Single Semiconductor Nanostructure Extinction ... - ACS Publications

Review Article Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

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Single Semiconductor Nanostructure Extinction Spectroscopy Rusha Chatterjee, Ilia M. Pavlovetc, Kyle Aleshire, and Masaru Kuno* Department of Chemistry and Biochemistry, University of Notre Dame, 251 Nieuwland Science Hall, Notre Dame, Indiana 46556, United States S Supporting Information *

ABSTRACT: While emission-based, single-particle microscopies and spectroscopies have been highly successful in revealing the properties of matter hidden by ensemble averages, their limits have now become apparent. To address recognized future needs and, in particular, the need to go beyond fluorescent specimens, single-particle extinction techniques have been developed. Motivating this has been the desire to acquire information about the electronic structure of nanoscale materials difficult to obtain otherwise using either ensemble or emission-based, single-particle measurements. These techniques are, however, nontrivial since single nanostructures attenuate only 0.0001−1% of the incident light. This Review Article describes the challenges associated with overcoming the low signal-to-noise ratios inherent to lowdimensional semiconductor nanostructure extinction measurements. It simultaneously describes the fundamental operating principles and achievements of photothermal heterodyne imaging (PHI) and spatial modulation spectroscopy (SMS), two of the most popular approaches to measuring single-particle extinction. It then reviews what exactly we have learned about the fundamental physics of a model system, viz., low-dimensional CdSe, via single-particle extinction measurements. The Review Article finally describes the development of a new single-particle extinction methodology, infrared photothermal heterodyne imaging, which portends future successes in revealing the detailed physics of nanostructures beyond both ensemble averages and corresponding single-particle, emission-based insights.

1. INTRODUCTION

that single-molecule detection is now readily achieved using scientific-grade CCD cameras. Guyot-Sionnest9 and Bawendi10 were the first to conduct single CdSe quantum dot emission measurements. They found that single-particle emission line widths were narrow and that there was evidence of spectral wandering much like that observed in the single-molecule community.11 There was also the realization of unexpected emission intensity blinking,12 again analogous to that seen with single molecules.13−15 These early measurements have since been complemented by other lowtemperature studies showing biexciton and charged exciton (i.e., trion) emission,16−18 all adding to our general understanding about fundamental differences between the optical/electrical response of nanomaterials and their parent bulk. Reference 19 provides an overview of recent work involving single-particle emission measurements on CdSe QDs, a model nanosystem.20 The problem with single-particle emission measurements is that they are restricted to emissive species.21 Such specimens must also possess relatively high emission quantum yields (QYs). For CdSe and a few other colloidal nanocrystal systems (e.g., PbSe,22 CsPbBr323), this has not been an issue given their high asmade QYs. Synthetic advances such as overcoating2,24,25 also mean that QYs can be further enhanced. As a result, ensemble

Research in modern nanoscience is driven by the fact that, when length scales are below certain critical dimensions, nanoscale materials exhibit properties different from the corresponding bulk system. The best illustration of this is quantum confinement effects that occur in semiconductors.1 A classic example involves the size-dependent absorption and emission of colloidal CdSe quantum dots (QDs).2 This realization was only made possible by significant advances in the synthesis of nanomaterials to reduce residual ensemble size distributions to values on the order of 5%.3,4 Unfortunately, even today there are no syntheses that yield perfectly monodisperse samples. Hence all ensemble measurements, which focus on elucidating the electronic and optical properties of nanomaterials, are inherently contaminated by size-related inhomogeneities. The most obvious way to circumvent this problem entails single-particle measurements. The traditional approach for conducting single-particle microscopy/spectroscopy has been to carry out emission-based measurements using technical approaches first developed by researchers in the single-molecule community.5−8 These emission-based measurements involve diluting samples to statistically have only one molecule within a diffraction-limited illumination volume. Appropriate excitation and filtering to block the incident laser light are equally crucial. Though detection of the emission early on entailed single element devices, CCD technology has improved sufficiently such © XXXX American Chemical Society

Received: January 23, 2018 Revised: March 8, 2018 Published: March 27, 2018 A

DOI: 10.1021/acs.jpcc.8b00790 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Review Article

The Journal of Physical Chemistry C emission efficiencies now routinely range from 30 to 60%.26−29 In the best cases, near unity (i.e., 85−95%) QYs have been reported.30−37 However, many other systems exist which are nonemissive or which do not have high QYs. Consequently, there are entire classes of nanoscale materials that cannot be studied using current emission-based, single-particle microscopies/spectroscopies. Additionally, emission-based measurements probe only the lowest excited state of the system. They do not directly reveal information about the excited-state progressions since nanostructure emission arises following thermalization. Furthermore, nanostructure emission is readily influenced by the local surroundings as well as by defect states.38 Finally, both intrinsic and surface-related Stokes shifts can complicate conclusions derived from single-particle emission spectroscopies.39,40 Emission-based measurements are therefore restricted in their applicability. The apparent solution to this dilemma involves conducting single-particle extinction measurements. In this regard, extinction is a universal phenomenon and directly affords information about a nanostructure’s intrinsic electronic structure and photophysics. From a historical standpoint, single chromophore extinction measurements are not new. In fact, the earliest single-molecule experiments conducted by Moerner41 were extinction, not emission, measurements. For molecules and small nanostructures, extinction and absorption are synonymous.42 The problem with measuring the extinction of a single particle or chromophore and the reason why emission-based techniques have dominated single-particle microscopy/spectroscopy for the last three decades is that extinction cross sections are small. Typical single-particle absorption cross sections range from 10−15 to 10−11 cm2.43 Consequently, the attenuation of an incident laser beam only ranges from 0.0001 to 1%. Most of the incident light is unattenuated and obscures the absorption given its substantially larger background noise. By contrast, with appropriate spectral filtering single-particle emission measurements are effectively zero background studies, explaining their ease.44 Single-particle extinction measurements are, however, possible and have been demonstrated.45−47 Success stems from the use of appropriate strategies to overcome the inherent signal-to-noise problem in these measurements. In what follows, we review the use of single-particle extinction/absorption measurements to unravel the fundamental optical and electrical properties of nanostructures under conditions free from ensemble averaging. We summarize what these measurements have taught us and what they may tell us in the future. Our focus will be on semiconductor nanostructures as there already exists a significant body of work on extinction studies with corresponding metal nanostructures.46 Additionally, an attempt will be made to delineate what exactly these extinction/absorption measurements have taught us which would otherwise have been difficult to discern through corresponding single-particle emission measurements. 1.1. Modeling Absorption Cross Sections. We begin by illustrating why single-particle extinction measurements are inherently difficult. At the heart of the problem is the intrinsically small extinction/absorption cross sections of individual nanostructures. Early on there were no measured absorption cross sections for semiconductor nanostructures. This has since changed with a plethora of values now available for common semiconductor nanomaterials possessing different dimension-

alities. Tables 1 and 2 below provide a partial compilation of this literature. Table 1. Reported Colloidal QD Band-Edge Absorption Cross Sectionsa

a

The horizonal shaded region for each entry denotes the range of cross sections observed for the associated range of QD radii (a, nm).

In the absence of experimental values, theoretical expressions were therefore used to estimate the absorption cross sections (σ) of relevant materials. Formally speaking, it is the extinction that one measures in an experiment where the extinction is the sum of absorption and scattering.42 However, the volume scaling of Table 2. Reported Nanorod (NR)/Nanowire (NW) BandEdge Absorption Cross Sectionsa

a

The horizonal shaded region for each entry denotes the range of cross sections observed for the associated dimensions (NR/NW radius: a, nm; NR/NW length: l).

B


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The Journal of Physical Chemistry C

Tables 1 and 2 provide tabulated experimental (room temperature) absorption cross sections for a few representative QD and NW systems. More detailed tables, which include references for the depicted values, can be found in the SI. A common order of magnitude values is 10−15 cm2 and 10−11 cm2 for QDs and NWs, respectively. As a point of comparison, the absorption cross section of a single molecule takes values on the order of 10−16−10−15 cm2.6,59−61 Table 2 makes it evident that NWs are, in an absolute sense, stronger absorbers than QDs. This primarily stems from their larger volumes. However, even when cross sections are volume normalized, NWs absorb more efficiently than QDs due to their more favorable local field factors. A detailed discussion of the inherently better absorptive properties of NWs can be found in ref 43. Irrespective of system, what these order of magnitude cross sections mean in an extinction experiment is that only 0.0001− 1% of the incident light is attenuated. As an explicit illustration, assume an excitation intensity Iexc associated with an input power Pexc. The fraction of Pexc extinguished by the nanostructure is σIexc σ = A where A is the area of the focused diffraction-limited P

absorption and scattering cross sections (i.e., linear versus quadratic) means that for small nanostructures extinction and absorption are effectively equivalent.42 The first expression employed was48−50 ω V |f (ω)|2 (2nsks) nmc

σ(ω) =

(1)

where ω is the angular frequency of the incident light; c is the speed of light; nm is the surrounding medium’s refractive index; V is the particle’s volume; f(ω) is a local field factor that reflects the ratio of the electric field inside the particle relative to that outside; and ns (ks) is the real (imaginary) part of the particle’s complex refractive index. In the absence of single nanostructure dielectric constants and refractive indices, eq 1 implicitly assumes bulk parameters.51 Consequently, it is only formally appropriate for evaluating nanostructure absorption cross sections far to the blue of the band edge where confinement effects are less pronounced and where the joint density of states becomes bulk-like.

