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non - equilibrium current noise in mesoscopic structures is a consequence of the discreteness of the charge carriers ( for reviews , see refs . @xcite ) . for conductors with open channels the fermionic statistics of electrons results in a suppression of shot noise below the classical schottky limit @xcite . this was first noted by khlus @xcite and lesovik @xcite for single channel conductors . subsequently , bttiker generalized this suppression for many - channel conductors @xcite . mesoscopic conductors are often probed by two or more leads . the quantum statistics induces cross - correlations between the currents in different terminals . since these cross - correlations vanish in the classical limit , even their sign is not obvious a priori . using only the unitarity of the scattering matrix , bttiker proved that cross - correlations for non - interacting fermions are _ always negative _ for circuits with leads maintained at constant potentials @xcite . note that this also holds in the presence of a magnetic field . it has also been found that an interacting paramagnetic dot shows negative cross - correlations in the absence of a magnetic field @xcite . spin - dependent cross - correlations in a non - interacting 4-terminal spin valve were studied @xcite and found to be negative . on the experimental side negative cross - correlations were measured by henny _ . _ @xcite and oliver _ et al . _ @xcite in mesoscopic beam splitters . several ways to produce positive cross - correlations in fermionic systems have been proposed ( see e.g. @xcite for a recent review ) . among these possibilities are sources which inject correlated electrons @xcite and finite - frequency voltage noise @xcite . the question of the existence of intrinsic mechanisms , i.  e. due to interactions occuring in the beam - splitter device itself , has been answered positively by us @xcite . surprisingly , a simple quantum dot connected to ferromagnetic contacts can lead to positive cross - correlations due the so - called _ dynamical spin - blockade_. simply speaking , up- and down - spins tunnel through the dot with different rates . in the limit where the coulomb interaction prevents a double occupancy of the dot , the spins which tunnel with a lower rate modulate the tunneling of the other spin - direction , which leads to an effective _ bunching _ of tunneling events . in a three terminal geometry with one input and two outputs , this results in positive cross - correlation between the two output currents . independently , sauret and feinberg proposed a slightly different setup of a ferromagnetic quantum dot , which also produces positive cross - correlations @xcite . experimentally , it is more difficult to fabricate quantum dots with ferromagnetic leads . however , quantum dots with paramagnetic leads have shown to exhibit spin - dependent transport . a magnetic field lifts the spin - degeneracy and a spin - polarized current with nearly 100% efficiency can be created @xcite . in this letter , we will address the current correlations in a few - electron quantum dot connected to three paramagnetic leads . we will show below that positive cross - correlations can be produced in this device simply by applying a _ magnetic field_. furthermore , this system also shows a super - poissonian shot noise .    to arrive at these conclusions we consider a quantum dot with one orbital energy level @xmath0 connected to three terminals by tunnel contacts . the junctions are characterized by bare tunneling rates @xmath1 ( @xmath2 ) and capacitances @xmath3 . we assume that a magnetic field @xmath4 is applied to the dot , which leads to a zeeman splitting of the level according to @xmath5 , where @xmath6 is the bohr magneton . the double occupancy of the dot costs the charging energy @xmath7 , with @xmath8 . the energy spacing to the next orbital is @xmath9 . we will assume @xmath10 according to these inequalities , the dot can be only singly occupied and we have to take into account only one orbital level .    in the sequential - tunneling limit @xmath11 , the time evolution of the occupation probabilities @xmath12 of states @xmath13 is described by the master equation : @xmath14 where @xmath15{ccc}% -\gamma_{\uparrow}^{-}-\gamma_{\downarrow\uparrow } & \gamma_{\uparrow \downarrow } & \gamma_{\uparrow}^{+}\\ \gamma_{\downarrow\uparrow } & -\gamma_{\downarrow}^{-}-\gamma_{\uparrow \downarrow } & \gamma_{\downarrow}^{+}\\ \gamma_{\uparrow}^{- } & \gamma_{\downarrow}^{- } & -\gamma_{\uparrow}% ^{+}-\gamma_{\downarrow}^{+}% \end{array } \right ]   \,.\label{matrixm}%\ ] ] the rate for an electron to tunnel on / off the dot ( @xmath16 ) through junction @xmath17 is given by @xmath18)$ ] , where @xmath19 and @xmath20 . here , we took the fermi energy @xmath21 for lead 2 as a reference . the total tunneling rates are @xmath22 and @xmath23 . spin flips on the dot are described by rates @xmath24 , which obey the detailed balance rule @xmath25 . from eq . ( [ masterequation ] ) the stationary occupation probabilities @xmath26 are @xmath27 and @xmath28 . these probabilities can be used to calculate the average value @xmath29 of the tunneling current @xmath30 through junction @xmath17 as @xmath31 where @xmath32 is the state of the dot after the tunneling of an electron with spin @xmath33 in the direction @xmath34 , i.  e. , @xmath35 and @xmath36 . the frequency  spectrum of the noise correlations can be defined as @xmath37 where @xmath38 is the deviation from the average current in terminal @xmath39 . it can be calculated using the method developed in refs . @xcite as:@xmath40 where the first term is the schottky noise produced by tunneling through junction @xmath17 , and @xmath41 with @xmath42 .    