We previously demonstrated that we can image electronic excitations of quantum dots by single-molecule absorption scanning tunneling microscopy (SMA-STM). With this technique, a modulated laser beam periodically saturates an electronic transition of a single nanoparticle, and the resulting tunneling current modulation Δ*I*(*x*_{0}, *y*_{0}) maps out the SMA-STM image. In this paper, we first derive the basic theory to calculate Δ*I*(*x*_{0}, *y*_{0}) in the one-electron approximation. For near-resonant tunneling through an empty orbital “*i*” of the nanostructure, the SMA-STM signal is approximately proportional to the electron density $\phi ix0,y02$ of the excited orbital in the tunneling region. Thus, the SMA-STM signal is approximated by an orbital density map (ODM) of the resonantly excited orbital at energy *E*_{i}. The situation is more complex for correlated electron motion, but either way a slice through the excited electronic state structure in the tunneling region is imaged. We then show experimentally that we can nudge quantum dots on the surface and roll them, thus imaging excited state electronic structure of a single quantum dot at different orientations. We use density functional theory to model ODMs at various orientations, for qualitative comparison with the SMA-STM experiment. The model demonstrates that our experimentally observed signal monitors excited states, localized by defects near the surface of an individual quantum dot. The sub-nanometer super-resolution imaging technique demonstrated here could become useful for mapping out the three-dimensional structure of excited states localized by defects within nanomaterials.

## I. INTRODUCTION

On the size scale of a few atoms to hundreds of atoms, molecules and small clusters can be synthesized or assembled so that virtually all copies are identical.^{1,2} A distinguishing characteristic of molecules and nanostructures in the nanometric size regime, such as large proteins, DNAs, or quantum dots, is that nominally identical molecules or nanostructures are likely to differ from one another because of defects. For example, translation errors in *E. coli* bacteria are estimated to lie in the 10^{−4}–10^{−3} range,^{3} so a typical protein of 300 amino acids has a ca. 10% chance of differing from the “average” sequence. Likewise, while quantum dots can be synthesized with a narrow size distribution,^{4} nominally identical dots still differ in diameter and in the location and types of defects, which affect their spectra, bandgaps, and other properties.^{5} There is furthermore the possibility of many more isomers (e.g., two different misfolded structures of a peptide or two different surface atomic arrangements of stoichiometrically identical quantum dots) than for smaller molecules.

Defects are not necessarily a bane. In bulk semiconductors, for example, doping is critical to function. Likewise in nanostructures, defects will become desirable once they can be controlled with atomic precision because they confer specific electronic properties.^{2} Thus techniques have been developed to determine the structure of individual nanoparticles, allowing the study of defects. For example, transmission electron microscopy (TEM) has been used to determine and reconstruct the 3-D structure of metal nanoparticles in their ground state.^{6,7}

Complementary to the study of ground state structure, excited states of nanostructures have been visualized directly in real space. For example, the localization of gap states on individual PbS quantum dots has been studied by measuring current-voltage (I-V) curves at many locations on a dot (scanning tunneling spectroscopy, STS).^{8} Highly excited S- and P-like states of quantum dots with defects have been imaged with sub-nanometer resolution^{9} by using single-molecule laser-absorption detected by scanning tunneling microscopy (SMA-STM), which also has been applied to carbon nanotubes to study defects and exciton size.^{9–11}

In other studies, the STM tip has been successfully used to manipulate individual atoms,^{12} small molecules,^{13} graphene nanoribbons,^{14} and carbon nanotubes^{15} on a surface. Quantum dots have been manipulated into chains to allow energy transfer down the chain.^{16} Manipulation by rolling could be very useful for looking at defects in nanostructures. For example, quantum dots are candidates for applications in solar cells and photocatalysis because of their broadband absorption and high fluorescence quantum yield.^{17–20} In these applications, the quantum dot excitation energy is either transferred on or used to break chemical bonds. Understanding how the presence of defects localizes excited electronic states of quantum dots will help engineer quantum dots for improving the efficiency of energy transfer or catalytic processes.

