## Abstract

Excitons of tetrapod-shaped nanocrystals made of CdTe, CdS, CdSe, ZnTe, and ZnSe were investigated systematically by numerical diagonalization of configuration interaction Hamiltonian based on single-particle states obtained by finite-element method. Both one-particle and exciton wave functions have high spatial symmetries due to the tetrahedral symmetry of the nanocrystals, which leads to a distinct selection rule for optical absorption and emission. The absorption spectra thus calculated were compared with available experimental data and good agreement was found.

© 2013 Optical Society of America

## 1. Introduction

Since the first report on their synthesis in 2000 [1], tetrapod-shaped nanocrystals of II–VI semiconductors have been attracting a great deal of attention due to their unique structure and interest in the chemical process of their synthesis. Studies on tetrapods made of CdSe [1–5], CdS [2, 6], CdTe [2, 7–15], ZnTe [2, 16], ZnSe [17], and their core/shell combinations [18–22] have been reported. In addition to the synthesis and characterization studies, applications to single-electron transistors [23] and photovoltaic cells were reported [24–28]. Because the energy levels of electron and hole in tetrapod-shaped nanocrystals are quantized due to the three-dimensional confinement of their wave functions, we call the tetrapod-shaped nanocrystals “quantum tetrapods” hereafter.

On the theoretical analysis of the electronic states of quantum tetrapods, single-particle states were discussed by using semiempirical pseudopotential method [29, 30] and effective-mass approximation [9], while exciton states were investigated by Hartree approximation [31–33] and pseudopotential method [34]. On the other hand, we recently analyzed the exciton states of CdTe quantum tetrapods by numerical diagonalization of configuration interaction Hamiltonian [35]. This method is superior to the Hartree approximation if sufficiently converged eigenvalues are obtained, because the latter does not take into consideration the full correlation energy.

In Ref. [35], we showed for CdTe quantum tetrapods that

- one-particle states were irreducible representations of point group
*T*, and the lowest twenty electron and hole states had_{d}*A*_{1}or*T*_{2}symmetry, - exciton states were also characterized by the symmetry of point group
*T*, and the lowest twenty exciton states had_{d}*A*_{1}or*T*_{2}symmetry as well, - there was a distinct selection rule for the electric-dipole transition due to the tetrahedral symmetry of quantum tetrapods and only
*A*_{1}excitons were optically active, - the binding energy of the lowest spin-triplet exciton was exceptionally large (≈ 100 meV) because of the large Coulomb interaction between electron and hole due to their efficient confinement into the small central spherical-core region with a diameter around 2 nm,
- the wavelength of the longest absorption band agreed well with available experimental data.

In the present study, we applied the same method to CdTe, CdS, CdSe, ZnTe, and ZnSe quantum tetrapods with a wider range of sample parameters and examined the above five properties. To the best of our knowledge, no systematic investigation has been reported on the electronic and optical properties of those quantum tetrapods in spite of the rapid progress of their synthesis studies.

This paper is organized as follows. In Section 2, the symmetry properties of one-particle and exciton states of quantum tetrapods are briefly summarized together with the formulation of the method of numerical diagonalization of configuration interaction Hamiltonian. In Section 3, we present numerical results of exciton energies and absorption spectra and show good agreement with available experimental data. Summary of the present study is given in Section 4.

## 2. Theory

We assume the tetrahedral symmetry for quantum tetrapods to clarify their unique electronic and optical properties caused by their structural symmetry, although it is known that there are various types of non-symmetric deformation in actual specimens. Then, Fig. 1 shows the structure and band diagram of the quantum tetrapods that we analyze in this paper. They consist of a spherical central core and four cylindrical arms. Because early experimental studies showed that the core had a zinc blende structure whereas the arms had a wurtzite structure [7–9], we generally assume non-zero band offsets between the core and arms as shown in Fig. 1(b) except ZnSe and ZnTe tetrapods for which both the core and arms have the zinc blende structure [16, 17].

We deal with both electron and hole states of quantum tetrapods by the single-band effective-mass approximation, which is justified when we only treat relatively low energy states close to the absorption edge. The valence band actually consists of heavy and light hole states. Their energy difference is 16 meV for Wurzite CdS, for example, and they are mixed by the confinement potential. However, the kinetic energy of the light hole is generally larger than the heavy hole due to the smaller effective mass of the former, so the light hole states are less important when we discuss the low energy part of the absorption spectrum. This was proven, for example, by Ref. [36] that showed a good agreement between the single-band and the multi-band calculations of low-energy exciton states in CdTe/CdSe core/shell quantum dots. As we will show in the following, the single-band calculation for quantum tetrapods also gives a good agreement with available experimental data. As for the selection rule, on the other hand, we do not think that there is a difference even when we take the light hole into consideration, since the structural symmetry is *T _{d}* and is sufficiently low. So, we assume the following forms for the 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 electron and hole coordinates are denoted by

_{h}**r**

*and*

_{e}**r**

*. The envelope functions are obtained by solving the 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 eigenvalue. The numerical calculation was conducted by the finite element method using commercial software COMSOL Multiphysics. Material parameters assumed in this study are listed in Table 1.