(

For spherical particles, f (ω) =

3εm εs̃ (ω) + 2εm

) such that

exc

2

3εm ω σ(ω) = V (2nsks) nmc εs̃ (ω) + 2εm

spot. Using nominal values for visible wavelengths, up to a 106 difference in areas exists. Although it would be preferable to improve this ratio, one is constrained by σa property of the absorberon one hand and by the optical diffraction limit on the other. Since only a very small change in light intensity arises due to extinction, the absorption of light by a nanostructure is readily obscured by laser noise which can easily be orders of magnitude larger. Single nanostructure experiments are thus inherently difficult. To be successful, they require techniques that effectively mitigate laser intensity fluctuations and other sources of noise that would otherwise obscure this small fractional extinction of the incident light.

(2)

4

with V = 3 πa3, where a is the nanoparticle (NP) radius. An explicit derivation of eq 2 can be found in the Supporting Information (SI). For cylindrical NWs, two expressions for f(ω) exist, one for light polarized parallel to the NW long axis and another for the polarization orthogonal to it. Expressions for relevant parallel and perpendicular polarization local field factors 2ε are therefore43 f∥ = 1 and f ⊥ = ε ̃ (ω) +m 2ε . What results are the s

m

following parallel and perpendicular NW absorption cross sections ω σNW = V (2nsks) nmc (3a)

2. COMMON APPROACHES TO SINGLE-PARTICLE EXTINCTION MEASUREMENTS Two common approaches exist for measuring the extinction/ absorption of individual nanostructures. The first is called photothermal heterodyne imaging (PHI).46,62−65 The second is referred to as spatial modulation spectroscopy (SMS).46,66 The key to both techniques is the modulation of the extinction so that lock-in detection can be used to discriminate the signal of interest against noise inherent to the unattenuated incident light as well as to any noise coming from the detection electronics. 2.1. Photothermal Heterodyne Imaging (PHI). Photothermal heterodyne imaging/detection specifically refers to a technique for measuring the absorption of an analyte, based on the local heating of its surrounding photothermal media. The key selling point of PHI, within the context of single-particle extinction spectroscopy, is that it exclusively measures absorption. It is insensitive to scattering. This feature stems from the fact that only light absorbed by a sample leads to local heating and has been exploited extensively in the study of single metal nanostructure plasmon resonances.67−69 As with all photothermal measurements,62 PHI works best with nonemissive samples where the absorption of incident light leads to near unity efficiency heat dissipation of the absorbed energy into the surrounding environment. What results is a temperature rise, which alters the surrounding medium’s refractive index. This refractive index change can then be used to indirectly infer the analyte’s absorption efficiency.

and 2

⊥ σNW =

2εm ω V (2nsks) nmc εs̃ (ω) + 2εm

(3b)

where now V = πa2l and a(l) is the NW radius (length). Equations 3a and 3b indicate that NWs should exhibit sizable polarization anisotropies due to their varying local field factors.52−54 To illustrate, assuming nominal semiconductor dielectric constants/refractive indices,51 up to 20-fold differences in absorption strength exist. These absorption differences can be captured using an absorption polarization anisotropy defined as

ρabs =

⊥ σNW − σNW ⊥ σNW + σNW

(4)

where, using nominal values of nm as well as bulk semiconductor dielectric constant/refractive indices, predicted polarization anisotropies are ρabs ∼ 0.9. These near unity values suggest that sizable NW polarization anisotropies can be exploited for use in polarization-sensitive applications.52,55−58 Note that eq 4 is classical in that it does not account for quantum mechanical polarization-specific transition selection rules that arise from a detailed evaluation of the NW electronic structure. Intrinsic NW polarization selection rules will be discussed later below. C


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Figure 1. (a) PHI setup schematic showing backward and forward detection schemes. (b) Back reflection PHI image of individual 10 nm Au nanoparticles acquired over a 6 × 6 μm region. Reprinted figure with permission from ref 72. Copyright 2006 by the American Physical Society.

In particular, when the excitation (i.e., heating) of the analyte is made periodic via intensity modulation of the pump, the surrounding medium’s refractive index also changes periodically. A (nonresonant) probe beam, focused on the analyte and its immediate surroundings, then undergoes scattering65,70,71 and, in turn, interferes with the incident reflected or transmitted probe beam. What results is a heterodyned signal at the frequency of the modulated pump with an intensity proportional to the analyte’s absorption cross section. By scanning the pump’s wavelength, the corresponding absorption spectrum of the analyte is obtained. References 64, 72, and 73 provide more detailed descriptions of the PHI technique. Having seen the apparent advantages of PHI, what are some of its disadvantages? First, as noted above, specimens must be nonemissive or have low emission QYs. This is because radiative recombination directly competes with nonradiative relaxation/ heat dissipation. Next, the experiment generally requires that specimens be immersed in a photothermal medium. Here issues arise if there are chemical or physical incompatibilities with the photothermal media. Additionally, relatively large pump intensities must be used to obtain PHI data since a sizable local heating is required. These large intensities can damage samples or saturate their transitions. Finally, PHI is an indirect method. As will be seen in the next section, the PHI signal expression cannot be used to directly yield an analyte’s absorption cross section. To make PHI quantitative, comparisons to standards having well-known σ-values must therefore be made. 2.1.1. Modeling the PHI Signal. A theory for the PHI signal was first developed by Lounis and co-workers.72 In it, one models probe light scattering due to periodic, pump-induced changes to the surrounding medium’s susceptibility. The scattered light is then assumed to interfere with the unscattered back-reflected or forward-transmitted probe beam. What results is the following expression for the power of the heterodyned signal measured in the far field Phetero

F(Ω) =

G(Ω) =

∫θ

1 Ω

∫θ

θmax

f (θ , Ω)[1 + cos2 θ ]sin θ dθ

(6a)

min

θmax

g (θ , Ω)[1 + cos2 θ ]sin θ dθ

(6b)

min

π

π

where (θmin = 0, θmax = 2 ) in forward transmission; (θmin = 2 , θmax = π) in back reflection; f (θ , Ω) =

1 u

g(θ , Ω) =

1 u

( (

u+1 (u + 1)2 + 1

+

u−1 (u − 1)2 + 1

1 (u + 1)2 + 1

−

1 (u − 1)2 + 1

); ); u =

4n m π λ

2D Ω

( θ2 );

sin

and D is the medium’s thermal diffusivity. If desired, the transmissivity (η) of the optical path in either forward or backscattered directions can be accounted for by multiplying Phetero with η. A complete derivation of eq 5 can be found in the SI. 2.1.1.1. Experimental Lock-in Signal. In an experiment where a lock-in is employed to read the PHI signal, the lock-in output voltage is Vhetero = RdetGdetPhetero where Rdet is the detector’s responsivity and Gdet is its gain. Consequently, the resulting lockin voltage is Vhetero = R detGdet

2π 2α Pprobe ⎡ ⎛ ∂nm ⎞ Pabs ⎤ ⎥ ⎢n m ⎜ ⎟ ⎢⎣ ⎝ ∂T ⎠ Cpλ 2 ⎥⎦ w

[F(Ω)cos Ωt − G(Ω)sin Ωt ]

(7)

Subsequent demodulation using an internal lock-in reference yields ⎡ π 2α Pprobe ⎤⎡ ⎛ ∂n ⎞ P ⎤ ⎥⎢nm⎜ m ⎟ abs ⎥ Vlock‐in = R detGdetVref ⎢ ⎢⎣ ⎥⎦⎢⎣ ⎝ ∂T ⎠ Cpλ 2 ⎥⎦ w F(Ω)2 + G(Ω)2

(8)

where Vref is the lock-in internal reference amplitude. Since Pabs is proportional to the analyte’s absorption cross section, the measured PHI signal is linked to σ. However, eq 8 depends on a number of parameters as well as model assumptions. Consequently, it does not readily lend itself to inversion in order to evaluate σ directly. PHI therefore relies on comparisons with standards having known σ-values in order to make quantitative assessments of an analyte’s absorption efficiency. Here gold NPs serve as a convenient photothermal “ruler” since Mie theory provides robust extinction cross-section estimates across size. A compilation of experimental size-dependent Au NP extinction cross sections along with Mie theory predictions can be found in the SI.

