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many devices exhibit negative differential conductance ( @xmath3 ) as multiple quantum wells , double barrier , double quantum dots , etc ndc1,ndc2,ndc3,ndc4,dqd1,dqds . the @xmath3 has applications as amplifiers and oscillators in the microwave , mm - wave and terahertz frequency range . extensive experimental and theoretical investigation have been devoted to the study of the @xmath0-@xmath1 characteristics and @xmath3 phenomenon in double quantum dot molecules @xcite . moreover , there is a wide literature on transport through double quantum dots ( @xmath4 ) in different geometries , e.g , @xmath4 in series and in parallel @xcite . two aspects of electronic transport through quantum dots have attracted great attention in the last years , the coulomb blockade effect and the kondo effect . recently kondo effect has been studied in side attached @xcite and parallel quantum dots @xcite . recent electron transport experiments showed that kondo and fano resonances occur simultaneously @xcite . multiple scattering of traveling electronic waves on a localized magnetic state are crucial for the formation of both resonances . the condition for the fano resonance is the existence of two scattering channels : a discrete level and a broad continuum band @xcite . an alternative configuration consists of a double quantum dot molecule side coupled to a perfect quantum wire ( qw)@xcite . this structure is reminiscent of the t - shaped quantum wave guides @xcite . in this case , the qds act as scattering centers in close analogy with the traditional kondo effect @xcite . although the electron - electron interaction does play an important role in many systems that exhibit negative differential conductance , as for instance producing bi - stabilities in the current , it is not the driving force of the negative conductance itself . in this work instead , we study a system in the kondo regime with a @xmath0-@xmath1 characteristics that possesses a @xmath3 region that is induced by the electronic correlation itself , tuned by the external potential and the parameters of the system . the system is constituted by a quantum dot molecule side coupled to a quantum wire , as shown in fig . [ f : esquema ] . we use the finite-@xmath2 slave boson mean - field approach , which was initially developed by kotliar and ruckenstein @xcite and used later by bing dong and x. l. lei to study the transport through coupled double quantum dots connected to leads dl01 . this approach enforces the correspondence between the impurity fermions and the auxiliary bosons to a mean - field level to release the @xmath5 restriction . this allows to treat non - perturbatively the dot - lead coupling for an arbitrary strength of the coulomb interaction @xmath2 @xcite .    in a previous work we study this system in a thermodynamic equilibrium situation . we found that in the weak interaction regime , when the direct antiferromagnetic interaction between the dots is less than the kondo temperature associated to the internal dot , the transmission spectrum shows a structure with two anti - resonances localized at the renormalized molecular energies of the double quantum dot @xcite . the ldos of the system shows that when the kondo correlations are dominant the system is in a two stage kondo regime with two different temperatures @xmath6 , each one associated to a dot .    in the present paper we study this system under the effect of a finite external field , which takes it out of the thermodynamic equilibrium , modifying the kondo regime and even destroying it , for enough large fields . this process has fundamental consequences on the transport properties of the system , and in particular , creates a remarkable @xmath3 in the @xmath0-@xmath1 characteristics . this @xmath3 phenomenon can be understood realizing that the applied bias destroys the lower temperature kondo state of the external dot modifying the capability of the other to interfere on the current that goes along the system . let us consider a quantum dot molecule side coupled to a perfect quantum wire ( qw ) ( see fig .  [ f : esquema ] ) . we describe it by the two - impurity anderson hamiltonian . each dot has a single level energy @xmath7 ( with @xmath8 ) and equal intra - dot coulomb repulsion @xmath2 . the side attached quantum - dot molecule is coupled to the qw with coupling @xmath9 . the qw sites have zero local energies and a hopping parameter @xmath10 . ( 220,100)(0,0 ) ( 40,70)(1,0)140 ( 110,70)(0,-1)60 ( 110,10 ) ( 110,30 ) ( 100,30)(0,0)@xmath11 ( 100,10)(0,0)@xmath12 ( 120,50)(0,0)@xmath9 ( 120,20)(0,0)@xmath13 ( 50,75)(0,0)[b ] ( 70,75)(0,0)[b ] ( 90,75)(0,0)[b]-1 ( 110,75)(0,0)[b]0 ( 130,75)(0,0)[b]1 ( 150,75)(0,0)[b ] ( 170,75)(0,0)[b ] ( 110,90)(0,0)[b]qw ( 150,20)(0,0)[l]dqd ( 80,0)(60,40 ) ( 10,70)(0,0)[r]@xmath14 ( 210,70)(0,0)[l]@xmath15 ( 40,70)(0,0)[r]@xmath16 ( 180,70)(0,0)[l]@xmath16 ( 50,70)(20,0)7 ( 60,65)(20,0)6(0,0)[t ] ( 0,50)(40,70)(0,90 ) ( 220,50)(180,70)(220,90 )    the corresponding model hamiltonian is ,    @xmath17   + \sum_{l=1,2,\sigma } \left [ \left ( \varepsilon_{l}+\frac{u}{2}\hat{n}_{l,-\sigma}\right ) \hat{n}_{l,\sigma }        \right ] \,\ ] ]    where @xmath18 ( @xmath19 ) is the creation ( annihilation ) operator of an electron with spin @xmath20 at the @xmath21-th site of the quantum wire ; @xmath22 ( @xmath23 ) is the creation ( annihilation ) operator of an electron with spin @xmath20 in the l - th qd , @xmath24 is the corresponding number operator , @xmath13 is the hopping matrix element between the dots and @xmath7 correspond to the energy of the local states at the dots . to find the solution of this correlated fermions system for finite u , we appeal to an analytical approach where , generalizing the infinite-@xmath2 slave - boson approximation @xcite the hilbert space is enlarged at each site , to contain in addition to the original fermions a set of four bosons @xcite represented by the creation ( annihilation ) operators @xmath25 ( @xmath26 ) , @xmath27 ( @xmath28 ) , and @xmath29 ( @xmath30 ) for the @xmath31-th dot . they act as projectors onto empty , single occupied ( with spin up and down ) and doubly occupied electron states , respectively . then , each creation ( annihilation ) operator of an electron with spin @xmath32 in the l - th qd , is substituted by @xmath33 ( @xmath34 ) where :    @xmath35    as the problem is solved adopting the u - finite slave boson mean field approximation ( sbmfa ) , the operator is chosen to reproduce the correct @xmath36 limit in the mean - field approximation without changing neither the eigenvalues nor the eigenvector.@xcite    the constraint , i.e. , the completeness relation @xmath37 and the condition among fermions and bosons @xmath38 , is incorporated with lagrange multipliers @xmath39 and @xmath40 into the hamiltonian . also in the mean - field approximation all the boson operators are replaced by their expectation values @xmath41 and @xmath42 which can be chosen , without loss of generality , as real numbers . the hamiltonian in this new and enlarged hilbert space , is , @xmath43 , where    @xmath44    depends explicitly only upon the boson expectation values ( @xmath45 and equivalently for the others operators ) and the lagrange multipliers . the hamiltonian @xmath46 can be written ,    @xmath47    the tight - binding hamiltonian depends implicitly on the boson expectation values through the parameters : @xmath48 , @xmath49 , where @xmath50is the value assumed by the operator @xmath51 when the four boson operators are substituted by their mean values in equation [ e : eq2 ] . @xmath52    the boson operator expectation values and the lagrange multipliers are determined by minimizing the energy @xmath53 with respect to these quantities . it is obtained in this way a set of nonlinear equations for each quantum dot , relating the expectation values of the four bosonic operators , the three lagrange multipliers and the electronic expectation values ,    [ e : minimizing ] @xmath54    where in the absence of external magnetic field the solutions are spin independent .    to obtain the electronic expectation values @xmath55 , the hamiltonian , @xmath56 is diagonalized . their stationary states can be written as    @xmath57    where @xmath58 and @xmath59 are the probability amplitudes to find the electron at the site @xmath60 and at the @xmath31-th qd respectively , with energy @xmath61 . as we study the paramagnetic case the spin index is neglected . the amplitudes @xmath58 and @xmath59 obey the following linear difference equations    [ e : diferencias ] @xmath62    in order to study the solutions of eqs .  , we assume that the electrons are described by an incident , a reflected and a transmitted plane waves with unitary , @xmath63 and @xmath64 amplitudes , respectively.@xcite that is ,    [ e : solut ] @xmath65    inserting eqs .   into eqs . , we get an inhomogeneous system of linear equations for @xmath64 , @xmath63 , @xmath58 and @xmath66 , leading to the following expression in equilibrium ( @xmath67 )    @xmath68    where the bonding ( @xmath69 ) and antibonding energy ( @xmath70 ) are defined by @xmath71 and @xmath72 is the renormalized coupling between the double quantum - dot and the quantum wire and @xmath73 is the density of states of the leads at the fermi level . in spite of the apparent simplicity of the expression , it is necessary to remember that the quantities @xmath74 and @xmath75 implicitly depend on the expectation values of the boson and fermion operators .    the transmission probability is given by @xmath76 ,    @xmath77^{2}}{[(\omega -\tilde{\varepsilon}_{-})(\omega -\tilde{% \varepsilon}_{+})]^{2}+[(\omega -\tilde{\varepsilon}_{d2})\tilde{\gamma}]^{2}% } \,.\ ] ]    from the amplitudes @xmath78 and @xmath79 we obtain the local density of states ( ldos ) at the quantum dot @xmath31 ( with @xmath80 ) . @xmath81^{2}+[(\omega -\tilde{\varepsilon}_{d2})\tilde{\gamma}]^{2}}\ , , \\ \rho _ { 2 } & = \frac{1}{\pi } \frac{\tilde{\gamma}\tilde{t}_{c}^{2}}{[(\omega -% \tilde{\varepsilon}_{-})(\omega -\tilde{\varepsilon}_{+})]^{2}+[(\omega -% \tilde{\varepsilon}_{d2})\tilde{\gamma}]^{2}}\,.\end{aligned}\ ] ]    in the nonequilibrium case , we suppose a finite source - drain biased with a symmetric voltage drop . the incident electrons from the left side ( @xmath82 ) are in equilibrium with the thermodynamical potential @xmath83 and those from the right side ( @xmath84 ) with @xmath85    once the amplitudes @xmath86 and @xmath87 are known , the electronic expectation values are obtained from ,    @xmath88    where @xmath89 the current given by , @xmath90    where @xmath91 it is the fermi function for incident electrons from the @xmath92 side and the sum on @xmath93 is taken up to the maximum value @xmath94     ( solid line ) , @xmath95 for various values of on site energy , @xmath96(solid line ) , @xmath97(dotted line ) @xmath98(dashed line ) ]     ( solid line ) , @xmath99 for various values of the gate voltage,@xmath95(solid line),@xmath100(dotted line ) and @xmath101(dashed line ) ]    the quantities @xmath102 , the energies of the local states at the dots , are taken to be equal to @xmath103 , where @xmath104 is the gate voltage applied to the quantum dots . in order to obtain a more clear insight , we study first the system in thermodynamical equilibrium . the dos at each dot of the quantum - molecule is shown in figure [ f : fig2 ] , for various values of @xmath2 , for the case @xmath105 and @xmath106 . as u increases , @xmath107 , the system passes from the intermediate valence regime into a kondo regime . this process is clearly seen to occur for qd2 where the resonance shifts and becomes sharper as the system enters into the kondo regime . the same process takes place for the qd1 although the sharpening is less accentuated . it is noticeable that as qd2 develops a kondo resonance , a dip appears at the fermi energy in the ldos of qd1 . this is produced because the spin of the local electron at qd2 is kondo correlated with the conduction band spins through the mediation of the intermediate dot , that is as well at the kondo regime . this coupling creates a sharp peak at the ldos of qd2 with width @xmath108 and a depletion of this same width at the fermi level of the kondo peak of width @xmath109 corresponding to qd1 . this result implies the existence of a two stage kondo regime that appears in this system in the weak interacting limit , when @xmath110 @xcite , being @xmath109 and @xmath108 the kondo temperatures associated to each dot . it is the depletion of the ldos of qd1 at the fermi level that permits the transmission to be unity . in this case , although the side attached qd1 , is in the kondo regime , it does not provide an alternative path for the conducting electrons . as a consequence , there is no destructive interference because there is only the direct path available for them .    