Intuitive expectations, based on a framework of non-interacting particles, suggest that reducing the length of a nanoscale system’s confining potential gradually induces the emergence of quantum confinement1,2,3 effects in its spectroscopic response. This crossover is expected to be smooth, without the appearance of a critical length. However, the presence of interactions fundamentally alters this picture of dimensional crossover in many-particle systems. Several low-dimensional systems, in fact, show interaction-induced phases of matter absent in higher dimensions. They include fractional quantum Hall states in two dimensions4, Tomonaga–Luttinger liquids in one-dimension (1D)5 and Kondo effects in zero-dimensional (0D) quantum dots (QDs)6. Here we demonstrate that analogous many-particle interactions dictate the 1D-to-0D dimensional crossover in low-dimensional semiconductors (Fig. 1). Our spectroscopic measurements and supporting theoretical calculations indicate that this transition occurs at a critical length determined by the delicate balance between carrier confinement and electrostatic interaction energies.

Figure 1: One-dimensional to zero-dimensional crossover of a semiconductor nanowire’s electronic structure.
figure 1

Top row: structural evolution of a nanowire into a quantum dot. Middle row: corresponding evolution of nanowire and nanorod electron and hole wavefunctions. Bottom row: a plot depicting the interplay between aspect ratio-dependent carrier confinement, Ek(b) and dielectric contrast/dielectric confinement electrostatic energies, |U|.

While past studies have attempted to investigate 100 meV bandgap (Eg) differences between 1D and 0D CdSe nanostructures by probing nanorods (NRs) of controlled length7,8,9 no consensus exists as to when a 1D object exhibits 0D character7,9. Although general trends have been gleaned, they are largely the result of ensemble measurements, which suffer from inhomogeneous broadening due to inherent size-and-shape distributions10,11,12. Importantly, they rely on photoluminescence and tunnelling spectroscopies7,9. However, dark/bright exciton splitting13, trap-induced Stokes shifts14 and enhanced exciton binding energies15 can significantly alter perceived bandgaps, complicating accurate Eg estimates.

Here we directly probe this 1D-to-0D transition using single nanowire (NW)/NR absorption spectroscopy to eliminate ambiguities as to the actual evolution of electronic structure across dimensionality. This is accomplished using spatial modulation microscopy12,14,16,17, which entails modulating a NW/NR’s position in and out of a focused laser beam. Subsequent lock-in detection measures the transferred transmitted laser power modulation, which relates to individual nanostructure extinction cross-sections (Supplementary Note 1 and Supplementary Fig. 1)17. As obtained experimental data are found to be in good agreement with an effective mass model that explicitly includes dielectric contrast/confinement effects, and, consequently, provides a critical length at which 1D nanostructures become 0D.


Spectroscopic observations

Figure 2a shows the absorption spectrum of an individual CdSe NW from a diameter (d) 6.8±1.2 nm (b≥5 μm) ensemble. Figure 2b,c illustrate corresponding absorption spectra of single CdSe NRs from d=6.7±1.1 nm (length; b=160±55 nm) and d=6.8±0.7 nm (b=30.4±2.6 nm) ensembles (Supplementary Note 2 and Supplementary Fig. 2). Three to four transitions (labelled α, β, γ, and δ) are apparent in each spectrum and are excitonic in nature as predicted by a model which explicitly accounts for both spatial confinement and enhanced electrostatic interactions in NRs15. These states can be explicitly linked to analogous α, β, γ, and δ transitions in individual CdSe NWs12,14,16. The data in Fig. 2 therefore represent the first direct measurements of single CdSe NR absorption spectra.

Figure 2: Absorption spectra of individual CdSe nanowires and nanorods.
figure 2

Absorption spectrum of an individual (a) nanowire from a d=6.8±1.2 nm (b≥5 μm) ensemble; (b) nanorod from a d=6.7±1.1 nm (b=160±55 nm) ensemble; and (c) nanorod from a d=6.8±0.7 nm (b=30.4±2.6 nm) ensemble. Blue open symbols represent measured extinction values plotted as a function of wavelength (λ). Corresponding peak α extinction cross-sections are σext 7 × 10−13 cm2 (b≥5 μm), σext2 × 10−13 cm2 (b160 nm), and σext8 × 10−14 cm2 (b30 nm). Spectra are fit to a sum of Gaussians (black dashed line) from where individual transitions (grey dashed lines) are extracted. Each sample’s sizing histogram is inset in (ac). S.d. reported.

Most notable is an overall 30 meV average α blueshift in b30 nm NRs (Fig. 2c) relative to those of longer, equi-diameter particles (Fig. 2a,b) (b30 nm: α=1.904±0.025 eV; b160 nm, α=1.872±0.016 eV; NW: α=1.874±0.022 eV). In fact, probing 25 individual b30 nm NRs reveals α blueshifts up to 64 meV. Figure 3 illustrates this, showing absorption spectra of three different b30 nm NRs.