2π 2α Pprobe ⎡ ⎛ ∂nm ⎞ Pabs ⎤ ⎢n m ⎜ ⎥[F(Ω)cos Ωt ⎟ = ⎢⎣ ⎝ ∂T ⎠ Cpλ 2 ⎥⎦ w − G(Ω)sin Ωt ]

1 Ω

(5)

In eq 5, Pabs = σIexc is the power absorbed by the analyte; Ω is the pump modulation angular frequency; Pprobe is the incident probe power; λ(w) is its wavelength (beam waist at focus); nm is the refractive index of the surrounding medium; Cp is its volumetric heat capacity; and α is the reflectivity or transmissivity at the sample/substrate interface. Additionally D


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The Journal of Physical Chemistry C 2.1.1.2. PHI Signal-to-Noise Ratio. Given eq 8, a measure of the signal-to-noise ratio (SNR) of a PHI measurement can be made. Measurement noise is expressed as74 Vnoise =

2 2qR detGdet PprobeΔf

reflected onto a photodetector. Orrit has referred to this signal isolation scheme as the “cat’s eye” geometry/reflector.64 In the latter transmission geometry, the collected heterodyned light is simply filtered to remove any pump light. It is then focused onto a photodetector. Given its ease, the transmission geometry has been used in the majority of PHI measurements.60,64,73,75,76,80−86 Furthermore, despite the back-reflection (cat’s eye reflector) configuration possessing a higher theoretical SNR,72 the transmission geometry often yields comparable results. As suggested by ref 64, this may stem from depolarization of the signal due to the objective or additional optical elements in the back-reflection setup. Critical to PHI is that the sample be embedded in a photothermal medium such as glycerol.60,64,76 Other chemical media used include mineral/silicone oil65,72,75,78,87 and pentane.64 Orrit and co-workers have also demonstrated PHI imaging of 5 nm Au NPs in Xe at critical temperature and pressure conditions with 2 orders of magnitude improvements in SNR over other solvents.88 Linking all these media are their good photothermal responses wherein large refractive index changes arise due to small temperature variations. This solvent property is

(9)

where q is the fundamental charge and Δf is the lock-in equivalent noise bandwidth. Consequently, the predicted signalto-noise ratio of a PHI measurement is ⎡ R det SNR = ⎢Vref ⎢⎣ qΔf

⎤⎡ π ( α ) ⎤⎡ ⎛ ∂n ⎞ Pabs Pprobe ⎤ m ⎥ ⎥⎢ ⎟ ⎥⎢n ⎜ ⎥⎦⎣ w ⎦⎢⎣ m⎝ ∂T ⎠ Cpλ 2 ⎥⎦

F(Ω)2 + G(Ω)2

(10)

Equation 10 is analogous to the SNR expression seen in ref 64 and leads to identical experimental SNR dependencies. Namely, increasing both the pump and probe powers improves SNR. Furthermore, increasing the lock-in bandwidth or alternatively reducing the probe beam waist increases SNR. Finally, in all cases the SNR is sensitive to the photothermal parameters of the surrounding medium and can be improved through appropriate choice of media. A more detailed description of the influence of the medium’s photothermal parameters can be found in ref 64. 2.1.2. PHI Experimental Setup. Figure 1a shows a schematic of the PHI experiment. In general, two lasers are involved with the first referred to as the “pump” and operating at a wavelength on resonance with one of the analyte’s transitions. The pump (or heating) laser’s intensity is modulated using an acousto optical modulator. Modulation frequencies range from 100 kHz75 to 15 MHz65 to reject electronic and mechanical noise. To acquire PHI-based absorption spectra, a tunable pump is used. Here, acousto optical tunable filter (AOTF)-dispersed supercontinuum white light sources,76 optical parametric oscillators,77 titanium sapphire,78 and even dye lasers72,77−79 have been employed. The second, continuous wave (CW) laser (called the “probe”) operates at a frequency off resonance with the analyte. The probe is often spectrally to the red of the pump. This is not an absolute requirement, and in fact the opposite is true in infrared PHI measurements to be discussed later. Figure 1a shows that both pump and probe beams are made collinear and are focused onto the specimen using a high numerical aperture (NA) microscope objective. The resulting back-reflected or forward-transmitted heterodyned signal is collected. The heterodyned signal results from interference between the scattered probe field and the unscattered backreflected or forward-transmitted probe light. In the former backreflection case, the same high NA microscope objective is used to capture the signal. In the latter, forward-transmission case, a second microscope objective, arranged in a collinear transmission geometry, collects the light. To better isolate the heterodyned signal in the back-reflection geometry, the incident probe is linearly polarized and is completely transmitted through a polarizing beamsplitting cube. The resulting output is then sent through a quarter waveplate to make the polarization circular. Following interaction with the analyte, the back-reflected and heterodyned signal passes through the same quarter waveplate, which makes its polarization linear again. This time, though, the polarization is orthogonal to that of the initial incident probe such that on entering the polarizing beamsplitting cube it is preferentially

characterized by the product

nm Cp

∂nm ∂T

( ) seen in eq 5. Ref 64 refers

to this parameter as the solvent’s “photothermal strength” and provides a compilation of values for common PHI solvents in its Supporting Information. To obtain PHI images, samples are placed atop a piezo stage. The specimen is then raster scanned under pump and probe foci with lock-in data of the PHI signal recorded at each position. A two-dimensional image is then generated point by point. Within the context of single-particle measurements, this is referred to as sample scanning.60,63,65,72,73,75−79,86,87,89 Figure 1b shows a representative 6 × 6 μm PHI image of individual 10 nm gold NPs acquired using sample scanning in the back-reflection collection geometry. 2.1.3. PHI Detection Limits. As shown by Orrit and coworkers, PHI can be used to detect the absorption of a single molecule.60 It is consequently very sensitive. But what are its practical constraints? Here, eq 10 shows that the SNR in a PHI measurement possesses a linear dependence with both pump and probe power. Increasing both thus improves the observed PHI response. Furthermore, since the probe is off resonance with the analyte, very large probe intensities can be used. In practice, limits exist to experimental pump and probe intensities. The literature shows that pump intensities generally range from Iexc = 1 kW/cm260,72to 5 MW/cm2.72 Beyond Iexc = 4 MW/cm275 saturation of the transition and/or sample photodamage becomes likely.61,65,90 Iprobe values are only limited by the sample’s damage threshold with employed values ranging from Iprobe = 20 kW/cm275 to Iprobe ∼ 300 MW/cm2.60 The PHI detection limit is also constrained by the surrounding medium’s photothermal parameters. As described in ref 64, the heat capacity, Cp, as well as sensitivity of the solvent’s refractive index to temperature change (i.e.,

( ∂∂Tn )) directly influence the

PHI signal. These parameters are not immediately tunable. Hence, one is restricted to photothermal media such as glycerol, mineral/silicone oil, or even pentane to obtain the best PHI response in a measurement. Such solvent restrictions may limit measurements especially if there are chemical incompatibilities with the analyte. 2.2. Spatial Modulation Spectroscopy (SMS). The next commonly used technique to measure the extinction of E