in figure [ f : fig3 ] we show the ldos of each qd for different values of @xmath111 in the two stage kondo regime . as expected , the peaks are pinned at the fermi energy independent of the values of the @xmath104 , a clear signal of the kondo regime . here it is clearly seen at the fermi energy , the antiresonance at it qd2 and the concomitant resonance at the qd1 . -@xmath1 characteristics for @xmath112 ( solid line ) and @xmath113 ( dashed line ) for various values of @xmath2 with @xmath114      the previous discussion was restricted to the thermodynamical equilibrium situation and was presented with the purpose of clarifying the concepts involved . when an external potential is applied the scenario changes completely . the physics of this new situation can be explained analyzing the current and the differential conductance @xmath115 , two significant and experimentally measurable quantities , as a function of the applied field . figure [ f : fig4 ] displays the @xmath0-@xmath1 characteristics ( solid line ) for two values of @xmath116 and different values of @xmath2 , with @xmath117 . for @xmath98 , the @xmath0-@xmath1 characteristics shows a plateau that is related to the fano antiresonances in the transmission spectrum . when the applied potential is of the order of the interdot interaction , the transmission , due to the fano destructive interference , is almost zero in the bonding and antibonding regions giving no additional contribution to the total current as the applied potential is increased . this is the origin of the plateau behavior , shown in the figure [ f : fig4 ] , when u=0 . as @xmath2 is increased , a negative differential conductance appears in the @xmath0-@xmath1 characteristics that gets more important as @xmath118 is augmented . ( solid line ) and @xmath119 ( dashed line ) , for various values of @xmath2 with @xmath117 ]    figure [ f : fig5 ] depicts the differential conductance for the same parameters of fig.[f : fig4 ] . for @xmath98 it reflects essentially the transmission spectrum . as @xmath2 is increased the differential conductance becomes negative in a region of the applied potential , reflecting the fact that the @xmath3 is a consequence of the coulomb interaction . it is a small effect in the fluctuating valence regime @xmath121 and develops completely in the kondo regime @xmath122 , increasing with @xmath123    in order to get insight into these results it is convenient to write the transmission ( eq.11 ) as the superposition of a fano and a breit - wigner line shapes , a good approximate expression for large values of @xmath2@xcite . the results is ,    @xmath124    where @xmath125 and @xmath126 corresponds to the two kondo temperatures @xmath127 and @xmath128    an analytical expression can be obtain for the current by integrating over @xmath129 the transmission probability given in eq .  . @xmath130 ,   \label{current}\ ] ]    we identify each term of the eq . as follows . the first term in the right side is the contribution arising from an ideal one dimensional conductor . the second term arises from the kondo - fano state with temperature @xmath109 giving rise to a quasi plateau for the current and almost zero differential conductance when @xmath131 . the third term arises from the kondo state weakly coupled to the wire and it is responsible for the rapid increase of the current in the region of small applied potentials . it is important to emphasize that in this expression the quantities @xmath132 and @xmath133 are functions of @xmath1 obtained through the self - consistent calculation presented above . the case of @xmath134 is shown in the inset of fig.