Here we combine two capabilities of STM: manipulating quantum dots on a surface by rolling and imaging excited electronic states. Such orientational manipulation of quantum dots combined with electronic excitation allows us to view an excited state electron density from different angles. Figure 1(a) illustrates the simple idea: a nanostructure is excited by an evanescent light wave from below the surface (green arrows), producing an excited state with an electron density distribution localized by a defect (green). The STM tip is used simultaneously to image the quantum dot [topography image Fig. 1(b)] and to detect laser-induced optical excitation [SMA-STM image Fig. 1(c)]. The tip then nudges the quantum dot (blue arrows) to roll on the surface, and the new orientation [Figs. 1(e) and 1(f)] is imaged, showing a different projection of the excited state image. Figures 1(d) and 1(g) show two projections, 45° apart, of a localized surface state on a small model quantum dot computed by DFT for qualitative comparison.

We begin by outlining in more detail a one-electron, second-order perturbation theory model for the SMA-STM signal. Our model extends the standard Bardeen theory for a single-tunneling junction (tip-surface) to a double-tunneling junction (tip-nanostructure-surface).^{21} The model predicts that modulated laser excitation, of an orbital resonant with the tunneling energy, allows the STM tip to trace out a current modulation map Δ*I*(*x*_{0}, *y*_{0}) proportional to the density in the tunneling region of the orbital *φ*_{i} being excited. We call this an orbital density map (ODM). The situation is more complex for excitons or other cases where the one-electron, single-orbital excitation picture breaks down. By comparison, STM topography images *I*(*x*_{0}, *y*_{0}) do not generally map out orbitals even in the one-electron approximation.^{22}

Next, we show by electronic excitation of individual quantum dots on a surface how SMA-STM images a localized defect. We then roll quantum dots on the surface with respect to a surface marker and show how the appearance of the excited state SMA-STM image changes. Finally, we use density functional theory (DFT) calculations of a disordered small model quantum dot to show that different excited state projections seen in the experiment can be modeled computationally. Although still a large step away from single particle tomography because projections at multiple angles without damaging the nanoparticle are required, orientation-dependent SMA-STM could reveal the uniqueness of an individual excited particle, complementing standard tomography methods that capture only averaged ground state structure.^{6,7}

## II. METHODS

Experiments were performed using a home-built STM with a base pressure ≤7 × 10^{−9} Pa^{23} and mechanically cut Pt/Ir (80/20) tips. PbS (1.01 eV bandgap, Evident Thermoelectrics) or CdSe/ZnS (2.33 eV bandgap, Ocean NanoTech) quantum dots were deposited either onto a clean, ultra-thin conductive Au surface (10 nm of Au over 5 nm of Pt on sapphire)^{24} or onto an amorphous SiC surface (∼1.8 eV bandgap). We used matrix-assisted dry contact transfer to deposit quantum dots onto the surfaces.^{9} Quantum dots were excited at a wavelength of 532 nm (∼2.33 eV). Excitation was from the back of the sample, in a total internal reflection geometry that avoids sample heating. The laser was amplitude-modulated at 2.2 kHz, and the modulated absorption signal was detected via modulation of the tunneling current by a lock-in amplifier. The laser power density was ca. 1200-2600 mW/mm^{2}. More details of sample preparation are described in Ref. 16.

To validate our experimental observations, we performed DFT calculations on a smaller PbS model quantum dot. Calculations were performed using the development version of the Gaussian program.^{25} PbS quantum dot structures were computed using the PBE1PBE hybrid DFT functional,^{26,27} with the Los Alamos double-zeta pseudocore potential (LANL2DZ) and associated basis set.^{28–30} Cubic PbS quantum dots were constructed with a principal diagonal of 1.495 nm. The cubic PbS quantum dots were constructed in stoichiometric ratios with two Pb atoms and two S atoms on an edge, which has been shown to result in a semiconducting behavior with no midgap states.^{31} The geometry of the quantum dots was then optimized. To produce a slightly disordered structure that would result in localized excited electronic states, each nucleus was displaced at random. Since the comparison of small model dots with experiments is currently only qualitative, we chose positional disorder over atomic deletion or addition defects. We previously showed that position randomization over a ±0.05 nm range in x, y, and z is the smallest amount sufficient to produce highly localized orbitals of the type we see here in our experiments.^{9}

Orbital density maps (ODMs) to approximately simulate the experimentally observed SMA-STM signal were generated from the transition dipole moments between occupied and unoccupied DFT orbitals and their energy differences, which were scaled to match experimental values by the ratio of the computed (3.02 eV) and experimental (1.01 eV) band gap energies. A single strong excitation for the distorted quantum dot near 532 nm was explored. The transition density was rotated to several angles and then projected into a 2-D plane parallel to the rotation axis, in order to simulate the observed signal response of the experimentally rolled quantum dots. For simplicity, we projected the entire density $|\phi ix0,y0|2=\u222bdz|\phi ix0,y0,z|2$ rather than the density in the tunneling region as derived in Sec. III. Because the spatial resolution of the simulated quantum dots is much higher than the experimental resolution of around 0.5 nm, a Gaussian blur with a full-width-half-maximum of 0.25 nm was applied to the ODM.