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the change in the power of the beam interacting with the analyte can be expressed as

individual nanostructures is called spatial modulation microscopy/spectroscopy.46,66 As the name suggests, the operating principle behind SMS is the spatial displacement of an analyte relative to the focus of a diffraction-limited laser beam. Alternative implementations involve beam scanning. However, since the majority of studies employs sample scanning, the following discussion assumes this approach. When modulating the sample, the particle moves completely in and out of the focus spot at a modulation frequency, f. Figure S3 of the SI shows a conceptual illustration of this. Practical modulation frequencies range from 90 Hz91 up to kHz46,92,93 with sample displacement often achieved using a piezo element. To optimize the observed signal, the modulation amplitude is adjusted to approximately 0.8 of the beam waist.66 The SMS signal arises as a direct consequence of light extinguished by the sample while in the beam. During this time, the analyte attenuates a small amount of incident light, yielding a transmitted intensity I. While out of focus, the observed intensity is Iexc. The intensity difference ΔI = Iexc − I (related to the analyte’s absorption cross section) is then encoded in the light detected by a photodiode. Since the spatial modulation is periodic, ΔI can be extracted using lock-in detection. As with PHI, there are advantages and disadvantages to SMS. Among advantages, SMS is a direct/quantitative technique that immediately yields the absorption cross section of an analyte. In this regard, the physical origin of the SMS signal is clear, unlike PHI where there are competing explanations for the response.72,73 SMS is also a single beam technique and can work in conjunction with single-particle emission measurements since it is equally applicable to emissive and nonemissive nanostructures. It also does not require the specimen to be immersed in a solvent. Among the disadvantages, SMS requires that the specimen’s spatial position (or beam focal point) be modulated. This adds complexity to the experiment. Beyond this, the need to modulate imposes a bandwidth to the measurement which is, in principle, smaller than that of PHI. This and other reasons have led some to search for modulation free schemes as reported in refs 94 and 95. Finally, SMS measures the extinction of an analyte. It is therefore sensitive to scattering so that when studying larger particles significant scattering contributions can exist in their measured SMS response. 2.2.1. Modeling the SMS Signal. In theory, SMS quantitatively assesses an analyte’s absorption efficiency with no prior knowledge about its morphology. In practice, linking the SMS lock-in signal to an analyte’s absorption cross section requires assuming a profile for the focused laser beam. Here, a two-dimensional Gaussian line shape is often invoked84,90,96−98 2

Iexc(x , y) = Ioe−2[(x − xo)

− (y − yo )2 ]/ w 2

Ptrans = Po − σIexc(x , y)

Two approaches now exist for arriving at model expressions for the SMS signal. The first invokes a Taylor expansion of Iexc to account for the effects of the sample spatial modulation. The second uses a Fourier series to do the same thing. In the latter case, what ultimately results are expressions immediately applicable to acquiring SMS extinction spectra. 2.2.1.1. Modeling the SMS Signal through a Taylor Series Expansion. In the first model, Iexc(x,y) in eq 11 is expanded as a Taylor series.66,97 What results are the following 1f and 2f voltages from a lock-in amplifier

− (y − yo )2 ]/ w 2

(14a)

⎛ 2P δ 2 ⎞⎡ 4x 2 ⎤ 2 2 2 VAC,2f (x , y) = −σ ⎜ o 4 ⎟⎢ 2 − 1⎥e−2(x + y )/ w ⎦ ⎝ π w ⎠⎣ w

(14b)

2

VDC(x , y) ≅ R detGdetP0e−2(y − yo )

/w2

(15)

as well as 1f and 2f AC voltages of 2 2 ⎛ 2R G P ⎞ VAC,1f (x , y) = ⎜σ det 2det 0 b1⎟e−2(y − yo ) / w ⎝ ⎠ πw

(16a)

2 2 ⎛ 2R G P ⎞ VAC,2f (x , y) = ⎜σ det 2det 0 a 2⎟e−2(y − yo ) / w ⎝ ⎠ πw

(16b)

where b1 and a2 are 1f and 2f Fourier coefficients, respectively. Equations 15 and 16a are explicitly derived in the SI and are accompanied by simulated 1f and 2f images. Near identical lineshapes to those obtained from the Taylor expansion modeling result. Next, the Fourier series approach readily produces expressions amenable to acquiring SMS extinction spectra. Namely, the 1f RMS amplitude of eq 16a is b1/ 2 . Consequently,

(11)

2P

2

⎛ 8P δ ⎞ 2 2 2 VAC,1f (x , y) = −σ ⎜ o 4 ⎟xe−2(x + y )/ w ⎝ πw ⎠

Equation 14 is explicitly derived in the SI with Figure S4 showing model 1f or 2f SMS images. In the former case, 1f images adopt a derivative profile with a nodal plane at the physical location of the particle. In the latter case, 2f images show a peak at the particle’s position with negative side lobes extending to either side. In practice, 1f or 2f x-axis line scans are taken across the center of experimental images and are fit with eqs 14a or 14b to obtain the analyte’s absorption cross section at a given excitation wavelength.66 2.2.1.2. Modeling the SMS Signal through a Fourier Series Expansion. In a second approach, Iexc (eq 12) is expanded using a Fourier series.98 What results is an expression for the detector voltage, containing the following DC term

where Io = o2 ; Po is the incident power; w is the beam waist; and πw (xo,yo) is the origin. Since the analyte’s spatial position relative to the beam focus is modulated during the SMS measurement (assumed here to occur along the x direction), Iexc can be expressed as Iexc(x , y) = Ioe−2[(x + δ sinΩt − xo)

(13)

1f Vlock ‐in,RMS

VDC

=

2 b1 πw 2

σ (17)

where V1flock‑in,RMS ∝ VAC,1f. By using the lock-in to read only the 1f signal, specifically its maximum RMS reading, V1flock‑in,RMS(xmax), the analyte’s absorption cross section at a given excitation frequency/wavelength is

(12)

In eq 12, δ is the modulation amplitude, and a corresponding displacement is Δx = δ sin Ωt. Implicitly assumed is that the analyte’s size is much smaller than the beam focus. Consequently,

σ(ω) = F

1f 2 Vlock ‐in,RMS(xmax )πw

2 VDCb1

(18) DOI: 10.1021/acs.jpcc.8b00790 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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Figure 2. (a) Schematic of an SMS experiment. (b) Extinction image of an individual d ∼ 40 nm Au NP. Reprinted from ref 98 with permission of AIP Publishing.

Finally, to acquire images, a second piezo stage is used to scan the position of the sample relative to the beam focus. Often this means that the second piezo stage translates both the specimen and the first piezo element used to spatially modulate samples. A two-dimensional image is then acquired in a point by point fashion with the bandwidth of the measurement being limited by mass-dependent mechanical resonances of the second piezo stage. Figure 2b shows a representative 1f SMS image acquired on an individual 40 nm diameter (d) gold NP.98 2.2.3. SMS Detection Limits. SMS has been used to measure the absorption of a variety of single analyte systems. As with PHI, most studies have focused on metal NPs.66,99−113,121−125 Significantly fewer measurements have been conducted on low-dimensional semiconductors.90−93,96,98,114−116,118 Among metal nanostructures, the smallest reported absorption cross section is σ = 2 × 10−13 cm2 for a d = 5 nm Au NP.66 For semiconductors, σmin is σ = 8.0 × 10−14 cm2 for a ∼ 7 × 30 nm CdSe NR.118 Although these detection limits are similar, there are important differences between metal and semiconductor analyte measurements that should be noted. For metals large excitation intensities can be used without damaging samples or saturating their absorption. This is because their transitions are plasmon resonances such that saturation effects only occur at very large intensities.126 The freedom to use larger Iexc values means that shot-noise-limited performance is readily achieved. This is evidenced by the relatively large Iexc values employed in most metal NP SMS measurements. As an example, ref 107 uses an intensity of Iexc = 100 MW/cm2, while refs 111 and 112 use equally large intensities of Iexc ∼ 50 MW/ cm2. For semiconductor nanostructures such as NWs, one deals with an electronic transition influenced by band filling, exciton screening, band gap renormalization, and other many-body effects.127−130 Saturation, especially for cross sections of the order of 10−11 cm2, is therefore problematic. To illustrate, Figure 3 shows the absorption spectrum of a single a ∼ 4.5 nm CdSe NW. Two spectra are shown, one acquired with Iexc = 2.4 kW/ cm2 (dashed red line). A second spectrum has been acquired using Iexc = 320 W/cm2 (solid blue line). The various resolved transitions will be explained shortly. For now, focus on the transition labeled α.

