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Excitonic Character in Optical Properties of Tetrahedral CdX (X = S, Se, Te) Clusters Xi Zhu,† Gregory A. Chass,‡ Leong-Chuan Kwek,†,§,∥,⊥ Andrey L. Rogach,*,# and Haibin Su*,† †

Institute of Advanced Studies, Nanyang Technological University, 60 Nanyang View, Singapore 639673, Singapore School of Biological and Chemical Science, Queen Mary University of London, London E1 4NS, U.K. § Center for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore ∥ National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616, Singapore ⊥ MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit, UMI 3654, Singapore # Department of Physics and Materials Science and Centre for Functional Photonics (CFP), City University of Hong Kong, Kowloon, Hong Kong SAR ‡

S Supporting Information *

ABSTRACT: Cadmium-based quantum dots incorporating group 16 elements show promise of superior optical properties. The excitonic properties of three small size CdX (X = S, Se, Te) tetrahedral clusters with structures mirroring the real chemical systems (Cd17X4(XH)28Na2, Cd32X14(XH)40Na4, Cd54X32(XH)52Na8) were investigated by density functional theory (DFT) in conjunction with quasi-particle corrections. Hallmarks of these systems’ specific excitation properties were resolved with the Bethe−Salpeter equation (BSE) approach. Results showed that the strong electron−hole coupling leads to the exciton state, which strongly modulates the optical properties of CdX clusters. The lowest excitonic excitations observed involve the mixing of multiple single-level transitions, while the size-dependent exciton binding energy exhibits power-law scaling characteristics. The absolute value of its exponent is much larger than those in both 0D and 1D nanostructures, as manifested by stronger screening in these clusters, emphasizing the 3D character of the cluster cores.

1. INTRODUCTION Quantum dots (QDs) are semiconductor materials of nanometer scale that exhibit remarkable quantum phenomena due to the confinement of charge carriers.12,3 Intensive efforts have been made in the past three decades to optimize synthetic methods and rationally fabricate QDs with controllable size and shape for a wide range of II−VI materials, most notably Cd and Zn chalcogenides.4−7 The high extinction coefficient and narrow emission spectra of QDs are highly desirable for applications in optoelectronics8,9 and biological imaging.10−13 Particularly, the fluorescence of colloidal QDs at room temperature with perfect antibunching under continuous or pulsed excitation is highly desirable to generate single photons.14 The confinement-induced augmented Coulomb coupling in QDs plays an important role in enabling multiple-exciton generation, which is very attractive in photovoltaics technology.15−18 It is thus of fundamental importance to gain understanding of the optical properties of semiconductor QDs in the strong quantum confinement regime. The successful syntheses of CdnXm (X = S, Se, Te) clusters with pyramidal shape, with interrelated (n, m) ratios of (17, 32), (32, 54), and (54, 84),19−24 have invoked a need for systematic analysis of electronic and optical properties of these ultrasmall QDs. The bulk Bohr exciton radii for CdX (X = S, Se, Te) systems are 28,25 56,25 and 73 Å, respectively, which are © 2015 American Chemical Society

a few times larger than the mean radii of these clusters. Hence, these clusters are in the strong quantum confinement regime and therefore are perfectly suitable as model systems to study the fundamental physical properties of Cd-based QDs of welldefined composition and structure. Despite extensive experimental and theoretical studies on (CdSe)m clusters,19,26−33 the literature is relatively devoid of atomistic-level computational analyses of the fundamental electronic and optical properties of ligand-capped supertetrahedral CdnXm (X = S, Se, Te) clusters.34 For instance, the excitonic effects on optical properties of passivated CdnSem clusters have been studied by Chelikowsky et al. using both time-dependent local density approximations and many-body approaches involving the combined Green’s function−screened Coulomb interaction approximation (G0W0) and Bethe− Salpeter equation (BSE).35,36 The importance of many-body effects on the first absorption peak was also addressed with G0W0−BSE, which was beyond time-dependent local density approximation. Frenzel et al. employed tight-binding-based time-dependent density functional response theory to investigate the surface states and their passivation in addition to the Received: October 6, 2015 Revised: December 6, 2015 Published: December 7, 2015 29171