## III. RESULTS

### A. Model for the orientation-dependent SMA-STM signal

The true SMA-STM signal, or change $\Delta Ix0,y0$ of the tunneling current upon optical excitation, can be evaluated only from first principles electronic structure calculations that take into account electron correlation and the entire tip-nanostructure-surface junction. Nonetheless, it is instructive to consider a simplified one-electron analytical model for two-step tunneling from a tip through a nanostructure into a surface. The key result is in Eqs. (12) and (14), which approximately relate the observed SMA-STM signal to the spin orbitals *φ*_{j} (occupied before laser excitation) and *φ*_{i} (occupied after laser excitation).

The tip-nanostructure-surface system is best described by a double-barrier tunnel junction. Figure 2(a) illustrates the case of positive sample bias although similar equations apply for negative bias. The tunneling rate is given approximately by the Fermi golden rule if the density of final states is high enough (e.g., for a metal surface)

Here *V* is the bias voltage applied on the STM sample. $M(x\u21c00)$ is the coupling matrix element of the tip-nanostructure-surface junction, which depends on the tip’s position $x\u21c00=(x0,y0,z0)$. *ρ*_{S} is the density of states of the surface, and *ρ*_{T} is the density of states of the tip. The tunneling rate value has the same sign as the sample bias, where (+) indicates tunneling from the tip to the sample. The dotted line at *E* = 0 in Fig. 2(a) corresponds to tunneling from the Fermi level of the tip, and the bottom equation in (1) makes the approximation that the tunneling probability for tip states below the Fermi level at *E* = 0 decreases rapidly, so *ρ*_{T}(*E*) may be approximated by a δ-function.

A nanostructure with defects, such as a quantum dot or a carbon nanotube, has a much smaller density of states than the metal tip or metal surface and is best described in a localized orbital basis rather than a Bloch basis. This is evident from the discrete peaks with relatively narrow widths seen by STS spectroscopy of such nanostructures.^{32} We formulate our model for the nanostructure in terms of a basis of one-electron spin orbitals |*n*⟩ with orbital energies *E*_{n} so that the true overall electronic state is approximated by the best Hartree-Fock wavefunction formed from the orbitals |*n*⟩. Only one-electron excitation and one-electron tunneling are considered.

Second order perturbation theory is required to describe the coupling across the double junction in the one electron picture

The first term in this expression describes the direct tip-surface tunneling. The wave function $\psi Tx\u21c00=\psi T(x\u2212x0,y\u2212y0,z\u2212z0)$ of the tip depends parametrically on the positioning (*x*_{0}, *y*_{0}, *z*_{0}) of the tip over the surface, with (0, 0, 0) corresponding to a position centered on the nanostructure and within the tunneling range. The second term describes the tunneling via orbitals of the nanostructure, as indicated by the blue-colored *z*-axis cross section of wave function $\phi nx\u21c0=x\u21c0n$ in Fig. 2(a). $\Delta En$ is the energy gap between the tip Fermi level (at *E* = 0) and state *n*. The energies $\Delta En=\Delta En(r)\u2212i\u210f\Gamma n$ may be complex to reflect the lifetime of state *E*_{n}. The summation over *n*′ is a sum over spin orbitals $n$ available for tunneling, i.e., those unfilled with electrons. Spin orbitals that are occupied by an electron are inaccessible due to Fermi statistics. The matrix element $\psi Tx\u21c00\u0124tunneln$ vanishes rapidly once $|x0|\u226b0$ and $y0\u226b0$: at positions far from the center of the nanostructure, the state $n$ has vanishing probability and only direct tip-surface tunneling is observed. The filled state $\psi Tx\u21c00$ of the metal tip and the empty state $\psi S$ of the metal surface are degenerate, but the state $n$ (a specific excited state of the nanostructure) is not necessarily degenerate with either.

