Magnetoresistance of a quantum dot with spin-active interfaces

We study the zero-bias magnetoresistance (MR) of an interacting quantum dot connected to two ferromagnetic leads and capacitively coupled to a gate voltage source $V_g$. We investigate the effects of the spin-activity of the contacts between the dot and the leads by introducing an effective exchange field in an Anderson model. This spin-activity makes easier negative MR effects, and can even lead to a giant MR effect with a sign tunable with $V_g$. Assuming a twofold orbital degeneracy, our approach allows one to interpret in an interacting picture the $\mathrm{MR}\left(V_g\right)$ measured by S. Sahoo et al. [Nature Phys. 1, 99 (2005)] in single wall carbon nanotubes with ferromagnetic contacts. If this experiment is repeated on a larger $V_g$ range, we expect that the $\mathrm{MR}\left(V_g\right)$ oscillations are not regular like in the presently available data, due to Coulomb interactions.

Figure 1

Mesoscopic element M connected to ferromagnetic leads $L$ and $R$. The magnetic polarizations ${\vec{p}}_L$ and ${\vec{p}}_R$ of leads $L$ and $R$ can be parallel (configuration P) or antiparallel (configuration AP). The element M is capacitively coupled to a gate voltage source $V_g$.

Figure 2

(Color online) Panels (a), (c), and (e): Conductance $G^P$ in the parallel configuration (red full lines) and conductance $G^{\mathit{AP}}$ in the antiparallel configuration (black dotted lines) as a function of the gate voltage $V_g$, for the circuit shown in Fig. 1, with M a one-orbital quantum dot. We have used ${\Gamma }_L=0.005U$, ${\Gamma }_R=0.07U$, $P_{L\left(R\right)}=0.2$, $U∕k_{B}T=30$, and $h_{\mathit{SDIPS}}^{\mathit{AP}}=0$. Panels (b), (d), and (f): Magnetoresistance $\mathrm{MR}=(G^P-G^{\mathit{AP}})∕(G^P+G^{\mathit{AP}})$ (pink curves) corresponding to the left conductance plots. The results are shown for $g{\mu }_B{h}_{\mathit{SDIPS}}^P=0$ [panels (a) and (b)], $g{\mu }_B{h}_{\mathit{SDIPS}}^P=0.06U$ [panels (c) and (d)] and $g{\mu }_B{h}_{\mathit{SDIPS}}^P=0.4U$ [panels (e) and (f)].

Figure 3

(Color online) Average occupations $⟨n_{d\uparrow }⟩$ (blue lines) and $⟨n_{d\downarrow }⟩$ (black lines) of level $d$ by spins ↑ and ↓ as a function of $V_g$, for the one-orbital quantum dot circuit of Fig. 1. The results are shown for the same parameters as in Fig. 2 and lead polarizations in the parallel configuration $(c=P)$, with $h_{\mathit{SDIPS}}^P=0$ [panel (a)], $g{\mu }_B{h}_{\mathit{SDIPS}}^P=0.06U$ [panel (b)], and $g{\mu }_{B\mathit{SDIPS}}^P=0.4U$ [panel (c)]. For $h_{\mathit{SDIPS}}^P=0$, $⟨n_{d\uparrow }⟩$ and $⟨n_{d\downarrow }⟩$ remain very close, simply showing two steps corresponding to the two conductances peaks visible in Fig. 2a. In the case of a finite $h_{\mathit{SDIPS}}^P$, $⟨n_{d\uparrow }⟩$ rises more strongly than $⟨n_{d\downarrow }⟩$ at the first conductance peak, revealing that current transport is due in majority to ↑ spins for this first peak. Then, $⟨n_{d\uparrow }⟩$ and $⟨n_{d\downarrow }⟩$ become closer when both ${\xi }_{d\uparrow }$ and ${\xi }_{d\downarrow }$ are below the Fermi level. At the second conductance peak $⟨n_{d\downarrow }⟩$ rises more strongly than $⟨n_{d\uparrow }⟩$ because current transport is now dominated by down spins. The asymmetry between the behaviors of spins ↑ and ↓ increases with $h_{\mathit{SDIPS}}^P$ (from left to right panels).

