Abstract
Atomic spin centers in 2D materials are a highly anticipated building block for quantum technologies. Here, we demonstrate the creation of an effective spin1/2 system via the atomically controlled generation of magnetic carbon radical ions (CRIs) in synthetic twodimensional transition metal dichalcogenides. Hydrogenated carbon impurities located at chalcogen sites introduced by chemical doping are activated with atomic precision by hydrogen depassivation using a scanning probe tip. In its anionic state, the carbon impurity is computed to have a magnetic moment of 1 μ_{B} resulting from an unpaired electron populating a spinpolarized ingap orbital. We show that the CRI defect states couple to a small number of local vibrational modes. The vibronic coupling strength critically depends on the spin state and differs for monolayer and bilayer WS_{2}. The carbon radical ion is a surfacebound atomic defect that can be selectively introduced, features a wellunderstood vibronic spectrum, and is charge state controlled.
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Introduction
For decades, defects in semiconductors and insulators have been widely utilized to control the electronic, optical, and catalytic properties of solids. More recently, the electron spin associated with atomic defects in crystals has attracted enormous interest in light of their potential applications in quantum technologies^{1,2,3}. The strong confinement of atomicscale quantum systems imposes a large characteristic energy range that is amenable for pushing quantum technologies beyond cryogenic environments^{4}, while the limited system size promotes favorable coherence properties^{4}.
However, generating identical defects with the necessary atomic precision, designing them to be tunable by external fields, and often even knowing their exact identities have remained unsolved challenges in the field. In this respect, twodimensional (2D) materials offer new opportunities for the atomicallyprecise generation and control of defect spins^{5,6}. There have been extensive efforts to utilize boron nitride or transition metal dichalcogenides (TMDs) as a platform for socalled quantum emitters, that is spincarrying defects that can be optically detected. In boron nitride, quantum emission has been attributed to carbon impurities^{7} among other defects^{8,9,10}. In TMDs similar behavior has been reported but is mostly ascribed to mesoscopic strain profiles^{11}. Atomic defects in TMDs have been generated previously by transmission electron microscopy^{12}, and ion beam lithography^{13,14}. However, the generated defects are often not identical and have mostly nonspinpolarized states.
Here, we introduce the carbon radical ion (CRI) in tungsten disulfide (WS_{2}) as an effective spin1/2 system that can be created with atomic precision while keeping the surrounding atomic structure virtually unchanged. We precisely characterize the coupling of the spinpolarized local defect states generated by the CRI with its host lattice. Inherent to any solidstate atomic quantum system, the defectspecific electronphonon interaction is often a dominant decay and decoherence channel^{15}. Here, we demonstrate that the electronphonon coupling associated with the two discrete electronic CRI defect states is limited to a only few vibrational modes and exhibits a distinct spin and layer dependence.
Carbon impurity defects at chalcogen sites (C_{X}, X = S, Se) are created by scanning tunneling microscopy (STM)induced hydrogen desorption from carbon–hydrogen (CH) complexes. Such CH impurities are frequently found in synthetic WS_{2} and WSe_{2}^{16} but can also be deliberately created by postsynthetic methane plasma treatment, as shown here for WS_{2}^{16,17}. We induce the hydrogen depassivation of CH_{X} by a voltage pulse from the STM tip, which is highly reproducible and atomically precise. For WS_{2}, the Fermi level alignment with the graphene substrate results in a negatively charged carbon impurity with a radical character that we refer to as a CRI and denote as C\({}_{\,{{\mbox{S}}}\,}^{\bullet }\). The CRI has an occupied spinpolarized defect state with a net magnetic moment of 1 μ_{B}, as predicted by our density functional theory (DFT) calculations. In WSe_{2}, we find C_{Se} is neutral and thus has no magnetic moment. Moreover, we quantify the vibronic coupling of a single CRI to the host lattice by inelastic transport spectroscopy and using DFT calculations. We find that the CRI defect orbitals couple predominantly to just a few vibrational modes. However, the coupling strength critically depends on the constituent state of the CRI twolevel system as well as on the number of TMD layers.
Results and discussion
In the following, we discuss our three primary conclusions: the hydrogen depassivation of the CH impurity and formation of the CRI, the two spinpolarized defect states associated with the CRI, and the vibronic coupling of the CRI defect states with the TMD host lattice (see Fig. 1).
