Dimensional crossover in semiconductor nanostructures

Abstract

Recent advances in semiconductor nanostructure syntheses provide unprecedented control over electronic quantum confinement and have led to extensive investigations of their size- and shape-dependent optical/electrical properties. Notably, spectroscopic measurements show that optical bandgaps of one-dimensional CdSe nanowires are substantially (approximately 100 meV) lower than their zero-dimensional counterparts for equivalent diameters spanning 5–10 nm. But what, exactly, dictates the dimensional crossover of a semiconductor’s electronic structure? Here we probe the one-dimensional to zero-dimensional transition of CdSe using single nanowire/nanorod absorption spectroscopy. We find that carrier electrostatic interactions play a fundamental role in establishing dimensional crossover. Moreover, the critical length at which this transition occurs is governed by the aspect ratio-dependent interplay between carrier confinement and dielectric contrast/confinement energies.

Introduction

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 confinement^1,2,3 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.

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, E_k(b) and dielectric contrast/dielectric confinement electrostatic energies, |U|.

While past studies have attempted to investigate ∼100 meV bandgap (E_g) differences between 1D and 0D CdSe nanostructures by probing nanorods (NRs) of controlled length^7,8,9 no consensus exists as to when a 1D object exhibits 0D character^7,9. 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 distributions^10,11,12. Importantly, they rely on photoluminescence and tunnelling spectroscopies^7,9. However, dark/bright exciton splitting¹³, trap-induced Stokes shifts¹⁴ and enhanced exciton binding energies¹⁵ can significantly alter perceived bandgaps, complicating accurate E_g 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 microscopy^12,14,16,17, 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.

Results

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 NWs^12,14,16. 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.

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 σ_ext ∼7 × 10⁻¹³ cm² (b≥5 μm), σ_ext∼2 × 10⁻¹³ cm² (b∼160 nm), and σ_ext∼8 × 10⁻¹⁴ cm² (b∼30 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 (a–c). S.d. reported.

Most notable is an overall ∼30 meV average α blueshift in b∼30 nm NRs (Fig. 2c) relative to those of longer, equi-diameter particles (Fig. 2a,b) (b∼30 nm: α=1.904±0.025 eV; b∼160 nm, α=1.872±0.016 eV; NW: α=1.874±0.022 eV). In fact, probing 25 individual b∼30 nm NRs reveals α blueshifts up to 64 meV. Figure 3 illustrates this, showing absorption spectra of three different b∼30 nm NRs.

Figure 3: Absorption spectra of individual CdSe nanorods.

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 b∼30 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 NWs^12,15,16. 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 NWs^12,14,16.

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, J_m(x) are Bessel functions of the first kind and α_n,m is the nth root of the mth order Bessel function. Parameters (n, n_z, 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; F_z 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, k_z≠0)^9,15,18. Full expressions and derivations can be found in Supplementary Note 3.

Corresponding quantum size level energies for b∼30 nm NRs only increase ∼3 meV over the k_z=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 b∼30 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 surroundings^12,15,19. The second is dielectric confinement, which arises due to repulsive ‘mirror’ forces at the particle/medium dielectric interface^12,15. 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 NWs^12,15,16,20. In QDs, electrons/holes effectively screen each other at every point such that their electrostatic contributions to overall carrier energies are vanishingly small^15,21. 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 r_e=(ρ_e,ϕ_e,z_e) (r_h=(ρ_h,ϕ_h,z_h)) 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(r_e,r_h) 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 (1Σ_3/21Σ_e) and second (1Σ_1/21Σ_e) 1D-excitons as functions of NR aspect ratio. Superimposed are extracted (average) α-energies, as well as tabulated equi-diameter QD literature values^2,22,23. 1Σ_1/21Σ_e (1Σ_3/21Σ_e) is predicted to be bright under parallel (perpendicular) polarized light. Consequently, we have previously assigned α in CdSe NWs to the 1Σ_1/21Σ_e exciton^12,14,16. In QDs, α arises from 1Σ_3/21Σ_e (1S_3/21S_e in QD literature)^13,24. The α assignment therefore shifts from 1Σ_1/21Σ_e to 1Σ_3/21Σ_e with decreasing aspect ratio due to a (b/d)-dependent transition strength. Figure 4 denotes this through 1Σ_1/21Σ_e/1Σ_3/21Σ_e curve transparencies.

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

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 b∼30 nm ensemble spectrum α energy. Superimposed over the data are theory lines for the first (1Σ_3/21Σ_e; solid blue line) and second (1Σ_1/21Σ_e; dashed green line) one-dimensional excitons¹⁵. Relative transition strengths for each are indicated by the transparency of the lines. Average nanowire and b∼160 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, b∼160 nm NR and QD results. Of note is that average b∼30 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 b∼30 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.

Discussion

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 E_g are its longitudinal confinement (E_k(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 E_k(b) and |U| as functions of aspect ratio. From it, we determine where E_k(b) balances |U| and define this to be the critical point where a 1D-to-0D transition occurs (Supplementary Note 8). In d∼6.8 nm CdSe, this length is b∼8.5 nm, which is just above its bulk exciton Bohr radius (a_B=5.6 nm)²⁵. The transition point is additionally sensitive to diameter and occurs at b∼6 nm (b∼11 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 b∼2a_B (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.

Methods

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 syntheses^12,27. 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 (x–y, 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 x–y grid (400 nm increments) while simultaneously detecting the corresponding absorption signal (lock-in time constant, τ=30 ms; integration time, t_int=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 t_int=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 σ_ext and compared with expected extinction cross-sections^12,16,17. 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.