Tuning spin–orbit coupling in (6,5) single-walled carbon nanotube doped with sp³ defects

Semiconducting single-walled carbon nanotubes (SWCNTs) containing $sp3$ quantum defects are a promising class of materials for a wide range of potential applications in optoelectronics. While the carbon nanotube consists of a carbon $sp2$ lattice, the defect introduces an $sp3$ bond (see Fig. 1), creating a localized defect potential that acts as an exciton trap,^1–4 resulting in increased photoluminescense emission intensity^5–11 with little disruption to the tube electrical conductivity.¹² The introduced exciton trap can result in single-photon emission with purity of up to 99% at room temperature.^13–15 For this class of materials, the compatibility of SWCNT with micro-fabrication techniques suggests facile integration with existing nano- photonic^16–20 and plasmonic^21–23 cavity technologies.

The $sp3$ defect can be incorporated into the SWCNT by a diazonium reaction,^24–26 which provides molecular control over the attached functional group. Thus, the defect provides a degree of freedom to tune the electronic and optical properties of the SWCNT in addition to its diameter and chirality.^3,27,28 For example, the electron-withdrawing/-donating capability of the functional group can tune the photoluminescence energy and modify the emission intensity of the SWCNT.^9,10,29,30 In addition, the choice and density of the functional group can impact the excited state lifetime^31–33 and exciton–phonon coupling in the emission process.³⁴ 

Here, we investigate the ability to control the spin–orbit coupling (SOC) and thus the intersystem crossing (ISC) dynamics of the SWCNTs using $sp3$ defects. ISC dynamics are important for many potential SWCNT applications (e.g., in quantum information science,^35–37 spintronics,^38,39 triplet–triplet upconversion,^40–45 and triplet energy transfer^46,47). Despite carbon’s low intrinsic SOC of $≈1$–10 $μ$eV, the curvature-induced $σ$–$π$ orbital mixing within SWCNTs leads to increased coupling of $≈1$–10 meV.^48–52 The significant SOC strength combined with the small singlet–triplet energy gap, as predicted^53,54 and measured,^55,56 suggests efficient ISC. Indeed, the ISC time constant has been measured in (6,5) SWCNT to be $≈20$ ps,⁴¹ competitive with the singlet exciton recombination lifetime of $≈100$ ps.^57,58 Moreover, up to 10% singlet–triplet ISC yield has been measured in SWCNTs.^41,42,59 Furthermore, the presence of defects in SWCNTs has been predicted to enhance SOC and ISC^60–64 and suggested by the observed brightening of previously dark triplet exciton emission in H-functionalized SWCNT.⁶⁵ With the addition of the $sp3$ defect containing a heavy element with large intrinsic SOC, it is possible that SOC, and, therefore, ISC yield, can be tuned by the addition of metal-containing defects as has been predicted for graphene.⁶⁶ 

We utilize first-principles density functional theory (DFT) in conjunction with photoluminescence measurements to study the electronic properties, including SOC, of defective (6,5) SWCNT with a variety of defect chemical structures. Previous studies of pristine SWCNT have shown that the ISC rate is limited by SOC;⁶⁷ hence, we focus on the magnitude of SOC-induced spin–orbit splitting as a measure of ISC. For four aryl functionalized SWCNTs that have been synthesized, both theory and experiment find a localized defect band containing a single electron at the $sp3$ site, the energy of which is insensitive to the functional group chosen. We then test three other Pd-containing functional groups in an effort to increase spin–orbit coupling of the near-gap states in the SWCNT. Importantly, we propose a new reaction mechanism to decrease the Pd–SWCNT distance such that there is non-negligible Pd density in the near-gap states, predicting an increase in the photoluminescence redshift and spin–orbit coupling. This study suggests that a direct metal–$sp3$–carbon bond can increase SOC in the SWCNT.

We studied seven aryl functional defects within the semiconducting (6,5) SWCNT; six are shown in Fig. 1: (1) acacPd(II)-4-carboxylatephenyl, (2) 4-carboxyl phenyl, (3) 4-nitrophenyl, (4) 4-bromophenyl, (5) acacPd(II)carboxylate, (6) acacPd(II)carboxylatemethyl, and (7) chlorodiphosphanepalladium(II) [Cl(PH$3$⁠)$2$Pd(II)–], is shown in Fig. 5.

