## Abstract

Excitons of CdTe tetrapod-shaped nanocrystals are theoretically analyzed. Individual electron and hole states are calculated by solving one-particle Schrödinger equation by the finite element method with the single-band effective-mass approximation and exciton states are obtained by numerical diagonalization of the configuration interaction Hamiltonian. Spatial symmetries of the exciton states are related to those of the one-particle states by group theory and verified by numerical calculation. It is shown that the lowest exciton state is an optically active *A*
_{1} exciton. Optical absorption spectra are calculated and compared with available experimental data.

© 2011 OSA

## 1. Introduction

Since the first report on their chemical synthesis in 2000 [1], tetrapod-shaped nanocrystals made of II–VI semiconductors have attracted a great deal of attention. Studies on tetrapods made of CdSe [1,2], CdS [2], CdTe [2–10], ZnTe [2], and their core/shell combinations [11–15] have been reported. In addition to these synthesis and characterization studies, application to photovoltaic cells, for example, has also been reported [16, 17]. As for theoretical analysis, calculations of one-particle states by the semiempirical pseudopotential method [18, 19] and exciton states by the Hartree approximation [20–22] were reported.

Similar to other semiconductor nanostructures, energy levels of electrons and holes in tetrapod-shaped nanocrystals are quantized due to the three-dimensional confinement of their wave functions, and so the tetrapods have discrete absorption and emission bands with an apparent quantum size effect; there is a blue shift of the bands as their size decreases, although the discreteness of the absorption bands in high frequency ranges is often hidden by a large inhomogeneous broadening due to their size distribution. Because of these quantum-confinement properties, we call the tetrapod-shaped nanocrystals “quantum tetrapods” hereafter.

The optical properties of the quantum tetrapods also depend on their shape. In contrast to quantum dots with more or less spherical shapes, ideal quantum tetrapods literally have the shape of a tetrapod. Because the optical properties of quantum nanostructures are crucially influenced by the nature of electron and hole wave functions, which in turn strongly depend on the structural symmetry, it is of fundamental importance to investigate the consequence of the unique tetrapod symmetry.

In this paper, we report on the calculation of their exciton states that are responsible for the absorption and emission bands assuming the ideal tetrahedral symmetry. In order to concentrate on the consequence of their structural symmetry, we deal with the CdTe tetrapod, which is one of the simplest and most investigated structures, and apply the single-band effective-mass approximation for both conduction-band electrons and valence-band (heavy) holes. CdTe has the narrowest energy bandgap (1.5 eV) in the series of cadmium chalcogenides, which is important for photovoltaic applications. We first calculate the one-particle energy levels by the finite element method. Next, we evaluate the matrix elements of the Coulomb and exchange interactions for the electron-hole pair states, and thus obtain the configuration interaction Hamiltonian. We numerically diagonalize it and obtain the wave functions and energy eigenvalues of excitons. We show that the lowest spin-singlet exciton level is of the *A*
_{1} symmetry of point group *T _{d}*, which contributes to dipole-allowed optical transition to and from the ground state. We also present absorption spectra of the tetrapods assuming several different structural parameters and compare these calculations with available experimental data.

To the best of our knowledge, this is the first calculation of exciton states of quantum tetrapods by the rigorous diagonalization of the configuration interaction Hamiltonian. This method is superior to the Hartree approximation [20–22] if sufficiently converged eigenvalues are obtained, because the latter does not take into consideration the full correlation energy.

## 2. Theory

We deal with the problem of excitons in the CdTe tetrapod by single-band effective-mass approximation. We assume the following forms for the spatial part of electron and hole wave functions:

*φ*(

_{e}*φ*) and

_{h}*u*(

_{e}*u*) are the envelope function and atomic wave function of the conduction band electron (heavy hole), respectively. The envelope functions are obtained by solving the time-independent Schrödinger equation assuming an isotropic effective mass for both the electron ( ${m}_{e}^{*}$) and heavy hole ( ${m}_{h}^{*}$):

_{h}*V*is the confinement potential, and

*E*is the energy eigenvalues.