In the above expression, the tunneling Hamiltonian is approximately proportional to the momentum operator in the tunneling *z* direction.^{21} This approximation is best when the tip-to-nanostructure tunneling gap *z*_{0} − (*r* + *d*_{NS}) and the nanostructure-to-surface gap *r* [see Fig. 2(a) for variable definitions] are sufficiently large so that the tip, nanostructure, and surface can still be treated as separate sub-systems

Here $P^$ is a permutation operator that switches the functions in the bracket surrounding $\u0124tunnel$. This operator, when inserted into the first term of Eq. (2), and integrated over any surface S_{a} in the tunneling region,

produces the conventional Bardeen tunneling rate

for a perfect tip and uniform surface. Here *W* is the work function of the tip (ca. 4.5 eV for tungsten). In the case of the second term in Eq. (2), the two matrix elements in the product in the numerator look the same as Eq. (4), but the respective surfaces for evaluation in the Bardeen approximation must lie between *z*_{0} > *z* > (*r* + *d*_{NS}) for the $MTn=\psi Tx\u21c00\u0124tunneln$ matrix element, and between $r>z>0$ for the $MnS=n\u0124tunnel\psi S$ matrix element.

The tunneling matrix element $M(x\u21c00)$ depends on $x\u21c00=(x0,y0,z0)$, but here the (*x*_{0}, *y*_{0}) dependence will be of main interest. The tip is not perfectly sharp, and its electric field is not perfectly perpendicular to the surface, so $\psi Tx\u21c00$ does not fall off to zero immediately when $x\u2212x0>0$ or $y\u2212y0>0$. For our tips, we find empirically (under laser illumination, see Subsection III B) that this limits the (*x*_{0}, *y*_{0}) resolution to about 0.5 nm. Features <0.5 nm cannot be interpreted without detailed modeling of the tip because *ψ*_{S} is not a simple plane wave state, and *ψ*_{T} does not drop off infinitely rapidly with *x*, *y* distance from the tip apex. As seen below, the corrugations of the surface on which the nanostructure sits (*ψ*_{S}) have little effect on the image.

Now consider one-electron excitation from orbital $j\u2192i$. The matrix element *M* when $i$ becomes occupied and $j$ becomes empty (laser has excited transition) is then given by

Here, $\Delta Mij(2)=\u2212Mi(2)+Mj(2)$ is the difference between the 2nd order matrix elements for the two tunneling paths that are opened/closed by excitation. When optical excitation has occurred, the tunneling rate is given by [Fig. 2(b)]

When the laser has not excited the transition, the matrix element is given by [Fig. 2(b)]

and the tunneling rate by

The tunneling current is *I*(*x*_{0}, *y*_{0}) = *ek*. Thus, the maximum possible difference in tunneling current (when the laser has Rabi-cycled the transition to the excited state $i$, *vs*. when the electron is in ground state $j$) is given by

In practice, the current modulation is half that value if the system is excited by a continuous laser and a time-averaged SMA-STM current image $\Delta Ix0,y0$ is collected. Rabi-cycling with a pulsed laser and using ultrafast gating could, in principle, yield the maximum current signal. However, as discussed in Ref. 33, the tip-enhanced laser field easily drives the transition $j\u2192i$ into saturation for transition dipole moments obtainable with quantum dots or carbon nanotubes, even if the excited state relaxes very rapidly and 1/Γ_{i} < 100 fs.

Equation (10) is exact within the Bardeen formalism and one-electron approximation. With some simplifying assumptions, it can be seen that Eq. (10) corresponds to imaging the structure of the two orbitals *i* and *j* connected by the optical transition, which can be interpreted intuitively even without assistance of exact electronic structure modeling of the tip, nanostructure, and surface as a multi-electron system. Consider the following three cases, all of which assume that the initial state of the system is the ground state with ∞ lifetime:

Case (1): The tip Fermi level is on-resonance with orbital $i$, and the excited state is sufficiently long-lived so that the small energy denominator $\u2212i\u210f\Gamma i$ makes $Mi(2)$ large. In that case, $|\Delta Mij(2)|\u2248|Mi(2)|,$ and $Mi(2)2\u226b2\u2009Re\u2009M0\Delta Mij(2)$. The integral in Eq. (10) at saturation power then reduces to

where

In Eq. (11a), the factor of two reduction noted below Eq. (10) has been added. The second of the squares in Eq. (11b), the nanostructure-surface tunneling contribution, is independent of (*x*_{0}, *y*_{0}) and contributes only a constant proportionality factor to the tunneling current modulation $\Delta Ix0,y0$. For a perfect tip with wavefunction $\psi T\u223c\delta (x\u2212x0)\delta (y\u2212y0)e+\gamma (za\u2212z0)$ on surface S_{a} in the tunneling region, and for a finite-sized nanostructure whose real wave functions in the tunneling region can be factored as $\phi ix\u21c0\u223c\phi ix,ye\u2212\gamma (z\u2212z0)$, the first square in Eq. (11b), when inserted into Eq. (4), yields

Thus, the tunneling current approximately maps out the probability density of the resonant orbital $i$ as it appears at the interface between the tip and nanostructure, with a lateral resolution limited by the spatial extent of the tip wave function. We refer to the signal in Eq. (12) as the “orbital density map” or ODM. In Fig. 1, our tip bias is −2.4 V, and the 532 nm excitation corresponds to 2.33 eV, so case (1) approximately obtains due to Fermi level equilibration.^{34}

Case (2): The tip Fermi level (set by the applied bias voltage *V*) is off-resonant with both the orbitals $i$ and $j$ so that $M0\u226b\Delta Mij$. According to Eq. (10),

The *M*_{kS} (*k* = *i*, *j*) matrix elements in Eq. (13) are again independent of the tip position $x\u21c00$. Thus, in the *x*, *y* plane, the signal maps out the weighted projection of the states $i$ and $j$ onto the tip wave function. If *M*_{0}, which sums over a large number of states with different nodal structure, is weakly position-dependent and $\psi T$ is again a perfect tip wave function, then the projection becomes approximately

Here we assumed again that the orbitals *φ*(*x*, *y*, *z*) can be factored into a *z* and (*x*, *y*) component in the tunneling region between the tip and nanostructure. Note that this real part of the difference between two orbitals, observed as an interference term between the unperturbed and laser-perturbed tunneling, can be either positive or negative, as seen in Figs. 3(d) and 3(f) in Sec. III B.

Unlike in scanning tunneling spectroscopy (STS), resonance of the system states in the tunneling gap between the two metal surfaces (tip and substrate) is not required to observe an SMA-STM signal although the SMA-STM signal is a weaker current perturbation than is the STS signal (typically 0.1 pA for SMA-STM at 10 pA tunneling currents, as expected from the above equations for off-resonance excitation).

Case (3): After excitation, the electron and hole form a strongly coupled exciton (i.e., with a binding energy that would make it energetically unfavorable for another electron to occupy state $j$ in Fig. 2). In that case, optical excitation still reduces tunneling through state $i$ but will have no effect on state $j$, which remains inaccessible to tunneling. In the resonant case, Eq. (12) is recovered, and the observed signal reflects the position dependence of the state occupied by the excited electron in the coupled electron-hole pair.

In summary, in the simplest cases where one-electron excitation is a good description of the electronic structure of the molecule or nanostructure in the junction, the STM-SMA signal is related to the shape of the orbitals involved in the transition. For strongly bound excitons, the approximation still holds if the tunneling electron cannot occupy the hole state. Of course, for highly correlated electronic states of the nanostructure, this approximation will fail, and one cannot approximate the SMA-STM signal by the density of an orbital *φ*_{i} at energy *E*_{i} that solves the Hartree-Fock self-consistent field equations for a wave function that is a product state of one-electron orbitals. However, even in more complex cases, we expect that full electronic structure calculations of the tunneling current will reveal that the SMA-STM signal maps out localized structure in the excited state due to defects arising from disorder, missing, or added atoms in the nanostructure.

