Overcoming the Sticking Point: Electrical Conductivity of Carbon Nanotube Networks

: Carbon nanotubes have excellent electrical conductivity along the length of the tubes. Yet, the electrical conductivity across the nanotube − nanotube intersections is weak and severely limits device performance. Here, we show that the incorporation of 3d metal (period 4) atoms into networks of semiconducting (8,0) carbon nanotubes significantly enhances the electrical conductivity within the network. Our calculations using quantum mechanical methods and semiclassical Boltzmann transport theory predict the changes to the electronic structure and provide directional information about the flow of electrons within the network. The ligand field splitting of the transition metals exerts strong effects on the conductivity. Interestingly, networks doped with Sc, V, or Fe can become insulating along certain directions or have higher conductivity across the junction than along the tubes. This finding suggests that doping with transition metals removes a bottleneck of charge transport within carbon nanotube films.


■ INTRODUCTION
−3 Transition metal atoms sandwiched between the aromatic rings form haptic, covalent bonds.The versatile d electron configuration strongly interacts with the delocalized πelectron system and leads to organometallic complexes with adaptable catalytic or optoelectronic properties. 2,4The integration of metallocenes into polymer networks 5,6 or multipledecker structures 7−9 broadens their applications to materials science areas, such as energy conversion and storage, optics, and electrical conductivity.
Carbon nanotubes (CNTs) are model platforms to explore the organometallic coordination and interaction of extended πelectrons with dopants.−19 Previous studies investigated the electronic structure of CNTs containing analytes or dopants using density functional theory (DFT) methods.−30 Atomic gold clusters at the CNT intersection increase the electrical conductivity if the clusters have an odd number of atoms. 31ransition metal doping is an avenue to access organometallic properties in CNT films.−36 Improved conductivity and sheet resistance were also observed experimentally in films containing 3d metals (Ti, V, Mn, and Fe). 37,38The experimental findings were supported by calculations based on Green's function methods that show an improved charge transport through the junction of intersecting nanotubes doped with group 6 transition metal atoms. 39,40In a computational study using Landauer formalism, the quantum conductance across two parallel CNT ends is vastly enhanced when the nanotubes are bridged by a metallocene linker atom (Sc, Ti, V, Cr, Co, or Ni). 41However, the calculations only extract the quantum conductance parallel to the nanotubes.Comprehensive DFT computational studies of the directional electrical conductivity of CNT networks doped with transition metal atoms are lacking.
Recently, the electrical conductivity of CNT networks containing Group 11 or s−p metal dopants were calculated using DFT methods and semiclassical Boltzmann transport theory. 42The three-dimensional periodicity enabled the electrical conductivity to be calculated both along the nanotubes and across the intersection.The dopants strengthen the conductivity in all directions and remove the bottleneck of charge transport within networks of mixed chiralities.
Here, we investigate carbon nanotube networks doped with 3d metals, K, or Ca using DFT and semiclassical Boltzmann transport theory.The focus is on homogeneous networks of (8,0) semiconducting nanotubes.Charge transport within CNT thin films with heterogeneous chirality is expected to be limited by the semiconducting tubes.First, the geometry of carbon nanotube networks containing 3d metal atoms are analyzed.Then, the effects of the dopants on the electronic structure and its influence on the electrical conductivity along the nanotubes and across the intersections are examined.

