Figure 1

Sketch of differential cotunneling conductance as a function of bias voltage, $V$, and control parameter, $x$. Thick blue (solid) lines indicate inelastic cotunneling thresholds with an avoided crossing near zero bias, caused by a finite mixing, $\lambda$, of the two involved states. The red (dashed) line indicates the line along which the levels cross (possibly resulting in a conductance peak), including the possibility of a slight bias dependence of the level splitting. The two different sides of this line may have a difference in $dI/dV$ (indicated by different shades of blue) due to a difference in elastic cotunneling amplitudes for the two states.Reuse & Permissions

Figure 2

Sketch of double-dot model. Electrons on different dots interact via their charge (Coulomb repulsion, $U_{12}$) as well as via their spin (exchange, $J_{12}$). Tunneling between electrode $r$ and dot $i$ takes place with amplitude $t_{ir}$. The gate electrode is not shown.Reuse & Permissions

Figure 3

Results of GME calculation presented as conductance maps, $|dI/dV|$ plotted on a logarithmic color scale as a function of $V$ and $V_g$ (the absolute value is plotted since negative values, occurring in certain regions because of the tunnel-rate asymmetry, are not well represented on a logarithmic scale). The red arrows point out the singlet-triplet line. Here the tunnel amplitudes are $t_{1R}=30,t_{1L}=0.7,t_{2R}=0.8,t_{2L}=0.5$ (arbitrary units), the Coulomb charging energies are $U_{11}=U_{22}=2000T$, $U_{12}=400T$, and we choose ${\epsilon }_2={\epsilon }_1+2T$. $B=0$ in (a) and $B=50T$ in (b).Reuse & Permissions

Figure 4

(a) The lower density plot shows $d^{3}I/d{V}^3$ vs. bias voltage, $V$, and gate voltage, $V_g$, for the Mn complex measured in Ref. 4. An external magnetic field of 10 Tesla clearly reveals a singlet ground state with an excited triplet towards the left, and a triplet ground state with an excited singlet towards the right. The zoom in shows the same data now plotted as $dI/dV$. A steep faint line is seen to be pinned to the ground state crossing. Cuts at various bias voltages, marked by dashed lines, reveal a distinct line shape as a function of $V_g$ developing at large applied bias: dip-peak at $V<0$ and peak-dip at $V>0$. The two nearly vertical dashed lines indicate the range in $V_g$ used for each of the cuts. The vertical range is the same in all the cuts, and has been centered at the mean value within each cut. (b) Density plot of differential conductance as calculated below in Sec. 4b, with cuts at bias voltages $V/T=\pm 269,\pm 76,0$ (parameters are chosen as in Fig. 6). The peak-dip (dip-peak) structure from panel (a) is clearly reproduced. All density plots are on a gray scale where darker indicates higher values.Reuse & Permissions

Figure 5

(a) Differential conductance $dI/dV$ vs $V/T$ at $V_g/T=-300$ (blue upper line) and at $V_g/T=200$ (red lower line). The curves are offset for clarity. The tunneling amplitudes are very asymmetric, $t_{1R}=30,t_{1L}=0.7,t_{2R}=0.8,t_{2L}=0.5$ (arbitrary units). We have furthermore chosen $B/T=50$, $\alpha =0.5$, and $\beta =0.1$. (b) Occupation probabilities of the different spin states at fixed bias $V/T=250$ with parameters as in (a). The state $|+\rangle$ refers to the lowest-lying state (i.e., $|S\rangle$ on the left-hand side and $|T_{-1}\rangle$ on the right-hand side). $|-\rangle$ refers to the next lowest (i.e., $|T_{-1}\rangle$ on the left-hand side and $|S\rangle$ on the right-hand side).Reuse & Permissions

Figure 6

(a) Differential conductance $dI/dV$ vs $V/T$ at $V_g/T=-300$ (blue upper line) and at $V_g/T=200$ (red lower line). The curves are offset for clarity. The tunnel amplitudes are now chosen more symmetric than in Fig. 5, with $t_{1R}=0.3$, and a finite spin-orbit coupling, $D/T=2\sqrt{2}$ has been included. All other parameters are the same as in Fig. 5. (b) Corresponding occupation probabilities of the different spin states at fixed bias $V/T=250$. Notice the occupation inversion between the two states $|-\rangle$ and $|+\rangle$ at their would-be degeneracy point at $V_g/T=-150$.Reuse & Permissions