The structural parameters assumed in this study are summerized in Fig. 1. Early experimental studies [3–5] showed that the four arms of the CdTe tetrapod have a wurzite (WZ) crystal structure, whereas the central region that connects the four arms, which we call the “core” hereafter, has a zinc blende (ZB) structure. In the present study, we follow these observations and assume the band parameters shown in Fig. 1(b).

We assume a perfect tetrahedral symmetry for the CdTe tetrapod structure, so that the confinement potential is invariant for any symmetry operation *R* of point group *T _{d}*. Since the Laplace operator is invariant for

*R*as well, the single-particle Hamiltonian

*ℋ*commutes with

_{e,h}*R*:

*φ*and

_{e}*φ*must be irreducible representations of point group

_{h}*T*. It has two one-dimensional representations (

_{d}*A*

_{1}and

*A*

_{2}), one two-dimensional representation (

*E*), and two three-dimensional representations (

*T*

_{1}and

*T*

_{2}). Their characters are listed in Table 1 [26]. In the next section, we will give the results of numerical culculations of the envelope functions by the finite element method and show that each of them is attributed to an irreducible representation in Table 1.

We calculate the exciton wave functions by exact diagonalization of the configuration interaction Hamiltonian. We solve the following two-body Schrödinger equation based on the expansion of the total wave function Ψ by the linear combination of pair states of electron and hole envelope functions:

*e*

_{0}denotes the elementary charge,

*ε*

_{0}is the permittivity of free space, and

*ε*(= 10.6) is the dielectric constant of CdTe [25].

Because the Coulomb term is also invariant for any symmetry operation *R* of *T _{d}*, the exciton Hamiltonian

*ℋ*commutes with

_{X}*R*:

*T*as well. Since we expand Ψ by pair states

_{d}*φ*(

_{e}**r**

*)*

_{e}*φ*(

_{h}**r**

*), it is convenient to know the symmetry of the pair states in advance. It can be obtained by the standard reduction procedure [26] and is summarized in Table 2, which will be used in the next section.*

_{h}It is worth noting that only pair states of the *A*
_{1} symmetry contribute to the dipole-allowed optical transition, since the transition from the valence band to the conduction band is dipole-allowed for CdTe and the following overlap integral (*I*
_{o}) of the electron and hole envelope functions is non-zero only for the *A*
_{1} symmetry:

To evaluate the Coulomb term, we should take into consideration the exchange interaction for different spin configurations. For spin-singlet pair states, whose total wave function may be denoted by

*ℋ*

_{2}) of the exciton Hamiltonian (

*ℋ*) is given by

_{X}## 3. Results and Discussion

#### 3.1. One-particle states

We solved one-particle Schrödinger equations given in Eqs. (3) and (4) by the finite element method with the commercial software COMSOL [27]. As an example, energy eigenvalues and wave function symmetries for the case of *L* = 9.0 nm and *D* = 2.2 nm are listed in Table 3. Although the effective mass and confinement potential are different between electron and hole, the symmetry of the envelope functions are the same for the two as far as the lowest twenty energy levels are concerned. In this energy range, we only have envelope functions of the *A*
_{1} and *T*
_{2} symmetries.

Figure 2 shows two examples of the electron envelope functions, that is, the lowest two *A*
_{1} states. It is apparent that the lowest *A*
_{1} state is localized in the core, whereas the second lowest *A*
_{1} state is distributed equally in the four arms. These features agree with previous calculations for larger CdTe [5], CdSe [18], and CdSe/CdS [22] tetrapods.

If we regard the localized distributions of the electron in each arm as independent wave functions and denote them by ϕ_{1} to ϕ_{4}, then the second lowest *A*
_{1} state is expressed by the following symmetric combination of ϕ’s:

*T*

_{2}state:

*R*of

*T*on any one of these three basis functions results in their linear transformation and the trace of the transformation matrix has the property of the

_{d}*T*

_{2}representation.

For higher *A*
_{1} and *T*
_{2} energy levels listed in Table 3, they have the similar property. The only difference is the number of nodes of the envelope function along the cylinder axis. That is, whereas either the second lowest *A*
_{1} or the lowest *T*
_{2} envelope function does not have a node along the cylinder axis, envelope functions of higher energy levels have an increasing number of nodes.

