High-Resolution Measurement of Molecular Internal Polarization Structure by Photoinduced Force Microscopy

: Near-ﬁeld interactions between metallic surfaces and single molecules play an essential role in the application of metamaterials. To reveal the near-ﬁeld around a photo-irradiated single molecule on the metallic surface, high-resolution photo-assisted scanning microscopy is required. In this study, we theoretically propose photoinduced force microscopy (PiFM) measurements of single molecules at the atomic resolution. For experimental demonstration, we performed a numerical calculation of PiFM images of various transition states, including optical forbidden transitions, and interpreted them in terms of the interaction between the molecular internal polarization structures and localized plasmon. We also clariﬁed the critical role of atomic-scale structures on the tip surface for high-resolution PiFM measurements.


Introduction
A fascinating feature of metamaterials is their ability to manipulate and enhance chiral fields, leading to peculiar interactions with chiral matter systems. Several metamaterial structures have been proposed and have demonstrated large circular dichroism (CD) [1][2][3][4]. Interestingly, the superchiral fields originated by such metamaterials boost the molecular CD signals and enable highly sensitive enantioselective detection of chiral molecules [5][6][7][8].
These results indicate that the elucidation of the interaction between metamaterials and molecular systems is an important factor for the future development of metamaterials for relevant applications. Although in [5][6][7][8], the authors have shown that near-field interactions of metallic surfaces and molecules should play an essential role in the above-mentioned metamaterial effects, it is still challenging to reveal the spatial structures of the near-field around a single molecule because of the limited resolution of existing photo-assisted microscopic techniques.
Photoinduced force microscopy (PiFM), which detects photoinduced force (gradient force) using atomic force microscopy, is a promising approach to address this problem [9,10]. To observe the optical response of single molecules, various types of scanning microscopic techniques have been developed, such as scanning tunneling microscopy luminescence (STML) [11,12], tip-enhanced Raman spectroscopy (TERS) [13,14], and the atomic force microscopy (AFM) with CO-terminated tip [15]. PiFM can be used as a complement to these microscopic techniques because unlike these methods, PiFM does not detect propagating light but directly obtains information of the localized near-field associated with molecules as the force. In particular, PiFM has the advantages of proving vertical polarization of the near-field, detection of buried objects, and no propagation loss [16][17][18][19]. Recently, heterodyne frequency modulation, an innovative measurement technique that enables high-precision and high-resolution PiFM measurements to eliminate photothermal oscillations, has been developed [20,21]. Further development is expected for atomic resolution PiFM measurements of single molecules. By utilizing such properties of PiFM, we can elucidate the near-field interaction between metamaterials and single molecules. For this purpose, it is important to theoretically clarify how the intrinsic optical response of each molecule can be "seen" by PiFM.
Thus, in this study, we theoretically propose PiFM measurements for a single molecule with a size smaller than 1 nm. In the PiFM system, because the molecules strongly interact with the localized plasmon enhanced at the gap between the tip and substrate, the interaction between the total field created by the entire system, including the molecule, and the polarization of all materials, must be treated self-consistently. By using the discrete dipole approximation (DDA) method, we numerically analyzed the total response field in the PiFM measurements and evaluated the radiation force acting on the tip. We show PiFM images for molecular excitation, including the optical forbidden transition beyond the long-wavelength approximation, and provide an interpretation in terms of the microscopic interaction between the molecule and the localized plasmon. In particular, under the electronic resonance, experimentally obtained images of PiFM become complicated, and hence, the theoretical simulations will be useful for interpreting the images. We also investigated how atomic-scale structures on the tip surface, picocavity [22], play an important role in the atomic resolution measurement of PiFM, which detects the field gradient of localized plasmons.

Discrete Dipole Approximation
We calculated the self-consistent total response electric field induced in a nanogap made of a tip, single molecule, and substrate by using the DDA method: where E(r i , ω) and E 0 (r i , ω) represent the total response field and incident field, respectively; i is the index of cells at the coordinate r i ; and ω is the angular frequency of the electric field. G(r i , r j , ω) is the free-space green function propagating in both the transverse and longitudinal electromagnetic fields. P(r j , ω) is the polarization of the j-th cell, and the integral of the second term in Equation (1) represents the field at the i-th cell propagated from the polarization at the j-th cell. V is the volume of the unit cell. Polarization is described by local susceptibility χ(r i , ω). In the DDA method, because the entire space must be divided into cells of the size necessary to describe microscopic structures, it is not realistic to perform calculations that consider the atomic-scale structures of the molecules and macroscopic PiFM system. To solve the problem of this computational size, we employed an extended DDA method that incorporates partial subdivision [23]. The photoinduced force acting on the tip was obtained as follows [24]: where the summation is performed on the cells that configure the tip.