DOI: 10.1021/acs.jpcc.5b09751 J. Phys. Chem. C 2015, 119, 29171−29177

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The Journal of Physical Chemistry C influence of particle shape on the optical spectra of CdnSm clusters.37 Therein, they showed that oscillation strength of the first excitonic excitation was significantly reduced in a tetrahedral particle with respect to those observed for spherical or cuboctahedral particles with a comparable Cd atom content. Effective mass approximations (EMA) predicted the exciton binding energy as being inversely proportional to cluster size,3 although sublinear scaling of the Coulomb energy with inverse particle size has been proposed by Franceschetti and Zunger.38 Such deviation from particle-in-a-box type models was further supported by analogous trends uncovered by empirical determinations of exciton binding energy in CdSe QDs employing X-ray absorption (XAS) and photoemission spectroscopy (XPS).39 Considering the difficulties in determination of exact atomic-scale geometries used in such experiments, further studies on exciton properties are highly desirable to gain better understanding of the scaling characteristics of QD clusters.

2. METHODS For the simulations of the clusters, the ground state wave functions were first obtained using the Perdew−Burke− Ernzerhof (PBE)40 exchange-correlation functional using the Quantum Espresso code,41 employing scalar-relativistic normconserving pseudopotentials42 with a kinetic energy cutoff of 60 Ry. Each structure was fully geometrically relaxed until force convergences of 0.01 eV Å−1 was obtained. To mimic an isolated cluster structure, we used a 30 Å × 30 Å × 30 Å cubic cell, and only the Γ point was included in all the cluster simulation. The G0W0 approximation was used for the selfenergy operator to incorporate requisite quasi-particle corrections to the band gaps obtained from the PBE functional. The plasmon-pole approximation43 was introduced to treat the screening. The electron−hole interactions were resolved by the BSE (Bethe−Salpeter equation) approach for the two-particle Green function,44 as below S (Eck − Evk )A vck +

∑

Figure 1. (a) Zinc-blende and wurtzite structures for the CdS crystal, along the ⟨111⟩ and ⟨0001⟩ direction, respectively; the yellow and light blue arrows indicate the different internal arrangement in these two phases. (b) A Cd54Se84 cluster with the internal zinc-blende phase and the corner wurtzite phase as represented by the yellow and light blue arrows. (c) Atomic structure of Cd(S) clusters with chemical formula indicated; the white, blue, pink, and green colors represent the hydrogen, sulfur, cadmium, and sodium, respectively. The black arrows marked by letters L1 and L2 represent the short and long lengths of the cluster. R is defined as the square root of the product of L1 and L2. The Cd(Se) and Cd(Te) clusters were constructed by replacing all the S atoms by Se and Te.

S S ⟨vck|K eh|v′c′k′⟩A vck = ΩSA vck

k′v′c′

enclosed by barrelanoid cages with a WZ-type lattice.34 These differing structural moieties are represented by yellow and light blue arrows, respectively, in Figure 1b. In the small size limit, the Cd(S) clusters composed of 17 Cd atoms in one unit cell are negatively charged with double diamond superlattices20 and are charge-balanced by cations from the encompassing solution. Geometry analysis shows the Cd(X) clusters being stable in the following stoichiometries: [Cd17X4(XR)28]2−, [Cd32X14(XR)40]4−, and [Cd54X32(XR)52]8−. Synthetic solution-phase routes are capable of generating the analogous Cd(S) clusters including [Cd17S4(SR)28]2−,20,46 [Cd32S14(SR)36],21,22 and [Cd54S32(SR)48]4−;47 the latter differing from the perfect proposed [Cd54S32(SR)52]8− structure; the discrepancy comprising the missing thiols at the four corners of the tetrahedral clusters. The structures as well as size- and composition-dependent electronic properties of the smallestand intermediate-sized Se analogues ([Cd17Se4(SR)28]2−,23,33,46 [Cd32Se14(SR)36]23,33) were characterized by X-ray diffraction (XRD) and optical measurements. Structures of the three corresponding Cd(Te) clusters ([Cd 17 Te 4 (SR) 28 ] 2− , [Cd32Te14(SR)40]4−, [Cd54Te32(SR)52]8−) were investigated by temperature-dependent X-ray absorption fine structure (EXAFS) measurements, revealing molecular-cluster sizes ranging from 1.3 to 2.4 nm.24,48,49 As demonstrated in recent

