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Abstract
Near-field images of molecules provide information about their excited orbitals, giving rise to photonic and chemical functions. Such information is crucial to the elucidation of the full potential of molecules as components in functional materials and devices at the nanoscale. However, direct imaging inside single molecules with a complex structure in the near-field is still challenging because it requires in situ observation at a higher resolution than the molecular scale. Here, using a proven theoretical method that has demonstrated sub-nanoscale resolution based on photoinduced force microscopy (PiFM) experiment [Nat. Commun. 12, 3865 (2021) [CrossRef] ], we propose an approach to obtaining the near-field imaging with spatial patterns of electronic transitions of single molecules. We use an extended discrete dipole approximation method that incorporates microscopic nonlocal optical response of molecules and demonstrate that PiFM can visualize circular-dichroism signal patterns at sub-nanometer scale for both optically allowed and forbidden transitions. The result will open the possibility for the direct observation of complex spatial patterns of electronic transitions in a single molecule, providing insight into the optical function of single molecules and helping realize new functional materials and devices.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Single molecules exhibit a variety of unique physical properties that are usually obscured by the statistical behavior of molecular ensembles. These include electrical conduction owing to orbital hybridization with metals [1,2], molecular magnetism control [3], and the local chemical properties of internal and surface structures [4,5]. These single-molecule properties were discovered in part because of the development of new single-molecule measurement techniques. For example, surface-enhanced Raman scattering (SERS) detects stronger and more stable vibrational Raman signals than single-molecule fluorescence [6,7]. Recent cutting-edge technologies such as tip-enhanced Raman spectroscopy (TERS) acquire target images at single-molecule scale and identify target species based on their Raman spectral properties. Many subsequent studies have reported chemically significant single-molecule optical responses with sub-nanometer spatial resolution [8–10]. Scanning tunneling microscopy (STM) luminescence spectroscopy has successfully revealed the energy-transfer properties and spectra of single molecules at micro-electron-volt energy resolution [11,12].
The chiral properties, and especially the chiral selective optical responses, of molecules are important in many optical applications [13,14], but measuring the chiroptical response of a single molecule is highly challenging. For example, although the measurement of circular dichroism (CD) in a single molecule has been reported [15], a slight change in the polarization state of the incoming laser beam in a conventional fluorescence microscope might cause artifacts through linear dichroism in single immobilized molecules [16]. However, recently, scanning near-field optical microscopy (SNOM) has been successfully utilized to capture CD images of metal nanostructures [17–20]. These images show that the optical selection rule for far-field optical activity is invalid for nanoscale near-field measurements; thus, near-field optical activities can elucidate the poorly understood potential of chirality-related functions.
Single molecules should exhibit a greater variety of optical activities compared to metal nanostructures because they perform various electronic transitions, including optically forbidden ones. The polarization induced by optically forbidden transitions has several atomic-scale nodes and couples to spatially modulated fields at the sub-nanometer scale. Thus, near-field CD images showing such transitions are important for exploring new chirality-induced optical functions. However, to the best of our knowledge, there are no published reports on the direct observation of fine structures of the electronic transitions inside single molecules.
In this report, we propose a method for imaging single-molecule near-field CDs using photoinduced force microscopy (PiFM), an application of atomic force microscopy (AFM) that can detect the photoinduced force produced when a localized surface plasmon is generated by laser-illumination of a probe tip and sample [21,22]. PiFM can be used to observe the near field directly as a force on the probe, with no propagation loss of the signal [23].
A crucial advantage of this scheme is its robustness (compared with conventional CD measurement of single molecules using a fluorescence microscope) against changes in the polarization state of the incident field: the force is detected not directly through the incident field, but through the superchiral field excited by the localized surface-plasmon resonance in the vicinity of the probe tip [24,25]. This robustness is demonstrated by the simulations in section 4 in Supplemental Document.