Model Parameters
We assumed that the tip is made of gold and the substrate is a gold film. The metallic structures are assumed to have a Drude-type susceptibility with the parameters of Au where Au is the background dielectric constant of the metal; 0 is the dielectric constant of vacuum; ω Au is the bulk plasma frequency; γ bulk is the electron relaxation constant of bulk gold; V F is the electron velocity at the Fermi level; and L eff is the effective mean free path of electrons, which is comparable to the size of the tip diameter. The following parameters were used for the gold structures: Au = 12.0, 0 = 1.0,hω Au = 8.958 eV, γ bulk = 72.3 meV,hV F = 0.922 nm · eV, and L eff = 60 nm [25,26].

PiFM Measurement for a Single-Molecule
We demonstrate PiFM measurements of a single molecule using a simplified model to understand the microscopic interaction between the molecule and the localized plasmon. We consider a PiFM system and six-membered ring molecule by DDA with a discretized unit cell size of 1 Å, as shown in Figure 1a,b. In the present study, we put stress on discussing the relationship between the geometrical structure of a molecule and the PiFM image having spatial structures inside the single molecule. That is why we use the simplified model of a molecule. The results obtained by using the orbital functions of a more realistic model will be published elsewhere. The tip is assumed to be a 60 nm diameter gold sphere with several protrusions, forming picocavity, on the apex, which is known in the field of tip-enhanced Raman spectroscopy [14,22]. The substrate is assumed to be a gold thin film with a thickness of 9 nm. The susceptibility of the metallic substrate is given by the Drude model with the parameter of Au, and each of the six elements of the molecule are determined by the Lorentz model as follows: where np , µ, V np ,hω np , and γ np are the background dielectric constant, dipole moment, volume, resonant energy, and damping constant of the elements of the six-membered ring molecule, respectively. For simplicity, we considered that the polarization is restricted in the molecular plane and the susceptibility is local. We assume elements with resonant energies ofhω np = 2.0, and the dipole moment of 1 Debye, which is similar to the value for porphyrin-based dye molecules [18,27,28], with parameters np = 1.5, γ np = 5 meV, and V np = 1 Å 3 , which is the same as the unit cell V. We assume a gap of 3 Å between the molecule and substrate, considering the experimental situation. A few atomic layers of a dielectric, such as NaCl, are laid down as spacers to avoid hybridization between the molecular orbitals of the sample and the metal substrate in the measurement of molecules, such as STM [11,12]. The six dipoles of the molecule can take on various spatial structures, each of which has a specific resonance state. Figure 1c,d show the absorption cross-section spectrum in a vacuum and the polarization structures of the molecule for each resonant energy, respectively. These peaks correspond to optical allowed transitions under the long-wavelength approximation.

Microscopic Interaction between Molecule and Plasmon
We discuss the PiFM (attractive force) spectra for the molecule, which are shown in Figure 2 as solid colored lines when the PiFM tip placed at each position. The dashed line represents the spectrum of the gold substrate, that is, in the absence of the molecule. The PiFM spectra show that PiFM has a high sensitivity for the wavelength of the incident laser, and different resonance states can be observed depending on the position of the tip, resulting in a high spatial resolution. This is due to the significant localization and enhancement of localized surface plasmons by the atomic-level spatial structure called picocavity at the tip apex [22]. In addition, a Fano-like structure with an asymmetric peakdip structure appears near the molecular resonance energy in the PiFM spectrum. This can be caused by the interference effect between the broad plasmon resonance peak and the sharp molecular resonance peak, which has been explained in our previous work [18]. When light with energy lower than the resonance energy of the molecule is incident, the polarization of the plasmon and the polarization of the molecule are in phase. As these enhance each other, the photoinduced force is clearly observed in the direction of attraction. In contrast, when light with an energy higher than the resonance energy of the molecule is incident, the polarization of the molecule and the plasmon are in anti-phase, and they suppress each other. In this case, the polarization of the molecule was very small and only slightly weakened the plasmon. Therefore, the peak in the attraction direction was large, and the dip in the repulsion direction was small.
We calculated the PiFM images and polarization structures at each peak of the PiFM spectrum shown in Figure 2. As shown in Figure 3a-e, completely different PiFM images were obtained for each resonance energy of the molecule. The calculated spatial polarization structures for the respective PiFM images are shown in Figure 3f-j. Figure 3f,g are optically allowed transitions that can be excited under the long-wavelength approximation, whereas Figure 3h-j are forbidden transitions, where the average of the polarization inside the molecule vanishes. The difference between these PiFM images can be understood by the interaction with the electric field vector of the plasmon spreading isotropically from the tip apex. The features of these images agree with the discussion of a previous study on molecular imaging by TERS [13]. In the allowed transition, shown in Figure 3a,b, when the tip is at the center of the molecule, two robe-like PiFM images and the nodes are observed. Because the electric field vectors of the localized plasmon spread radially around the tip apex, when the polarizations of plasmons and molecules are in the same line, the molecule is excited and the force signal is obtained, as shown in Figure 4a. In contrast, when the polarizations of plasmons and molecules are orthogonal, the molecule cannot be excited, and the node appears in the PiFM image, as shown in Figure 4b.
For optically forbidden transitions, the molecule can be excited when the tip is at a position where the dipoles are opposed to each other, as shown in Figure 4c. In Figure 3c, when the tip is at the center of the molecule, the dipoles of the two diagonal elements in the y-axis direction are in phase with the plasmon but the other four elements are in the opposite phase; therefore, they cancel each other out and the molecule cannot be excited. In contrast, in Figure 3d, when the tip is at the center of the molecule, all dipoles of the molecule are in phase with the plasmon, so the PiFM image appears at the center of the molecule. In Figure 3e, the structure of the dipoles is similar to Figure 3d, but when the tip is at the center of the molecule, the four dipoles on the sides from the y-axis are orthogonal with the plasmon; thus, the PiFM image is a little weaker at the center and shows a torus-like structure.
In this section, although we analyzed the PiFM measurements of a simple model that mimics a six-membered ring molecule, we obtained a qualitative understanding from the perspective of the interaction between the polarization structure inside the molecule and the localized plasmon. The results we obtained facilitate the understanding of PiFM measurements in a realistic analysis and experiments.

