Resonance effects in correlated multilayer heterostructures

We study the occurrence of negative differential conductance induced by resonance effects in a model for a multilayer heterostructure. In particular, we consider a system consisting of several correlated and noncorrelated monatomic layers sandwiched between two metallic leads. The geometry confines electrons in wells within the heterostructures, which are connected to each other and to the leads by tunneling processes. The nonequilibrium situation is produced by applying a bias voltage to the leads. Our results show that, for specific values of the parameters, resonance tunneling takes place. We investigate in detail its influence on the current-voltage characteristics. Our results are obtained via nonequilibrium real-space dynamical mean-field theory. As an impurity solver we use the so-called auxiliary master equation approach, which addresses the impurity problem within an auxiliary system consisting of a correlated impurity, a small number of uncorrelated bath sites, and two Markovian environments described by a generalized master equation.

Figure 1

Schematic representation of the triple-well system studied in this paper, consisting of a central region of 12 layers sandwiched between two semi-infinite metallic leads (blue), with chemical potentials ${\mu }_{l/r}$ and onsite energies ${\varepsilon }_{l/r}={\mu }_{l/r}$, respectively. A Hubbard interaction is only present in the boundary layers (red, $z=1,12)$ with interaction strength $U_1=U_{12}=U=8$ and onsite energies ${\varepsilon }_1^{\left(0\right)}={\varepsilon }_{12}^{\left(0\right)}=-U/2$. They form the outer walls of the quantum wells. The inner walls are the layers 5 and 8 (brown) and are caused by discontinuous onsite energies ${\varepsilon }_5^{\left(0\right)}=-{\varepsilon }_8^{\left(0\right)}=-4$, while all other layers have ${\varepsilon }_z^{\left(0\right)}=0$. Energies are measured in units of the nearest-neighbor hopping $t_c$ within the central region. For the nearest-neighbor hopping within the leads we use $t_l=t_r=2$, and the hybridization between the left (right) lead and the central region is $v_l=v_r=1$. A bias voltage $\mathrm{\Phi }:={\mu }_l-{\mu }_r$ is applied to the leads, which linearly shifts the onsite energies of each layer: ${\varepsilon }_z={\varepsilon }_z^{\left(0\right)}+{\mu }_l-z({\mu }_l-{\mu }_r)/(L+1)$.

Figure 2

Current $J$ vs bias voltage $\mathrm{\Phi }$. Solid, dashed, and dotted lines are obtained by solving the impurity problem with $N_b=6,N_b=4$, and $N_b=2$, respectively (see text). Parameters are the same as in Fig. 1.

Figure 3

Steady-state spectral function $A_z(\omega ,{\varepsilon }_{\mathbf{k}})$ for different values of bias voltage $\mathrm{\Phi }$ and $\varepsilon \left(\mathbf{k}\right)$. To illustrate the resonance effect, we present results for bias voltages for which the current displays a maximum $(\mathrm{\Phi }\simeq 2.5$ and $\mathrm{\Phi }\simeq 5.25)$, a minimum $\left(\mathrm{\Phi }\simeq 4\right)$, and for a value in between $\left(\mathrm{\Phi }=1\right)$. The shaded area emphasizes the fact that, for $\mathrm{\Phi }\simeq 5.25$, the peak maxima of layers $z=3$ and $z=10$ overlap. Results are obtained with $N_b=6$. Here, ${\varepsilon }_{\mathbf{k}}=-2$ (green), ${\varepsilon }_{\mathbf{k}}=0$ (red), and ${\varepsilon }_{\mathbf{k}}=2$ (blue). Other parameters are the same as in Fig. 1.

Figure 4

Current $J$ as a function of the Hubbard interaction $U$ at the resonance. On-site energies in the first and the last layers are fixed to ${\varepsilon }_1^{\left(0\right)}={\varepsilon }_{12}^{\left(0\right)}=-4$. Results are obtained with $N_b=4$. Other parameters are the same as in Fig. 1.