#### 3.2. Exciton states

Before showing numerical results for the exciton states, let us first give a qualitative prediction by the group theory. When we neglect the Coulomb potential, we have energies and symmetries of excitons, or electron-hole pair states, shown in Table 4, which was obtained by listing the pair states in the ascending order of their energy and assigning their symmetry by consulting Table 2. From Table 4, we may expect that low energy excitons have the *A*
_{1} or *T*
_{2} symmetry.

The exciton states with the Coulomb potential was obtained by numerically diagonalizing the *ℋ _{X}* matrix evaluated with |

*ij*(

*s*)〉 or |

*ij*(

*t*)〉. To check the convergence of the calculation, we examined the variation of energy eigenvalues with respect to the number of basis pair functions to calculate the

*ℋ*matrix elements. As shown in Fig. 3, the convergence is fast, and 400 pair states give sufficiently converged results.

_{X}
Figure 4 shows the energy eigenvalues of the lowest 20 spin-singlet excitons. We assume three values for both *D* and *L* according to observation by electron microscope in Refs. [10] and [15]. In all cases, the lowest exciton state is of the *A*
_{1} symmetry, and so it is optically active. The *D* dependence is much stronger than the *L* dependence as far as the analyzed parameter ranges are concerned, which agrees with the results of previous experiments that did not show apparent dependence of absorption peaks on *L* [10, 15]. The symmetries of the lowest 20 excitons are *A*
_{1} or *T*
_{2}, which agree well with the group theoretical prediction given in Table 4.

As an example, let us describe the case of *L* = 9.0 nm and *D* = 2.2 nm in more detail. The energy eigenvalue of the lowest spin-singlet *A*
_{1} exciton is 2.234 eV, which mainly (92 %) consists of the lowest *A*
_{1} electron-hole pair listed in Table 4. The exciton binding energy is 11 meV, if we define it by the energy difference between the absence and presence of the Coulomb attraction in the *A*
_{1} state. The lowest *T*
_{2} exciton at 2.254 eV mainly consists of the lowest two *T*
_{2} electron-hole pairs at 2.286 (86 %) and 2.307 eV (12 %) in Table 4. Since its overlap integral (*I*
_{o}) is equal to zero by symmetry, it does not contribute to optical transition.

Next, Fig. 5 shows the results for the lowest 20 spin-triplet excitons. Again the lowest state is of the *A*
_{1} symmetry. For the case of *L* = 9.0 nm and *D* = 2.2 nm, it mainly (96 %) consists of the lowest *A*
_{1} electron-hole pair. Its binding energy is 98 meV, so it is much more stabilized than the lowest singlet exciton. The lowest *T*
_{2} spin-triplet exciton at 2.231 eV mainly consists of the lowest three *T*
_{2} electron-hole pairs at 2.286 eV (55 %), 2.307 eV (32 %), and 2.341 eV (11 %). Note that optical transition between these spin-triplet excitons and the ground state is, of course, spin-forbidden.

Let us give a comment on the exceptionally large binding energy of the lowest *A*
_{1} spin-triplet exciton. The averaged value of 1/|**r**
_{1} – **r**
_{2}| in Eq. (12) is 0.67 × 10^{−9} [1/m] for the lowest *A*
_{1} electron-hole pair. If we regard this value as the inverse of the averaged electron-hole distance, the latter is 1.5 nm. Since the wave functions of both the lowest electron and hole states are localized in the core region whose diameter is 2.2 nm, the calculated value of 1.5 nm looks reasonable. The Coulomb matrix element corresponding to this averaged distance is −100 meV. As we mentioned before, the lowest *A*
_{1} spin-triplet exciton mainly (96 %) consists of the lowest *A*
_{1} electron-hole pair. So, its binding energy of 98 meV, which is close to 100 meV, is again very reasonable. On the other hand, in the case of the lowest *A*
_{1} spin-singlet exciton, we have to take into consideration the exchange term whose sign is opposite to the Coulomb term. Thus they nearly cancel out and give a moderate binding energy of 11 meV.

