Electronic transport and dynamics in correlated heterostructures

We investigate by means of the time-dependent Gutzwiller approximation the transport properties of a strongly correlated slab subject to Hubbard repulsion and connected with to two metallic leads kept at a different electrochemical potential. We focus on the real-time evolution of the electronic properties after the slab is connected to the leads and consider both metallic and Mott insulating slabs. When the correlated slab is metallic, the system relaxes to a steady state that sustains a finite current. The zero-bias conductance is finite and independent of the degree of correlations within the slab as long as the system remains metallic. On the other hand, when the slab is in a Mott insulating state, the external bias leads to currents that are exponentially activated by charge tunneling across the Mott-Hubbard gap, consistent with the Landau-Zener dielectric breakdown scenario.

Figure 1

Sketch of the correlated slab sandwiched between semi-infinite metallic leads. $v_L$ and $v_R$ represent respectively left and right slab-leads hybridization coupling.

Figure 2

Layer-resolved dynamics of the local quasiparticle weights $|R_z{\left(t\right)|}^2$ for a slab of $N=20$ layers and two values of the interaction $U$. The slab-lead hybridization is equal to the interlayer hopping amplitude $v_{\text{hyb}}=1.0$.

Figure 3

(a) Dynamics of the local quasiparticle weight for the first layer. Dashed lines are the fitting curves obtained with Eq. (21). Arrows represent the hybridized slab equilibrium values. Inset: dead layer awakening time as a function of $U$. Dashed lines represents the fitting curve $\tau =\alpha /|U-U_c{|}^{{\nu }_{*}}$ with ${\nu }_{*}\approx 0.4895$. (b) Dynamics of the local quasiparticle weight for the bulk $\left(z=10\right)$ layer. Arrows represent the hybridized slab equilibrium values. (c) Quasiparticle weight profiles at times $t=0$ (dotted lines) and $t=50$ (lines).

Figure 4

Top: Real-time dynamics of the currents computed at the slab-lead contact (thick lines) and between two neighboring layers (light grey lines) after a bias quench with $\mathrm{\Delta }V=0.5$ and $U=12$. for a $N=10$ slab and different values of the slab-leads coupling $v_{\text{hyb}}$. Inset: Current dynamics for a ramplike switching protocol $r\left(t\right)=[1-3/2cos(\pi t/{\tau }_{*})+1/2cos{(\pi t/{\tau }_{*})}^3]/4$ compared to the sudden quench limit $\left({\tau }_{*}=30\right)$. Bottom: Dynamics of the local electronic densities $n_z\left(t\right)$ for the first, third, and fifth layer and $v_{\text{hyb}}=1.0$. Inset: stationary density profile showing an almost flat density distribution with slightly doped regions near the left and right contacts.

Figure 5

Relative variation of the slab internal energy as defined in Eq. (22) for the same set of parameters of Fig. 4. Inset: Stationary current for $\mathrm{\Delta }V=0.5$ as a function of the hybridization with the leads with a fitting curve $j\left(v_{\text{hyb}}\right)=j_0{v}_{\text{hyb}}^2$ (dashed line).

Figure 6

(a) Real-time dynamics for the contact currents for the same parameters and values of hybridization coupling of Fig. 4 and $\mathrm{\Delta }V=2.0$. (b) Blowup of the currents dynamics for $v_{\text{hyb}}=0.1$ and $v_{\text{hyb}}=0.25$. (c) Dynamics of the current time average $\langle j\left(t\right)\rangle$ as defined in the main text for $v_{\text{hyb}}=0.5$ and $v_{\text{hyb}}=1.0$.

Figure 7

Dynamics of the relative energy variation for the same parameters in Fig. 6.

Figure 8

Dynamics of the mean quasiparticle weight for the same parameters in Fig. 6 and $v_{\text{hyb}}=0.1$ (black line) and $v_{\text{hyb}}=1.0$ (red line).

Figure 9

Current (a) and internal energy dynamics (b) for $U=12,v_{\text{hyb}}=1.0$, and three values of the applied bias. The occurrence of the breakdown of the stationary dynamics due to the dynamical transition is highlighted by the vertical arrows.

Figure 10

(a) Current-bias characteristics for a $N=10$ slab for different values of $U$ and $v_{\text{hyb}}=0.1$. (b) Blowup of the linear part of the current-bias characteristics. (c) Differential conductance measured with respect to the quantum conductance $G_0=1/2\pi$ in our units. Grey line represents the universal zero-bias value.

Figure 11

Left: current-bias characteristics for a $N=10$ slab, different values of $U$, and $v_{\text{hyb}}=1.0$. Plus, cross, and star symbols represents stationary currents values, while circles represent converged currents time averages values. Right: blowup of the linear part of the current-bias characteristics for $v_{\text{hyb}}=0.5$ and $v_{\text{hyb}}=1.0$, showing universal zero-bias conductivity $G/G_0\approx 0.452$ and $1.203$ respectively.

Figure 12

Real-time dynamics for the quasiparticle weights from layer 1 to 5 (from top to bottom) of a $N=10$ Mott insulating slab suddenly coupled to the metallic leads $\left(v_{\text{hyb}}=1.0\right)$. $U=16.5$ and $U=17.5$. Inset: inverse of the characteristic time for the exponential quasiparticle formation ${\tau }^{-1}\sim {(U-U_c)}^{-{\nu }_{*}},{\nu }_{*}\approx 0.4753$.

Figure 13

Time-averaged currents for three slabs with applied bias $\mathrm{\Delta }V=0.5$. $U=16.5>U_c,v_{\text{hyb}}=1.0$, and $N=4,12,20$ (from top to bottom).

Figure 14

Layer dependent quasiparticle weight dynamics for layers 1 (a), 5 (b), and 10 (c) of a $N=20$ slab, $U=16.5,v_{\text{hyb}}=1.0$ and two values of the applied bias $\mathrm{\Delta }V=1.0$ (red lines) and $\mathrm{\Delta }V=4.0$ (blue lines). (d) Time-averaged stationary quasiparticle weight profile.

Figure 15

Time-averaged currents for the same parameters in Fig. 14 and $\mathrm{\Delta }V=4.0,3.0,2.0$, and $1.0$ (from top to bottom). Dashed lines are fitting curves from Eq. (26). Inset: Real-time dynamics of the current for $\mathrm{\Delta }V=4.0$.

Figure 16

Current bias characteristics for $U=16.5$ and different values of the slab length $N=8,10,12,16$, and 20 (from top to bottom). Dashed lines represent fitting curves with Eq. (27). Insets: threshold bias $\mathrm{\Delta }{V}_{\text{th}}$ and electric field $E_{\text{th}}=\mathrm{\Delta }{V}_{\text{th}}/N$ as a function of the slab size. The horizontal line is the extrapolation for the large-$N$ size independent electric field.

Figure 17

Layer-dependent quasiparticle effective chemical potential profile. Parameters are the same as in Fig. 15 for $\mathrm{\Delta }V=1$ and $\mathrm{\Delta }V=4$. The grey dashed line represents the applied bias linear profile. Inset: Real-time dynamics of the quasiparticle chemical potential on the tenth layer. All data are plotted with respect to the leads' chemical potential absolute value $\mathrm{\Delta }V/2$.