Sources of negative tunneling magnetoresistance in multilevel quantum dots with ferromagnetic contacts

We analyze distinct sources of spin-dependent energy level shifts and their impact on the tunneling magnetoresistance (TMR) of interacting quantum dots coupled to collinearly polarized ferromagnetic leads. Level shifts due to virtual charge fluctuations can be quantitatively evaluated within a diagrammatic representation of our transport theory. The theory is valid for multilevel quantum dot systems and we exemplarily apply it to carbon nanotube quantum dots, where we show that the presence of many levels, among them of excited states, can qualitatively influence the TMR effect.

Figure 1

(a) Transport setup of a CNT quantum dot with ferromagnetic leads. If an electron enters the barrier region at the tube ends, it can either tunnel to the lead or be reflected at the interface. Tunneling processes in the contact region are described by even powers of the tunneling Hamiltonian $H_T$. (b) and (c) Conductances $G_P$ and $G_A$ versus gate voltage for two situations exhibiting negative TMR. (d) The negative TMR mechanism related to panel (c), where thick/thin lines describe processes that are favored/disfavored being associated with majority/minority spins.

Figure 2

Structure of (a) a sequential tunneling diagram and (b) a level renormalization diagram for a sequential tunneling event. Each “bubble” describes a charge fluctuation. We consider $k$ charge fluctuations (to states $c_j$) in the final state $b$ and $n-k$ charge fluctuations (to states $c_i^{\prime }$) in the initial state $a$. We have indicated the time order, as well as the intermediate states on the contours.

Figure 3

Parallel ($G_P$) and antiparallel ($G_A$) conductance along with the resulting tunneling magneto-resistance (TMR) for a CNT of 500 nm in length (${\varepsilon }_0=3.35$meV) and charging energy $E_c=6.7$ meV. The thermal energy was set to $k_{B}T=0.3$ meV, the lead polarization to $P=0.4$, the tunnel broadening to $\hslash {\Gamma }_s=3$ $\mu$eV, and the bandwidth to $W=3$ eV. (a) For a full fourth-order calculation a mirror-symmetric TMR slightly oscillating around a value of $20\%$ is obtained. (b) An equal splitting $h_{\mathrm{ext}}^P=h_{\mathrm{ext}}^A=0.4$ meV causes gate asymmetry and negative values of the TMR by the mechanism shown in Fig. 1c. (c) For $h_{\mathrm{ext}}^P=2h_{\mathrm{ext}}^A=0.8$ meV, the regularity of the curve is broken and the negative TMR mechanism shown in Fig. 1b comes into play. (d) TMR and conductance for ${\Delta }_{\mathrm{St}}=0$. We find electron-hole symmetry perfectly preserved, also upon inclusion of virtual excited states (with cutoff set to ${\varepsilon }_0$ and $2{\varepsilon }_0)$. (e) Same as panel (d) but for ${\Delta }_{\mathrm{St}}=0.2W$, which breaks electron-hole symmetry. Including the influence of the virtual excited states negative TMR can be reached. (f) Combining intrinsic shifts with external splitting. An asymmetric coupling to the leads can enlarge ($\gamma <1$) or diminish ($\gamma >1$) the TMR effect.

Figure 4

Gate-voltage dependence of the intrinsic level shift $h_{\mathrm{int}}^{ba}$, Eq. (7), for exemplary electron-hole symmetric transitions, see state labels in panel (a). The electron-hole symmetry is nicely reflected in the curves for ${\Delta }_{\mathrm{St}}=0$. Moreover, it becomes obvious that ${\Delta }_{\mathrm{St}}$ acts opposite to an external Zeeman splitting, i.e., increases (decreases) the energy difference for $\uparrow$ ($\downarrow$) transitions. Raising the energy cutoff for the inclusion of excited states, more and more transitions to virtual states will be contributing to the charge fluctuations for the exemplary transitions. (a) Only fluctuations to ground states. (b) Fluctuations to all states within the range ${\varepsilon }_0$. (c) All states within the range $2{\varepsilon }_0$.

Figure 5

Gate-voltage dependence of the difference of energy shifts $\delta E_b^P-\delta E_b^A$ [see Eq. (8)] for states with spin 0 (a), spin $\pm 1/2$ (b), and spin $\pm 1$ (c). Notice the pronounced influence of virtual states including bosonic excitations.

Figure 6

Gate-voltage dependence of the difference of energy shifts $\delta E_b-\delta E_{b^{\prime }}$ for parallel configuration for the two states $b=|\uparrow ,·\rangle$ and $b^{\prime }=|\downarrow ,·\rangle$ ($N=4n+1$, upper panel) and $b=|\uparrow \downarrow ,\uparrow \rangle$ and $b^{\prime }=|\uparrow \downarrow ,\downarrow \rangle$ ($N=4n+3$, lower panel). Notice the pronounced influence of virtual states including bosonic excitations. For the legend, see Fig. 5.