## Abstract

We report our theoretical investigation onto the shape dependence of two-photon absorption (TPA) in semiconductor nanocrystals (NCs). Based on a four-band model under effective mass approximation, we have developed a simple analytical theory capable of providing a quantitative explanation of the recent TPA measurement on CdS nanorods [Appl. Phys. Lett. 94, 103117 (2009)]. With this theory, we have systematically revealed the characteristics of TPA in CdSe and ZnO NCs with four different shapes: sphere, cube, cylinder and cuboid. Due to the splitting of degenerate energy levels caused by the decreased degree of symmetry, nanocuboids and nanocubes exhibit greater TPA cross-sections than nanocylinders and nanospheres of similar sizes, respectively. Similarly, nanocuboids and nanocylinders possess larger TPA cross-sections than nanocubes and nanospheres of similar lateral dimension, respectively. Given TPA-allowed transitions, nanocuboids show stronger size dependence than nanocylinders. The size dependence of TPA cross-section is more sensitive to the lateral size than the longitudinal size in the cases of nanocylinders and nanocuboids.

© 2009 Optical Society of America

## 1. Introduction

Over the past decade, semiconductor nanocrystals (NCs) have received considerable attention due to their potential applications in optical switching for optical communications, three-dimensional optical data storage, optical limiting for protection of optics sensors from laser-induced damages, three-dimensional confocal imaging for biological specimens, and photodynamic therapy [1–6]. In these applications, semiconductor NCs play an essential role in laser excitation through a nonlinear optical process: two-photon absorption (TPA).

Many novel phenomena in semiconductor NCs result from quantum confinement effects, which are strongly related to two crucial nanoscale geometrical factors: size and shape. Extensive studies have been carried out toward revealing the size-dependent optical properties of semiconductor NCs due to the fact that their optical and electronic properties can be tuned simply by altering their size [7–9]. Recent advances in chemical synthesis have enabled the achievement of semiconductor NCs with an increased degree of structural complexity and shape. For example, there are reports on semiconductor NCs with shapes of rod, bullet pyramid, cube and tetrapod as well as hyperbranched dendritic geometries [10–17]. Whereas the variation of NC size mainly changes energy level spacing, the difference in shape might cause more subtle changes in the electronic state structure, symmetry of wave functions, polarization, and localization of the electronic states [18–21]. Thus, semiconductor NCs with various shapes offer a platform which not only allows researchers to study the fundamental aspect of material physics, for example, how the nonlinear optical properties of semiconductor NCs depend on the shape, but also provides ones with rich opportunities of engineering the certain optical property of semiconductor NCs for desired applications.

The alteration in the electronic state structure and symmetry of wave functions caused by varying size or shape of semiconductor NCs should be expected to moderate the linear optical properties as well as nonlinear optical properties such as two-photon absorption (TPA). Up to now, size-dependent TPA in semiconductor NCs in the shape of spheres or quantum dots (QDs) has been studied intensively, in the context of both theory and experiment. Fedorov et al. [22] established the frequency-degenerate TPA theory in CdS QDs under the framework of both effective mass approximation and well-known four-band model. In their model, there are a doubly spin degenerate conduction band, a heavy-hole band, a light-hole band and a spin-orbit-split band; and all the four bands are treated mathematically as being parabolic with constant effective masses. Padilha et al. not only extended it to frequency-nondegenerate TPA [23] but also developed it into a *k.p* model [24]. They also measured the TPA spectra in CdSe and CdTe QDs with different dot sizes and size distributions. The TPA cross-sections of CdSe QDs were measured to be approximately ~10^{−46} cm^{4} s/photon [25, 26]. The TPA in ZnS QDs was observed at wavelengths of 532 nm and 520 nm by using picosecond laser pulses [27]. Both nonlinear refractive index and TPA coefficient of Mn-doped ZnSe QDs were determined by Z-scan technique at 800 nm wavelength [28, 29]. The TPA of ZnO QDs was also characterized in the wavelength range from 700 nm to 900 nm [30]. The TPA cross-section was observed as high as 10^{4} GM (1 GM=10^{-50}cm^{4} s/photon) for 4.9-nm-sized CdTe QDs [31]. All the above-mentioned research efforts were focused on the size dependence. Very recently, Gu’s group measured the TPA cross-sections of CdS nanocrystal rods and dots by Z-scan technique at 800 nm and found that the TPA cross-section of CdS quantum rods was one order of magnitude larger than CdS QDs of similar diameters [32]. Up to now, however, no theoretical studies have been reported on the TPA of semiconductor NCs with non-spherical shape.