2. Theoretical method
One of the present authors previously developed a heterodyne frequency-modulation technique that significantly reduced the influence of photothermal oscillations [26,27]. The technique enabled high-precision and high-resolution measurements, achieving a resolution of 0.7 nm at room temperature [28]. At the same time, it revealed that a plasmonic picocavity (an atomic-scale protrusion on the probe-tip facet) can play an essential role in sub-nanometer PiFM measurements by enabling confinement of light to single-molecule volumes through localized surface-plasmon resonance [29]. Our numerical calculations reproduced these results, which encouraged us to investigate applying PiFM measurement to single molecules.
Given that atomic-scale structures are more stable under cryogenic conditions, an improvement of the resolution beyond the single-molecule scale is possible. To guide further experimental work on this, it is important to clarify theoretically how information of electronic transitions in single molecules with the chirality in the PiFM image.
In connection with the sub-molecular scale of TERS imaging, quantum-chemical calculations of the influence of the near field around the tip on the molecular orbitals have been performed in previous works [30,31]. However, to obtain PiFM images under electronic resonance, the self-consistent total-field calculations are essential, because the local field in itself is strongly affected by the molecular induced polarizations. Thus, it is necessary to consider the microscopic electric-field variation within the coherent volume of the molecular wavefunctions, together with the cavity effect that arises from the metal tip and substrate on a size-scale several orders of magnitude larger. For this treatment, we used an extended discrete dipole approximation (DDA) that we have developed [32], in which different cell sizes were utilized according to the fineness of the spatial structures of the electric field. In developing the constitutive equation to describe the nonlocal relation between the induced polarization and the electric field, the molecular susceptibility was calculated using wavefunctions obtained with the quantum-chemistry calculation software GAMESS (US) [33]. The results found by the self-consistent method automatically included the energy shift and spectral width of the molecular resonance arising from the interaction between the molecule and the localized surface-plasmon polaritons. Note that we avoided quantum mixing between the molecular orbitals and metal orbitals by carefully controlling the molecule-metal distance as discussed in Ref. [34].
The calculation procedure is as follows: First, we calculate the self-consistent total response electric field by solving the following equation.
In the discretized space, we assume the interior of the discretized cell to be uniform and obtain the polarization
The photoinduced force acting on the tip is obtained using the following formula: [36]
The details of the calculation method are provided in sections 1–3 of Supplement 1.
3. Results and discussions
3.1 PiFM image of single molecules
We numerically simulated the PiFM imaging of a single phthalocyanine molecule and an azulenocyanine molecule [37]. Both the molecular structures are achiral in situations where they can freely change their orientation in three dimensions, such as in vacuum or in solution. However, when the orientation of the molecules is fixed by the substrate, one of them, azulenocyanine, is considered chiral in the two-dimensional plane parallel to the substrate [20]; see Fig. 3(c).
First, we examined PiFM images of a single phthalocyanine molecule. Figure 1(a) shows a schematic of the PiFM measurements. We assumed that the tip was a metal hemisphere with a diameter of 30 nm and a protrusion of approximately 1 nm on the tip’s apex, as shown in Fig. 1(b). The picocavity has attracted much attention for atomic resolution measurements in the field of STM luminescence and TERS [10,29,38]. It can be stabilized at cryogenic temperatures when the diffusion coefficient of the metal is small. Naturally, sharper tips result in PiFM images with higher resolutions. However, it is not clear whether the tip apex can maintain a stable atomic structure during PiFM measurement. To prove the reality of single-molecule measurements via PiFM, we show that even a protrusion much larger than the atomic scale (approaching the nanometer scale) is sufficient for PiFM measurements of molecules with sizes of a few nanometers, such as phthalocyanine. The laser’s output was a p-polarized plane wave with an intensity of 10 kW/cm$^2$ and an incident angle of 70$^\circ$. This was determined with reference to the experimental setup of the previous study [28]. The magnitude of photoinduced force is proportional to the incident intensity as long as the optical response is within the linear response regime. The spectra of the photoinduced force between the gold probe and gold substrate and between the aluminum probe and aluminum substrate are shown in Fig. 1(c): it is appropriate to use gold in the visible-light region and aluminum in the ultraviolet region [39,40].