Hydrogen depassivation of a CH impurity
In Fig. 2, STM and COtip noncontact atomic force microscopy (ncAFM) images of CH\({}_{\,{{\mbox{S}}}\,}^{}\) defects in deliberately CHdoped monolayer WS_{2} (0.6% atomic doping concentration) are presented. In the ncAFM images, CH\({}_{\,{{\mbox{S}}}\,}^{}\) defects appear as a small protrusion at a sulfur site (Fig. 2c), in excellent agreement with the simulated AFM contrast obtained from the relaxed geometry using DFT calculations (Supplementary Fig. 20). In STM, CH\({}_{\,{{\mbox{S}}}\,}^{}\) is imaged as a large, circular depression at positive bias resulting from upwards band bending due to the negative charge^{18,19}. After scanning the tip over the defect at high applied sample biases and high tunneling current setpoints (~2.5 V and ~15 nA), a dramatic change in the STM and AFM contrast is observed. In AFM, the small protrusion disappears (see Fig. 2d). In STM, a threefold symmetric, bright orbital structure on a dark background is observed at positive voltage. Based on the prior knowledge of the precursor defect by targeted doping, the ncAFM simulations, and the characteristic electronic fingerprint of the converted defect (discussed below), we show that the conversion process is controlled desorption of the hydrogen atom from the CH complex. The hydrogen desorption by the STM tip is likely a resonant process where tunneling into an unoccupied CH_{S} defect state weakens the C–H bond. This defect state as identified by DFT calculations exhibits a local antibonding character with a nodal plane between the carbon and hydrogen atom, supporting this hypothesis^{16}. While the process is stochastic in nature, we find a minimum value of 2.3 V, required for the C–H dissociation. Tunneling at negative bias with a comparable magnitude does not result in Hdesorption.
Hydrogen desorption by STM has been reported for hydrogenterminated silicon surfaces^{20,21,22} and organic molecules^{23,24}. As with these systems, removal of the hydrogen creates a dangling bond with a radical character, hence the defect becomes “depassivated”. The dehydrogenation of the CH defect is very reliable (success rate of about 95%), and can be performed with singleatom precision as seen in Fig. 2e. Larger patterns can be written by scanning the surface at elevated bias and currents, shown in Fig. 2f. Occasionally, a hydrogen atom reattaches from a previously depassivated C_{S} defect while scanning at high bias (see Supplementary Fig. 3). We speculate that an H atom residing close to the tip apex transfers back and passivates the defect again, as previously suggested^{25}. This shows that the process is reversible. Even more rarely, the entire CH complex is removed irreversibly, creating a sulfur top vacancy.
Hydrogen depassivation of CHdoped TMDs can generate single defects in a 2D material with atomic precision. This is a soughtafter capability for defectbased quantum systems not yet demonstrated for a 2D material. In analogy to the single transistor technologies based on hydrogen resist lithography in silicon^{21,26}, we also envision that the dangling bonds of C_{X} could be used as a predefined reactive docking site for other atoms or molecules. This approach will enable embedding functional atoms in a 2D manifold in a spatially controlled manner.
Electronic and magnetic properties of the CRI
The drastic change in STM contrast after dehydrogenation suggests a significant reconfiguration in the defect electronic structure. Scanning tunneling spectra across the CH\({}_{\,{{\mbox{S}}}\,}^{}\) defect in monolayer WS_{2} are shown in Fig. 3a. The negative charge localized at the defect gives rise to a strong upwards band bending, explaining why it appears as a dark extended depression in STM images at positive bias voltage. At negative sample bias, multiple defect states are observed, which we attribute to hydrogenic bound and resonant states of the screened Coulomb potential, as we reported recently^{18}.
After hydrogen desorption, two prominent defect states emerge deep in the WS_{2} bandgap, one at positive (~0.6 V) and the other at negative (~−0.3 V) sample bias, as seen in Fig. 3b, d. These highly localized defect states are well decoupled from the dispersive bulk WS_{2} band structure. Each state exhibits an oscillatory fine structure that is a signature of the vibronic coupling to the TMD lattice, which will be discussed later in detail. Spatial imaging of these defect states (Fig. 3f, g) reveals that they have threefold symmetry and nearly identical orbital shapes, strongly suggesting that the two resonances originate from a single openshell defect state^{27}. The gap between the defect states is slightly larger on bilayer (Δ = 770 meV) than on monolayer (Δ = 705 meV) WS_{2}. Note that these numbers were corrected for the ~11% voltage drop^{28} across the TMD—Gr/SiC interface in our doublebarrier tunneling junction geometry. The same upwards band bending before and after dehydrogenation indicates that the defect is still negatively charged, consistent with the persisting dark halo in STM images of C_{S}.
