Abstract
The intrinsic light–matter characteristics of transitionmetal dichalcogenides have not only been of great scientific interest but have also provided novel opportunities for the development of advanced optoelectronic devices. Among the family of transitionmetal dichalcogenide structures, the onedimensional nanotube is particularly attractive because it produces a spontaneous photocurrent that is prohibited in its higherdimensional counterparts. Here, we show that WS_{2} nanotubes exhibit a giant shift current near the infrared region, amounting to four times the previously reported values in the higher frequency range. The walltowall charge shift constitutes a key advantage of the onedimensional nanotube geometry, and we consider a Janustype heteroatomic configuration that can maximize this interwall effect. To assess the nonlinear effect of a strong field and the nonadiabatic effect of atomic motion, we carried out direct realtime integration of the photoinduced current using timedependent density functional theory. Our findings provide a solid basis for a complete quantum mechanical understanding of the unique light–matter interaction hidden in the geometric characteristics of the reduced dimension.
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Introduction
Atomically thin transitionmetal dichalcogenides (TMDs), which consist of transitionmetal atoms sandwiched between layers of chalcogen atoms, have attracted extensive interest as a novel platform hosting various quantum phenomena^{1,2,3,4}. Singlelayer TMDs exhibit a direct bandgap with a unique spin–valley coupling induced by the broken inversion symmetry^{5,6,7}, and the lessscreened Coulomb interaction gives rise to the strong excitonic transitions including higherorder excitonic states^{8,9,10,11,12}. In addition, large photoluminescence^{13,14}, fast photocurrent switching^{15,16,17}, and high photoresponsivity^{18,19,20} have been demonstrated and various advanced optoelectronic applications (e.g., photodetectors^{21,22,23}, photovoltaic devices^{24,25}, and lightemitting diodes^{26,27}) have been envisioned for van der Waals (vdW)stacked TMD layers.
Despite the various intriguing light–matter characteristics of TMD layers, the spontaneous photocurrent without external bias, which has been referred to as the bulk photovoltaic effect (BPVE), is absent because of the inherent mirror symmetries. Intensive efforts have been devoted to unleashing the intrinsic optoelectronic advantages confined under the symmetry constraints of TMDs, such as by using electrostatic gating^{25,28}, an inplane/outofplane p–n junction^{29}, the broken symmetry of an edge^{30,31}, and vdW heterostacks^{32,33,34}. Moreover, an innovative approach to the TMDbased BPVE was initiated by a recent study of onedimensional (1D) WS_{2} nanotubes^{35}. The obtained shortcircuit current of a WS_{2} nanotube was found to be a few orders of magnitude larger than that of conventional photovoltaic materials. This dimensionality reduction, from a twodimensional (2D) layer to a 1D nanotube, is now attracting extensive attention as a breakthrough that overcomes the limitations of conventional devices based on junctions or heterointerfaces of materials^{36,37}.
Nevertheless, the physical mechanism underlying the large BPVE in TMD nanotubes has not yet been elucidated. A spontaneous photocurrent (shift current) is induced by the spatial charge shift between occupied and unoccupied states via optical excitation of polar materials^{38}. As atomic structures, electronic band dispersions, and electric polarizations are closely intertwined in this phenomenon^{39,40}, the fundamental question is whether any particular geometrical advantages are inherent to the 1D structure^{35}. In these inspections, the justification of the perturbationbased theories and the consideration of the nonlinear effect of strong fields also need to be addressed to achieve complete understanding of the behavior of TMD nanotubes under practical application conditions.
In the present work, we address the aforementioned questions through secondorder perturbation analysis^{41} based on static density functional theory (DFT), together with realtime calculations using timedependent density functional theory (TDDFT). We find that the BPVE characteristics of nanotubes can be decomposed into intratube and intertube effects. In WS_{2} singlewalled nanotubes (SWNTs), the shift current is dominated by the intratube effect triggered by d–d transitions of W atoms^{42}, leading to a shift current three times larger than that of photovoltaic perovskites^{43}. For WS_{2} doublewalled nanotubes (DWNTs), the walltowall charge shift becomes substantial, which leads to a shift current near the infrared region four times greater than that previously reported^{35}. By carrying out realtime integration of the current using realtime timedependent density functional theory (rtTDDFT) calculations, we find that the nonlinear effect of a strong field and the nonadiabatic effects of atomic motions can be critically effective for generating photocurrent. We propose that our results can be used as a guiding principle for designing advanced optoelectronic applications in reduced dimensions.