Here we report a simple analytical theory on the shape-dependent TPA of semiconductor NCs. The simplicity and validity of Fedorov’s TPA model for larger semiconductor QDs [23, 24] make a foundation for our modeling. Following Fedorov’s theoretical approach for the shapes of spherical NCs, we extend it to NCs with the shapes of cube, cylinder and cuboid, whose size descriptions are defined in Fig. 1. Under the effective mass approximation and four-parabolic-band model, the analytical expressions of TPA cross-section are derived for the three differently shaped NCs in the quantum confinement regime. The comparisons of TPA spectra among nanosphere, nanocube, nanocylinder and nanocuboid are carried out. In order to verify our theoretical model, TPA cross-sections of CdS NCs with the shapes of sphere and cylinder are calculated firstly. With the theoretical results agreeable to the measurement by Gu’s group [32], we then make prediction for shape-dependent TPA in CdSe and ZnO NCs of different shapes. These two semiconductors are selected because their NCs of various shapes have been successfully synthesized and reported [10, 33–35]. We find, for the first time, that TPA cross-section is dependent on the NC shape with the following characteristics: (1) lower degree of symmetry and anisotropy give rise to larger TPA cross-sections; and (2) size dependence of TPA cross-section is more sensitive to the lateral size (e.g. width or diameter) than the longitudinal size (e.g. length).

## 2. Theory

The two-photon generation rate with incident light frequency *ω* can be represented in second-order perturbation theory with respect to the electron-photon interaction as [36]

$${M}_{{v}_{1},{v}_{0}}=\underset{{v}_{2}}{\Sigma}\genfrac{}{}{0.1ex}{}{{H}_{{v}_{1},{v}_{2}}^{\mathrm{int}}{H}_{{v}_{2},{v}_{0}}^{\mathrm{int}}}{{E}_{{v}_{2}}-{E}_{{v}_{0}}-\hslash \omega -i\hslash {\gamma}_{{v}_{2}}}.$$

where *E*
_{v0}, *E*
_{v1} and *E*
_{v2} represent the energies of the initial, final and intermediate states of an electron, respectively. *H ^{int}*=(

*e*/

*m*

_{0}c)

**A·p**describes the electron-photon interaction and

*m*

_{0}is the mass of free electron,

**A=Ae**is the vector potential of the light wave with the amplitude

*A*and the polarization vector

**e, p**is the electron momentum operator, and

*γ*

_{v2}is the inverse of the lifetime in each excited state. Since TPA is a process wherein two photons are absorbed simultaneously from the initial state through one virtual state to the final state, for semiconductor NCs, each integrated TPA process from the valence band to the conduction band contains one intraband transition and one interband transition. Noted that the energy levels and electronic states are sensitive to the NC shape [18], the parameters

*E*and

_{v}*H*

^{int}in Eq. (1) are dependent on both shape and size. Similar to the approach taken by Fedorov et al. [22], we also adopt their theoretical model wherein there are four independent bands, explicitly including the doubly degenerate conduction band and two-fold degenerate bands of heavy, light, and spin-orbit-split holes; all the effective masses are constant; and there is no band mixing between the light and heavy holes in the valance band. Because of shape and size independence, the matrix elements of the one-photon interband transitions in cube, cylinder and cuboid can also be expressed as the formula in Ref. [22], which is the same as sphere. However, a different situation occurs in the case of intraband transition whose matrix elements are strongly dependent on the NC shape and size. The general expression of matrix elements for one-photon transition in the

*a*band (

*a*=conduction band or heavy hole, light hole, and spin-orbit-split holes) can be written generally as

where *nlm* (*n’l’m’*) are the quantum numbers which denote the electronic states, and *m _{a}* is the electron effective mass in the

*a*band. The function

*B*differs greatly for different NC shapes. For cylinder, it is given by

$$+\genfrac{}{}{0.1ex}{}{1}{L}m{\delta}_{n,n\prime}{\delta}_{l,l\prime}\left(1-{\delta}_{m,m\prime}\right)\left[\genfrac{}{}{0.1ex}{}{1-\mathrm{cos}\left(m+m\prime \right)\pi}{m+m\prime}+\genfrac{}{}{0.1ex}{}{1-\mathrm{cos}\left(m\prime -m\right)\pi}{m\prime -m}\right]{e}_{z}\},$$

where *nlm* (*n’l’m’*) are the quantum numbers in three directions of the cylindrical coordinates, *µ _{n}^{l}* is the

*n*th root of the

*l*th-order Bessel function,

*e*(

_{j}*j=x, y, z*) are the Cartesian components of the polarization vector, and

*D*and

*L*are the diameter and length of the cylinder as shown in Fig. 1.