In WSe_{2} on the other hand, the carbon defect becomes chargeneutral after dehydrogenation (CH\({}_{\,{{\mbox{Se}}}\,}^{}\,\) → C\({}_{\,{{\mbox{Se}}}\,}^{0}\)). The charge neutrality can be deduced from the absence of band bending thus leading to a disappearance of the dark halo around the defect when imaged at positive sample biases upon dehydrogenation (see Supplementary Fig. 8), and the disappearance of associated hydrogenic states as seen in Fig. 3c. Moreover, the carbon impurity exhibits only a single, fully unoccupied defect state in the WSe_{2} bandgap, and no state at negative bias is observed (Fig. 3e). This neutral charge is a consequence of the different band alignments of WS_{2} and WSe_{2} with the Gr/SiC substrate. In the WSe_{2}/Gr/SiC heterostructure, the Fermi level lies roughly in the center of the WSe_{2} bandgap, whereas it is 443 meV higher for WS_{2}^{29}. Accordingly, the underlying substrate does not donate an electron to the dangling bondlike defect state of C_{Se} in WSe_{2} and thus remains empty. Alternatively, the charge state of a C_{X} could be controlled by changing the graphene Fermi level electrostatically.^{30,31} Prior DFT calculations of a neutral carbon impurity in WS_{2} featured a single unoccupied defect state in the energy gap^{17}. Addition of one electron to form a negatively charged carbon impurity in WS_{2} results in a spin polarization and exchangeinduced splitting of the ingap state, as shown in Fig. 4b. The total energy gain of the spinpolarized ground state as compared to the less favorable nonmagnetic configuration is 142 meV per single carbon atom (see Supplementary Fig. 16 for details). Hence, the negatively charged carbon impurity is characterized by a spinsplit twolevel system of which the lower level is occupied by one electron. Spatial maps of these defect states computed from DFT show they are strongly localized and almost identical to each other, in agreement with experimental observations (Supplementary Fig. 9). Orbital projected density of states from our DFT calculations reveals that the defect states are strongly hybridized and possess C 2p, W 5d, and S 3p orbitals character. The charged carbon impurity is computed to possess a magnetic moment of 1 μ_{B} from our DFT calculations, with a spin distribution closely localized at the carbon atom as shown in Fig. 4d. Based on these calculations, we conclude that the defect can be described as a CRI C\({}_{\,{{\mbox{S}}}\,}^{\bullet }\), or CRI; and we assign the two experimentally observed ingap states as the occupied and unoccupied spinsplit defect states associated primarily with the carbon anion.
While the two defect states closely resemble a spin1/2 system, the hybrid character of the states, with contributions from mainly four atoms, in conjunction with SOC, leads to a small but finite magnetocrystalline anisotropy, as reported before in other solidstate spin systems^{32}. Our DFT calculations predict magnetocrystalline anisotropy energy (MAE) of 0.1 meV with an easyaxis perpendicular to the WS_{2} plane. Because the crystal field breaks the rotational symmetry of the impurity, the total angular momentum J is strictly speaking not a good quantum number. Rather it is more appropriate to classify defect states by the irreducible representation of the symmetry group associated with the defect site, in this case, the C_{3v} point group. Our DFT calculations indicate that the spatial part of both the occupied and unoccupied defect states transform symmetrically under the C_{3v} point group and can be labeled by the A_{1} irreducible representation. Since the dimension of A_{1} is one, the two defect states are nondegenerate. The Mo_{S} antisite defect in MoS_{2} with the same symmetry has been recently predicted to feature a giant MAE of about 500 meV^{33}. The comparably small MAE of C\({}_{\,{{\mbox{S}}}\,}^{\bullet }\) might be expected given the small spin–orbit coupling (SOC) of carbon, similar to other light element color centers.