Fig. 1. Calculation model used for the PiFM. (a) Schematic model of the PiFM tip, substrate, and molecule. The metal-coated tip is a 30-nm-diameter metal hemisphere, and the substrate is a metal thin film. The p-polarized laser beam has an incident angle of 70$^\circ$. (b) Enlarged cross-sectional image near the tip. A protrusion of approximately 1 nm is assumed to exist at the tip apex. The grid lines in 1 Å increments represent the discretized space in the discrete dipole approximation. (c) Spectra of the photoinduced force between the Au tip and Au substrate (red line) and between the Al tip and Al substrate (blue line) in the absence of the molecule. (d) The energy scheme and orbitals of phthalocyanine obtained based on quantum-chemistry calculations using B3LYP/6-31G*.
As shown in Fig. 1, we assumed that a phthalocyanine molecule was on the substrate. Because several layers of NaCl were used as spacers in the STM luminescence experiments involving single molecules [8,41], we chose 4 Å as distance between the molecule and substrate (that is the width of the NaCl layer)which is large enough to prevent hybridization of the electronic orbitals of the molecule with those of the substrate. The tip scanned 4 Å above the molecule in parallel with the substrate. For metal nanogaps, it is known that if the distance between the gaps is greater than 3.5 Å, quantum effects owing to the overlap of wave functions do not appear, and classical electromagnetic analysis can be utilized [34]. It is known that the closer the tip is to the sample, the more sensitive it is, and the signal is proportional to $z^{-4}$ with tip-sample distance $z$ [42].
PiFM images of a single phthalocyanine are shown in Figs. 2(a), (d), (g), and (j). We considered four transition modes from HOMO to LUMO, LUMO+1, LUMO+2, and LUMO+3, referred to as modes 0, 1, 2, and 3, respectively. The spatial distribution of molecular polarization was obtained, as shown in Figs. 2(b), (e), (h), and (k). Resonance energies of 2.2041, 2.2667, 3.9919, and 4.1579 eV were obtained for the various modes via B3LYP/6-31G*, as shown in Fig. 1(d). Given that the resonance energies were red-shifted owing to interactions with plasmons [12,43], the incident laser energies were tuned to 2.1917, 2.2549, 3.9880, and 4.1555 eV, respectively.
Fig. 2. PiFM images of phthalocyanine: (a) calculated PiFM image; (b) molecular internal-dipole structure; (c) schematically illustrated polarization structure of mode 0. Similarly, (d)–(f), (g)–(i), and (j)–(l) correspond to modes 1, 2, and 3, respectively. PiFM images were obtained using incident-laser energies of 2.1917, 2.2549, 3.9880, and 4.1555 eV for the resonance energies of modes 0–3, respectively.
Modes 0 and 1 are optically allowed transitions in which the sum of the dipole moments in the molecules is directional. The spatial structures of the total polarization are depicted in Figs. 2(c) and (f), respectively. We assumed a gold probe and gold substrate for these modes with transition energies in the visible region. Modes 2 and 3 are optically forbidden transitions in which the dipole moments in the molecule cancel out. However, they have local polarization structures inside the molecule and can be excited by the steep electric field gradient associated with the localized plasmon. To make the molecular polarization structure more obvious, Figs. 2(i) and (l) [44] include arrows that represent the polarization in the four regions of the molecule. For optically forbidden transitions, we considered an aluminum tip and an aluminum substrate with a plasmon resonance in the ultraviolet region to covers the resonance energy of the molecule [39,40].
