Atomic transition region at the crossover between quantum dots to molecules

Semiconducting structures restricting the electron motion to the nanometer scale in three dimensions are defined as nanocrystallites or quantum dots (QDs). The variety of QDs includes lithographically engineered electrostatically confined systems of electrons [1–4], nano-sized crystallites in glasses [5, 6] and self-assembled colloids [7, 8]. Lead sulfide (PbS) QDs, a typical representative of the latter case, possesses an attractive direct band gap range (specifically with respect to optical fiber application) and a relatively large Bohr radius (∼ 20 nm) [9]. Consequently, due to these advantageous intrinsic properties, PbS QD colloids have been vigorously studied in order to explore their relevance to potential technological applications, such as tunable infrared electroluminescent devices [10], biological imaging [11], intense light emitters with tunable chromaticity [12] and solar cells [13], to mention a few.

Our work here focuses on absorption properties of PbS QDs in the 'small-QD' limit (also called the strong quantum confinement limit), i.e. we investigated QDs with diameters clearly below 20 nm in the range of 2.7–4.8 nm. For such small diameters, hybrid electronic properties not characteristic of either the molecular or solid-state limits are expected [7]. The number of atoms comprising QDs determines their belonging to the different phases of matter: it was pointed out that ∼10⁴ atoms are required to form crystallites with bulk properties [8], while the critical number of atoms at which QDs collapse into the molecular state had not been sought after. The knowledge and mastery of this critical accumulation of atoms exactly at the boundary between molecular and solid-state appearances would allow the realization of extremely sensitive bistable-like devices—switching between confined and non-confined states—due to the removal or attachment of a group of atoms to the QDs in controlled one-step ways.

The pursuit of the smallest possible QD (equivalent to the smallest number of atoms forming a QD) raised first the following question: what method and parameter can be used to record evidence of the descent of a solid to its atomic components? We approached this matter by fitting the temperature dependence of the band gap energy E_g(T) for PbS QDs of decreasing diameters with Fan's formula [14–17]

$$\begin{equation} E_{\rm g} (T) = E_{\rm g} (0) + \frac{A}{{\exp\left( {\left\langle {E_{\rm{p}} } \right\rangle /kT} \right) - 1}}, \end{equation} \tag{ 1 }$$

describing the band gap variation via phonon–electron coupling, where E_g(0) is the band gap value toward T = 0 K, A is the Fan parameter (or self-energy), 〈E_p〉 is the average phonon energy responsible for the thermal band gap alteration and k is the Boltzmann constant. Parameter A has been analyzed with Fan's microscopic expression [18] for colloidal 2.0 nm [19] and 4.7 nm [16, 20] PbS QDs, and 5.1 nm PbS QD embedded in a glass matrix [6]. In this paper, we demonstrate that 〈E_p〉 is the decisive 'order' parameter tracking the gradual crossover of crystalline PbS QDs to the molecular phase, i.e. it is demonstrated that 〈E_p〉 depends linearly on the number of atoms forming the QDs, pointing to the 'QD collapse toward the molecule' at 90 atoms.

In order to check this profound result, we present here a second independent quantum mechanical argument based on the uncertainty principle for calculating the minimum QD size. By means of the virial theorem, we start with the general definition of the kinetic exciton energy in a Coulomb potential⁵ [21]

$$\begin{eqnarray} \frac{{(\Delta p)^2 }}{{2\mu }} &=& \frac{1}{2}\left( {\frac{{e^2 }}{{4\pi \varepsilon _{\rm r} a_{\rm x} }}} \right) = \frac{1}{2}\left( {\frac{{e^2 }}{{4\pi \varepsilon _{\rm r} a_0 }}} \right)\left( {\frac{{a_0 }}{{a_{\rm x} }}} \right)\nonumber\\& =& \left| {E_1 } \right|\left( {\frac{{a_0 }}{{a_{\rm x} }}} \right), \end{eqnarray} \tag{ 2 }$$

where p is the momentum, μ is the reduced effective electron hole mass, e is the elementary charge, ε is the corresponding permittivity, a_x is the exciton Bohr radius, a₀ is the Bohr radius of the hydrogen atom (0.0529 nm) and E₁ (− 13.6 eV) is the ground state energy of the hydrogen atom. Employing the uncertainty relation, we insert Δ p  = ℏ /(2Δ x) in equation (2), where the coordinate uncertainty Δx becomes the measure for the minimum QD diameter d_min and ℏ is the Planck constant over 2π, and find

$$\begin{equation} d_{min}^2 = \frac{{\hbar ^2 }}{{8\mu \left| {E_1 } \right|(a_0 /a_x )}} = \left( {\frac{{\hbar ^2 }}{{8m_0 a_0^2 \left| {E_1 } \right|}}} \right)\left( {\frac{{m_0 }}{\mu }} \right)(a_0 a_x ), \end{equation} \tag{ 3 }$$

where m₀ is the free electron mass. By inserting all the constants, the relation

$$\begin{equation} d_{min}^{} = 0.5 \times \sqrt {\left( {\frac{{m_0 }}{\mu }} \right)(a_0 a_x )} \end{equation} \tag{ 4 }$$

is derived. As the foregoing discussion below will illustrate, expression (4) reasonably matches the result attained with the fits using the Fan formula (1).