At ultralow temperatures (where MAE ≫ k_{B}T), the defect magnetic moment will be fixed in space. At a finite defect density, a net magnetic order could induce a spindependent shift of the spinpolarized WS_{2} valence band electrons as shown in Fig. 4c, lifting the energy degeneracy between the K and K’ valleys due to timereversal symmetry breaking. This energy shift λ is calculated to be 16 meV for an ordered array of ferromagnetically coupled defects with a density of 4.7 × 10^{13} cm^{−2} (5 × 5 unit cell) with the PBE functional^{34} (see Supplementary Fig. 19 for density dependence).
The unpaired electron of the negatively charged CRI constitutes an effective spin1/2 system. In organic chemistry, CRIs have been studied for decades in the context of reaction intermediates. Unpaired spins of organic compounds can be detected by electron spin resonance (ESR)^{35}. Owing to their high reactivity, free radicals are usually very shortlived. In our experiments, the UHV conditions stabilize the CRIs, but alternatively, an inert capping layer such as hBN could be employed to protect the carbon dangling bond. It is also worth noting that isotopically pure CRIs can be easily prepared by using commercially available ^{12}C or ^{13}C clean variants of methane in the plasma treatment. Moreover, the low abundance of nonzero nuclear spin isotopes in certain TMDs and the intrinsically reduced spin densities in lowdimensional materials makes the TMD matrix a great host for defect spins^{36}. While spinbath fluctuations can be expected to act favorably in 2D TMDs, electron–phonon coupling could pose another significant source of spin decoherence, which will be discussed next.
Vibronic coupling of the CRI
Each C_{S} defect resonance above and below the Fermi energy features characteristic, equidistant peaks, a consequence of a strong electronphonon interaction which can be probed by the transient attachment of an electron (at positive bias voltage) or a hole (at negative bias voltage) associated with the tunneling process. Understanding the vibronic coupling of solidstate atomic quantum systems is critical as it can limit the attainable coherence times^{15}. However, phonon sidebands can be effectively suppressed by the frequencyselective emission enhancement of a resonant cavity^{15,37}. Moreover, lowloss local vibrations or surface acoustic phonons are analogous to nanomechanical resonators that can be used to store or coherently transmit quantum information between remote quantum systems^{38,39,40}. Electron–phonon coupling is particularly relevant for very localized states, which in general lead to larger lattice relaxations. This applies to any deep center in widebandgap semiconductors. The theoretical framework of electron–phonon interactions at bulk defects was developed by Huang and Rhys, as well as Pekar in the 1950s and 1960s^{41}. More recently, ab initio calculations of vibronic coupling to defect states in diamond ^{42} and in 2D materials have been a subject of active research^{43,44}.
In scanning tunneling spectroscopy (STS) measurements, inelastic scattering between localized charged excitations and vibrational modes is well known to lead to characteristic sideband structures. Such phenomena have been observed for molecules on surfaces^{45,46,47}, color centers in dielectrics^{48}, and semiconductor quantum well tunneling devices^{49}, where particular vibrational modes have been found to couple to localized electronic states. As mentioned in the previous section, when a CRI is introduced, we are able to clearly resolve this sideband structure in our dI/dV measurements (Fig. 3d). Interestingly this sideband structure differs substantially depending on the spinstate of the CRI, with the electron attachment (positive bias) exhibiting a clean Franck–Condonlike vibronic profile while the hole attachment (negative bias) exhibits a more complex fine structure, possibly involving multiple phonons. That the vibronic structure of the spinsplit defect state is sensitive to the spin state is somewhat unintuitive given that these states derive from the same nonspinsplit parent state and have essentially the same orbital structure. We note that the interaction between the spinsplit localized defect state and bath of harmonic phonons can be described by an effective independent spinboson Hamiltonian^{50,51}, as detailed in the Methods. From the exact solution of this model Hamiltonian, we can derive the electron spectral function A_{σ}(ω), that is
where ω_{ν} is the frequency of the vibrational mode ν, and \({S}_{\nu \sigma }={({g}_{\nu \sigma }/{\omega }_{\nu })}^{2}\) is the Huang–Rhys factor, related in turn to the defect statephonon coupling strength g_{νσ}; and l_{ν} is an integer. Note that the Huang–Rhys factors and defectphonon coupling strengths have a spin index, σ, while the vibrational frequencies do not, reflecting the fact that the frequencies are insensitive to the spin states while the coupling, in general, is not. Finally, \({\bar{\epsilon }}_{\sigma }\) denotes the electronic defect state energy, including the vibrational selfenergy.