Because we assume the tetrahedral symmetry for the quantum tetrapod structure, the confinement potential *V* is invariant for any symmetry operation *R* of point group *T _{d}*. Since the Laplace operator is also invariant for

*R*, the single-particle Hamiltonian ℋ

*commutes with*

_{e,h}*R*:

*φ*and

_{e}*φ*are 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}) [53].

For exciton energy levels and wave functions, we calculate them by numerical diagonalization of 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*is the dielectric constant of the quantum tetrapod. Because the Coulomb term is also invariant for any symmetry operation

*R*of

*T*, the exciton Hamiltonian ℋ

_{d}*also commutes with*

_{X}*R*:

*T*as well.

_{d}It is important to note that only pair states of the *A*_{1} symmetry contribute to the dipole-allowed optical transition, since the following overlap integral (*I*_{o}) is non-zero only for the *A*_{1} symmetry:

*A*

_{1}symmetry contribute to the dipole-allowed optical transition. As a consequence, if the lowest exciton has another symmetry, quantum tetrapods are essentially non-luminescent.

In our calculation of the Coulomb energy, we did not take into consideration the surface polarization charge induced by the discontinuity of the dielectric constant as was done for spherical quantum dots in Ref. [54]. This is because we cannot obtain its analytical solution for the tetrapod geometry in contrast to the spherical geometry, so we have to rely on the numerical solution of Poisson’s equation to evaluate the Coulomb term, which is quite time-consuming and impractical in the theoretical framework of the present study. The neglect of the surface polarization charge may result in the underestimation of the binding energy of excitons. This matter remains a future problem.

To evaluate the Coulomb term, we should take into consideration the exchange interaction for different spin configurations. For spin-singlet pair states, we can easily prove that the matrix element of the two-body part (ℋ_{2}) of the exciton Hamiltonian (ℋ* _{X}*) is given by

*D*and

*L*was about 24 hours by parallel computing with 32 CPUs. Because the Bohr radius (

*a*

_{B}) of the semiconductor materials analyzed in the present study is from 2.6 nm (CdS) to 6.5 nm (CdTe), the structural size of the tetrapod that is represented by

*D*satisfies

*D*/2 ≤

*a*

_{B}in most cases, which means that the system is in the strong confinement regime. So, the Coulomb energy is relatively small compared with the kinetic energy, which justifies our numerical method of the diagonalization of the configuration interaction Hamiltonian. It also brings about distinct peaks in the absorption spectra in spite of the large inhomogeneous broadening as will be shown in Section 3.

We found that most of the electron and hole wave functions in the low energy range are characterized by the angular momentum around the arm axis, *m*, and the number of nodes along the arm axis, *n*, since four arms are approximately independent being separated by the central core and the arms have a cylindrical symmetry. We also found that as far as the low energy range close to the absorption edge is concerned, important contribution is made by one-particle states with *m* = 0. Then, it is convenient to know their possible symmetry in advance, since we can judge whether they can contribute to the dipole-allowed optical transition. When we denote the electron or hole wave function localized in the *i* th arm by *ϕ _{i}*, it is easy to see that their symmetric combination gives a wave function of the

*A*

_{1}symmetry of point group

*T*:

_{d}*T*

_{2}-symmetric energy level:

On the other hand, the lowest electron wave function, which has the *A*_{1} symmetry, is localized in and around the central core. It is worth noting that we can tell the symmetries of low energy excitons from this fact. Actually, the pair states composed of the lowest electron level and low energy hole levels, the latter of which have *A*_{1} or *T*_{2} symmetry as mentioned above, can only have *A*_{1} or *T*_{2} symmetry, which can be easily verified by using the well-known reduction formula of group theory [53]. Therefore, excitons in the low energy range close to the absorption edge are characterized by the *A*_{1} and *T*_{2} symmetries, among which only *A*_{1} excitons contribute to dipole-allowed optical transitions. As a consequence, if the lowest exciton has the *T*_{2} symmetry, quantum tetrapods are non-luminescent unless thermal excitation induces a non-vanishing population of higher energy *A*_{1} excitons.

## 3. Results and discussion

Figure 2 shows the *D* (arm diameter) dependence of the lowest twenty exciton energies for CdTe, CdS, CdSe, ZnTe, and ZnSe quantum tetrapods. From our previous study [35], we found that the *L* (arm length) dependence of the exciton energy was small, so we fixed it to 9 nm, which is a typical value observed in experiments. On the other hand, we varied *D* from 2.2 nm to 7 nm in order to compare our numerical results with available experimental data. We also examined the wave functions of the lowest twenty electron and hole states, which govern the absorption and emission spectra in the vicinity of the absorption edge, and found that all of them were characterized by angular momentum of *m* = 0 and had *A*_{1} or *T*_{2} spatial symmetry. In a higer energy range, we also found *m* ≥ 1 states. But they were not important for the energy range that we deal with in this study.

