CNT Bands Learning Materials

by Denis Areshkin

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By completing the CNTBands, you will be able to:

a) understand the relationship between system geometry (roll vector of nano-tube or crystallographic direction of nano-ribbon) and its band structure.

b) Understand the importance of employing screw symmetry in calculations of electronic properties of nano-tubes (Homework problems and solutions).

The specific objectives of the CNT-Bands are:

CNTBands-Scheme_Small.png

Recommended Reading

  • [1] C.T. White, D.H. Roberrtson, and J.W. Mintmire, “Helical and rotational symmetries of na-noscale graphitic tubules”, PRB v. 47, p.5485 (1993)
  • [2] D.A. Areshkin, D. Gunlycke, C.T. White, “Ballistic Transport in Graphene Nanostrips in the Presence of Disorder: Importance of Edge Effects”, NANO LETTERS v.7, p.204 (2007)
  • [3{{{]}}} D.A. Areshkin, B.K. Nikolic, “I-V curve signatures of nonequilibrium-driven band gap collapse in magnetically ordered zigzag graphene nanoribbon two-terminal devices”, PRB v.79, 205430 (2009)

Demo

CNTBands : First Time User Guide

Exercises:

Historical Note

There is a large amount of online material on the history of nanotubes discovery (e.g. time line for nanotube research in Wikipedia). However, one interesting, but not widely known historical fact is worth to be mentioned here. The possibility of nanotubes existence was predicted with the help of computer Density Functional simulations by C.T. White (Naval Research Lab, Washington DC, USA) simultaneously with their experimental observation by S. Iijima (NEC Corporation, Tsukuba, Japan). The research of C. White was conducted in attempt to find truly metallic one-dimensional systems. Such structures should not be susceptible to Jahn-Teller distortion, which breaks symmetry of the system and opens a finite (though sometimes very small) band gap. Fifteen years later C. White privately supposed that (carbon) nanotubes are the only truly metallic 1D systems and the existence of other 1D systems is unlikely. This exceptional property of nanotubes is due to the high symmetry of nanotubes and high strength of carbon-carbon bonds, which make the temperature of Jahn-Teller transition (from distorted to undistorted state) very low. Initially C. White considered nanotubes purely hypothetical structures with low chances to be synthesized, but has reported his findings at the last day of APS March meeting in 1991 (when most attendees have already left). After that he was contacted by S. Iijima who made a report on experimental observations of nanotubes at the same conference. One of the reasons for explosive scientific interest in nanotubes was due to the existence of comprehensive theoretical explanation of nanotubes properties at the time of their observation by S. Iijima. That partly explains why multiple previous observations of nanotubes (starting in 1952;USSR) and even US patent on nanotubes synthesis (by H.G. Tennett from 1987) have not created such inter-est as S. Iijima’s work.

Key Properties of Carbon Nanotubes

One of the most important early theoretical advances was the explanation of why CNT conductance is very stable with respect to miscellaneous distortions such as phonons and static defects. The average mean free electron path in CNT at room temperature is of the order of microns, which by far exceeds the mean free paths in nano-wires or in systems of higher dimensionality.


The absence of Jahn-Teller distortion allows bands to cross at the Fermi level (EF). To illustrate the importance of that fact let us first consider what happened if CNTs were susceptible to Jahn-Teller distortion. Such distortion opens a band gap; electronic dispersion of above mentioned hypothetical CNT is depicted in Fig. 1a and the blowup in the vicinity of EF is in Fig. 1b. Electron-phonon scattering must comply with energy and momentum conservation laws. Due to high strength of C-C bonds, the energy of optical phonons is rather high – about 0.15 eV. Therefore, for electrons with energies near EF there is virtually no scattering on optical phonons, and only acoustic phonons should be considered. Dispersion of acoustic phonons can be approximated with a linear function $$\varepsilon=a\mid q\mid$$ within a wide region around gamma point, where $$\varepsilon$$ is the phonon energy, q is the phonon wave number, and a is a positive constant (red curve in Fig. 1b). Hence, the conservation laws are:

Conservation_01.png

Here k is the wave number of electron, and indexes 1 and 2 denote respectively initial and final states. For any k1 and q1 these two equations always have non-trivial solutions with $$k_{1}\neq k_{2}$$ and $$q_{1}\neq q_{2}$$, i.e. conservation laws do not prohibit electron-phonon scattering.