■ METHODS
All calculations presented in this work are performed with CRYSTAL17. 43Networks of (8,0) carbon nanotubes were constructed with periodicity in three dimensions.The nanotubes lie along a⃗ and b ⃗ , and α = β = 90°.The CNT intersection angle is equivalent to γ. Semiconducting nanotubes with (8,0) chirality are examined because of the limiting effect of semiconducting nanotubes on charge transport within networks. 42−38 Therefore, two 3d metal atoms were situated between the nanotubes at the intersection (Figure 1).The atomic positions and lattice constants of the pristine and doped networks were relaxed to their minimum energy geometric conformation.During preliminary calculations on a doped network containing Ti atoms without any symmetry constraints (space group P1), the optimization ended up in a minimum geometry in space group Cccm (66).Thereafter, all pristine and doped networks were optimized in the centrosymmetric space group Cccm.The consistent space group of all networks enabled systematic comparison of the qualitative trends of the metallocene-like coordination across the period.One dopant atom per nanotube occupies both junctions and is necessary to examine the behavior of networks with periodicity in three dimensions.Other concentrations were not considered.The networks are overall neutral.
We applied the PBE0 hybrid density functional method with 25% Hartree−Fock and 75% PBE exchange 44,45 and utilized Gaussian-type basis sets based on the molecular Karlsruhe basis sets. 46Triple-ζ-valence + polarization (TZVP) level basis sets were used for C 47 and 3d metals. 48Split-valence + polarization (SVP) level basis sets were used for K 49 and Ca. 42 The main aim was to characterize the qualitative trends in electrical conductivity resulting from doped CNT networks.Thus, the trends were investigated without van der Waals effects, and the conductivity was analyzed relative to the pristine network.The electrical conductivity, σ, at temperature, T, equal to 300 K was obtained using Boltzmann transport theory, as implemented in CRYSTAL17. 50The transport coefficient was solved within the constant relaxation time and rigid band approximations, such that where the energy is E; the Fermi−Dirac distribution is f 0 ; the Fermi level is μ; and the transport coefficients, Ξ qr (E), are defined as υ i,q was the velocity of the i th band along q; N k is the number of kpoints over the cell volume, V; and the lifetime, τ, was 1 fs for all studied systems.The conductivity is shown as the summation of the α and β spin states.The in-plane conductivity and junction conductivities at the Fermi level are reported for the doped networks relative to the pristine network.The relative in-plane conductivity (σ plane * ) is the average along the x and y axes, and the junction conductivity (σ junc * ) is along z, i.e., across the junction.The reciprocal space paths for the electronic band structures were obtained from the SeeK-path service 51 (see Figure S1 for reciprocal space and Brillouin-zone information).The density of states and electrical conductivities were obtained with a dense kpoint mesh of 16 × 16 × 16 and 8 × 8 × 8, respectively.
■ RESULTS AND DISCUSSION Network Geometry.The primitive cell lattice constants of the minimum energy geometries of the pristine and doped carbon nanotube networks are shown in Table 1.Interestingly, the transition metal can shrink the cell volume and the length of the unit cell across the junction, c.The contraction occurs because the transition metal atoms are situated between the nanotubes in a bis-hexahapto (η 6 ) metallocene-like coordination in the Cccm (66) centrosymmetric space group (Figure 2).The nanotubes are relaxed, and the CNT intersection angle, γ, is free to rotate during the optimization.The carbon atoms across The Journal of Physical Chemistry C the junction are eclipsed when γ equals 60°and staggered when γ equals 60 ± 30°.Networks doped with midperiod transition metals (Fe, Co, and Ni) have a more closely bound networks in a nearly staggered conformation.Furthermore, the distance c reflects the competition between the ionic radii 52 and electronegativity 53 of the doping element across the periodic row.The larger, more diffuse atoms, such as Sc, form relatively long bonds with the carbon nanotubes.The more compact atoms are generally more electronegative and less willing to donate electrons.Consequently, the competition leads to a minimum value of c in the Mn-doped network.The trends in c are consistent with the interatomic distance between the C atoms in the metallocene-like coordination (M-C a , M-C b , and M-C c ).Each C is 4-fold degenerate because of the space group symmetry.The unequal distances are due to the curvature of the nanotube.The interatomic distances are similar to that between Cr and the C atoms in bis(η 6 -benzene) chromium (2.13 Å) calculated with the same exchange-correlation functional.
The binding energy and local environment of each dopant are summarized in Table 2.The partial Mulliken charge of each dopant (Z M ) is obtained by subtracting its Mulliken population from its atomic number (Z).As expected, electron density is shifted toward the carbon nanotube network, which results in a slightly positive dopant.s−p metals and Zn dopants have small Mulliken bond overlap populations between the metals and the nearest C atom (P MC ), thereby indicating ionic (K, Ca) or weak (Zn) interactions.The bond overlap populations of other 3d metals are rather similar.Covalent bonds form between the transition metal atoms and the carbon nanotube networks.The binding energy of Mn is positive, which indicates the interaction is energetically unfavorable.Other 3d dopant atoms bind favorably to the network, and the binding energies range from −102 to −706 kJ/mol.There is a discontinuous decline across the period.The binding becomes generally less favorable from Ca to Mn and from Co to Zn. Cr, despite being known to form stable (bis)-hapto structures with polycyclic aromatics, has a relatively weak binding energy.This may be due to the stability of atomic Cr in the half-filled 4s 1 3d 5 electron configuration.The Mn atom also has an energetically stable electron configuration (4s 2 3d 5 ), and networks containing Mn are less favorable.Similar binding energies were observed elsewhere. 41Zn is weakly bound to the network because of its fully filled d shell.The atom is not covalently bound to the nanotubes, and the network has the largest cell volume and c.We note that the binding energies of K, Ca, and Cu in these networks are different from those reported in our previous work 42 because of the different space group symmetry.
Electronic Structure and Conductivity of the Doped Networks.Covalent bonds are formed between the carbon nanotubes and the transition metal atoms in the metallocenelike coordination.The 3d orbitals of the transition metal dopants are energetically positioned to strongly interact with the delocalized π-electron system of the (8,0) nanotube.The number of electrons and ligand field splitting during the bishexahapto bond formation lead to changes in the electronic properties of the networks.
In the pristine network, the electrical conductivity is weaker across the intersection than along the tubes, which is in agreement with previous work that did not enforce symme-Table 1. Lattice Constant across Junction (c), Dopant-C Inter-Atomic Distance (M-C), CNT Intersection Angle (γ), and Cell Volume (V) of Pristine and Doped (8,0) CNT Networks   The Journal of Physical Chemistry C try. 39,42,54−56 This implies that the flow of electrons across the junction are a bottleneck of charge transport in pristine networks.The inclusion of transition metals can cause the semiconducting networks to become n-type or induce a gap state.A summary of the electronic properties and electrical conductivities of networks containing dopants along the 3d period are shown in Table 3.The electrical conductivities at the Fermi level across the junction and in the plane of the nanotubes are reported relative to the pristine network.Notably, the charge transport across the junction is enhanced in all dopants considered except K. Transition metal linker atoms (Sc, Ti, V, Cr, Co, or Ni) were previously shown to enhance the electrical transport along two parallel CNTs. 41The current results show that the improvement is several orders of magnitude greater across the intersection than along the tubes.The relative conductivity increases along the nanotubes of all the n-type networks except the one containing Co.The shift of the Fermi level into the conduction band enhances the charge transfer.Of the networks doped with transition metals, Mn improves the conductivity along the nanotubes the most (8.2fold).The network containing Co is unique because of the d character and discrete order at the conduction band edge.Similar improvements were previously reported in n-type networks containing other dopants including Group 11 and s−p metals (K, Ca, and Al). 42By contrast, the relative electrical conductivity along the tubes worsens in all doped networks with a band gap or gap state.These networks cannot overcome the localization of the extended π-electron system by the 3d states, thereby reducing the electron flow.
The dopant facilitates charge transport across the junction.The 3d π orbital mixing localizes the electron density at the intersection bridging the nanotubes.The bridging is confirmed by visualizing the band-projected charge density.For example, in the Sc-doped network, the charge density of band 789 extends over the transition metal and across the junction (Figure 3).This network had the second largest relative conductivity across the junction (4166-fold).Furthermore, the charge density extends evenly along the tubes, which suggests the charge transfer is not preferential on a particular tube.By comparison, the pristine carbon nanotube network has no bridging.
The enhanced electrical conductivity of the doped networks is also apparent in the band velocity near the Fermi level (Figure  The Journal of Physical Chemistry C 3).The band dispersion across the junction (Γ−Z) is significantly larger in the doped network than the pristine networks.The Sc-doped network is metallic, and its overlapping midgap states (bands 789 and 790) have an occupancy of 1.5 and 0.5, respectively.By contrast, the bands in the pristine network are highly dispersive along Γ−Y and Γ−S (in the plane of the nanotube axes) but not along Γ−Z (across the junction).
The variation in the properties of different transition metal dopants is in part explained by the number of d electrons.It is most noticeable in networks containing early period transition metals where the ligand field splitting is similar and near the    The Journal of Physical Chemistry C and V-doped networks are metallic with partially filled midgap states, and their band structures resemble each other.The degeneracy at the R special point in the network containing Sc is lifted in the V-doped network.This contributes to the slightly reduced conductivity compared with the Sc-doped network.
The conductivities along each nanotube and across the junction are seen in Figure 5.In stark contrast to pristine networks, the conductivity at the Fermi level is smaller along either nanotube than across the junction containing Sc or V. Explicit conductivities at the Fermi level are provided in Table S1.The doping alters the easiest direction of electron flow from along the tubes to across the junction.This suggests that it is possible for the bottleneck of charge transport to switch to along the nanotubes.The asymmetry of the electrical conductivity along the nanotubes is due to the rotation of the unit cell vectors from the Cartesian coordinate axis.This asymmetry occurs when the tubes intersect at angles other than 90°and has been discussed elsewhere. 42etworks containing late period elements (Mn and beyond) or s−p metal dopants are n-type metallic networks, with the exceptions of Fe and Zn.The networks have bands with high velocity along the nanotube (Figure 6).The regular band structure of the pristine network is disrupted by the doping, which can increase dispersion across the junction.The Fe-doped network has a half-filled zone.Its total spin multiplicity is 4.0 (cf.Table 3).The high spin state is due to the worsening Fermi level alignment and weakening of the ligand field splitting.Across the period, the d states lower in energy, as seen in the orbital projected densities of states in Figures 5, 7, and S2.The top of the valence band of the network containing Zn has contributions from s and sp orbitals of the dopant because of its filled 3d subshell.High-spin states are typical in Fe organometallics. 57imilar effects are seen in the network containing Ni, which has a total spin multiplicity of 2.2.We note that the relative conductivities of the networks containing K, Ca, or Cu are different from those reported in our previous work 42 because of the different space group symmetry.

