On the electronic and transport properties of semiconducting carbon nanotubes: the role of $\hbox {sp}^3$-defects

Abstract

The effect of $\hbox {sp}^3$-defects on the electronic and transport properties of semiconducting carbon nanotubes has been systematically studied on the basis of a quantum mechanical tight-binding model. We have calculated the band structure for carbon nanotubes with ordered defect patterns showing a large impact on the bandgap energy whereas for randomly distributed defects the band structure remains relatively robust. The transport behavior has been studied on the basis of the Green’s function method. The results indicate that the conductance of defective carbon nanotubes strongly depends on the number of defects and the tube diameter. We further show that the transport properties can be classified, depending on the number of defects, into two regimes which are either characterized by the mean-free path or the localization length. For both, analytical equations describing the impact of the tube diameter as well as the number of defects are derived. Comparing these values with the channel length indicates the dominant transport regime.

5. Appendix

1.1 5.1 Construction of the Hamiltonian matrices

A schematic representation of how to construct the Hamiltonian for a CNT structure is shown in Fig. 8. The whole CNT structure can be simplified by representing the unit cells with $N_{\text {cell}}$ layers, starting with the first layer connected to the left contact and ending up with the n-th layer coupled to the right contact. Every layer can be further divided into four sublayers, each described by an Hamiltonian matrix $H_{\text {S}} \in \mathbb {R}^{4n \times 4n}$, where n determines the chiral index of the (n,0)-CNT. Because none of the carbon atoms in a sublayer is connected to direct neighbors, $H_{\text {S}}$ is equal to identity

$$\begin{aligned} H_{\text {S}} = \left( \begin{matrix} 1 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad \cdots &{}\quad 0 \\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad 1 \end{matrix} \right) \cdot \varepsilon _{\text {2p}} \end{aligned}$$

(16)

The sublayers are connected to each other by two different coupling matrices $V_{\text {1}}, V_{\text {2}} \in \mathbb {R}^{4n \times 4n}$, which have the form

$$\begin{aligned} V_{\text {1}}= & {} \left( \begin{matrix} 1 &{}\quad 1 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 1 &{}\quad \cdots &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 &{}\quad \cdots &{}\quad 0 \\ \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ 1 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 1 \end{matrix} \right) \cdot t_{\text {hopp}},\nonumber \\ V_{\text {2}}= & {} \left( \begin{matrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 &{}\quad \cdots &{}\quad 0 \\ \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 1 \end{matrix} \right) \cdot t_{\text {hopp}} \end{aligned}$$

(17)

The Hamiltonian $H_{\text {C}} \in \mathbb {C}^{4nN_{\text {cell}} \times 4nN_{\text {cell}}}$ of the whole CNT structure can then be constructed out of those matrices. The block-diagonal form is given by

$$\begin{aligned} H_{\text {C}} = \left( \begin{matrix} H_{\text {S}} &{}\quad V_{\text {1}} &{}\quad &{}\quad &{}\quad &{}\quad A \\ V_{\text {1}}^{\dagger } &{}\quad H_{\text {S}} &{}\quad V_{\text {2}} &{}\quad &{}\quad &{}\quad \\ &{}\quad V_{\text {2}}^{\dagger } &{}\quad H_{\text {S}} &{}\quad V_{\text {2}} &{}\quad &{}\quad \\ &{}\quad &{}\quad V_{\text {2}}^{\dagger } &{}\quad H_{\text {S}} &{}\quad \ddots &{}\quad \\ &{}\quad &{}\quad &{}\quad \ddots &{}\quad \ddots &{}\quad V_{\text {1}} \\ A^{\dagger } &{}\quad &{}\quad &{}\quad &{}\quad V_{\text {1}}^{\dagger } &{}\quad H_{\text {S}} \end{matrix} \right) , \end{aligned}$$

(18)

where the submatrix A in the upper right and lower left corner determines whether a periodic or semi-infinite structure is considered

$$\begin{aligned} A = {\left\{ \begin{array}{ll} V_{\text {2}} &{} \text { if periodic }, \\ 0 &{}\text { if semi-infinite }. \end{array}\right. } \end{aligned}$$

(19)

Note that for a periodic structure the coupling matrix elements have to be multiplied by the phase factor exp$( i k R_{ij} )$ where $R_{ij}$ denotes the distance of connected C-atoms.

1.2 5.2 Recursive Green’s function method

If the Hamiltonian of the device structure is defined in a block-diagonal form according to (18), the device Green’s function can be computed by a recursive procedure. Let $V_{0,1}$ be the coupling matrix of the first principal layer to left contact. The Green’s function of the first principal layer is then calculated through coupling of the left surface GF to the unperturbed Hamiltonian $h_1$ following

$$\begin{aligned} G_{1,1}^{\text {L}} = \left( E - h_1 - V_{1,0} g_{\text {L}} V_{0,1} \right) ^{-1} \end{aligned}$$

(20)

which leads then to the GF matrix

$$\begin{aligned} G_{0,1}^{\text {L}} = g_{\text {L}} V_{0,1} G_{1,1}^{\text {L}}. \end{aligned}$$

(21)

This enables the calculation of the GF of the i-th layer by using the recurrence formula

$$\begin{aligned} G_{i,i}^{\text {L}} = \left( E - h_i - V_{i,i-1} G_{i-1,i-1}^{\text {L}} V_{i-1,i} \right) ^{-1} \end{aligned}$$

(22)

and thus

$$\begin{aligned} G_{0,i}^{\text {L}} = G^{\text {L}}_{0,i-1} V_{i-1,i} G_{i,i}^{\text {L}}. \end{aligned}$$

(23)

Finally, the full Green’s function $G_{n+1,n+1}$ is obtained through the connection to the right lead following

$$\begin{aligned} G_{n+1,n+1} = \left( g_{\text {R}}^{-1} - V_{n+1,n} G_{n,n}^{\text {L}} V_{n,n+1} \right) ^{-1} \end{aligned}$$

(24)

and

$$\begin{aligned} G_{0,n+1} = G_{0,n}^{\text {L}} V_{n,n+1} G_{n+1,n+1}. \end{aligned}$$

(25)

Eventually, the transmission probability is given by

$$\begin{aligned} T(E) = \mathrm {Tr} \left[ \varGamma _{\text {L}} G_{0,n+1}(E) \varGamma _{\text {R}} G_{0,n+1}^{\dagger }(E) \right] . \end{aligned}$$

(26)