Chapter 3 Method
3.2 The transmission of the polyene junctions in between
3.2.1 Introduction to the system
We etch a (𝑛, 𝑛) CNT bulk to become two angled-cut (𝑛, 𝑛) CNT leads. The tube edges of the two leads are terminated by the hydrogen atoms. There are 2𝑛 contact sites along each lead edge. These contact sites are labeled by 2𝑙 + 1 and 2(𝑛 + 𝑙 + 1), where 𝑙 = 0,1, ⋯ , 𝑛 − 1, as shown in Fig. 3.4. The gap between two leads is then bridged by the polyene molecule(s). In our study, we consider one- and two-polyene cases for the molecular junction, and for the convenience of the ab initio calculation, we mainly consider 𝑛 = 8 for the CNT leads.
Fig. 3.4: The labels of the sites on the rim of an angled-cut (8,8) CNT lead. The contact sites are emphasized by solid circles.
We first consider the one-polyene junctions. We use one polyene which possesses 17 carbon atoms to bridge the gap between two leads. Because of the symmetry of the leads, there are effectively 8 cases. The contact combination can be represented as 2𝑙 + 1, where 𝑙 = 0,1, ⋯ ,7. These cases are illustrated in Fig. 3.5.
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Fig. 3.5: One-polyene junctions. We use the labels of the contact sites to represent each case.
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As for the two-polyene junctions, we use two identical polyenes, each with 17 carbon atoms, to bridge the gap between the two leads. In our study, the two bridging polyenes are parallel to the z direction of the leads. With the contact site labels just introduced, each combination of the two contact sites is labeled by a pair of numbers.
All independent combinations are illustrated in Fig. 3.6. Note that we have excluded the cases of combinations (15,18) and (1,32), where the two bridging polyenes are too close and cannot be realized in the ab initio calculations.
Fig. 3.6: Two-polyene junctions. We use the labels of the contact sites to represent each case.
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3.2.2 Tight-binding model calculations
In the simplest tight-binding model, we use the same hopping energy 𝑡 for both the angled-cut (8,8) CNT leads and the polyenes, and assume the same 𝑡 for the coupling 𝜏 between the lead and the molecule. The on-site energy of each carbon atom, in the angled-cut (8,8) CNT leads and in the polyenes, is equal to the Fermi energy (𝐸𝐹) of the CNT leads, and 𝐸𝐹 ≡ 0.
Fig. 3.7: (a) One-polyene junction. (b) Two-polyene junction.
As illustrated in Fig. 3.7, each junction can be divided into three parts: (1) the device including one or two polyenes, (2) the left and (3) the right angled-cut (8,8) CNT leads. Therefore, the Hamiltonian of the junction can be represented as
In the iterative calculation, we choose a small value of 𝜂 (10−7) and the criterion on
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the norm of the recursion error (10−10) to achieve the desired accuracy of the calculation. Once the surface Green’s function of each lead is obtained, the self-energies from two leads are calculated by Eq. (16). Then, we can obtain the Green’s function of the device from
] 1
)
[(
E i I HD L R
G .
We use Eq. (9) to calculate the broadening. Finally, we use Eq. (30) to obtain the transmission of the polyene junctions in between angled-cut armchair CNT leads.
3.2.3 Ab initio calculations
Fig. 3.8: (a) One-polyene junction. (b) Two-polyene junction.
We first optimize the geometry structure of the (8,8) CNT bulk with the SIESTA package. Each lead consists of four unit cells of the (8,8) CNT. We choose an energy cutoff of 200 Ryd, the Double- plus polarization (DZP) basis, and the Ceperley-Alder local density approximation (LDA) for the exchange and correlation functional to implement this relaxation calculation. A rectangular supercell must be large enough to pass the convergence test of the bond length, so we use a rectangular supercell of 20Å × 20Å × 𝑎, where 𝑎 is the length of the lead in the z direction. The geometry relaxation process continues until the force acting on each atom of the (8,8) CNT lead is less than 0.05 𝑒𝑉/Å . Then, the relaxed structure is used for the self-consistent field (SCF) calculation of the lead in Nanodcal.
We optimize the geometry structure of the scattering region in the similar way. In the scattering region, 3 and 3.5 unit cells from each lead are contained at the left and
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the right sides of the polyenes, respectively. We use a rectangular supercell of 20Å × 20Å × 𝐷, where 𝐷 is the length of the scattering region in the z direction. In the geometry relaxation process, we fix the atomic coordinates of the unit cell that is directly connected to either lead so that they can maintain the original bulk structure, while the coordinates of the remaining atoms in the scattering region are relaxed with no geometry constraint. Then, the relaxed structure is used for the SCF calculation of the two-probe system in Nanodcal.
After the relaxed structure is obtained, we can perform the transport calculation with the Nanodcal package. We use the DZP basis and the LDA_PZ81 for the exchange and correlation functional in Nanodcal calculation. To make the vacuum area sufficiently large, 30Å × 30Å × 𝑎 and 30Å × 30Å × 𝐷 are taken as the supercells of the lead and the scattering region, respectively. We first carry out the SCF calculation of the lead. Once the calculation result of the lead is obtained, the SCF calculation of the two-probe system can be performed. When the SCF calculation is complete, we can calculate the transmission coefficients by the transmission calculation.
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In the previous study, it has been shown that the transmission of the two-polyene junctions can be separated into the contributions from the even and odd channels.
[30,31] We next calculate the transmission of the even and odd channels in the cases of two-polyene junctions.
For the two-polyene junctions, the transmission only depends on the two contact sites on the rim of the CNT lead. Therefore, the surface Green’s function of the CNT lead can be reduced to a 2 × 2 matrix effectively: equation to obtain the surface Green’s function of a system that contains atom 1 and atom 18:
Next, we apply the same strategy to obtain the surface Green’s function of a system that includes two atoms, namely, atom 2 and atom 19:
1 1
2[( ) D s]
s E i I H g
g .
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The procedure is repeated until we reach the pair of atom 16 and atom 33, as shown in the following equations:
.
Finally, the last pair of atom 17 and atom 34 is considered as the device, whose Green’s function reads
] 1
)
[(
E i I HD L R
G .
Note that the self-energies from the left and right leads are
L Lg16sL.
R Rgs0R.
We use Eq. (9) to calculate the broadening. Then, we obtain the transmission matrix from
G G T L R .
In the above procedure, we only need to deal with a two-dimensional space, either for the surface or for the device. Due to the symmetry of the Hamiltonian of the surface and the device, an orthogonal matrix 𝑈 can be used to rotate and therefore diagonalize the space:
TU U T' ,
where this matrix 𝑈 is exactly constructed by the two vectors representing the even and odd channels.
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