For the optically allowed states, two lobe-like images appeared in the PiFM image, as shown in Figs. 2(a) and (d). The plasmon field vector induced around the probe tip spread isotropically from the tip’s apex, and these images can be understood in terms of the microscopic interaction between the molecule and the plasmon field. The photoinduced force signal was strong when the tip was at a position such that the direction of the plasmon field vector spreading isotropically from the tip coincided with the polarization direction of the molecule. When the tip was at the center of the molecule, the molecule was not excited because the vector of the plasmon field was totally vertical and did not match the polarization of the molecule.
These predicted PiFM images were in good agreement with the photon images acquired via STM luminescence [8,10]. Note that in the STM emission studies, zinc phthalocyanine with a quadruple symmetry was used; the LUMO and LUMO+1 levels were degenerate, and there were two equivalent polarization states in the plane. For zinc phthalocyanine, four lobes can be observed as a superposition of Figs. 2(a) and (d).
Figures 2(g) and (j) show PiFM images of modes 2 and 3, which are the optically forbidden transitions. These PiFM images have more nodes than the optically allowed cases. The plasmon field localized at the tip’s apex with a steep electric field gradient allowed the observation of optically forbidden transitions using PiFM, supplying microscopic information on the spatial structures of induced polarizations beyond the long-wavelength approximation.
3.2 PiFM circular-dichroism measurement of single molecules
Next, we illustrate theoretically the chirality measurement of a single molecule using PiFM. The chirality-measurement model in the PiFM is shown in Fig. 3(a). To avoid asymmetry due to the direction of the incident laser, we consider a PiFM system in which a left or right circularly polarized laser is incident from the back of a substrate. We considered a glass substrate for the chirality measurement, assuming a refractive index $n_{\rm glass} = 1.45$. Phthalocyanine and azulenocyanine belong in a family of phthalocyanines [37], as shown in Fig. 3(b) and (c). Although both molecules are chemically achiral, the structure of azulenocyanine, including the substrate and incident laser, is chiral in two dimensions [20]. Here, we define the asymmetric $g$-factor of the photoinduced force as [45]
where $F_{\rm L}$ ($F_{\rm R}$) represents the optically induced force detected by the tip when it is irradiated with left (right) circularly polarized light. The range of $g_{\rm F}$ is from $-2$ to $2$.Fig. 3. Schematic model of chiral measurement using photoinduced force microscopy (PiFM). (a) Schematic of a PiFM chiral measurement. A circularly-polarized laser is used to irradiate the back of the glass substrate. (b) and (c) Phthalocyanine and azulenocyanine molecules; they have achiral and two-dimensional chiral structures, respectively. (d) Energy schemes and orbitals of azulenocyanine obtained from quantum-chemistry calculations using B3LYP/6-31G*. (e) and (g) Molecular internal dipole structures of azulenocyanine in modes 2 and 3, respectively. (f) and (h) Polarization structures of (e) and (g), respectively.
Note that the randomness or anisotropy of the probe tip might create artifacts in a single CD map that is deformed from that obtained by a tip with perfect cylindrical symmetry. However, such deformation can be eliminated by averaging over multiple images. This issue is thoroughly discussed in section 5 of Supplement Document. Thus, here, we consider a probe tip without randomness or anisotropy to concentrate on the essential aspect of the problem.
PiFM images of phthalocyanine, an achiral molecule, are shown in Figs. 4(a)–(c) for mode 0, an optically allowed transition, and (d)–(f) for mode 2, an optically forbidden transition. The incident laser energies were 2.1992 and 3.9900 eV, respectively. Figure 4(a) and (d) are PiFM images for left circular polarization, (b) and (e) for right circular polarization. Figure 4(c) and (f) show the localized PiFM chiral images, which are the maps of the $g$-factor of the photoinduced force. The red (blue) color indicates that the photoinduced force was detected more strongly for the left (right) circularly-polarized laser. Local chirality can be detected even for an achiral molecule at the atomic scale. The PiFM maps for the left and right circular polarizations are mirror images of each other, which indicates that the average of $g_{\rm F}$ over the entire molecule is 0, and the local chirality is canceled out.