### B. Experimental re-orientation of quantum dots to visualize excited states

Figure 3 illustrates the type of excited state imaging possible by SMA-STM. Three different PbS quantum dots (bandgap 1.01 eV) were excited at 532 nm on two different surfaces. The orange images on the left are the STM topography scan in constant current mode. The images on the right in grayscale are the SMA-STM signal, with black encoding a strong SMA-STM signal (reduction in tunneling current when laser is on) at positive sample bias, reversely for white. Figures 3(a) and 3(b) show a quantum dot on the Au surface that absorbs only weakly. Figures 3(c) and 3(d) show a quantum dot on an amorphous SiC surface that absorbs strongly at resonance [the laser energy offsets the bias voltage as seen in Fig. 2(a)], exciting a P-like state that appears as a donut. Figures 2(e) and 2(f) show a similar quantum dot on the Au surface with a defect on its right side leading to localization of the excited state into a “C” shape. Note that the STM-SMA current signals obtained at negative tip bias [Fig. 2(d)] *vs*. positive tip bias [Fig. 2(f)] are 180° out of phase and encode as white/black with respect to the gray background (no laser-induced current modulation). A detailed comparison of this type of data with DFT calculations has been reported previously.^{9}

Next, quantum dots were manipulated by rolling them on a Au surface, using a previously reported manipulation method.^{15} The STM tip first scans a line (∼15–20 nm) over the quantum dot at topographic tip-surface separation (typically 10 pA, 1-1.5 V for PbS quantum dots and 2.5-3.5 V for CdSe/ZnS quantum dots). The tip then scans the same recorded line again but is moved down by a preset value (typically 0.7 nm, still smaller than the separation between the tip and the quantum dot of ca. 1-2 nm). The closer separation increases the interaction between the tip and the quantum dot, and the quantum dot is rolled along the predefined direction. During this step, the STM current feedback loop is turned off. This method has been shown to efficiently move carbon nanotubes on the silicon surface without damaging the molecules.^{15}

Using the STM tip, we roll quantum dots on the surface to image two different 3-D orientations of the excited state projected into the surface (*x*, *y*) plane. In the topography image of Figs. 1(b) and 1(e), a CdSe/ZnS quantum dot (bandgap 2.3 eV) is rolled on the Au surface along the direction marked by the blue arrow. Thanks to a marker on the surface [the dark horizontal line on the left of the quantum dot in Figs. 1(b) and 1(e) is a narrow gap between two Au islands], the amount of rolling can be quantified by the formula *θ* = (*L*/*d*)·(360°/π) ≈ (0.4)·(114°) ≈ 45°, where *L* is the dot center-to-center displacement along the surface (size marker in Fig. 1 is 5 nm) and *d* is the apparent dot diameter in the rolling direction. This formula only holds when (i) no slipping occurs and (ii) the apparent diameter reflects the actual rotational diameter. Our angle is likely a lower bound because the actual rolling diameter is likely less than the imaged diameter due to lateral STM resolution. The corresponding absorption images excited at 532 nm in Figs. 1(c) and 1(f) show how the SMA-STM signal changes. The excited state is rotated off to the edge of the quantum dot, revealing that it must be surface-localized. Here the excitation is near-resonant, so the observed signal corresponds approximately to the orbital density map (ODM) of case 1, Eq. (12).

Figure 4 shows the same two orientations of the CdSe/ZnS dot at positive and negative bias voltages. Voltages shown are bias of the surface with respect to the tip. As expected, the SMA-STM current modulation signal changes sign with bias voltage (black signal at positive V vs. white signal at negative V): when electrons tunnel from the tip to surface (positive surface bias), excitation hinders their tunneling through the newly occupied spin state [negative Δ*I*(*x*_{0}, *y*_{0}) is coded as black]; when electrons tunnel to the tip, the excited electron in the occupied spin state tunnels more easily than any other electrons [positive Δ*I*(*x*_{0}, *y*_{0}) is coded as white].

Next we studied a non-spherical quantum dot, so rolling can be unambiguously distinguished from slipping over the surface. Figures 5(a) and 5(b) show a non-spherical ∼4 nm × 5 nm PbS quantum dot excited at 532 nm on an Au surface before and after rotation along the blue arrow. The tip’s torque rotated the quantum dot mainly in-plane (rotation axis perpendicular to the Au surface plane). The non-spherical shape of the quantum dot is used to determine the rotation angle in this case. The quantum dot was translated up and left by ∼5 nm and rotated ∼90°, as indicated in the dotted outline in Fig. 5(c) (features on the step edge of the Au island on the upper right were used as reference). The corresponding absorption SMA-STM absorption images are shown in Figs. 5(d)–5(f). The absorption shape is non-uniform and rotates with the quantum dot, showing different views of the orbital density map obtained by SMA-STM.