Since we now know energy eigenvalues and wave functions of excitons, we can calculate their absorption spectra by using Fermi’s golden rule. Figure 6 shows spectra for three different *D* values. The blue shift due to the quantum size effect is clearly observed. In each spectrum, the longer wavelength peak mainly consists of the lowest and the second lowest *A*
_{1} excitons, whereas the shorter wavelength peak mainly consists of the eighth lowest *A*
_{1} exciton. Contributions from other *A*
_{1} excitons are relatively small.

Reference [10] and [15] reported synthesis and optical absorption spectra of CdTe tetrapods. Calculated values in the present study agree with the experimental observation of [10] quite well. Their deviation is less than 20 nm. On the other hand, Ref. [15] reported that the longest peak wavelength of the CdTe tetrapod was 620 nm for the case of *D* = 2.2 ± 0.3 nm and *L* = 9.0 ± 0.7 nm, which is appreciably larger than the calculated value. The reason is not necessarily clear, but the specimen may have suffered from aggregation of tetrapods. As for the second longest peak found by the calculation, they were not observed by the experiments presumably due to large inhomogenious broadening of the absorption bands. However, such peaks are frequently observed for larger tetrapods with smaller size distribution. Their detailed comparizon remains to be examined in future works.

Finally, Fig. 7 shows absorption spectra for three different *L* values. Since the exciton energy does not depend strongly on *L* as far as the analyzed parameter range is concerned, the three spectra in Fig. 7 are not so different from each other. The shorter wavelength peak shows a small normal blue shift with decreasing *L*, whereas the longer wavelength peak shows a slight red shift. The latter, which may look strange at first glance, is actually caused by increased absorption strength of the lowest *A*
_{1} exciton, which results in the red shift of the center-of-mass of the longer wavelength peak.

Generally speaking, for larger variation of the arm length *L*, we expect both the larger shift of the absorption peaks, which is caused by the energy shift of one-particle states due to the quantum size effect, and the larger change in the peak height, which is caused by the change in the overlap integral of the electron and hole wave functions. However, although the exciton energy was calculated within a variation of only 15 % of *L* in the present study, we examined the electron and hole state energies for *L* = 6 and 12 nm (67 % variation around *L* = 9 nm) and found only minor variation of energy for the first *A*
_{1} state (2.3 meV for electron and 0.01 meV for hole) and little bit larger variation for the first *T*
_{2} state (54 meV for electron and 12 meV for hole). As electron and hole energies constitute the main part of the exciton energy, we may expect the lengthening effect on the low-energy exciton states to be small in this range of *L*.

Such a small lengthening effect on the exciton energy agrees well with previous experimental observations for CdTe tetratpods [3] and suggests that the band gap is mainly determined by the lateral confinement, whereas spectra of tetrapods with comparable diameters but different arm lengths are almost identical.

In this study, we assumed the perfect tetrahedral symmetry for the CdTe tetrapods, although the present method is applicable to tetrapods with broken symmetries as well without an increase in the computational cost. For tetrapods with broken symmetries, localization of wave functions in particular arms and anisotropic emission polarization for individual tetrapods are known [22]. However, for an ensemble of tetrapods, the inhomogeneously broadened absorption and emission bands are usually isotropic due to the stochastic distribution of the tetrapod orientation. So, the absorption spectrum of the ensemble reflects the nature of the symmetric tetrapod structure.

## 4. Conclusion

We calculated the exciton states of CdTe quantum tetrapods with tetrahedral symmetry by numerical diagonalization of the configuration interaction Hamiltonian for the first time. For the basis electron-hole pair states, we calculated them by the finite element method with single-band effective-mass approximation. Spatial symmetry of the envelope functions thus obtained agreed with the prediction of group theory. For all combinations of assumed structural parameters, we found that the lowest exciton state is an optically active *A*
_{1} exciton. The culculated absorption spectra showed a good agreement with previous experimental results. Thus we clarified the size effect of the energy bandgap of CdTe tetrapods by this systematic theoretical study.

## Acknowledgments

This study was supported by Grant-in-Aid for Scientific Research (B) from Japan Society for the Promotion of Science (Grant number 23340090).

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