ASvck

where is the amplitude of exciton, Keh is the electron−hole interaction kernel, |ck⟩ and |vk⟩ are the quasi-particle states (electron and hole), respectively. ΩS is the excitation energy, Eck and Evk are the quasi-particle energies. A box-shape truncated Coulomb interaction was applied to the calculation cell to avoid problematic image effects caused by proximate supercells. All the G0W0 and BSE calculations were performed with YAMBO code.45

3. RESULTS AND DISCUSSION We conducted first-principles density functional theory (DFT) determinations with quasi-particle corrections and electron− hole interactions toward quantitatively resolving optical properties of capped tetrahedral CdX (X = S, Se, Te) clusters with inclusion of explicit counterions. For the bulk CdS materials, the following two phases are stable in ambient conditions: the cubic zinc-blende type (ZB, also known as sphalerite) and the hexagonal wurtzite (WZ) type. The ZB structure exhibits ABC stacking along the ⟨111⟩ direction, while the WZ structure displays AB stacking along the ⟨0001⟩ direction (Figure 1a). The corresponding Cd(X) QDs, taking the Cd54Se82 QD as an example (Figure 1b), have crystal structures that could be described as consisting of a ZB-type core, with its four corners 29172

DOI: 10.1021/acs.jpcc.5b09751 J. Phys. Chem. C 2015, 119, 29171−29177

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The Journal of Physical Chemistry C calculations,50 the ligand effects on the optical properties are significant especially for tetrahedral chalcogen-rich nonstoichiometric clusters. Here, for a better understanding of this effect, we replaced each organic group ligands (R) with a hydrogen atom, while sodium atoms were used to charge-balance the clusters to be neutral overall. The resultant chemical formulas of the three Cd(S) tetrahedral clusters characterized in this work were Cd17S4(SH)28Na2, Cd32S14(SH)40Na4, and Cd54S32(SH)52Na8. The optimized Cd−S, Cd−Se, and Cd−Te bond lengths in the core part of the clusters are 2.56, 2.66, and 2.79 Å, respectively, in excellent agreement with their respective experimentally determined lengths of 2.55, 2.62,27 and 2.77 Å.24 In the bulk phase, the bond length are 2.52, 2.62, and 2.81 Å.51 At the nanoscale, Cd(Te) QDs structurally deviate from their corresponding Cd(S) and Cd(Se) structures. Specifically, Cd−Te bond lengths are reduced in QDs compared to those in the bulk, while the Cd−S and Cd−Se clusters show an inverse trend with respect to their corresponding bulk values, with increased bond lengths. This observation is further supported by empirical determinations, wherein the Debye temperature in the Cd(Te) QDs is higher than that in the Cd(Te) bulk, while the opposite trend is observed for the Cd(S) QDs and Cd(S) bulk.42 The band gaps of the zinc-blende CdS, CdSe, and CdTe systems were experimentally determined to be 2.55, 1.90, and 1.60 eV, respectively.52 On going from the bulk CdX to the [Cd17X4(XR)28]2− cluster, the band gap increased by 124, 192, and 225%, respectively. For the larger [Cd54X36(SR)48]8− clusters, more moderate band gap increases of 61, 75, and 88%, respectively, were observed. These results translate to a strength-of-confinement trend sequence of CdS < CdSe < CdTe, increasing from S through Se to Te and following the size of the bulk Bohr exciton radii from CdS (28 Å),25 through CdSe (56 Å),25 to CdTe (73 Å).53 The effective sizes of the clusters corroborate nicely with their optical gap values. For the Cd17(S) cluster, the experimental value of the optical gap is 4.20 eV,46 as measured by UV−vis absorption spectroscopy at room temperature, the computed value of 4.21 eV being in good agreement (Table 1). One of the Cd(S) structures exhibits a geometry of two superlattices interlaced by a sulfur bridge, with the single cluster’s stoichiometry of Cd17S4(SCH2CH2OH)26 exhibiting a 4.28 eV optical gap,20 arising from the interplay between the charge state of the cluster and sulfurbridge-induced delocalization. For the Cd32(S) cluster, the computed optical band gap of 3.72 eV is within ∼2.4% of the experimental value (3.81 eV) obtained for the synthetic analogue [Cd32S14(SCH2CH(OH)CH3)36·4H2O].21 Both values are higher than the 3.46 eV first excitation energy of [Cd32S14(SPh)36], likely due to the accumulation of superfluous electronic density in the aromatic rings of the thiophenol ligands.22 For the Cd54(S) cluster, synthesis in dimethylformamide solution produced the [Cd54S32(SPh)48(H2O)4]4− system with an optical transition energy of 3.51 eV,47 four electrons short of the [Cd54S32(SR)52]8− structure and its reduced 3.21 eV transition energy; the difference is due to the balance between charge state and the substituent thiophenol aromatic rings. The computed optical transition energy of the Cd17(Se) cluster is 4.13 eV, while the measured excitation energy is 3.60 eV for [Cd17Se4(SePh)28]2− in dimethylformamide solution.46 The observed red-shift is again due to electron accumulation effects induced by the aromatic rings of the thiophenol ligands. For the Cd32(Se) cluster, the calculated