As for cuboid, the function *B* is as follows,

$$+\genfrac{}{}{0.1ex}{}{1}{D}l{\delta}_{m,m\prime}{\delta}_{n,n\prime}\left(1-{\delta}_{l,l\prime}\right)\left(\genfrac{}{}{0.1ex}{}{1-\mathrm{cos}\left(l+l\prime \right)\pi}{l+l\prime}+\genfrac{}{}{0.1ex}{}{1-\mathrm{cos}\left(l\prime -l\right)\pi}{l\prime -l}\right){e}_{y}$$

$$+\genfrac{}{}{0.1ex}{}{1}{L}m{\delta}_{n,n\prime}{\delta}_{l,l\prime}\left(1-{\delta}_{m,m\prime}\right)\left(\genfrac{}{}{0.1ex}{}{1-\mathrm{cos}\left(m+m\prime \right)\pi}{m+m\prime}+\genfrac{}{}{0.1ex}{}{1-\mathrm{cos}\left(m\prime -m\right)\pi}{m\prime -m}\right){e}_{z}].$$

Here we introduce the definition of aspect ratio* V, V=L/D*. When *D=L*, Eq. (4) is adapted to nanocube. From Eqs. (3–4) and Eq. (6) in Ref. [22], we can find that the matrix elements of one-photon intraband transition differ greatly for different NC shapes. For the same shape, the intraband matrix elements depend explicitly upon the NC size. However, the current state-of-the-art nanotechnology gives very little scope to manufacture NCs with a uniform size. Hence, one has to study the NCs with an inhomogeneous size dispersion, which is characterized by a size-distribution function *f* (*a*) in the experimental studies. Under the groundwork of interband and intraband transition matrix elements of the electron-photon interaction in NCs, considering orientations and size distribution, the TPA coefficient *α*
_{2} for an ensemble of NCs is related to the average two-photon generation rate *W*̄^{(2)} by

where *N* is the NC concentration and *I* is the incident light intensity *I*=*ε _{ω}*

^{1/2}

*ω*

^{2}

*A*

^{2}(2

*πc*)

^{-1}, and

*ε*is the dielectric constant of the semiconductor at the light frequency. The size distribution results from the conditions of sample preparation. Usually, the Gaussian function [37] is mostly used. For an arbitrary function

_{ω}*f*(

*a*), the TPA coefficient can be expressed as follows,

$$\u3008{F}_{c,{h}_{j}}\u3009=\underset{\mathrm{nlm},n\prime l\prime m\prime}{\Sigma}\int {\mid B\mid}^{2}Tf\left(a\right)\delta \left({E}_{{v}_{0}}^{{h}_{j}}-{E}_{{v}_{1}}^{c}-2\hslash \omega \right)\mathrm{da},$$

$$T=\mid \genfrac{}{}{0.1ex}{}{1}{{m}_{c}}\genfrac{}{}{0.1ex}{}{1}{\left({E}_{{v}_{0}}^{c}-{E}_{{v}_{0}}^{{h}_{j}}-\hslash \omega -i\hslash {\gamma}_{{v}_{0}}\right)}+\genfrac{}{}{0.1ex}{}{1}{{m}_{{h}_{j}}}\genfrac{}{}{0.1ex}{}{1}{\left({{E}_{{v}_{0}}^{{h}_{j}}-E}_{{v}_{1}}^{{h}_{j}}+\hslash \omega +i\hslash {\gamma}_{{v}_{1}}\right)}\mid ,$$

where *P*=ħ^{2}
*p _{c,h}/m*

_{0}=

*ħ*

^{2}<

*S|∂/∂*>/

_{Z}|Z*m*

_{0},

*p*is the interband matrix element of the electron momentum.

_{c,h}*h*(

_{j}*j*=1,2,3) refers to the light hole, heavy hole and spin-orbit-split hole bands, respectively. The important feature of the given two-photon transitions is the selection rules. According to Eqs. (3–4), TPA transitions can occur only if quantum numbers of the electrons (

*n’l’m’*) and holes (

*nlm*) satisfy the relation that makes Eqs. (3–4) nonzero. For NCs with different shapes, the intraband transition selection rules are different, while for interband transitions, the selection rule is independent of shape.

## 3. Result and discussion

Since the TPA measurements of CdS NCs with the shapes of sphere and cylinder have been carried out by the technique of Z-Scan and two-photon-induced fluorescence [32], we first calculate the TPA cross-sections for CdS NCs to verify our theoretical model by comparison. And then, we perform the calculations to predict the TPA cross-sections for CdSe and ZnO NCs with the shapes of sphere, cube, cylinder and cuboid. The material parameters listed in Table 1 are used in the following calculations and discussion.