■ CONCLUSIONS
The incorporation of transition metal atoms into carbon nanotube networks changes its electronic structure and electrical conductivity.At the junction of two CNTs, 3d metal atoms bind favorably in a metallocene-like coordination.Organometallic networks are found by the covalent bonding between the d electrons with the delocalized π-electron system.The versatile d electron configuration across the period changes the electrical transport of networks of semiconducting (8,0) carbon nanotubes.
−41 This work reveals the anisotropic effects of the electrical conductivity in different directions within the network using quantum mechanical methods and semiclassical Boltzmann transport theory.The largest improvement occurs across intersections bridged by a transition metal atom.The localization of electron density across the junction enhances the charge transport in all networks containing transition metals.Interestingly, dopants can enhance the intertube conductivity or suppress the conductivity along the tube to change the bottleneck of electrical conductivity within networks.Because of the changes to the adjacent sp 2 C atoms and delocalized π-system, the conductivity along the nanotubes remained large only for the ntype networks except the one containing Co.The conductivity along the nanotubes can even become insulating when the networks contain Fe or Zn.This offers a possibility to form CNT thin films with adaptable catalytic or optoelectronic properties.Although concentration and local geometries would affect the performance of films, this work examines the trends of 3d metals in a bis-hexahapto (η 6 ) metallocene-like coordination.Quantification of the changes in different directions could improve models of network conductivity that lack a detailed electronic structure, such as percolation methods.
■ ASSOCIATED CONTENT * sı Supporting Information Spin polarization was taken into account in all networks because of the open-shell nature of some of the dopants.The reciprocal space was sampled using a Monkhorst−Pack-type 2 × 2 × 2 kpoint mesh.The calculations were carried out with Coulomb and exchange integral tolerance factors (TOLINTEG) set to tight values of 8, 8, 8, 8, and 16.

