Modeling extended contacts for nanotube and graphene devices

Carrier injection into carbon nanotubes and graphene nanoribbons, contacted by a metal coating over an arbitrary length, is studied by various means: Minimal models allow for exact analytic solutions, which can be transferred to the original system with high precision. Microscopic ab initio calculations of the electronic structure at the carbon-metal interface allow us to extract—for Ti and Pd as contacting materials—realistic parameters, which are then used in large scale tight-binding models for transport calculations. The results are shown to be robust against nonepitaxially grown electrodes and general disorder at the interface, as well as various refinements of the model.

Figure 1

(Color online) Minimal model for extended contacts solvable analytically: a linear chain of identical atoms (one orbital per atom) with hopping integral $\gamma$ between nearest neighbors. To the left, the chain continues infinitely; at the right end, $N$ atoms are contacted, each by an independent wideband lead of strength $\Delta$. For defining the transmission, the system is virtually split into three regions: the conductor $C$ and the leads $L$ and $R$.

Figure 2

(Color online) Breit–Wigner resonance in the conductance of an extended molecule sandwiched between two leads. The internal hopping is fixed as $2\gamma ={\Gamma }_{\mathrm{L}}$ to allow an optimal match to the left contact. In the case $N=1$, the system is identical to the molecular junction described by Eq. (1). The transmission shows the shift of the Breit–Wigner peak toward lower ${\Gamma }_{\mathrm{R}}$ with growing $N$. The functional form of this shift can be approximated for large $N$ as ${\Gamma }_{\mathrm{R}}={\Gamma }_{\mathrm{R}}\ln N∕N$ [see Eq. (7)].

Figure 3

(Color online) Transmission through the system displayed in Fig. 1, as given in Eq. (A8). Top panel: for a fixed contact length $N$, starting from low $\Delta$, the transmission first improves, reaches an optimum, and then degrades again at high $\Delta$. Bottom panel: For fixed contact strength $\Delta$, transmission improves with growing $N$ and saturates for large $N$.

Figure 4

(Color online) Contact reflection in the system displayed in Fig. 1, as given in Eq. (2). At fixed contact strength $\Delta$, with growing $N$, the contact becomes more transparent and saturates at an $N$-independent value.

Figure 5

(Color online) Transmission $T$ (top panel) and contact reflection $R$ (bottom panel) in the system displayed in Fig. 1 at fixed energy $E=\epsilon$ for varying contact length $N$ and selected values of the contact strength $\Delta$.

Figure 6

(Color online) $\Delta$ dependence of the contact reflection $R$ for various even (upper panel) and odd (lower panel) contact lengths $N$. Solid: the exact value as given in Eq. (2). Dashed: the limit $R_{N\to \infty }$ given in Eq. (4), along with its approximation ${\Delta }^2∕64{\gamma }^2$, valid for $\Delta ⪡1$. Dotted: the approximation $\exp (-N\Delta ∕\gamma )$, valid for even $N$ in the $N$-resonant regime. The inset shows the identical data in the semilogarithmic scale, further illustrating the precision of the $\exp (-N\Delta ∕\gamma )$ approximation in the $N$-resonant regime.

Figure 7

(Color online) The contact reflection for even $N$ obtained from Eq. (5). Quite visible are the two regimes separated by the minimal line ${\Delta }_{\mathrm{opt}}\left(N\right)$. The “exact” value for ${\Delta }_{\mathrm{opt}}$ is the true minimum for fixed $N$. The “approximate” value comes from Eq. (7).

Figure 8

(Color online) Generalized model including nondiagonal terms: the individual wideband leads for each atom in the extended contact region are replaced by a metal with an internal structure, here modeled as a 2D-square lattice. The new parameters are ${\gamma }_{\parallel }$ and ${\gamma }_{\perp }$, describing the internal hopping in the lattice parallel and perpendicular to the contact surface, as well as ${\gamma }_c$, describing the hopping at the contact. For simplicity, we consider only the isotropic case ${\gamma }_{\parallel }={\gamma }_{\perp }$. This leaves us with the single effective parameter $\Delta ={\gamma }_c^2∕{\gamma }_{\perp }$.

