Shape Matters: Effect of 1D, 2D, and 3D Isovolumetric Quantum

Article pubs.acs.org/JPCC

Shape Matters: Effect of 1D, 2D, and 3D Isovolumetric Quantum Confinement in Semiconductor Nanoparticles Jeremy A. Scher,† Jennifer M. Elward,‡ and Arindam Chakraborty*,† †

Department of Chemistry, Syracuse University, Syracuse, New York 13244 United States Army Research Laboratory, Aberdeen Proving Ground, Aberdeen, Maryland 21005, United States

‡

S Supporting Information *

ABSTRACT: Semiconductor nanoparticles (NPs) are a class of nanoscopic materials with highly tunable optical and electronic properties. The electronic density of states of these NPs depends strongly on both shape and size and has allowed semiconductor NPs to be tailored for applications in various fields including photovoltaics, solid-state lighting, and biological labeling. This work presents investigation of the effect of shape on excitonic properties of electronically excited NPs. Specifically, this work focuses on isovolumetric NPs and addresses the question of how optical properties of NPs are impacted by isovolumetic deformation of NP shapes. The effects of three shapes, representing 1D, 2D, and 3D quantum confinement, for three sizes and four semiconductor materials (CdSe, CdS, CdTe, and PbS) were studied. The electronic excitation in these NPs was described using electron−hole (eh) quasiparticle representation, and exciton binding energies, ehjoint probabilities, and eh-separation distances were calculated using the eh explicitly correlated Hartree−Fock method. The calculations demonstrated that increased anisotropy in the confinement potential resulted in decreased exciton binding energy in the NPs. Within a specific volume, it was found that nanorods exhibited lower exciton binding energies than did nanodisks and that nanodisks exhibited lower exciton binding energies than nanospheres of identical volume. In contrast, the trend for eh-joint probability was found to be opposite that of exciton binding energies. These results demonstrate that a relatively small change in NP structure can result in a substantial change in the excitonic properties of these nanomaterials. exhibit 2D confinement,80−83 and nanorods and wires exhibit 1D confinement.1,64,84 For 1D quantum-confined materials, theoretical and experimental studies have shown that modifying the aspect ratio in quantum rods can substantially change their optical properties.66,72,76,77,85−88 Nanorods have been used to enhance carrier multiplicity,64 optimize resonance energy transfer,39 and generate linear polarized light.89 The properties and shape effects of 2D quantum-confined materials such as disks and platelike nanoparticles have also been investigated.59,90,91 Song and Ulloa have studied excitons confined to quantum disks and calculated the geometrical effects on exciton binding energy, eh-separation, and the resulting linear optical properties observed.27 Biadala et al. investigated ehrecombination behavior in CdSe nanoplates and observed that the behavior of this 2D material was different than that of their corresponding 3D spherical CdSe NPs.92,93 In this work, we have used a computational approach to investigate the effect of quantum confinement in a series of semiconductor NPs. To differentiate the effect of shape from size on the NP excitonic properties, we performed calculations on semiconductor nanorods, nanodisks, and nanospheres under

1. INTRODUCTION Semiconductor nanoparticles (NPs) have tunable optical and electronic properties that make them attractive in applications such as photovoltaics,1−5 light-emitting devices,6−8 charge and energy transfer processes,9−12 and biological labeling.13,14 This tunability is a consequence of differences in the quantum confinement effect with variations in the size and shape of the NPs. The size of the NP not only determines its density of states but also restricts the spatial localization of excitons when the NP is electronically excited. Adjusting the NP’s size has been used extensively to manipulate excitonic interactions.15−19 For example, one wellstudied phenomenon is the change in photoluminescence spectra of quantum dots (QDs) as a function of dot size.18,20−28 The effect of dot size on Auger recombination rates,20,21,29−32 electron-transfer processes,2,9,33−38 energy-transfer processes,21,39−49 carrier multiplicity,32,50−62 and dissociation of excitons26,63,64 has been and continues to be extensively studied using both experimental and theoretical methods. In contrast to size, NP shape controls the spatial asymmetry of the confinement volume, which leads to shape-dependent effects on exciton behavior. The shape of a NP is associated with a number of different electron−hole (eh) confinement regimes.64−79 Specifically, spherical NPs such as quantum dots exhibit 3D quantum confinement, nanoplatelets and nanodisks © 2016 American Chemical Society

Received: July 5, 2016 Revised: September 23, 2016 Published: October 7, 2016 24999

