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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 15 — Jul. 20, 2009
  • pp: 13140–13150
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Shape-dependent two-photon absorption in semiconductor nanocrystals

Xiaobo Feng and Wei Ji  »View Author Affiliations


Optics Express, Vol. 17, Issue 15, pp. 13140-13150 (2009)
http://dx.doi.org/10.1364/OE.17.013140


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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

1. X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, “Quantum qots for live cells, in vivo imaging, and diagnostics,” Science 307, 538–544 (2005). [CrossRef] [PubMed]

6

6. G. S. He, K. T. Yong, Q. D. Zheng, Y. Sahoo, A. Baev, A. I. Ryasnyanskiy, and P. N. Prasad, “Multi-photon excitation properties of CdSe quantum dots solutions and optical limiting behavior in infrared range,” Opt. Express 15, 12818–12833 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-20-12818. [CrossRef] [PubMed]

]. In these applications, semiconductor NCs play an essential role in laser excitation through a nonlinear optical process: two-photon absorption (TPA).

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

22. A. V. Fedorov, A. V. Baranov, and K. Inoue, “Two-photon transitions in systems with semiconductor quantum dots,” Phys. Rev. B 54, 8627–8632 (1996). [CrossRef]

] 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

23. L. A. Padilha, J. Fu, D. J. Hagan, E. W. Van Stryland, C. L. Cesar, L. C. Barbosa, and Cruz C H B, “Two-photon absorption in CdTe quantum dots,” Opt. Express 13, 6460–6467 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-17-6460. [CrossRef] [PubMed]

] but also developed it into a k.p model [24

24. L. A. Padilha, J. Fu, D. J. Hagan, E. W. Van Stryland, C. L. Cesar, L. C. Barbosa, C. H. B. Cruz, D. Buso, and A. Martucci, “Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots,” Phys. Rev. B 75, 075325–075332 (2007). [CrossRef]

]. 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 cm4 s/photon [25

25. M. E. Schmidt, S. A. Blanton, M. A. Hines, and P. Guyot-Sionnest, “Size-dependent two-photon excitation spectroscopy of CdSe nanocrystals,” Phys. Rev. B 53, 12629–12632 (1996). [CrossRef]

, 26

26. Y. Qu, W. Ji, Y. Zheng, and J. Y. Ying, “Auger recombination and intraband absorption of two-photon-excited carriers in colloidal CdSe quantum dots,” Appl. Phys. Lett. 90, 133112–133114 (2007). [CrossRef]

]. The TPA in ZnS QDs was observed at wavelengths of 532 nm and 520 nm by using picosecond laser pulses [27

27. V. V. Nikesh, A. Dharmadhikari, H. Ono, S. Nozaki, G. S. Kumar, and S. Mahamuni, “Optical nonlinearity of monodispersed, capped ZnS quantum particles,” Appl. Phys. Lett. 84, 46024604–46024606 (2004). [CrossRef]

]. Both nonlinear refractive index and TPA coefficient of Mn-doped ZnSe QDs were determined by Z-scan technique at 800 nm wavelength [28

28. C. Gan, M. Xiao, D. Battaglia, N. Pradhan, and X. G Peng, “Size dependence of nonlinear optical absorption and refraction of Mn-doped ZnSe nanocrystals,” Appl. Phys. Lett. 91, 201103–201105 (2007). [CrossRef]

, 29

29. G. C. Xing, W. Ji, Y. G. Zheng, and J. Y. Ying, “Two- and three-photon absorption of semiconductor quantum dots in the vicinity of half of lowest exciton energy,” Appl. Phys. Lett. 93, 241114–241116 (2008). [CrossRef]

]. The TPA of ZnO QDs was also characterized in the wavelength range from 700 nm to 900 nm [30

30. W. Ji, Y. Qu, J. Mi, Y. Zheng, and J. Y. Ying, “Wavelength scaling for multiphoton absorption in semiconductor quantum dots,” Chinese Opt. Lett. 3, S203–S204 (2005).

