Full counting statistics for transport through a molecular quantum dot magnet: Incoherent tunneling regime

Full counting statistics (FCS) of transport through a molecular quantum dot magnet is studied. Our analysis is theoretical, and its range of validity is restricted here to the incoherent tunneling regime. One of the original points is our Hamiltonian describing a single-level quantum dot, magnetically coupled to an additional local spin, the latter representing the total molecular spin $s$. We assume that the system is in the strong Coulomb blockade regime, i.e., double occupancy on the dot is forbidden. The master-equation approach to FCS is applied to derive a generating function yielding the FCS of charge and current. In the application of the master-equation approach to our system, Clebsch-Gordan coefficients appear in the transition probabilities, whereas the derivation of generating function reduces to solving the eigenvalue problem of a modified master equation with counting fields. The latter needs de facto only the eigenstate which collapses smoothly to the zero-eigenvalue stationary state in the limit of vanishing counting fields. Our main discovery is that in our problem with arbitrary spin $s$, some quartic relations among Clebsch-Gordan coefficients allow us to identify the desired eigenspace without solving the whole problem. Thus, the FCS generating function is derived analytically and exactly in the framework of the master-equation approach for an arbitrary value of spin $s$. By considering more specific cases, some contour plots of the joint charge-current probability distribution function are obtained numerically. The obtained FCS generating function is spin independent in the large bias regime, whereas for a small bias voltage, it suggests transport through our molecular quantum dot magnet looks exactly like that of a spinless fermion in the limit of large $s$. This rather counterintuitive consequence is subject to a direct experimental check.

Figure 1

The dot states and the Fermi window: (i) two spin subsectors; (ii) only one spin subsector $(j=s-1∕2)$ in the bias window, corresponding to AFM exchange interaction, $g>0$; and (iii) similar to (ii), but here the tripletlike $(j=s+1∕2)$ subsector is in the bias window. This situation can be realized for a FM exchange interaction, $g<0$. The figure is drawn for $s=1∕2$.

Figure 2

The probability distributions of (a) charge and (b) current for the symmetric coupling ${\Gamma }_L={\Gamma }_R$. The bias voltage is large enough and both of the two spin subsectors are in the bias window [Fig. 1(i)]. (c) The contour plot of the nonequilibrium charge-current joint distribution $\log P(N,I)$. The horizontal (vertical) axis corresponds to current $I$ (charge $N$). The contour interval is $(2{\Gamma }_L+{\Gamma }_R)\tau ∕20$.

Figure 3

Contour plot of the joint probability distribution $\log P(N,I)$ in the case of only one spin subsector in the bias window [Figs. 1(ii) and 1(iii)]. The coupling to reservoirs is assumed to be symmetric, ${\Gamma }_L={\Gamma }_R$. Panels (a) and (b) correspond to a small local spin $s=1∕2$, whereas panel (c) corresponds to a large spin $s\to \infty$. The exchange coupling between the local and an itinerant spin is FM $(g<0)$ in panel (a), whereas it is AFM $(g>0)$ in panel (b). The contour interval is $(z{\Gamma }_L+{\Gamma }_R)∕20$.

Figure 4

Contour plots of $\log P$ for a small local spin $s=1∕2$ for the asymmetric coupling ${\Gamma }_R=10{\Gamma }_L$. The FM $(g<0)$ and AFM $(g>0)$ coupling cases are shown in panels (a) and (b), respectively. The contour interval is $({\Gamma }_L+{\Gamma }_R)∕20$. (c) Fano factor as a function of the asymmetry factor ${\Gamma }_L∕{\Gamma }_R$ and (d) probability distributions of charge for AFM and FM exchange interactions.

Figure 5

(Color online) The absolute value of the lowest three cumulants, $C_{01}$, $C_{02}$, and $C_{03}$, for a molecular spin $s=10$ and an asymmetric tunnel coupling $({\Gamma }_R=5{\Gamma }_L)$. We also assumed that the exchange interaction is AFM $(g>0)$. The origin of energy is taken to be at the singlet level, ${ϵ}_{s-1∕2}=0$. Panels (a-1) and (a-2) corresponds, respectively, to a positive and to a negative bias voltage. In panels (b-1) and (b-2), the corresponding Fano factors $C_{02}∕C_{01}$ and $C_{03}∕C_{01}$ are plotted.