Fig. 4. photoinduced force microscopy (PiFM) chiral images for (a)–(f) phthalocyanine and (g)–(l) azulenocyanine. For both molecules, PiFM images obtained using left circularly polarized (LCP) and right circularly polarized (RCP) laser sources and a chiral map of the asymmetric factor of the photoinduced force are shown for optically allowed and forbidden transitions.
The CD signal appears by the plasmonic field that contains mainly the vertical component of the field for the same reason that lobe structures are observed in PiFM images: the horizontal component of the inclined field excites the molecule when the tip position is shifted from the center of the molecule. Circular polarizations in such horizontal components in the strongly enhanced field play an essential role in generating the CD signal.
The PiFM images and the localized chiral image for azulenocyanine, a two-dimensional chiral molecule, are shown in Figs. 4(g)–(l). We selected transition modes 2 (optically forbidden) and 3 (optically allowed), as shown in Figs. 4(g)–(i) and (j)–(l), respectively. Transition modes 0 and 1 of azulenocyanine are in the near-infrared region [37], and modes 2 and 3 are in the visible-light region. In this study, we assumed PiFM experiments using only a source in the visible region [28] to avoid complex issues related to thermal expansion [46,47]. The resonance energies for transition modes 2 and 3 were determined to be 2.1579 and 2.7837 eV, respectively, based on quantum-chemistry calculations. For these calculations, we used gold and aluminum tips with plasmon resonance bands that matched the respective resonance energies. The incident laser energies were set to the shifted energies, 2.1562 and 2.7834 eV. The PiFM chiral images are shown in Figs. 4(i) and (l). As shown in Figs. 3(f) and (h), the overall structures of the internal molecular dipole of modes 2 and 3 of azulenocyanine are similar to those of modes 2 and 0 of phthalocyanine (Figs. 2(i) and (c)), respectively. (Note that the direction of the molecular frame for azulenocyanine is tilted by 45° from that of phthalocyanine.) However, the PiFM images of azulenocyanine shown in Figs. 4(g)–(k) differ strikingly from those of phthalocyanine shown in Figs. 4(a)–(e). For phthalocyanine, the images associated with the right and left circular polarization of the incident light are mirror-symmetric. (Compare Figs. 4(a) with (b) and Figs. 4(d) with (e).) Thus, the average $g$-value of the entire molecule vanishes, although the local $g$-value is finite. However, for azulenocyanine, there is no such mirror symmetry between the images for different directions of incident circular polarization. Instead, we identified stronger (brighter) regions in the maps for the parallel combination of chirality of the molecules and incident light. (Fig. 4(h) has brighter regions than Fig. 4(g), and Fig. 4(k) has brighter regions than h Fig. 4(j).) The images for azulenocyanine are twisted due to the chirality of the polarization structure inside the molecule, which leads to a stronger optical response selectively for a parallel combination of chiralities. Because of this situation, the red and blue in the chiral images in Figs. 4(i) and (l) are no longer symmetrical; local chirality is not canceled out in the entire molecule. The $g$-values are remarkably large compared to those observed in the far-field. This holds even for the optically forbidden transition. These results provide information on the macroscopic optical functions due to the molecular chirality originating in the electronic transition properties of individual single molecules. Furthermore, the near-field optical functions, including those from optically forbidden transitions of the chiral molecules, are revealed.
These PiFM chiral images reflect the local CD of the molecules. For the allowed transition, we plotted the CD field maps on the planes of 4, 6, and 8 Å above the molecule in Fig. 5. We also calculated the response electric field without the tip. The CD field was obtained using $\Delta I\equiv I_{\rm L}-I_{\rm R}$, where $I_{\rm L}$ ($I_{\rm R}$) is the intensity of the response field normalized by an incident field with a left (right) circularly polarized source. The chiral field maps for mode 0 of phthalocyanine and mode 3 of azulenocyanine are shown in Fig. 5(a)–(c) and (d)–(f), respectively. The PiFM chiral images, Fig. 4(c) and (l), well reflect the spatial distribution of the local CD. As shown by the distance dependence in Fig. 5, the local CD structure disappears farther away from the molecule.
