Interplay between interference and Coulomb interaction in the ferromagnetic Anderson model with applied magnetic field

We study the competition between interference due to multiple single-particle paths and Coulomb interaction in a simple model of an Anderson-type impurity with local-magnetic-field-induced level splitting coupled to ferromagnetic leads. The model along with its potential experimental relevance in the field of spintronics serves as a nontrivial benchmark system where various quantum-transport approaches can be tested and compared. We present results for the linear conductance obtained by a spin-dependent implementation of the density-matrix renormalization-group scheme which are compared with a mean-field solution as well as a seemingly more advanced Hubbard-I approximation. We explain why mean field yields nearly perfect results while the more sophisticated Hubbard-I approach fails even at a purely conceptual level since it breaks hermiticity of the related density matrix. Furthermore, we study finite bias transport through the impurity by the mean-field approach and recently developed higher-order density-matrix equations. We found that the mean-field solution fails to describe the plausible results of the higher-order density-matrix approach both quantitatively and qualitatively, as it does not capture some essential features of the current-voltage characteristics such as negative differential conductance.

Figure 1

Sketch of our model, where the magnetic field $B$ in the central region is tilted by an angle $\varphi$ with respect to the magnetization of the contacts. The setup resembles the experiment of Ref. 37. The lead magnetizations must be pinned so that they are not affected by a moderate magnetic field, which might be accomplished by using thin anisotropic films in the lead contacts.

Figure 2

The linear conductance of Eq. (2), which shows the antiresonances for noninteracting electrons (full line, $U=0$) and the spin-valve behavior for strongly interacting electrons (dashed line, $U=100B$) as a function of the angle between the magnetizations of the leads and the applied magnetic field.

Figure 3

Schematic energy spectrum in the linear conductance regime for the noninteracting case. The bare resonant level is split due to a magnetic field, and the angle between the magnetizations of the leads and the applied magnetic field is denoted by $\varphi$. The widths of the two resonances depend on $\varphi$ as ${\text{cos}}^2\left(\varphi /2\right)$ and ${\text{sin}}^2\left(\varphi /2\right)$, respectively.

Figure 4

Sketch of the different parameter regimes. For vanishing interaction $U$, nonequilibrium Green’s functions provide the complete solution of the transport problem [see Eq. (1)]. For large level splittings, the scattering formalism allows for a quantitative description of the cotunneling events [see Eqs. (2, A1)]. In this work we provide results for the more intricate region of moderate level splitting and a finite on-site Coulomb interaction.

Figure 5

(Color online) Sketch of the DMRG setup for the FAB model. Notice the implementation of the polarization through the hopping matrix elements, where $t_{\alpha ,\sigma }$ and the on-site Coulomb interaction $U$ are indicated in the figure. In the DMRG evaluation a single lead mapping is used in combination with a momentum-space representation of the single tight-binding lead (Ref. 54). Here $t_k$ indicates a discretization dependent hopping to all states in the momentum space.

Figure 6

(Color online) The linear conductance vs the angle $\varphi$ for different values of the magnetic field $B$, Coulomb interaction $U$, and the bare level position ${\xi }_0$. The $+$ symbols are the DMRG results and the lines are the nonequilibrium Green’s function results using the Hartree-Fock approximation. We apply an elliptic density of states, using Eq. (15) for the coupling constant with $D=6.1{\Gamma }^0$. The polarizations are $P=P_L=P_R=0.8$.

Figure 7

(Color online) The mean-field occupations vs $\varphi$ for $B={\Gamma }^0/2$, and four different values of the on-site Coulomb interaction and with the bare level at resonance, ${\xi }_0=0$. The upper bunch of curves are for $⟨n_{\tilde{\uparrow }}⟩$ and the lower ones are for $⟨n_{\tilde{\downarrow }}⟩$. The rest of the parameters are as in Fig. 6a.

Figure 8

(Color online) The current versus bias voltage for five different angles $\varphi$ obtained using (a) the 2vN density-matrix formalism, (b) the mean-field Hartree-Fock approach, and (c) the master-equation approach. The parameters are ${\xi }_0=0$, $B=2{\Gamma }^0$, $U=8{\Gamma }^0$, $k_{B}T=0.1{\Gamma }^0$, and $P_L=P_R=0.8$, and the bias is applied symmetrically. For the 2vN method we used a constant density of states with a half bandwidth $D=20{\Gamma }^0$ (see the main text) while for the Hartree-Fock calculation the wide-band limit is applied, i.e., the real part of the self-energy was neglected.