cea / grenoble / drfmc / sp2m / nm for fellowships , a.  bagrets is grateful to laboratoire louis nel / cnrs ( grenoble , france ) for hospitality . this work was partially supported by russian foundation for basic research ( grant no . 98 - 02 - 16806 ) . in this appendix we briefly describe how to obtain eq . ( [ ward ] ) ( ward identity ) , and using a simple example we will show how this identity can be proved .    starting from eq . ( [ current0 ] ) , one first has to compute the derivation of the `` bubble '' conductivity . assuming , that the self - energy @xmath277 is symmetric with respect to the rearrangement of band indices , i.e.@xmath458 , and using the fact that functions @xmath459 are the solutions of the schrdinger equation ( [ schrod ] ) , for the derivation of the current matrix we obtain : @xmath460^{\alpha\beta},\ ] ] where @xmath461 and the matrix @xmath462^{\alpha\beta}$ ] is defined by eq . ( [ rho ] ) . then , from eq . ( [ s_0 ] ) , ( [ lambda ] ) we have @xmath463 . \nonumber\end{aligned}\ ] ] to proceed further , let us compute the derivation of the vertex correction to conductivity . the initial expression for the matrix @xmath218 ( see diagram in fig . 2 ) is written as @xmath464 taking into account that the green s functions are the solution of the eq . ( [ green_eq ] ) , we obtain @xmath465 substituting the obtained expression to formula ( [ s_vertex ] ) for the vertex correction , we get @xmath466 . \nonumber\end{aligned}\ ] ] the summation here is also performed over indices @xmath229 , @xmath467 , and @xmath228 . a similar expression can be written for the derivation of the vertex correction @xmath468 at interface @xmath83 . from ( [ app_s0 ] ) and ( [ app_s_vertex ] ) one can obtain the final expression ( [ ward ] ) for ward identity . let s prove identity for the simple case of only @xmath272 scattering , when there are no spin - flip processes , and @xmath359 . this situation was considered in sec . iii , and there were introduced the notations @xmath469 , @xmath470 , and @xmath471 ( @xmath472 ) for the components of the @xmath128matricies @xmath47 , @xmath179 , and @xmath367 . within these notations , the identity ( [ ward ] ) , which has to be proved , can be written as @xmath473.\ ] ] here we omitted spin suffixes , all values associated either with the interface @xmath216 or @xmath217 , @xmath474 and @xmath469 are defined by exps . ( [ g_a ] ,  [ cpa_50 ] ) . from ( [ d ] ,  [ gamma_eq ] ) it follows that @xmath475 using these expressions , ward identity can be written in the form : @xmath476_{\alpha\beta } + \delta_{\alpha\beta}|g^{\alpha\alpha}|^2 \bigr\ } { \rm im}\,\sigma^{\beta\beta } = 0.\ ] ] using ( [ t_sd ] ) , we then can find that @xmath477 where @xmath478 the same expessions for @xmath479 and @xmath480 follow directly from the cpa equations ( [ cpa_50 ] )  that finishes the proof . in this appendix we will derive the cpa equations for our particular case using the augmented - space formalism ( asf ) @xcite . as was described previously , we assume the f / o interface to be a random binary alloy of the type a@xmath9b@xmath87 , where a are ions of the ferromagnet and b are impurities . following the asf , we associate each random variable @xmath89 and @xmath274 with the self - conjugate operators @xmath481 and @xmath482 , respectively , which are determined in the auxiliary 2-dimensional vector space @xmath483 such a way , that the spectrum of these operators coincides with the set of possible values of random variables . for the sake of clarity , hereafter , the sign `` tilda '' is ascribed to any operator acting on the auxiliary space . we also define the orthonormalized basis @xmath484 , where @xmath485 or @xmath486 , which are eigenvectors of @xmath481 and @xmath482 , so that @xmath487 