in the following we study the dot in a beam - splitter configuration , in which a bias voltage @xmath43 is applied between terminal 2 and terminals 1 and 3 . we consider the case @xmath44 , so that it is energetically more favorable for electrons to go from lead 2 to leads 1 and 3 . we will limit our discussion to the case in which the two zeeman sublevels are below the fermi energy at equilibrium ( _ i .  e. _ @xmath45 ) . the opposite case was discussed in ref . @xcite for a two - terminal dot . we are mostly interested in the total zero - frequency current noise @xmath46 and the cross - correlations @xmath47 between the two output leads . it is useful to define the fano factor @xmath48 and , correspondingly , @xmath49 .    , @xmath50 , @xmath51 , @xmath52 , @xmath53 , and different values of @xmath54 . the average current @xmath55 through lead @xmath56 is plotted in units of @xmath57 ; the voltage is in units of @xmath0 . the positions of @xmath58 and @xmath59 are indicated in dotted lines . , width=377 ]    in the following , we will first assume that @xmath60 . transport through the down level is energetically allowed for @xmath61 . however , for @xmath62 , the dot is blocked by an up spin , thus down spins can not cross the dot . around @xmath63 , the lower zeeman level is close to the fermi level of leads 1 and 3 , as represented by the level diagram in the lower right inset of fig . [ fig : current ] . the blockade of the dot is then partially lifted and transport through both levels is allowed . in this regime , we can write the tunneling rates as @xmath64 , @xmath65 , @xmath66 , @xmath67 , @xmath68 , and @xmath69 , where @xmath70)$ ] ranges from 0 to 1 with increasing voltage . furthermore , taking @xmath71 and @xmath72 , we find for the current @xmath73 for the fano factor @xmath74 and for the cross correlations @xmath75 we observe that the current increases with voltage ( i.e. with @xmath76 ) around the voltage step @xmath58 . note that this current is _ not spin - polarized _ because up and down spin have the same probability to enter the dot , regardless of what happens at the output . the fano factor @xmath77 and the cross - correlations @xmath78 deviate from their poissonian values depending on the applied voltage . our main results are that @xmath77 can be super - poissonian and @xmath78 positive for @xmath79 , as can be cleary seen from ( [ f2para ] ) or ( [ eq : c2para ] ) in the limit @xmath80 . these features are a consequence of dynamical spin blockade : up spins leave the dot with a rate smaller than down spins , leading to a bunching of tunneling events @xcite . in the limit @xmath81 , the fano factor is always sub - poissonian and the cross - correlations always negative . this is due to the fact  that the tunnel rates of up and down spins are equal , thus the zeeman splitting plays no role and the dot is equivalent to a simple quantum dot with a spin - degenerate level . in the limit @xmath82 , one could also expect the super - poissonian nature of @xmath77 and the positivity of @xmath78 to be lost since the transport is enabled only by thermally activated processes . however , below the voltage threshold @xmath58 , the fano factor tends to : @xmath83 which is always super - poissonian . if the coupling to terminal 2 dominates , i.  e. @xmath80 , the fano factor takes a maximal value of 3 . in the opposite limit @xmath84 , @xmath77 approaches the poisson limit of uncorrelated single charge transfer . it is interesting to note that a symmetric junction @xmath85 produces twice the poisson noise level . the cross - correlations in the same limit have the form @xmath86 in the three cases discussed above , the cross - correlations thus take the limiting values @xmath87 for @xmath84 , @xmath88 for @xmath85 and @xmath89 for @xmath90 . hence , both the super - poissonian nature of @xmath77 and the positivity of @xmath78 can persist for @xmath82 . even if the transport is enabled only by thermally activated processes , dynamical spin blockade already results in a correlated transfer of electrons .     of the total current as a function of voltage , for the same circuit parameters as in fig . [ fig : current ] . left panel : data for different values of @xmath91 and @xmath71 . right panel : effect of spin flip scattering for @xmath92 and different values of @xmath93 . the curves displayed in both panels are independent of the asymmetry between the output leads . ] we now turn to the discussion of the general results displayed in figs . ( [ fig : current])-([fig : cross ] ) , obtained from an exact treatment of the full master equation . fig . [ fig : current ] shows the full voltage dependence of the average current . as expected , the current shows a single step at @xmath94 @xcite . the step width is about @xmath95 , whereas its position varies only slightly with the asymmetry of the junctions ( the maximal variation is about @xmath96 ) . the left panel of figure [ fig : fano ] shows the voltage dependence of the fano factor in the absence of spin - flip scattering , for some values of @xmath97 . the divergence @xmath98 of the fano factor at zero voltage is simply a result of the dominating thermal noise in the limit @xmath99 . note that similarly to @xmath55 , the fano factor @xmath77 shows one single step at @xmath100 . the right panel of fig . [ fig : fano ] shows the effect of spin - flip scattering on @xmath101 for the case @xmath102 . for @xmath103 spin flips become effective when @xmath104 , see eq . ( [ matrixm ] ) . the sensitivity to @xmath93 thus increases with b. below @xmath59 , even smaller spin - flip rates suppress the super - poissonian noise because the dwell time of electrons on the dot is very long . far above @xmath58 , spin - flip scattering has no effect on the sub - poissonian noise .     