]. The TPA cross-section was observed as high as 104 GM (1 GM=10-50cm4 s/photon) for 4.9-nm-sized CdTe QDs [31

31. L. Y. Pan, N. Tamai, K. Kamada, and S. Deki, “Nonlinear optical properties of thiol-capped CdTe quantum dots in nonresonant region,” Appl. Phys. Lett. 91, 051902–051904 (2007). [CrossRef]

]. 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

32. X. Li, J. Embden, W. M. Chon, and M. Gu, “Enhanced two-photon absorption of CdS nanocrystal rods,” Appl. Phys.Lett. 94, 103117–103119 (2009). [CrossRef]

]. 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

23. L. A. Padilha, J. Fu, D. J. Hagan, E. W. Van Stryland, C. L. Cesar, L. C. Barbosa, and Cruz C H B, “Two-photon absorption in CdTe quantum dots,” Opt. Express 13, 6460–6467 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-17-6460. [CrossRef] [PubMed]

, 24

24. L. A. Padilha, J. Fu, D. J. Hagan, E. W. Van Stryland, C. L. Cesar, L. C. Barbosa, C. H. B. Cruz, D. Buso, and A. Martucci, “Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots,” Phys. Rev. B 75, 075325–075332 (2007). [CrossRef]

] 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

32. X. Li, J. Embden, W. M. Chon, and M. Gu, “Enhanced two-photon absorption of CdS nanocrystal rods,” Appl. Phys.Lett. 94, 103117–103119 (2009). [CrossRef]

], 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

10. S. Kumar and T. Nann, “Shape control of II–VI semiconductor nanomaterials,” Small 2, 316–329 (2006). [CrossRef] [PubMed]

, 33

33. N. L. Thomas, E. Herz, O. Schöps, and U. Woggon, “Exciton Fine Structure in Single CdSe Nanorods,” Phys. Rev. Lett. 94, 016803–016806 (2005). [CrossRef] [PubMed]

35

35. T. Yatsui, S. Sangu, K. Kobayashi, T. Kawazoe, M. Ohtsu, J. Yoo, and G. C. Yi, “Nanophotonic energy up conversion using ZnO nanorod double-quantum-well structures,” Appl. Phys. Lett. 94, 083113–083115 (2009). [CrossRef]

]. 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).

Fig. 1. Schematic diagrams of semiconductor NCs with four shapes.

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

36. V. Nathan, A. H. Guenther, and S. S. Mitra, “Review of multiphoton absorption in crystalline solids,” J. Opt. Soc. Am. B 2, 294–316 (1985). [CrossRef]

]

W(2)=2πΣv1,v0Mv1,v02δ(Ev1Ev02ω),
Mv1,v0=Σv2Hv1,v2intHv2,v0intEv2Ev0ωiγv2.
(1)

where E v0, E v1 and E v2 represent the energies of the initial, final and intermediate states of an electron, respectively. Hint=(e/m 0c)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

18. J. Li and L. W. Wang, “Shape effects on electronic states of nanocrystals,” Nano Lett. 3, 1357–1363 (2003). [CrossRef]

], the parameters Ev and H int in Eq. (1) are dependent on both shape and size. Similar to the approach taken by Fedorov et al. [22

22. A. V. Fedorov, A. V. Baranov, and K. Inoue, “Two-photon transitions in systems with semiconductor quantum dots,” Phys. Rev. B 54, 8627–8632 (1996). [CrossRef]

], 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

22. A. V. Fedorov, A. V. Baranov, and K. Inoue, “Two-photon transitions in systems with semiconductor quantum dots,” Phys. Rev. B 54, 8627–8632 (1996). [CrossRef]

], 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

a,nlmHinta,nlm=ieAmac×B,
(2)

where nlm (n’l’m’) are the quantum numbers which denote the electronic states, and ma is the electron effective mass in the a band. The function B differs greatly for different NC shapes. For cylinder, it is given by

