O$_{2}$ Frozen Layers Frozen Layers Intertube Transport Transport Intratube Leads Leads Central Region Δ G$_{inter}$ (%) d [ ˚ A] SWNT Junction O$_{2}$ N$_{2}$ SWNT-O$_{2}$ SWNT-N$_{2}$ (7,7) Armchair 30 6 2.3 2.8 (12,0) Zigzag 140 14 2.6 2.8 (13,0) Zigzag 1800 130 2.5 2.8 TABLE I: Change in intertube conductance Δ G$_{inter}$ at the valence- band maximum ε$_{VB}$ relative to the pristine junction for O$_{2}$ and N$_{2}$ physisorbed in SWNT junctions of (7,7), (12,0), and (13,0) SWNTs, with the respective SWNT-O$_{2}$ and SWNT-N$_{2}$ separations d in Angstrom. FIG. 4: (Color online) Schematic of (13,0) SWNT junction consist- ing of four SWNT leads (light gray) coupled to the central scattering region via four SWNT primitive unit cells or layers (gray) “frozen” at their relative positions in the isolated SWNT. The atomic positions of the remaining eight SWNT layers (dark gray) and the physisorbed O$_{2}$ molecule have been relaxed. A SWNT separation consistent with experiments of approximately 3.4 ˚ A has been used. The intratube transport through the SWNT and the intertube transport between the SWNTs has been shown schematically. 3 of each of the four SWNT layers adjacent to the boundaries H prin C , to be within 0.1 eV of the Hamiltonian for the respective leads H$_{α}$ , so that max | H prin C − H$_{α}$ | < 0.1 eV. In this way the electronic structure at the edges of the central region was ensured to be converged to that in the leads. The Landauer-B¨utticker conductance for a multi-terminal system can be calculated from the Green’s function of the cen- tral region, G$_{C}$ , according to the formula 32–34 scription of the SWNT network’s work function, which may be significantly different for smaller tubes. where the trace runs over all localized basis functions in the central region. To describe the conductance at small bias for semiconducting systems, the Fermi energy ε$_{F}$ should be taken as the energy of the valence-band maximum ε$_{VB}$ or conduction-band minimum ε$_{CB}$ for p -type and n -type semi- conductors, respectively. The central region Green’s function is calculated from where z = ε + i 0 $^{+}$, S$_{C}$ and H$_{C}$ are the overlap matrix and Kohn-Sham Hamiltonian matrix of the central region in the localized basis, Σ$_{α}$ is the self-energy of lead α , and the coupling elements between the central region and lead α for the overlap and Kohn-Sham Hamiltonian are S$_{Cα}$ and H$_{Cα}$ respectively. The coupling strengths of the input and output leads are then given by Γ$_{in}$$_{/}$$_{out}$ = i (Σ$_{in}$$_{/}$$_{out}$ − Σ † in / $_{out}$) . For a four- terminal SWNT junction, the intratube and intertube trans- mission functions are calculated by choosing the appropriate output lead, as depicted in Fig. 4. For each of the three types of SWNT junctions considered, we find that both O$_{2}$ and N$_{2}$ are physisorbed with binding en- ergies of ∼ 0.2 eV, as depicted in Fig. 1. Further, the SWNT– O$_{2}$ and SWNT–N$_{2}$ equilibrium separation distance d is in the IV. THEORETICAL RESULTS & DISCUSSION III. BASIC THEORY Our DFT calculations have been performed with the SIESTA DFT code 27,28 using a double-zeta polarized (DZP) basis set for the physisorbed molecules (O and N), and a single-zeta polarized (SZP) basis set for the SWNTs (C), and the Perdew- Burke-Ernzerhof exchange-correlation functional. 29 We note here that the DZP and SZP SIESTA basis sets have recently been shown to yield transmission functions in quantitative agreement with plane-wave codes and maximally localized Wannier functions. 30 When modeling O$_{2}$ we have performed spin-polarized calculations 12 but have performed spin- unpo- larized calculations otherwise. We have modeled the junction system using 6(11) primi- tive unit cells or layers for each zigzag(armchair) SWNT per supercell, with a separation of approximately 3.4 ˚ A, as de- picted for a (13,0) SWNT junction in Figs. 1 and 4. 31 The four SWNT layers at the boundaries of the central region, shown in gray in Fig. 4, were kept fixed at their relaxed positions in the isolated SWNT. At the same time the central 4(9) primi- tive unit cells from each tube, shown in dark gray in Fig. 4, and the physisorbed molecules were relaxed until a maximum force of less than 0.1 eV/ ˚ A was obtained. Since the super- cell has dimensions of ≳ 25 ˚ A for each SWNT junction, a Γ point calculation was sufficient to describe the periodicity of the structure. Such a large supercell was necessary for the Hamiltonian Σ$_{α}$( ε ) = [ zS$_{Cα}$ − H$_{Cα}$ ] [ zS$_{α}$ − H$_{α}$ ] − $^{1}$[ zS † Cα − H † $_{Cα}$] , (3) G$_{C}$ ( ε ) = [ zS$_{C}$ − H$_{C}$ − ∑ α Σ$_{α}$( ε )$^{]}$$^{−}$$^{1}$ , (2) G = G$_{0}$ Tr[ G$_{C}$ Γ$_{in}$ G † $_{C}$Γ$_{out}$ ] ∣ ∣ ε = ε$_{F}$ , (1)