[f : fig6 ] . it is clear that increasing @xmath1 , both the kondo temperature of the external dot and the current reduce to zero simultaneously with the disappearance of the @xmath3 phenomenon . as mentioned above , this behavior can be understood realizing that the effect of the external quantum dot is to reduce the intermediate dot interference effect on the current circulating along the leads . this role is exercised by the external quantum dot as far as it is at the kondo regime . when the external potential is large enough as to disrupt its kondo ground state ( @xmath135)the interference is reestablished and the current goes to zero . as @xmath108 increases with @xmath13 , for larger values of @xmath13 this disrupting process requires bigger values of the applied potential as shown in fig.[f : fig4 ] . when @xmath136increases still further it destroys as well the kondo regime of the intermediate quantum dot and the current rises as depicted in this same figure . this seems to be the behavior of a two stage kondo system under the effect of an external potential . the kondo regime of the outside quantum dot , not directly connected to the continuum , depends upon the kondo effect of the intermediate quantum dot . as its kondo temperature is lower , its kondo ground state is disrupted by lower values of @xmath1 than the other dot . this process manifests in the transport properties by the appearance of a @xmath3 region in the current . the fig.[f : fig6 ] displays a comparison between the @xmath0-@xmath1 characteristics of the numerical calculation and the approximation ( eq . ( 17 ) ) for @xmath137 and @xmath138 . the approximation over estimates the peak of the current however , qualitatively , it maintains the form . deriving the current in eq.([current ] ) we obtain    @xmath139 \right\ } .\ ] ]    @xmath1 characteristics between the numerical calculation ( solid line)and the approximation ( eq.4)(dashed line ) for @xmath140 , @xmath112 with @xmath141 . @xmath142 in the inset ]    in the above equation , the last term is responsible for the negative differential conductance . this term is negative because @xmath143 . this expression is able to reproduce very well the results of the differential conductance shown in fig.[f : fig5 ] .    regarding the observability of the @xmath3 in the side - coupled double quantum dot molecule , we consider the value of @xmath144 given by sato et al . sato , @xmath145 . for @xmath146 and @xmath147 the maximum and minimum of the current are @xmath148 na and @xmath149 na , respectively , giving a peak to valley ratio of 4:1 . the lower kondo temperature of the external dot for these same parameters is of the order of 2 k. these values of the current and temperature are well above the experimental limits of present day techniques . in summary we have studied the nonequilibrium transport through a double quantum dot molecule side - coupled to a quantum wire using the finite-@xmath2 slave boson mean field approach at @xmath150 as a function of the parameters that define the system . we find that the @xmath0-@xmath1 characteristics shows a remarkable @xmath3 , different from case reported in the literature , is induced by the electronic correlation . this @xmath3 behavior is a consequence of the properties of a two stage kondo system under the effect of an external applied potential that takes the system out of equilibrium when the applied potential is large enough to destroy the kondo regime characterized by the lower kondo temperature . we would like to thank m.a . davidovich for her very useful comments of the manuscript . this work was partially financial supported by conicyt / programa bicentenario de ciencia y tecnologia ( cenava , grant act27 ) , ciam ( conicyt - cnpq - nsf ) and fondecyt under grant 1040385 . thank u.a . ( pei-1305 - 04 ) . e.v.a . acknowledges support from the brazilian agencies , cnpq and faperj .
electron tunneling through a two stage kondo system constituted by a double quantum - dot molecule side coupled to a quantum wire , under the effect of a finite external potential is studied . 
 we found that @xmath0-@xmath1 characteristic shows a negative differential conductance region induced by the electronic correlation . 
 this phenomenon is a consequence of the properties of the two stage kondo regime under the effect of an external applied potential that takes the system out of equilibrium . 
 the problem is solved using the mean - field finite-@xmath2 slave - boson formalism .