B={2Dμnlμnl(μnl)2(μnl)2[δm,m(δn,n+1+δn,n1)ex+iδm,m(δn,n1δn,n+1)ey]
+1Lmδn,nδl,l(1δm,m)[1cos(m+m)πm+m+1cos(mm)πmm]ez},
(3)

where nlm (n’l’m’) are the quantum numbers in three directions of the cylindrical coordinates, µnl is the nth root of the lth-order Bessel function, ej (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,

B=[1Dnδm,mδl,l(1δn,n)(1cos(n+n)n+n+1cos(nn)πn'n)ex
+1Dlδm,mδn,n(1δl,l)(1cos(l+l)πl+l+1cos(ll)πll)ey
+1Lmδn,nδl,l(1δm,m)(1cos(m+m)πm+m+1cos(mm)πmm)ez].
(4)

α2=4ωNI2Wˉ(2)f(a)da,
(5)

where N is the NC concentration and I is the incident light intensity I=εω 1/2 ω 2 A 2(2πc)-1, and εω 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

37. E. P. Pokatilov, V. A. Fonoberov, V. M. Fomin, and J. T. Devreese, “Development of an eight-band theory for quantum dot heterostructures,” Phys. Rev. B 64, 245328–245343 (2001). [CrossRef]

] is mostly used. For an arbitrary function f (a), the TPA coefficient can be expressed as follows,

α2=8πωNεω(2πe2Pcω2)2Σj=13Fc,hj,
Fc,hj=Σnlm,nlmB2Tf(a)δ(Ev0hjEv1c2ω)da,
T=1mc1(Ev0cEv0hjωiγv0)+1mhj1(Ev0hjEv1hj+ω+iγv1),
(6)

where P2 pc,h/m 0=ħ 2<S|∂/∂Z|Z>/m 0, pc,h is the interband matrix element of the electron momentum. hj (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

32. X. Li, J. Embden, W. M. Chon, and M. Gu, “Enhanced two-photon absorption of CdS nanocrystal rods,” Appl. Phys.Lett. 94, 103117–103119 (2009). [CrossRef]

], 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.

Table 1. Parameters used in the Calculations for the Investigated Materials

table-icon
View This Table

3.1 TPA Cross-sections of CdS nanosphere and nanocylinder

Fig. 2. TPA cross-sections of cylindrical and spherical CdS NCs plotted as a function of excitation wavelength. The blue circles and red squares are the experimental data in Ref. [32].

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

Fig. 3. Comparisons of TPA spectra between CdSe NCs and ZnO NCs with four shapes: (a) sphere vs. cylinder; (b) cube vs. cuboid; and (c) cube vs. sphere.
Fig. 4. (a) Lowest transition energy versus diameter (or width); and (b) TPA cross-section contributed by the lowest transition versus diameter (or width). The aspect ratios for cylinder and cuboid are V=3.

Fig. 5. Size dependence of TPA cross-section for (a) CdSe NCs at 780 nm and (b) ZnO NCs at 532 nm.

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 cm4 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 cm4 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. D. L. Yao, G. Zhang, and B. W. Li, “A universal expression of band gap for silicon nanowires of different cross-section geometries,” Nano Lett. 8, 4557–4561 (2008). [CrossRef]

48

48. D. L. Yao, G. Zhang, and B. W. Li, “A universal expression of band gap for silicon nanowires of different cross-section geometries,” Nano Lett. 8, 4557–4561 (2008). [CrossRef]

]. 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

47. L. S. Li, J. T. Hu, W. D. Yang, and A. P. Alivisatos, “Band gap variation of size- and shape-controlled colloidal CdSe quantum rods,” Nano Lett. 1, 349–351 (2001). [CrossRef]

].

Fig. 6. TPA cross-section vs. width (or diameter) and length at 780 nm for CdSe NCs and at 532 nm for ZnO NCs with the shapes of (a) cylinder and (b) cuboid.