Equation 18 is therefore immediately applicable to measuring SMS extinction spectra. Corresponding 2f expressions can be found in the SI. 2.2.2. SMS Experimental Setup. Figure 2a summarizes the general components of an SMS measurement. In brief, light from a tunable source is focused down to a diffraction-limited spot on a substrate. The analyte’s position is modulated in and out of the focus spot with a piezo stage. Alternatively, the beam’s image at the focus is modulated using a galvo mirror. Transmitted light during the modulation is then detected using an autobalanced photodiode or other detectors. Different light sources have been used in SMS measurements. The literature reveals that SMS has been conducted using incandescent lamps99−102 as well as with diode-pumped solid state,92,96 titanium sapphire,93,103−110 optical parametric oscillator,91,93,104 and supercontinuum laser sources.90,98,107−109,111−113 In all cases, the one desirable feature of a source is the ease of wavelength tunability as this enables one to acquire absorption spectra of individual analyte particles. To date, most SMS spectroscopy measurements have employed supercontinuum sources90,98,111−118 where the supercontinuum is dispersed using either prisms90 or commercial AOTFs.98,114,116−118 Advantages of current, commercial supercontinuum laser/AOTF systems (e.g., Fianium or NKT systems) include near collimated outputs, broad tuning ranges from 450 to 2400 nm, and sizable spectral power densities on the order of ∼5 mW/nm. Although supercontinuum sources are pulsed, they operate with repetition rates between 10 and 80 MHz, making them quasi-continuous. The next key component in most SMS experiments is a piezo element to modulate the analyte’s position relative to the focused laser beam. Numerous experiments use open-loop piezo stages with modulation frequencies between f = 90 and 1500 Hz.90−93,98,115,116,118 The range of f-values that can be used is limited by mechanical resonances of the stage. Galvo mirrors have also been used as described in refs 119 and 120. The goal, in all cases, is to modulate the signal at a frequency above the 1/f noise inherent to all electronics.98 In practice, frequencies above a few hundred Hz are generally sufficient to isolate the SMS signal of interest. G


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In what follows, we highlight single-particle extinction/ absorption results acquired on low-dimensional CdSe since most work has been conducted with this material. 3.1. System: CdSe. Historically, CdSe was one of the first low-dimensional semiconductors to be produced in high quality.3,4,132,133 Although its most popular form today is as colloidal QDs, near equivalent quality NRs,134 NWs,135−139 nanosheets,140−143 and other low-dimensional morphologies such as tetrapods144 have been made. It is consequently a wellstudied system, with over three decades of work on synthesizing it145,146 as well as on characterizing it optically and electrically.49,53,54,57,58,75,134−139,146−168, We now focus on what has been learned by conducting single CdSe nanostructure extinction measurements. Our discussion begins with colloidal CdSe NWs, a material first synthesized in good quality between 2003 and 2004.135−137 These early colloidal syntheses produced NWs with diameters between 5 and 10 nm, dimensions smaller than twice CdSe’s bulk exciton Bohr radius (aB ∼ 5.6 nm147). Consequently, quantum confinement effects were seen in ensemble linear absorption spectra135,154 as progressive blueshifts of both the band-edge absorption and emission with decreasing NW radius, a. Figure 4 plots acquired (ensemble) size-dependent linear absorption spectra from ref 135 that illustrate these apparent size-dependent effects.

Figure 3. Iexc-dependent extinction spectra of an individual a ∼ 4.5 nm CdSe NW. Solid blue (dashed red) line illustrates the low (high) intensity spectrum. Adapted with permission from ref 90. Copyright 2011 American Chemical Society.

In Figure 3 the low Iexc spectrum exhibits a clearly resolved band-edge excitonic (α) feature. By contrast, this transition is washed out in the high Iexc spectrum. The data thus illustrate that individual NR/NW optical transitions are highly susceptible to saturation/many-body effects.130 Accurate NR/NW extinction measurements therefore require using low excitation intensities on the order of ∼100 W/cm2. In some cases, intensities as low as ∼50 W/cm2 have been needed.115 These restrictions necessitate the use of sensitive autobalanced photodetection. Here a commercial autobalanced photodiode (Newport, Nirvana 2007)90,98,114,115 has often been used. We have also developed116,118 a custom autobalanced photodiode capable of operating at intensities up to Iexc ∼ 4 kW/cm2 with near shot noise limited performance. A schematic, parts list, and performance comparison to the Nirvana 2007 can be found in the SI.

3. WHAT HAVE WE LEARNED? We now describe the physics gleaned from individual semiconductor nanostructure extinction measurements. Recall that motivating these studies has been the desire to acquire knowledge difficult to obtain otherwise using either ensemble or emission-based single-particle measurements. Circumstances where single-particle extinction measurements offer superior alternatives to traditional spectroscopies include: 1. Cases where the material of interest is nonemissive. 2. Cases where existing syntheses yield ensembles with large size distributions and/or compositional or structurally induced electronic disorder. 3. Cases where the desire is to go beyond emission studies to learn more about the underlying electronic structure of a system, as seen through clearly resolved excited-state progressions. In this regard, emission-based measurements monitor a material’s response after photogenerated carriers have thermalized to the semiconductor band edge. Consequently, the information gathered from single-particle emission measurements primarily centers on the emitting state,131 its interaction with near-edge defect states,38 and Coulombic interactions between thermalized carriers.18,49 Information about the underlying electronic structure of the material, something directly encoded in the absorption, is less forthcoming and is at best inferred indirectly.

Figure 4. Size-dependent (a) absorption and (b) emission spectra of CdSe NWs with diameters (radii) between d = 5.2 and 18.8 nm (a = 2.6−9.4 nm). Reprinted with permission from ref 135. Copyright 2003 American Chemical Society.

Although the CdSe NW syntheses developed at the time yielded crystalline wires with lengths exceeding 1 μm, resulting radial size distributions were on the order of 20%.135−137 This should be contrasted to the 5% standard deviations typical of comparable CdSe QD ensembles.3 Consequently, Figure 4 shows inhomogeneously broadened absorption spectra with unresolved shoulders only suggestive of discrete transitions at higher energies. Beyond the inability to resolve higher energy electronic transitions, the inhomogeneously broadened spectra masked a larger question with the acquired data. Namely, they did not immediately resemble the van Hove joint density of states expected for a 1D nanostructure.169 In fact, CdSe NW absorption spectra more closely resembled those of colloidal CdSe QDs albeit with band edges further to the red and with more apparent spectral broadening. The obvious question then was why NW spectra so closely resembled those of QDs? H


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The Journal of Physical Chemistry C 3.1.1. Single CdSe NW Extinction Spectra. Figure 5 now shows the (room temperature) SMS-acquired, single NW linear

Figure 6. (a) Extinction and emission spectra for three NW ensembles with corresponding average radii of ⟨a⟩ ∼ 2.5, ∼3.7, and ∼5.0 nm. (b) Size-dependent progression of single CdSe NW extinction spectra. Inset: corresponding emission peak position as a function of a. Adapted with permission from ref 90. Copyright 2011 American Chemical Society.

Figure 5. Extinction spectrum of a single a ∼ 3.7 nm CdSe NW (open circles). Corresponding emission spectrum shown by the dash−double dotted curve. An associated ensemble extinction spectrum (⟨a⟩ ∼ 2.5 nm) shown (solid line) along with the emission spectrum (dasheddotted line). Adapted with permission from ref 90. Copyright 2011 American Chemical Society.

absorption spectrum of an a ∼ 3.7 nm CdSe NW.90 In the experiment, light polarized parallel to the NW long axis has been used to maximize the observed extinction signal. The wire’s emission spectrum is also shown. Corresponding ensemble data are reproduced above the single wire data where a comparison between the two reveals dramatic improvements to the single wire spectral resolution. At least three transitions, labeled α, γ, and δ, are now resolved, whereas they were only hinted at before in the ensemble spectrum. Furthermore, the single wire emission line width narrows from ∼63 to ∼31 meV. Of note in Figure 5 is that the absorption y-axis is quantitative. As discussed earlier when describing SMS microscopy/spectroscopy, quantitative absorption efficiencies are obtained using this technique. Consequently, the NW in Figure 5 has a peak α absorption cross section of σ∥ ∼ 1.4 × 10−11 cm2/μm. Further to the blue at δ, its peak absorption cross section increases to σ∥ ∼ 3.2 × 10−11 cm2/μm. 3.1.2. Size-Dependent Single CdSe NW Optical Transitions. By subsequently acquiring individual spectra from within three CdSe NW ensembles with average radii of ⟨a⟩ = 5.0, 3.7, and 2.5 nm (Figure 6a), the size-dependent linear absorption of CdSe NWs has been revealed for the first time.90 This is shown in Figure 6b. Clearly evident are blueshifts of the absorption edge with decreasing size. An additional observation is that whereas three discrete transitions were resolved for smaller sizes, up to five transitions become apparent in a ∼ 5.0 nm NWs. These latter transitions have likewise been labeled with Greek letters (α, β, γ, δ, ζ). Assignment of the various transitions will be discussed shortly. 3.1.3. Origin of CdSe NW Optical Transitions. The spectrally resolved transitions in Figure 6b can now be studied to reveal the underlying electronic structure of confined CdSe NWs. To this end, Figure 7 plots α (open black circles), β (gray crosses), γ (open blue inverted triangles), δ (open green squares), and ζ

Figure 7. NW excited-state transition energies plotted relative to α. Solid (dotted) lines represent the strongest (weakest) predicted transitions from an effective mass model. Dashed lines denote transitions with intermediate strength. Solid black × symbols represent transitions extracted from current and prior ensemble transient absorption data.164 Adapted with permission from ref 90. Copyright 2011 American Chemical Society.