#### 3.1 TPA Cross-sections of CdS nanosphere and nanocylinder

According to the expressions in the previous section, the TPA cross-section spectra of CdS spherical NCs with the diameter of 4.45 nm and cylinder NCs with the diameter of 4.4 nm and the length of 43 nm (aspect ratio of ~10) are plotted as the function of excitation wavelength in Fig. 2. The experimental data in Ref. [32] are also inserted for comparison. It is found that the TPA cross-section for the spherical CdS NCs is on the order of ~10^{3} GM, while for the cylindrical CdS NCs, it is 10^{4}~10^{5} GM, demonstrating a more significant enhancement in cylinder than sphere. This can be explained by the lower degree of symmetry and the anisotropy induced splitting of degenerate energy states in cylindrical NCs, which increases the density of energy state. Compared the theoretical curves with the experimental data, it is consistent that the TPA cross-section decreases with light wavelength. Our theoretical curves are in excellent agreement with the experimental data, especially for CdS nanospheres. In the case of CdS nanorods, discrepancy exists at longer wavelengths for CdS nanorods. It is plausible that the discrepancy is caused by the fact that the quantum efficiency of two-photon-excited photoluminescence (PL) becomes larger at longer wavelengths due to the elongated geometry, but it was assumed to be independent of light wavelength in the data analysis [32]. In addition, scattering effects are expected to play a more important role in larger NCs (nanorod length ~43 nm) than smaller NCs (nanodot diameter ~4.4 nm). It is also possible that scattering effects becomes less significant at longer wavelengths and more incident light is absorbed by CdS nanorods to convert PL, which effectively increased the measured TPA cross-section if scattering effects were not included in the data analysis [32]. Having said the above, it should be emphasized that the discrepancy is small (within three-folds at worst) and is expected to become negligible as rod length decreases or at shorter wavelengths.

#### 3.2 TPA Cross-sections of CdSe and ZnO NCs with the four shapes

Fig. 3 shows the TPA spectra of CdSe and ZnO NCs with the shapes of sphere, cube, cylinder and cuboid, calculated by the TPA model presented in the previous section with a Gaussian size distribution (FWHM=10%). The comparisons between sphere and cylinder, cube and cuboid, sphere and cube are illustrated in Fig. 3 (a–c) respectively, with the same diameter (or width) *D*=3.2 nm. For both CdSe and ZnO NCs, it is found that with the same width or diameter, the TPA cross-sections of nanocylinders are larger than those of nanospheres at the same wavelength. Similarly, the TPA cross-section of nanocuboid is greater than that of nanocube. Attention should be paid to the TPA cross-section of nanocube which is remarkably larger than nanosphere. For CdSe nanocube, its TPA cross-section is nearly three times as large as its nanosphere counterpart at light wavelength ~780 nm, and even 10 times at shorter wavelengths.

Such shape dependence could be understood by the predicted splitting of energy levels upon elongation of dots into rods. Calculations based on the effective mass approximation have been performed to study the quantum confinement of semiconductor NCs whose shape is different from usually spherical shapes [39–45]. The decreased degree of symmetry and the anisotropy effect lead to the splitting of degenerate energy states. As a result, an increased density of energy states results in the increase of the electron transition possibilities, which are related to the TPA cross-section. Therefore, in the cases of cylinder and cuboid, which are of lower degree of symmetry, TPA cross-sections are greater than cube and sphere. In addition, shape dependence of other properties in semiconductor NCs, including exciton, relaxation dynamics and bandgap variation, has been reported recently [19, 46, 47]. Cantele et al. [42, 43] calculated the confined state and infrared optical transitions in semiconductor ellipsoidal QDs. And they also attributed their results to a consequence of the geometry-induced effect. For cylinder and cuboid with a diameter (or width) of 3.2 nm, we calculate the TPA spectra with three different aspect ratios, as displayed in Fig. 3 (a) and (b). It is obvious that the larger the aspect ratio, the greater the TPA cross-section. The energy level spacing (i.e. bandgap) of nanorods decreases with increasing aspect ratio towards the value expected of true one-dimension quantum wires [19].