We calculate the vibrational frequencies, ω_{ν}, using density functional perturbation theory (DFPT). Subsequently, the spindependent Huang–Rhys factors, S_{νσ}, are extracted from a spinpolarized finite difference DFT calculation of the electronphonon coupling, using eigendisplacements from DFPT. In this manner, all quantities appearing in Eq. (5) are determined from the first principles with no adjustable parameters (for details see Methods).
In our DFT calculations of a negatively charged carbon impurity, we find two vibrational modes spatially localized around the defect that exhibit significant coupling to the defect state, with frequencies ℏω = 22 and 75 meV. The 22 meV mode is located 0.3 meV below the top of the acoustic branch of the pristine WS_{2} monolayer and corresponds to an outofphase breathing motion involving the C–S bond and the neighboring three W atoms (Fig. 5d). Interestingly, the defect statephonon coupling associated with this mode is very sensitive to spin and occupation of the CRI state: while we compute S = 4.5 for the unoccupied CRI state, we obtain S = 0.5 for the occupied state. This nearly order of magnitude difference can be attributed to the spindependence of the change in XC potential in response to specific phonon perturbations and distinct spatial parts of the wavefunctions of the two spinpolarized defect states.
To illustrate the origin of this spindependent coupling, we partition the KohnSham Hamiltonian into two terms, a kinetic + ionic + Hartree term (T + V_{ext} + V_{H}) and an XC term (\({V}_{\,{{\mbox{xc}}}\,}^{\sigma }\)), and express the total defectstate electronphonon coupling as a sum of the two contributions, namely
where ψ_{σ} labels the defect state of interest and ∂_{ν} denotes a derivative with respect to the amplitude of displacements of phonon mode ν. The utility of the above decomposition is that it isolates the spindependent part of the KS Hamiltonian \({V}_{{{{{{{{\rm{xc}}}}}}}}}^{\sigma }\). In Supplementary Fig. 13, we explicitly show how the kinetic + ionic + Hartree (KIH) term and exchangecorrelation (XC) term contribute to the overall defect statephonon coupling. For the 75 meV mode, we find for the occupied state the KIH and XC contributions add constructively while for the unoccupied state they approximately cancel out, leading to significant differences in vibrational coupling for the two states. In contrast, for the 22 meV mode, contributions from \({V}_{\,{{\mbox{xc}}}\,}^{\sigma }\) dominate the coupling strength for the unoccupied state while it is marginal for the occupied state.
Separately, in Supplementary Fig. 13, we show the KIH term also exhibits some spindependence, associated with the different character of the spinsplit defect state wavefunction ψ_{σ}. To further explore this point, we plot in Fig. 4e, f the difference of the defect state electron densities, Δρ = ρ_{o} − ρ_{u}, where ρ_{o/u} = ∣ψ_{o/u}∣^{2}, clearly revealing the difference of the two defect states. The occupied defect state features more charge density in the vicinity of the carbon atom, which increases the coupling to the 75 meV mode, corresponding to a local outofplane C vibration (Fig. 5e). The unoccupied defect state has more charge density in the W plane, which leads to stronger coupling to the 22 meV mode (see Fig. 4e, f). We find that the vibronic coupling to a defect state becomes significant as the degree of the localization of the electronic wave function is sizable at the lattice sites where the vibration occurs.
From this analysis, we conclude the difference in vibrational coupling strength for the two defect states results from a combination of spindependent exchange and the distinct defect wavefunction character of the two states. These results are summarized in Fig. 5. As will be discussed shortly, the difference in coupling strengths is what ultimately gives rise to the very different sideband structure shown in Fig. 6a, b.
Repeating our calculations for a CRI defect in a WS_{2} bilayer identifies the same modes with significant coupling strengths as for monolayer, but with generally smaller S values (see Fig. 5a, b). This indicates that the defect states change less as the local vibration is excited in the bilayer system. In the bilayer, the defect states are delocalized into both layers, as shown in Supplementary Fig. 14, reducing the coupling to the lattice vibration. For the monolayer, our calculations also reveal strong coupling to lowfrequency resonant flexural modes (Fig. 5c) that involve the CRI defect (ℏω ≈ 5 meV), particularly for the occupied CRI state. We note that a hybrid acousticCRI defect vibrational mode is more sensitive to supercell size, increasing the uncertainty of our calculated S values as detailed in the Supplementary Materials.