according to that definition , @xmath481 and @xmath482 commutate with each other . let now @xmath488 be a function or an operator of random variables @xmath89 and @xmath274 . then , the operator in the auxiliary space @xmath483 , associated with function @xmath489 , is defined as @xmath490 and according to  ( [ phi ] ) , e.g. @xmath491 is a value of @xmath489 , if the site @xmath270 is occupied by an ion a. one can introduce another orthonormal basis in @xmath483 @xmath492 so that the operators @xmath481 and @xmath482 in this representation are written as @xmath493 here @xmath92 and @xmath494 are the concentration of a - type ions ( ferromagnet s ions ) and b - type ions ( impurities ) on the interface , respectively , and @xmath495 then one can prove that the average value of @xmath489 is given by @xmath496 . together with eq . ( [ operators ] ) this property is the way to evaluate the average of any given operator , depending on the random variables , and we can apply this method to average the @xmath281-matrix  ( [ t_matrix ] ) .    following the general scheme of the asf , outlined above , the random effective potential @xmath280 is associated with the operator @xmath497 acting in the augmented vector space @xmath498 , where @xmath499 denotes the 4-dimensional space of orbital ( @xmath500 ) and spin ( @xmath501 ) electron degrees of freedom . in accordance with  ( [ v_gamma ] ) , ( [ v_spin ] ) and ( [ operators ] ) this operator has the form @xmath502 where @xmath503 is defined by  ( [ self_e ] ) , operators @xmath504 are defined similar to  ( [ v_gamma ] ) , and other operators are given by @xmath505    let us also introduce the nonrandom averaged propagator acting in the augmented space @xmath506 and associated with potential @xmath507 ( [ u ] ) the augmented scattering @xmath281-matrix @xmath508 its projection onto the zero - level @xmath509 of the augmented space @xmath510 coincides with the average from the `` physical '' random @xmath281-matrix  ( [ t_matrix ] ) . the subsequent averaging over magnon degrees of freedom @xmath511 must vanish due to condition ( [ cpa_system ] ) .    to proceed further , let s introduce the electron propagator @xmath512 @xmath513 which is associated with the `` propagation '' of an electron on the first level of the augmented space in the potential @xmath514 . taking into account the explicit form of @xmath515 with respect to spin - up and spin - down subspaces , @xmath516 where operators @xmath517 and @xmath518 are @xmath519 one gets @xmath520^{-1}\hat g^{\uparrow } ,   \nonumber \\ \label{g_1 } \hat g_1^{\downarrow } & = & \left [ 1 - \hat g^{\downarrow } \left(\hat\gamma_1 - \sigma^{\downarrow } + \hat j_1\hat n_{-}\hat g_1^{\uparrow}\hat j_1 \right ) \right]^{-1}\hat g^{\downarrow } , \\ \hat g_1^{+ } & = & \hat g_1^{\uparrow}\hat j_1 \hat s_{- } \hat g_1^{\downarrow } , \quad \hat g_1^{- } = \hat g_1^{\downarrow}\hat j_1 \hat s_{+ } \hat g_1^{\uparrow } , \nonumber\end{aligned}\ ] ] where @xmath521^{-1 } \hat g^{\uparrow(\downarrow)},\ ] ] and @xmath522 , @xmath523 . the physical meaning of these formulae is rather transparent . the green s function @xmath524 corresponds to the propagation of the electron in a spin conserving part of the potential @xmath514 which is @xmath525 , while @xmath526 corresponds to scattering on the potential @xmath527 , renormalized with respect to the spin - conserving potential due to the interaction with surface magnons on the interface . coming back to the evaluation of scattering matrix element @xmath528 , let introduce the `` denominator '' @xmath529 , corresponding to the whole augmented potential @xmath497 @xmath530 again , using the technique of the inversion of a matrix in the block form and taking into account the elements @xmath507  ( [ u ] ) with