Carbon nanotubes


Two-probe systems	The Magnetic Tunnel Junction Builder

Table of Contents

Creating a nanotube two-probe system

One of the hottest research areas within nanotechnology is the topic of carbon nanotubes. For that reason, Virtual NanoLab (VNL) also comes with a Nanotube Grower tool that can be used to construct and design carbon nanotubes. To demonstrate some of the principal capabilities of the Nanotube Grower, we will describe the steps needed for building and computing the band structure of a perfect carbon nanotube.

1. Open the Nanotube Grower

The Nanotube Grower is opened by double-clicking the associated icon on the VNL Toolbar. An active Nanotube Grower window is shown in Figure 33.

A nanotube is essentially characterized by just two integer parameters, known as the tube indices (n,m). The tube indices and distance between the carbon atoms can be specified in the left panel, which also shows some general information about the chosen tube.

Just as for any 3D plot in VNL, the object in the Preview window can be transformed by right-clicking the preview window and choosing Properties. Similar to the Nanoscope, you may use the dialog for changing the radius of the atoms, the background color, as well as other properties of the 3D scene.

Geometry preview of a (4,1) carbon nanotube. The size of the atoms has been reduced to emphasize the tube bond structure. The entire tube has been repeated 6 times along the tube axis. carbon nanotube (4,1)

Figure 33: Geometry preview of a (4,1) carbon nanotube. The size of the atoms has been reduced to emphasize the tube bond structure. The entire tube has been repeated 6 times along the tube axis.

2. Inspect the tight-binding band structure

Click the Band Structure tab to get a preview of the band structure of the nanotube in the energy area around the Fermi energy. Like the 3D preview, this plot usually updates automatically when the tube indices are changed. Sometimes, however, if there are many atoms/bands, the program waits to re-display the plots until the button Show (or any point inside the plot window) is clicked, in order not to slow down the interface too much. This is automatically handled by VNL (in fact, all 3D windows behave the same way) by continuously measuring the time it takes to set up the plots.

Figure 34: Tight-binding band structure of a (4,1) carbon nanotube.

If we stick to the default (4,1) tube for the moment, VNL predicts that this particular tube is semi-metallic. At the same time, the band structure plot appears to show a metallic structure. This is due to the fact that the preview of the band structure is generated on-the-fly using an analytic third-nearest-neighbor tight-binding formula. In this approximation, all tubes with indices n and m offset by an integer multiple of 3 will turn out metallic. This happens because the tight-binding formula does not take the proper hybridization of s and p orbitals into account. The hybridization effects are due to the fact that the graphene sheet is “rolled up” into a tube. On the other hand, the band gap displayed in the left side of the panel is extracted from a semi-empirical interpolation formula, which does include the curvature effect. The measurements on which this method is based reveal that only armchair tubes (n = m) are truly metallic – which also follows from a more careful theoretical analysis.

3. Create a metallic (4,4) tube

Here we will construct a metallic nanotube, which would is more suitable for building conductive electrodes, which we will do in a later step. Therefore, change the tube indices to (4,4) and click Save/Save As.

The configuration stored in the resulting NanoLanguage script is of bulk type. In the following, we demonstrate how to create a perfect nanotube by setting this up as a two-probe system.

Creating a nanotube two-probe system

Constructing a perfect carbon nanotube as a two-probe system will serve as a starting point for studying transport properties of the tube under high bias voltages as well as in the presence of defects. In this tutorial, we will limit ourselves to the first necessary step: the construction of a perfect nanotube not as a bulk, but as a two-probe system. For this purpose, we need to define

one part of the infinite, perfect nanotube as the left electrode
one part as the central region
one part as the right electrode.

As a consequence, we first need to “cleave” the tube in half and then “glue” it back together. All these operations are carried out using the Atomic Manipulator.

In this case, it would appear that a central region is redundant; by just gluing the two electrodes together, we would immediately have a perfect nanotube. For numerical reasons, this is not recommended. If we did set up the system this way, there would be direct interactions between the two electrodes and this is not allowed in the algorithms used for calculating two-probe systems. This is a general consideration that always must be observed: Always make the number of surface layers large enough, even if the central region is identical to the electrodes.

1. Cleave the nanotube

The first thing you must do is to locate the NanoLanguage script that contains the nanotube configuration created above. Then drop the file onto an open Atomic Manipulator tool.

Since the Nanotube Grower creates tubes that are aligned along the Z-axis, the tubes can immediately be turned into electrodes by right-clicking the left panel and choose Cleave from the context menu. In this case, there are no surface parameters, since its definition makes no sense for one-dimensional systems; cleaving a nanotube just means “cutting it in half”.