(open dark green stars) energies, all as functions of size, where specific transition energies have been extracted by fitting single NW spectra.90 As done previously for CdSe QDs,147 peak transition energies are plotted relative to the energy of the lowest excited state (α) since its energy is better known than the corresponding NW radius. The α transition therefore appears as a horizontal line across NW sizes with β, γ, δ, and ζ plotted above it in sequence and each monotonically increasing in energy with decreasing NW size. Corresponding band-edge emission energies are plotted below α as open red triangles. Although clear size-dependent trends exist in the CdSe NW electronic structure, the identity and origin of α, β, γ, δ, and ζ are unknown. Like CdSe QDs, however, such highly resolved spectra and apparent size dependencies provide an opportunity to reveal the electronic structure of NWs using suitable theoretical modeling. In this regard, an effective mass model developed in the late 80s151,170 was instrumental in providing the first I


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of an explicit accounting of Coulombic terms (dashed line), the model and experiment agree poorly. Only when calculated direct and indirect Coulomb interactions are considered does the model become semiquantitative, as seen through the better overlap of the solid theory line with experiment. Within the context of 1D nanomaterials, such direct and indirect Coulomb interactions arise due to so-called dielectric contrast and dielectric confinement effects.114,176 The former describes how the direct electron−hole Coulomb interaction is influenced by the surrounding dielectric environment where it is generally assumed that εm < εs. Consequently, depending on the degree of contrast, electron−hole binding energies vary, being enhanced (suppressed) with increasing (decreasing) dielectric contrast. In general, dielectric contrast leads to increased electron and hole binding energies.176 The latter dielectric confinement refers to interactions between an electron (or hole) and its mirror image charges in the surrounding medium. Dielectric confinement generally increases electron and hole energies in the NW. This counteracts the energy-minimizing effects of dielectric contrast. The most important realization to arise from the NW modeling in Figures 7 and 8 is that dielectric contrast dominates dielectric confinement. The net effect is a substantial increase of electron and hole binding energies in NWs. For the CdSe wires described here, binding energies as large as ∼300 meV have been predicted.90,176 This realization has, in turn, explained our original question for why NW spectra seen in Figure 4 so closely resemble those of colloidal CdSe QDs. The large binding energies in confined CdSe NWs mean that the extinction spectra being measured are excitonic, not free carrier, in nature. Consequently, one does not expect to see the van Hove lineshapes, predicted for a textbook 1D material.169 Instead, discrete excitonic transitions like those in colloidal QDs arise. That room-temperature 1D excitons exist has also been realized in the SWCNT community.91 A final point on terminology, NW excitons are of the formal Coulombic variety, whereas QD “excitons” are in name only since they simply arise from the spatial confinement of carriers. 3.1.4. Single-Wire Polarization Anisotropies. Beyond explaining observed transitions in single NW extinction spectra, the effective mass model yields a semiquantitative explanation of their polarization sensitivities. To illustrate, Figure 9 shows the linear absorption spectrum of an a ∼ 4.0 nm CdSe NW acquired

quantitative insights into the fundamental optical response of CdSe nanocrystals. Its successful application to explaining their experimentally observed size dependencies, transition energy differences, and avoided crossings has since made CdSe QDs the hydrogen atom of modern nanoscience.20 For the single NW extinction results in Figure 7, extensions of this effective mass model to NRs and NWs have since been used to reveal the origin of observed size-dependent electronic transitions.171−175 For CdSe, the resulting NR/NW effective mass model is described in detail in refs 90, 114, 115, and 173 These references contain explicit expressions for resulting electron and hole wave functions, evaluated by assuming that the wires adopt CdSe’s zincblende (cubic) lattice. In brief, what results are confined electron and hole states, described by the symmetry labels Σ, Π, and Δ (akin to the atomic s, p, d labels used for CdSe QDs147), and with principal quantum numbers 1, 2, and 3 etc. to denote levels with increasing energy. Angular momentum subscripts indicate the angular momentum projection of electron and hole states onto the NW c-axis where, in the former case, the subscript “e” denotes 1/2. By evaluating electron and hole wave function overlap integrals, optical transitions between confined NW electron and hole levels are found. Figure 7 shows these transitions where overall transition strengths are denoted using solid (strong) and dashed (weak) lines. As with CdSe QDs, there is remarkably good qualitative agreement between experiment and theory. This extends beyond reproducing both the overall energies, and trends of the transitions as transition strengths, notably for β, have been captured as well. Each transition in Figure 7 is subsequently given a symmetry label. The band-edge α transition is assigned to 1Σ1/21Σe, while β is assigned to either 1Σ3/21Σe or 2Σ1/21Σe. Above them, γ is assigned to 1Π1/21Πe, while a variety of possible transitions account for δ and ζ. It is therefore apparent that the effective mass model, in its current form, captures the general electronic structure of CdSe NWs. A key reason for the model’s success is the explicit inclusion of theoretically derived Coulombic interactions between carriers and their corresponding image charges in the local environment. To illustrate, Figure 8 shows the model’s prediction of the NW size-dependent (ensemble) α transition energy (solid line) in comparison to the experimental data (symbols). In the absence

Figure 9. Parallel (blue open circles) and perpendicular (red open triangles) extinction spectra of a single a ∼ 4.0 nm CdSe NW. The perpendicular spectrum is scaled by a factor of 3 for clarity. Adapted with permission from ref 115. Copyright 2012 American Chemical Society.

Figure 8. Comparison of ensemble α transition energies tabulated from the literature (○ [ref 135], △ [ref 155], □ [ref 154]) to bare (dashed line) and Coulomb-corrected (solid line) 1Σ1/21Σe transitions. J


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excited-state electronic structure of confined CdSe NWs. It does not, however, explain their emission. In this regard, the theory predicts no energy difference between α and the associated bandedge emission energy. Figures 5, 7, and 11a, however, reveal that experimental peak emission energies always lie further to the red of α. Energy differences range from 20 to 60 meV and generally increase with decreasing NW radius.117 Figure 11b provides an explicit illustration where experimental absorption/emission energy differences have been plotted versus size. The effective mass model, as developed, thus does not fully capture the electronic structure of CdSe NWs. Such band-edge Stokes shifts39 have been seen before within the context of colloidal CdSe QDs and were originally attributed to emission arising from surface states.40,132 Over subsequent years, refinements in both experiment and theory have led to the general conclusion that such shifts are intrinsic to the particles and that their origin can be explained within the context of intrinsic QD electronic states.39 For NWs, we have made analogous refinements to the effective mass model used in Figure 7 to better capture the underlying electronic structure of CdSe NWs.117 Here an important consideration has been the inclusion of crystal field splitting within the Luttinger Hamiltonian used to describe CdSe’s valence band.177 Inclusion of crystal field splitting stems from the fact that CdSe NWs do not fully adopt their cubic, zincblende lattice in practice. Instead, high-magnification transmission electron microscopy measurements reveal that colloidal CdSe NWs possess both zincblende and (hexagonal) wurtzite phases.137,158 Consequently, an intrinsic asymmetry exists in the effective crystal lattice of the wires which should be accounted for by incorporating crystal field splitting terms into CdSe’s valence band Hamiltonian. Figure 12 now shows results of the refined (wurtzite) effective mass model. Several immediate differences emerge. First, β is assigned to either 2Σ1/21Σe or 2Σ3/21Σe instead of 1Σ3/21Σe. The assignment is supported by the better agreement between theoretical α and β anisotropy values to those seen experimentally.115 Next, overall energies for γ (i.e., 1Π1/2LH1Πe, 1Π3/21Πe, and 1Π1/2HH1Πe) are in closer agreement to experiment. Finally, 1Σ3/21Σe extends below α for a range of NW sizes. This is seen by the dashed line extending below 1Σ1/21Σe in Figure 12. An evaluation of its transition strength also reveals that 1Σ3/21Σe is weak under parallel polarized excitation. Consequently, the existence of a relatively dark (1Σ3/21Σe) state below an optically strong 1Σ1/21Σe transition, responsible for the absorption, suggests an intrinsic origin for the single wire Stokes shifts seen in Figures 7 and 11. Figure 11b now plots the predicted intrinsic effect mass model Stokes shift (solid black line) across size and compares its magnitude to the experimental data. Two things stand out. First, the predicted magnitude is smaller than that seen experimentally. Next, below a ∼ 3.0 nm, no Stokes shift is predicted. In contrast, the data show large Stokes shifts between 10 and 60 meV in this region. The refined model thus does not fully account for the data. To address this discrepancy, we have therefore additionally considered the role of potential energy fluctuations that could lead to exciton localization at the NW band edge.117 In principle, such potential energy fluctuations could arise from incomplete surface passivation of the colloidal CdSe NWs. Local minima for exciton localization would then result in additional energy shifts that would add to the intrinsic shifts predicted by the refined effective mass model.