Table 1. Computed Values of Cluster Radii (R/nm), PBE GW Band Gap (EPBE g /eV), G0W0 Corrected Band Gap (Eg /eV), Exciton Binding Energy (Eb/eV), and Computed First Absorption Energy (Eabs/eV) with Experimental Values (eV) inside Parentheses of the Cd(S), Cd(Se), and Cd(Te) Clusters Characterized with the PBE + G0W0−BSE Method G W0

Ra

EPBE g

Eg 0

Cd17S4(SH)28Na2 Cd32S14(SH)40Na4

0.65 0.81

1.51 1.32

5.71 4.93

1.50 1.21

Cd54S32(SH)52Na8 Cd17Se4(SeH)28Na2 Cd32Se14(SeH)40Na4 Cd54Se32(SeH)52Na8 Cd17Te4(TeH)28Na2 Cd32Te14(TeH)40Na4 Cd54Te32(TeH)52Na8

1.15 0.69 0.87 1.21 0.72 0.93 1.28

0.81 1.51 1.22 0.71 1.31 1.12 0.58

4.10 5.56 4.21 3.32 5.16 3.87 3.11

0.89 1.43 1.12 0.85 1.35 1.05 0.80

clusters

Eb

Eabs 4.21 (4.2046 b) 3.72 (3.81,21 c 3.4622 d) 3.21 (3.5147 e) 4.13 (3.6046 f) 3.09 (3.3054 g) 2.47 (N/A) 3.81 (N/A) 2.82 (N/A) 2.31 (2.6149 h)

a

Defined as the square root of the product of long and short lengths of the cluster as shown in Figure 1c. b[Cd17S4(SPh)28]2− in dimethoxyethane solution. cCd32S14(SCH2CH(OH)CH3)36·4H2O in dimethylformamide solution. dCd32S14(SPh)36·4DMF in dimethylformamide solution. eCd54S32(SPh)48·(H2O)44− in dimethylformamide solution. f [ C d 1 7 S e 4 ( S e P h ) 2 8 ] 2 − i n d im e t h y l f o r m a m i d e s o l u t io n . g Cd 3 2 Se 14 (SePh) 36 ·(OC 4 H 8 ) 4 in tetrahydrofuran solution. h [Cd54Te32(SCH2CH2OH)48]8− in aqueous solution.