While tempting to use the DFT values for ω_{ν} and S_{νσ} in conjunction with Eq. (5) and compare directly with experimental STS data, we find that in practice A_{σ}(ω) is quite sensitive to small numerical uncertainties in the Huang–Rhys factors introduced by our approximate calculations, e.g., choice of XC functional. Instead, guided by the small number of phonon modes with significant coupling strength identified by theory, we fit the measured STS spectra using Eq. (5) and subsequently compare the fit parameters to those obtained from DFT. For the unoccupied CRI state (electron attachment), three modes were sufficient for a good fit to the data; while for the occupied CRI state (hole attachment), four modes were used to largely reproduce all vibronic peaks. The fits to the tunneling spectra are shown in Fig. 6, while the refined values for ω_{ν} and S_{νσ} used in these fits are reported in Table 1. All vibronic resonances exhibit a homogeneous Gaussian line broadening (σ = 3–4 meV). Hence, this broadening is likely not temperature or lifetime limited but may be induced by coupling of the local vibrations to lattice acoustic modes^{52}.
We pause to note that the values reported in Table 1 are not entirely unique in that different sets of ω_{ν} and S_{νσ} may be able to reproduce the STS data equally well. Thus care must be taken in interpreting the results. Nevertheless, there are some robust features independent of how the fit is performed which we can confidently compare with the DFT frequencies and couplings. Notably, we find that the pristine Frank–Condonlike sideband structure observed for the unoccupied defect state (Fig. 6a) derives primarily from strong defect coupling (S ≈ 5) to a single low frequency (ℏω ≈ 20 meV) vibrational mode. Conversely, the more complex sideband structure seen in the occupied defectstate STS spectra (Fig. 6b) primarily originates from two modes, a lowfrequency mode (ℏω ≈ 20 meV) with moderate coupling (S ≈ 2) and a highfrequency mode (ℏω ≈ 80 meV) with a weaker coupling (S ≈ 1), the latter manifesting a beating pattern in the STS data. Remarkably, our DFT results are consistent with these results.
In short, we identify two vibrational modes, involving the CRI defect, that couple strongly to its defect states. While all mode energies are similar for both states and layer independent, their coupling strength is greatest for monolayer WS_{2} and is sensitive to the CRI defect state, indicating strong spinphonon coupling in this system.
In summary, we demonstrate the selective and atomically precise generation of CRIs (C\({}_{\,{{\mbox{S}}}\,}^{\bullet }\)) in a TMD host crystal. This is achieved by atomic editing of the TMD surface via STMinduced hydrogen depassivation. In its anionic state, the CRI forms an effective spin1/2 system in the bandgap with a calculated magnetic moment of 1 μ_{B}. Complementary measurements would be of use to confirm the calculated spin character of CRI in a magnetic field, for instance using ESRSTM. Synthetically introduced CH impurities are depassivated by H desorption using an STM tip. The resulting dangling bond introduces a deep defect state in the TMD bandgap that can be populated by electrons from the graphene substrate. For WS_{2} on Gr/SiC, the Fermi level alignment leads to a negative charge state of the C_{S} impurity, resulting in an openshell, spinpolarized ingap state. We also demonstrate that the atomic defect couples predominantly to two vibronic modes. While the vibrational frequencies largely defect state and layer independent, we find that the electron–phonon coupling strength is stronger for monolayer WS_{2} as compared to bilayer WS_{2}. The different coupling strengths to the spinpolarized defect states are a manifestation of the spindependent vibronic coupling in CRIs.
Methods
Sample preparation
Metalorganic chemical vapor deposition synthesis of monolayer tungsten diselenide (WSe_{2}) on graphene on silicon carbide (SiC) was performed from tungsten hexacarbonyl [W(CO)_{6}, 99.99%, SigmaAldrich] and hydrogen selenide (H_{2}Se, 99.99%, Matheson) precursors in a hydrogen gas atmosphere as previously reported.^{53} Monolayer islands of tungsten disulfide (WS_{2}) on graphene/SiC were grown on graphene/SiC substrates with an ambient pressure CVD approach from tungsten oxide (WO_{3}) and sulfur powder under argon gas.^{54,55} An inductively coupled plasmaenhanced CVD system was used to carry out the postgrowth CH doping under an Ar/H_{2} mixture and with 1 sccm methane CH_{4} flow.^{17} Raman spectroscopy was used to characterize the sample, as discussed previously.^{16,17} After exsitu growth, samples were transferred to ultrahigh vacuum (<2 × 10^{−10} mbar) and annealed at 200 °C to remove adsorbates.