Figure 4

(Color online) Panels (a), (c), and (e): Conductance $G^P$ in the parallel configuration (red full lines) and conductance $G^{\mathit{AP}}$ in the antiparallel configuration (black dotted lines), for the circuit of Fig. 1, with M a two-orbitals quantum dot. We have used identical tunnel rates to the two orbitals, i.e., ${\Gamma }_L=0.0043U$, ${\Gamma }_R=0.0725U$, and $P_{L\left(R\right)}=0.4$. We have also used $U∕k_{B}T=30$ and $h_{\mathit{SDIPS}}^{\mathit{AP}}=0$. Panels (b), (d), and (f): Magnetoresistance MR (pink full lines) corresponding to the left conductance plots. The results are shown for $g{\mu }_B{h}_{\mathit{SDIPS}}^P=0$ [panels (a) and (b)], $g{\mu }_B{h}_{\mathit{SDIPS}}^P=0.05U$ [panels (c) and (d)], and $g{\mu }_B{h}_{\mathit{SDIPS}}^P=0.3U$ [panels (e) and (f)].

Figure 5

(Color online) Average occupations $⟨n_{\uparrow }⟩=⟨n_{K\uparrow }⟩=⟨n_{K^{\prime }\uparrow }⟩$ (blue lines) and $⟨n_{\downarrow }⟩=⟨n_{K\downarrow }⟩=⟨n_{K^{\prime }\downarrow }⟩$ (black lines) of levels $K$ and $K^{\prime }$ by spins ↑ and ↓ as a function of $V_g$, for a two-orbitals quantum dot circuit with the same parameters as in Fig. 4. The results are shown for lead polarizations in the parallel configuration $(c=P)$, with $h_{\mathit{SDIPS}}^P=0$ [panel (a)], $g{\mu }_B{h}_{\mathit{SDIPS}}^P=0.05U$ [panel (b)], and $g{\mu }_B{h}_{\mathit{SDIPS}}^P=0.3U$ [panel (c)]. For $h_{\mathit{SDIPS}}^P=0$, $⟨n_{\uparrow }⟩$ and $⟨n_{\downarrow }⟩$ remain very close, showing four steps corresponding to the four conductances peaks visible in Fig. 4a. In the case of a finite $h_{\mathit{SDIPS}}^P$, $⟨n_{\uparrow }⟩$ rises more strongly than $⟨n_{\downarrow }⟩$ for the two first conductance peaks, which shows that current transport is due in majority to up spins for these two peaks. The opposite situation occurs for the two last conductance peaks. The asymmetry between the behaviors of spins ↑ and ↓ increases with $h_{\mathit{SDIPS}}^P$ (from left to right panels).

Figure 6

(Color online) Comparison between the data of Ref. 17 (squares) and the two-orbitals theory. We show the conductance $G^P$ in the parallel configuration (top panel) and the corresponding magnetoresistance ${\mathrm{MR}}^{\prime }=(G^P-G^{\mathit{AP}})∕G^{\mathit{AP}}$ (bottom panel). The theory is shown for parameters consistent with the experiment, i.e., $U=5\mathrm{meV}$, $U∕k_{B}T=30$, and $\alpha =0.0986$. We also use relatively low values of polarization $P_{L\left(R\right)}=0.4$ because usual ferromagnetic contact materials are not fully polarized. (Ref. 43) Assuming identical tunnel couplings for the two orbitals, the values of tunnel rates ${\Gamma }_L=0.0043U$ and ${\Gamma }_R=0.0725U$ are imposed by the width and height of the conductance peaks. Then, $h_{\mathit{SDIPS}}^{P\left[\mathit{AP}\right]}$ are the only free fitting parameters which remain for interpreting the MR curve. We have assumed $g{\mu }_B{h}_{\mathit{SDIPS}}^P=0.05U$ for all the theoretical curves shown in this figure. We have plot the MR curves of the bottom panel for $h_{\mathit{SDIPS}}^{\mathit{AP}}=0$ [pink curve, corresponding to Fig. 4d], $h_{\mathit{SDIPS}}^{\mathit{AP}}=-0.01U$ (green dashed curve) and $h_{\mathit{SDIPS}}^{\mathit{AP}}=0.01U$ (blue dot-dashed curve). Note that in this figure, we show ${\mathrm{MR}}^{\prime }$ instead of $\mathrm{MR}=(G^P-G^{\mathit{AP}})∕(G^P+G^{\mathit{AP}})$ in order to be consistent with Ref. 17. A strong gate voltage offset jump occurred at $V_g=4.331\mathrm{V}$, therefore we show the data at the left/right of this jump with gray/black symbols.