In order to investigate the contribution of a specific two-photon transition to the overall measured TPA in NCs with different shapes, the lowest transition is studied as shown in Fig. 4. According to the TPA transition selection rules, the lowest transitions for CdSe and ZnO NCs with the shapes of sphere, cylinder, cube and cuboid are revealed as follows, respectively: |*h*
_{1},1,1>→|*c*,1,0>, |*h*
_{1},1,0,2>→|*c*,1,0,1>, |*h*
_{1},1,0,1>→|*c*,2,0,1>, and |*h*
_{1},1,0,1>→|*c*,1,1,1>. Fig. 4 (a) demonstrates the size dependence of the lowest transition energy for CdSe and ZnO NCs with the above-said four shapes. For CdSe NCs with a fixed size, the nanocube’s transition energy is the highest and the cuboid’s is the lowest. This means that if the incident photon energy is varied from a lower to a higher value, TPA in cuboid occurs first. In ZnO NCs, attention should be paid to the lowest transition energies in sphere and cube which are very close to each other. It is also observed that with the increase in the diameter (or width), the lowest transition energy decreases. The smaller the size is, the more rapidly the energy decreases. Compared cylinder with cuboid, the lowest transition energy decreases slightly faster in cylinder with the increase in the diameter (or width). The contribution of the lowest transition to the amplitude of TPA is shown in Fig. 4 (b). It is clear that with the increase in the size, the contribution of the lowest transition to TPA increases accordingly. In cubic NCs, the lowest transition contributes more to the TPA cross-section than the other three shapes and the difference between cylinder and cuboid is insignificant. By comparison between Fig. 4(b) and Fig. 3(c), it is interesting to note that the contribution from the lowest transition accounts for ~10% of the total TPA at the lowest peak of TPA spectrum for ZnO nanocube.

In Fig. 5, we display the size-dependent TPA for CdSe and ZnO NCs with the four shapes at 780-nm and 532-nm wavelength, respectively. As *D* increases from 2 nm to 5 nm, the TPA cross-sections for CdSe NCs with the four shapes increase in different extent due to the different degree of quantum confinement. In order to generate comparable results for cylinders and cuboids, we set the aspect ratio as *V*=3, making them have the same size. It should be pointed out that the TPA cross-sections of cylinder and cuboid show significantly different change with respect to the cross-sectional shape. Compared with cylinder, TPA cross-section of cuboid shows stronger size dependence. For instance, in Fig. 5 (a), the TPA cross-section of CdSe cuboid increases to 3×10^{-45} cm^{4} s photon^{-1} as *D* increases from 3 to 5 nm. Within the same range, however, the TPA cross-section of cylinder only reaches to 0.75×10^{-45} cm^{4} s photon^{-1}. This is indicative of the importance of shapes to the TPA properties. Recently, it has been reported that the sharp corner structure of geometrical cross-section produces a larger bandgap [48–48]. From our modeling, similar conclusions can be drawn by comparison between sphere and cubes. TPA of cube shows stronger size dependence than sphere.

The two geometrical parameters characterize the sizes of cylinder and cuboid: diameter (or width) *D* and length *L*. In Fig. 5, the aspect ratio is fixed at 3, which means that *L* increases with the increase of *D*. In order to investigate which parameter plays a dominant role in TPA, we display the calculated TPA cross-section as a function of both *L* and *D* in Fig. 6. As expected from quantum confinement considerations, the general tendency is that the TPA cross-section increases with an increase in either width or length. However, the TPA cross-section depends more sensitively on width than length, as indicated by the slopes of two directions in Fig. 6. This suggests that the confinement should be determined mainly by the lateral dimension (not the longitudinal size), which is consistent with the conception in Ref. [47].

## 4. Conclusion

In conclusion, based on a four-band model under effective mass approximation, we have developed a simple analytical theory capable of providing a quantitative explanation of the recent TPA measurement on CdS nanorods [Appl. Phys. Lett. **94**, 103117 (2009)]. With this theory, we have systematically revealed the characteristics of TPA in CdSe and ZnO NCs with four different shapes: sphere, cube, cylinder and cuboid. As a result of the splitting of degenerate energy levels caused by the decreased degree of symmetry, nanocuboids and nanocubes exhibit greater TPA cross-sections than nanocylinders and nanospheres of similar sizes, respectively. Similarly, nanocuboids and nanocylinders possess larger TPA cross-sections than nanocubes and nanospheres of similar lateral dimension, respectively. Given TPA-allowed transitions, nanocuboids show stronger size dependence than nanocylinders. More importantly, the size dependence of TPA cross-section is more sensitive to the lateral size than the longitudinal size in the cases of nanocylinders and nanocuboids.

## Acknowledgment

We are grateful to the financial support from the National University of Singapore (Research Grant # R-144-000-213-112).

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