between leads 1 and 3 as a function of voltage , for the same circuit parameters as in fig . [ fig : current ] and different values of @xmath54 . in all curves of the main frame @xmath105 the inset shows the effect of spin flip scattering for @xmath92 and different values of @xmath93 . right : @xmath78 as a function of voltage for different values of @xmath4 . the curves are shown for @xmath106 . a cross indicates the position of @xmath59 for each case.,title="fig:",width=272 ]   between leads 1 and 3 as a function of voltage , for the same circuit parameters as in fig . [ fig : current ] and different values of @xmath54 . in all curves of the main frame @xmath105 the inset shows the effect of spin flip scattering for @xmath92 and different values of @xmath93 . right : @xmath78 as a function of voltage for different values of @xmath4 . the curves are shown for @xmath106 . a cross indicates the position of @xmath59 for each case.,title="fig:",width=257 ]    the left panel of fig . [ fig : cross ] shows the voltage dependence of the cross - correlations factor @xmath78 between the two output terminals , for the same parameters as in fig . [ fig : fano ] . first , around the voltage threshold @xmath58 , we observe the features discussed above . the cross - correlations develop from a positive or negative level below @xmath58 depending on the ratio @xmath97 to the usual negative cross - correlations above @xmath58 , where the spin - splitting plays no role anymore . remarkably , in contrast to @xmath77 , the cross - correlations also show a step around the lower voltage threshold @xmath59 . this illustrates clearly that @xmath78 and @xmath77 are qualitatively different . the absence of the lower step for @xmath77 can be interpreted as a consequence of the unidirectionnality of tunneling through junction 2 . indeed , @xmath107 means that @xmath77 depends only on @xmath108 and @xmath109 [ see ( [ eq : current ] ) and ( [ eq : noise2 ] ) ] . now , for @xmath110 @xmath111 implies that the contribution of these terms is independent of @xmath43 . on the contrary , @xmath78 also depends on @xmath112 and @xmath113 with @xmath114 . for @xmath111 , these last terms depend strongly on @xmath115 which varies itself significantly with @xmath43 around @xmath59 . note that the absence of a step in @xmath77 implies a redistribution of the noise between @xmath116 , @xmath117 and @xmath118 when the threshold @xmath59 is crossed ( due to charge conservation , @xmath119 ) . the extra step of @xmath78 disappears for @xmath80 . in this limit , the cross - correlations display a single low voltage plateau @xmath89 , which is an upper bound for the two low voltages plateaux found in the general case . the inset in the left panel of fig . [ fig : cross ] shows the effect of spin - flip scattering on the cross - correlations . as expected , they suppress all spin - effects and the positive cross - correlations become finally negative . like for the fano factor , very small spin - flip scattering rates @xmath93 are already sufficient to modify @xmath78 for @xmath120 . since the positive cross - correlations found in this work are intimately related to the dynamical spin - blockade , we expect a strong dependence on the magnetic field . the right panel of fig . [ fig : cross ] shows the voltage dependence of @xmath78 around the step @xmath58 , for a fixed temperature and various magnetic fields . just below @xmath58 the limiting value of @xmath78 is determined by formula ( [ eq : cross_below_step ] ) . thus , for a constant voltage @xmath121 we predict a cross - over from negative to positive cross - correlations with increasing magnetic field . one can see a qualitative change in the curves , which can be understood by a gradual splitting of the voltage steps @xmath59 and @xmath58 . the lower step is at @xmath122 . as long as @xmath123 , the two voltage steps are indistinguishable however , positive cross - correlations are already expected for @xmath124 . for @xmath125 the two steps still overlap , resulting in a broad peak , whereas for the higher magnetic field @xmath126 the lower threshold at @xmath59 is outside the plotting region . the regime @xmath100 has the advantage that the current is not exponentially small ( c.f . [ fig : current ] ) and thus observable more easily in an experiment . for @xmath127 , the maximum value obtained for the cross - correlations is @xmath128 at @xmath129 , i.  e. @xmath130 in fig . [ fig : cross ] . with @xmath131 ghz , this corresponds to @xmath132a@xmath133s , a noise level accessible experimentally @xcite .    in conclusion we have studied current correlations for a three terminal quantum dot with unpolarized leads , placed in a magnetic field . below the voltage threshold @xmath58 , as a result of dynamical spin - blockade , the fano factor of the input current shows an interesting super - poissonian behavior and the cross - correlations in the two output leads can be positive . at higher voltages the fano factor becomes sub - poissonian and the cross - correlations negative , as usual . the effect we predict should be observable in semiconductor quantum dots of ref .
we investigate current fluctuations in a three - terminal quantum dot in the sequential tunneling regime . 
 dynamical spin blockade can be induced when the spin - degeneracy of the dot states is lifted by a magnetic field . 
 this results in super - poissonian shot noise and positive zero - frequency cross - correlations . 
 our proposed setup can be realized with semiconductor quantum dots .