Scanning probe microscopy (SPM) measurements
SPM measurements were acquired with a Createc GmbH scanning probe microscope at liquid helium temperatures (T < 7 K) under an ultrahigh vacuum (p < 2 × 10^{−10} mbar). The quartz crystal cantilever (qPlus based) sensor^{56} tip apex was prepared by indentations into a gold substrate and verified as metallic on the Au(111) surface. Noncontact AFM images were taken with a carbon monoxide functionalized tip^{57} in constant height mode at zero bias. STM topographic measurements were taken in constant current mode with the bias applied to the sample. STS measurements were recorded using a lockin amplifier with a resonance frequency of 683 Hz and a modulation amplitude between 2 and 10 mV.
Vibronic model
Our STS measurements of the defect states feature a characteristic sideband structure as shown in Fig. 6, arising from strong electron–phonon interactions^{51,58}. In our experimental setup, resonant tunneling between the tip and sample will occur when the applied bias voltage is aligned with the defect energy levels. The defectphonon system can be described by an effective independent spinboson Hamiltonian^{50}, that is
where ϵ_{σ} is the energy level of the defect with spin state σ, \({c}_{\sigma }({c}_{\sigma }^{{{{\dagger}}} })\) the defect annihilation (creation) operator, ω_{ν} phonon frequency of mode ν, and \({a}_{\nu }({a}_{\nu }^{{{{\dagger}}} })\) phonon annihilation (creation) operator, and g_{νσ} the defectphonon coupling matrix element.
The Hamiltonian in Eq. (3) can be solved exactly via a canonical transformation to arrive at the T = 0 electronic Green’s function^{50}
where \({{{\Phi }}}_{\sigma }(t)={\sum }_{\nu }{S}_{\nu \sigma }(1{e}^{i{\omega }_{\nu }t})\) is the phonon contribution, \({S}_{\nu \sigma }={({g}_{\nu \sigma }/{\omega }_{\nu })}^{2}\) the coupling constant (also known as Huang–Rhys factor), and Ω_{σ} = ∑_{ν}S_{νσ}ω_{ν} the selfenergy. We can derive the defect energy level spectral function by taking the imaginary part of the retarded Green’s function, obtaining
where n denotes the number of the phonon modes that couple to the defect. In the main text, we define \({\bar{\epsilon }}_{\sigma }={\epsilon }_{\sigma }{{{\Omega }}}_{\sigma }\) since the Ω_{σ} only gives a relative shift of the defect level, and is not responsible for the fine structure of the sidebands.
To more closely model the experimental setup, we note that Wingreen et al.^{51} added the hopping terms coupling of the contact leads with the defect state to the Hamiltonian in Eq. (3) and derived an expression for the resonant tunneling transmission probability. They found that when the hopping rate between contact and defect is assumed constant, the aforementioned transmission probability is proportional to the spectral function reported above, justifying a direct comparison of spectral function (5) with the transmission probability and ultimately the experimentally measured dI/dV curves in the main text.
Firstprinciples calculations
In this work, the spectral function (5) is constructed in a completely ab initio manner with no adjustable parameters. We calculate ϵ_{σ}, ω_{ν}, and S_{νσ} from first principles, as follows. First, for a fully optimized supercell, we perform a spinpolarized density functional theory (DFT) calculation to determine the Eigen energy associated with the defect state, ϵ_{σ}. Using the same supercell, we then calculate ω_{ν} at the Γpoint of the Brillouin zone using densityfunctional perturbation theory (DFPT)^{59}. After obtaining ω_{ν} and the corresponding (mass reduced) eigenmodes ξ^{ν}, we construct modulated structures freezing in displacement patterns associated with all phonon modes using PHONOPY^{60}. The defectphonon coupling matrix elements are then calculated with the following the finite difference method
where ϵ_{σ}(R_{0}) is the defect energy level with nuclei fixed at their equilibrium configuration, R_{0}, while ϵ_{σ}(R_{0} + Aξ^{ν}) denotes the defect energy when nuclei are displaced, by amplitude A, along the phonon eigenmode ξ^{ν}. For each phonon mode, we then compute the change in energy of the defect levels as a function of amplitude, A, considering a few different amplitudes. We perform a linear fit of the defect energy level with amplitude, determining g_{νσ} from the slope [Eq. (6)]. We have explicitly confirmed, for nonspin polarized calculations, that this approach gives the same result (within 2 meV) as explicit extraction of the coupling from DFPT. In this analysis, we neglect SOC. Additional details regarding the firstprinciples calculation are provided below.