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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L. A. Padilha, J. Fu, D. J. Hagan, E. W. Van Stryland, C. L. Cesar, L. C. Barbosa, and Cruz C H B, “Two-photon absorption in CdTe quantum dots,” Opt. Express 13, 6460–6467 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-17-6460. [CrossRef] [PubMed]

24.

L. A. Padilha, J. Fu, D. J. Hagan, E. W. Van Stryland, C. L. Cesar, L. C. Barbosa, C. H. B. Cruz, D. Buso, and A. Martucci, “Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots,” Phys. Rev. B 75, 075325–075332 (2007). [CrossRef]

25.

M. E. Schmidt, S. A. Blanton, M. A. Hines, and P. Guyot-Sionnest, “Size-dependent two-photon excitation spectroscopy of CdSe nanocrystals,” Phys. Rev. B 53, 12629–12632 (1996). [CrossRef]

26.

Y. Qu, W. Ji, Y. Zheng, and J. Y. Ying, “Auger recombination and intraband absorption of two-photon-excited carriers in colloidal CdSe quantum dots,” Appl. Phys. Lett. 90, 133112–133114 (2007). [CrossRef]

27.

V. V. Nikesh, A. Dharmadhikari, H. Ono, S. Nozaki, G. S. Kumar, and S. Mahamuni, “Optical nonlinearity of monodispersed, capped ZnS quantum particles,” Appl. Phys. Lett. 84, 46024604–46024606 (2004). [CrossRef]

28.

C. Gan, M. Xiao, D. Battaglia, N. Pradhan, and X. G Peng, “Size dependence of nonlinear optical absorption and refraction of Mn-doped ZnSe nanocrystals,” Appl. Phys. Lett. 91, 201103–201105 (2007). [CrossRef]

29.

G. C. Xing, W. Ji, Y. G. Zheng, and J. Y. Ying, “Two- and three-photon absorption of semiconductor quantum dots in the vicinity of half of lowest exciton energy,” Appl. Phys. Lett. 93, 241114–241116 (2008). [CrossRef]

30.

W. Ji, Y. Qu, J. Mi, Y. Zheng, and J. Y. Ying, “Wavelength scaling for multiphoton absorption in semiconductor quantum dots,” Chinese Opt. Lett. 3, S203–S204 (2005).

31.

L. Y. Pan, N. Tamai, K. Kamada, and S. Deki, “Nonlinear optical properties of thiol-capped CdTe quantum dots in nonresonant region,” Appl. Phys. Lett. 91, 051902–051904 (2007). [CrossRef]

32.

X. Li, J. Embden, W. M. Chon, and M. Gu, “Enhanced two-photon absorption of CdS nanocrystal rods,” Appl. Phys.Lett. 94, 103117–103119 (2009). [CrossRef]

33.

N. L. Thomas, E. Herz, O. Schöps, and U. Woggon, “Exciton Fine Structure in Single CdSe Nanorods,” Phys. Rev. Lett. 94, 016803–016806 (2005). [CrossRef] [PubMed]

34.

C. Zhang, F. Zhang, S. Qian, N. Kumar, J. Hahm, and J. Xu, “Multiphoton absorption induced amplified spontaneous emission from biocatalyst-synthesized ZnO nanorods,” Appl. Phys. Lett. 92, 233116–233118 (2008). [CrossRef]

35.

T. Yatsui, S. Sangu, K. Kobayashi, T. Kawazoe, M. Ohtsu, J. Yoo, and G. C. Yi, “Nanophotonic energy up conversion using ZnO nanorod double-quantum-well structures,” Appl. Phys. Lett. 94, 083113–083115 (2009). [CrossRef]

36.

V. Nathan, A. H. Guenther, and S. S. Mitra, “Review of multiphoton absorption in crystalline solids,” J. Opt. Soc. Am. B 2, 294–316 (1985). [CrossRef]

37.

E. P. Pokatilov, V. A. Fonoberov, V. M. Fomin, and J. T. Devreese, “Development of an eight-band theory for quantum dot heterostructures,” Phys. Rev. B 64, 245328–245343 (2001). [CrossRef]

38.