In order not to neglect any matrix elements, you may still set the number of tube Periods within the electrode cell. Normally, however, a single period is sufficient for including all second-order matrix elements – even for armchair tubes, which have the shortest possible period (they consist of just two layers).

2. Check the electrode cell

Although the nanotube is one-dimensional, the electrode unit cell is kept three-dimensional for numerical reasons. As discussed earlier, the size of the electrode cell is chosen sufficiently large to make sure that the nanotube does not interact with its own repeated copies (due to the periodic boundary conditions in the transverse plane). Where you not able to control the size of the electrode unit cell in the X-Y plane, this would lead to problems for two-probe systems: Imagine a molecule inserted between the two tube electrodes; if the molecule is larger than the electrode unit cell, the two-probe system will not be valid due to overlaps between the repeated copies of the molecule caused by the periodic boundary conditions in the transverse plane .

So, when nanotubes are used as electrodes , an additional parameter, called the Padding Factor, is available for extending the electrode unit cell in the X-Y plane. By default, the padding factor is zero; a value of 1 corresponds to twice as large a cell, whereas a negative value (which is not recommended) makes the cell smaller than the default size.

3. Represent the perfect nanotube as a two-probe system

As described above, we are forced to add a few additional nanotube periods to the central region in order to avoid any direct interactions between the electrodes. Therefore, drop the (4,4) carbon nanotube again on the opened Manipulator. If a nanotube is dropped on a Manipulator that already contains a two-probe system, it does not replace the electrodes. Instead, it is imported as part of the central region where it can be positioned and repeated (but not rotated). This feature makes it possible to set up e.g. metal-nanotube-metal two-probe systems.

We also need to take make sure that the surfaces correspond to an integer number of tube periods. Let us use two layers (i.e. one nanotube period) both to the left and the right, and two periods in the middle. In total, there will be 8 layers (four periods) and thereby 64 atoms in the central region. To obtain one full period in both the left and right segments, we need to set the left/right surface layers to 2 (see the VNL Manual for more details).

	Note
It is actually not the number of repetitions or the number of surface layers which is relevant, but rather the total number of layers in the central region, which must be of a sufficient size to achieve proper screening for a perfectly periodic system. This is simply a requirement in the algorithms used to calculate two-probe systems. So, alternatively, we could have used a single period in the central region, and compensate for this by increasing the surface layers accordingly; in fact, we could have skipped the nanotube in the central region completely and just used the surfaces to define the central region. The motivation for setting up the system as we did was pedagogical (rather than physical or numerical) with a primary purpose of illustrating how to include a nanotube in the central region. It should also be noted that a surface layer is defined as atoms that have a unique Z-value. For the (4,4) nanotube, it is pretty convenient, but for other tube indices it means that a layer can contain a single atom, which might require a bit more attention to get the correct alignment.

Note

It is actually not the number of repetitions or the number of surface layers which is relevant, but rather the total number of layers in the central region, which must be of a sufficient size to achieve proper screening for a perfectly periodic system. This is simply a requirement in the algorithms used to calculate two-probe systems. So, alternatively, we could have used a single period in the central region, and compensate for this by increasing the surface layers accordingly; in fact, we could have skipped the nanotube in the central region completely and just used the surfaces to define the central region. The motivation for setting up the system as we did was pedagogical (rather than physical or numerical) with a primary purpose of illustrating how to include a nanotube in the central region.
It should also be noted that a surface layer is defined as atoms that have a unique Z-value. For the (4,4) nanotube, it is pretty convenient, but for other tube indices it means that a layer can contain a single atom, which might require a bit more attention to get the correct alignment.

A perfect carbon nanotube, set up as a two-probe system in order to prepare it for a calculation. For clarity, the unit cells of the left electrode and the nanotube in the central region have been hidden.

Figure 35: A perfect carbon nanotube, set up as a two-probe system in order to prepare it for a calculation. For clarity, the unit cells of the left electrode and the nanotube in the central region have been hidden.

Finally, to obtain a perfect match for the periodicity of the tube, we need to position the nanotube in the central region and set the central region width. If the periodicity is broken just slightly, you will typically see a gap in the transmission spectrum around the Fermi level . The same will happen if the electrode cell is too small in the transverse directions resulting in residual electrostatic interactions (or direct basis set overlaps) between the repeated copies of the electrodes.

A perfect match is achieved by setting the origin of the nanotube 3 times the nanotube period length (7.389 Å), whereas the central region width should be one layer (half a period) shorter, that is 6.1575 Å.

Click Save/Save As to create aNanoLanguage script file containing a perfect nanotube, but now stored as a two-probe configuration. The configuration is now ready for setting up electronic transport calculations.