We performed firstprinciples DFT calculations using the QUANTUM ESPRESSO package.^{61} We employed the PBE generalized gradient approximation for XC functional^{34} and used scalar relativistic optimized normconserving Vanderbilt (ONCV) pseudopotentials for W, S, C, and H atoms from the PseudoDojo library^{62} in all cases unless explicitly including SOC within nonlinear DFT; in these cases, the fully relativistic pseudopotentials of the same type were used. We used an 80 Ry planewave cutoff energy. In order to model a charged system, we added one additional electron to the cell, along with compensating background charge. We used the experimental lattice parameter for WS_{2}, 3.15 Å, throughout this work unless otherwise stated. We generated 3 × 3, 5 × 5, 6 × 6, and 7 × 7 supercells including a single point defect with a vacuum of ~16 Å. For our total energy (electron–phonon) calculations, we used Γcentered uniform 4 × 4(3 × 3), 3 × 3(2 × 2), 2 × 2(2 × 2), and 2 × 2(1 × 1)kgrids for 3 × 3, 5 × 5, 6 × 6, and 7 × 7 supercells, respectively, with Marzari–Vanderbilt smearing with the spread of 0.001 Ry^{63}. Our convergence thresholds for ionic minimization on forces and total energy are 10^{−4} and 10^{−6} in atomic units. We considered spin polarization, in a collinear framework, throughout our calculations.
As the phonon frequencies of 2D materials are quite sensitive with respect to lattice parameters, we used the experimental lattice parameter of 3.15 Å since the PBE optimized lattice parameter (3.19 Å) is larger than the experimental value, which results in lower phonon frequencies. Our convergence threshold for selfconsistency in our DFPT calculations is set to 1.0 × 10^{−18} a.u.
Data availability
The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
We thank Andreas Schmid, Frank Ogletree, and Liang Z. Tan for helpful discussions. This work was supported as part of the Center for Novel Pathways to Quantum Coherence in Materials, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Basic Energy Sciences. Scanning probe measurements were performed at the Molecular Foundry supported by the Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DEAC0205CH11231. K.A.C. was supported by the University of California—National Lab Collaborative Research and Training (UCNL CRT) program. A.W.B. was supported by the U.S. Department of Energy Early Career Award. B.S. appreciates funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 948243). Calculations and analysis performed by J.H.L. and J.B.N. were supported by the Air Force Office of Scientific Research Hybrid Materials MURI under award number FA95501810480, and the development of analytic models by J.B.H. and J.B.N. was supported by the Theory FWP, funded by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Sciences, and Engineering Division under Contract DEAC0205CH11231. A.K. and J.A.R. acknowledge funding from Intel through the Semiconductor Research Corporation (SRC) Task 2746.001, the Penn State 2D Crystal Consortium (2DCC)Materials Innovation Platform (2DCCMIP) under NSF cooperative agreement DMR1539916, and NSF CAREER Award 1453924. T.Z. and M.T. were supported by the Basic Office of Science of the Department of Energy under award DESC0018025. The authors thank Zhuohang Yu for their technical support. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under Contract No. DEAC0205CH11231.
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B.S. and A.W.B. conceived the original concept and initiated the work. B.S. and K.A.C. conducted the scanning probe measurements and analysis. J.H.L., J.B.H., and J.B.N. performed the density functional theory calculations. T.Z., A.K., J.A.R., and M.T. grew the TMD samples. C.K. and J.R. helped to analyze the data. All authors contributed to the paper.
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Cochrane, K.A., Lee, JH., Kastl, C. et al. Spindependent vibronic response of a carbon radical ion in twodimensional WS_{2}. Nat Commun 12, 7287 (2021). https://doi.org/10.1038/s4146702127585x
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DOI: https://doi.org/10.1038/s4146702127585x
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