excitation energy is 3.09 eV, which is 0.21 eV below the experimental value of 3.30 eV for the Cd 32 Se 14 (SePh)36(OC4H8)4 structure in tetrahydrofuran (THF) solution,54 likely due to the counterbalance between the aromatic rings and the polar THF solvent. For Cd(Te) clusters, the sole empirical optical transition energy reported is for the [Cd54Te32(SCH2CH2OH)488−] cluster (2.61 eV).49 This represents a 0.30 eV blue-shift relative to the computed value of 2.31 eV for the [Cd54Te32(TeH)528−] cluster, rationalized by the stronger electron-withdrawing potential of the sulfur ligand relative to that of tellurium. The computed binding energies of excitons in these clusters are listed in Table 1 and reveal pronounced coupling between the electron and hole in these structures. With respect to relevant geometry, Meulenberg et al. have used X-ray spectroscopy to reliably measure cluster-size-dependent exciton binding energies in Cd(Se) QDs: 1.0 and 0.7 eV for QDs of 0.6 and 1.2 nm radii, respectively.39 Despite the variations in composition and structure of the clusters studied here, exciton binding energies are in satisfactory agreement at 1.43 and 0.85 eV for the [Cd17Se4(SeH)28Na2] (0.69 nm) and [Cd54Se32(SeH)52Na8] (1.21 nm) systems, respectively. The radii and electronic properties of all Cd(X) clusters characterized in this work are listed in Table 1. We present the available experimental data together with the data stimulated in this work on optical transition energies of QDs as a function of radius in Figure 2. The band gap values obtained by EMA are also plotted as reference. Both pseudopotential55 and tight-binding56,57 based calculations clearly correct the overestimation of quantum confinement arising in EMA. Thus, the electron and hole binding strengths will be overestimated, especially for structures with radii less than 1.5 nm, when using EMA-determined energy gap values. Since the Cd(X) clusters are finite in size, the quantum confinement dominates the binding strength of the exciton.58 The enhancement of exciton binding strength due to the reduction in size results from the size-dependent dielectric constants as well. The computed size-dependent dielectric 29173

DOI: 10.1021/acs.jpcc.5b09751 J. Phys. Chem. C 2015, 119, 29171−29177

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Figure 2. Computed (with the G0W0−BSE method, represented by circles) and experimentally determined size-dependent optical absorption energy of the lowest exciton state for Cd(S), Cd(Se), and Cd(Te) clusters. The band gaps calculated by the effective mass approximation (EMA) are plotted as dashed lines; their strong deviations from empirical trends highlight the limitations of that method.

constants using the first-principles method can be well described by Penn’s model,59 i.e., ε(R) = 1 + (ϵb − 1)/(1 + (α/R)l), where ϵb is the bulk dielectric constant and α and l are material-dependent parameters (here choosing l to be 1.4 as used previously for the silicon quantum dots;60 the values of α are 11.6, 12.1, and 12.6 Å for the Cd(S), Cd(Se), and Cd(Te) clusters, respectively). We find that the confinement plays an important role in size-dependent dielectric constants. For example, the dielectric constants decrease by 19% from Cd54S32(SH)52Na8 (ε = 4.81) to Cd17S4(SH)28Na2 (ε = 3.29), while the larger reduction in the dielectric constant, i.e. 34%, occurs from Cd54Te32(SH)52Na8 (ε = 5.81) to Cd17Te4(SH)28Na2 (ε = 3.86), indicating the strongest confinement in the Cd(Te) clusters. Hence, the exciton in Cd17Te4(SH)28Na2 binds most strongly in these nanoclusters. We plot exciton binding energies of Cd(X) clusters, Eb (listed in Table 1), as a function of the inverse of radius (1/R) in Figure 3. The sizedependent Eb ∼ 1/R0.87 relationship, provided by Franceschetti and Zunger38 with direct pseudopotential calculation, is likewise presented in Figure 3. The exponent shows clear dependence on the nature of nanostructures. For example, the graphene QD structures exhibit an exponent of 0.38,61 while, the scaling exponents of graphene nanoribbons are 0.6, 0.73, and 0.55 for the 3p, 3p + 1, and 3p + 2 families, respectively.62