Figure 3: Absorption spectra of individual CdSe nanorods.
figure 3

Three individual CdSe nanorod absorption spectra obtained from the same ensemble (d=6.8±0.7 nm, b=30.4±2.6 nm). The solid (red, green and blue) circles are experimental data points with corresponding (red, green and blue) dashed lines as their sum of Gaussians fit. The dashed vertical line represents the average α energy of 25 individual b30 nm rods.

The simplest explanation for this behaviour stems from an increase in electronic confinement as NR lengths decrease. In particular, carriers experience additional confinement along the NR z-axis, adding to the radial (ρ) confinement present exclusively in NWs12,15,16. To assess such confinement effects, we have constructed a modified version of an effective mass model previously used to explain absorption spectra of individual CdSe NWs12,14,16.

Effective mass model

In the model, NW/NR electron wavefunctions are given by

u±1/2 is the electron Bloch function, a is the NW/NR radius, b is the corresponding length, Jm(x) are Bessel functions of the first kind and αn,m is the nth root of the mth order Bessel function. Parameters (n, nz, m) are radial, longitudinal and angular quantum numbers, respectively. Hole wavefunctions are linear combinations of effective heavy-hole and light-hole states, given by18

with , , and their relative weights; Fz is the angular momentum projection onto the NW/NR z-axis. Importantly, longitudinal kinetic energy terms are not assumed to be negligible in comparison to radial confinement (that is, kz≠0)9,15,18. Full expressions and derivations can be found in Supplementary Note 3.

Corresponding quantum size level energies for b30 nm NRs only increase 3 meV over the kz=0 case, consistent with previous modelling9 (Supplementary Fig. 3). Clearly, simply accounting for longitudinal carrier confinement cannot explain the 30 meV average blueshift seen in Figs 2 and 3. Additionally, even though Supplementary Fig. 4 shows that obtained b30 nm NR spectra are blueshifted (18 meV) relative to α of the corresponding ensemble spectrum, the residual 12 meV blueshift cannot be explained via confinement alone.

Electrostatic contributions

A more complete explanation must therefore consider the dimensional evolution of carrier electrostatic effects wherein two contributions exist. The first is dielectric contrast, which stems from dielectric constant (ɛ) differences between a nanoparticle and its immediate surroundings12,15,19. The second is dielectric confinement, which arises due to repulsive ‘mirror’ forces at the particle/medium dielectric interface12,15. In 1D systems, dielectric contrast outweighs dielectric confinement, lowers α’s predicted energy by 60 meV and leads to the formation of 1D-excitons in CdSe NWs12,15,16,20. In QDs, electrons/holes effectively screen each other at every point such that their electrostatic contributions to overall carrier energies are vanishingly small15,21. Aspect ratio-dependent electrostatic effects therefore rationalize average α shifts observed in Figs 2 and 3.

To explicitly account for how CdSe’s electronic structure transitions from 1D-to-0D, we model these electrostatic effects as functions of NW/NR aspect ratio (b/d). In practice, this entails solving Poisson’s equation inside a finite length (b) dielectric cylinder to find the potential (V(r, r0)) at an arbitrary position r due to a point charge at r0. The corresponding electrostatic energy is given by15

where re=(ρee,ze) (rh=(ρhh,zh)) is the electron (hole) position, and q is the elementary charge. The first two terms in equation (3) represent direct and indirect attractive forces between an electron and a hole in a NR. The last two terms correspond to repulsive forces induced by mirror charges at NR/medium dielectric boundaries. Equation (3) is spatially averaged over electron/hole wavefunctions, reducing U(re,rh) to a 1-dimensional potential. This representation is subsequently used to calculate exciton binding and self-interaction energies which, together with quantum size levels, describe the evolution of CdSe’s electronic structure across dimensionality (Supplementary Note 4).

Figure 4 shows calculated transition energies for the first (1Σ3/2e) and second (1Σ1/2e) 1D-excitons as functions of NR aspect ratio. Superimposed are extracted (average) α-energies, as well as tabulated equi-diameter QD literature values2,22,23. 1Σ1/2e (1Σ3/2e) is predicted to be bright under parallel (perpendicular) polarized light. Consequently, we have previously assigned α in CdSe NWs to the 1Σ1/2e exciton12,14,16. In QDs, α arises from 1Σ3/2e (1S3/21Se in QD literature)13,24. The α assignment therefore shifts from 1Σ1/2e to 1Σ3/2e with decreasing aspect ratio due to a (b/d)-dependent transition strength. Figure 4 denotes this through 1Σ1/2e/1Σ3/2e curve transparencies.

Figure 4: Evolution of CdSe nanoparticle optical properties across dimensionality.
figure 4

Extracted (average) α energies plotted as a function of aspect ratio (b/d) for all three wire/rod samples (solid red circles) as well as tabulated quantum dot literature values (open square) (d=6.8 nm)2,22,23. S.D. reported. The open red triangle is the b30 nm ensemble spectrum α energy. Superimposed over the data are theory lines for the first (1Σ3/2e; solid blue line) and second (1Σ1/2e; dashed green line) one-dimensional excitons15. Relative transition strengths for each are indicated by the transparency of the lines. Average nanowire and b160 nm nanorod α energies were obtained using weighted individual wire/rod energies with weighting factors obtained from a literature-compiled sizing curve and transmission electron microscopy-derived diameter distributions (Supplementary Note 5, Supplementary Note 6 and Supplementary Fig. 2).