DFT within the local spin density functional approximation (LSDA)^68,69 was applied to study the electronic structure and spin properties of pristine and defective SWCNT within the VASP package.^70–73 Core electrons and nuclei were described by the scalar relativistic projector-augmented wave (PAW) method^74,75 with 1, 4, 5, 6, 5, 7, and 10 electrons considered as valence for H, C, N, O, P, Cl, and Pd, respectively. The plane wave cutoff energy was 400 eV with a $Γ$-centered k-point mesh of $1×1×2$⁠, which converged the total energy to less than 0.005 meV/atom. Each self-consistent cycle electronic energy was converged to below 10$−6$ eV. For the pristine SWCNT, the structure was optimized such that forces between atoms were smaller than 0.01 eV/Å. For the with-defect structures, the lattice vectors of the SWCNT were fixed and the atoms allowed to relax in the presence of the defect. The predicted lattice vectors were (19.92, 19.92, 40.45) Å for the pristine and (32.37, 19.92, 40.45) Å for the with-defect SWCNT, where the c axis defines the tube direction, and 10 Å of vacuum was added around the tube to isolate it from its periodic images. The predicted lattice vector along the nanotube direction is in good agreement with a tight binding analysis^76,77 and previous DFT calculations.^78,79 Band structures were calculated non-self-consistently on a $Γ$-centered $1×1×16$ k-point mesh. The magnetic moment along the tube direction was determined to be 1 bohr magneton for all defects studied.

According to Fermi’s golden rule and the Marcus theory, the ISC rate can be described as^80,81

where $HSOC$ is the spin–orbit Hamiltonian within the zero-order regular approximation⁸² treated perturbatively through the second-variation perturbation method^83,84 and $ρfcwd$ is the Franck–Condon coupling between the transition (e.g., singlet and triplet) states. Since the first optical transition of SWCNTs is dominated by the transition from valence to conduction bands,²⁸ the spin–orbit splitting of bands near the gap serves as a measure of the singlet–triplet coupling strength, $⟨SHsocT⟩2$⁠. Thus, here, we aim to tune spin splitting as a result of SOC in the near-gap bands. For the singly occupied defect band, we define spin-splitting due to SOC (hereafter referred to as “the spin-splitting”) as the change in the energy difference between the two spin states as a result of turning on SOC Hamiltonian.

The four aryl functionalized SWCNT samples (functional groups 1–4 in Fig. 1) were prepared through reaction of the corresponding aryl diazonium dopants with (6,5) SWCNTs that are encapsulated in a sodium deoxycholate (DOC) surfactant environment^9,15 as outlined in our previous work.⁸⁵ Specifically, chirality enriched (6,5)-SWCNTs dispersed in 1% wt/v sodium dodecyl sulfate (SDS) solutions were prepared following the two-step phase separation method.⁸⁶ To introduce functional groups to the sidewalls of the (6,5)-SWCNTs, diazonium salt solutions were added to the SWCNT suspensions for reaction.⁹ The reaction progress was monitored by measuring the photoluminescence spectra of the suspensions. Once the designated doping levels were achieved, the reactions were quenched by adding sufficient amount of concentrated sodium deoxycholate (DOC) solutions to the SWCNT suspensions. The functionalized SWCNTs were then transferred back to 1% wt/v DOC solution using pressure filtration. To ensure that all the residual diazonium reactants had been removed from the SWCNT suspensions, the pressure filtration step was repeated four to five times. The absorption spectra of the samples were measured on a UV–VIS CARY-50 spectrometer. The PL spectra of the samples were measured using a home-built laser microscope in which an InGaAs array was mounted on a 300 mm spectrometer for detecting the PL spectra of the solution samples. A continuous-wave diode laser with a wavelength of 560 nm was used to excite the samples.

Figure 2(a) presents the predicted band structure of the pristine (6,5) SWCNT. The pristine SWCNT is an indirect band gap material, with an LDA-predicted fundamental gap of 0.94 eV and a direct gap of 0.97 eV at the $k=0$ (⁠$Γ$⁠)-point, in good agreement with previous DFT-based studies.^78,79 While LDA tends to underestimate the fundamental gap,⁸⁷ we expect trends to be well-described. In order to better understand the SWCNT energetics, we also performed DFT calculations with the functional of Perdew, Burke, and Erzenhof (PBE)⁸⁸ and that of Heyd, Scuseria, and Erzenhof (HSE06)⁸⁹ as shown in the supplementary material, which predict a gap at $Γ$ of 0.94 eV and 1.23 eV, respectively. The HSE-predicted gap, which has empirically been found to reproduce semiconductor optical gaps,⁹⁰ agrees well with the photoluminescence gap of 1.27 eV measured experimentally.⁹¹ 