Results
Atomic structure and spontaneous polarization of SWNTs
As electric polarization is an essential ingredient for the BPVE, we investigate the electric polarization of SWNTs with various diameters. The geometry of the nanotubes is determined by the rollingup direction, which can be characterized by the socalled chiral vector^{44,45}, which is given as \(\vec{C}={n\vec{a}}_{1}+{m\vec{a}}_{2}\equiv (n,m)\), where \({\vec{a}}_{1}\) and \({\vec{a}}_{2}\) are the unit vectors of the hexagonal 2H phase of WS_{2} and n and m are integers. In the present study, we consider zigzag, armchair, and chiral SWNTs, corresponding to \(m=0\), \(m=n\), and \(m\ne n\), respectively.
As illustrated in Fig. 1a, mirror symmetry (M_{xy}), which is inherited from twodimensional WS_{2}, is present in the armchair SWNTs, removing the electric polarization (Fig. 1c)^{35,46,47}. On the other hand, the mirror symmetry is broken for the zigzag SWNTs (Fig. 1b and Supplementary Fig. 1), leading to spontaneous polarization along the zaxis (Fig. 1d). When we assume the closepack triangular lattice of nanotube bundles, the estimated polarization density (0.82–1.52 C/m^{2}) of the zigzag SWNT is comparable to or even greater than that of representative polar materials such as BaTiO_{3} (0.26 C/m^{2}) and BiFeO_{3} (0.9 C/m^{2})^{48,49} (see Supplementary note 1). The main contributor to this large electric polarization in the axial direction is the ionic part, and the polarization normalized by the tube diameter (the polarization divided by n) approaches the polarization density of a TMD monolayer (Supplementary Fig. 2)^{50,51,52}. The electric polarization of chiral SWNTs is intermediate between those of the zigzag and armchair SWNTs, depending on the rollingup direction (Supplementary Fig. 3). Although various chiral nanotubes have been speculated to be coexist in the condensed phases of actual synthesized samples^{35}, the zigzag nanotube is likely to be the greatest contributor to the photovoltaic property because of its apparent axial polarization. Hereafter, we focus on the electronic and optoelectronic properties of the zigzag nanotube to elucidate the fundamental origin of the experimentally observed large BPVE in WS_{2} nanotubes.
Electronic and optoelectronic properties of the zigzag SWNT
To identify the characteristics of the electronic structure of the zigzag SWNTs relevant to the shiftcurrent generation, we calculate its band structures and shiftcurrent spectra. As presented in Fig. 2a, the zigzag (10,0) SWNT has a direct bandgap at the Brillouin zone center (Γ point)^{42,53}. The band structures of other zigzag SWNTs have similar overall features, and the size of the bandgap increases with the tube diameter (Supplementary Fig. 4) in good agreement with previous results^{50,51}. As the bandgap increases with the tube size, the onset frequency slightly shifts up and the absorption strength marginally decreases^{54}. Nevertheless, the overall shape of the absorption spectrum is maintained irrespective of the tube size (Fig. 2b and Supplementary Fig. 4)^{47}. The bandedge excitation is almost negligible, and dominant absorption peaks are observed in the higherenergy region above 1.0 eV (Fig. 2b). On the basis of the electronic and optical properties, we evaluate the shiftcurrent spectra (i.e., the response function between the shiftcurrent density and external electric fields) using secondorder perturbation theory^{41} (Fig. 2c). The spectrum of the zigzag (10,0) SWNT shows large shiftcurrent peaks: 6.2 mA Å^{2}/V^{2} at ℏω = 1.0 eV and −14.2 mA Å^{2}/V^{2} at 2.2 eV. When the SWNTs are assumed to form a bundle closely packed into the 2D hexagonal lattice, these shiftcurrent values correspond to 22.1 and 50.6 μA/V^{2}, respectively, which are much larger than the maximum values of the known photovoltaic perovskite oxides, such as PbTiO_{3} (16.9 μA/V^{2}) and BaTiO_{3} (16.1 μA/V^{2}), near the visiblelight region^{44}. The shift current spectra of various \((n,{\!}0)\) SWNTs are presented in Supplementary Fig. 4 in the supplementary data. The overall size of the shift current somewhat decreases with increasing tube size, which depends on the optical absorption associated with the bandgap of the SWNTs^{51,55}. (Supplementary Fig. 4)
