Applied Physics of Carbon Nanotubes

1 From Quantum Models to Novel Effects to New Applications: Theory of Nanotube Devices by S.V. Rotkin (p. 3-4)

Classical and quantum effects in the physics of nanotube devices are presented. In particular, weak screening in one–dimensional systems is shown to essentially modify textbook theory of field–effect devices and lead to an interesting dependence of the device characteristics on geometrical factors.

The capacitance of a nanoscale device has two main components: a classical geometric capacitance and a quantum term. The latter is related to a finite density of states of the nanosystem. Derivation of this density of states in the presence of external perturbations is a difficult task. We present some examples of the modification of the nanotube bandstructure by external perturbations.

Electric fields can be used for band gap engineering in nanotubes, which may be translated into the device function. The concept of the Metallic Field–Effect Transistor is proposed. This device shows, at least theoretically, metallic conductance in the ON state and insulating behavior in the OFF state, which may be important for applications.

1.1 Introduction: Classical vs. Quantum Modelling

One of the expectations of nanotechnology, an area foreseen by Dr. Richard P. Feynman in 1959 [1], is that we may be able to access quantum properties of materials, which may ultimately lead to new applications and new device operations, which are not possible at the macroscale. To enable this new technology a theory that can make both qualitative and quantitative predictions is needed, whether it be a classical or quantum theory or a combination of both. In this chapter we give a few examples and present the quantum vs. the classical approach using recent results from our modelling of nanotube based devices.

Carbon nanotubes (NTs), discovered in 1991 [2], nowadays represent a new class of electronic materials. The electronic properties of NTs depend on their symmetry [3]. This is not unusual but for a single–wall nanotube (SWNT) there are just a couple of geometrical parameters: a curvature radius, R, and a helicity angle (the measure of chirality of the SWNT lattice), which solely define transport [4], optical [5] and even, to some extent, chemical [6,7] properties of a SWNT.

Knowledge of these two parameters will allow us to divide all possible NTs into several distinct classes. Two thirds of SWNTs have a forbidden band gap, which makes them semiconductors. The band gaps of the semiconductor SWNTs are in the optical region (near–IR/visible), depending on the value of R. The experimental fact that, over a wide range of R, the energy gap is proportional to the curvature, 1/R [5], is a clear manifestation of a simple quantum effect of a space quantization of an electron. When winding around the tube circumference, the electron acquires a phase. After making a full turn the phase has to be 2π, which results in a so–called quantization condition.

The quantization energy sets the separation between the conduction and valence bands, and therefore the optical gap. In a similar way, as the atomic size (atomic number) of an element in the Periodic Table solely defines the properties of the substance, the curvature radius and the helicity angle of a SWNT define its electronic material properties. One third of SWNTs are either metals or narrow gap semiconductors (often called “quasi–metals” in NT literature). The difference between the gap size in the last two SWNT classes appears in the second order of the curvature, 1/R2. The gap in the quasi–metallic SWNT scales as γ/R2, where γ  2.7 eV is the hopping integral which gives the NT energy scale. In the same second order of the curvature, 1/R2, the first SWNT class (semiconductors) splits into two sub–classes by their chirality. All this constitutes a specific “Periodic Table” of nanotubes.

The other important property of nanotubes relates to their third dimension: we have already considered the radius and chirality of the tube but not the length. Recent success in NT synthesis (see also [8]) allowed experimental study of NTs with R ∼ 1 nm and lengths of about several hundreds to thousands of microns, which implies an aspect ratio of 1:100,000 and greater. Certainly, this object must show physics similar to the physics of a one– dimensional (1D) wire, for example, a weak screening. Below we demonstrate that the weak 1D screening properties of NTs have important consequences for electronic devices.