Predicted transition energies are in excellent agreement with experimental NW, b160 nm NR and QD results. Of note is that average b30 nm NR α energies are higher than theoretically-derived energies. This stems from sampling slightly smaller rods within the ensemble’s residual size-distribution (Supplementary Note 6). Figure 4 plots α obtained from the b30 nm NR ensemble spectrum relative to the single NR α-average. Approximations in the model14 also potentially contribute to deviations between experiment and theory (Supplementary Note 7). In general though, observed experimental and theoretical trends are in qualitative and, in some cases, quantitative agreement.


Given that the general evolution of a semiconductor’s dimensionality is described by (Fig. 4), at what point does the 1D-to-0D transition occur? Since the only aspect ratio-dependent energies contributing to a nanostructure’s overall Eg are its longitudinal confinement (Ek(b)) and electrostatic (|U|) energies, the 1D-to-0D evolution is characterized by the interplay between these two terms (Fig. 1). Supplementary Fig. 6 plots Ek(b) and |U| as functions of aspect ratio. From it, we determine where Ek(b) balances |U| and define this to be the critical point where a 1D-to-0D transition occurs (Supplementary Note 8). In d6.8 nm CdSe, this length is b8.5 nm, which is just above its bulk exciton Bohr radius (aB=5.6 nm)25. The transition point is additionally sensitive to diameter and occurs at b6 nm (b11 nm) for d=4 nm (d=10 nm) NRs. These findings differ from previous studies which suggest transition lengths of b≥30 nm for d=3–6 nm CdSe NRs7. The discrepancy exists because experimental blueshifts in NR absorption spectra arise from both carrier and dielectric confinement (Supplementary Fig. 6). At large b, dielectric confinement is the predominant source of blueshifts to NR extinction spectra. Hence, only at lengths below b2aB (ref. 26) does carrier confinement matter, and where a crossover in nanostructure dimensionality occurs.

In summary, we directly probe the 1D-to-0D transition in CdSe using single NW/NR absorption spectroscopy. These measurements expand the limits of conventional single particle microscopies by providing the first direct absorption spectra of individual CdSe NRs. Beyond this, they clearly show how excitonic blueshifts in the linear absorption depend exquisitely on the delicate balance between confinement and dielectric contrast/confinement interactions, the latter being fundamental interactions which govern the dimensional crossover in semiconductors.


Sample synthesis

CdSe NWs (d=6.8±1.2 nm) and NRs (d=6.7±1.1; b=160±55 nm and d=6.8±0.7 nm; b=30.4±2.6 nm) were made using previously established wet chemical syntheses12,27. A detailed description of these methods, as well as information on sample characterization can be found in Supplementary Note 2.

NW/NR absorption spectroscopy

Individual CdSe NWs and NRs were probed on a homebuilt system constructed around a commercial inverted microscope body (Nikon). A schematic of the experimental setup is provided in Supplementary Fig. 1. Samples were prepared by drop casting dilute NW/NR-toluene suspensions onto methanol cleaned, flamed fused silica microscope coverslips (40/20 scratch/dig; UQG Optics). The suspensions were allowed to dry with sample coverages of 1 particle per 5 μm2.

Loaded coverslips were then affixed to an open-loop 3-axis piezo stage (Nanonics) coupled to a closed-loop 3-axis piezo stage (Physik Instumente) and a 2-axis mechanical stage (Semprex) for fine and coarse particle positioning, respectively. The open-loop piezo stage supplied the spatial modulation of individual particles (peak-to-peak particle displacement 360 nm) at 750 Hz. A 2D survey (xy, lateral) absorption map was first obtained to identify and locate single NWs/NRs. This was accomplished by scanning the sample through the focused excitation point-by-point over an xy grid (400 nm increments) while simultaneously detecting the corresponding absorption signal (lock-in time constant, τ=30 ms; integration time, tint=100 ms). These 25 × 25 μm survey absorption images were then used to locate individual particles from where likely candidates were positioned in the laser focus by maximizing the signal from the lock-in amplifier. Absorption spectra were subsequently acquired by scanning the excitation wavelength (λ) through the visible (450–750, 2 nm steps) while synchronously detecting the absorption signal (τ=1 s and tint=10 s). Data acquisition, particle positioning, and wavelength scanning were all controlled using home-written software (C++).

Identification of individual NWs/NRs was established through a three-point vetting process. First, the absorption image was examined to ensure that no extraneous absorbers were within 3 μm of the particle. Next, the absorption signal was converted into σext and compared with expected extinction cross-sections12,16,17. Finally, the absorption spectrum was obtained and examined for the presence of clear transitions.

Data availability

The data that support the findings of this study are available from the corresponding author (M.K.) upon request.

Additional information

How to cite this article: McDonald, M. P. et al. Dimensional crossover in semiconductor nanostructures. Nat. Commun. 7:12726 doi: 10.1038/ncomms12726 (2016).