To elucidate the electronic origin of the enhanced shift current, we compute the shiftcurrentweighted density of states (SDOS), which is defined by the orbitalprojected density of states (Supplementary Fig. 5) weighted by the contribution of each state to the shift current (Fig. 2d)^{40}. The orbital character of hole (electron) carriers contributing to the shift current is represented by the spectral density below (above) the Fermi level in the SDOS of the zigzag (10,0) SWNT. We investigate the transitions of three different energy regions (ℏω = 1.0, 1.5, 2.2 eV) corresponding the vertical dotted lines in Fig. 2c. We find that, rather than the p–d or p–p transitions, the transitions between different d orbitals are dominant in the generation of the shift current, irrespective of the light frequency (Supplementary Fig. 6)^{53}. In all these transitions, a substantial amount of the axial orbital (\({d}_{{z}^{2}}\)) of the initial state is excited into another d orbital. For the lowenergy transition (ℏω = 1.0 eV), the excited electron states mainly consist of the planar orbital (\({d}_{{x}^{2}{y}^{2}}+{d}_{{xy}}\)), whereas the hole states are derived from a mixture of various d orbitals (Fig. 2e). The positive SDOS indicates that the shift current flows in only one direction (the top panel in Fig. 2d). With increasing excitation frequency (ℏω = 1.5 eV), a small negative SDOS with a large planar orbital contribution begins to appear above the Fermi level (the middle panel in Fig. 2d), which contributes to the shift currents developing in the opposite direction. As a result of the cancellation, the shiftcurrent peak becomes smaller near ℏω = 1.5 eV (Fig. 2c). Under the higherfrequency light radiation (ℏω = 2.2 eV), the positive SDOS is strongly suppressed, leading to a large negative shiftcurrent peak. On the basis of the shift current and SDOS result, we can control the direction and the magnitude of the shift current in SWNTs by tuning the applied light.
To examine possible artifacts coming from the approximated density functions, we carry out the same calculations with the hybrid functional and confirm that essentially the same electronic and optoelectronic properties of the zigzag SWNTs are obtained (Supplementary Fig. 7). We also confirmed that various other TMD nanotubes result in qualitatively the same shift current characteristics (Supplementary Fig. 8).
Electronic and optoelectronic properties of the zigzag DWNT
The results discussed above clearly indicate that the optoelectronic nature of TMD materials can be activated by the dimensionality reduction to the 1D nanotube structure. Here, we extend the study to multiwall structures, with particular focus on the interwall effect, given that the samples synthesized in the experiment are thought to be mostly multiwalled concentric tubes^{35}. To investigate whether the interwall interaction gives rise to an additional enhancement beyond the mere sum of individual tubes, we selected the (7,0)@(18,0) DWNT as a minimal example of multiwalled tubes. As shown in Fig. 3a, the bandgap of the zigzag (7,0)@(18,0) DWNT (0.18 eV) is slightly smaller than that of the inner SWNT (0.21 eV), leading to a marginal downshift of the absorption peak (Fig. 3b). Overall, the absorption intensity of the DWNT is comparable to the summation of the absorption intensities of the individual SWNTs (Fig. 3b). However, the DWNT exhibits an orderofmagnitude larger shift current compared with that of the constituent SWNTs (Fig. 3c). As indicated by the downward arrow in Fig. 3c, the lowfrequency peak value (193 mA Å^{2}/V^{2}) is almost 20 times larger than that of the SWNTs (5.9–16.6 mA Å^{2}/V^{2}, Fig. 2c). This giant shiftcurrent peak indicates that the DWNT possesses a new photoactive conduction channel that is not inherent to the individual SWNTs. This enhancement owing to the interwall effect is persistent even for the large walltowall distance. In the supplementary data, Supplementary Fig. 9, we show that the interwall effect decays with the walltowall distance, but still remains until the distance reaches 11 Å.