using light polarized parallel and perpendicular to the NW long axis. It is apparent that overall absorption cross sections as well as relative α and β transition strengths change with incident polarization. For the NW shown, α has a parallel (perpendicular) absorption cross section of σ∥ ∼ 4.4 × 10−11 cm2/μm (σ⊥ ∼ 4.0 × 10−12 cm2/μm). Observed order-of-magnitude differences in cross-section are immediately predicted and rationalized by corresponding order-of-magnitude differences in local field factor-derived absorption cross sections (eq 3a). More relevantly, the effective mass model explains why α and β transition strengths reverse between parallel and perpendicularly polarized excitation conditions.115 This stems from changes to the transition dipole moment where in the perpendicularly polarized case Bloch parts of constituent electron/hole wave functions are better coupled by the incident light than in the corresponding parallel-polarized case. What results are polarization-specific transition selection rules, stemming from the intrinsic NW electronic structure. Thus, beyond extrinsic, localfield factor-induced polarization anisotropies (i.e., eq 4), intrinsic quantum mechanical selection rules emerge when the NW electronic structure is explicitly considered. Details of these calculations can be found in ref 115. By explicitly accounting for local field factor differences between parallel and perpendicularly polarized excitation conditions, selection rule contributions to the overall polarization anisotropy, ρ, of the NWs can be revealed. This is illustrated in Figure 10 which shows experimentally averaged single a = 2.5−5.0 nm ρ-values for α, β, γ, η, and δ optical transitions.

Figure 10. Average anisotropy ρ-values for several single CdSe NW traces. Symbols correspond to 1Σ1/21Σe (α, red triangle), 1Σ3/21Σe (β, black star), 1Π1/2HH1Πe (γ, blue circle), 1Π5/21Πe (η, purple square), and 1Π1/2LH1Πe (δ, pink diamond) transitions. Inlayed across the data is the anisotropy spectrum of a single a ∼ 4.6 nm NW (black dots/black dashed line). Reprinted with permission from ref 115. Copyright 2012 American Chemical Society.

Although not quantitative, the effective mass model qualitatively captures the observed state-specific polarization anisotropies. Specifically, for an a = 4.0 nm NW the model shows that the α transition has an anisotropy value of ρ = 0.961. For β, ρ = −1.0. A clear reversal of α and β transition strengths is therefore predicted under perpendicularly polarized excitation conditions. This is exactly what is seen in the data. 3.1.5. Existence of a Stokes Shift. Figures 7, 8, and 10 show that the effective mass model successfully captures the general K


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Figure 11. (a) Intrawire absorption (open blue circles) and emission (solid red lines) spectra taken from three different positions on a single a ∼ 3.9 nm CdSe NW. Inset: Associated Stokes shift for each position. (b) Size-dependent single CdSe NW Stokes shifts (open red circles) and model-predicted shifts, which contain intrinsic (solid black line) and extrinsic (N = 1, 2, 3) contributions (dashed blue lines). Reprinted figure with permission from ref 117. Copyright 2015 by the American Physical Society.

Several possible origins exist for these intrawire energy differences. In one scenario, variations in the dielectric environment (ε) surrounding the NW lead to fluctuations of the exciton binding energy.159 In this regard, colloidal NWs are generally coated with organic surfactant molecules to electronically passivate their surface defects. ε-Variations could stem from uneven surface coverage.135,137 Alternatively, observed absorption/emission energy differences stem from intrawire radius variations. Attesting to this latter possibility are high-resolution TEM images that reveal up to 5% fluctuations in a across the length of individual NWs.137,179 By comparing experimental and effective mass modelpredicted energy deviations, it has been concluded that intrawire size variations are the most likely explanation for observed intrawire absorption/emission spectral shifts. Predicted energy changes due to dielectric contrast fluctuations are much too small to explain the data. Additional details of these studies can be found in ref 116. 3.1.7. Evolution of Dimensionality: Toward Quantum Dots. Finally, what about the results of extinction measurements conducted on other low-dimensional CdSe nanostructures? Here, single nanostructure extinction measurements could address an important question: How does a system’s electronic structure evolve with dimensionality? To illustrate, the abovediscussed CdSe NWs are nominally 1D in terms of their carrier confinement. At the other end of the dimensionality spectrum are zero-dimensional (0D) QDs of the same radius where confinement occurs across all three dimensions. One asks how CdSe’s electronic structure evolves as a NW transitions into a QD. Figure 14 illustrates this transition across dimensionality where Figure 14a shows size-dependent, ensemble (1Σ1/21Σe) absorption energies for CdSe NWs and compares them to the lowest-energy 1S3/21Se transition of colloidal CdSe QDs. Evidently, a sizable energy gap exists between the two curves. From theoretical modeling,90,114,117,147 the origin of this energy difference must stem from changes to the CdSe NW length/ aspect ratio as well as from varying dielectric contrast/ confinement contributions to overall electron/hole energies. This is illustrated schematically in Figure 14b. Few studies have probed this dimensionality-dependent evolution of a semiconductor’s electronic structure.135,180,181

Figure 12. Wurtzite effective mass model for CdSe NWs. Reprinted from ref 116 with permission of AIP Publishing.

Reference 117 describes how such potential energy fluctuations are described within the context of a model first applied to rationalize Stokes shifts in amorphous Si.178 Resulting (intrinsic) Stokes shift predictions of the refined effective mass model plus Stokes shift predictions that include extrinsic potential energy fluctuations (dashed blue lines) are shown in Figure 11b. There is significantly improved agreement between theory and experiment. Of note is that the model rationalizes why Stokes shifts exist in the narrowest NWs despite the wurtzite effective mass model predicting no shift in this size regime. 3.1.6. Intrawire Absorption and Emission Energy Variations. The SMS-based single NW microscopy/spectroscopy described above also enables intrawire absorption and emission measurements.116 Figure 13a illustrates this by showing linear absorption and emission spectra acquired from a single a = 2.4 nm CdSe NW. Data have been taken from five locations, each separated by ∼1 μm. Figures 13b and 13c are zoomed in plots, revealing up to ∼50 meV absorption/emission energy variations along the NW length. L


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Figure 13. (a) Absorption (open blue circles) and emission (solid red curves) spectra obtained from five different sections of a single a ∼ 2.4 nm CdSe NW. Corresponding Stokes shifts shown alongside each spectrum. (b) Variations in band-edge absorption for positions marked X on the adjacent absorption image. (c) Band-edge emission energy variations for positions marked X in the adjacent emission image. In all cases, vertical dashed lines serve as guides to the eye. Reprinted from ref 116 with permission of AIP Publishing.

Figure 14. (a) Experimental ensemble room-temperature QD and NW band-edge transition energies along with theoretical predictions of a 6-band effective mass model. (b) Schematic of aspect-ratio-dependent single NR band-edge transition energies across the 0D/1D divide.

Average extracted α energies of several single NRs/NWs, with dimensions similar to those mentioned above, show similar blueshifts. These average energies have then been compared to theoretical effective mass predictions of the two lowest NR/NW (1Σ1/21Σe and 1Σ3/21Σe) exciton states. Although the theory is only semiquantitative, it captures the general trend of increasing band-edge energies with decreasing aspect ratio.118 More relevantly, the model shows that for a ∼ 3.4 nm NR and NW dimensionality changes at an aspect ratio of ∼1.3 where the corresponding physical length is b = 8.5 nm. This differs significantly from the >30 nm length previously reported using results from ensemble emission-based measurements.135 In effect, the model and derived critical length indicate that dimensionality changes when b becomes approximately twice the bulk exciton Bohr radius (i.e., ∼2aB) of the semiconductor. Additional details of the model can be found in ref 118.