DOI: 10.1021/acs.jpcc.6b06728 J. Phys. Chem. C 2016, 120, 24999−25009

Article

The Journal of Physical Chemistry C isovolumetric conditions. This constraint ensures that the overall amount of material in the NPs is equivalent while allowing the spatial distribution of the material to differ with shape. This isovolumetric condition is very difficult to reproduce experimentally, and control over the NP structure in this way provides us with unprecedented theoretical insight into the structure−property relationship of these materials. For each NP shape, three representative particle sizes were selected (Figure 1), and each volume/shape combination was

+∑ ⟨i| ij

−ℏ2 2 h ∇ + vext |j⟩hi†hj 2mh

(2)

−1 + ∑ ⟨iji′j′|ϵ−1reh |iji′j′⟩ei†ejhi†hj ′ ′

(3)

iji ′ j ′

ee † † +∑ wijkl ei ej elek + ijkl

∑ wijklhhhi†hj†hlhk (4)

ijkl

The effective electron−hole Hamiltonian provides a computationally efficient means of calculating the optical properties of interest in these NPs. The external potential used in this investigation is a parabolic potential for both the electron and hole. Parabolic potentials have been used extensively to approximate the confinement potential of semiconductor NPs.102−105 α vext =

1 kα|rα|2 2

α = e, h

(5)

The magnitude of the force constants in this external confining potential is important for the accurate calculation of exciton binding energy in these NPs. The force constants are calculated such that the electron and hole are confined to the physical dimensions of the NP min = (Nα − min kα

kα a =

investigated for four different materials: CdSe, CdS, CdTe, and PbS. These compositions were chosen because many of these NPs have been successfully synthesized,94,95 which allowed us to benchmark and validate the computational procedures used. The nanoscale dimensions (3D, 2D, and 1D) combined with the isovolumetric constraint of these systems led to a number of significant questions about the excitonic interactions in these NPs. To elucidate the relationship between shape and the excitonic properties of select material compositions, we conducted investigations on (1) the binding energy of the eh-interaction, (2) the spatial separation between the eh-pair, and (3) the eh-joint probability.

ij

∫ dΩ ρα (r)[vextα ])2

(6)

2 kα spherersphere

a2

(7)

Ψeh − XCHF = Ĝ ΦeΦh

(8)

The geminal operator Ĝ is a two-body operator and introduces an explicit dependence on the electron−hole separation distance to the wave function as well as variational parameters bk and γk. This geminal operator has the form of a linear combination of Gaussian functions96 Ne

Ĝ =

Nh

Ng

2 k eh

∑ ∑ ∑ b k e −γ r i=1 j=1 k=1

(9)

The eh-XCHF wave function is obtained variationally by minimizing the total energy with respect to the parameters of the electron and hole reference wave functions as well as the geminal parameters bk and γk

2

−ℏ 2 e ∇ + vext |j⟩ei†ej 2me

dr r 2

where a is the length of the semiaxis and rsphere is the radius of a spherical NP with a corresponding force constant kαsphere. The ansatz of the eh-XCHF wave function is a product of electron and hole reference wave functions and a correlation function.96 The electron and hole reference wave functions are Slater determinants, and the correlation function is a Gaussiantype geminal operator

2. METHODS The NPs in this study were modeled using the electron−hole explicitly correlated Hartree−Fock method. While the complete details of this method are outlined in the work by Elward et al.,96 a brief overview of the method is discussed here. To solve the electron−hole Schrödinger equation for these systems, the effective electron−hole Hamiltonian97−101 was used:

∑ ⟨i|

Ddot /2

where α = e, h, dΩ = sin dθ dφ, Ddot is the diameter of the corresponding spherical quantum dot, and kα is the smallest force constant that satisfies the above minimization conditions. An appropriate force constant for the spherical CdSe NP was obtained using this approach, and then the force constants corresponding to all other semi axis lengths were scaled using this quantity

Figure 1. Representative ellipsoid showing the A, B, and C semiaxes. The nanoparticles in this investigation are ellipsoids with semiaxis lengths constructed so that there are three sets of differently shaped NPs of equal volume. Nanorods are constructed to have length equal to eight times their diameter, and nanodisks are constructed to have diameter equal to eight times their thickness.

Ĥ =

∫0

(1) 25000


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The Journal of Physical Chemistry C

Figure 2. Exciton binding energies (in meV) of (a) CdSe, (b) CdS, (c) CdTe, and (d) PbS nanoparticle spheres, disks, and rods. Experimental results for CdSe nanospheres provided for comparison. Dashed lines correspond to the exciton Rydberg energy (in meV) for each material.