P. Lawaetz, “Valence-band parameters in cubic semiconductors,” Phys. Rev. B 4, 3460–3467 (1971). [CrossRef]

39.

A. L. Efros and A. V. Arondina, “Band-edge absorption and luminescence of nonspherical nanometer-size crystals,” Phys. Rev. B 47, 10005–10007 (1993). [CrossRef]

40.

D. Ninno, G. Iadonisi, and F. Buonocore, “Carrier localization and photoluminescence in porous silicon” Solid State Commun. 112, 521–525 (1999). [CrossRef]

41.

F. Buonocore, D. Ninno, and G. Iadonisi, “Localized states in arbitrarily shaped quantum wire: a variation-perturbation technique,” Phys. Stat. Sol. (b) 225, 343–352 (2001). [CrossRef]

42.

G. Cantele, D. Ninno, and G. Iadonisi, “Confined states in ellipsoidal quantum dots,” J. Phys.: Condens. Matter 12, 9019–9036 (2000). [CrossRef]

43.

G. Cantele, D. Ninno, and G. Iadonisi, “Calculation of the infrared optical transitions in semiconductor ellipsoidal quantum dots,” Nano Lett. 1, 121–124 (2001). [CrossRef]

44.

Y. Kayanuma, “Wannier excitons in low-dimensional microstructures: Shape dependence of the quantum size effect,” Phys. Rev. B 44, 13085–13088 (1991). [CrossRef]

45.

S. L. Goff and B. Stébé, “Influence of longitudinal and lateral confinements on excitons in cylindrical quantum dots of semiconductors,” Phys. Rev. B 47, 1383–1391 (1993). [CrossRef]

46.

M. B. Mohamed, C. Burda, and M. A. El-Sayed, “Shape dependent ultrafast relaxation dynamics of CdSe nanocrystals: Nanorods vs Nanodots,” Nano Lett. 1, 589–593 (2001). [CrossRef]

47.

L. S. Li, J. T. Hu, W. D. Yang, and A. P. Alivisatos, “Band gap variation of size- and shape-controlled colloidal CdSe quantum rods,” Nano Lett. 1, 349–351 (2001). [CrossRef]

48.

D. L. Yao, G. Zhang, and B. W. Li, “A universal expression of band gap for silicon nanowires of different cross-section geometries,” Nano Lett. 8, 4557–4561 (2008). [CrossRef]

49.

J. X. Cao, X. G. Gong, J. X. Zhong, and R. Q. Wu, “Sharp corners in the cross section of ultrathin Si nanowires,” Phys. Rev. Lett. 97, 136105–136108 (2006). [CrossRef] [PubMed]

OCIS Codes
(190.4180) Nonlinear optics : Multiphoton processes
(190.5970) Nonlinear optics : Semiconductor nonlinear optics including MQW

ToC Category:
Nonlinear Optics

History
Original Manuscript: June 1, 2009
Revised Manuscript: July 12, 2009
Manuscript Accepted: July 12, 2009
Published: July 16, 2009

Citation
Xiaobo Feng and Wei Ji, "Shape-dependent two-photon absorption in semiconductor nanocrystals," Opt. Express 17, 13140-13150 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13140


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  42. G. Cantele, D. Ninno, and G. Iadonisi, "Confined states in ellipsoidal quantum dots," J. Phys.: Condens. Matter 12, 9019-9036 (2000). [CrossRef]
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  46. M. B. Mohamed, C. Burda, and M. A. El-Sayed, "Shape dependent ultrafast relaxation dynamics of CdSe nanocrystals: Nanorods vs Nanodots," Nano Lett. 1, 589-593 (2001). [CrossRef]
  47. L. S. Li, J. T. Hu, W. D. Yang, and A. P. Alivisatos, "Band gap variation of size- and shape-controlled colloidal CdSe quantum rods," Nano Lett. 1, 349-351 (2001). [CrossRef]
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