Recently, there have been significant advances in the fabrication and demonstration of individual molecular electronic wires and diode switches. This paper reviews those developments and shows how demonstrated molecular devices might be combined to design molecular-scale electronic digital computer logic. The design for the demonstrated rectifying molecular diode switches is refined and made more compatible with the demonstrated wires through the introduction of intramolecular dopant groups chemically bonded to modified molecular wires. Quantum mechanical calculations are performed to characterize some of the electrical properties of the proposed molecular diode switches. Explicit structural designs are displayed for AND, OR, and XOR gates that are built from molecular wires and molecular diode switches. The diode-based molecular electronic logic gates are combined to produce a design for a molecular-scale electronic half adder and a molecular-scale electronic full adder. These designs correspond to conductive monomolecular circuit structures that would be one million times smaller in area than the corresponding micron-scale digital logic circuits fabricated on conventional solid-state semiconductor computer chips. It appears likely that these nanometer-scale molecular electronic logic circuits could be fabricated and tested in the foreseeable future. At the very least, such molecular circuit designs constitute an exploration of the ultimate limits of electronic computer circuit miniaturization.

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Architectures for molecular electronic computers. I. Logic structures and an adder designed from molecular electronic diodes

We investigate curvature effects on geometric parameters, energetics, and electronic structure of zigzag nanotubes with fully optimized geometries from first-principle calculations. The calculated curvature energies, which are inversely proportional to the square of radius, are in good agreement with the classical elasticity theory. The variation of the band gap with radius is found to differ from simple rules based on the zone folded graphene bands. Large discrepancies between tight binding and first-principles calculations of the band gap values of small nanotubes are discussed in detail.

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Systematic ab initio study of curvature effects in carbon nanotubes

Electronic states of carbon nanotubes in a magnetic field perpendicular to the axis are studied in a tight-binding model and the validity of k · p theory is examined. The energy bands near the Fermi level calculated in the tight-binding model agree with those of k · p in a wide magnetic-field range except in extremely high fields where the flux passing through the unit cell of a two-dimensional graphite is comparable to the flux quantum, i.e. up to ∼4000 T. A slight deviation from the k · p result in weak magnetic fields can be explained by the inclusion of higher order terms in the k · p expansion leading to trigonal warping of bands.

Energy bands of carbon nanotubes in magnetic fields

The current methods of synthesis and investigation of carbon nanotubes are reviewed. The interconnection between structural peculiarities and electronic characteristics of carbon nanotubes has been studied. The recently developed methods of growth of similar nanotubes with pre-assigned characteristics are described. The development of these methods offers the possibility of mass production and application of relevant devices. The phenomenon of electron field emission from carbon nanotubes and possibilities of its usage in flat panel displays, gas discharge devices, etc. is analyzed. Prospects and real achievements in the area of application of nanotubes as cold electron emission sources are discussed.

Carbon nanotubes and their emission properties

We present first-principles calculations on the band structures of 40 different small diameter $(d)$ single-wall carbon nanotubes (SWCNTs), including 14 chiral ones, employing density functional theory (DFT) within the local density approximation (LDA), using the Vienna ab initio simulation package (VASP). The band gaps are calculated and discussed for all of the tubes. From small to large diameters, the gap of semiconducting zigzag tubes first increases, then reaches a maximum of about $1\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ for (11,0), after which it decreases, approximately as $1∕d$, showing a ``buckling'' around this average behavior. The smallest diameter zigzag tubes are all metallic, due to $\ensuremath{\sigma}\ensuremath{-}\ensuremath{\pi}$ mixing caused by high curvature. The Fermi wave-vector of armchair tubes shows a downshift from its ideal, zone folding expected value; this shift is proportional to $1∕{d}^{\phantom{\rule{0.1em}{0ex}}2}$. Eight of the ZF-metallic tubes (4 zigzags, 4 chirals) show a small gap in the band structure. For the zigzag tubes, this band gap is roughly given by the formula $\ensuremath{\Delta}=1.99∕{d}^{\phantom{\rule{0.1em}{0ex}}2}+140.9∕{d}^{4}$ ($\ensuremath{\Delta}$ in eV, $d$ in \AA{}). The appearance of the $1∕{d}^{4}$ higher order correction term is due to high curvature at small diameters, however it's apparently overestimated in our calculations. This overestimation can likely be eliminated by considering many-electron effects.

First-principles calculations for the electronic band structures of small diameter single-wall carbon nanotubes

The effect of the mixing between \ensuremath{\pi} and \ensuremath{\sigma} orbitals upon the energy gaps (${\mathit{E}}_{\mathit{g}}$'s) of semiconducting carbon nanotubules has been theoretically studied within the tight-binding approximation. The magnitude of a chirality-dependent energy gap (${\mathit{E}}_{\mathrm{\ensuremath{\theta}}}$) has been estimated. This term cannot be neglected for tubules with a small radius. The relation of ${\mathit{E}}_{\mathit{g}}$ to R, ${\mathit{E}}_{\mathit{g}}$\ensuremath{\propto}${\mathit{R}}^{\mathrm{\ensuremath{-}}1}$, no longer holds for small-radius tubules. An approximate expression for ${\mathit{E}}_{\mathit{g}}$, which can be applicable even to small-radius semiconducting tubules, is presented. In addition, it is shown that the ${\mathit{E}}_{\mathrm{\ensuremath{\theta}}}$ is well correlated with a fraction of the \ensuremath{\sigma} component in the wave functions for the conduction-band bottom and valence-band top.

Energy gaps of semiconducting nanotubules.

The electronic structure of graphitic tubules with general chirality is thoroughly studied by means of a recursion tight-binding method. A detailed investigation of densities of states around the Fermi level for various types of tubules confirms the previous theoretical prediction that tubules become metals (M), narrow-gap semiconductors (NS's), or moderate-gap semiconductors (MS's), depending on an integer index k specifying the geometrical structure of the tubules. We point out that the MS's are systematically classified into two types, MS1 and MS2, according to this index. Namely, tubules correspond to M for k=0; otherwise they belong to semiconductors, being NS's, MS1, or MS2, respectively, for k modulo 3=0, 1, or 2. Our result is compared with an approximate expression for energy gaps of MS's, derived by White et al. and we discuss the deviation of our results from this expression. An empirical expression for the energy gap, which elucidates such a deviation, is suggested.

H. Yorikawa

Papers

Energy gaps of semiconducting nanotubules.

Electronic properties of semiconducting graphitic microtubules.