Figure 6(a) shows a PbS quantum dot on an Au surface, with its corresponding absorption image shown in Fig. 6(f). In this case, as in Fig. 5, the combination of a tunneling voltage of 1.5 V and 532 nm excitation means that there are significant off-resonant contributions to the SMA-STM image, given by case 2, Eq. (14) in the simplest approximation. The SMA-STM signal Δ*I*(*x*_{0}, *y*_{0}) of this quantum dot is therefore more complicated and shows both positive and negative Δ*I* (lighter and darker than the gray background) in different locations (*x*_{0}, *y*_{0}) across the excited state. While the actual signal for an excited quantum dot is not necessarily quantitatively explained by the one-electron picture in case 2, Eq. (14), the experimental result and model both highlight that defects will lead to localized spatial variations of Δ*I*(*x*_{0}, *y*_{0}) that image a slice through the excited state electronic structure of the quantum dot.

The topography and absorption images of this PbS dot were retaken without nudging and are shown in Figs. 6(b) and 6(g). The topography and the absorption shape of the quantum dot remain nearly unchanged, indicating the reproducibility of the SMA-STM measurements. The quantum dot then was rolled multiple times on the surface along the directions indicated by the arrows. Figures 6(c)–6(e) show the quantum dot after three such manipulations. As in Fig. 5, the non-spherical shape of this quantum dot verifies rolling as opposed to pure translation. The shape of the quantum dot is significantly different when it is viewed from different angles: it appears nearly spherical in Figs. 6(a) and 6(b), slightly elongated in Figs. 6(c) and 6(e) and even more elongated with nearly parallel edges in Fig. 6(d). These topographic images allow us to roughly determine the 3D shape of the quantum dot, as shown in the cartoons below Figs. 6(a)–6(e). The corresponding SMA-STM absorption images are shown in Figs. 6(f)–6(j). The absorption shape is re-oriented upon rolling, revealing a more complete picture of the excited state structure of the quantum dot from several angles. Thus, using simultaneous topography and SMA-STM detection, both the shape and the excited states of the quantum dots are visualized. We note that with our tip-nudging procedure, we expect that the torque of the tip will mainly induce rolling and not translation, to the extent that the concept of a smaller rolling friction than sliding friction still applies at the nanoscopic scale.

Ground state tomographic data, such as from cryo-electron micrography, can be interpreted with a model that simply uses a classical density. Modeling of SMA-STM data minimally requires quantum mechanical calculation of the density of an orbital unoccupied prior to excitation [case (1), Eq. (12)] or of orbital differences [case (2), Eq. (14)] projected into the 2-D imaging plane. To simulate the resonant case [Eq. (12)] from Figs. 1(c) and 1(f), we performed a DFT calculation on a model PbS quantum dot of a much smaller size than in our experiments and scaled the model energies assuming particle-in-a-box scaling.^{9} Figure 7(a) shows the 2-D projected orbital density map (ODM) of a strongly dipole-active transition near 532 nm for a PbS quantum dot with defects simulated by random displacement of the atomic center positions (see Sec. II). For excitation well above the bandgap, the calculated transition connects delocalized valence states to a localized surface state, hence the non-zero transition moment. Figure 7(a) illustrates rolling of the computed ODMs at five angles 0°, 22.5°, 45°, 67.5°, and 90°. The rotated ODMs show that the orbital density map is localized on one side of the defected quantum dot surfaces. For reference, Fig. 7(c) shows an orbital at a similar energy of the undistorted quantum dot.

To account for the experimental resolution of ≈0.5 nm and the larger experimental quantum dot size of 4.2 nm (*vs.* ca. 2 nm in simulation), a Gaussian spatial filter with FWHM of 0.25 nm is applied to the ODM [Figs. 7(b) and 7(d)]. The electron density is centered on the left half of the dot at 45° and is localized at the left edge of the quantum dot at 90°. Thus, by rolling the quantum dot, different 2-D projected shapes are obtained. The localized orbital DFT projections in Fig. 7(b) (45°, 90°) are also reproduced in Figs. 1(d) and 1(g) for qualitative comparison with experiment. The computed projections strongly resemble our experimental data in Figs. 1(c) and 1(f) for a rotation from ca. 45° to ca. 90°. This result illustrates the resonant case 1 of Eq. (12), where the electronic state excited by the laser corresponds to the excitation of a surface-localized orbital near the tunneling energy.