As we described in the previous section, all these excitons have *A*_{1} or *T*_{2} symmetry. They show an apparent blue shift with decreasing *D* as a consequence of quantum confinement of exciton wave functions. Most of the lowest spin-singlet excitons have the *A*_{1} symmetry, so those quantum tetrapods are luminescent. Exceptions are CdS for all *D*, CdTe with *D* ≥ 4 nm, and CdSe with *D* ≥ 5 nm. Their lowest spin-singlet excitons have the optically inactive *T*_{2} symmetry, so they are basically non-luminescent. However, the energy difference between these lowest *T*_{2} and the lowest *A*_{1} excitons is less than 2 meV. So, the room-temperature thermal energy (26 meV) mixes the population of these two exciton states and luminescence from the *A*_{1} exciton level must be observed.

In Fig. 2(f), the energy of spin-triplet excitons is shown. As we found for CdTe tetrapods with 1.9 nm ≤ *D* ≤ 2.2 nm in our previous study [35], the binding energy of the lowest spin-triplet exciton, which is defined by the energy difference between the lowest pair state and the lowest spin-triplet exciton state, is exceptionally large due to the strong confinement of both electron and hole wave functions to a small central-core region, and so, the large negative Coulomb energy. This binding energy decreases with increasing *D* because of the delocalization of the wave functions. As shown in Fig. 3, this tendency is common to all tetrapods that we dealt with in this study.

Because we obtained the energy levels and wave functions, we could calculate the absorption spectra of quantum tetrapods according to Fermi’s golden rule. The absorption spectra of CdTe tetrapods are shown in Fig. 4. Because the single-band approximation is appropriate for the low energy range around the absorption edge, we focused on the lowest and second lowest absorption bands in our calculation, for which the single-band approximation can safely be applied. A continuum of higher energy absorption bands should follow these two bands. The actual specimens of quantum tetrapods have a fairly large size distribution. So, we rather arbitrarily assumed an inhomogeneous width of 60 meV (FWHM) for each absorption line. This value corresponds to the diameter-size (*D*) distribution of about 0.5 nm on average, which was deduced from the *D* dependence of the exciton energy shown in Fig. 2. This value (0.5 nm) of the diameter-size distribution is among the typical values found in experimental studies. In addition to an obvious quantum confinement effect, the relative decrease of the lowest band intensity with increasing *D* is observed in Fig. 4, which is caused by the decrease of the overlap integral of the lowest electron and hole wave functions mainly due to the delocalization of the latter with increasing *D*. This point will be discussed again later in connection with Fig. 6.

Figure 5 shows the absorption spectra of five kinds of quantum tetrapods, where *D* is fixed to 3 nm. Their spectral shift is mainly caused by the change in the band gap energy.

Finally, Fig. 6 shows the *D* dependence of the lowest absorption peak energy and its comparison with experiments, where the amount of the *D* dependence is mainly governed by the kinetic energy of carriers, and so, by their effective mass. Experimental data are mainly available for CdTe and CdSe tetrapods. If we take into consideration the fairly large size distribution and structural deformation from the perfect tetrahedral symmetry in actual specimens, we may conclude that the agreement between our calculation and the experimental observation is good.

However, there is somewhat systematic deviation between them for the CdTe tetrapod with *D* ≥ 3 nm. This deviation may be explained by the relative decrease of the lowest band intensity compared with the second lowest band. If the inhomogeneous width is large, the lowest band may be difficult to identify experimentally and the second band may be regarded as the absorption edge. From this view point, the calculated peak energy of the second band is also plotted in Fig. 6(a), which shows considerably good agreement with the experimental data for *D* ≥ 4 nm.

## 4. Conclusion

We systematically investigated exciton states of CdTe, CdS, CdSe, ZnTe, and ZnSe quantum tetrapods by numerical diagonalization of configuration interaction Hamiltonian with the single-band effective-mass approximation. We found five main features of their electronic and optical properties: (1) the lowest twenty electron and hole states, which govern the absorption and emission spectra in the vicinity of the absorption edge, have *A*_{1} or *T*_{2} symmetry; (2) the lowest twenty exciton states have the *A*_{1} or *T*_{2} symmetry as well; (3) most of the lowest spin-singlet excitons have the *A*_{1} symmetry, so they are optically active and luminescent. Even when it is of *T*_{2} symmetry, the room-temperature thermal energy induces non-vanishing population in the lowest *A*_{1} exciton, so the tetrapod can essentially be luminescent; (4) the binding energy of the lowest spin-triplet exciton is exceptionally large, for small *D* in particular, because of the large Coulomb interaction between electron and hole due to their efficient confinement into the small central-core region; (5) the wavelength of the lowest absorption band agrees well with available experimental data.

## Acknowledgments

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

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