Bands_Touching_vs_Crossing_Small.jpg Figure 1



Density of electronic states in 1D systems exhibits van-Hove singularities at the bands inflection points, including band edges. Figure 1-c schematically depicts two such singularities near EF. The rate of elastic scattering on static random uncorrelated defects (e.g. potential distortion imposed by a substrate supporting CNT) is proportional to the fourth power of DOS [Ref. 1]. Hence the presence of even small band gap near EF substantially deteriorates conductance of 1D system due to both elastic and inelastic scattering. In the absence of Jahn-Teller distortion bands cross each other. That makes the dispersion linear (Figs 2a-b). The conservation laws

Conservation_02.png

have only trivial solution $$k_{1}=k_{2}$$ and $$q_{1}=q_{2}$$ for $$a\neq b$$ (for CNT a << b), which means the scattering on acoustic phonons in metallic CNTs is prohibited to the first order. The DOS in the vicinity of EF is constant and small (just one open conduction channel), which makes elastic scattering on static defects low, and gives rise to extremely robust conductance and long electron mean free path.


All metallic CNT’s have two dispersion branches crossing the Fermi energy, each brunch corresponds to one conductance quanta for a given spin. Quantum conductance in 1D periodic structures follows from the trivial identity:

ConductanceQuantization.png,

where v is group velocity of electron. As it is demonstrated in Problem 4, in the nearest neighbor π-orbital tight-binding approximation the value of ‘v’ at EF is the same for all metallic CNTs. That means the larger is the diameter of CNT, the smaller is the density of states in the energy window near EF where only two conduction channels are available. Therefore conductance robustness in CNTs with respect to distortion potential is a trade of the two factors. CNT diameter increase reduces DOS in the two-band conduction window and hence reduces the scattering probability. The second factor is the mechanical stability of CNT, which got compromised if the CNT diameter becomes too large. Figures 2a demonstrates critical angle for {8,8} CNT. If the bending angle is less than critical, the CNT transmission within a two-bands window near EF remains nearly ideal (Figure 2b). Only when a kink forms (Figure 2c) and sp2 bonding is disrupted, the transmission within a two-bands window deviates from the ideal value.

Bended_8x8_CNT.png Figure 2a $$~ ~~$$ {8,8} CNT bended by 56 degrees
ConductanceEvolution_vs_BendingAngle_7_10_CNT_Small.png Figure 2b $$~ ~~$$ Transmission for (7,10) CNT bent to 55 degrees in a Self-Consistent Environment-Dependent Tight Binding approximation
PotentialScan_HelicalCNT.jpg Figure 2c $$~ ~~$$ Electrostatic potential in a Self-Consistent Environment-Dependent Tight Binding approximation in the plane raised above {7,10} CNT by a distance slightly exceeding its radius. Only kinked portion of CNT (solid blue lines) appears above the plane.

CNT Challenge Problem

In many experimental reports on electrical or optical properties of CNTs it is essential to estimate the influence of supporting substrate. The crystal lattice of the substrate is usually (unless graphite is used) out of registry with CNT, i.e. the influence of the potential associated with the atoms of the substrate may be viewed as a random potential. The localization length of a randomly distorted CNT is defined as its length, for which the logarithm of the ratio of ideal transmission to the transmission in the distorted CNT equals 2. Equations. Figure 3 plots analytical expression for localization length along with the

Fig_9_1110_Localization_Length_One_Channel_01.png

Key Properties of Graphene Nano-Ribbons

Graphene – Geometry dependent band gap – Disorder-induced mobility gap in A-GNRs – Spin polarization along the edges in Z-GNRs – The possibility to remove spin polarization, and hence to reduce spin-induced band gap in Z-GNRs by passing current. – Simple computational models suggest that Z-GNR 60° turns are highly reflective and 120° are virtually transparent. These computations, however, do not include spin-polarization effect. Therefore, if the effect will be observed experimentally, the transmis-sion differences between 60° and 120° turns is expected to be less pronounced.

Intra-valley vs. Inter-valley Scattering in Z-GNRs

GNR Challenge Problem

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