Gate coupling to nanoscale electronics

The realization of single-molecule electronic devices, in which a nanometer-scale molecule is connected to macroscopic leads, requires the reproducible production of highly ordered nanoscale gaps in which a molecule of interest is electrostatically coupled to nearby gate electrodes. Understanding how the molecule-gate coupling depends on key parameters is crucial for the development of high-performance devices. Here we directly address this, presenting two- and three-dimensional finite-element electrostatic simulations of the electrode geometries formed using emerging fabrication techniques. We quantify the gate coupling intrinsic to these devices, exploring the roles of parameters believed to be relevant to such devices. These include the thickness and nature of the dielectric used, and the gate screening due to different device geometries. On the single-molecule $\left(\sim 1\text{nm}\right)$ scale, we find that device geometry plays a greater role in the gate coupling than the dielectric constant or the thickness of the insulator. Compared to the typical uniform nanogap electrode geometry envisioned, we find that nonuniform tapered electrodes yield a significant 3 orders of magnitude improvement in gate coupling. We also find that in the tapered geometry the polarizability of a molecular channel works to enhance the gate coupling.

Figure 1

(Color online) Electrodes with uniform cross-section—role of the dielectric constant. (a) Schematic of a typical three-terminal molecular-scale electronic device; (b) Vertical cross section through the device shown in A, with calculated equipotentials (light blue) and electrostatic potential (heat map); (c) normalized potential for the two-dimensional device shown in A with $L_{ox}=3\text{nm}$, $t=6\text{nm}$, including the analytic form of the potential (thick black line, see Sec. 3A of main text for details). Throughout this paper, the left dotted line indicates the interface between the gate dielectric and the electrodes, while the right dotted line indicates the top of the electrodes; (d) Variation of the gate coupling with dielectric constant for nanogaps of two different sizes, for the two-dimensional device shown in A with $L_{ox}=3\text{nm}$, $t=6\text{nm}$.

Figure 2

(Color online) Electrodes with uniform cross-section—role of the dielectric thickness. (a) Normalized potential for the two-dimensional device shown in Fig. 1a with $t=6\text{nm}$, $s=2\text{nm}$, $\kappa =4$, values of $L_{ox}$ from top to bottom are 1, 2, 3, 5, 10, 25, and 50 nm; (b) Gate coupling as a function of dielectric thickness for the two-dimensional device shown in Fig. 1a with $t=6\text{nm}$, $\kappa =4$.

Figure 3

(Color online) Electrodes with uniform cross-section—roles of the electrode thickness and spacing. (a) Normalized potential for the two-dimensional device shown in Fig. 1a with $L_{ox}=3\text{nm}$, $s=2\text{nm}$, $\kappa =4$; (b) Normalized potential for the two-dimensional device shown in Fig. 1a with $L_{ox}=3\text{nm}$, $s=5\text{nm}$, $\kappa =4$; (c) gate coupling as a function of nanogap size for the two-dimensional device shown in Fig. 1a with $L_{ox}=3\text{nm}$, $t=6\text{nm}$, $\kappa =4$.

Figure 4

(Color online) Electrodes with nonuniform cross section—role of the electrode thickness. (a) Schematic of a typical three-terminal molecular-scale electronic device fabricated using feedback controlled electromigration; (b) normalized potential for the three-dimensional device shown in A with $L_{ox}=3\text{nm}$, $s=2\text{nm}$, $\kappa =4$, $\theta =120°$, values of $t$ from top to bottom are 3, 6, and 9 nm; (c) ratio between gate coupling factor $\Gamma$ for 3D electrodes with $\theta =120°$ and for 2D electrodes, both with $L_{ox}=3\text{nm}$, $\kappa =4$.

Figure 5

(Color online) Electrodes with nonuniform cross section—role of the electrode angle. (a) Normalized potential for the three-dimensional device shown in Fig. 4a with $L_{ox}=3\text{nm}$, $t=3\text{nm}$, $s=2\text{nm}$, $\kappa =4$, values of $\theta$ from top to bottom are $0°$, $30°$, $60°$, $90°$, $120°$, $140°$, $160°$, and $180°$; (b) gate coupling as a function of electrode angle for the three-dimensional device shown in Fig. 4a with $L_{ox}=3\text{nm}$, $t=3\text{nm}$, $\kappa =4$.

Figure 6

(Color online) Electrodes with nonuniform cross section—role of the dielectric constant. (a) Normalized potential for the three-dimensional device shown in Fig. 4a with $L_{ox}=3\text{nm}$, $t=3\text{nm}$, $s=2\text{nm}$, $\theta =120°$, values of $\kappa$ from bottom to top are 1, 4, and 80; (b) Normalized potential for the three-dimensional device shown in Fig. 4a with $L_{ox}=3\text{nm}$, $t=3\text{nm}$, $s=5\text{nm}$, $\theta =120°$, values of $\kappa$ from bottom to top are 1, 4, and 80; (c) Gate coupling for the three-dimensional device shown in Fig. 4a with $L_{ox}=3\text{nm}$, $t=3\text{nm}$, $\theta =120°$.

Figure 7

(Color online) Electrodes with nonuniform cross section—role of the dielectric thickness. (a) Normalized potential for the three-dimensional device shown in Fig. 4a with $t=3\text{nm}$, $s=2\text{nm}$, $\theta =120°$, $\kappa =4$, values of $L_{ox}$ from top to bottom are 2, 3, 5, 10, 25, and 50 nm; (b) Gate coupling as a function of dielectric thickness for the three-dimensional device shown in Fig. 4a with $t=3\text{nm}$, $\theta =120°$, $\kappa =4$.

Figure 8

(Color online) Electrodes with uniform and nonuniform cross section—effect of molecular polarizability. (a) Normalized potential for the two-dimensional device shown in Fig. 1a with $L_{ox}=3\text{nm}$, $t=6\text{nm}$, $\kappa =4$, for bare electrodes and electrodes with a polarizable sphere of diameter $s$ and $\kappa ={10}^6$ positioned at the top of the nanogap; (b) Normalized potential for the three-dimensional device shown in Fig. 4a with $L_{ox}=3\text{nm}$, $t=3\text{nm}$, $\kappa =4$, for bare electrodes and electrodes with a polarizable sphere of diameter $s$ and $\kappa ={10}^6$ positioned at the top of the nanogap; (c) derivative of the normalized potential with respect to vertical position for the three-dimensional device shown in Fig. 4a with $L_{ox}=3\text{nm}$, $t=3\text{nm}$, $\kappa =4$.