As shown in Fig. 2(b), the addition of a single covalently bonded $sp3$ defect (functional group 1 in Fig. 1) leads to a midgap state (“defect band”), which is singly occupied, similar to what was reported previously for $sp3$-functionalized SWCNT,⁹² graphene,⁹³ and C$60$ fullerene.⁹⁴ The finding that the defect introduces a single localized unpaired electron is in agreement with electron spin resonance (ESR) measurements by us that found a Landé g-factor of $2.0037±0.0003$ for this system.⁸⁵ The defect band is non-dispersive, suggesting negligible defect–defect interaction with its periodic images or with the SWCNT bands. There is a slight perturbation to the pristine SWCNT’s valence and conduction bands with the degeneracy at high symmetry points in the pristine-like valence and conduction bands lifted and a direct band gap at $Γ$ of 0.97 eV. The defect band energy is 0.30 eV above the pristine-like SWCNT’s valence band with an exchange splitting [the energy difference between spin up (⁠$α$⁠) and down (⁠$β$⁠) states] of 0.09 eV. We note that improving the functional with HSE06 leads to qualitatively similar conclusions, with the defect band 0.20 eV above the valence band and a larger exchange splitting of 0.5 eV presumably due to the inclusion of partial exact exchange (see Table SII in the supplementary material).

For all four experimentally synthesized defects (groups 1–4), we predict a qualitatively and quantitatively similar electronic structure. The relative defect band to valence band and exchange splitting vary by less than 20 meV and 3 meV, respectively, across the aryl functional groups (see Table SI in the supplementary material). As discussed below, the fact that the defect state energies are essentially independent of the functional group is because the defect introduces a single electron around the $sp3$ carbon, with little spatial overlap of the electron with the aryl substitutional group. We note that previous studies of $sp3$ defects in (6,5) SWCNT^95–98 considered hydrogen passivation of the $sp3$ site and so found different electronic than the present work. For our systems, ESR measurements suggest that hydrogen passivation does not occur.⁸⁵ 

The unpaired electron within the occupied defect band is localized near the defect site as shown in Fig. 2(c). For the pristine (6,5) SWCNT, the valence and conduction band densities are consistent with $π$ orbitals delocalized over the entire tube. The addition of group 1 defect does not qualitatively modify the pristine-like SWCNT valence and conduction band density but introduces an orbital with a localized density mostly composed of out-of-plane p orbitals on the tube near the defect site and an $sp3$ orbital from the C atom on the functional group directly connected to the $sp3$ carbon. The spin density of the defective system is entirely contained within the defect orbital and is qualitatively unchanged across aryl functional groups 1–4.

The finding that the excited state energetics are insensitive to the functional group is consistent with photoluminescence measurements. Figure 3(a) shows a sketch of the measured photoluminescence spectrum of (6,5) SWCNT containing the functional group 3 $sp3$ defect (see Fig. S1 in the supplementary material for the raw spectra). All four defects show similar features; therefore, only one representative is shown. The spectrum shows one peak associated with the pristine SWCNT (labeled as E$11$⁠), which is measured to be around 1.25–1.26 eV for the different defects considered. This number agrees well with the predicted HSE06 gap for the pristine SWCNT. Another peak at $≈1.09$ eV is associated with the defect (labeled as E$11∗$⁠), indicating that the lowest energy transition is reduced by 0.16–0.17 eV with the introduction of the defect. This finding agrees well with our predictions that a new defect band is introduced at 0.3 (0.2) eV above the valence band maximum within DFT-LDA (HSE06). There are differences in the measured E$11∗$ peak position on the order of 10 meV between the various defects considered, possibly related to the different affinities of the dopant;⁹ obtaining such energy differences is beyond the accuracy of the calculations.

The pristine (6,5) SWCNT displays a significant spin–orbit coupling. The asymmetric mixing of $σ$–$π$ bonds^49–52 due to the tube’s curvature leads to spin–orbit induced splitting of near-gap states, which we predict to be 0.8 and 1.7 meV in the pristine-like valence and conduction bands, respectively [Fig. 2(d)]. These values are similar to previously reported theoretical values of 0.3–0.5 meV and 1.3–1.7 meV, respectively.^99,100 With the introduction of a Pd atom via the $sp3$ defect, we may expect an increase in spin–orbit coupling and, therefore, spin–orbit splitting of near-gap states. The strong d-orbital mixing of an isolated Pd$2+$ ion leads to a strong spin–orbit splitting of $≈400$ meV;¹⁰¹ therefore, we might expect large splittings in the defective material as well. However, for group 1 defect, the spin–orbit splitting in the defect band is 0.025 meV, of a similar order of magnitude as predicted for methyl-functionalized graphene.¹⁰² Failure to increase SO splitting is likely due to the lack of Pd d-orbital characters in these states [see projected density of states (PDOS) in Fig. 4]. Additionally, in the presence of the defect, spin–orbit splitting in both valence and conduction bands is reduced by 0.2 meV. The decreased splitting may be understood as being due to the reduced delocalization of the electron density around the circumference of the tube, which for the pristine (6,5) SWCNT results in a radial electric field that induces SOC.³⁸ 