To elucidate the effect of the walltowall transitions more explicitly, as depicted in Fig. 3d, we analyze the origin of the large shiftcurrent peak using the SDOS projected into d orbitals of each inner and outer tube. The spatial distributions of the hole and electron carrier densities associated with the two distinct interwall excitations, denoted as α and β in Fig. 3d, are presented in Fig. 3e, which obviously confirms that each carrier belongs to each individual tube. This result clearly indicates that newly emergent interwall excitation channels give rise to the giant BPVE. Notably, the interwall chargeshifting mechanism is not necessarily confined to the coaxial nanotubes. We verify the appearance of the interwall effect in a SWNT bundle (Supplementary Fig. 10) and an MoS_{2} DWNT (another member of the TMD nanotube family) (Supplementary Fig. 11). Our results imply that the TMD multiwalled nanotubes, which are equipped with interwall chargeshifting channels, have excellent potential for lowfrequency optoelectronic applications. Compared with conventional solarenergyharvesting devices, which are centered around the visible or higherfrequency regime,^{35} the TMD nanotubes have an advantage in the lowerfrequency region.
The interwall chargeshifting mechanism, shown to yield the dramatic increase in the shift current, is not limited to the DWNTs but extensively applicable to general multiwalled nanotubes^{35}. The effect in a thick multiwalled tube can be estimated from the comparison between the DWNT and the triplewalled nanotube (TWNT), as summarized in Supplementary Fig. 12. While the DWNT presents apparent enhancements over the SWNT, the shift current and absorption strength of the TWNT is largely comparable to that of DWNT in the lowenergy region (<2 eV); the effect of triple walls is appreciable only in the higherenergy region (> 2 eV). The larger tube, in the outer shell, possess a larger bandgap, and the interwall mechanisms between the pairs of outer walls are manifested in higherenergy region. This indicates that, even in thick multiwalled tubes, the interwall effect is mostly governed by the same effect in the DWNT. Furthermore, this intriguing advantage of the interwall charge shift is distinct from the effect of dimensionality lowering, as it cannot be obtained from the monolayers; the effect of dimensional crossover, from 2D layers to 1D nanotubes, was highlighted in the discussion in ref. ^{35}. For a side discussion, condensed aggregates of realistic samples certainly include portions of chiral nanotubes. As we present in Supplementary Fig. 9, the interwall chargeshifting mechanism is quite persistent over large walltowall separations, and the main optical absorption responsible for the large shift current is not driven by the bandedge configurations but by the interwall effect between the d orbitals states of neighboring tubes. Thus, the interwall effect we discussed for the zigzag DWNTs could be extended to the multiwalled TMDC nanotubes with various chiralities.
Photovoltaic effect of Janustype WSSe nanotubes
The important contribution of the interwall charge shift in multiwalled TMD nanotubes, as discussed in the preceding section, inspired us to consider whether Janustype multiwalled tubes could lead to a further enhancement of the intertube chargeshifting mechanism because of their builtin radial electric field^{56,57}. The Janustype nanotube, whose inner and outer chalcogen layers are composed of S and Se atoms^{58}, respectively, is illustrated in the inset of Fig. 4a and is hereafter referred to as WSSe nanotube. We first examine the optoelectronic properties of the (10,0) SWNT of the WSSe. Although the overall shiftcurrent spectrum exhibits similar features as that of the WS_{2} (10,0) SWNT, the spectrum of the Janustype SWNT has a prominent lowenergy peak, as indicated by the blue arrow in Fig. 4a. This large shiftcurrent peak is attributable to the increased carrier population (optical absorption: Fig. 4b) and the larger shift vector (Fig. 4c). Shift vector is the chargecenter difference between initial and final states on the excitation^{41}. The SDOS of the (10,0) WSSe SWNT confirms that the lowenergy excitation is dominated by the d–d transition (Supplementary Fig. 13); various other combinations of the Janustype TMD nanotubes exhibit similar trends (Supplementary Fig. 14).