E = min e h G,̂ Φ , Φ

⟨Ψeh‐XCHF|Ĥ |Ψeh‐XCHF⟩ ⟨Ψeh‐XCHF|Ψeh‐XCHF⟩

3. RESULTS AND DISCUSSION Spheres, disks, and rods were selected to represent the 3D, 2D, and 1D quantum confinement effect, respectively. We calculated three excitonic properties of each NP: exciton binding energy (EEB), eh-joint probability (Peh), and ehseparation distance (⟨reh⟩). We performed these calculations using the eh-explicitly correlated Hartree−Fock (eh-XCHF) method.96 The effect of size, shape, and NP material (i.e., chemical composition) on the calculated excitonic properties was analyzed and is presented here. 3.1. Effect of Shape and Size on Exciton Binding Energy. An important first step for the evaluation of excitonic properties is to benchmark and validate the computational procedure. In this work, we benchmarked the eh-XCHF method by comparing the calculated E EB values with experimental data106 for spherical CdSe NPs, and the results are presented in Figure 2. The calculated eh-XCHF EEB values were found to recover between 86 and 99% of the experimental binding energies. The effects of isovolumetric deformation on EEB for all other NPs are also presented in Figure 2. For CdSe NPs, we found that the nanorods exhibited lower EEB than isovolumetric nanodisks and nanospheres. This trend is observed at all three representative volumes. The EEB values showed significant variation with change in material type. Specifically, EEB was found to follow the trend of EEB(PbS) < EEB(CdTe) < EEB(CdSe) < EEB(CdS), which was consistently observed across all NP shapes. In addition to the eh-XCHF method, an approximate description of eh-interaction can also be obtained by modeling the exciton as a hydrogenic system.107 The advantage of this approach is that one can obtain an analytical expression for the exciton binding energy (called the exciton Rydberg energy107) in terms of the material parameters. A comparison between the hydrogenic and eh-XCHF exciton binding energies is presented in Figure 2. In all cases, it was found that the hydrogenic model underestimated the exciton binding energies. Although the hydrogenic model fails to capture the size dependence of the

(10)

To improve the efficiency of the calculation in eq 10, the effect of the geminal operator is moved from the wave function to the Hamiltonian. This congruent transformation of the Hamiltonian has the form seen in eq 11: † ̂ ̂ H̃ = Ĝ HG

(11)

† 1̃ = Ĝ 1Ĝ

(12)

This transformation changes the form of the eh-XCHF energy, such that the wave function used is composed only of Slater determinants for the electron and hole E=

⟨ΦeΦh|H̃ |ΦeΦh⟩ ⟨ΦeΦh|1̃|ΦeΦh⟩

(13)

The three optical properties of interest in this study are the exciton binding energy, the electron−hole joint probability, and the electron−hole separation distance. The exciton binding energy is defined as the difference between the noninteracting and interacting electron−hole pair. E EB = ⟨Enon − interacting ⟩ − ⟨Eexciton⟩

(14)

The electron−hole separation distance is calculated as the expectation value of the absolute difference between the electron and hole coordinates: ⟨reh⟩ = ⟨Ψeh‐XCHF∥re − rh∥Ψeh‐XCHF⟩

(15)

The electron−hole joint probability is calculated as the probability of finding a hole within a small spherical region of radius Δ around the electron coordinate Peh =

1 NeNh

∫ dre ∫ drh ρeh (re, rh)θ(Δ − reh)

(16)

where θ is the Heaviside step function. 25001


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Figure 3. Material parameter η vs exciton binding energy. Data points are grouped into four sets, each corresponding to the labeled material type with volumes of V = 524 nm3.

Figure 4. Electron−hole joint probabilities of (a) CdSe, (b) CdS, (c) CdTe, and (d) PbS nanoparticle spheres, disks, and rods. Peh is presented relative to a referenced P0eh with P0eh = 2.9945 × 10−18).

exciton binding energies, it was able to capture the general trend with respect to material types. The variation in observed EEB across the different NP morphologies, as compared to the static exciton Rydberg energies, highlights the differences in the quantum confinement effect in these systems. The EEB values depend on the fundamental interactions between eh-pairs in these NPs and are strongly influenced by material parameters such as effective masses and dielectric screening parameters. To understand the roles of these material parameters on EEB, we investigated how the excitonic wave function depends on them. Specifically, analysis of the excitonic wave function in the vicinity of the eh-coalescence point (reh → 0) shows that, up to first order, the exciton wave function can be expressed as108 ⎛ ∂Ψeh ⎞ ⎜ ⎟ ⎝ ∂reh ⎠r

