Current-voltage response for unipolar funnel-shaped nanochannel diodes

Permselective nanochannels can rectify the electric current transported through them similar to solid-state diodes. The rectification is due to symmetry breaking related to distribution of the nanochannels’ surface charge as well as the geometry. Thus far, most of the works related to the asymmetric current response have been primarily experimental. Here, we theoretically model a funnel-shaped nanochannel with a nonhomogeneous surface charge from which we derive a current-voltage relation (I-V). If the effects of the adjacent microchannels are ignored, the I-V is shown to behave like a unipolar diodes. When the effects of the adjacent microchannels are accounted for, the channel behaves like a diode only in a small voltage domain, while at larger voltages, the response is determined by the microchannels. The theoretical results are confirmed by numerical simulations.

Figure 1

Schematic of a 2D funnel-shaped channel of angle $\alpha$ under an applied potential $V$. The top surface in region 1 (marked in blue) is charged by a negative surface charge ${\sigma }_s$.

Figure 2

1D current density–voltage ($j-V$) curve for varying values of $N$ (simulations—markers; theory—solid line). Color coding of theory corresponds to the according simulations. (Inset) Ratio of $j^{\mathrm{Simulations}}/j^{\mathrm{Theory}}$ vs $V$. Simulation parameters are $L_1=L_2=1,\varepsilon ={10}^{-4}$.

Figure 3

Current-voltage ($I-V$) curves for varying values of $h_2$ (simulations—markers; theory—solid line). (Inset) Ratio of $I^{\mathrm{Simulations}}/I^{\mathrm{Theory}}$ vs $V$. Simulation parameters are $L_1=L_2=1, h_1=0.1, N_h\left(h(x=0)=h_1\right)={10}^3, \varepsilon ={10}^{-3}$.

Figure 4

Current-voltage $(I-V)$ response curve comparing three different scenarios (simulations—markers; theory—solid line): unipolar diode without [Eq. (21)] and with [Eq. (28)] microchannels and uniformly charged nanochannel with microchannels [Eq. (32)]. (Inset) Funnel-shaped channel placed between two reservoirs: Left and Right. The height $H$ and length $L$ of each region are subscripted with $L$ and $R$ accordingly. Simulation parameters are $H_{L,R}=L_{1,2,L,R}=1,h_{1,2}={10}^{-2},N_h={10}^3,\varepsilon ={10}^{-3}$.

Figure 5

Current-voltage response curve comparing the two different numerical methods for the geometry given for the following parameters: $L_1=0.03,L_2=1,h_1={10}^{-4},h_2={10}^{-3},N_h\left(h_1\right)=150,{\sigma }_s=7500,\varepsilon ={10}^{-3}$.