For the case of the Janustype coaxial DWNT, as expected, the shift current is dramatically enhanced, particularly in the lowenergy region (Fig. 4d). Whereas the optical absorption does not substantially differ from that of the WS_{2} DWNT (Fig. 4e), the value of the shift vector for the Janus DWNTs is more than three times greater than that for the pristine DWNTs (Fig. 4f). The large shift vector of the Janus DWNT is obviously ascribed to the radial builtin potential due to the compositional asymmetry between the interfacing walls. Various other compositions of the Janustype DWNTs are also considered, which exhibit similar features, as summarized in Supplementary Fig. 15.
Realtime ab initio study of the chargeshifting dynamics: the nonlinear effect of a strong field and the nonadiabatic effect of atomic motions
The shiftcurrent results presented in the preceding section are all based on perturbation theory. To extend our understanding of the shiftcurrent generation mechanism beyond the limitations of the linear response theories, we here investigate the timedependent photocurrent using rtTDDFT calculations, focusing on the effects of a strong field and atomic motion. The details of our rtTDDFT calculation and the definition of the currentrelated quantities, including all evenorder responses (\({J}_{{{{{{\rm{even}}}}}}}\left(t\right)\)), are summarized in the Methods section. As predicted by our shiftcurrent calculation (Fig. 5a), the zigzag (7,0) SWNT exhibits distinct responses depending on the frequency of the light (ℏω = 0.9 and 1.7 eV) for a given intensity (6.05 × 10^{10 }W/cm^{2}). In this scheme, the timedependent carrier population is obtained through the projection of the timeevolving wavefunctions onto the ground states (Fig. 5b). For both frequencies, the electron/holecarrier density of states monotonically increases with time, implying a charge shift through carrier excitation, in agreement with the secondorder perturbation theories discussed above. To estimate the direct constant current unambiguously, as presented in Fig. 5, the evenorder realtime currents are employed (\({J}_{{{{{{\rm{even}}}}}}}\left(t\right)=\frac{({J}_{+}(t)+{J}_{}(t))}{2}\)). In this computation, we focus on the linearly polarized light and the injection current is not reflected in the time profile of the current^{41}. When the light with frequency ℏω = 0.9 eV is applied, the carrier excitation occurs mainly near the Γ point in secondorder perturbation theory (Supplementary Fig. 16), leading to the two pronounced peaks in Fig. 5b. However, when the light frequency is ℏω = 1.7 eV, the absorption arises over a wide range of the momentum space (Supplementary Fig. 16), enabling substantial spread of electron/hole carriers beyond the Γ point (Fig. 5b). The consistency between the perturbation results and the direct realtime integration of the photocurrent using the rtTDDFT approach is verified in this section. As the external fields are implemented into the vector potential of the density functional Hamiltonian, the latter method can naturally be applied in the case of a strong field; we focus on this approach in the following paragraphs.