Figure 3. Scaling relationship between the exciton binding energy (Eb) and the reciprocal of radii (R) of CdX clusters. The black squares, red circles, and blue triangles represent Cd(S), Cd(Se), and Cd(Te) clusters, respectively. The green dashed lines are plotted with Eb = AL(1/R0.87) dependence proposed by Franceschetti and Zunger.38

For the Cd(X) clusters considered here, the interior is 3D-like with the zinc-blende lattice in the core area, and the scaling exponent is 0.87much larger than those in both aforementioned nanostructures, clearly indicating remarkably strong screening effects. The computed absorption spectra including excitonic effects of Cd(S) clusters are plotted in Figure 4a. The first absorption peaks exhibit a distinct blue-shift with decrease of cluster size. Selected molecular orbitals of Cd17S4(SH)28Na2 are presented 29174

DOI: 10.1021/acs.jpcc.5b09751 J. Phys. Chem. C 2015, 119, 29171−29177

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Figure 4. (a) Computed absorption spectra with G0W0−BSE of Cd(S) clusters; (b) computed wave functions of HOMO−1, HOMO, and LUMO of Cd17S4(SH)28Na2 cluster; (c) the wave function of the lowest exciton state in Cd17S4(SH)28Na2 structure.

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in Figure 4b. Both the highest-occupied molecular orbital (HOMO, 4-fold degenerate) and the HOMO−1 (3-fold degenerate) are mainly distributed on the anionic corners of the tetrahedral cluster, while the nondegenerate lowestunoccupied molecular orbital (LUMO) is predominantly distributed over the cations in this structure. Differing from time-dependent DFT (TDDFT), where the excitation is dominated by single-level to single-level transitions, the G0W0−BSE method invokes mixing among differing transitions.35 Our calculations demonstrate that the lowest excition has strong multiple transition character, namely, 85% of this excition state is from HOMO → LUMO, while the remaining 15% arises from other transitions, including HOMO−1 → LUMO. The probability of the exciton wave function is shown in Figure 4c for the Cd17S4(SH)28Na2 structure. For this cluster, the ligand-to-metal charge transfer is observed in the optical excitation. This results in the relatively weak oscillation strength of the first excitonic peak. The charge transfer character is also featured in the other two clusters as shown in Figure S1 (see Supporting Information). The relative appreciable overlap between occupied states’ wave functions and unoccupied ones in Cd32S14(SH)40Na4 results in larger oscillator strength of the first excitonic peak.

ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.5b09751. Density of states (DOS) of the Cd17S4(SH)28Na2, Cd32S14(SH)40Na4, and Cd54S32(SH)52Na8 structures (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected] (H.S.). *E-mail [email protected] (A.L.R.). Notes

The authors declare no competing financial interest.

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4. CONCLUSIONS

ACKNOWLEDGMENTS This work is partially supported by AOARD Grant (FA238613-1-4074), MOE Tier-2 grant (no. MOE2013-T2-2-049), and by the Research Grants Council of the Hong Kong S.A.R., China (project CityU 11302114). G.A.C. thanks RSC, UK, and STFC, UK, for support. The authors are grateful for Ruth Pachter for the important advice in formulating this study, sharing ideas, and data before publication and Efrat Lifshitz and Kenneth Caster for fruitful discussions.

In conclusion, we have conducted a series of first-principles calculations augmented by the G0W0−BSE method on the optical properties of tetrahedral Cd(X) (X = S, Se, Te) clusters ([Cd 17 X 4 (XH) 28 Na 2 ], [Cd 32 X 14 (XH) 40 Na 4 ], [Cd 54 X 32 (XH)52Na8]) with radii from 0.65 to 1.28 nmstructures mirroring the real chemical systems. The exciton state, originating from the strong electron−hole interaction, modulates the optical properties of these systems. The lowest excitonic energies involve the mixing of multiple single-level transitions, while the size-dependent exciton binding energy exhibits power-law scaling characteristics. The analysis of sizedependent exciton binding energy shows that the screening in Cd(X) clusters is much stronger than that in 0D and 1D materials.

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