η=

(18)

where me and me are effective electron and hole masses and ε is the dielectric constant. The dependence of EEB as a function of η is presented in Figure 3 for NPs with volumes V = 524 nm3. For the present set of materials, each material type has a unique value of η that is independent of NP size or shape. In general, we found that EEB increased with increasing η. Figure 3 also shows the variation in binding energies of differently shaped NPs made of the same material. The results showed that PbS, which had a total range of only 15 meV, is least sensitive to changing EEB with respect to shape deformation. In contrast, CdSe was the material most sensitive to changing shape, with a total range in EEB values of 39 meV. 3.2. Effect of Shape on Electron Hole Joint Probability. Figure 4 presents results for the calculated eh-joint probability (Peh) for the NPs investigated at the three volumes and three morphologies of interest. As expected, smaller NPs had higher Peh values than larger NPs, since smaller NPs confine the electron and hole to a lesser total volume.

∝ ηΨeh(reh = 0) (in atomic units)

eh = 0

memh 1 × ε(me + mh ) m0

(17)

where the unitless parameter η is defined as (eq 18) 25002


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Figure 5. Material parameter η vs electron−hole joint probability. Data points are grouped into four sets, each corresponding to the labeled material type with volumes of V = 524 nm3. Peh is presented relative to a referenced P0eh with P0eh = 2. 9945 × 10−18).

Figure 6. Electron−hole separation distance of (a) CdSe, (b) CdS, (c) CdTe, and (d) PbS nanoparticle spheres, disks, and rods. The separation distance increases as both nanoparticle size and anisotropy increases. Dashed lines correspond to the exciton Bohr radius (in nm) for each material.

Figure 5 shows the relationship between Peh vs η. As observed with EEB, increases in η resulted in increased Peh. The range of Peh values across spheres, disks, and rods was again smallest for PbS, but largest for CdS, rather than CdSe as it was with EEB. PbS was again the least sensitive to changing Peh with respect to changing NP shape. The total range of Peh values was only 0.0302 relative units. CdS was most sensitive to changing Peh with respect to changing shape, with a total range of 0.4434 relative units. This property of PbS to be highly resistant to changing optical properties with respect to shape deformation is likely a result of its large dielectric screening of the ehinteraction.107 3.3. Effect of Shape on Electron Hole Separation Distance. The eh-separation distance was obtained from the expectation value of the eh-distance ⟨ρeh reh⟩. Figure 6 presents the calculated values of ⟨reh⟩ for the investigated NPs. For comparison to the bulk material, the dashed lines on each plot indicate the exciton Bohr radius107 of these materials (in nm).

Focusing on the effect of shape, we found that nanospheres exhibited the smallest Peh values, followed by disks, and then rods, which had the largest joint probabilities. The CdS nanorods had the largest Peh, and the PbS nanospheres had the smallest Peh. While Peh did become larger the smaller the NP volume became, unlike EEB, it also increased as the anisotropy of the NP increased. This result suggests that Peh is dominated not by the average eh-separation distance in the NP but rather by the smallest confining dimension of the system. Since the isovolumetric nanorods have a comparatively small radial axis relative to their length, this small confining dimension forces the electron and hole close together. Although on average the electron and hole can be further apart than in other NP morphologies, they still have more opportunities to join than they do in nanospheres. Therefore, for NPs of identical volume, nanospheres exhibit the least amount of joint probability. Increasing the NP volume will further decrease the joint probability of all morphologies. 25003


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Figure 7. (Top) Nanoparticle volume V vs A(V)RMSD for both A = EEB (left) and A = Peh (right). (Bottom) Nanoparticle material type M vs A(M)RMSD for both A = EEB (left) and A = Peh (right).

Figure 8. Exciton binding energies (in meV) of (a) CdSe, (b) CdS, (c) CdTe, and (d) PbS nanoparticle spheres, disks, and rods calculated using the independent particle approximation.

As expected, ⟨reh⟩ increased with increasing volume. The ⟨reh⟩ distance was inversely correlated with EEB, and thus, NPs with large ⟨reh⟩ distances exhibited smaller EEB. This phenomenon was true for all materials investigated and suggests that, unlike Peh, the largest structural dimension of the system controls ⟨reh⟩. We see that as with exciton Rydberg energies, our ⟨reh⟩ results followed a similar trend to the exciton Bohr radii of these materials. However, ⟨reh⟩ is a much better description of

the spatial separation of the bound eh-pair because, unlike the exciton Bohr radius which treats the exciton approximately as a hydrogenic atom, ⟨reh⟩ is a system-dependent property that will change on the basis of the NP size, shape, and composition. 3.4. Effect of Isovolumetric Deformation. Upon obtaining all of these calculated optical properties of each NP, we were interested in determining the structural parameters that make a particular NP more or less sensitive to changes in its observed optical properties. To do this, we 25004


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Figure 9. Electron−hole joint probability of (a) CdSe, (b) CdS, (c) CdTe, and (d) PbS nanoparticle spheres, disks, and rods calculated using the independent particle approximation. Peh is presented relative to a referenced P0eh with P0eh = 2.9945 × 10−18).