Critical Role of Picocavity in Resolution
In this section, we discuss the feasibility of atomic resolution measurements using PiFM. We revealed that the atomic structure on the tip surface, which has recently been called picocavity [22] in the field of TERS, plays an important role in PiFM resolution. We consider a tip model that imitates the gathering of atoms, as shown in Figure 5a,b. The alignment of the protrusions is not important for resolution. Rather, it is easier to consider only one protrusion in terms of resolution. Considering the atomic structure on the tip surface, it is necessary to consider whether ghost images caused by other protrusions can be avoided.  Figure 5c,d show the response field intensity map and line profile at the plane of 3 Å below from the tip apex, respectively, irradiated with the incident laser. The intensity was determined as a ratio to the incident field, |E| 2 /|E 0 | 2 . The enhanced field is extremely localized in the vicinity of the picocavity. Naturally, the sharpness of the tip contributes to high-resolution imaging. However, in scanning optical microscopic techniques, such as TERS and PiFM, the enhancement and localization of the field are also determinant factors for high-resolution imaging. The force curve of the PiFM is steep [21], and the contribution from the other protrusions is negligible.
In our previous study, we experimentally and theoretically demonstrated that the nanometer-scale structure on the tip surface enabled us to realize a resolution of PiFM measurement of less than 1 nm at room temperature [19]. In this study, we considered a tip with picocavity; however, our study is based on the assumption that the diffusion coefficient of gold atoms is small at cryogenic temperatures [22,29].

Conclusions
In this study, we numerically demonstrated a PiFM measurement for single molecules on an atomic scale. By analyzing the simplified six-membered ring molecular model, we clarified the excitation conditions. We obtained PiFM images at an atomic-scale resolution, corresponding to the molecular transition states including optically allowed and forbidden transitions beyond the long-wavelength approximation. We interpreted PiFM images of single molecules in terms of the microscopic interaction between the molecular internal polarization structures and localized plasmons. In contrast, the local structure of the excited state of a single molecule can be determined from the PiFM image based on the above-type of analysis. We have also discussed the feasibility of atomic resolution measurement by PiFM. The calculated results indicate that the enhancement and localization of the electric field near the atomic-scale structure, picocavity, allows for PiFM measurements at the atomic resolution.
It is important to clarify the optical response of the local polarization structure of single molecules at the atomic resolution for the elucidation of molecular chemical reactions, development of functional metamaterials, and kinetic manipulation of molecules. PiFM is advantageous because multiple information of the localized near-field can be obtained without propagation loss, unlike other photo-assisted scanning microscopic techniques. Furthermore, it has been reported that PiFM can detect the longitudinal polarization of local chiral fields induced by chiral nanomaterials [30], which cannot be detected by SNOM [31]. PiFM measurements at the atomic resolution can allow the detection of single-molecular chirality through the near field. This will be a new capability that will outperform other types of photo-assisted scanning microscopies.  Acknowledgments: The authors thank Y. Sugawara and J. Yamanishi for their fruitful discussions.

Conflicts of Interest:
The authors declare no conflicts of interest.