Three oleic acid capped colloidal PbS QD samples with diameters of 2.7, 3.1 and 4.8 nm were investigated. The QD deposition on glass was accomplished with the supercritical fluid CO₂ method described in detail in [22]. In order to visualize the QD size distribution and the homogeneity of the substrate surface coverage, scanning electron microscopy has been performed. The result for the 2.7 nm QDs is displayed in figure 1. The QD diameter fluctuations were established to be 2.7 (± 0.4), 3.1 (± 0.5) and 4.8 (± 0.6) nm.

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The transmittance (TR) of the samples was measured with a Fourier transform infrared (FTIR) BOMEM spectrometer in conjunction with nitrogen-cooled InGaAs or InSb detectors. By illuminating the sample with the appropriate internal light source of the BOMEM, the thermally provoked E_g variation, which is defined by the shift of the lowest energy TR minimum [16], was measured employing a closed cycle optical cryostat.

Transmittance spectra and their shape analysis of PbS QDs with a diameter around 3 nm and below are not found in the literature, while figures 2 and 3 show such an example at 10 and 300 K for the 2.7 and 3.1 nm QDs, respectively. The symbols indicate the measurements and the broken and solid lines represent Gaussian fits superimposed on the linear background caused by the ligands. The theoretically expected discrete narrow lines of the absorptive QD transitions [1, 23] are diluted in rather broad (∼ 100 meV) inhomogeneous distributions with a fairly temperature invariant full-width at half-maximum (FWHM) pointing to a line broadening based on size (band gap) fluctuations of the individually absorbing QDs in the illuminated ensemble. Figure 4 displays the E_g(T) trend for both samples versus temperature over the range of 10–300 K. The symbols show the experimental result for various temperatures and the lines are fits using equation (1). The same measurements and fits were performed with the 4.8 nm QDs.

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The results undoubtedly demonstrate that E_g(T) becomes more and more linear over the entire temperature range for a shrinking QD diameter. Indeed, the 2.7 nm QDs in figure 4 clearly deviate from the typical nonlinear 'Fan-like' characteristic, which is split into two regions, i.e. dE_g/dT (< 50 K) <  dE_g/dT (> 50 K). However, figure 5 demonstrates that the fit using the Fan equation (1) prevails over the linear expression E_g(T) = E_g(0) + AkT/〈E_p〉, which determines the ratio A/〈E_p〉 rather than the parameters themselves. The slightly positively curved solid line fit achieved by equation (1) results in an improved goodness of fit (χ² = 0.834) in comparison to the linear fit (χ² = 0.774), by describing the subtle tendency of dE_g/dT to flatten below ∼50 K. Table 1 displays the parameters used in equation (1) best fitting the E_g(T) trends for the samples investigated, to which the parameters for 4.7 nm QDs from [16] have been added. The table reveals the decrease in 〈E_p〉 for smaller QD diameters, pointing to the diminution of phonons in QDs with decreasing size. Calculating the number of atoms in QDs with [24],

$$\begin{equation} N = (4\pi /3) \times (D/l)^3 , \end{equation} \tag{ 5 }$$

where D is the QD diameter and l is the PbS lattice constant (0.5936 nm), we were able to plot 〈E_p〉 = f(N). The result is shown in figure 6, whereas the linear fit—displayed by the broken line—revealed N = 90 (corresponding to D = 1.7 nm) for 〈E_p〉 = 0 eV and the slope of d〈E_p〉/dN = 8 μeV per atom, suggesting that a range could exist and extend to a lower limit of 44 atoms (corresponding to D = 1.3 nm, i.e. the gathering of only 16 simple unit cells of the PbS structure) and upper limit of 500 atoms (corresponding to D = 2.9 nm). For comparison, with μ = 0.039m₀ [16] and a_x = 20 nm [9], equation (4) gives d_min = 2.6 nm being in the above determined range.

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It is of importance to note that the QDs formed with supercritical fluid deposition maintain the cubic rock salt (galena) structure of bulk PbS [22]⁶, justifying the employment of the Fan formula, the effective mass approximation in equation (4) and the use of equation (5). However, as the size of the PbS QDs gets smaller and smaller (∼ 1 nm), it is expected that a larger fraction of the atoms is located on the surface causing hybridization and the consequent formation of monomers rather than QDs with a defined lattice structure [25]. Consequently, the result in figure 6 points to the smallest possible size of cubic crystalline PbS QDs. However, besides the diameter of the QD, the stoichiometry is also highly influential. It was pointed out by Fernée et al [26] that PbS QDs with a size of 1–2 nm can be influenced by an unbalanced stoichiometry, causing variations in the optical properties. Thus, referring to these particles as PbS QDs could be misleading as well. Therefore, we carefully verified a balanced stoichiometry (Pb/S = 1) during the QD preparation⁷ [27].

We showed that the average vibronic energy as part of the classical Fan formula can be used to monitor the gradual transition from crystalline quantized matter to molecular structures without characteristic phonon energy due to the lack of coherent atomic movements. The work reveals the notable conclusion that the Fan formula, which was designed for materials with macro-dimensions, leads to results comparable with calculations based on quantum mechanics. The topic investigated here raises the question of whether it might be possible in the future to realize 'quasi-matter' at the transition threshold from QDs to molecules in order to allow device operations initiated by the repeatable and revertible flip between molecules and quantized matter due to the controlled fine tuning of atomic accumulations.

The work was partially supported by the DGAPA-UNAM PAPIIT project TB100213-RR170213, PI Bruno Ullrich. We also acknowledge R Garcia and U Amaya for technical support. BU dedicates this work to Susi for showing unwavering commitment.