## IV. DISCUSSION

There is currently much interest in using quantum dots or other nanostructures in molecular electronic devices,^{35,36} or in light-harvesting films.^{18,37} The challenge of nanomaterials is that even nominally identical nanostructures differ in composition from one another. Defects represent a challenge and an opportunity, as they can also be used to tune the properties of nanomaterials on a particle-by-particle basis. In particular, defects can be used to achieve desirable excited state properties, such as localizing and therefore enhancing electron density near a catalytic surface site.^{38,39}

It would be advantageous to be able to image excited states of individual nanoparticles directly so that one can differentiate the electronic structure of defects. It would be even more advantageous if such imaging could be carried out for different projections of a nanoparticle, so a better view of the three-dimensional structure of the excited state can be obtained. This is what SMA-STM combined with single-particle manipulation offers. The experiment and the DFT calculation in Fig. 1 illustrate how rolling a nanoparticle on the surface provides different views of the excited state. In the simplest on-resonant case, the interpretation of the STM SMA signal is fairly intuitive in terms of the orbital density map $\phi ix0,y02$ of the excited state orbital being occupied by the optical transition [case 1, Eq. (12)], thus hindering the tunneling process through the nanostructure. In the more complex case of off-resonant excitation, additional phase information becomes available: the signal in case 2, Eq. (14), is signed and reports on the real part of an orbital difference map between ground and excited states. Such positive and negative phase signals are evident in the off-resonant excitation case of Figs. 6(f)–6(j), where both enhanced tunneling and reduced tunneling are observed. Of course when the one-electron picture of excitation from one spin orbital to another breaks down, it may not be possible to relate the SMA-STM signal to the simple results in case 1, Eq. (12) and case 2, Eq. (14), and a more complete quantum mechanical calculation using electron correlation is required to model the experimental observations.

The approach presented here has the potential to be used for true single particle tomography of excited states, as opposed to the common approach of depositing an ensemble of nominally identical particles on a surface at random orientations, and reconstructing a 3-D structure for the ground state of the ensemble. However, a number of improvements would be required over the results presented here: First, one has to ensure that the particle and tip are not altered by the manipulation process so that manipulation can be repeated many times without affecting the particle being analyzed. Moreover, quantitative modeling of excited states will require calculation for actual-size quantum dots with different types of defects for comparison with experiment. Both of these requirements may be satisfied in the near future. For instance, recently developed HfB_{2}-coated tungsten tips offer the hardness and sharpness for highly repeatable manipulation.^{40} In addition, DFT simulations are progressing to the point where a series of calculations for >2 nm quantum dots containing heavy atoms (like Pb, Cd or S) is becoming feasible in a reasonable amount of time.^{41}

In conclusion, we have demonstrated that SMA-STM can be used to investigate the orientation dependence of excited-states of quantum dots. Both topography and electronically excited states can be simultaneously recorded by measuring STM current *I*(*x*_{0}, *y*_{0}) and laser modulation of the current Δ*I*(*x*_{0}, *y*_{0}). The experiment agrees qualitatively with a size-scaled DFT calculation that assumes a one-electron model [case 1, Eq. (12)] for a measurement where the defect localizes the excited state density near the quantum dot surface. We speculate that in the future, it may be possible to do single particle tomography of excited states, if damage to quantum dots can be avoided during manipulation, and if accurate excited state calculations for multiple dot structures can be compared directly with experiment. As in conventional tomography, accurate rotation angles need not be known to fit a 3-D model, if enough views can be measured. Unlike conventional tomography (e.g., via cryo-EM), single particle tomography could highlight defects in individual nanoparticles, rather than just an average structure, and SMA-STM could enable excited state imaging, rather than just ground state imaging.

## ACKNOWLEDGMENTS

The experimental work at UIUC was supported by the National Science Foundation CHE Directorate (D.N., J.L., and M.G.) and the James R. Eiszner Chair (H.A.N. and M.G.). The computational work at UW was supported by a U.S. National Science Foundation Graduate Research Fellowship (No. DGE 1256082 to J.J.G.) and grants from the National Science Foundation, NSF Nos. CHE 1464497 and CHE 1565520 (J.J.G. and X.L.). D.N. thanks the Beckman Institute for a Beckman Graduate Fellowship while this work was carried out.

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