respect to the auxiliary space @xmath483 , the blocks of @xmath529 can be expressed in terms of the propagator @xmath512 as follows @xmath531^{-1 } ,   \nonumber \\ \hat d_{10 } & = & \hat g_1 \hat\delta \hat d_{00 } , \quad \hat d_{01 } = \hat g_0 \hat\delta \hat d_1 ,   \\ \hat d_{11 } & = & \left(1 + \hat g_0 \hat\delta \hat g_1 \hat\delta   \right)\hat d_{00 } , \nonumber\end{aligned}\ ] ] where we define the propagator @xmath532 , @xmath533^{-1 } \hat g = \left ( \begin{array}{cc } \hat g_0^{\uparrow } & \hat g_0^{+ } \\ \hat g_0^{- } & \hat g_0^{\downarrow } \end{array } \right),\ ] ] corresponding to the propagation of the electron in the effective potential @xmath534 , where @xmath535 the potential @xmath536 can be regarded as a renormalization of the `` virtual '' crystal potential @xmath537 of the zero level of the augmented space , representing the average of the random potential on a site . the renormalization comes from the `` interaction '' @xmath538 in the auxiliary space with the first level , being described by the propagator @xmath512 . using the explicit form of @xmath538 , @xmath539 one can write down the elements @xmath540 and @xmath541 : @xmath542 @xmath543 one can regard @xmath540 as the effective spin - conserving potential , taking into account the effects of a disorder , and @xmath541  as the renormalized electron - magnon interaction . finally , the simple algebra using  ( [ u ] ) ( [ den ] ) , yields @xmath544^{-1 } \nonumber \\ & = & \left ( \begin{array}{cc } \hat t_0^{\uparrow } & \hat t_0^{+}\hat s_{- } \\ \hat t_0^{-}\hat s_{+ } & \hat t_0^{\downarrow } \end{array } \right ) . \nonumber\end{aligned}\ ] ] thus , we have obtained the logical result , that the @xmath281-matrix , averaged over configurations , corresponds to scattering on the effective potential @xmath534 . to evaluate the spin - conserving @xmath545 and spin - flip @xmath546 parts of scattering matrix , we introduce the propagators @xmath547^{-1}\hat g^{\uparrow(\downarrow ) } , \label{g0_up_down}\ ] ] corresponding to the potentials @xmath548 and define the `` denominator '' @xmath549^{-1 } = \left ( \begin{array}{cc } \hat d_0^{\uparrow } & \hat d_0^{+ } \\ \hat d_0^{- } & \hat d_0^{\downarrow } \end{array } \right).\ ] ] related with the effective potential @xmath550 . the elements of @xmath551 and propagator @xmath532 can be expressed via @xmath552 using the form of potential @xmath536  ( [ w_eff ] ) , namely @xmath553^{-1}\hat g^{\uparrow } ,   \nonumber \\ \nonumber \hat g_0^{\downarrow } & = & \left [ 1 - \hat g^{\downarrow } \left(\hat w^{\downarrow } - \sigma^{\downarrow } + \hat j_{-}\hat n_{-}\hat g_0^{\uparrow}\hat j_{+}\right ) \right]^{-1}\hat g^{\downarrow } , \\ \hat g_0^{+ } & = & \hat g_0^{\uparrow}\hat j_{+}\hat s_{- } g_0^{\downarrow } , \quad \hat g_0^{- } = \hat g_0^{\downarrow}\hat j_{-}\hat s_{+ } g_0^{\uparrow } ,   \label{g_0}\end{aligned}\ ] ] and @xmath554^{-1 } , \nonumber\\ \hat d_0^{\downarrow } & = & \left [ 1 - \hat g^{\downarrow } \left(\hat w^{\downarrow } - \hat \sigma^{\downarrow } + \hat j_{-}\hat n_{-}\hat g_0^{\uparrow}\hat j_{+ } \right ) \right]^{-1 } , \nonumber \\ \hat d_0^{+ } & = & \hat g_0^{\uparrow}\hat j_{+ } \hat s_{- } d_0^{\downarrow } , \quad \hat d_0^{- } = \hat g_0^{\downarrow}\hat j_{- } \hat s_{+ } d_0^{\uparrow } ,   \label{d_0}\end{aligned}\ ] ] where `` denominators '' @xmath555 are given by @xmath556^{-1}.