Here, we consider the effect of field strength on the generation of the photocurrent. For a more comprehensive comparison, we calculate the normalized timeaveraged currents, defined as \({\widetilde{J}}_{{{{{{\rm{avg}}}}}}}(t)={J}_{{{{{{\rm{avg}}}}}}}(t)/I\), where I is the intensity of the external electric field. For the given field (ℏω = 0.9 eV), the normalized photocurrent of the zigzag (7,0) SWNT is presented in Fig. 5c. Interestingly, the current direction is reversed when the field strength is increased beyond I ≥ 10^{12 }W/cm^{2}. To analyze this effect of field strength, we compare the contribution from the zonecenter states near the Γ point (k < 0.007 Å^{−1}) and other states of the Brillouin zone (k > 0.007 Å^{−1}), as summarized in Fig. 5d. For the weak field (I = 6.05 × 10^{10 }W/cm^{2}), as shown in Fig. 5e, the photocurrent is dominated by the zonecenter state. However, under the strong field (I = 1.51 × 10^{12 }W/cm^{2}), the contribution of the nonΓ region becomes substantial and the current direction is flipped. By projecting the timeevolving Kohn–Sham states into the given ground state, we traced the bandresolved timedependent carrier populations (Fig. 5e). The bandresolved carriers are depicted by colored spheres, whose scale is normalized by the maximum value of the weakfield case within the time window of 0 ≤ t ≤ 30 fs. The strong field enables the carrier excitations to be spread over nonΓ regions through a virtual multilevel transition, resulting in the flipping of the photocurrent direction (Fig. 5c, d).
To examine the effect of atomic motions, accompanied by the carrier excitation on the shiftcurrent generation mechanism, we repeat the same realtime photocurrent calculations through the Ehrenfest dynamics, in which atomic motions are progressed through the instantaneous forces^{59}. We tested with the light frequencies corresponding to the peaks in the shift current spectra (the inset of Fig. 5a), and hereafter, we mainly present the results with the frequency of ℏω = 0.9 eV. The electronic excitation causes the spontaneous displacement of outer sulfur atoms along the radial direction (inset of Fig. 5f), which greatly enhances the generated photocurrent. To understand the large enhancement of the photocurrent induced by the atomic motion, we examine the time evolution of the projected density of states for W atoms. When the atoms are fixed in the groundstate configuration, the states near the Fermi level, which are mostly attributable to W orbitals, are primarily responsible for the carrier excitation (Fig. 5b and Supplementary Fig. 17). The overall timeevolving characters of the main electron/hole peaks are preserved, even in the presence of Ehrenfest atomic movements; however, the carriers are dispersed over a wider energy range (Fig. 5g). The main electron and hole peaks are located at the local maximum and minimum points of the crystal orbital Hamilton populations (COHPs), as denoted by the vertical red and blue lines, respectively, in Fig. 5h^{60}. Notably, the positive and negative values in the plot of the COHP correspond to the bonding and antibonding character, respectively. The aforementioned excitations weaken the bond strength by increasing antibonding character (Fig. 5h), which triggers the outward displacement of the S atoms. The atomic motion promotes orbital mixing, resulting in a wider dispersion of the electron/hole carriers (Fig. 5b, g). This increased antibonding character induces segregation of the outer S atoms, which further promotes the charge transfer to the W and the inner S atoms (Supplementary Fig. 18). To visualize this redistribution of the excited carrier, we plot the variation of the instantaneous charge density from the ground state in Fig. 5i. The number of excited carriers is substantially increased by the elongation of the W–S distance, and they oscillate back and forth in synchronization with the frequency of the external field. As the carriers are more loosely bound to the W atoms, they become more vulnerable to the shifting in response to the field, resulting in the large photocurrent observed in Fig. 5f. We repeated the same calculations with the geometries fixed at several instantaneous positions; the enhancement is negligible, and the obtained photocurrents are all similar to that of the equilibrium geometry (Supplementary Fig. 19). This Ehrenfest dynamics represents a substantial nonadiabatic effect of atomic motions on the photocurrent. We also carry out the same calculations with ℏω = 1.7 eV and find a similar enhancement of the photocurrent (Supplementary Fig. 20).