Fig. 5. The chiral field map for mode 0 of phthalocyanine (a)-(c) and mode 3 of azulenocyanine (d)-(f). The maps plot the intensity of the response electric field on planes 4, 6, and 8 Å above the molecule, respectively.
Thus, we have shown that PiFM chiral measurements can be used to image the local CD structure near the molecule. It should be noted that optically forbidden transitions cannot be excited under the conditions described by the long-wavelength approximation, and such CD maps cannot be obtained. However, using the PiFM system, local chirality can be obtained even for optically forbidden transitions, as shown in Figs. 4(f) and (i).
Certain issues related to the magnitude of the calculated photoinduced force merit discussion. As we have discussed in detail elsewhere [28], our model does not predict the absolute values of the detected force in the PiFM experiment because the force strongly depends on the size and shape of the probe tip. In the proposed model, the tip size was determined by considering the computational load; if a more realistic size of the entire probe tip is considered, the force becomes much stronger than the calculated one. In Ref. [28], the experimentally detected force was several orders of magnitude stronger than the calculated value. No negative effect on resolution of the measurement exists, even if the large tip is considered, because the resolution of PiFM is determined by the atomic-level spatial structure at the tip apex [28]. For chirality detection in the present demonstration, the calculated force may seem weak. However, this issue can be overcome by increasing the incident laser intensity. The observed PiFM signal is proportional to the intensity as long as the optical response is within the linear response regime. Measurement techniques such as the heterodyne frequency modulation method [26], which separates photothermal oscillations from photoinduced force signals, have been proposed. It is also expected that measurements at cryogenic temperatures and in ultrahigh vacuum allow precise PiFM using the higher intense laser. Those techniques would assist the chirality detection by our proposed method even for high intensity excitation.
4. Summary and conclusion
Finally, we summarize the main results. We proposed that PiFM can be used to observe the internal structures of the induced polarization of single molecules for both optically allowed and forbidden electronic transitions. Information on optically forbidden transitions is especially important because it is closely related to chemical processes and optical effects involving multi-photon processes. However, the spatial structures of the induced polarization of forbidden transitions were finer than those of allowed transitions, making it difficult to observe this phenomenon via conventional photo-assisted scanning spectroscopy. The present study proposed that such information is accessible using a state-of-the-art PiFM scheme [28]. Further, we demonstrated that PiFM can reveal the chiral spatial structures of single molecules that arise from the intrinsic nature of their wavefunctions. We showed that, even for achiral molecules, chirality appears in the local optical response at the atomic scale and can be detected via PiFM. For a molecule with a chiral structure, the chiral image revealed that left–right circular polarization symmetry is broken, which can be observed using PiFM with high detection sensitivity.
We have shown that for the respective electronic transitions of a molecule, including optically forbidden transitions, PiFM can produce images that reflect the internal wavefunction of the molecule; these chiral images reflect the CD of the localized electric field induced by single molecules. However, our demonstrations were performed using a simplified model of the probe and substrate; a more sophisticated model based on feedback from the experimental results would be preferred. Such a study including experimental results will be published in the near future.
Funding
Japan Society for the Promotion of Science (JP16H06504, JP21H05019, JP21K14554, JP22H05132).
Acknowledgments
This work was supported in part by JSPS KAKENHI Grant Number JP16H06504 for Scientific Research on Innovative Areas "Nano-Material Optical-Manipulation", JSPS KAKENHI (Grant Number: JP21H05019), JSPS KAKENHI (Grant Number: JP22H05132), and by JSPS KAKENHI (Grant Number: JP21K14554).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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