Eightfold shell-filling patterns in spin-dependent transport through double-wall carbon nanotube quantum dots

We analyze theoretically tunneling through double-wall carbon nanotubes weakly coupled to external nonmagnetic as well as ferromagnetic leads. Using the real-time diagrammatic technique and taking into account both the sequential and cotunneling processes, we calculate the differential conductance, shot noise and the corresponding Fano factor, as well as tunnel magnetoresistance in the case of ferromagnetic leads. The analysis is focused mainly on the transport regime where the conductance spectra reveal eightfold periodicity. Some specific cases where deviations from this eightfold periodicity appear are also analyzed. In addition, we show that when the double-wall carbon nanotube is coupled to leads with different spin polarizations the system behaves as a gate-controlled spin diode.

Figure 1

(Color online) The differential conductance (a) and the Fano factor (c) as a function of the bias and gate voltages, and the gate voltage dependence of the linear conductance (b). The parameters are: $U=11.0\text{meV}$, $k_{\text{B}}T=0.5\text{meV}$, $t=1\text{meV}$, $\delta =0$, $\Gamma =0.2\text{meV}$, and $p_L=p_R=0$. The numbers in part (a) indicate the average charge in the double wall nanotube when sweeping the gate voltage. Part (b) is plotted on logarithmic scale to emphasize the difference between the sequential and sequential plus cotunneling results. The white stripe in (c) marks the region where the thermal noise dominates (the Fano factor is divergent).

Figure 2

The linear conductance as a function of the gate voltage for indicated values of the intertube hoping parameter $t$: (a) $t=0.5\text{meV}$, (b) $t=1\text{meV}$, and (c) $t=2\text{meV}$. The other parameters as in Fig. 1.

Figure 3

(Color online) The linear conductance as a function of the gate voltage for $t=0.5\text{meV}$ and indicated values of subband mismatch ${\delta }_{i\left(o\right)}$ and mismatch between the levels of each tube $\tilde{\Delta }$. The other parameters are the same as in Fig. 1. For comparison the doted line shows the corresponding results from Fig. 2a, i.e., for ${\delta }_{i\left(o\right)}=0$ and $\tilde{\Delta }=0$.

Figure 4

(Color online) The linear conductance in (a) the parallel and (b) antiparallel magnetic configurations as well as the resulting (c) TMR as a function of the gate voltage for different values of the intertube hoping $t$ and for $p_L=p_R=0.5$. The other parameters are the same as in Fig. 1.

Figure 5

(Color online) Differential conductance in (a) the parallel and (b) antiparallel magnetic configuration, (c) total TMR and (d) sequential TMR as well as the Fano factor in (e) the parallel and (f) antiparallel magnetic configurations of the system, plotted as a function of the bias and gate voltages for $t=1\text{meV}$. The other parameters are the same as in Fig. 4. The Fano factor in the low-bias voltage regime is divergent, therefore this transport regime is marked with a white stripe.

Figure 6

(Color online) Tunnel magnetoresistance as a function of the bias and gate voltages for different hoping between the inner and outer tubes: (a) $t=0.5\text{meV}$, (b) $t=1\text{meV}$, and (c) $t=2\text{meV}$. The other parameters are the same as in Fig. 4. The same scale is used in all density plots.

Figure 7

(Color online) The linear response TMR as a function of the gate voltage for $t=0.5\text{meV}$ and indicated values of ${\delta }_{i\left(o\right)}$ and $\tilde{\Delta }$. The other parameters are the same as in Fig. 4. For comparison the doted line shows the linear TMR calculated for ${\delta }_{i\left(o\right)}=0$ and $\tilde{\Delta }=0$.

Figure 8

(Color online) Bias and gate voltage dependence of (a) the absolute value of the current, (b) differential conductance, (c) current spin polarization $P_{\text{cur}}=\left(I^{\uparrow }-I^{\downarrow }\right)/\left(I^{\uparrow }+I^{\downarrow }\right)$, and (d) the Fano factor for $p_L=0$ and $p_R=0.9$. The other parameters are the same as in Fig. 1. Results were obtained using only sequential tunneling processes. The current is plotted in the units of $I_0$ with $I_0=e\Gamma /\hslash$.