Figure 10. Electron−hole separation distance (in nm) of (a) CdSe, (b) CdS, (c) CdTe, and (d) PbS nanoparticle spheres, disks, and rods calculated using the independent particle approximation.

define the root-mean-squared deviation A(α)RMSD below in eq 19 as

material composition. These results are plotted in Figure 7. For a given NP size, we see that A(α)RMSD for both EEB and Peh were smaller for larger NPs. Therefore, larger NPs are, on average, less susceptible to changing EEB and Peh with respect to changing shape or material type. Compared to EEB, A(α)RMSD for Peh is smaller across all corresponding NP sizes. We conclude that, in general, NPs are more resistant to changes in Peh than they are to EEB when the particle shape or composition is changed. With respect to material type, PbS NPs were, on average, more resistant to changing EEB and Peh with respect to changing NP size and shape. CdS nanoparticles were more

n

A(α)RMSD =

∑i = 1 (Ai (α) − ⟨A(α)⟩)2 n

(19)

where α is a structural property of the NP being fixed, Ai(α) is the optical property of a particular NP satisfying α, ⟨A(α)⟩ is the average of that optical property across all NPs satisfying α, and n is the total number of NPs that satisfy α. A(α)RMSD was calculated for both EEB and Peh at fixed NP volume and fixed 25005


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The Journal of Physical Chemistry C sensitive to changing EEB and Peh with respect to changing NP size and shape. Understanding the relationship between EEB and Peh can help researchers make informed decisions about the types of NP systems to use when designing semiconductor devices. In general, as EEB increased, Peh also increased, but this trend was not strictly followed by all NPs. The actual relationship between these quantities is impacted largely by material type. Overall, Peh scaled fastest with EEB in spheres and slowest in rods. This scaling again varies based on NP material type. The conclusion from this analysis is that, in general, larger rodlike PbS NPs may be the best candidate for semiconductor applications where small binding energies and low Peh values are desired. 3.5. Comparison with Independent Particle Approximation. In addition to the eh-XCHF calculation, we have also performed calculations using the independent particle approximation (IPA). The goal of these calculations is to understand the effect of electron−hole correlation on excitonic properties. The results from these calculations are presented in Figures 8−10. The results are very encouraging and demonstrate that the IPA is able to capture most of the overall trends for shape and size variations in the NPs. However, there are noticeable differences in the results as compared to the eh-XCHF method. The IPA was found to underestimate exciton-binding energy in all NPs. To a more extreme degree, the IPA also dramatically underestimates the eh-joint probabilities, typically by 10 orders of magnitude as compared to the eh-XCHF method. Interestingly, it was also found that the IPA results for ehjoint probability did not exhibit any variation with respect to isovolumetric deformation for any material type. This was true for NPs of all four material types investigated. While the IPA underestimated exciton binding energy and joint probability, it overestimated the eh-separation distances of all NP systems. It was also found that material composition had little effect on the eh-separation distance for the IPA calculations.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This material is based upon work supported by the National Science Foundation under Grant No. CHE-1349892. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant No. ACI-1053575. We are also grateful to Syracuse University for additional computational and financial support.

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REFERENCES

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4. CONCLUSIONS The eh-XCHF method was used to study optoelectronic properties of CdSe, CdS, CdTe, and PbS NP ellipsoids by performing isovolumetric transformations. This general method can be used to study a variety of electron−hole systems. In this work, the electron and hole were treated as an interacting pair confined to a three-dimensional parabolic potential. The ehXCHF wave function was used to calculate the ground-state exciton binding energy, electron−hole separation distance, and electron−hole joint probability of the NPs. We found that all three properties have a strong dependence on ellipsoid shape, even at constant volumes. We also found that exciton binding energy was at a maximum in the case of sphere and minimum in rods, which had the largest aspect ratio. The electron−hole separation distance exhibited the opposite trend. These trends were observed consistently across all semiconductor NPs, regardless of size. Under the independent particle approximation, the overall trends for the investigated optical properties were the same, but the absolute magnitudes were greatly inaccurate.

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Table of material parameters used in the calculation of nanoparticle optical properties (Table S1) (PDF)

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.6b06728. 25006


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