Discussion
In summary, we examined the underlying physics of the giant BPVE of TMD nanotubes by using perturbation theory analysis, together with ab initio realtime simulations of the photocurrent. We found that the unique interwall charge shift of the multiwalled coaxial tubes is the primary factor responsible for the large BPVE of the nanotubes. We predicted that the shift current near the infrared region can be four times larger than the maximum value known in the highfrequency region. As an example structure that possesses a strong intrinsic advantage in the walltowall charge shift, we considered Janustype nanotubes. Beyond the limitations of perturbation methods, we addressed the nonlinear effect of strong fields and the nonadiabatic effect induced by atomic motion by performing rtTDDFT calculations. We found that the direction and the magnitude of the photocurrent could be controlled by tuning the field frequency and intensity. Through simulations of the Ehrenfest dynamics, we found that the electronic high excitations were relaxed into the radial breathing motions of atoms, which constitutes an additional unique advantage of the nanotube geometry. Our results suggest that the unique geometrical advantage of TMD nanotubes is not simply caused by the intrinsic symmetry lowering of their 1D structure but is mainly attributable to the unique walltowall charge shift combined with the nonadiabatic effect of atomic relaxation.
Methods
Electronic structure calculation
We performed density functional theory (DFT) calculations via the projectoraugmented planewave method^{61}, as implemented in the Vienna Ab initio Simulation Package (VASP)^{62}. The Perdew–Burke–Ernzerhof (PBE) functional of the generalized gradient approximation^{63} was used to describe the exchange–correlation interactions among electrons. We also used the Heyd–Scuseria–Ernzerhof (HSE) hybrid functional^{64} to crosscheck the calculation of the PBE functional. The spin–orbit coupling effect was included in our electronic structure calculations. Atomic relaxations that included the effect of the vdW interactions were carried out until the maximum forces were less than 0.001 eV Å^{−1}. The vacuum layer was set to >15 Å to simulate isolated WS_{2} nanotubes. The energy cutoff for the planewavebasis expansion was selected to be 500 eV. We used a 1 × 1 × 10 kpoint grid for singlewalled WS_{2} nanotubes and a 1 × 1 × 5 kpoint grid for doublewalled WS_{2} nanotubes. The electric polarization of the nanotubes was estimated using the Berry phase method^{65}
Secondorder optical response using perturbation theory
The shiftcurrent spectra were evaluated from the tightbinding Hamiltonian based on the maximally localized Wannier functions^{66} using the secondorder optical response formalism^{39,40,41}. For the shiftcurrent estimation, a 3 × 3 × 100 kpoint grid was used in our calculations, which provide a sufficiently dense mesh, leading to the well converged value of the shiftcurrent spectra. To remove the ambiguity in the presentation of the dielectric constant of nanotube caused by the vacuum region in the plane direction, we normalize the optical/optoelectrical quantities by the unit length in the axial direction instead of the volume of a supercell^{54,67}. In this regard, the shift current spectra for the 1D nanotube, in the present work, is presented in the unit of A Å^{2}/V^{2}. The calculation of the shift currents, in the scope of the present work, does not include excitonic effects. Since the excitonic effects usually dominate in the absorption peaks near bandedge transitions^{8,9}, our analysis on the shift current generation mechanism in higherenergy regions, particularly for the interwall charge shift mechanism—the main focus of the present study—is still valid. Note that previous theoretical works without considering the excitonic effects also successfully reproduced the experimental results^{43}.