Figure 1 .
Figure 1.Network of (8,0) carbon nanotubes either (a,b) doped with transition metal atoms or (c) pristine, as viewed from the side or top-down.The doped networks have one transition metal atom per nanotube.The carbon atoms are brown, and the transition metal atoms (Mn) are purple.The networks belong to the Cccm (66) centrosymmetric space group.

Figure 2 .
Figure 2. (8,0) CNT network doped with (a) Ni in a staggered conformation and (b) Zn in the eclipsed conformation.The CNT intersection angle, γ, is 33.8°and 67.8°, respectively.There is one dopant atom per nanotube.

Table 3 .
Electrical Conductivity of Pristine and Doped (8,0) CNT Networks at the Fermi Level Reported Relative to the Pristine (8,0) Network Intersecting at 49°(2 S/m across the Junction and 9477 S/m in the Plane of the Tubes) Type" means that the pseudogap lies below the Fermi level, and "gap state" means that the Fermi level lies at a state in the pseudogap.

Figure 3 .
Figure 3. Projected charge density of the topmost band of the (a) pristine (band 768) and (b) Sc-doped (band 789) (8,0) CNT network.The isovalue is 4 × 10 −4 e/a 0 3 .The band occupancy is 2.0 electrons in the pristine networks and 1.5 electrons in the doped network (α + β = 1.5 and α − β = 0.0).The band structures of the (c) pristine and (d) Sc-doped networks.The energy is reported relative to the Fermi level, and there is one dopant atom per nanotube.

Figure 4 .
Figure 4. Bands of (a) pristine and (8,0) CNT networks doped with (b) Sc, (c) Ti, (d) V, (e) Cr, and (f) Mn.There is one dopant atom per nanotube.The energy is reported relative to the Fermi level.The band gap of the pristine network is 1.49 eV, and the network doped with Mn is n-type.The networks doped with Sc or V have a gap state, and the networks doped with Ti or Cr have a band gap of 0.98 and 1.08 eV, respectively.

Figure 5 .
Figure 5. Electrical conductivity and density of states of (a) pristine (with symmetry imposed, 49°intersection angle) and (8,0) CNT networks doped with (b) Sc, (c) Ti, (d) V, (e) Cr, and (f) Mn.There is one dopant atom per nanotube.The energies are reported with respect to the Fermi level.

Figure 6 .
Figure 6.Bands of (a) pristine and (8,0) CNT networks doped with (b) Fe, (c) Co, (d) Ni, (e) Cu, and (f) Zn.There is one dopant atom per nanotube.The energy is reported relative to the Fermi level.The band gap of the pristine network is 1.49 eV, and the network doped with Fe is a halffilled metal.The networks doped with Co, Ni, or Cu are n-type, and the network doped with Zn has a band gap of 1.21 eV.

Table 2 .
Binding Energy per Dopant, Mulliken Partial Charge of the Dopant (Z M ), and Mulliken Bond Overlap Population between the Dopant and the Nearest C Atom (P MC ) in (8,0) CNT Networks