\ ] ] now , using  ( [ g_0 ] ) and ( [ d_0 ] ) and taking into account the obvious relation @xmath557 , one finds @xmath558 according to the asf , @xmath559 and @xmath545 represent the configurationally averaged quantities , and after the averaging over magnon degrees of freedom one must get @xmath560 and @xmath561 . it means that due to the cpa self - consistency conditions the averaged scattering @xmath281-matrix must vanish and averaged electron propagator has to be equal to the effective green s function  ( [ g_int ] ) , determined by the self - energy operator @xmath503 . carrying out the averaging procedure over magnon degrees of freedom the similar way as it was proposed earlier in sec . iic , it is possible to satisfy both of these condition if one assumes that @xmath562 we have pointed out the explicit dependence of matrices @xmath563 , @xmath564 and @xmath565 on the average magnon number @xmath270 ( eq .  [ n_t ] ) . this dependence is assumed to be the same as it comes from the initial definition of operators @xmath566 , @xmath541 , and @xmath552 as functions on @xmath309 [ eqs . ( [ w_up_down])([g0_up_down ] ) ] , that is the consequence of the adopted approximate averaging procedure . the matrices @xmath566 , @xmath541 and @xmath552 functionally depend on @xmath567 . due to that , the system  ( [ cpa_iter ] ) represents the alternative to eq . ( [ cpa_system ] ) form of the cpa conditions and it can be simply solved by means of successive numerical iterations . j.  s.  moodera , l.  r.  kinder , t.  m.  wong , and r.  meservey , phys . * 74 * , 3273 ( 1995 ) ; j.  s.  moodera , l.  r.  kinder , j.  appl . phys .  * 79 * , 4724 ( 1996 ) ; j.  s.  moodera , j.  nowak , and r.  j.  m.  van de veerdonk , phys .  rev . 80 * , 2941 ( 1998 )                                        t.  l.  monchesky , a.  enders , r.  urban , j.  f.  cochran , b.  heinrich , w.  wulfhekel , m.  klaua , f.  zavaliche , and j.  kirschner , in _ the physics of low dimensions _ , edited by jose luis moran lopez ( plenum , new york , 2000 )    o.  wunnicke , n.  papanikolaou , r.  zeller , p.  h.  dederichs , v.  drchal , and j.  kudrnovsky , communication on the _ european graduate school on condensed matter ( egscm prague01 ) _ ( unpublished ) ; m.  zwierzycki , k.  xia , p.  h.  kelly , g.  e.  bauer , i.  turek , ibid.(unpublished )
we propose the two - band @xmath0-@xmath1 model to describe theoretically a diffuse regime of the spin - dependent electron transport in magnetic tunnel junctions ( mtj s ) of the form f / o / f where f s are 3@xmath1 transition metal ferromagnetic layers and o is the insulating spacer . 
 we aim to explain the strong interface sensitivity of the tunneling properties of mtj s and investigate the influence of electron scattering at the nonideal interfaces on the degradation of the tmr magnitude . 
 the generalized kubo formalism and the green s functions method were used to calculate the conductance of the system . 
 the vertex corrections to the conductivity were found with the use of `` ladder '' approximation combined with the coherent - potential approximation ( cpa ) that allowed to consider the case of strong electron scattering . 
 it is shown that the ward identity is satisfied in the framework of this approximation that provides the necessary condition for a conservation of a tunneling current . 
 based on the known results of ab - initio calculations of the tmr for ballistic junctions , we assume that exchange split quasi - free  @xmath0-like electrons with the density of states being greater for the majority spin sub - band give the main contribution to the tmr effect . 
 we show that , due to interfacial inter - band scattering , the tmr can be substantially reduced even down to zero value . 
 this is related to the fact that delocalized quasi - free electrons can scatter into the strongly localized @xmath1  sub - band with the density of states at the fermi energy being larger for minority spins compared to majority spins . 
 it is also shown that spin - flip electron scattering on the surface magnons within the interface leads to a further decrease of the tmr at finite temperature .