Direct realtime calculation of the current \({{{{{\boldsymbol{J}}}}}}({{{{{\boldsymbol{t}}}}}})\) using the rtTDDFT
To examine the electric current of the excited state, we performed rtTDDFT calculations using the planewavebased realtime evolution^{42,68,69}. In our calculations, the Kohn–Sham wavefunction, the density, and the Hamiltonian were selfconsistently evolved through the timedependent equation:
where n and \({{{{{\bf{k}}}}}}\) denote the band index and the Bloch momentum vector, respectively. \({{{{{{\bf{A}}}}}}}_{{{{{{\rm{ext}}}}}}}\) and \({V}_{{{{{{\rm{DFT}}}}}}}\) indicate the timedependent vector potential and DFT potential, respectively. The discretized time step for the time integration (\(\triangle t\)) was set to 2.415 as. In our calculations, the electric field was expressed using the velocity gauge of the vector potential via the relation \({{{{{\bf{E}}}}}}\left(t\right)=\frac{1}{c}\partial {{{{{{\bf{A}}}}}}}_{{{{{{\rm{ext}}}}}}}/\partial t\). The initial wavefunctions \([{\psi }_{n,{{{{{\bf{k}}}}}}}(t=0)]\) were obtained from the staticgroundstate DFT calculations using the QUANTUM ESPRESSO package^{68} with the PBE exchange–correlation functional^{70}. Spin–orbit coupling was not considered in our rtTDDFT calculation to treat the large cell of TMD nanotubes. Using the timeevolving Bloch wavefunctions, we evaluated the time profile of the current density as follows^{70}:
where n is the band index, \({f}_{\!\!n,{{{{{\bf{k}}}}}}}\) is the initial occupation of the Bloch state, m is the mass of an electron, and the gaugeinvariant mechanical momentum is defined as \(\hat{{{{{{\boldsymbol{\pi }}}}}}}=\frac{m}{i\hslash }[\hat{{{{{{\bf{r}}}}}}},\hat{H}]=\hat{{{{{{\bf{p}}}}}}}+\frac{e}{c}{{{{{{\bf{A}}}}}}}_{{{{{{\rm{ext}}}}}}}\left(t\right)+i\frac{m}{\hslash }\left[{V}_{\!\!{{{{{\rm{NL}}}}}}},\hat{{{{{{\bf{r}}}}}}}\right]\). We considered two different polarized lights in our calculation: \({{{{{{\bf{E}}}}}}}_{\pm }\left(t\right)={\pm {{{{{\bf{E}}}}}}}_{0}{{\sin }}\left(\omega t\right)\). \({{{{{{\bf{J}}}}}}}_{\pm }\left(t\right)\) refers to the calculated current under external field \({{{{{{\bf{E}}}}}}}_{{{{{{\boldsymbol{\pm }}}}}}}\left(t\right)\). To extract the secondorder response of the estimated current, we used the evenorder currents \({{{{{{\bf{J}}}}}}}_{{{{{{\rm{even}}}}}}}\left(t\right)=\frac{({{{{{{\bf{J}}}}}}}_{+}(t)+{{{{{{\bf{J}}}}}}}_{}(t))}{2}\). To mitigate the noisy rapid oscillations, the evenorder realtime currents were averaged over time: \({{{{{{\bf{J}}}}}}}_{{{{{{\rm{avg}}}}}}}\left(t\right)=\frac{1}{T}{\int }_{0}^{T}{{{{{{\bf{J}}}}}}}_{{{{{{\rm{even}}}}}}}\left(\tau \right){{{{{\rm{d}}}}}}\tau\). The atomic motion under a timedependent potential (i.e., Ehrenfest dynamics) was determined by the timedependent Hellmann–Feynman force, which was calculated as follows^{69}:
where \({M}_{\lambda }\) and \({{{{{{\bf{R}}}}}}}_{\lambda }(t)\) are the atomic mass of the λth atom and the position of the λth atom, respectively.
Data availability
The data presented in the main text are provided in the Source Data file. Additional data necessary for extension of the present work can be available from the authors on request. Source data are provided with this paper.
Code availability
The tightbinding code for computing the shift current spectra is partially available from the github: https://github.com/BSKim12/Shift_current.git.
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Acknowledgements
B.K. and N.P. were supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF2019R1A2C2089332). J.K. was supported by an NRF grant funded by the Korea government (MSIT) (No. 2020R1F1A1048143). This work was supported by the National Supercomputing Center with supercomputing resources including technical support (KSC2020CRE0101).
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J.K. conceived the idea of this study, B.K. performed firstprinciples calculations. B.K., N.P., and J.K. analyzed the data and wrote the manuscript together.
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Kim, B., Park, N. & Kim, J. Giant bulk photovoltaic effect driven by the walltowall charge shift in WS_{2} nanotubes. Nat Commun 13, 3237 (2022). https://doi.org/10.1038/s41467022310188
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DOI: https://doi.org/10.1038/s41467022310188
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