To understand the role that Pd might play in increasing spin–orbit splitting for the near-gap states, we considered two additional functional groups that have not yet been synthesized (5) and (6). By either eliminating the benzene ring in group 1 defect or replacing it with a methylene group, the Pd-SWCNT distance is systematically reduced. We expect that decreasing the distance will increase the wavefunction overlap between the metal d-orbitals and the SWCNT near-gap states (pristine-like valence and conduction bands and the defect band), increasing their spin–orbit splitting. However, we predict that even with the proximity of the Pd to the SWCNT, there is very little d-orbital mixing with near-gap states. PDOS analysis shows that the highest energy with a significant d-orbital contribution is located at about 0.5, 1.0, and 0.8 eV below the Fermi energy for functional groups 1, 5, and 6, respectively, with little d-orbital mixing in the near-gap states. Consequently, we do not predict any increase in the spin–orbit splitting of these states as shown in Table SIII and Fig. S2 of the supplementary material but rather energy shifts in bands with more significant Pd d-orbital mixing as shown in Fig. 4. This suggests that connecting the heavy metal directly to the $sp3$ C is necessary for a significant d-orbital character and hence spin–orbit splitting of near-gap states.

To achieve direct bonding between Pd and the SWCNT’s $sp3$ carbon, we propose a possible route shown in Fig. 5(a) to synthesis of functional group 7. This proposed oxidative addition reaction is a redox reaction between an alkyl chloride (R–Cl) and tetrakis(trialkylphosphine)palladium(0) [Pd(PR$3$⁠)$4$] to create alkylchlorodi(trialkylphosphine)palladium(II) [R–Pd(PR$3$⁠)$2$–Cl].¹⁰³ While this reaction pathway has not been attempted to our knowledge for doping of SWCNT, it may be possible if (1) a chlorinated (6,5) SWCNT with one unpaired electron per site can be synthesized, (2) steric hindrance from the SWCNT provides enough of an energetic barrier resulting in a kinetic (rather than thermodynamic) controlled product, and (3) a suitable solvent is found. Here, the calculated ligands are phosphine (PH$3$⁠) as a proof of concept, but other ligands may be utilized as well. The resulting band structure, frontier orbital densities, and spin–orbit splitting for this defect are shown in Figs. 5(b)–5(d). We again predict a midgap defect band, now located 0.65 eV above the valence band, suggesting the presence of a significantly lower energy photoluminescence peak. The exchange splitting is 0.11 eV, close to that predicted for defects 1–6 above. The pristine-like valence and conduction band density is again qualitatively similar to the pristine SWCNT, while the defect-centered midgap state now shows a significant density on the Pd atom and its ligands [Fig. 5(c)]. The percentage of the Pd d-character within the defect orbital is $≈24$%, while it is 0% for the valence band and 7% for the conduction band, respectively. Consequently, the spin–orbit splitting is increased to $≈0.7$ meV in the defect band; interestingly, the spin–orbit splitting increases in the pristine-like conduction band as well (by 1.2 meV). The increase is in contrast with other Pd-containing functional groups, which had no significant spin–orbit contribution to the near-gap states. This finding suggests that the SOC of the near-gap states can be further tuned by varying the heavy element directly bound to the $sp3$ site.

In summary, we performed a theoretical study, complemented by experiment, of the electronic structure of $sp3$-functionalized (6,5) SWCNT. The defects studied introduce a single unpaired electron localized on the p orbitals of the nearest neighbor atom on the functional group resulting in a midgap defect band. We determined that introducing a functional group containing a Pd atom via a diazonium reaction did not necessarily result in an increase of spin–orbit splitting of the near-gap bands because of the lack of overlap of these states with the Pd atom and thus will not increase ISC. We proposed a redox reaction to introduce a new Cl(PH$3$⁠)$2$Pd(II)– defect with a heavy metal attached to the $sp3$ carbon on the SWCNT and predicted an increase of the Pd character in the near-gap states, which resulted in increased spin–orbit splitting. Our study shows that it is possible to tune SOC and hence ISC in SWCNT by the introduction of an $sp3$ defect containing a heavy element.

See the supplementary material for experimental PL spectra, the electronic band structure of all defective (6,5) SWCNT studied, and spin–orbit splitting of defect groups 1, 5, and 6.

We acknowledge financial support from the National Science Foundation (NSF) under Grant No. DMR-19005990 and computational resources from the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. Use of the Center for Nanoscale Materials, an Office of Science user facility, was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. In addition, we appreciate a meaningful discussion with Professor Linda Doerrer at Boston University regarding organometallic chemistry.

The data that support the findings of this study are available from the corresponding author upon reasonable request.