content_image/387034.jpg
<img src='content_image/387034.jpg'>

On the other hand, in the N = 2 Coulomb blockade 

valley, the d iagonal and off-diagonal elements of A p =0 are rather po sitive and always smaller than π Γ. In con- sequence, the full spectral function is properly normal- ized A p =0 = A p =0 11 + A p =0 22 + 2 A p =0 12 = 2 π Γ for ω = 0, see Fig. 3(d) . Furthermore, the full spectral function clearly exhib its the resonance corresponding to the un- derscreened Kondo effect in the Coulomb blockade valley N = 2 ( ε < − 0 . 2), while for N = 1 ( ε > − 0 . 2) the S = 1 2 Kondo effect develops. 

The spin a nd level-resolved normalized spectral func- tions π Γ A p = 0 . 4 jj ′ σ in the case of ferromagnetic lead with p = 0 . 4 are displayed in Fig. 4. Now the spectral func- tion is differe nt for each spin component due to the spin- dependence of the coupling parameters. Furthermore, when the lea d is ferromagnetic, both Kondo resonances become com pletely suppressed. This is directly associ- ated with sp in-dependent level renormalization, which lifts the spin degeneracy in the dot, suppressing the spin- flip cotunne ling processes driving the Kondo effect. In other words, the ferromagnetic lead exerts an effective exchange fie ld on the dot, which destroys the Kondo ef- fect. There is only a very narrow resonance at ε = − 0 . 6 when the sp in splitting of the dot levels generated by the spin imb alance of the conduction electrons vanishes, which happe ns exactly in the middle of the triplet valley, see also Fig. 2(b). 

As mentio ned above, the suppression of the Kondo peaks is asso ciated with the exchange field which devel- ops in the pr esence of spin-dependent couplings. How- ever, as sho wn in the case of single level quantum dots, 18,19 on e may try to compensate for the presence of the exchange field by applying properly tuned external magnetic fie ld. In Fig. 5 we show the spin-resolved spec- tral functions calculated in the regime of S = 1 2 Kondo effect for ε = 0 for different values of magnetic field B . It can be clear ly seen that by tuning the magnetic field it is possible t o f ully restore the Kondo resonance at ω = 0. This happ en s when B = B$_{c}$ = 0 . 009215, see Fig. 5, with B$_{c}$ being t he compensating field, i.e. a field at which the previously -s plit dot level becomes degenerate again. 

The sit u at ion, however, becomes more complicated for ε = − 0 . 4, w hich corresponds to the Coulomb valley with N = 2 an d S = 1, when the dot is described by the un- derscreene d Kondo model. The respective spin-resolved spectral f unctions are shown in Fig. 6 using the loga- rithmic sc ale to emphasize the distinct dependence on magnetic field. Because the dot is coupled only to one conductio n channel, only a half of the dot’s spin can be screened by conduction electrons and the ground state consists of a Fermi liquid and a decoupled spin S = 1 $_{2}$. Conseque n tly, at T = 0 an infinitesimally small mag- netic field is enough to polarize the residual spin-a-half in the dot . This leads to an extremely high sensitivity of transpo rt properties with respect to magnetic field, as can be see n in Fig. 5. In consequence, in order to com- pensate fo r the presence of exchange field in the case of underscre ened Kondo model, one needs to perform a fine- tuning of magnetic field. Furthermore, it turns out that 