The ones that do have mostly monitored changes to ensemble band-edge emission energies. An exception is the Katz study which conducts scanning tunneling spectroscopy measurements of individual NRs.180 Consequently, in most cases acquired data are complicated by inhomogeneous broadening, exciton fine structure,149,173,182 size-dependent Stokes shifts,116,117 and aspect-ratio-dependent Coulombic interactions.118,173 To avoid these complications, direct, SMS-based extinction measurements have been conducted on individual CdSe NWs and equiradii NRs.118 By acquiring aspect-ratio-dependent absorption spectra, we have followed how the band-edge α energy changes. Then through application of the refined NR/ NW effective mass model, we have followed how these energy changes arise from the evolution of dielectric contrast/ confinement across dimensionality. Figure 15 shows representative extinction spectra of individual a ∼ 3.4 nm CdSe NWs and equiradii NRs with lengths of b ∼ 160 nm and b ∼ 30 nm. Apart from the structured absorption seen in all cases, the most important observation is that the band-edge absorption exhibits a progressive blueshift with decreasing length. This qualitatively agrees with what is expected from Figure 14.

4. FUTURE DIRECTIONS: INFRARED PHI MEASUREMENTS What next? One future avenue of single nanostructure extinction microscopy/spectroscopy entails extending extinction measurements to the infrared (IR). Motivating this is the desire to go M


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absorption measurements therefore represent an unexplored frontier in single-particle microscopy and spectroscopy. Toward this goal, we have recently developed an infrared absorption technique, called infrared photothermal heterodyne imaging (IR-PHI).89,183−190 IR-PHI is based on the PHI technique described earlier. A major technical difference is the use of a tunable OPO-based infrared pump laser in tandem with a visible probe laser. Consequently, the spatial resolution of IRPHI is limited by the visiblenot inf rareddiffraction limit. Figure 16 is a schematic of the IR-PHI technique.

Figure 16. Schematic of the IR-PHI experiment. Adapted from ref 89. Further permissions related to ref 89 should be directed to the American Chemical Society.

IR-PHI has been successfully implemented for the imaging and spectroscopy of single cells,89,183,184,188,189 liquid crystals,186,187 single polymeric beads,89,183,188 and perovskite thin films.190 Furthermore, instead of focusing both the pump and probe onto the specimen using the same objective (for IR measurements this would mean using a lower NA reflective Cassegrain objective),183−189 we employ a counter propagating pump/probe configuration89,190 to separately focus the probe using a high NA (refractive) objective and have achieved spatial resolutions of 300 nm, the highest to date.89 4.1. Mixed Cation Hybrid Perovskites. Although IR-PHI is only at its inception, we have begun applying it to semiconductor systems. For example, we have studied mixed cation, hybrid perovskite thin films of composition FAxMA1−xPbI3 [formamidinium (FA = HC(NH2)2+), methyalammonium (MA = CH3NH3+)].190 These are important systems for the development of next-generation solar cells191 due to their exceptional optical and electrical properties, which include optimal direct band gaps and long carrier diffusion lengths.192−195 At present, mixed cation perovskite power conversion efficiencies (PCEs) are still far from their theoretical Shockley−Quiesser limit.196 Addressing their current PCE limitations is therefore of significant interest. Here it is known that variations in MA and FA stoichiometry impact observed

Figure 15. Absorption spectrum of an individual a ∼ 3.4 nm CdSe (a) NW, (b) NR with b ∼ 160 and (c) NR with b ∼ 30 nm. Reprinted from ref 118.

beyond interband transitions to measure nanostructure intraband transitions, infrared resonances of plasmonic materials, and vibrational transitions of chemically disordered semiconductor nanostructures. In the latter case, the mid-IR “fingerprint” region of the spectrum possesses characteristic vibrational transitions, which can be used to identify and characterize chemical heterogeneity at the nanoscale. Single nanostructure infrared N


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Figure 17. Normalized (a) MA- and (b) FA-specific IR-PHI maps for 10 × 10 μm areas in FA0.1MA0.9PbI3 (x = 0.1). (c) Corresponding MA/FA ratio map. (d) IR-PHI spectra obtained at the labeled points in (c). All spectra normalized to the 1465 cm−1 MA transition. Adapted with permission from ref 190. Copyright 2018 American Chemical Society.

band gaps197,198 and, in turn, device performance through variations of the local open-circuit voltage.199 Until now, no study has spatially resolved cation-specific compositions in mixed cation perovskite thin films, quantified their nonuniformities, and directly linked their existence to local photophysical variations. Figures 17a and 17b show 10 × 10 μm absorption images of a ∼400 nm thick FA0.1MA0.9PbI3 (x = 0.1) film acquired at 1465 and 1710 cm−1. The first frequency is on resonance with MA’s symmetric NH3+ bend and the second probes FA’s CN stretch. In either case, apparent intensity differences in the images arise from intrafilm thickness variations. Figure 17c is a ratio map created by dividing the above MA and FA images. Clearly evident is heterogeneity within the film’s cation distribution. Observed compositional differences are supported by infrared absorption spectra acquired at two points within the images (labeled) (Figure 17d). These IR-PHI measurements establish that MA/FA heterogeneities exist in FAxMA1−xPbI3 thin films. They are not stoichiometrically uniform. The degree of heterogeneity is significant with up to 3-fold differences from the ideal stoichiometric value. Furthermore, corresponding band gap variations exist and correlate positively with the observed local cation stoichiometry (not shown).190 The developed IR-PHI technique is thus capable of revealing local chemical disorder in a semiconductor. At this point, an open question about IR-PHI is its underlying operating mechanism. Whereas virtually all PHI measurements to date have used a solvent to generate photothermal contrast via local heating,64,72 the above IR-PHI measurements have been conducted under solvent-free conditions. Preliminary estimates we have made suggest that local heating does not lead to any photothermal contrast.89 Instead, our computational simulations suggest that the IR-PHI contrast mechanism originates from absorption and subsequent relaxation-induced changes to the material’s scattering cross section.89 Consequently, if true this would represent a new IR contrast mechanism. Further work, though, is required to establish and verify this model.

means to understand semiconductor nanostructure photophysics. Although inherently difficult, a number of strategies have been developed to overcome the practical issues of measuring the ∼0.0001% extinction of light by an analyte. PHI and SMS are two of the most popular of these newly developed techniques, and through their use, the physics of individual semiconductor nanostructures such as 1D CdSe NWs and NRs have been explored at an unprecedented level. These achievements together with the development of new single-particle extinction methodologies such as IR-PHI portend future successes in revealing the detailed physics of nanostructures beyond both ensemble averages and corresponding single-particle, emissionbased insights.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.8b00790. Derivation of extinction cross-section expressions. Compilation of semiconductor QD absorption cross sections. Compilation of semiconductor NR and NW absorption cross sections. Derivation of the PHI signal. Plot of experimental Au NP extinction cross sections along with Mie theory predictions. Conceptual illustration of SMS. Derivation of the SMS signal via a Taylor series expansion. Simulated SMS images using Taylor series-derived expressions. Derivation of the SMS signal via a Fourier series expansion. Simulated SMS images using Fourier series-derived expressions. Expressions for SMS extinction spectra. Schematic of a custom autobalanced photodetector and associated parts list. Comparative performance plots of a custom vs commercial Nirvana 2007 autobalanced photodetector (PDF)

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5. CONCLUSIONS AND OUTLOOK There has been 30 years of emission-based, single-particle microscopy and spectroscopy since Moerner’s seminal study. While widely used and highly successful in revealing the properties of matter hidden by ensemble averages, emissionbased, single-particle measurements are not universally applicable to all nanosystems. In addition, they offer only limited insight into the intrinsic electronic structure of nanomaterials. Single-particle extinction measurements therefore offer a better

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Masaru Kuno: 0000-0003-4210-8514 Notes

The authors declare no competing financial interest. O


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The Journal of Physical Chemistry C Biographies

Masaru Kuno did his undergraduate studies in chemistry at Washington University in St. Louis. He then pursued graduate studies at MIT with Moungi Bawendi between 1993 and 1998. This was followed by a NRC postdoctoral stint with David Nesbitt and Alan Gallagher at JILA/ NIST/University of Colorado, Boulder. Since 2003, he has worked on the synthesis and optical characterization of low-dimensional materials at Notre Dame.

Rusha Chatterjee received her B.Sc. (Honors) in chemistry from Delhi University in 2009. She completed her masters in chemistry at IIT Delhi in 2011. She is currently completing a Ph.D. in physical chemistry at Notre Dame. Her research interests include single-particle microscopies and spectroscopies of nanomaterials.

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ACKNOWLEDGMENTS M. K. thanks his current and former group members as well as collaborators for the work described herein. M. K. also thanks the National Science Foundation (CHE1208091 and CHE1563528) for financial support. M. K. additionally thanks the Army Research Office for DURIP Award W911NF1410604, which was used to purchase the IR OPO system.

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REFERENCES

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