Figure 9

(Color online) Nondiagonal contacts: Reflection $R$ of the generalized model displayed in Fig. 8. Relating the parameter $\Delta ={\gamma }_c^2∕{\gamma }_{\perp }$ to the parameter $\Delta$ of the wideband leads, the results are qualitatively similar to those of the original model (Figs. 5, 6). One prominent difference is the enlarged reflection $R\left(\Delta \right)$ for $\Delta ∕\gamma$ between 1 and 10: While the diagonal self-energy was uniform for every atom along the contact, the nondiagonal self-energy now is sensitive to the edge of the contact. For large values of $\Delta$, where only the atoms near the edge contribute to the transport, this causes the visible deviation from the $R={\Delta }^2∕{\alpha }_2^2$ law. The same reason is behind the visible irregularities in the resonant oscillations of $R\left(N\right)$.

Figure 10

(Color online) Extended contacts to a (6,6) CNT (left) and the corresponding graphene nanoribbon. Top panel: density of states with characteristic van Hove singularities. The zigzag-edge state in the nanoribbon causes a peak at $E=E_F$. Center panel: The value ${\alpha }_1=-2N\Delta ∕\ln R$, here computed for $N=40$ and $\Delta ={10}^{-5}\mathrm{eV}$, lies already very near to the limiting case ${\alpha }_1=N_{\mathrm{ch}}∕\pi \rho$. Bottom panel: The value ${\alpha }_2=\Delta ∕\sqrt{R}$ is well converged for $N\to \infty$ and $\Delta ={10}^{-2}\mathrm{eV}$. Note the suppression of both ${\alpha }_1$ and ${\alpha }_2$ in the ribbon at $E_F$ where the presence of the localized edge state suppresses the conductance in the contact region.

Figure 11

(Color online) Differential conductance through a two-terminal setup of a linear chain with symmetric, infinite-length extended contacts. The central region consists of $N_{\text{conductor}}=200$ atoms. The distance between the Fabry–Pérot oscillations in the gate voltage $V_g$ depends on the total gate capacitance $C_g$ as $\delta V_g=e∕C_g$. The extent of the diamonds in direction of the bias voltage $V_b$ depends directly on the level spacing $\delta E=v_{F}h∕2L_{\text{conductor}}=2\gamma \pi ∕N_{\text{conductor}}$ as $\delta V_b=e\delta E$. Top: strong coupling $\Delta =10\gamma$, leading to an extremely short effective contact length producing sharp resonances and a distinct diamond pattern. Bottom: moderate coupling $\Delta =1\gamma$ leading to an effective contact length of about four unit cells. The oscillations have a sinoidal shape with strongly reduced amplitude. For yet weaker coupling as it is to be expected for Pd or Ti, the amplitude is rapidly reduced even further making the oscillations undetectable. In this case, a stong defect within the contact region may act as a point of scattering and recover sharp resonances.

Figure 12

(Color online) Effects of two different kinds of disorder on extended contacts: relatively weak fluctuation disorder with varying contact strength ${\Delta }_i$ on each atom $i$ and stronger dilution disorder with only a randomly selected fraction of the atoms in the contact region attached to a lead. In each case, the parameter $\Delta$ refers to the average contact strength. For details, see the text.

Figure 13

(Color online) Analysis of the hybridization between a graphene sheet and a metal monolayer. As visible in the upper scheme, the hybrid band structure matches well with an overlay of the band structures of the two individual systems rigidly shifted in energy and hybridization at some band crossings. Highlighted in the $\mathrm{Pd}∕\mathrm{C}$ and $\mathrm{Ti}∕\mathrm{C}$ band structures are the regions of interest, i.e., those hybridizations that contribute most to the electron injection.